1.
Euclid
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Euclid, sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. In the Elements, Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, rigor. Euclid is the anglicized version of the Greek Εὐκλείδης, which means "renowned, glorious". Very original references to Euclid survive, so little is known about his life. The place and circumstances of both his birth and death are unknown and may only be estimated roughly relative to other people mentioned with him. He is usually referred to as" ὁ στοιχειώτης". The historical references to Euclid were written centuries after he lived by Proclus c. 450 AD and Pappus of Alexandria c. 320 AD. Proclus introduces Euclid only briefly in his Commentary on the Elements. This anecdote is questionable since it is similar to a story told about Alexander the Great. 247–222 BC. A detailed biography of Euclid is given by Arabian authors, mentioning, for example, a town of Tyre. This biography is generally believed to be completely fictitious. However, there is little evidence in its favor.
Euclid
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Euclid by Justus van Gent, 15th century
Euclid
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition 5.
Euclid
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Statue in honor of Euclid in the Oxford University Museum of Natural History
2.
Calipers
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A caliper is a device used to measure the distance between two opposite sides of an object. A caliper can be as simple as a compass with inward or outward-facing points. It is used in many fields such as mechanical engineering, metalworking, forestry, medicine. A plurale sense of the word "calipers" coexists with the regular noun sense of "caliper". Also existing colloquially but not in formal usage is referring to a caliper as a "pair of verniers". In colloquial usage, some speakers extend this even to dial calipers, although they involve no scale. In machine-shop usage, the term "caliper" is often used in contradistinction to "micrometer", even though outside micrometers are technically a form of caliper. In this usage, "caliper" implies only the factor of the vernier or caliper. The earliest caliper has been found in the Greek Giglio wreck near the Italian coast. The ship find dates to the 6th century BC. The wooden piece already featured a fixed and a movable jaw. Although rare finds, caliper remained in use by the Greeks and Romans. A caliper, dating from 9 AD, was used during the Chinese Xin dynasty. The calipers included pin" and "graduated in tenths of an inch." The modern vernier caliper, reading to thousandths of an inch, was invented by American Joseph R. Brown in 1851.
Calipers
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Caliper with graduated bow 0–10 mm
Calipers
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Two inside calipers
Calipers
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Three outside calipers.
Calipers
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A pair of dividers
3.
Raphael
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Raffaello Sanzio da Urbino, known as Raphael, was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, visual achievement of the Neoplatonic ideal of human grandeur. Together with Leonardo da Vinci, he forms the traditional trinity of great masters of that period. Raphael was enormously productive, running an unusually large workshop and, despite his death at 37, leaving a large body of work. Many of his works are found in the Vatican Palace, where the frescoed Raphael Rooms were the the largest, work of his career. The best known work is The School of Athens in the Vatican Stanza della Segnatura. After his early years in Rome much of his work was executed by his workshop with considerable loss of quality. He was extremely influential in his lifetime, though outside Rome his work was mostly known from his collaborative printmaking. His poem to Federico shows him as keen to show awareness of Early Netherlandish artists as well. In the very small court of Urbino he was probably more integrated into the central circle of the ruling family than most court painters. Under them, the court continued for literary culture. Social skills stressed by Vasari. They became good friends. Raphael mixed easily in the highest circles throughout one of the factors that tended to give a misleading impression of effortlessness to his career. He did not receive a humanistic education however; it is unclear how easily he read Latin.
Raphael
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Presumed Portrait of Raphael
Raphael
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Self-portrait with a friend (1518 circa), Louvre, Paris
Raphael
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Giovanni Santi, Raphael's father; Christ supported by two angels, c.1490
Raphael
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The Mond Crucifixion, 1502–3, very much in the style of Perugino
4.
The School of Athens
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The School of Athens is one of the most famous frescoes by the Italian Renaissance artist Raphael. The picture has long been seen as "the perfect embodiment of the classical spirit of the Renaissance". The School of Athens is one of a group of four main frescoes on the walls of the Stanza that depict distinct branches of knowledge. Accordingly, the figures on the walls below exemplify Law. The traditional title is not Raphael's. Indeed, Plato and Aristotle appear to be the central figures in the scene. However, all the philosophers depicted sought knowledge of first causes. Hardly a third were Athenians. The general semi-circular setting having Plato and Aristotle at its centre might be alluding to Pythagoras' circumpunct. Compounding the problem, Raphael had to allude to various figures for whom there were no traditional visual types. For example, while the Socrates figure is immediately recognizable from Classical busts, the alleged Epicurus is far removed from his standard type. Few such interpretations are unanimously accepted among scholars. The popular idea that the rhetorical gestures of Plato and Aristotle are kinds of pointing is very likely. Aristotle, with his four-elements theory, held that all change on Earth was owing to motions of the heavens. In the painting Aristotle carries his Ethics, which he denied could be reduced to a mathematical science.
The School of Athens
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The School of Athens
The School of Athens
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An elder Plato walks alongside Aristotle.
The School of Athens
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Architecture
The School of Athens
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Epicurus
5.
Ancient Greek
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Ancient Greek includes the forms of Greek used in ancient Greece and the ancient world from around the 9th century BC to the 6th century AD. It is often roughly divided into the Archaic period, Hellenistic period. It is antedated by Mycenaean Greek. The language of the Hellenistic phase is known as Koine. Prior to the Koine period, Greek of earlier periods included several regional dialects. Ancient Greek was the language of Homer and of classical Athenian historians, philosophers. It has been a standard subject of study in educational institutions of the West since the Renaissance. This article primarily contains information of the language. Ancient Greek was a pluricentric language, divided into many dialects. The main dialect groups are Doric, many of them with several subdivisions. Some dialects are found in literary forms used in literature, while others are attested only in inscriptions. There are also historical forms. Homeric Greek is a literary form of Archaic Greek used by other authors. Homeric Greek had significant differences in pronunciation from Classical Attic and other Classical-era dialects. The early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence.
Ancient Greek
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Inscription about the construction of the statue of Athena Parthenos in the Parthenon, 440/439 BC
Ancient Greek
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Ostracon bearing the name of Cimon, Stoa of Attalos
Ancient Greek
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The words ΜΟΛΩΝ ΛΑΒΕ as they are inscribed on the marble of the 1955 Leonidas Monument at Thermopylae
6.
Number
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A number is a mathematical object used to count, measure, label. The original examples are 1, 2, 3, so forth. A notational symbol that represents a number is called a numeral. In addition to their use in measuring, numerals are often used for labels, for ordering, for codes. In common usage, number may refer to a symbol, a mathematical abstraction. Calculations with numbers are done with the most familiar being addition, subtraction, multiplication, division, exponentiation. Their usage is called arithmetic. The same term may also refer to the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world. For example, "a million" may signify "a lot." Though it is now regarded as pseudoscience, the belief in a mystical significance of numbers, permeated ancient and medieval thought. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of many problems in theory which are still of interest today. During the 19th century, mathematicians may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from the symbols used to represent numbers.
Number
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The number 605 in Khmer numerals, from an inscription from 683 AD. An early use of zero as a decimal figure.
Number
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Subsets of the complex numbers.
7.
Space
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Space is the boundless three-dimensional extent in which objects and events have relative position and direction. The concept of space is considered to be to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, part of a conceptual framework. Many of these philosophical questions were discussed in the Renaissance and then reformulated in the 17th century, particularly during the early development of classical mechanics. In Isaac Newton's view, space was absolute -- in the sense that it existed independently of whether there was any matter in the space. Kant referred to the experience of "space" as being a subjective "pure a priori form of intuition". In the 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. In the seventeenth century, the philosophy of time emerged as a central issue in epistemology and metaphysics. At its heart, the English physicist-mathematician, set out two opposing theories of what space is. Unoccupied regions are those that could have objects in them, thus spatial relations with other places. Space could be thought in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people. According to the principle of sufficient reason, any theory of space that implied that there could be these two possible universes must therefore be wrong.
Space
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Gottfried Leibniz
Space
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A right-handed three-dimensional Cartesian coordinate system used to indicate positions in space.
Space
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Isaac Newton
Space
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Immanuel Kant
8.
Calculus
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It has two major branches, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed by Isaac Newton and Gottfried Leibniz. Calculus has widespread uses in science, engineering and economics. Calculus is a part of modern education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". Calculus is also used for naming theories of computation, such as propositional calculus, calculus of variations, lambda calculus, process calculus. The method of exhaustion was later reinvented by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th AD, Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. The infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal term.
Calculus
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Isaac Newton developed the use of calculus in his laws of motion and gravitation.
Calculus
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Gottfried Wilhelm Leibniz was the first to publish his results on the development of calculus.
Calculus
–
Maria Gaetana Agnesi
Calculus
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The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus
9.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, change. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. It was the Pythagoreans who coined the term "mathematics", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Mathematics in the Islamic world during the Middle Ages followed various modes of funding varied based primarily on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics, maths and astronomy of Ibn al-Haytham. The Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many mathematicians gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.”
Mathematician
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Euclid (holding calipers), Greek mathematician, known as the "Father of Geometry"
Mathematician
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In 1938 in the United States, mathematicians were desired as teachers, calculating machine operators, mechanical engineers, accounting auditor bookkeepers, and actuary statisticians
Mathematician
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Archimedes, c. 287 – 212 BC
Mathematician
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Brahmagupta, c. 598 - 670
10.
Patterns
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A pattern, apart from the term's use to mean "Template", is a discernible regularity in the world or in a manmade design. As such, the elements of a pattern repeat in a predictable manner. Typically repeating like a wallpaper. Any of the five senses may directly observe patterns. Conversely, abstract patterns in science, language may be observable only by analysis. Direct observation in practice means seeing visual patterns, which are widespread in art. Visual patterns in nature are often never exactly repeating, often involve fractals. Those created by symmetries of rotation and reflection. Patterns have an mathematical structure; indeed, mathematics can be seen as the search for regularities, the output of any function is a mathematical pattern. Similarly in the sciences, theories predict regularities in the world. In architecture, decorations or visual motifs may be combined and repeated to form patterns designed to have a chosen effect on the viewer. In science, a software design pattern is a known solution to a class of problems in programming. In fashion, the pattern is a template used to create any number of similar garments. Nature provides examples including symmetries, trees and other structures with a fractal dimension, spirals, meanders, waves, foams, tilings, cracks and stripes. Symmetry is widespread in living things.
Patterns
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Tilings, such as these from Igreja de Campanhã, Porto, Portugal, are visual patterns used for decoration.
Patterns
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Snowflake sixfold symmetry
Patterns
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Aloe polyphylla phyllotaxis
Patterns
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Vortex street turbulence
11.
Conjecture
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In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found. The successful proof was released by Andrew Wiles, formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. Möbius mentioned the problem in his lectures as early as 1840. A number of false counterexamples have appeared in 1852. The four theorem was proven by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. Since then the proof has gained wider acceptance, although doubts remain. It was originally formulated in 1908, by Steinitz and Tietze.
Conjecture
–
The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. The Riemann hypothesis, a famous conjecture, says that all non-trivial zeros of the zeta function lie along the critical line.
12.
Mathematical proof
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In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. An unproved proposition, believed to be true is known as a conjecture. Proofs usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. Formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics. The philosophy of mathematics is concerned with mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren. The early use of "probity" was in the presentation of legal evidence. A person such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as analogies preceded strict mathematical proof.
Mathematical proof
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One of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
Mathematical proof
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Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
13.
Logic
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Logic, originally meaning "the word" or "what is spoken", is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a specific relation of logical support between the assumptions of its conclusion. Historically, recently logic has been studied in computer science, linguistics, psychology, other fields. The concept of logical form is central to logic. The validity of an argument is determined by its logical form, not by its content. Modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments. The study of fallacies is an important branch of informal logic. Since informal argument is not strictly speaking deductive, on some conceptions of logic, informal logic is not logic at all. See'Rival conceptions', below. Formal logic is the study of inference with purely formal content. The works of Aristotle contain the earliest formal study of logic. Modern formal logic expands on Aristotle. In many definitions of logic, logical inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language.
Logic
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Aristotle, 384–322 BCE.
14.
Counting
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Counting is the action of finding the number of elements of a finite set of objects. The related enumeration refers to uniquely identifying the elements of a finite set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos", or "counting by fives". There is archeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track such as number of group members, prey animals, property, or debts. The development of counting led to the development of mathematical notation, writing. Counting can occur in a variety of forms. Counting can be verbal;, speaking every number out loud to keep track of progress. This is often used to count objects that are present already, instead of counting a variety of things over time. Counting can also be in the form of tally marks, making a mark for each number and then counting all of the marks when done tallying. This is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting; normal counting is done in base 10. Computers use base 2 counting. Counting can also be in the form of counting, especially when counting small numbers. This is often used by children to facilitate simple mathematical operations.
Counting
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Counting using tally marks at Hanakapiai Beach
15.
Measurement
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Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context and discipline. Measurement is a cornerstone of quantitative research in many disciplines. Historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators. Since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a mathematical combination of seven base units. The science of measurement is pursued in the field of metrology. The measurement of a property may be categorized by the following criteria: uncertainty. They enable unambiguous comparisons between measurements. The type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by preference. The type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the numerical value of the characterization, usually obtained with a suitably chosen measuring instrument. An uncertainty represents the systemic errors of the procedure; it indicates a level in the measurement.
Measurement
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A typical tape measure with both metric and US units and two US pennies for comparison
Measurement
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A baby bottle that measures in three measurement systems, Imperial (U.K.), U.S. customary, and metric.
Measurement
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Four measuring devices having metric calibrations
16.
Shape
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Psychologists have theorized that humans mentally break down images into geometric shapes called geons. Examples of geons include spheres. Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles, pentagons, etc.. Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares, etc.. Common shapes are points, lines, planes, conic sections such as ellipses, circles, parabolas. Among the most 3-dimensional shapes are polyhedra, which are shapes with flat faces; ellipsoids, which are egg-shaped or sphere-shaped objects; cylinders; and cones. If an object falls into one of these categories even approximately, we can use it to describe the shape of the object. Thus, we say that the shape of a manhole cover is a disk, because it is approximately the geometric object as an actual geometric disk. Similarity: Two objects reflections. Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. Sometimes, only the external boundary of the object is considered to determine its shape. For instance, an hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used to state that two shapes are approximately the same.
Shape
–
A variety of polygonal shapes.
17.
Motion (physics)
–
In physics, motion is a change in position of an object over time. Motion is typically described in terms of displacement, distance, velocity, acceleration, speed. An object's motion can not change unless it is acted by a force, as described. Momentum is a quantity, used for measuring motion of an object. As there is no absolute frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving. One can also speak of motion of boundaries. So, the motion in general signifies a continuous change in the configuration of a physical system. In physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of familiar objects in the universe are described by classical mechanics. Whereas the motion of sub-atomic objects is described by quantum mechanics. It is one of the oldest and largest in science, engineering, technology. Classical mechanics is fundamentally based on Newton's laws of motion. These laws describe the relationship between the forces acting on the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 1687.
Motion (physics)
–
Motion involves a change in position, such as in this perspective of rapidly leaving Yongsan Station.
18.
History of Mathematics
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Before the worldwide spread of knowledge, written examples of mathematical developments have come to light only in a few locales. The most mathematical texts available are Plimpton 322, the Moscow Mathematical Papyrus. All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly expanded the matter of mathematics. Chinese mathematics made early contributions, including a system. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. The origins of mathematical thought lie in the concepts of form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of a six-month calendar. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. Undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.
History of Mathematics
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A proof from Euclid 's Elements, widely considered the most influential textbook of all time.
History of Mathematics
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The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
History of Mathematics
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Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
History of Mathematics
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One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
19.
Greek mathematics
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Greek mathematicians were united by language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word "mathematics" itself derives from the ancient Greek μάθημα, meaning "subject of instruction". The origin of Greek mathematics is not well documented. The earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations were capable of advanced engineering, including four-story palaces with beehive tombs, they left behind no mathematical documents. Though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. Thales' theorem and theorem are attributed to Thales. It is for this reason that Thales is often hailed as the true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed. Another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, but settled in Croton, Magna Graecia. And since in antiquity it was customary to give all credit to the master, Pythagoras himself was given credit for the discoveries made by his order.
Greek mathematics
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Statue of Euclid in the Oxford University Museum of Natural History
Greek mathematics
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An illustration of Euclid 's proof of the Pythagorean Theorem
Greek mathematics
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The Antikythera mechanism, an ancient mechanical calculator.
20.
Euclid's Elements
–
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, mathematical proofs of the propositions. The books cover the ancient Greek version of elementary number theory. It is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. According to Proclus, the term "element" was used to describe a theorem that helps furnishing proofs of many other theorems. The element in the Greek language is the same as letter. This suggests that theorems in the Elements should be seen as standing as letters to language. Euclid's Elements has been referred to as the most influential textbook ever written. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, supplemented by some original work. The Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions. Papyrus Oxyrhynchus 29 only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated prior to Boethius in the fifth or sixth century. The Arabs received the Elements around 760; this version was translated into Arabic under Harun al Rashid circa 800. The Byzantine scholar Arethas commissioned the copying of the extant Greek manuscripts of Euclid in the late ninth century.
Euclid's Elements
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The frontispiece of Sir Henry Billingsley's first English version of Euclid's Elements, 1570
Euclid's Elements
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A fragment of Euclid's "Elements" on part of the Oxyrhynchus papyri
Euclid's Elements
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An illumination from a manuscript based on Adelard of Bath 's translation of the Elements, c. 1309–1316; Adelard's is the oldest surviving translation of the Elements into Latin, done in the 12th-century work and translated from Arabic.
Euclid's Elements
–
Euclidis – Elementorum libri XV Paris, Hieronymum de Marnef & Guillaume Cavelat, 1573 (second edition after the 1557 ed.); in-8, 350, (2)pp. THOMAS-STANFORD, Early Editions of Euclid's Elements, n°32. Mentioned in T.L. Heath's translation. Private collection Hector Zenil.
21.
Giuseppe Peano
–
Giuseppe Peano was an Italian mathematician. The author of papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of this effort, he made key contributions to the modern systematic treatment of the method of mathematical induction. He spent most of his teaching mathematics at the University of Turin. Peano was raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. Due to Genocchi's poor health, Peano took within 2 years. A textbook on calculus, was published in 1884 and was credited to Genocchi. A few years later, Peano published his first book dealing with mathematical logic. Here the modern symbols for the intersection of sets appeared for the first time. In 1887, Peano married the daughter of the Turin-based painter Luigi Crosio, known for painting the Refugium Peccatorum Madonna. In 1886, he was promoted to Professor First Class in 1889. The University of Turin also granted him his full professorship. Peano's space-filling curve appeared in 1890 as a counterexample. He used it to show that a continuous curve can not always be enclosed in an small region.
Giuseppe Peano
–
Giuseppe Peano
Giuseppe Peano
–
Aritmetica generale e algebra elementare, 1902
Giuseppe Peano
–
Giuseppe Peano and his wife Carola Crosio in 1887
22.
David Hilbert
–
David Hilbert was a German mathematician. He is recognized as one of universal mathematicians of the 19th and early 20th centuries. Hilbert developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of the foundations of functional analysis. Hilbert warmly defended Georg Cantor's set theory and transfinite numbers. His students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium; but, after an unhappy period, he graduated from the more science-oriented Wilhelm Gymnasium. In autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In Hermann Minkowski, returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the gifted Minkowski". In 1884, Adolf Hurwitz arrived as an Extraordinarius. Hilbert obtained his doctorate with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg from 1886 to 1895. As a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen.
David Hilbert
–
David Hilbert (1912)
David Hilbert
–
The Mathematical Institute in Göttingen. Its new building, constructed with funds from the Rockefeller Foundation, was opened by Hilbert and Courant in 1930.
David Hilbert
–
Hilbert's tomb: Wir müssen wissen Wir werden wissen
23.
Truth
–
Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth may often be used in modern contexts to refer to an idea of "truth to self," or authenticity. The commonly understood opposite of truth is falsehood, which, correspondingly, can also take on a logical, ethical meaning. The concept of truth is debated in several contexts, including philosophy, art, religion. Commonly, truth is thought to an independent reality, in what is sometimes called the correspondence theory of truth. Other philosophers take this common meaning to be secondary and derivative. Various views of truth continue to be debated among scholars, philosophers, theologians. The English word truth is derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, Old Norse tryggð. Like troth, it is a - nominalisation of the adjective true. Old Norse trú, "word of honour; religious faith, belief". Thus, ` truth' involves that of "agreement with fact or reality", in Anglo-Saxon expressed by sōþ. All Germanic languages besides English have introduced a terminological distinction between truth "factuality". To express "factuality", North Germanic opted for nouns derived from sanna "to assert, affirm", while continental West Germanic opted for continuations of wâra "faith, pact". Romance languages use terms following the Latin veritas, while the Greek aletheia, South Slavic istina have separate etymological origins. Each presents perspectives that are widely shared by published scholars.
Truth
–
Time Saving Truth from Falsehood and Envy, François Lemoyne, 1737
Truth
–
Truth, holding a mirror and a serpent (1896). Olin Levi Warner, Library of Congress Thomas Jefferson Building, Washington, D.C.
Truth
–
An angel carrying the banner of "Truth", Roslin, Midlothian
Truth
–
Walter Seymour Allward 's Veritas (Truth) outside Supreme Court of Canada, Ottawa, Ontario Canada
24.
Definition
–
A definition is a statement of the meaning of a term. Definitions can be classified into two large categories, extensional definitions. Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have multiple meanings, thus require multiple definitions. In mathematics, a definition is used to give a precise meaning instead of describing a pre-existing term. Axioms are the basis on which all of mathematics is constructed. In modern usage, a definition is something, typically expressed in words, that attaches a meaning to a group of words. Note that the definiens is not the meaning of the word is instead something that conveys the same meaning as that word. There are many sub-types of definitions, often specific to a given field of study. An intensional definition, also called a connotative definition, specifies the sufficient conditions for a thing being a member of a specific set. Any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition. An extensional definition, also called a denotative definition, of a term specifies its extension. It is a list naming every object, a member of a specific set. An extensional definition would be the list of greed, sloth, pride, lust, envy, gluttony. A genus–differentia definition is a type of intensional definition that takes a large category and narrows it down to a smaller category by a distinguishing characteristic.
Definition
–
A definition states the meaning of a word using other words. This is sometimes challenging. Common dictionaries contain lexical, descriptive definitions, but there are various types of definition - all with different purposes and focuses.
25.
Renaissance
–
This new thinking became manifest in art, politics, literature. Early examples were the development of perspective in oil painting and the recycled knowledge of how to make concrete. The Renaissance first began in Florence, in the 14th century. Major centres were Italian city-states such as Venice, Genoa, Milan, Bologna, finally Rome during the Renaissance Papacy. The word Renaissance, literally meaning "Rebirth" in French, first appeared in English in the 1830s. The word also occurs in Jules Michelet's 1855 work, Histoire de France. The word Renaissance has also been extended to other historical and cultural movements, such as the Carolingian Renaissance and the Renaissance of the 12th century. The Renaissance was a cultural movement that profoundly affected intellectual life in the modern period. Renaissance scholars searched in art. However, a subtle shift took place in the way that intellectuals approached religion, reflected in many other areas of cultural life. Political philosophers, most famously Niccolò Machiavelli, sought to describe political life as it really was, to understand it rationally. Others see more general competition between artists and polymaths such as Brunelleschi, Ghiberti, Donatello, Masaccio for artistic commissions as sparking the creativity of the Renaissance. Yet it remains much debated why the Renaissance began in Italy, why it began when it did. Accordingly, several theories have been put forward to explain its origins. During the Renaissance, money and art went hand in hand.
Renaissance
–
David, by Michelangelo (Accademia di Belle Arti, Florence) is a masterpiece of Renaissance and world art.
Renaissance
–
Renaissance
Renaissance
–
Leonardo da Vinci 's Vitruvian Man (c. 1490) shows clearly the effect writers of Antiquity had on Renaissance thinkers. Based on the specifications in Vitruvius ' De architectura (1st century BC), Leonardo tried to draw the perfectly proportioned man.
Renaissance
–
Portrait of a young woman (c. 1480-85) (Simonetta Vespucci) by Sandro Botticelli
26.
Timeline of scientific discoveries
–
The timeline below shows the date of publication of possible major scientific theories and discoveries, along with the discoverer. In many cases, the discoveries spanned several years. 4th century BCE - Mandragora was described by Theophrastus in the fourth century B.C.E. for treatment of wounds, gout, sleeplessness, as a love potion. First use of controlled experiments and reproducibility of its results. 1020s -- Avicenna's The Canon of Medicine 1054 -- early astronomers observe supernova, later correlated to the Crab Nebula. Shen Kuo: Discovers the concepts of true north and magnetic declination. In addition, he develops the first theory of Geomorphology. 1821 – Thomas Johann Seebeck is the first to observe a property of semiconductors. 1873 – Frederick Guthrie discovers thermionic emission. 1873 - Willoughby Smith discovers photoconductivity. 1887 – Albert A. Michelson and Edward W. Morley: lack of evidence for the aether 1888 – Friedrich Reinitzer discovers liquid crystals. 1997 – Roslin Institute: Dolly the sheep was cloned. 1997 – CDF and DØ experiments at Fermilab: Top quark. 1998 – Supernova Cosmology Project and the High-Z Supernova Search Team: discovery of the accelerated expansion of the Universe / Dark Energy. 2000 – The Tau neutrino is discovered by the DONUT collaboration 2001 – The first draft of the Human Genome Project is published.
Timeline of scientific discoveries
27.
Galileo Galilei
–
Galileo Galilei was an Italian polymath: astronomer, physicist, engineer, philosopher, mathematician, he played a major role in the scientific revolution of the seventeenth century. Galileo has been called the "father of the "father of science". Galileo also worked in applied science and technology, inventing an improved military compass and other instruments. Galileo's championing of heliocentrism and Copernicanism was controversial during his lifetime, when most subscribed to either geocentrism or the Tychonic system. He met with opposition from astronomers, who doubted heliocentrism because of the absence of an observed stellar parallax. He was tried by the Inquisition, found "vehemently suspect of heresy", forced to recant. He spent the rest of his life under house arrest. Three of Galileo's five siblings survived infancy. Michelangelo, also became a noted composer although he contributed during Galileo's young adulthood. Michelangelo would occasionally have to support his musical excursions. These financial burdens may have contributed to Galileo's early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence, but he was left with Jacopo Borghini for two years. Galileo then was educated at 35 southeast of Florence. The Italian male given name "Galileo" derives from the Latin "Galilaeus", meaning "of Galilee", a biblically significant region in Northern Israel. The biblical roots of Galileo's name and surname were to become the subject of a famous pun.
Galileo Galilei
–
Portrait of Galileo Galilei by Giusto Sustermans
Galileo Galilei
–
Galileo's beloved elder daughter, Virginia (Sister Maria Celeste), was particularly devoted to her father. She is buried with him in his tomb in the Basilica of Santa Croce, Florence.
Galileo Galilei
–
Galileo Galilei. Portrait by Leoni
Galileo Galilei
–
Cristiano Banti 's 1857 painting Galileo facing the Roman Inquisition
28.
Carl Friedrich Gauss
–
Carl Friedrich Gauss was born on 30 April 1777 in Brunswick, as the son of poor working-class parents. He was confirmed in a church near the school he attended as a child. Gauss was a prodigy. A contested story relates that, when he was eight, he figured out how to add up all the numbers from 1 to 100. He made his first ground-breaking mathematical discoveries while still a teenager. He completed his magnum opus, in 1798 at the age of 21, though it was not published until 1801. This work has shaped the field to the present day. While at university, Gauss independently rediscovered important theorems. Gauss was so pleased by this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle. The 1796 was most productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in theory. On April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic.
Carl Friedrich Gauss
–
Carl Friedrich Gauß (1777–1855), painted by Christian Albrecht Jensen
Carl Friedrich Gauss
–
Statue of Gauss at his birthplace, Brunswick
Carl Friedrich Gauss
–
Title page of Gauss's Disquisitiones Arithmeticae
Carl Friedrich Gauss
–
Gauss's portrait published in Astronomische Nachrichten 1828
29.
Benjamin Peirce
–
Benjamin Peirce was an American mathematician who taught at Harvard University for approximately 50 years. Peirce made contributions to celestial mechanics, statistics, number theory, the philosophy of mathematics. Peirce was the son of Benjamin Peirce, Lydia Ropes Nichols Peirce. After graduating from Harvard, Peirce was subsequently appointed professor of mathematics in 1831. Peirce remained as Harvard professor until his death. Benjamin Peirce is often regarded as the earliest American scientist whose research was recognized as class. Peirce was an apologist for slavery opining that it should be condoned if it was used to allow an elite to pursue scientific enquiry. In theory, Peirce proved there is no odd perfect number with fewer than four prime factors. In algebra, Peirce was notable for the study of associative algebras. He also introduced the Peirce decomposition. In the philosophy of mathematics, Peirce became known for the statement that "Mathematics is the science that draws necessary conclusions". Peirce's definition of mathematics was credited as helping to initiate the consequence-oriented philosophy of pragmatism. Like George Boole, he believed that mathematics could be used to study logic. These ideas were further developed by Charles Sanders Peirce, who noted that logic also includes the study of faulty reasoning. In contrast, the later logicist program of Bertrand Russell attempted to base mathematics on logic.
Benjamin Peirce
–
Benjamin Peirce
Benjamin Peirce
–
With Louis Agassiz
30.
Albert Einstein
–
Albert Einstein was a German-born theoretical physicist. Einstein developed the general theory of one of the two pillars of modern physics. Einstein's work is also known on the philosophy of science. Einstein is best known in popular culture for his mass -- energy equivalence E = mc2. This led him to develop his special theory of relativity. Einstein continued to deal with problems of statistical mechanics and theory, which led to his explanations of particle theory and the motion of molecules. Einstein also investigated the thermal properties of light which laid the foundation of the theory of light. In 1917, he applied the general theory of relativity to model the large-scale structure of the universe. Einstein settled in the U.S. becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project. He largely denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, he signed the Russell -- Einstein Manifesto, which highlighted the danger of nuclear weapons. He was affiliated with the Institute until his death in 1955. He published more than 300 scientific papers along over 150 non-scientific works. On 5 universities and archives announced the release of Einstein's papers, comprising more than 30,000 unique documents.
Albert Einstein
–
Albert Einstein in 1921
Albert Einstein
–
Einstein at the age of 3 in 1882
Albert Einstein
–
Albert Einstein in 1893 (age 14)
Albert Einstein
–
Einstein's matriculation certificate at the age of 17, showing his final grades from the Argovian cantonal school (Aargauische Kantonsschule, on a scale of 1–6, with 6 being the highest possible mark)
31.
Natural science
–
Natural science is a branch of science concerned with the description, prediction, understanding of natural phenomena, based on observational and empirical evidence. Mechanisms such as peer repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two main branches: physical science. Physical science is subdivided into branches, including physics, astronomy, chemistry, science. These branches of natural science may be further divided into more specialized branches. Modern natural science succeeded more classical approaches to natural philosophy, usually traced to ancient Greece. Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, presuppositions, often overlooked, remain requisite in natural science. Systematic collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, so on. "natural history" suggests observational descriptions aimed at popular audiences. Philosophers of science have suggested a number including Karl Popper's controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity, quality control, such as peer review and repeatability of findings, are amongst the most respected criteria in the present-day global scientific community. This field encompasses a set of disciplines that examines phenomena related to living organisms. The scale of study can range from sub-component biophysics up to complex ecologies. However, it was not until the 19th century that biology became a unified science.
Natural science
–
The natural sciences seek to understand how the world and universe around us works. There are five major branches: Chemistry (center), astronomy, earth science, physics, and biology (clockwise from top-left).
Natural science
–
Space missions have been used to image distant locations within the Solar System, such as this Apollo 11 view of Daedalus crater on the far side of the Moon.
Natural science
–
Plato (left) and Aristotle in a 1509 painting by Raphael. Plato rejected inquiry into natural philosophy as against religion, while his student, Aristotle, created a body of work on the natural world that influenced generations of scholars.
Natural science
–
Isaac Newton is widely regarded as one of the most influential scientists of all time.
32.
Engineering
–
The term Engineering is derived from ingeniare, meaning "to contrive, devise". Engineering has existed as humans devised fundamental inventions such as lever, wheel, pulley. Each of these inventions is essentially consistent with the modern definition of engineering. The engineering is derived from the engineer, which itself dates back to 1390 when an engine'er originally referred to "a constructor of military engines." In this context, now obsolete, an "engine" referred to a military machine, i.e. a mechanical contraption used in war. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, e.g. the U.S. Army Corps of Engineers. The word "engine" itself is of even older origin, ultimately deriving from the Latin ingenium, meaning "innate quality, especially mental power, hence a clever invention." The earliest civil engineer known by name is Imhotep. Ancient Greece developed machines in both civilian and military domains. The mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed. The first engine was built by Thomas Savery. The development of this device gave rise to the Industrial Revolution in the coming decades, allowing for the beginnings of mass production. Similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of the Scottish engineer James Watt gave rise to mechanical engineering.
Engineering
–
The steam engine, a major driver in the Industrial Revolution, underscores the importance of engineering in modern history. This beam engine is on display in the Technical University of Madrid.
Engineering
–
Relief map of the Citadel of Lille, designed in 1668 by Vauban, the foremost military engineer of his age.
Engineering
–
The Ancient Romans built aqueducts to bring a steady supply of clean fresh water to cities and towns in the empire.
Engineering
–
The International Space Station represents a modern engineering challenge from many disciplines.
33.
Medicine
–
Medicine is the science and practice of the diagnosis, treatment, prevention of disease. The medicine is derived from Latin medicus, meaning "a physician". Medicine encompasses a variety of care practices evolved to maintain and restore health by the prevention and treatment of illness. Since the advent of modern science, most medicine has become a combination of art and science. Prescientific forms of medicine are now known as traditional medicine and medicine. They remain commonly are thus called alternative medicine. For example, evidence on the effectiveness of acupuncture is generally safe when done by an appropriately trained practitioner. In contrast, treatments outside the bounds of efficacy are termed quackery. Clinical practice varies across the world due to regional differences in culture and technology. In clinical practice, doctors personally assess patients in order to diagnose, treat, prevent disease using clinical judgment. Basic medical devices are typically used. After examination for interviewing for symptoms, the doctor may order medical tests, take a biopsy, or prescribe pharmaceutical drugs or other therapies. Differential diagnosis methods help to rule out conditions based on the information provided. During the encounter, properly informing the patient of all relevant facts is the development of trust. The medical encounter is then documented in the medical record, a legal document in many jurisdictions.
Medicine
–
Early Medicine Bottles
Medicine
Medicine
–
The Doctor, by Sir Luke Fildes (1891)
Medicine
–
The Hospital of Santa Maria della Scala, fresco by Domenico di Bartolo, 1441–1442
34.
Finance
–
Finance is a field that deals with the study of investments. It includes the dynamics of liabilities over time under conditions of different degrees of uncertainty and risk. Finance can also be defined as the science of management. Finance aims to price assets based on their expected rate of return. Finance can be broken into three different sub-categories: public finance, personal finance. Personal finance may also involve paying for a debt obligations. Household flow totals up all the expected sources of income within a year, minus all expected expenses within the same year. From this analysis, the financial planner can determine in what time the personal goals can be accomplished. Adequate protection: the analysis of how to protect a household from unforeseen risks. These risks can be divided into the following: liability, property, death, disability, health and long care. Some of these risks may be self-insurable, while most will require the purchase of an contract. Determining how much insurance to get, at the most cost effective terms requires knowledge of the market for personal insurance. Business owners, professionals, entertainers require specialized insurance professionals to adequately protect themselves. Since insurance also enjoys some tax benefits, utilizing investment products may be a critical piece of the overall investment planning. Tax planning: typically the tax is the single largest expense in a household.
Finance
–
London Stock Exchange, global center of finance.
Finance
Finance
–
Wall Street, the center of American finance.
35.
Social sciences
–
Social science is a major category of academic disciplines, concerned with society and the relationships among individuals within a society. It in turn has many branches, each of, considered a "social science". The social sciences include economics, political science, human geography, demography, sociology. In a wider sense, social science also includes some fields in the humanities such as anthropology, archaeology, linguistics. The term is also sometimes used to refer specifically to the field of sociology, the original'science of society', established in the 19th century. Social scientists use methods resembling those of the natural sciences as tools for society, so define science in its stricter modern sense. In modern academic practice, researchers are often eclectic, using multiple methodologies. The term social research has also acquired a degree of autonomy in its methods. The social sciences developed from prescriptive practices, relating to the social improvement of a group of interacting entities. The beginnings of the social sciences in the 18th century are reflected with articles from other pioneers. The growth of the social sciences is also reflected in other specialized encyclopedias. The modern period saw "social science" first used as a distinct conceptual field. Social science was influenced by positivism, focusing on knowledge based on actual positive sense experience and avoiding the negative; metaphysical speculation was avoided. Following this period, there were five paths of development that sprang forth in the social sciences, influenced by Comte on other fields. One route, taken was the rise of social research.
Social sciences
–
Buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala.
Social sciences
–
A depiction of world's oldest university, the University of Bologna, in Italy
Social sciences
–
This article is about the science studying social groups. For the integrated field of study intended to promote civic competence, see Social studies.
Social sciences
–
A trial at a criminal court, the Old Bailey in London
36.
Applied mathematics
–
Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. The activity of applied mathematics is thus intimately connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory; and applied probability. Quantitative finance is now taught in mathematics departments across universities and mathematical finance is considered a full branch of applied mathematics. Engineering and computer science departments have traditionally made use of applied mathematics. Today, the term "applied mathematics" is used in a broader sense. It includes the classical areas noted above as well as other areas that have become increasingly important in applications. There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the change over time, also by the way universities organize departments, degrees. Many mathematicians distinguish between "applied mathematics,", concerned with mathematical methods, the "applications of mathematics" within science and engineering. Mathematicians such as Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics." Similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called "industrial mathematics".
Applied mathematics
–
Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.
37.
Statistics
–
Statistics is the study of the collection, analysis, interpretation, presentation, organization of data. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of collection in terms of the design of surveys and experiments. Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed to each other". When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation. Inferences on mathematical statistics are made under the framework of theory, which deals with the analysis of random phenomena. Working from a null hypothesis, two basic forms of error are recognized: Type errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are also subject to error. Other types of errors can also be important. Specific techniques have been developed to address these problems. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data. Statistics is a mathematical body of science that pertains as a branch of mathematics.
Statistics
–
Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics
–
More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Statistics
–
Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistics
–
Karl Pearson, a founder of mathematical statistics.
38.
Game theory
–
Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers." Game theory is mainly used in psychology, as well as logic, computer science and biology. Originally, it addressed zero-sum games, in which one person's gains result in losses for the other participants. Modern theory began with the idea regarding the existence of mixed-strategy equilibria by John von Neumann. Von Neumann's original proof used Brouwer fixed-point theorem into compact convex sets, which became a standard method in mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. This theory was developed extensively in the 1950s by many scholars. Game theory was later explicitly applied to biology in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. With the Nobel Memorial Prize in Economic Sciences going to theorist Jean Tirole in 2014, game-theorists have now won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Early discussions of examples of two-person games occurred long before the rise of mathematical theory. James Madison made what we now recognize as a game-theoretic analysis of the ways states can be expected to behave under different systems of taxation. In 1913 Ernst Zermelo published Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels.
Game theory
–
An extensive form game
39.
Pure mathematics
–
Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in practice there is much overlap in the activity of pure and applied mathematicians. To develop accurate models for describing the real world, many applied mathematicians draw on tools and techniques that are often considered to be "pure" mathematics. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. Ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between "arithmetic", now called number theory, "logistic", now called arithmetic. The term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a separate discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. In fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training. One central concept in pure mathematics is the idea of generality; pure mathematics often exhibits a trend towards increased generality.
Pure mathematics
–
An illustration of the Banach–Tarski paradox, a famous result in pure mathematics. Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world.
40.
History of mathematics
–
Before the worldwide spread of knowledge, written examples of mathematical developments have come to light only in a few locales. The most mathematical texts available are Plimpton 322, the Moscow Mathematical Papyrus. All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. Greek mathematics greatly expanded the matter of mathematics. Chinese mathematics made early contributions, including a system. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. The origins of mathematical thought lie in the concepts of form. Modern studies of animal cognition have shown that these concepts are not unique to humans. Such concepts would have been part of everyday life in hunter-gatherer societies. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. Common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of a six-month calendar. Predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. Undisputed mathematical documents are from Babylonian and dynastic Egyptian sources.
History of mathematics
–
A proof from Euclid 's Elements, widely considered the most influential textbook of all time.
History of mathematics
–
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
History of mathematics
–
Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.
History of mathematics
–
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.
41.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so little reliable information is known about him. He travelled, visiting Egypt and Greece, maybe India. Around 530 BC, there established some kind of school or guild. In 520 BC, he returned to Samos. Pythagoras made influential contributions in the late 6th century BC. He is best known for the Pythagorean theorem which bears his name. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his successors. Some accounts mention that numbers were important. Burkert states that Aristoxenus and Dicaearchus are the most important accounts. Aristotle had written a separate work On the Pythagoreans, no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans. Dicaearchus, Aristoxenus, Heraclides Ponticus had written on the same subject. According to Clement of Alexandria, Pythagoras was a disciple of Soches, Plato of Sechnuphis of Heliopolis. Herodotus, other early writers agree that Pythagoras was the son of Mnesarchus, born on a Greek island in the eastern Aegean called Samos.
Pythagoras
–
Bust of Pythagoras of Samos in the Capitoline Museums, Rome.
Pythagoras
–
Bust of Pythagoras, Vatican
Pythagoras
–
A scene at the Chartres Cathedral shows a philosopher, on one of the archivolts over the right door of the west portal at Chartres, which has been attributed to depict Pythagoras.
Pythagoras
–
Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.
42.
Pythagorean theorem
–
In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework. Mesopotamian, Indian and Chinese mathematicians all, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including algebraic proofs, with some dating back thousands of years. He may well have been the first to prove it. In any event, the proof is called a proof by rearrangement. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem, Q.E.D. That Pythagoras originated this very simple proof is sometimes inferred from the writings of mathematician Proclus. This is known as the Pythagorean one. If the length of b are known, then c can be calculated as c = a 2 + b 2. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other; The Pythagorean Proposition contains 370 proofs.
Pythagorean theorem
–
The Plimpton 322 tablet records Pythagorean triples from Babylonian times.
Pythagorean theorem
–
Pythagorean theorem The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Pythagorean theorem
–
Geometric proof of the Pythagorean theorem from the Zhou Bi Suan Jing.
Pythagorean theorem
–
Exhibit on the Pythagorean theorem at the Universum museum in Mexico City
43.
Mayan numerals
–
The Maya numeral system is a vigesimal positional numeral system used by the Pre-Columbian Maya civilization. The numerals are made up of three symbols; zero, one and five. For example, thirteen is written as three dots in a horizontal row above two horizontal lines stacked above each other. Numbers after 19 were written vertically in powers of twenty. For example, thirty-three would be written as one dot above three dots, which are in turn atop two lines. The first dot represents "one twenty" or "1×20", added to three dots and two bars, or thirteen. Therefore, + 13 = 33. Upon reaching 202 or 400, another row is started. The number 429 would be written as one dot above one dot above four dots and a bar, or + + 9 = 429. The powers of twenty are numerals, just as the Hindu-Arabic numeral system uses powers of tens. Other than the bar and dot notation, Maya numerals can be illustrated by face type glyphs or pictures. The face glyph for a number represents the deity associated with the number. These face number glyphs were rarely used, are mostly seen on some of the most elaborate monumental carving. Addition and subtraction: Adding and subtracting numbers below 20 using Maya numerals is very simple. If four or more bars result, four bars are removed and a dot is added to the next higher row.
Mayan numerals
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Numeral systems
Mayan numerals
–
Maya numerals
Mayan numerals
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Detail showing three columns of glyphs from La Mojarra Stela 1. The left column uses Maya numerals to show a Long Count date of 8.5.16.9.7, or 156 CE.
44.
Tally sticks
–
A tally stick was an ancient memory aid device used to record and document numbers, quantities, or even messages. Tally sticks first appear as animal bones carved with notches during the Upper Paleolithic; a notable example is the Ishango Bone. Tallies have been used for numerous purposes such as messaging and scheduling, especially in financial and legal transactions, to the point of being currency. Principally, there are two different kinds of tally sticks: the tally. A common form of the same kind of primitive counting device is seen in various kinds of prayer beads. It is a brown length of the fibula of a baboon. It has a series of tally marks carved in three columns running the length of the tool. It was found in 1960 in Belgian Congo. The Lebombo Bone is a baboon's fibula with 29 distinct notches, discovered within the Border Cave in the Lebombo Mountains of Swaziland. The so-called Wolf bone is a prehistoric artefact discovered in 1937 in Czechoslovakia during excavations at Vestonice, Moravia, led by Karl Absolon. Dated to the Aurignacian, approximately 30,000 years ago, the bone is marked with 55 marks which some believe to be tally marks. The head of an ivory Venus figurine was excavated close to the bone. The single stick was an elongated piece of bone, ivory, stone, marked with a system of notches. The single tally stick serves predominantly mnemonic purposes. Related to the single tally concept are messenger sticks, the knotted cords, khipus or quipus, as used by the Inca.
Tally sticks
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Medieval English split tally stick (front and reverse view). The stick is notched and inscribed to record a debt owed to the rural dean of Preston Candover, Hampshire, of a tithe of 20 d each on 32 sheep, amounting to a total sum of £2 13s. 4d.
Tally sticks
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Single and split tallies from the Swiss Alps, 18th to early 20th century (Swiss Alpine Museum)
Tally sticks
–
Entrance gates to the UK National Archives, Kew, from Ruskin Avenue. The notched vertical elements were inspired by medieval tally sticks.
45.
Prehistoric
–
Prehistory means literally "before history", from the Latin word for "before," præ, Greek ιστορία. Neighbouring civilisations were the first to follow. Most other civilisations reached the end of prehistory during the Iron Age. The period when a culture is written about by others, but has not developed its own writing is often known as the protohistory of the culture. By definition, there are no written records from human prehistory, so dating of prehistoric materials is crucial. Clear techniques for dating were not well-developed until the 19th century. This article is concerned with human prehistory as defined here above. There are separate articles for the overall history of the Earth and the history of life before humans. The first use of the prehistory in English, however, occurred in 1836. Some scholars are beginning to make more use of evidence from the social sciences. This view has been articulated by advocates of deep history. Human population geneticists and historical linguists are also providing valuable insight for these questions. Restricted to artifacts rather than written records, prehistory is anonymous. Because of this, reference terms that prehistorians use, such as Neanderthal or Iron Age are modern labels with definitions sometimes subject to debate. "Palaeolithic" means "Old Stone Age," and begins with the first use of stone tools.
Prehistoric
–
Massive stone pillars at Göbekli Tepe, in southeast Turkey, erected for ritual use by early Neolithic people 11,000 years ago.
Prehistoric
–
A prehistoric man and boy.
Prehistoric
–
Dugout canoe
Prehistoric
–
Entrance to the Ġgantija phase temple complex of Hagar Qim, Malta, 3900 BC.
46.
Season
–
A season is a division of the year marked by changes in weather, ecology and hours of daylight. During May, June, July, the northern hemisphere is exposed to more direct sunlight because the hemisphere faces the sun. The same is true of the southern hemisphere in November, December, January. It is the tilt of the Earth that causes the Sun to be higher during the summer months which increases the solar flux. In subpolar regions, four calendar-based seasons are generally recognized: spring, summer, autumn and winter. In Canadian English, fall is sometimes used as a synonym for both autumn and autumnal. Ecologists often use a six-season model for temperate climate regions that includes late summer as distinct seasons along with the traditional four. Hot regions have three seasons; the rainy season and the dry season, in some tropical areas, a cool or mild season. In some parts of the world, special "seasons" are loosely defined based on important events such as a hurricane season, a wildfire season. The change of seasons was often attended by ritual. The seasons result from the Earth's axis of rotation being tilted with respect by an angle of approximately 23.5 degrees. Regardless of the time of year, the southern hemispheres always experience opposite seasons. For approximately half of the year, the northern hemisphere tips with the maximum amount occurring on about June 21. For the other half of the year, the same in the southern hemisphere instead of the northern, with the maximum around December 21. The two instants when the Sun is directly overhead at the Equator are the equinoxes.
Season
–
Red and green trees in spring
Season
–
A tree in winter
Season
–
The six ecological seasons
Season
–
The four calendar seasons, depicted in an ancient Roman mosaic from Tunisia.
47.
Before Christ
–
The terms anno Domini and before Christ are used to label or number years in the Julian and Gregorian calendars. The anno Domini is Medieval Latin, which means in the year of the Lord but is often translated as in the year of our Lord. It is occasionally set out more fully as anno Domini nostri Iesu Christi. Dionysius Exiguus of Scythia Minor introduced the AD system in AD 525, counting the years since the birth of Christ. There is no zero in this scheme, so the year AD 1 immediately follows the year 1 BC. This dating system was not widely used until after 800. The Gregorian calendar is the most widely used calendar in the today. Traditionally, English followed Latin usage by placing the "AD" abbreviation before the number. However, BC is placed after the number, which also preserves syntactic order. The abbreviation is also widely used after the number of a millennium, as in "fourth century AD" or "second millennium AD". Because BC is the English abbreviation for Before Christ, it is sometimes incorrectly concluded that AD means after the death of Jesus. ISO 8601 avoid words or abbreviations related to Christianity, but use the same numbers for AD years. The Anno Domini dating system was devised by Dionysius Exiguus to enumerate the years in his Easter table. The last year of Diocletian 247, was immediately followed by the first year of his table, AD 532. Thus Dionysius implied that Jesus' Incarnation occurred 525 years earlier, without stating the specific year during which his conception occurred.
Before Christ
–
Dionysius Exiguus invented Anno Domini years to date Easter.
Before Christ
–
Anno Domini inscription at a cathedral in Carinthia, Austria.
Before Christ
–
Statue of Charlemagne by Agostino Cornacchini (1725), at St. Peter's Basilica, Vatican, Italy. Charlemagne promoted the usage of the Anno Domini epoch throughout the Carolingian Empire
48.
Babylonia
–
Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia. A Amorite-ruled state emerged in 1894 BC, which contained at this time the minor city of Babylon. Babylon greatly expanded in the first half of the 18th century BC becoming a major capital city. During the reign of Hammurabi and afterwards, Babylonia was called Māt Akkadī "the country of Akkad" in the Akkadian language. It was often involved in northern Mesopotamia. During the 3rd millennium BC, an cultural symbiosis occurred between Sumerian and Akkadian-speakers, which included widespread bilingualism. This has prompted scholars to refer in the third millennium as a sprachbund. The empire eventually disintegrated due to economic decline, civil war, followed by attacks by the Gutians from the Zagros Mountains. Sumer ejected the Gutians from southern Mesopotamia. They also seem to have gained ascendancy over most of the territory of the Akkadian kings of Assyria for a time. The states of the south were unable to stem the Amorite advance. I purified their copper. There is no explicit record of that. More recently, the text has been taken to mean that Asshur supplied the south with copper from tax duties. These policies were continued by Ikunum.
Babylonia
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The extent of the Babylonian Empire at the start and end of Hammurabi's reign
Babylonia
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Old Babylonian Cylinder Seal, hematite, The king makes an animal offering to Shamash. This seal was probably made in a workshop at Sippar.
Babylonia
–
Geography
49.
Taxation
–
A failure to pay, or evasion of or resistance to taxation, is usually punishable by law. Taxes may be paid in money or as its labour equivalent. The economic definition of taxes differ in that economists do not regard many transfers to governments as taxes. For example, some transfers to the public sector are comparable to prices. Examples include tuition at public fees for utilities provided by local governments. Governments also obtain resources through voluntary gifts, by imposing penalties, by borrowing, by confiscating wealth. In modern taxation systems, governments levy taxes in money; but corvée taxation are characteristic of traditional or pre-capitalist states and their functional equivalents. The government expenditure of taxes raised is often highly debated in politics and economics. When taxes are not fully paid, the state may impose criminal penalties on the non-paying entity or individual. The levying of taxes aims to raise revenue to governing. Their functional equivalents throughout history have used money provided by taxation to carry out many functions. A government's ability to raise taxes is known as fiscal capacity. When expenditures exceed revenue, a government accumulates debt. A portion of taxes may be used to service past debts. Governments also use taxes to fund public services.
Taxation
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Pieter Brueghel the Younger, The tax collector's office, 1640
Taxation
–
Taxation
Taxation
–
Egyptian peasants seized for non-payment of taxes. (Pyramid Age)
50.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. The objects of interest include planets, moons, stars, comets; while the phenomena include supernovae explosions, gamma ray bursts, cosmic microwave background radiation. More generally all astronomical phenomena that originate outside Earth's atmosphere is within the perview of astronomy. Physical cosmology, is concerned with the study of the Universe as a whole. Astronomy is the oldest of the natural sciences. The early civilizations in recorded history, such as Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, Maya performed methodical observations of the night sky. During the 20th century, the field of professional astronomy split into theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, then analyzed using basic principles of physics. Theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the observational observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can still play an active role, especially in the observation of transient phenomena. Amateur astronomers have contributed to many important astronomical discoveries, such as finding new comets. Astronomy means "law of the stars". Astronomy should not be confused with the belief system which claims that human affairs are correlated with the positions of celestial objects.
Astronomy
–
A star -forming region in the Large Magellanic Cloud, an irregular galaxy.
Astronomy
–
A giant Hubble mosaic of the Crab Nebula, a supernova remnant
Astronomy
–
19th century Sydney Observatory, Australia (1873)
Astronomy
–
19th century Quito Astronomical Observatory is located 12 minutes south of the Equator in Quito, Ecuador.
51.
Trade
–
A network that allows trade is called a market. Barter, saw the direct exchange of goods and services for other goods and services. Barter is trading things without the use of money. Later one side of the barter started to involve precious metals, which gained symbolic well as practical importance. Modern traders generally negotiate through a medium such as money. As a result, buying can be separated from earning. The invention of money greatly promoted trade. Trade between two traders is called bilateral trade, while trade between more than two traders is called multilateral trade. As such, trade at market prices between locations can benefit both locations. Trade originated with human communication in prehistoric times. Trading was the main facility of prehistoric people, who bartered goods and services before the innovation of modern-day currency. Peter Watson dates the history of long-distance commerce from circa 150,000 years ago. Trade is believed to have taken place throughout much of recorded human history. There is evidence during the stone age. Trade in obsidian is believed to have taken place in Guinea from 17,000 BCE.
Trade
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A San Juan de Dios Market in Guadalajara, Jalisco.
Trade
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A trader in Germany, 16th century
Trade
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The Liberty to Trade as Buttressed by National Law (1909) by George Howard Earle, Jr.
Trade
–
A map of the Silk Road trade route between Europe and Asia.
52.
Land measurement
–
Surveying or land surveying is the technique, profession, science of determining the terrestrial or three-dimensional position of points and the distances and angles between them. A land surveying professional is called a surveyor. Surveyors work with elements of geometry, trigonometry, regression analysis, physics, engineering, metrology, the law. They use equipment like total stations, robotic total stations, GPS receivers, retroreflectors, 3D scanners, radios, handheld tablets, digital levels, drones, surveying software. Surveying has been an element in the development of the human environment since the beginning of recorded history. The execution of most forms of construction require it. It is also used in transport, communications, the definition of legal boundaries for land ownership. It is an important tool for research in many scientific disciplines. Basic surveyance has occurred since humans built the large structures. The prehistoric monument at Stonehenge was set out by prehistoric surveyors using peg and geometry. In ancient Egypt, a stretcher would use simple geometry to re-establish boundaries after the annual floods of the Nile River. North-south orientation of the Great Pyramid of Giza, built c. 2700 BC, affirm the Egyptians' command of surveying. The Groma instrument originated in Mesopotamia. The mathematician Liu Hui described ways of measuring distant objects in The Sea Island Mathematical Manual, published in 263 AD. The Romans recognized land surveyors as a profession.
Land measurement
–
A surveyor at work with an infrared reflector used for distance measurement.
Land measurement
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Table of Surveying, 1728 Cyclopaedia
Land measurement
–
A map of India showing the Great Trigonometrical Survey, produced in 1870
Land measurement
–
A German engineer surveying during the First World War, 1918
53.
Painting
–
Painting is the practice of applying paint, pigment, color or other medium to a solid surface. Other implements, such as knives, sponges, airbrushes, can be used. The forms are numerous. Drawing, gesture, composition, abstraction, among other aesthetic modes, may serve to manifest the expressive and conceptual intention of the practitioner. Paintings can be naturalistic and representational, photographic, abstract, narrative, symbolistic, political in nature. A portion of the history of painting in both Eastern and Western art is dominated by spiritual ideas. In art, the painting describes both the act and the result of the action. The painting is also used outside of art as a common trade among craftsmen and builders. What enables painting is the representation of intensity. Every point in space has different intensity, which can be all the gray shades between. In practice, painters can articulate shapes by juxtaposing surfaces of different intensity; by using just color one can only represent symbolic shapes. Thus, the basic means of painting are distinct from ideological means, such as geometrical figures, symbols. In technical drawing, thickness of line is also ideal, demarcating ideal outlines of an object within a perceptual frame different from the one used by painters. Tone are the essence of painting as pitch and rhythm are the essence of music. Color is highly subjective, but has psychological effects, although these can differ from one culture to the next.
Painting
–
The Mona Lisa, by Leonardo da Vinci, is one of the most recognizable paintings in the world.
Painting
–
Chen Hongshou (1598–1652), Leaf album painting (Ming Dynasty)
Painting
–
Circus Sideshow (French: Parade de cirque), Georges Seurat, 1887–88
Painting
–
Cave painting of aurochs, (French: Bos primigenius primigenius), Lascaux, France, prehistoric art
54.
Weaving
–
Similar methods are knitting, braiding or plaiting. The lateral threads are the weft or filling. The method in which these threads are inter woven affects the characteristics of the cloth. Cloth is usually woven on a device that holds the warp threads in place while filling threads are woven through them. A band which meets this definition of cloth can also be made using other methods, including tablet weaving, back-strap, or other techniques without looms. The way the filling threads interlace with each other is called the weave. The majority of woven products are created with one of three basic weaves: satin weave, or twill. Cloth can be plain, or can be woven in decorative or artistic design. One thread is called an end and one weft thread is called a pick. The warp threads are held taut and typically in a loom. There are many types of looms. Weaving can be summarized as a repetition of these three actions, also called the primary motion of the loom. Beating-up or battening: where the weft is pushed up against the fell of the cloth by the reed. Repeating these actions form a mesh but without beating-up, the final distance between the adjacent wefts would be irregular and far too large. The warp-beam is a wooden or metal cylinder on the back of the loom on which the warp is delivered.
Weaving
–
Warp and weft in plain weaving
Weaving
–
A satin weave, common for silk, each warp thread floats over 16 weft threads.
Weaving
–
A woman weaving with a free-standing loom (type of hand loom)
Weaving
–
An Indian weaver preparing his warp on a pegged loom (another type of hand loom)
55.
Babylonian mathematics
–
Babylonian mathematical texts are plentiful and well edited. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia. In contrast to the scarcity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed while the clay was moist, baked hard in an oven or by the heat of the sun. The Babylonian tablet YBC 7289 gives an approximation to 2 accurate to three significant sexagesimal digits. Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC. Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. The Babylonian system of mathematics was sexagesimal numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values. The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC.
Babylonian mathematics
–
Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296... The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888...
56.
Elementary arithmetic
–
Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, division. It should not be confused with elementary arithmetic. Elementary arithmetic starts with the written symbols that represent them. Elementary arithmetic also includes negative numbers, which can be represented on a number line. Digits are the entire set of symbols used to represent numbers. In modern usage, the most frequently used form of these digits is the Western style. Each single digit matches:0, zero. Used in the absence of objects to be counted. For example, a different way of saying "there are no sticks here", is to say one. Applied to a single item. For example, here is one stick: two. Applied to a pair of items. Here are two sticks: I I3, three. Applied to three items. Here are three sticks: I I I4, four.
Elementary arithmetic
–
The basic elementary arithmetic symbols.
57.
Addition
–
Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two whole numbers is the total amount of those quantities combined. In the picture on the right, there is a combination of three apples and two apples together, making a total of five apples. This observation is equivalent to the mathematical expression "+ 2 = 5" i.e. "3 add 2 is equal to 5". Besides counting fruits, addition can also represent combining physical objects. In arithmetic, rules for addition involving negative numbers have been devised amongst others. In algebra, addition is studied more abstractly. Addition has important properties. Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as multiplication. Performing addition is one of the simplest numerical tasks. In primary education, students are taught to add numbers in the decimal system, progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day. Addition is written using the plus sign" +" in infix notation. The result is expressed with an equals sign.
Addition
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Part of Charles Babbage's Difference Engine including the addition and carry mechanisms
Addition
–
The plus sign
Addition
–
A circular slide rule
58.
Subtraction
–
Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign. Therefore, 5 − 2 = 3. Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number. Subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, generalizing up beyond. General binary operations that continue these patterns are studied in abstract algebra. Performing subtraction is one of the simplest numerical tasks. Subtraction of very small numbers is accessible to young children. In primary education, students are taught to subtract numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Subtraction is written using the minus" −" in infix notation. The result is expressed with an equals sign.
Subtraction
–
Placard outside shop in Bordeaux advertising subtraction of 20% from the price of a second perfume
Subtraction
–
"5 − 2 = 3" (verbally, "five minus two equals three")
Subtraction
–
1 + … = 3
59.
Multiplication
–
Multiplication is one of the four elementary, mathematical operations of arithmetic; with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths. The area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The inverse operation of multiplication is division. For example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number. Multiplication is also defined for other types of numbers, such as complex numbers, more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter. A listing of the different kinds of products that are used in mathematics is given in the page. In arithmetic, multiplication is often written using the sign" ×" in notation. There are mathematical notations for multiplication: Multiplication is also denoted by dot signs, usually a middle-position dot: 5 ⋅ 5. When the character is not accessible, the interpunct is used. In other countries that use a comma as a decimal mark, either the period or a middle dot is used for multiplication. In algebra, multiplication involving variables is often written as a juxtaposition. The notation can also be used for quantities that are surrounded by parentheses.
Multiplication
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4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.
Multiplication
–
Four bags of three marbles gives twelve marbles (4 × 3 = 12).
60.
Division (mathematics)
–
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, multiplication. The division of two natural numbers is the process of calculating the number of times one number is contained within one another. Division can also be thought of as the process of evaluating a fraction, fractional notation is commonly used to represent division. Division is the inverse of multiplication; if a × b = c, then a = c ÷ b, as b is not zero. In division, the dividend is divided by the divisor to get a quotient. In the above example, 20 is the dividend, five is the divisor, the quotient is four. Besides dividing apples, division can be applied to other physical and abstract objects. Teaching division usually leads to the concept of fractions being introduced to school pupils. Unlike multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder. To complete the division of the remainder, the system is extended to include rational numbers as they are more generally called. Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. A fraction is a division expression where both dividend and divisor are integers, there is no implication that the division must be evaluated further. ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent itself, for instance as a label on a key of a calculator.
Division (mathematics)
–
This article is about the arithmetical operation. For other uses, see Division (disambiguation).
61.
Numeracy
–
Numeracy is the ability to reason and to apply simple numerical concepts. Basic numeracy skills consist of comprehending fundamental arithmetics like subtraction, multiplication, division. Substantial aspects of numeracy also include number sense, operation sense, computation, measurement, geometry, statistics. A numerically literate person can respond to the mathematical demands of life. By contrast, innumeracy can have a negative impact. Numeracy has risk perception towards health decisions. For example, innumeracy may negatively affect economic choices. Humans have evolved in two major ways from observation. They are: Precise representation of the quantity of individual items. Approximate representations of numerical magnitude imply that one can relatively comprehend an amount if the number is large. For example, one experiment showed arrays of many dots. After briefly observinging them, both groups could accurately estimate the approximate number of dots. However, distinguishing differences between large numbers of dots proved to be more challenging. Precise representations of distinct individuals demonstrate that people are more accurate in estimating distinguishing differences when the numbers are relatively small. In one experiment, an experimenter presented an infant with two piles of crackers, one with two crackers the other with three.
Numeracy
–
Children in Laos have fun as they improve numeracy with "Number Bingo." They roll three dice, construct an equation from the numbers to produce a new number, then cover that number on the board, trying to get 4 in a row.
62.
Writing
–
Writing is a medium of human communication that represents language and emotion through the inscription or recording of signs and symbols. In most languages, writing is a spoken language. Writing is not a language but a form of technology that developed as tools developed with human society. The recipient of text is called a reader. Motivations for writing include publication, storytelling, diary. Writing has been instrumental in maintaining culture, dissemination of knowledge through the media and the formation of legal systems. As human societies emerged, the development of writing was driven by pragmatic exigencies such as exchanging information, maintaining financial accounts, recording history. In Mesoamerica writing may have evolved through calendrics and a political necessity for recording historical and environmental events. H.G. Wells argued that writing has the ability to "put agreements, commandments on record. It made the growth of states larger than the old city states possible. It made a historical consciousness possible. The command of his seal could go far beyond his sight and voice and could survive his death". The major writing systems -- methods of inscription -- broadly fall into four categories: logographic, syllabic, featural. Another category, ideographic, has never been developed sufficiently to represent language. A sixth category, pictographic, often forms the core of logographies.
Writing
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Writing with a pen
Writing
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Olin Levi Warner, tympanum representing Writing, above exterior of main entrance doors, Thomas Jefferson Building, Washington DC, 1896.
Writing
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Globular envelope with a cluster of accountancy tokens, Uruk period, from Susa. Louvre Museum
Writing
–
Narmer Palette, with the two serpopards representing unification of Upper and Lower Egypt, 3000 B. C.
63.
Numeral system
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The number the numeral represents is called its value. Such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system. Two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the place-value notation in the 5th century and a century later Brahmagupta introduced the symbol for zero. The Arabs adopted and modified it. The Arabs call the Hindu numeral system. The Arabs spread them with them. The Western world modified them and called them the Arabic numerals, as they learned from them. Hence the western system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, still used in India and neighboring Nepal. The simplest system is the unary system, in which every natural number is represented by a corresponding number of symbols. If the / is chosen, for example, seven would be represented by / / / / / / /. Tally marks represent one such system still in common use. The unary system is only useful for small numbers, although it plays an important role in theoretical computer science.
Numeral system
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Numeral systems
64.
Ancient Egypt
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It is one of six civilizations to arise independently. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh Narmer. In the aftermath of Alexander one of his generals, Ptolemy Soter, established himself as the new ruler of Egypt. This Greek Ptolemaic Kingdom ruled Egypt until 30 BC, when, under Cleopatra, it became a Roman province. The success of Egyptian civilization came partly from its ability to adapt to the conditions of the Nile River valley for agriculture. The predictable flooding and controlled irrigation of the fertile valley produced surplus crops, which supported social development and culture. Egypt left a lasting legacy. Its antiquities carried off to far corners of the world. Its monumental ruins have inspired the imaginations of writers for centuries. The Nile has been the lifeline of its region for much of human history. Nomadic human hunter-gatherers began living in the Nile valley through the end of the Middle Pleistocene some 120,000 years ago. In Predynastic and Early Dynastic times, the Egyptian climate was much less arid than it is today. Large regions of Egypt were traversed by herds of grazing ungulates. The Nile region supported large populations of waterfowl. This is also the period when many animals were first domesticated.
Ancient Egypt
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The Great Sphinx and the pyramids of Giza are among the most recognizable symbols of the civilization of ancient Egypt.
Ancient Egypt
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A typical Naqada II jar decorated with gazelles. (Predynastic Period)
Ancient Egypt
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The Narmer Palette depicts the unification of the Two Lands.
65.
Middle Kingdom of Egypt
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Some scholars also include the Thirteenth Dynasty of Egypt wholly into this period as well, in which case the Middle Kingdom would finish c. 1650, while others only include it until Merneferre Ay c. 1700 BC, last king of this dynasty to be attested in both Upper and Lower Egypt. During the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards, centered on el-Lisht. After the collapse of the Old Kingdom, Egypt entered a period of decentralization called the First Intermediate Period. Towards the end of this period, two rival dynasties, known as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt to the Tenth Nome of Upper Egypt. To the north, Lower Egypt was ruled by the 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B.C. During Mentuhotep II's fourteenth he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. For this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south far as the Second Cataract in Nubia, which had gained its independence during the First Intermediate Period. He also restored Egyptian hegemony over the Sinai region, lost since the end of the Old Kingdom. Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all Egyptian king lists. The Turin Papyrus claims after Mentuhotep III came "seven kingless years."
Middle Kingdom of Egypt
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An Osiride statue of the first pharaoh of the Middle Kingdom, Mentuhotep II
Middle Kingdom of Egypt
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The head of a statue of Senusret I.
Middle Kingdom of Egypt
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Statue head of Senusret III
66.
Rhind Mathematical Papyrus
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The Rhind Mathematical Papyrus is one of the best known examples of Egyptian mathematics. It dates to around BC. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt. It was copied from a now-lost text from the reign of king Amenemhat III. Written in the hieratic script, this Egyptian manuscript consists of multiple parts which in total make it over 5m long. The papyrus began to be mathematically translated in the late 19th century. The mathematical aspect remains incomplete in several respects. The Ahmose writes this copy. A handful of these stand out. A more recent overview of the Rhind Papyrus was published by Robins and Shute. The first part of the Rhind papyrus consists of a collection of 21 arithmetic and 20 algebraic problems. The problems start out followed by completion problems and more involved linear equations. The first part of the papyrus is taken up by the 2/n table.
Rhind Mathematical Papyrus
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A portion of the Rhind Papyrus
Rhind Mathematical Papyrus
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Building
67.
Ancient Greeks
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Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages to c. 5th century BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in ancient Greece is the period of Classical Greece, which flourished during the 5th to 4th centuries BC. Classical Greece began with the era of the Persian Wars. Because of conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished from Central Asia to the western end of the Mediterranean Sea. Classical Antiquity in the Mediterranean region is commonly considered to have begun in the 8th century BC and ended in the 6th century AD. Classical Antiquity in Greece is preceded by the Greek Dark Ages, archaeologically characterised by the protogeometric and geometric styles of designs on pottery. The end of the Dark Ages is also frequently dated to 776 BC, the year of the first Olympic Games. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures in stiff, hieratic poses with the dreamlike "archaic smile". The Archaic period is often taken to end with the overthrow of the last tyrant of Athens and the start of Athenian Democracy in 508 BC. This period saw the Greco-Persian Wars and the Rise of Macedon. Following the Classical period was the Hellenistic period, during which Greek culture and power expanded into the Near and Middle East. This period begins with the death of Alexander and ends with the Roman conquest. Herodotus is widely known as the "father of history": his Histories are eponymous of the entire field. Herodotus was succeeded by authors such as Thucydides, Xenophon, Demosthenes, Plato and Aristotle.
Ancient Greeks
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The Parthenon, a temple dedicated to Athena, located on the Acropolis in Athens, is one of the most representative symbols of the culture and sophistication of the ancient Greeks.
Ancient Greeks
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Dipylon Vase of the late Geometric period, or the beginning of the Archaic period, c. 750 BC.
Ancient Greeks
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Political geography of ancient Greece in the Archaic and Classical periods
68.
Muhammad ibn Musa al-Khwarizmi
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In the 12th century, Latin translations of his work on the Indian numerals introduced the positional system to the Western world. The Compendious Book on Calculation by Completion and Balancing presented the systematic solution of linear and quadratic equations in Arabic. He is often considered one of the fathers of algebra. He wrote on astrology. Some words reflect the importance of al-Khwārizmī's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of guarismo and of algarismo, both meaning digit. Few details of al-Khwārizmī's life are known with certainty. He was born in a Persian family and Ibn al-Nadim gives his birthplace as Khwarezm in Greater Khorasan. Muhammad ibn Jarir al-Tabari gives his name as Muḥammad ibn Musá al-Khwārizmiyy al-Majūsiyy al-Quṭrubbaliyy. The al-Qutrubbulli could indicate he might instead have come from a viticulture district near Baghdad. Recently, G. J. Toomer... with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Ibn al-Nadīm's Kitāb al-Fihrist includes a short biography on al-Khwārizmī together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833.
Muhammad ibn Musa al-Khwarizmi
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A page from al-Khwārizmī's Algebra
Muhammad ibn Musa al-Khwarizmi
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A stamp issued September 6, 1983 in the Soviet Union, commemorating al-Khwārizmī's (approximate) 1200th birthday.
Muhammad ibn Musa al-Khwarizmi
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A 15th-century version of Ptolemy 's Geography for comparison.
69.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. The letter c is a constant, the speed of light in a vacuum. Algebra gives methods for expressing formulas that are much easier than the older method of writing everything out in words. The algebra is also used in certain specialized ways. A mathematician who does research in algebra is called an algebraist. The algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wa ` l-muḳābala by al-Khwarizmi. The word entered the English language from either Spanish, Italian, or Medieval Latin. It originally referred to the surgical procedure of setting dislocated bones. The mathematical meaning was first recorded in the sixteenth century. The word "algebra" has related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, "algebra" names a broad part of mathematics.
Algebra
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A page from Al-Khwārizmī 's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
Algebra
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Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.
70.
Islamic Golden Age
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A contemporary scholars place the end of the Islamic Golden Age as late as the end of 15th to 16th centuries. During the 20th century, the term was used only occasionally, often referred to the early military successes of the Rashidun caliphs. Definitions may still vary considerably. During this period, the Muslims showed a strong interest in assimilating the scientific knowledge of the civilizations, conquered. The Muslim government heavily patronized scholars. Notable translators, such as Hunayn ibn Ishaq, had salaries that are estimated to be the equivalent of professional athletes today. The House of Wisdom was a library, translation institute, academy established in Abbasid-era Baghdad, his son al-Ma ` mun. Islamic paper makers devised assembly-line methods of hand-copying manuscripts to turn out editions far larger than any available in Europe for centuries. It was from these countries that the rest of the world learned to make paper from linen. Other philosophers such as al-Kindi and al-Farabi combined Aristotelianism and Neoplatonism with other ideas introduced through Islam. Philosophic literature was translated into Latin and Ladino, contributing to the development of modern European philosophy. During this period, non-Muslims were allowed to flourish relative to treatment of religious minorities in the Byzantine Empire. The philosopher Moses Maimonides, who lived in Andalusia, is an example. In epistemology, Ibn Tufail in response Ibn al-Nafis wrote the novel Theologus Autodidactus. Both were concerning autodidacticism as illuminated through the life of a feral child spontaneously generated in a cave on a island.
Islamic Golden Age
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Scholars at an Abbasid library. Maqamat of al-Hariri Illustration by Yahyá al-Wasiti, Baghdad 1237
Islamic Golden Age
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A manuscript written on paper during the Abbasid Era.
Islamic Golden Age
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Islamic architecture in Alhambra, Al-Andalus, in modern-day Spain
Islamic Golden Age
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The eye, according to Hunain ibn Ishaq. From a manuscript dated circa 1200.
71.
Science
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Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The formal sciences are often excluded as they do not depend on empirical observations. Disciplines which use science, like medicine, may also be considered to be applied sciences. However, during the Islamic Golden Age foundations for the scientific method were laid by Ibn al-Haytham in his Book of Optics. In the 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws. It was during this time that scientific disciplines such as biology, physics reached their modern shapes. Science in a broad sense existed in many historical civilizations. Modern science is successful in its results, so it now defines what science is in the strictest sense of the term. Science in its original sense was a word for a type of knowledge rather than a specialized word for the pursuit of such knowledge. In particular, it was the type of knowledge which people can communicate to share. For example, knowledge about the working of natural things was led to the development of complex abstract thought. This is shown by the construction of techniques for making poisonous plants edible, buildings such as the Pyramids. They were mainly theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen as a more appropriate interest for lower class artisans. A clear-cut distinction between empirical science was made by the pre-Socratic philosopher Parmenides.
Science
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Maize, known in some English-speaking countries as corn, is a large grain plant domesticated by indigenous peoples in Mesoamerica in prehistoric times.
Science
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The scale of the universe mapped to the branches of science and the hierarchy of science.
Science
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Aristotle, 384 BC – 322 BC, - one of the early figures in the development of the scientific method.
Science
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Galen (129—c.216) noted the optic chiasm is X-shaped. (Engraving from Vesalius, 1543)
72.
Bulletin of the American Mathematical Society
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The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society. It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, short Mathematical Perspectives articles. It underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a journal for its members. The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, Current Contents/Physical, Chemical & Earth Sciences.
Bulletin of the American Mathematical Society
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October 2010 issue
73.
Theorem
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A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the statement. Mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. However, the conditional could be interpreted differently depending on the meanings assigned to the derivation rules and the conditional symbol. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". Its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically, many theorems are of the form of an conditional: if A, then B. Such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion. To be proved, a theorem must be expressible as a formal statement.
Theorem
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A planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
74.
Latin language
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Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from Greek alphabets. Latin was originally spoken in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, initially in Italy and subsequently throughout the Roman Empire. Vulgar Latin developed such as Italian, Portuguese, Spanish, French, Romanian. Latin, Italian and French have contributed many words to the English language. Ancient Greek roots are used in theology, biology, medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin. Vulgar Latin was the colloquial form attested in inscriptions and the works of comic playwrights like Plautus and Terence. Later, Early Modern Latin and Modern Latin evolved. Latin was used until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the Roman Rite of the Catholic Church. Many students, scholars and members of the Catholic clergy speak Latin fluently. It is taught around the world. The language has been passed down through various forms.
Latin language
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Latin inscription, in the Colosseum
Latin language
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Julius Caesar 's Commentarii de Bello Gallico is one of the most famous classical Latin texts of the Golden Age of Latin. The unvarnished, journalistic style of this patrician general has long been taught as a model of the urbane Latin officially spoken and written in the floruit of the Roman republic.
Latin language
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A multi-volume Latin dictionary in the University Library of Graz
Latin language
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Latin and Ancient Greek Language - Culture - Linguistics at Duke University in 2014.
75.
Pythagoreanism
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Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Through them, all of Western philosophy. According to tradition, pythagoreanism developed into two separate schools of thought, the mathēmatikoi and the akousmatikoi. There is the outer circle John Burnet noted Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. Memory was the most valued faculty. By musical sounds alone unaccompanied with words they healed the passions of the soul and certain diseases, enchanting in reality, as they say. It is probable that from i. e. "enchantment," came to be generally used. Each of these he corrected through the rule of virtue, attempering them through appropriate melodies, well as through certain salubrious medicine. However, intelligence is a part of virtue and of success, for we say that success either is it. According to historians like Thomas Gale, Cantor, Archytas became the head of the school, about a century after the murder of Pythagoras. According to August Böckh, who cites Nicomachus, Philolaus was the successor of Pythagoras. And according to Cicero, Philolaus was teacher of Archytas of Tarentum. According to the historian's of Philosophy, "Philolaus and Eurytus are identified by Aristoxenus as teachers of the last generation of Pythagoreans". A Echecrates is mentioned as a student of Philolaus and Eurytus. The mathēmatikoi were supposed to have developed the more mathematical and scientific work begun by Pythagoras.
Pythagoreanism
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Bust of Pythagoras, Musei Capitolini, Rome.
Pythagoreanism
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Pythagoreans celebrate sunrise by Fyodor Bronnikov
Pythagoreanism
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Excerpt from Philolaus Pythagoras book, (Charles Peter Mason, 1870)
76.
Saint Augustine
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Augustine was the bishop of Hippo Regius, located in Numidia. Augustine is viewed as one of the most important Church Fathers in the Patristic Era. Among his most important works are The City of God and Confessions. According to his contemporary, Jerome, Augustine "established anew the ancient Faith." In his early years, Augustine was heavily influenced afterward by the neo-Platonism of Plotinus. After his conversion to Christianity in 386, he developed his own approach to philosophy and theology, accommodating a variety of methods and perspectives. His thoughts profoundly influenced the medieval worldview. In the Anglican Communion, Augustine is a saint, a preeminent Doctor of the Church, the patron of the Augustinians. His memorial is celebrated on 28 August, the day of his death. Augustine is the patron saint of brewers, printers, theologians, a number of cities and dioceses. Especially Calvinists, consider him to be one of the theological fathers of the Protestant Reformation due to his teachings on salvation and divine grace. In the East, some of his teachings have in the 20th century in particular come under attack by such theologians as John Romanides. But other figures of the Eastern Orthodox Church have shown significant appropriation of his writings, chiefly Georges Florovsky. The most controversial doctrine surrounding his name is the filioque, rejected by the Orthodox Church. Disputed teachings include his views on original sin, the doctrine of grace, predestination.
Saint Augustine
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Saint Augustine from a 19th-century engraving
Saint Augustine
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The Saint Augustine Taken to School by Saint Monica. by Niccolò di Pietro 1413-15
Saint Augustine
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The earliest known portrait of Saint Augustine in a 6th-century fresco, Lateran, Rome.
Saint Augustine
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Angelico, Fra. "The Conversion of St. Augustine" (painting).
77.
Cicero
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Marcus Tullius Cicero was a Roman philosopher, politician, lawyer, orator, political theorist, consul, constitutionalist. Cicero is considered one of Rome's greatest orators and prose stylists. He created a Latin philosophical vocabulary distinguishing himself as a translator and philosopher. Though he was an accomplished successful lawyer, he believed his political career was his most important achievement. Following Julius Caesar's death, he became an enemy of Mark Antony in the ensuing struggle, attacking him in a series of speeches. His severed hands and head were then, as a final revenge of Mark Antony, displayed in the Roman Forum. Petrarch's rediscovery of Cicero's letters is often credited for initiating the 14th-century Renaissance in public affairs, classical Roman culture. He was born in 106 BC in a hill town 100 kilometers southeast of Rome. His father possessed good connections in Rome. However, being a semi-invalid, Cicero studied extensively to compensate. Cicero's brother Quintus wrote in a letter that she was a thrifty housewife. Personal surname, comes from the Latin for chickpea, cicer. Plutarch explains that the name was originally given to one of Cicero's ancestors who had a cleft in the tip of his nose resembling a chickpea. However, it is more likely that Cicero's ancestors prospered through the sale of chickpeas. Romans often chose personal surnames: the famous family names of Fabius, Lentulus, Piso come from the Latin names of beans, lentils, peas.
Cicero
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A first century AD bust of Cicero in the Capitoline Museums, Rome
Cicero
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The Young Cicero Reading by Vincenzo Foppa (fresco, 1464), now at the Wallace Collection
Cicero
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Cicero Denounces Catiline, fresco by Cesare Maccari, 1882–88
Cicero
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Cicero's death (France, 15th century)
78.
Aristotle
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Aristotle was a Greek philosopher and scientist born in the city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, whereafter Proxenus of Atarneus became his guardian. At eighteen years of age, he remained there until the age of thirty-seven. Shortly after Plato died, Aristotle left Athens and, at the request of Philip of Macedon, tutored Alexander the Great beginning in 343 BC. Teaching Alexander the Great gave an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books. He believed all peoples' concepts and all of their knowledge was ultimately based on perception. Aristotle's views on natural sciences represent the groundwork underlying many of his works. Aristotle's views on physical science profoundly shaped medieval scholarship. Some such as on the hectocotyl arm of the octopus, were not refuted until the 19th century. His works contain the earliest formal study of logic, incorporated into modern formal logic. Aristotle was well known among medieval Muslim intellectuals and revered as "The First Teacher". His ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotle's philosophy continue to be the object of active academic study today. Aristotle, whose name means "the best purpose", was born in 384 BC in Stagira, Chalcidice, about 55 km east of modern-day Thessaloniki.
Aristotle
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Roman copy in marble of a Greek bronze bust of Aristotle by Lysippus, c. 330 BC. The alabaster mantle is modern.
Aristotle
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Aristotelianism
Aristotle
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School of Aristotle in Mieza, Macedonia, Greece
Aristotle
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"Aristotle" by Francesco Hayez (1791–1882)
79.
Physics
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One of the main goal of physics is to understand how the universe behaves. Physics is one of perhaps the oldest through its inclusion of astronomy. The boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences while opening new avenues of research in areas such as philosophy. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. The United Nations named the World Year of Physics. Astronomy is the oldest of the natural sciences. The planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, these early observations laid the foundation for later astronomy. In the book, he was also the first to delved further into the way the eye itself works. Fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haytham's Optics ranks alongside that of Newton's work of the same title, published 700 years later. The translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the same devices as what Ibn Al Haytham understand the way light works. From this, important things as eyeglasses, magnifying glasses, telescopes, cameras were developed.
Physics
–
Further information: Outline of physics
Physics
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Ancient Egyptian astronomy is evident in monuments like the ceiling of Senemut's tomb from the Eighteenth Dynasty of Egypt.
Physics
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Sir Isaac Newton (1643–1727), whose laws of motion and universal gravitation were major milestones in classical physics
Physics
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Albert Einstein (1879–1955), whose work on the photoelectric effect and the theory of relativity led to a revolution in 20th century physics
80.
Metaphysics
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Metaphysics is a branch of philosophy investigating the fundamental nature of being and the world that encompasses it. Metaphysics attempts to answer two basic questions: Ultimately, what is there? What is it like? Topics of metaphysical investigation include existence, objects and their properties, space and time, cause and effect, possibility. A central branch of metaphysics is ontology, the investigation into the basic categories of being and how they relate to one another. Another central branch is metaphysical cosmology: which seeks to understand the origin and meaning of the universe by thought alone. There are two broad conceptions about what "world" is studied by metaphysics. Some philosophers, notably Kant, discuss both of these "worlds" and what can be inferred about each one. Some philosophers and scientists, such as the logical positivists, reject the entire subject of metaphysics as meaningless, while others disagree and think that it is legitimate. Ontology deals with the determination whether categories of being are fundamental and discusses in what sense the items in those categories may be said to "be". Most ontologies assume or assert the existence of categories including objects, properties, space and time. Immediate questions arising from this include the nature of objects. How can we be sure that such objects exist at all? The word "is" has two distinct uses in English, separated out in ontology. Some philosophers also include sub-classing as a third form of "is-ness" or being, as in "the elephant is a mammal".
Metaphysics
–
Plato – Kant – Nietzsche
81.
Leonardo Fibonacci
–
Fibonacci popularized the Hindu -- Arabic numeral system primarily through his composition in 1202 of Liber Abaci. He also introduced Europe to the sequence of Fibonacci numbers, which he used in Liber Abaci. Fibonacci was born around 1175 for Pisa. Guglielmo directed a trading post in North Africa. It was in Bugia that he learned about the Hindu -- Arabic numeral system. Fibonacci travelled extensively around the Mediterranean coast, learning about their systems of doing arithmetic. He soon realised the many advantages of the Hindu-Arabic system. In 1202, he completed the Liber Abaci which popularized Hindu–Arabic numerals in Europe. Fibonacci became a guest of Emperor Frederick II, who enjoyed mathematics and science. It has been estimated to be between 1240 and 1250, most likely in Pisa. In the Liber Abaci, Fibonacci introduced today known as Hindu-Arabic numerals. The book advocated 0 -- 9 and place value. The book had a profound impact on European thought. No copies of the 1202 edition are known to exist. The book also discusses prime numbers.
Leonardo Fibonacci
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Portrait by an unknown artist
Leonardo Fibonacci
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A page of Fibonacci's Liber Abaci from the Biblioteca Nazionale di Firenze showing (in box on right) the Fibonacci sequence with the position in the sequence labeled in Roman numerals and the value in Hindu-Arabic numerals.
Leonardo Fibonacci
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19th century statue of Fibonacci in Camposanto, Pisa.
82.
Italians
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Italians are a nation and ethnic group native to Italy who share a common Italian culture, ancestry and speak the Italian language as a mother tongue. Italians have greatly contributed to science, arts, technology, cuisine, sports, jurisprudence and banking both abroad and worldwide. Italian people are generally known to clothing and family values. The term Italian has a history that goes back to pre-Roman Italy. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. This period of unification was followed by one of conquest beginning with the First Punic War against Carthage. In the course of the century-long struggle against Carthage, the Romans conquered Sicily, Sardinia and Corsica. The final victor, was accorded the title of Augustus by the Senate and thereby became the first Roman emperor. Emperor Diocletian's administrative division of the empire into two parts in 285 provided only temporary relief; it became permanent in 395. In 313, churches thereafter rose throughout the empire. However, he also moved his capital to Constantinople greatly reducing the importance of the former. Romulus Augustulus, was deposed in 476 by a Germanic foederati general in Italy, Odoacer. His defeat marked the end of the western part of the Roman Empire. Odoacer ruled well after gaining control of Italy in 476. Then he was defeated by Theodoric, the king of another Germanic tribe, the Ostrogoths.
Italians
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Amerigo Vespucci, the notable geographer and traveller from whose name the word America is derived.
Italians
Italians
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Christopher Columbus, the discoverer of the New World.
Italians
–
Laura Bassi, the first chairwoman of a university in a scientific field of studies.
83.
Group theory
–
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of theory that have experienced advances and have become subject areas in their own right. Physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus the closely related representation theory have many important applications in physics, chemistry, materials science. Group theory is also central to key cryptography. The first class of groups to undergo a systematic study was permutation groups. An early construction due to Cayley exhibited any group as a group, acting on itself by means of the left regular representation. In many cases, the structure of a group can be studied using the properties of its action on the corresponding set. In this way one proves that for n ≥ 5, the alternating group An is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a algebraic equation of degree n' ≥ 5 in radicals. The important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given n over a field K, closed under the products and inverses. Such a group acts on the n-dimensional space Kn by linear transformations. In the case of permutation groups, X is a set; for matrix groups, X is a space.
Group theory
–
Water molecule with symmetry axis
Group theory
–
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups.
84.
Projective geometry
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Projective geometry is a topic of mathematics. Projective is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, a selective set of basic geometric concepts. The first issue for geometers is what kind of geometry is adequate for a situation. One source for projective geometry was indeed the theory of perspective. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a drawing. See projective plane for the basics of projective geometry in two dimensions. While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of complex space, the coordinates used being complex numbers. Major types of more abstract mathematics were based on projective geometry. Projective was also a subject with a large number of practitioners for its own sake, as synthetic geometry. Another topic that developed from axiomatic studies of projective geometry is finite geometry. The topic of projective geometry is itself now divided into two examples of which are projective algebraic geometry and projective differential geometry. Projective geometry is an non-metrical form of geometry, meaning that it is not based on a concept of distance. In two dimensions Projective begins with the study of configurations of lines.
Projective geometry
–
Growth measure and the polar vortices. Based on the work of Lawrence Edwards
Projective geometry
–
Projecting a sphere to a plane.
Projective geometry
–
Forms
85.
Formalism (mathematics)
–
They are syntactic forms whose locations have no meaning unless they are given an interpretation. Formalism is associated with rigorous method. In common use, a formalism means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed commonly enough within something formalisable with claims to be one. Complete formalisation is in the domain of science. Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert. A formalist is an individual who belongs to the school of formalism, a mathematical-philosophical doctrine descending from Hilbert. Formalists are relatively tolerant and inviting to logic, non-standard number systems, new set theories, etc.. The more games we study, the better. However, in all three of these examples, motivation is drawn from existing philosophical concerns. The "games" are usually not arbitrary. Because of their close connection with science, this idea is also advocated by mathematical intuitionists and constructivists in the "computability" tradition. Another version of formalism is often known as deductivism. In deductivism, the Pythagorean theorem is not a relative one. Under deductivism, the same view is held to be true for all other statements of formal logic and mathematics.
Formalism (mathematics)
–
David Hilbert
86.
Principia Mathematica
–
PM was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. Contemporary mathematics, however, avoids paradoxes such as Russell's such as the system of Zermelo -- Fraenkel set theory. PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM states: "The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... PM has long been known for its typographical complexity. Famously, several hundred pages are required in PM to prove the validity of the proposition +1 = 2. The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century. The Principia covered only set theory, cardinal numbers, real numbers. It was also clear how lengthy such a development would be. The authors admitted to intellectual exhaustion upon completion of the third. Another observation is that immediately in the theory, interpretations are presented in terms of truth-values for the behaviour of the symbols" ⊢"," ~", "V". Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw theory would not provide the meaning of the symbols that form a "primitive proposition" -- the symbols themselves could be absolutely arbitrary and unfamiliar. The theory would specify only how the symbols behave based on the grammar of the theory.
Principia Mathematica
–
The title page of the shortened version of the Principia Mathematica to *56
87.
Bertrand Russell
–
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. Russell was born into one of the most aristocratic families in the United Kingdom. In the early 20th century, Russell led the British "revolt against idealism". Russell is considered one of the founders of analytic philosophy with his predecessor Gottlob Frege, protégé Ludwig Wittgenstein. He is widely held to be one of the 20th century's premier logicians. With A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay "On Denoting" has been considered a "paradigm of philosophy". Russell mostly was a prominent anti-war activist; he championed anti-imperialism. Occasionally, Russell advocated nuclear war, before the opportunity provided by the atomic monopoly is gone, "welcomed with enthusiasm" government. He went to prison for his pacifism during World War I. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire, into an influential and liberal family of the British aristocracy. Viscountess Amberley, were radical for their times. Lord Amberley consented with the biologist Douglas Spalding. Both were early advocates of birth control at a time when this was considered scandalous. His atheism was evident when he asked the philosopher John Stuart Mill to act as Russell's godfather.
Bertrand Russell
–
Russell as a four year-old
Bertrand Russell
–
Bertrand Russell
Bertrand Russell
–
Childhood home, Pembroke Lodge
Bertrand Russell
–
Russell in 1907
88.
Alfred North Whitehead
–
Alfred North Whitehead OM FRS was an English mathematician and philosopher. In his early career Whitehead wrote primarily on mathematics, physics. His most notable work in these fields is the three-volume Principia Mathematica, which he wrote with former student Bertrand Russell. Beginning in the late early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science, finally to metaphysics. He developed a metaphysical system which radically departed from most of western philosophy. Today Whitehead's philosophical works -- particularly Reality -- are regarded as the foundational texts of process philosophy. Cobb, Jr. Alfred North Whitehead was born in Ramsgate, Kent, England, in 1861. Alfred Whitehead, was a minister and schoolmaster of Chatham House Academy, a school for boys established by Thomas Whitehead, Alfred North's grandfather. Whitehead himself recalled both of them that his grandfather was the more extraordinary man. Whitehead's mother was Maria Sarah Whitehead, formerly Maria Sarah Buckmaster. Whitehead was educated at Dorset, then considered one of the best public schools in the country. His childhood was described as over-protected, but when at school he was head prefect of his class. In 1880, Whitehead studied mathematics. His academic advisor was Edward John Routh. He graduated as fourth wrangler.
Alfred North Whitehead
–
Alfred North Whitehead
Alfred North Whitehead
–
Whewell's Court north range at Trinity College, Cambridge. Whitehead spent thirty years at Trinity, five as a student and twenty-five as a senior lecturer.
Alfred North Whitehead
–
Bertrand Russell in 1907. Russell was a student of Whitehead's at Trinity College, and a longtime collaborator and friend.
Alfred North Whitehead
–
The title page of the shortened version of the Principia Mathematica to *56
89.
Symbolic logic
–
Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, theoretical computer science. The unifying themes in mathematical logic include the study of the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, proof theory. These areas share basic results on logic, definability. In science mathematical logic encompasses additional topics not detailed in this article; see Logic in computer science for those. Since its inception, mathematical logic has been motivated by, the study of foundations of mathematics. This study began for geometry, arithmetic, analysis. In the 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, others provided partial resolution to the program, clarified the issues involved in proving consistency. Each area has a distinct focus, although many results are shared among multiple areas. The lines separating mathematical logic and other fields of mathematics, are not always sharp. Gödel's incompleteness theorem has also led to Löb's theorem in modal logic. The method of forcing is employed in set theory, recursion theory, as well as in the study of intuitionistic mathematics. These foundations use toposes, which resemble generalized models of theory that may employ classical or nonclassical logic.
Symbolic logic
–
Aristotle, 384–322 BCE.
Symbolic logic
–
Plato – Kant – Nietzsche
90.
L.E.J. Brouwer
–
He was the founder of the mathematical philosophy of intuitionism. Early in his career, Brouwer proved a number of theorems that were breakthroughs in the emerging field of topology. The most celebrated results were: his fixed point theorem, the topological invariance of dimension. The most popular of the three among mathematicians is the first one called the Brouwer Fixed Point Theorem. This one is the most popular among algebraic topologists. The third is perhaps the hardest. At age 31, he was elected a member of the Royal Netherlands Academy of Arts and Sciences. As a variety of constructive mathematics, intuitionism is essentially a philosophy of the foundations of mathematics. It is rather simplistically characterized by saying that its adherents refuse to use the law of excluded middle in mathematical reasoning. Brouwer was a member of the Significs group. It formed part of the early history of semiotics -- the study of symbols -- in particular. The original meaning of his intuitionism probably can not be completely disentangled from the intellectual milieu of that group. Arthur Schopenhauer had a formative influence on Brouwer, not least because he insisted that all concepts be fundamentally based on sense intuitions. Nevertheless, in 1908: "... It was then that "Brouwer felt free to return to his revolutionary project which he was now calling intuitionism ".
L.E.J. Brouwer
–
L. E. J. Brouwer
91.
German language
–
German is a West Germanic language, mainly spoken in Central Europe. Major languages which are most similar to German include other members of the West Germanic branch, such as Afrikaans, Dutch, English. It is the second most widely spoken Germanic language, after English. German derives most of its vocabulary from the Germanic branch of the Indo-European family. Fewer are borrowed from French and English. With slightly different standardized variants, German is a pluricentric language. Like English, German is also notable with many unique varieties existing in Europe and also other parts of the world. The history of the German language begins with the German consonant shift during the migration period, which separated Old High German dialects from Old Saxon. When Martin Luther translated the Bible, he based his translation primarily on the bureaucratic language used in Saxony, also known as Meißner Deutsch. Copies of Luther's Bible featured a long list of glosses for each region that translated words which were unknown in the region into the regional dialect. It was not until the middle of the 18th century that a widely accepted standard was created, ending the period of Early New High German. Until about 1800, standard German was mainly a written language: in northern Germany, the local Low German dialects were spoken. Standard German, markedly different, was often learned as a foreign language with uncertain pronunciation. German pronunciation was considered the standard in prescriptive pronunciation guides though; however, the actual pronunciation of Standard German varies from region to region. German was the language in the Habsburg Empire, which encompassed a large area of Central and Eastern Europe.
German language
–
Old Frisian (Alt-Friesisch)
German language
–
The widespread popularity of the Bible translated into German by Martin Luther helped establish modern German
German language
–
Examples of German language in Namibian everyday life
German language
–
German-language newspapers in the U.S. in 1922
92.
Scholasticism
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It originated at the earliest European universities. Scholastic thought is also known for the careful drawing of distinctions. Because of its emphasis on dialectical method, scholasticism was eventually applied to many other fields of study. The terms "scholastic" and "scholasticism" derive from the Greek σχολαστικός, which means "that belongs to the school". The "scholastics" were, roughly, "schoolmen". The significant renewal of learning in the West came with the Carolingian Renaissance of the Early Middle Ages. Charlemagne, advised by Peter of York, attracted the scholars of England and Ireland. By decree in AD 787, he established schools in his empire. These schools, from which the scholasticism is derived, became centers of medieval learning. Irish scholars had a considerable presence in the Frankish court, where they were renowned for their learning. Among them was Johannes Scotus Eriugena, one of the founders of scholasticism. Eriugena was an outstanding philosopher in terms of originality. He translated many works into Latin, affording access to the Cappadocian Fathers and the Greek theological tradition. The other three founders of scholasticism were the 11th-century scholars Peter Abelard, Archbishop Anselm of Canterbury. This period saw the beginning of the ` rediscovery' of Greek works, lost to the Latin West.
Scholasticism
–
14th-century image of a university lecture
Scholasticism
93.
Organon
–
The Organon is the standard collection of Aristotle's six works on logic. The Organon was given by Aristotle's followers, the Peripatetics. They are as follows: The order of the works was deliberately chosen by Theophrastus to constitute a well-structured system. Indeed, parts of them seem to be a scheme of a lecture on logic. The arrangement of the works was made around 40 BC. The Categories introduces Aristotle's 10-fold classification of that which exists: substance, quantity, quality, relation, place, time, situation, condition, passion. On Interpretation introduces the various relations between affirmative, negative, universal, particular propositions. Aristotle discusses the square of Apuleius in Chapter 7 and its appendix Chapter 8. Chapter 9 deals with the problem of future contingents. The Prior Analytics introduces his syllogistic method, discusses inductive inference. The Posterior Analytics deals with demonstration, scientific knowledge. The Topics treats inference, probable, rather than certain. It is in this treatise that Aristotle mentions the Predicables, later discussed by the scholastic logicians. The Sophistical Refutations provides a key link to Aristotle's work on rhetoric. So much so that after Aristotle's death, his publishers collected these works.
Organon
–
Aristotelianism
94.
Biology
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Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, evolution, distribution, identification and taxonomy. Modern biology is a eclectic field, composed of many branches and subdisciplines. The biology is derived from the Greek word βίος, bios, "life" and the suffix - λογία, - logia, "study of." The Latin-language form of the term first appeared in 1736 when Swedish scientist Carl Linnaeus used biologi in his Bibliotheca botanica. Biologie, was in a 1771 translation of Linnaeus' work. In 1797, Theodor Georg August Roose used the term in the preface of Grundzüge der Lehre van der Lebenskraft. Karl Friedrich Burdach used the term in a more restricted sense of the study of human beings from a morphological, physiological and psychological perspective. We will indicate by the name biology or the doctrine of life. Although modern biology is a relatively recent development, sciences included within it have been studied since ancient times. Natural philosophy was studied early as the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent, China. However, its approach to the study of nature are most often traced back to ancient Greece. While the formal study of medicine dates back to Hippocrates, it was Aristotle who contributed extensively to the development of biology. Scholars of the Islamic world who wrote on biology included al-Jahiz, Al-Dīnawarī, who wrote on botany, Rhazes who wrote on anatomy and physiology. Biology began to quickly grow with Anton van Leeuwenhoek's dramatic improvement of the microscope. It was then that scholars discovered spermatozoa, bacteria, the diversity of microscopic life.
Biology
Biology
Biology
Biology
95.
Chemistry
–
Chemistry is a branch of physical science that studies the composition, structure, properties and change of matter. Chemistry is sometimes called the central science because it bridges natural sciences, including physics, biology. For the differences between physics see comparison of physics. Scholars disagree about the etymology of the word chemistry. The history of chemistry can be traced to alchemy, practiced for several millennia in various parts of the world. The chemistry comes from alchemy, which referred to an earlier set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, medicine. An alchemist was called a'chemist' in popular speech, later the suffix "-ry" was added to this to describe the art of the chemist as "chemistry". The modern alchemy in turn is derived from the Arabic al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία. Alternately, al-kīmīā may derive from χημεία, meaning "cast together". In retrospect, the definition of chemistry has changed over time, as new discoveries and theories add to the functionality of the science. The term "chymistry", in the view of noted scientist Robert Boyle in 1661, meant the subject of the material principles of mixed bodies. In 1837, Jean-Baptiste Dumas considered the word "chemistry" to refer to the science concerned with the laws and effects of molecular forces. More recently, in 1998, Professor Raymond Chang broadened the definition of "chemistry" to mean the study of matter and the changes it undergoes. Early civilizations, such as the Egyptians Babylonians, Indians amassed practical knowledge concerning the arts of metallurgy, pottery and dyes, but didn't develop a systematic theory.
Chemistry
–
Solutions of substances in reagent bottles, including ammonium hydroxide and nitric acid, illuminated in different colors
Chemistry
–
Democritus ' atomist philosophy was later adopted by Epicurus (341–270 BCE).
Chemistry
–
Antoine-Laurent de Lavoisier is considered the "Father of Modern Chemistry".
Chemistry
–
Laboratory, Institute of Biochemistry, University of Cologne.
96.
Marcus du Sautoy
–
Formerly a Fellow of All Souls College, Wadham College, he is now a Fellow of New College. He was previously President of a Royal Society University Research Fellow. His academic work concerns mainly group theory and number theory. In October 2008, he was appointed to the Simonyi Professorship for the Public Understanding of Science, succeeding the inaugural holder Richard Dawkins. He went on to complete his DPhil in mathematics. He currently plays the trumpet. In March 2006, his article Prime Numbers Get Hitched was published by the online Seed magazine. He has also published an article in the scientific magazine New Scientist. In December 2006, du Sautoy delivered Christmas Lectures under The Num8er My5teries. The venue for the 2006 Christmas Lectures was Technology's headquarters at London. He has described his own religion as being "Arsenal - football," as he sees religion as wanting to belong to a community. Du Sautoy is a supporter of Common Hope, an organisation that helps people in Guatemala. He is known for his work popularising mathematics. He has been named by The Independent on Sunday as one of the UK's leading scientists. He has appeared several times on television.
Marcus du Sautoy
–
Marcus du Sautoy, 2007
97.
Falsifiability
–
Falsifiability or refutability of a statement, hypothesis, or theory is the inherent possibility that it can be proven false. A statement is called falsifiable if it is possible to conceive of an observation or an argument which negates the statement in question. In this sense, falsify is synonymous with nullify, meaning to invalidate or "show to be false". Thus, the term falsifiability is sometimes synonymous to testability. Some statements, such as It will be raining here in one million years, are falsifiable in principle, but not in practice. The concern with falsifiability gained attention by way of philosopher of science Karl Popper's scientific epistemology "falsificationism". Popper argued that this would require the inference of a general rule from a number of individual cases, inadmissible in deductive logic. However, if one finds one single swan, not white, deductive logic admits the conclusion that the statement that all swans are white is false. Falsificationism thus strives for questioning, for falsification, of hypotheses instead of proving them. For a statement to be questioned using observation, it needs to be at least theoretically possible that it can come into conflict with observation. Popper chose falsifiability as the name of this criterion. My proposal is based upon an asymmetry between verifiability and falsifiability; an asymmetry which results from the logical form of universal statements. For these are never derivable from singular statements, but can be contradicted by singular statements. Popper stressed that unfalsifiable statements are important in science. Contrary to intuition, unfalsifiable statements can be embedded in — and deductively entailed by — falsifiable theories.
Falsifiability
–
Are all swans white?
98.
Karl Popper
–
Sir Karl Raimund Popper CH FBA FRS was an Austrian-British philosopher and professor. Popper is generally regarded as one of the greatest philosophers of science of the 20th century. His political philosophy embraces ideas from attempts to reconcile them: socialism/social democracy, libertarianism/classical liberalism and conservatism. Karl Popper was born in Vienna to upper middle-class parents. The Popper family converted to Lutheranism before Karl was born, so he received Lutheran baptism. They understood this as part of their cultural assimilation, not as an expression of devout belief. His father was a bibliophile who took an interest in philosophy, the classics, social and political issues. He inherited both the disposition from him. Later, Popper would describe the atmosphere of his upbringing as having been "decidedly bookish." In 1919, he subsequently joined the Association of Socialist School Students. Popper also became a member of the Social Democratic Workers' Party of Austria, at that time a party that fully adopted the Marxist ideology. Popper was unable to cope with the heavy labour. Continuing to attend university as a student, Popper started an apprenticeship as cabinetmaker, which he completed as a journeyman. Popper was dreaming at that time of starting a facility for children, for which he assumed the ability to make furniture might be useful. After that Popper did voluntary service for children.
Karl Popper
–
Karl Popper c. 1980s
Karl Popper
–
Sir Karl Popper's gravesite in Lainzer Friedhof (de), in Vienna, Austria
Karl Popper
–
Sir Karl Popper, Prof. Cyril Höschl. K. Popper received the Honorary Doctor's degree of Charles University in Prague (May 1994)
99.
Hypothesis
–
A hypothesis is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories. Even though the words "hypothesis" and "theory" are often used synonymously, a scientific hypothesis is not the same as a scientific theory. A working hypothesis is a provisionally accepted hypothesis proposed for further research. P is the assumption in a What If question. Remember, the way that you prove an implication is by assuming the hypothesis. --Philip Wadler In its ancient usage, hypothesis referred to a summary of the plot of a classical drama. The English word hypothesis comes from the Greek word hupothesis, meaning "to put under" or "to suppose". In Plato's Meno, Socrates dissects virtue with a method used by mathematicians, that of "investigating from a hypothesis." In this sense, ` hypothesis' refers to a mathematical approach that simplifies cumbersome calculations. In common usage in the 21st century, a hypothesis refers to a provisional idea whose merit requires evaluation. For proper evaluation, the framer of a hypothesis needs to define specifics in operational terms. A hypothesis requires more work by the researcher in order to either confirm or disprove it. In due course, a confirmed hypothesis may become part of a theory or occasionally may grow to become a theory itself.
Hypothesis
–
Andreas Cellarius hypothesis, demonstrating the planetary motions in eccentric and epicyclical orbits
100.
Imre Lakatos
–
He was born Imre Lipschitz to a Jewish family in 1922. Lakatos received a degree from the University of Debrecen in 1944. In May of that year, the group was joined by a 19-year-old Jewish antifascist activist. Subsequently, a member of the group gave her cyanide. During the occupation, he avoided Nazi persecution of Jews by changing his name to Imre Molnár. His grandmother died in Auschwitz. Lakatos changed his surname again to Lakatos in honor of Géza Lakatos. From 1947 Lakatos worked as a senior official in the Hungarian ministry of education. Lakatos also also attended György Lukács's weekly Wednesday afternoon private seminars. Lakatos also studied under the supervision of Sofya Yanovskaya in 1949. More of Lakatos' activities in Hungary after World War II have recently become known. After his release, he returned to academic life, translating George Pólya's How to Solve It into Hungarian. After the Soviet Union invaded Hungary in November 1956, he later reached England. He received a doctorate in philosophy in 1961 from the University of Cambridge; his thesis advisor was R. B. Braithwaite.
Imre Lakatos
–
Imre Lakatos, c. 1960s
101.
Falsificationism
–
Falsifiability or refutability of a statement, hypothesis, or theory is the inherent possibility that it can be proven false. A statement is called falsifiable if it is possible to conceive of an observation or an argument which negates the statement in question. In this sense, falsify is synonymous with nullify, meaning to invalidate or "show to be false". Thus, the term falsifiability is sometimes synonymous to testability. Some statements, such as It will be raining here in one million years, are falsifiable in principle, but not in practice. The concern with falsifiability gained attention by way of philosopher of science Karl Popper's scientific epistemology "falsificationism". Popper argued that this would require the inference of a general rule from a number of individual cases, inadmissible in deductive logic. However, if one finds one single swan, not white, deductive logic admits the conclusion that the statement that all swans are white is false. Falsificationism thus strives for questioning, for falsification, of hypotheses instead of proving them. For a statement to be questioned using observation, it needs to be at least theoretically possible that it can come into conflict with observation. Popper chose falsifiability as the name of this criterion. My proposal is based upon an asymmetry between verifiability and falsifiability; an asymmetry which results from the logical form of universal statements. For these are never derivable from singular statements, but can be contradicted by singular statements. Popper stressed that unfalsifiable statements are important in science. Contrary to intuition, unfalsifiable statements can be embedded in — and deductively entailed by — falsifiable theories.
Falsificationism
–
Are all swans white?
102.
Theoretical physics
–
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science depends in general on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigor while giving little weight to experiments and observations. Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect, previously an experimental result lacking a theoretical formulation. A physical theory is a model of physical events. It is judged by the extent to which its predictions agree with empirical observations. The quality of a physical theory is also judged on its ability to make new predictions which can be verified by new observations. A physical theory similarly differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms. A physical theory involves one or more relationships between various measurable quantities. Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ empirical formulas to agree with experimental results, often without deep physical understanding. Some attempt to create approximate theories, called effective theories, because fully developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, formalise, reinterpret or generalise extant theories, or create completely new ones altogether.
Theoretical physics
–
Visual representation of a Schwarzschild wormhole. Wormholes have never been observed, but they are predicted to exist through mathematical models and scientific theory.
103.
Intuition (knowledge)
–
Intuition is the ability to acquire knowledge without proof, evidence, or conscious reasoning, or without understanding how the knowledge was acquired. The word "intuition" comes from late middle English word intuit, "to contemplate". Both Eastern and Western philosophers have studied the concept in great detail. Philosophy of mind deals with the concept of intuition. There are philosophers who contend that this concept is often confused with other concepts such as truth, meaning in philosophical discussion. Various meanings exist from different religious texts. In Hinduism various attempts have been made to interpret other esoteric texts. He finds that this process which seems to be a decent, is actual a circle of progress. As a lower faculty is being pushed to take up much from a higher way of working. Advaita vedanta takes intuition to be an experience through which one can experience Brahman. In parts of Zen Buddhism intuition is deemed one's individual, discriminating mind. In Islam there are various scholars with varied interpretation of intuition, sometimes relating the ability of having intuitive knowledge to hood. While Ibn Sīnā finds the ability of having intuition as terms it as a knowledge obtained without intentionally acquiring it. He finds regular knowledge is based on imitation while intuitive knowledge as based on intellectual certitude. In the West, early mention and definition can be traced back to Plato.
Intuition (knowledge)
–
A phrenological mapping of the brain – phrenology was among the first attempts to correlate mental functions with specific parts of the brain
Intuition (knowledge)
–
Papirus Oxyrhynchus, with fragment of Plato's Republic
Intuition (knowledge)
–
Girl with a Book by José Ferraz de Almeida Júnior
104.
Experiment
–
An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight by demonstrating what outcome occurs when a particular factor is manipulated. Experiments always rely on repeatable procedure and logical analysis of the results. There also exists experimental studies. Other types of hands-on activities are very important to student learning in the science classroom. Experiments can help a student become more engaged and interested in the material they are learning, especially when used over time. Experiments can vary from informal natural comparisons, to highly controlled. Uses of experiments vary considerably between the human sciences. Experiments typically include controls, which are designed to minimize the effects of variables other than the independent variable. This increases the reliability of the results, often through a comparison between the other measurements. Scientific controls are a part of the scientific method. Ideally, none are uncontrolled. In the scientific method, an experiment is an empirical procedure that arbitrates between hypotheses. Researchers also use experimentation to test new hypotheses to support or disprove them. An experiment usually tests a hypothesis, an expectation about how phenomenon works.
Experiment
–
Even very young children perform rudimentary experiments to learn about the world and how things work.
Experiment
–
Original map by John Snow showing the clusters of cholera cases in the London epidemic of 1854
105.
Liberal arts
–
Grammar, logic, rhetoric were the core liberal arts, while arithmetic, geometry, the theory of music, astronomy also played a part in education. In modern times, liberal arts education is a term that can be interpreted in different ways. For example, Harvard University offers a Bachelor of Arts degree, which covers the physical sciences well as the humanities. For both interpretations, the term generally refers to matters not relating to the professional, vocational, or technical curriculum. ` scientific' artes -- music, arithmetic, astronomy -- were known from the time of Boethius onwards as the Quadrivium. After the 9th century, the remaining three arts of the ` humanities' -- rhetoric -- were classed as well as the Trivium. It was in that two-fold form that the seven liberal arts were studied in the medieval Western university. During the Middle Ages, logic gradually came to take predominance over the other parts of the Trivium. In many respects continuing the traditions of the Middle Ages, reversed that process. The ideal of a liberal arts, or humanistic education grounded in classical languages and literature, persisted until the middle of the twentieth century. Analyzing and interpreting information is also included. The liberal arts education at the secondary school level prepares the student for higher education at a university. They are thus meant for the more academically minded students. In addition to the usual curriculum, students of a liberal arts education often study Latin and Ancient Greek. Some liberal arts education provide general education, others have a specific focus.
Liberal arts
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Philosophia et septem artes liberales, The seven liberal arts – Picture from the Hortus deliciarum of Herrad of Landsberg (12th century)
Liberal arts
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Page from Marriage of Mercury and Philology
106.
University
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A university is an institution of higher education and research which grants academic degrees in various subjects. Universities typically provide postgraduate education. The word "university" is derived from the universitas magistrorum et scholarium, which roughly means "community of teachers and scholars." Universities were evolved from Cathedral schools for the clergy during the High Middle Ages. The original Latin word "universitas" refers in general to "a number of persons associated into one body, a society, company, community, guild, corporation, etc." Like other guilds, they were self-regulating and determined the qualifications of their members. An important idea in the definition of a university is the notion of academic freedom. The first evidence of this comes from early in the life of the first university. This is claimed as the origin of "academic freedom". This is now widely recognised internationally - on September 1988, 430 university rectors signed the Magna Charta Universitatum, marking the 900th anniversary of Bologna's foundation. The number of universities signing the Magna Charta Universitatum continues drawing from all parts of the world. The earliest universities were developed by papal bull as studia generalia and perhaps from cathedral schools. It is possible, however, that the development of cathedral schools into universities was quite rare, with the University of Paris being an exception. Later they were also founded by municipal administrations. Many historians state that universities and cathedral schools were a continuation of the interest in learning promoted by monasteries.
University
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Degree ceremony at the University of Oxford. The Pro-Vice-Chancellor in MA gown and hood, Proctor in official dress and new Doctors of Philosophy in scarlet full dress. Behind them, a bedel, a Doctor and Bachelors of Arts and Medicine graduate.
University
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The University of Bologna is the oldest University in history, founded in 1088.
University
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Meeting of doctors at the University of Paris. From a medieval manuscript.
University
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Sapienza University of Rome is the largest university in Europe and one of the most prestigious European universities.
107.
Philosophy of mathematics
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The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, implications of mathematics. Itself makes this study both unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably. The latter, however, may be used to refer to several other areas of study. Another refers to the working philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include: What is the role of Mankind in developing mathematics? What are the sources of mathematical subject matter? What is the ontological status of mathematical entities? What does it mean to refer to a mathematical object? What is the character of a mathematical proposition? What is the relation between logic and mathematics? What is the role of hermeneutics in mathematics? What kinds of inquiry play a role in mathematics? What are the objectives of mathematical inquiry? What gives mathematics its hold on experience?
Philosophy of mathematics
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David Hilbert
108.
Mathematical beauty
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Mathematical beauty describes the notion that some mathematicians may derive aesthetic pleasure from their work, from mathematics in general. They express this pleasure by describing mathematics as beautiful. Mathematicians describe mathematics as an form or, at a minimum, as a creative activity. Comparisons are often made with poetry. Paul Erdős expressed his views on the ineffability of mathematics when he said, "Why are numbers beautiful? It's like asking why is Beethoven's Ninth Symphony beautiful. If you don't see why, someone can't tell you. I know numbers are beautiful. If they aren't beautiful, nothing is". Mathematicians describe an especially pleasing method of proof as elegant. Depending on context, this may mean: A proof that uses a minimum of previous results. A proof, unusually succinct. A proof that derives a result in a surprising way A proof, based on new and original insights. A method of proof that can be easily generalized to solve a family of similar problems. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs having been published.
Mathematical beauty
–
Diagram from Leon Battista Alberti 's 1435 Della Pittura, with pillars in perspective on a grid
Mathematical beauty
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An example of "beauty in method"—a simple and elegant proof of the Pythagorean theorem.
Mathematical beauty
–
Forms
109.
Isaac Newton
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His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations for classical mechanics. He shares credit with Gottfried Wilhelm Leibniz for the development of calculus. Newton's Principia formulated the laws of motion and universal gravitation, which dominated scientists' view of the physical universe for the next three centuries. This work also demonstrated that the motion of objects of celestial bodies could be described by the same principles. Newton formulated an empirical law of cooling, introduced the notion of a Newtonian fluid. He was the second Lucasian Professor of Mathematics at the University of Cambridge. In his later life, he became president of the Royal Society. He served the British government as Warden and Master of the Royal Mint. His father, also named Isaac Newton, had died three months before. Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a mug. Newton's mother had three children from her second marriage. Newton hated farming. Master at the King's School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a schoolyard bully, Newton became the top-ranked student, distinguishing himself mainly by building models of windmills. In June 1661, Newton was admitted on the recommendation of his uncle Rev William Ayscough who had studied there.
Isaac Newton
–
Portrait of Isaac Newton in 1689 (age 46) by Godfrey Kneller
Isaac Newton
–
Newton in a 1702 portrait by Godfrey Kneller
Isaac Newton
–
Isaac Newton (Bolton, Sarah K. Famous Men of Science. NY: Thomas Y. Crowell & Co., 1889)
Isaac Newton
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Replica of Newton's second Reflecting telescope that he presented to the Royal Society in 1672
110.
Gottfried Wilhelm Leibniz
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Leibniz's notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and Transcendental Law of Homogeneity found mathematical implementation. He became one of the most prolific inventors in the field of mechanical calculators. He also refined the binary number system, the foundation of virtually all digital computers. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th-century advocates of rationalism. Leibniz wrote works on philology. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, in unpublished manuscripts. He wrote in several languages, but primarily in Latin, French, German. There is no complete gathering of the writings of Leibniz. Gottfried Leibniz was born on July 1, 1646, toward the end of the Thirty Years' War, in Leipzig, Saxony, to Friedrich Leibniz and Catharina Schmuck. Friedrich noted in his family journal: 21. Juny am Sontag 1646 Ist mein Sohn Gottfried Wilhelm, post sextam vespertinam 1/4 uff 7 uhr abents zur welt gebohren, im Wassermann. In English: On Sunday 21 June 1646, my son Gottfried Wilhelm is born into the world a quarter after six in the evening, in Aquarius. Leibniz was baptized on July 3 of that year at St. Nicholas Church, Leipzig; his godfather was the Lutheran theologian Martin Geier. His father died when he was six and a half years old, from that point on he was raised by his mother.
Gottfried Wilhelm Leibniz
–
Portrait by Christoph Bernhard Francke
Gottfried Wilhelm Leibniz
–
Engraving of Gottfried Wilhelm Leibniz
Gottfried Wilhelm Leibniz
–
Stepped Reckoner
Gottfried Wilhelm Leibniz
–
Leibniz's correspondence, papers and notes from 1669-1704, National Library of Poland.
111.
Commerce
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Commerce is the activity of buying and selling of goods and services, especially on a large scale. The system includes legal, economic, political, social, technological systems that are in operation in any country or internationally. Thus, commerce is an environment that affects the business prospects of economies. It can also be defined as a component of business which includes functions involved in transferring goods from producers to consumers. Some commentators trace the origins of commerce in prehistoric times. Apart from traditional self-sufficiency, trading became a principal facility of prehistoric people, who bartered what they had for services from each other. Historian Peter Watson and Ramesh Manickam dates the history of long-distance commerce from circa 150,000 years ago. In historic times, the introduction of currency as a standardized money, facilitated a wider exchange of services. Numismatists have collections of these pokem tokens, which include coins from some World large-scale societies, although initial usage involved unmarked lumps of precious metal. For example, if a man who makes pots for a living needs a new house, he/she may wish to hire someone to build it for him/her. During the Middle Ages, commerce developed in Europe at trade fairs. Wealth became converted into movable capital. Banking systems developed where money on account was transferred across national boundaries. Hand to hand markets were regulated by town authorities.
Commerce
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The caduceus has been used today as the symbol of commerce with which Mercury has traditionally been associated.
112.
Architecture
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Architecture is both the process and the product of planning, designing, constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as works of art. Historical civilizations are often identified with their surviving architectural achievements. "Architecture" can mean: A general term to describe physical structures. The art and science of designing buildings and nonbuilding structures. The style of design and method of construction of buildings and other physical structures. Knowledge of art, science, technology, humanity. The practice of the architect, where architecture means offering or rendering professional services in connection with the design and construction of buildings, or built environments. The design activity of the architect, from the macro-level to the micro-level. Architecture has to reflect functional, technical, environmental, aesthetic considerations. It requires the creative coordination of materials and technology, of shadow. Often, conflicting requirements must be resolved. The practice of architecture also encompasses the pragmatic aspects of realizing structures, including scheduling, construction administration. The word "architecture" has also been adopted to describe other designed systems, especially in information technology. The earliest surviving written work on the subject of architecture is De architectura, by the Roman architect Vitruvius in the 1st AD.
Architecture
–
Brunelleschi, in the building of the dome of Florence Cathedral in the early 15th-century, not only transformed the building and the city, but also the role and status of the architect.
Architecture
–
Section of Brunelleschi 's dome drawn by the architect Cigoli (c. 1600)
Architecture
–
The Parthenon, Athens, Greece, "the supreme example among architectural sites." (Fletcher).
Architecture
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The Houses of Parliament, Westminster, master-planned by Charles Barry, with interiors and details by A.W.N. Pugin
113.
Physicist
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A physicist is a scientist who specializes or works in the field of physics. Physicists generally are interested in the root or ultimate causes of phenomena, usually frame their understanding in mathematical terms. Physicists can also apply their knowledge towards solving real-world problems or developing new technologies. Some physicists specialize in sectors outside the science of physics itself, such as engineering. The practice of physics is based on an intellectual ladder of discoveries and insights to the present. Used today found their earliest expression in Asian culture, as well as the Islamic medieval period. New knowledge in the 21st century includes a large increase in cosmology. The term physicist was coined by William Whewell in his 1840 book The Philosophy of the Inductive Sciences. Many physicist positions require an undergraduate degree in applied physics or a related science or a Master's degree like MSc, MPhil, MPhys or MSci. In a level, students tend to specialize in a particular field. Physics students also need training in mathematics, also in programming. For being employed as a physicist a doctoral background may be required for certain positions. The highest honor awarded to physicists is the Nobel Prize in Physics, awarded since 1901 by the Royal Swedish Academy of Sciences. The three major employers of career physicists are private industries, with the largest employer being the last. Physicists in academia or government labs tend to have titles such as Assistants, Professors, Sr./Jr.
Physicist
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Albert Einstein, physicist who developed the theory of general relativity.
114.
Richard Feynman
–
For his contributions to the development of quantum electrodynamics, Feynman, jointly with Sin ` ichirō Tomonaga, received the Nobel Prize in Physics in 1965. Feynman developed a widely used pictorial scheme for the mathematical expressions governing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In addition to his work in theoretical physics, Feynman has been credited with introducing the concept of nanotechnology. He held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology. By his youth Feynman described himself as an "avowed atheist". Like Edward Teller, Feynman was a late talker, by his third birthday had yet to utter a single word. He retained a Brooklyn accent as an adult. From his mother he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, delighted in repairing radios. When he was in school, he created a home burglar alarm system while his parents were out for the day running errands. Four years later, the family moved to Far Rockaway, Queens. Though separated by nine years, Joan and Richard were close, as they both shared a natural curiosity about the world. Their mother thought that women did not have the cranial capacity to comprehend such things.
Richard Feynman
–
Richard Feynman
Richard Feynman
–
Feynman (center) with Robert Oppenheimer (right) relaxing at a Los Alamos social function during the Manhattan Project
Richard Feynman
–
The Feynman section at the Caltech bookstore
Richard Feynman
–
Mention of Feynman's prize on the monument at the American Museum of Natural History in New York City. Because the monument is dedicated to American Laureates, Tomonaga is not mentioned.
115.
Path integral formulation
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The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. Unlike previous methods, the path-integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system. Possible downsides of the approach include that unitarity of the S-matrix is obscure in the formulation. The path-integral approach has been proved to be equivalent to the other formalisms of theory. Thus, by deriving either approach from the other, problems associated with one or the other approach go away. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 paper. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian as a starting point. In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time-translations. The Hamiltonian in classical mechanics is derived from a Lagrangian, a more fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. So the Hamiltonian is different in different frames, this type of symmetry is not apparent in the original formulation of quantum mechanics. The Hamiltonian is a function of the position and momentum at one time, it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later.
Path integral formulation
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These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time t 0 to point B at some other time t 1.
116.
Quantum mechanics
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Quantum mechanics, including quantum field theory, is a fundamental branch of physics concerned with processes involving, for example, atoms and photons. Systems such as these which obey quantum mechanics can be in a quantum superposition of different states, unlike in classical physics. Early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms. In one of them, the wave function, provides information about the probability amplitude of position, momentum, other physical properties of a particle. This experiment played a major role in the general acceptance of the theory of light. In 1838, Michael Faraday discovered cathode rays. Planck's hypothesis that energy is absorbed in discrete "quanta" precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a law of black-body radiation, known as Wien's law in his honor. Ludwig Boltzmann independently arrived by considerations of Maxwell's equations. However, it underestimated the radiance at low frequencies. Following Max Planck's solution to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohr's theory of atomic structure, introducing elliptical orbits, a concept also introduced by Arnold Sommerfeld. This phase is known as old theory.
Quantum mechanics
–
Max Planck is considered the father of the quantum theory.
Quantum mechanics
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Solution to Schrödinger's equation for the hydrogen atom at different energy levels. The brighter areas represent a higher probability of finding an electron
Quantum mechanics
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The 1927 Solvay Conference in Brussels.
117.
String theory
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In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate with each other. In theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force. Thus theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory was first studied in the late 1960s before being abandoned in favor of quantum chromodynamics. The earliest version of bosonic string theory, incorporated only the class of particles known as bosons. It later developed into superstring theory, which posits a connection called supersymmetry between the class of particles called fermions. One of the challenges of theory is that the full theory does not have a satisfactory definition in all circumstances. These issues have led some in the community to question the value of continued research on string theory unification. In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics. One of these frameworks was Albert Einstein's general theory of a theory that explains the force of gravity and the structure of space and time. The other was a radically different formalism for describing physical phenomena using probability. In spite of these successes, there are still many problems that remain to be solved. One of the deepest problems in modern physics is the problem of gravity.
String theory
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A cross section of a quintic Calabi–Yau manifold
String theory
–
String theory
String theory
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A magnet levitating above a high-temperature superconductor. Today some physicists are working to understand high-temperature superconductivity using the AdS/CFT correspondence.
String theory
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A graph of the j-function in the complex plane
118.
Fundamental interaction
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Fundamental interactions, also known as fundamental forces, are the interactions in physical systems that do not appear to be reducible to more basic interactions. There are four conventionally accepted fundamental interactions—gravitational, electromagnetic, strong nuclear, weak nuclear. Each one is understood as the dynamics of a field. The gravitational force is modelled as a continuous classical field. The other three are each modelled as discrete quantum fields, exhibit a measurable unit or elementary particle. The two nuclear interactions produce strong forces at minuscule, subatomic distances. The strong nuclear interaction is responsible for the binding of atomic nuclei. The weak nuclear interaction also acts on the nucleus, mediating radioactive decay. Electromagnetism and gravity produce significant forces at macroscopic scales where the effects can be seen directly in everyday life. . Other theorists seek to unite the electroweak and strong fields within a Grand Unified Theory. Thus Newton's theory violated the first principle of mechanical philosophy, as stated by Descartes, No action at a distance. Conversely, during the 1820s, when explaining magnetism, Michael Faraday inferred a field filling space and transmitting that force. Faraday conjectured that ultimately, all forces unified into one. The Standard Model of particle physics was developed throughout the latter half of the 20th century.
Fundamental interaction
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The Standard Model of elementary particles, with the fermions in the first three columns, the gauge bosons in the fourth column, and the Higgs boson in the fifth column
119.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations. One may also study real numbers in relation to rational numbers, e.g. as approximated by the latter. The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". In particular, arithmetical is preferred as an adjective to number-theoretic. The triples are too many and too large to have been obtained by brute force. The heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly flowered only later. It has been suggested instead that the table was a source of numerical examples for school problems. While Babylonian theory -- or what survives of Babylonian mathematics that can be called thus -- consists of this striking fragment, Babylonian algebra was well developed. Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt.
Number theory
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A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.
Number theory
–
The Plimpton 322 tablet
Number theory
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Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Number theory
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Leonhard Euler
120.
Cryptography
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Cryptography or cryptology is the practice and study of techniques for secure communication in the presence of third parties called adversaries. Modern cryptography exists at the intersection of the disciplines of mathematics, electrical engineering. Applications of cryptography include ATM cards, electronic commerce. Cryptography prior to the modern age was effectively synonymous with the conversion of information from a readable state to apparent nonsense. The literature often uses Alice for the sender, Bob for the intended recipient, Eve for the adversary. It is infeasible to do so by any known practical means. The growth of cryptographic technology has raised a number of legal issues in the age. In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation. Cryptography also plays a major role of digital media. Until modern times, cryptography referred exclusively to encryption, the process of converting ordinary information into unintelligible text. Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher is a pair of algorithms that create the reversing decryption. The detailed operation of a cipher is controlled by a "key". The key is a secret, usually a short string of characters, needed to decrypt the ciphertext. Historically, ciphers were often used directly without additional procedures such as authentication or integrity checks.
Cryptography
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German Lorenz cipher machine, used in World War II to encrypt very-high-level general staff messages
Cryptography
–
16th-century book-shaped French cipher machine, with arms of Henri II of France
Cryptography
–
Enciphered letter from Gabriel de Luetz d'Aramon, French Ambassador to the Ottoman Empire, after 1546, with partial decipherment
Cryptography
–
Whitfield Diffie and Martin Hellman, authors of the first published paper on public-key cryptography
121.
Eugene Wigner
–
Eugene Paul "E. P." Wigner, was a Hungarian-American theoretical physicist, engineer and mathematician. Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way Wigner performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. Wigner is also known into the structure of the atomic nucleus. In 1930, he moved to the United States. He was afraid that the nuclear weapon project would develop an atomic bomb first. During the Manhattan Project, Wigner led a team whose task was to design nuclear reactors to convert uranium into weapons plutonium. At the time, no reactor had yet gone critical. He was disappointed that DuPont was given responsibility for the detailed design of the reactors, just their construction. Wigner Jenő Pál was born in Budapest, Austria-Hungary on November 17, 1902, to middle class Jewish parents, Elisabeth and Anthony Wigner, a leather tanner. Wigner had a younger sister Margit, known as Manci, who later married British theoretical physicist Paul Dirac. Wigner was home schooled until the age of 9 when he started school at the third grade. During this period, he developed an interest in mathematical problems. At the age of 11, he contracted what his doctors believed to be tuberculosis.
Eugene Wigner
–
Eugene Wigner
Eugene Wigner
–
Signature
Eugene Wigner
–
Werner Heisenberg and Eugene Wigner (1928)
Eugene Wigner
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Wigner receiving the Medal for Merit for his work on the Manhattan Project from Robert P. Patterson (left), March 5, 1946
122.
Operations research
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Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions. Further, the term ` operational analysis' is used as an intrinsic part of capability development, management and assurance. In particular, operational analysis forms part of the Combined Operational Effectiveness and Investment Appraisals, which support British defence capability decision-making. It is often considered to be a sub-field of mathematics. Decision science are sometimes used as synonyms. Operations research is often concerned with determining the minimum of some real-world objective. Originating before World War II, its techniques have grown to concern problems in a variety of industries. Nearly all of these techniques involve the construction of mathematical models that attempt to describe the system. Because of the statistical nature of most of these fields, OR also has strong ties to computer science and analytics. In the decades after the two world wars, the techniques were more widely applied to problems in business, society. Early work in operational research was carried out by individuals such as Charles Babbage. Percy Bridgman would later attempt to extend these to the social sciences. Rowe conceived the idea as a means to improve the working of the UK's early warning radar system, Chain Home. Initially, he analysed the operating of its communication networks, expanding later to include the operating personnel's behaviour. This allowed remedial action to be taken.
Operations research
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A Liberator in standard RAF green/dark earth/black night bomber finish as originally used by Coastal Command
Operations research
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A Warwick in the revised RAF Coastal Command green/dark grey/white finish
123.
Computer science
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Computer science is the study of the theory, experimentation, engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating that scale. A scientist specializes in the design of computational systems. Its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational theory, are highly abstract, while fields such as computer graphics emphasize visual applications. Other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, universally accessible to humans. The earliest foundations of what would become science predate the invention of the digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division. Further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and theorist, among other reasons, documenting the binary number system. He started developing this machine in 1834 and "in less than two years he had sketched out many of the salient features of the modern computer". "A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable.
Computer science
–
Ada Lovelace is credited with writing the first algorithm intended for processing on a computer.
Computer science
Computer science
–
The German military used the Enigma machine (shown here) during World War II for communications they wanted kept secret. The large-scale decryption of Enigma traffic at Bletchley Park was an important factor that contributed to Allied victory in WWII.
Computer science
–
Digital logic
124.
Aesthetics
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Aesthetics is a branch of philosophy dealing with the nature of art, beauty, taste, with the creation and appreciation of beauty. It is more scientifically defined as the study of sensory or sensori-emotional values, sometimes called judgements of sentiment and taste. More broadly, scholars in the field define aesthetics as "critical reflection on art, nature". The aesthetic is derived from the Greek αἰσθητικός, which in turn was derived from αἰσθάνομαι. Aesthetics is for the artist as Ornithology is for the birds. Modern day aesthetics, especially among younger people, refers to the simplicity in beauty in art. Art is an autonomous entity for philosophy, because art deals with art is as such free of any moral or political purpose. Aesthetics is neither epistemology nor ethics. Any aesthetic doctrines that guided the interpretation of prehistoric art are mostly unknown. Western aesthetics usually refers as the earliest source of formal aesthetic considerations. Plato believed in beauty as a form which causes them to be beautiful. He felt that beautiful objects incorporated proportion, unity among their parts. Similarly, in the Metaphysics, Aristotle found that the universal elements of beauty were order, definiteness. From the late 17th to the 20th century Western aesthetics underwent a slow revolution into what is often called modernism. British thinkers emphasized beauty as the key component of art and of the aesthetic experience, saw art as necessarily aiming at absolute beauty.
Aesthetics
–
Bronze sculpture, thought to be either Poseidon or Zeus, National Archaeological Museum of Athens
Aesthetics
–
Cubist painting by Georges Braque, Violin and Candlestick (1910)
Aesthetics
–
William Hogarth, self-portrait, 1745
Aesthetics
–
Example of the Dada aesthetic, Marcel Duchamp 's Fountain 1917
125.
Beauty
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Beauty is a characteristic of an animal, idea, object, person or place that provides a perceptual experience of pleasure or satisfaction. Beauty is studied as part of aesthetics, social sociology. An "ideal beauty" is an entity which possesses features widely attributed to beauty for perfection. Because this can be a subjective experience, it is often said that "beauty is in the eye of the beholder." The adjective was καλός, kalos. However, kalos is also thus has a broader meaning than only beautiful. Similarly, kallos was used differently from the English beauty in that it bears an erotic connotation. The Greek word for beautiful was hōraios, an adjective etymologically coming from the word ὥρα, hōra, meaning "hour". In Koine Greek, beauty was thus associated with "being of one's hour". In Attic Greek, hōraios had many meanings, including "youthful" and "ripe old age". The earliest Western theory of beauty can be found in the works of Greek philosophers such as Pythagoras. The Pythagorean school saw a strong connection between mathematics and beauty. In particular, they noted that objects proportioned according to the golden ratio seemed more attractive. Greek architecture is based on this view of proportion. Plato considered beauty to be the Idea above all other Ideas.
Beauty
–
Rayonnant rose window in Notre Dame de Paris. In Gothic architecture, light was considered the most beautiful revelation of God.
Beauty
–
For beauty as a characteristic of a person's appearance, see Physical attractiveness. For other uses, see Beauty (disambiguation).
Beauty
–
The Birth of Venus, by Sandro Botticelli. The goddess Venus is the classical personification of beauty.
Beauty
–
Fresco of a Roman woman from Pompeii, c. 50 AD
126.
Proof (mathematics)
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In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies. An unproved proposition, believed to be true is known as a conjecture. Proofs usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. Formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics. The philosophy of mathematics is concerned with mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren. The early use of "probity" was in the presentation of legal evidence. A person such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as analogies preceded strict mathematical proof.
Proof (mathematics)
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One of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
Proof (mathematics)
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Visual proof for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
127.
Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1, not a prime number is called a composite number. The property of being prime is called primality. A slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n. Algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits. There are infinitely many primes, as demonstrated around 300 BC. There is no known simple formula that separates prime numbers from composite numbers. However, the statistical behaviour of primes in the large, can be modelled. Many questions regarding prime numbers remain open, such as the twin prime conjecture. Such questions spurred the development of various branches of theory, focusing on analytic or algebraic aspects of numbers. Prime numbers give rise to various generalizations in mainly algebra, such as prime elements and prime ideals. A natural number is called a prime number if it has itself.
Prime number
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The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
128.
Numerical method
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In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate check in a programming language is called a numerical algorithm. The problems of which the method consists need not be well-posed. If they are, the method is said to be well-posed. When F n = F, ∀ n ∈ N on S the method is said to be strictly consistent. One can easily prove that the point-wise convergence of n ∈ N to y implies the convergence of the associated method.
Numerical method
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Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296...
Numerical method
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Direct method
Numerical method
129.
Fast Fourier transform
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A fast Fourier transform algorithm computes the discrete Fourier transform of a sequence, or its inverse. Fourier analysis converts a signal from its original domain to a representation in the frequency versa. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse factors. Fast Fourier transforms are widely used for many applications in mathematics. Some algorithms had been derived early as 1805. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields but computing it directly from the definition is often too slow to be practical. The difference in speed can be enormous, especially for long data sets where N may be in the thousands or millions. The best-known FFT algorithms depend upon the factorization of N, but there are FFTs with O complexity for all N, even for prime N. While Gauss's work predated even Fourier's results in 1822, he did not analyze the computation time and eventually used other methods to achieve his goal. Between 1805 and 1965, some versions of FFT were published by other authors. Yates in 1932 published his version called interaction algorithm, which provided efficient computation of Hadamard and Walsh transforms. Yates' algorithm is still used in the field of statistical design and analysis of experiments. In 1942, Danielson and Lanczos published their version to compute DFT for x-ray crystallography, a field where calculation of Fourier transforms presented a formidable bottleneck. Cooley and Tukey published a more general version of FFT in 1965, applicable when N is composite and not necessarily a power of 2.
Fast Fourier transform
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Time and frequency domain for the same signal
130.
G.H. Hardy
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Godfrey Harold "G. H." Hardy FRS was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy -- a basic principle of population genetics. In addition to his research, he is remembered on the aesthetics of mathematics, entitled A Mathematician's Apology. Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan. Starting in 1914, he was the mentor of a relationship that has become celebrated. Hardy and Ramanujan became close collaborators. Hardy called their collaboration "the one romantic incident in my life." G. H. Hardy was born on 7 February 1877, in Cranleigh, Surrey, England, into a family. His father was Bursar and Art Master at Cranleigh School; his mother had been a senior mistress at Lincoln Training College for teachers. Both parents were mathematically inclined. Hardy's natural affinity for mathematics was perceptible at an early age. When taken to church he amused himself by factorising the numbers of the hymns. After schooling at Cranleigh, he was awarded a scholarship to Winchester College for his mathematical work. In 1896 Hardy entered Cambridge. After only two years of preparation under Robert Alfred Herman, he was fourth in the Mathematics Tripos examination.
G.H. Hardy
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G. H. Hardy
131.
A Mathematician's Apology
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A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It gives the layman an insight into the mind of a working mathematician. Hardy felt the need to justify his life's work at this time mainly for two reasons. Firstly, at age 62, Hardy felt the decline of his mathematical creativity and skills. By devoting time to writing the Apology, Hardy was admitting that his own time as a creative mathematician was finished. In Hardy's words, appreciation, is work for second-rate minds. It is a melancholy experience for a professional mathematician to find himself writing about mathematics. Hardy makes his justification not to God but to his fellow man. One of the main themes of the book is the beauty which Hardy compares to painting and poetry. If an application of theory were to be found, then certainly no one would try to dethrone the "queen of mathematics" because of that. This view reflects Hardy's increasing depression at the wane of his mathematical powers. For Hardy, real mathematics was essentially a creative activity, rather than an expository one. Hardy's opinions were heavily influenced between World War I and World War II. Some of Hardy's examples seem unfortunate in retrospect. Since then theory was used to crack German enigma codes and much later, figure prominently in public-key cryptography.
A Mathematician's Apology
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In A Mathematician's Apology, G. H. Hardy defined a set of criteria for mathematical beauty.
A Mathematician's Apology
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Forms
132.
Leonhard Euler
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He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. Euler is also known for his work in mechanics, music theory. Euler was one of the most eminent mathematicians of the 18th century, is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field. He spent most of his adult life in St. Petersburg, Russia, in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler later won this annual prize twelve times.
Leonhard Euler
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Portrait by Jakob Emanuel Handmann (1756)
Leonhard Euler
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1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.
Leonhard Euler
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Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, nowadays written as " v − e + f = 2".
Leonhard Euler
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Euler's grave at the Alexander Nevsky Monastery
133.
Musical notation
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Much information about ancient music notation is fragmentary. Although many ancient cultures used symbols to represent melodies and rhythms, this has limited today's understanding of their music. The seeds of what would eventually become western notation were sown in medieval Europe, starting with the Catholic church's goal for ecclesiastical uniformity. The church began notating plainchant melodies so that the same chants could be used throughout the church. Music notation developed in the Renaissance and Baroque music eras. The introduction of figured notation in the Baroque era marked the beginning of the first compositions based around chord progressions. In the Romantic music era, notation continued to develop as new musical instrument technologies were developed. Music notation has been adapted to many kinds including classical music, popular music, traditional music. The earliest form of musical notation can be found in a tablet, created at Nippur, in Sumer, in about 2000 BC. The tablet represents fragmentary instructions for performing music, that it was written using a diatonic scale. A tablet from about 1250 BC shows a more developed form of notation. Although they are fragmentary, these tablets represent the earliest notated melodies found anywhere in the world. The notation consists of symbols placed above text syllables. An example of a complete composition is the Seikilos epitaph, variously dated to the 1st century AD. Three hymns by Mesomedes of Crete exist in manuscript.
Musical notation
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A photograph of the original stone at Delphi containing the second of the two Delphic Hymns to Apollo. The music notation is the line of occasional symbols above the main, uninterrupted line of Greek lettering.
Musical notation
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Hand-written musical notation by J. S. Bach: beginning of the Prelude from the Suite for Lute in G minor BWV 995 (transcription of Cello Suite No. 5, BWV 1011) BR Bruxelles II. 4805.
Musical notation
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Music notation from an early 14th century English Missal
Musical notation
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Early music notation
134.
Open set
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In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line. They allow enormous flexibility in the choice of open sets. In the two extremes, no set can be open but the space itself and the empty set. In practice, however, open sets are usually chosen to be similar to the open intervals of the real line. Each choice of open sets for a space is called a topology. Intuitively, an open set provides a method to distinguish two points. In this manner, one may speak of whether two subsets of a topological space are "near" without concretely defining a metric on the topological space. Therefore, topological spaces may be seen as a generalization of metric spaces. In essence, points within ε of x approximate x to an accuracy of degree ε. However, with ε = 0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a greater degree of accuracy compared to when ε = 1. In particular, sets of the form give a lot of information about points close to x = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to x. In fact, one may generalize these notions to an arbitrary set; rather than just the real numbers. Given a point of that set, one may define a collection of sets "around" x, used to approximate x.
Open set
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Example: The points (x, y) satisfying x 2 + y 2 = r 2 are colored blue. The points (x, y) satisfying x 2 + y 2 < r 2 are colored red. The red points form an open set. The blue points form a boundary set. The union of the red and blue points is a closed set.
135.
Field (mathematics)
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In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a commutative ring, or equivalently a ring whose nonzero elements form an group under multiplication. As such it is an algebraic structure with notions of division satisfying the appropriate abelian group equations and distributive law. Any field may be used as the scalars for a vector space, the standard general context for linear algebra. In modern mathematics, the theory of fields plays an essential role in number theory and algebraic geometry. As an algebraic structure, every field is a ring, but not every ring is a field. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed is called a field. Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, such that a + = 0. Similarly, for a in F other than 0, there exists an element a 1 in F, such that a − 1 = 1. In other words, subtraction and division operations exist. Distributivity For all a c in the following equality holds: a · = +. The additive inverse of such a fraction is simply −a/b, the multiplicative inverse is b/a.
Field (mathematics)
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Given 0, 1, r 1 and r 2, the construction yields r 1 · r 2
136.
Homeomorphism
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Homeomorphisms are the isomorphisms in the category of topological spaces—, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, from a topological viewpoint they are the same. The homeomorphism comes from the Greek words ὅμοιος form. The homeomorphism is a continuous bending of the object into a new shape. Thus, a torus are not. A function with these three properties is sometimes called bicontinuous. If such a function exists, we say X and Y are homeomorphic. A self-homeomorphism is a homeomorphism of a topological space and itself. The homeomorphisms form an equivalence relation on the class of all topological spaces. The resulting equivalence classes are called homeomorphism classes. The open interval is homeomorphic to the real numbers R for any a < b.. The square in R2 are homeomorphic; since the unit disc can be deformed into the unit square. An example of a bicontinuous mapping from the square to the disc is, in polar coordinates, ↦. The graph of a differentiable function is homeomorphic to the domain of the function. A differentiable parametrization of a curve is an homeomorphism between the domain of the parametrization and the curve.
Homeomorphism
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A trefoil knot is homeomorphic to a circle, but not isotopic. Continuous mappings are not always realizable as deformations. Here the knot has been thickened to make the image understandable.
Homeomorphism
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A continuous deformation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse.
137.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total. Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written: F = ∫ f d x. The integrals discussed in this article are those termed definite integrals. A rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a limiting procedure which approximates the area of a region by breaking the region into vertical slabs. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space. A similar method was independently developed in China around the 3rd century AD by Liu Hui, who used it to find the area of the circle. This method was later used by father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere. The next significant advances in integral calculus did not begin to appear until the 17th century. Further steps were made in the 17th century by Barrow and Torricelli, who provided the first hints of a connection between differentiation. Barrow provided the first proof of the fundamental theorem of calculus.
Integral
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A definite integral of a function can be represented as the signed area of the region bounded by its graph.
138.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of theory was initiated in the 1870s. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, theory is a branch of mathematics in its own right, with an active community. Mathematical topics typically evolve among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began on theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of theory led to the article "Mengenlehre" contributed in 1898 to Klein's encyclopedia. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox.
Set theory
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Georg Cantor
Set theory
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A Venn diagram illustrating the intersection of two sets.
139.
Abacus
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The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus. The Latin word came from ἄβαξ abax which means something without base, improperly, any piece of rectangular board or plank. Alternatively, "drawing-board covered with dust". Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps word akin to Hebrew ʾābāq, "dust". The preferred plural of abacus is a subject of disagreement, with both abaci in use. The user of an abacus is called an abacist. Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument have not been discovered. During the Achaemenid Empire, around 600 BC the Persians first began to use the abacus. The earliest archaeological evidence for the use of the Greek abacus dates to the 5th BC. Also Demosthenes talked of the need to use pebbles for calculations too difficult for your head. The Greek abacus was a table of wood or marble, metal for mathematical calculations. This Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, until the French Revolution, the Western Christian world. A tablet found in 1846 AD, dates back to 300 BC, making it the oldest counting board discovered so far.
Abacus
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A Chinese abacus
Abacus
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Calculating-Table by Gregor Reisch: Margarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (inaccurately represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th.
Abacus
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Copy of a Roman abacus
Abacus
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Japanese soroban
140.
Arithmetic
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Arithmetic or arithmetics is the oldest and most elementary branch of mathematics. It consists of the study of numbers, especially the properties of the traditional operations between them -- subtraction, division. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, descended from tally marks used for counting. In both cases, this origin resulted in values that used a decimal base but did not include positional notation. Complex calculations with Roman numerals required the assistance of a counting board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the sexagesimal system for Babylonian numerals and the vigesimal system that defined Maya numerals. Because of this place-value concept, the ability to reuse the same digits for different values contributed to simpler and more efficient methods of calculation. The historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, although it originated later than the Babylonian and Egyptian examples. Prior to the works of Euclid around 300 BC, Greek studies in mathematics overlapped with philosophical and mystical beliefs. For example, Nicomachus summarized the viewpoint of the earlier Pythagorean approach to Arithmetic. Greek numerals were used from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols. One for the hundred's. Then for the thousand's place they would reuse the symbols for the unit's place, so on.
Arithmetic
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Arithmetic tables for children, Lausanne, 1835
Arithmetic
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A scale calibrated in imperial units with an associated cost display.
141.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures for dealing with lengths, areas, volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since then, into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience. While geometry has evolved significantly throughout the years, there are some general concepts that are less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, curves, as well as the more advanced notions of manifolds and topology or metric. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense. The educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, analytic geometry. Euclidean geometry also has applications in computer science, various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry.
Geometry
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Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Geometry
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An illustration of Desargues' theorem, an important result in Euclidean and projective geometry
Geometry
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Geometry lessons in the 20th century
Geometry
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A European and an Arab practicing geometry in the 15th century.
142.
Mathematical analysis
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, analytic functions. These theories are usually studied in the context of real and complex functions. Analysis evolved from calculus, which involves the elementary techniques of analysis. Many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of Greek mathematics. For instance, an infinite sum is implicit in Zeno's paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes' The Method of a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. The Indian mathematician Bhāskara II used what is now known as Rolle's theorem in the 12th century. His followers at the Kerala school of mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. During this period, techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Instead, Cauchy formulated calculus in terms of geometric infinitesimals.
Mathematical analysis
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A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications to science and engineering.
143.
Uncertainty
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Uncertainty is a situation which involves imperfect and/or unknown information. However, "uncertainty is an unintelligible expression without a straightforward description". It applies to predictions to the unknown. Uncertainty arises in partially observable and/or stochastic environments, as well as due to ignorance and/or indolence. A state of having limited knowledge where it is impossible to exactly describe the existing state, a future outcome, or more than one possible outcome. Risk A state of uncertainty where some possible outcomes have an undesired effect or significant loss. If probabilities are applied to the possible outcomes using weather forecasts or even just a calibrated probability assessment, the uncertainty has been quantified. Suppose it is quantified as a 90% chance of sunshine. Furthermore, if this is a business event and $100,000 would be lost if it rains, then the risk has been quantified. These situations can be made even more realistic by quantifying light rain vs. the cost of outright cancellation, etc.. Some may represent the risk in this example as the "expected opportunity loss" or the chance of the loss multiplied by the amount of the loss. That is useful if the organizer of the event is "risk neutral", which most people are not. Most would be willing to pay a premium to avoid the loss. An company, for example, would compute an EOL as a minimum for any coverage, then add onto that other operating costs and profit. Since many people are willing to buy insurance for many reasons, then clearly the EOL alone is not the perceived value of avoiding the risk.
Uncertainty
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We are frequently presented with situations wherein a decision must be made when we are uncertain of exactly how to proceed.
144.
Riemann surface
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In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. For example, they can look like several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. It contains more structure, needed for the unambiguous definition of holomorphic functions. A real manifold can be turned into a Riemann surface if and only if it is orientable and metrizable. The Möbius strip, Klein bottle and projective plane do not. They often provide the intuition and motivation for generalizations to other curves, manifolds or varieties. The Riemann–Roch theorem is a prime example of this influence. There are several equivalent definitions of a Riemann surface. A Riemann surface X is a complex manifold of complex one. The map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the transition maps between two overlapping charts are required to be holomorphic. A Riemann surface is an oriented manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any x of X, the space is homeomorphic to a subset of the real plane. Two such metrics are considered equivalent if the angles they measure are the same.
Riemann surface
–
Riemann surface for the function ƒ (z) = √ z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √ z. For the imaginary part of √ z, rotate the plot 180° around the vertical axis.
145.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a well-defined collection of distinct objects. The objects that make up a set can be anything: other sets, so on. Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description: A is the set whose members are the first four positive integers. B is the set of colors of the French flag. The second way is by extension –, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets: C = D =. One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D. There are two important points to note about sets.
Set (mathematics)
–
A set of polygons in a Venn diagram
146.
Category theory
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Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows. A category has the existence of an identity arrow for each object. The language of theory has been used to formalize concepts of other high-level abstractions such as sets, rings, groups. Several terms used in theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In theory, morphisms obey conditions specific to category theory itself. Category theory has practical applications in particular for the study of monads in functional programming. Categories represent abstraction of mathematical concepts. Many areas of mathematics can be formalised as categories. A basic example of a category is the category of sets, where the arrows are functions from one set to another. However, the arrows need not be functions. The "arrows" of theory are often said to represent a process connecting two objects, or in many cases a "structure-preserving" transformation connecting two objects. There are, however, many applications where much more abstract concepts are represented by morphisms. The most important property of the arrows is that they can be "composed", in other words, arranged in a sequence to form a new arrow. Linear algebra can also be expressed in terms of categories of matrices. Consider the following example.
Category theory
–
Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1 X, 1 Y and 1 Z, if explicitly represented, would appear as three arrows, next to the letters X, Y, and Z, respectively, each having as its "shaft" a circular arc measuring almost 360 degrees.)
147.
Theoretical computer science
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Work in this field is often distinguished by its emphasis on mathematical rigor. Despite this broad scope, the "theory people" in computer self-identify as different from the "applied people." Some characterize themselves as doing the "'science' underlying the field of computing." Other "theory-applied people" suggest that it is impossible to separate application. This means that the so-called "theory people" regularly use experimental science done in less-theoretical areas such as software research. It also means that there is more cooperation than mutually exclusive competition between application. These developments have led as a whole. Information theory was added by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of parallel distributed processing were established. In other words, one could compute functions on multiple states simultaneously. Modern theoretical computer research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed. An algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, automated reasoning. An algorithm is an effective method expressed for calculating a function.
Theoretical computer science
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An artistic representation of a Turing machine. Turing machines are used to model general computing devices.
148.
Computational complexity theory
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A problem is regarded inherently difficult if its solution requires significant resources, whatever the algorithm used. Other complexity measures are also used, such as the amount of communication, the number of processors. One of the roles of computational theory is to determine the practical limits on what computers can and can not do. Closely related fields in theoretical science are analysis of algorithms and computability theory. More precisely, computational theory tries to classify problems that can or can not be solved with appropriately restricted resources. A computational problem can be viewed together with a solution for every instance. The string for a computational problem is referred to as a problem instance, should not be confused with the problem itself. In computational theory, a problem refers to the abstract question to be solved. In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a problem. For example, consider the problem of primality testing. The solution is "yes" if the number is prime and "no" otherwise. The solution is the output corresponding to the given input. For this reason, theory addresses computational problems and not particular problem instances. When considering computational problems, a instance is a string over an alphabet. Usually, thus the strings are bitstrings.
Computational complexity theory
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A traveling salesman tour through Germany ’s 15 largest cities.
149.
Information theory
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Information theory studies the quantification, storage, communication of information. A key measure in information theory is "entropy". Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip provides less information than specifying the outcome from a roll of a die. Some important measures in theory are mutual information, channel capacity, error exponents, relative entropy. Applications of fundamental topics of theory include lossless data compression, channel coding. The field is at the intersection of mathematics, statistics, computer science, electrical engineering. Information theory studies the transmission, extraction of information. Abstractly, information can be thought of as the resolution of uncertainty. Information theory is a deep mathematical theory, amongst, the vital field of coding theory. These codes can be roughly subdivided into error-correction techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms. Results from coding theory and information theory are widely used in cryptography and cryptanalysis. See the article ban for a historical application.
Information theory
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A picture showing scratches on the readable surface of a CD-R. Music and data CDs are coded using error correcting codes and thus can still be read even if they have minor scratches using error detection and correction.
Information theory
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Entropy of a Bernoulli trial as a function of success probability, often called the binary entropy function,. The entropy is maximized at 1 bit per trial when the two possible outcomes are equally probable, as in an unbiased coin toss.
150.
Turing machine
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Given any computer algorithm, a Turing machine can be constructed, capable of simulating that algorithm's logic. The machine operates on an infinite tape divided into cells. The machine positions its head over a cell and "reads" the symbol there. The Turing machine was invented by Alan Turing, who called it an a-machine. Thus, Turing machines prove fundamental limitations on the power of mechanical computation. Turing completeness is the ability for a system of instructions to simulate a Turing machine. Assuming a black box, the Turing machine cannot know whether it will eventually enumerate any one specific string of the subset with a given program. This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. This is famously demonstrated through lambda calculus. A Turing machine, able to simulate any other machine is called a universal Turing machine. Studying their abstract properties yields many insights into complexity theory. At any moment there is one symbol in the machine; it is called the scanned symbol. However, the tape can be moved forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings. The Turing machine mathematically models a machine that mechanically operates on a tape.
Turing machine
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The evolution of the busy-beaver's computation starts at the top and proceeds to the bottom.
Turing machine
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An implementation of a Turing machine
Turing machine
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A Turing machine realisation in LEGO
Turing machine
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An experimental prototype to achieve Turing machine
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P = NP problem
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The P versus NP problem is a major unsolved problem in computer science. Informally speaking, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. The first mention of the underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether theorem-proving could be solved in quadratic or linear time. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution. The general class of questions for which some algorithm can provide an answer in polynomial time is called "class P" or just "P". The class of questions for which an answer can be verified in polynomial time is called NP, which stands for "nondeterministic polynomial time." Consider the subset sum problem, an example of a problem, easy to verify, but whose answer may be difficult to compute. Given a set of integers, does some nonempty subset of them sum to 0? For instance, does a subset of the set add up to 0? The answer "yes, because the subset adds up to zero" can be quickly verified with three additions. The most common resources are time and space. In such analysis, a model of the computer for which time must be analyzed is required. Typically such models assume that the computer is deterministic and sequential. Clearly, P ⊆ NP.
P = NP problem
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Diagram of complexity classes provided that P ≠ NP. The existence of problems within NP but outside both P and NP -complete, under that assumption, was established by Ladner's theorem.
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Data compression
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In signal processing, data compression, source coding, or bit-rate reduction involves encoding information using fewer bits than the original representation. Compression can be either lossless. Lossless compression reduces bits by eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing less important information. The process of reducing the size of a file is referred to as data compression. In the context of transmission, it is called source coding in opposition to channel coding. Compression is useful because it reduces resources required to transmit data. Computational resources are consumed in the process and, usually, in the reversal of the process. Data compression is subject to a space -- time trade-off. Lossless compression algorithms usually exploit statistical redundancy to represent data without losing any information, so that the process is reversible. Lossless compression is possible because most real-world data exhibits statistical redundancy. This is a basic example of run-length encoding; there are many schemes to reduce size by eliminating redundancy. The Lempel–Ziv compression methods are among the most popular algorithms for lossless storage. Compression can be slow.
Data compression
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Comparison of acoustic spectrograms of a song in an uncompressed format and lossy formats. That the lossy spectrograms are different from the uncompressed one indicates that they are, in fact, lossy, but nothing can be assumed about the effect of the changes on perceived quality.
Data compression
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Solidyne 922: The world's first commercial audio bit compression card for PC, 1990
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Entropy (information theory)
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In information theory, systems are modeled by a transmitter, channel, receiver. The transmitter produces messages that are sent through the channel. The channel modifies the message in some way. The receiver attempts to infer which message was sent. In this context, entropy is the expected value of the information contained in each message. 'Messages' can be modeled by any flow of information. In a more technical sense, there are reasons to define information as the negative of the logarithm of the probability distribution. The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources. For instance, the entropy of a coin toss is 1 shannon, whereas of m tosses it is m shannons. Generally, you need log2 bits to represent a variable that can take one of n values if n is a power of 2. If these values are equally probable, the entropy is equal to the number of bits. Equality between number of bits and shannons holds only while all outcomes are equally probable. If one of the events is more probable than others, observation of that event is less informative. Conversely, rarer events provide more information when observed. Since observation of less probable events occurs more rarely, the net effect is that the entropy received from non-uniformly distributed data is less than log2.
Entropy (information theory)
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2 shannons of entropy: Information entropy is the log-base-2 of the number of possible outcomes; with two coins there are four outcomes, and the entropy is two bits.
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Theory of computation
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In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. The most commonly examined is the Turing machine. So in principle, any problem that can be solved by a Turing machine can be solved by a computer that has a finite amount of memory. The theory of computation can be considered the creation of models of all kinds in the field of science. Therefore, logic are used. In the last century it was separated from mathematics. Some pioneers of the theory of computation were Alonzo Church, Kurt Gödel, Alan Turing, Stephen Kleene, Claude Shannon. Automata theory is the computational problems that can be solved using these machines. These abstract machines are called automata. Automata comes from the Greek word which means that something is doing something by itself. An automaton can be a finite representation of a formal language that may be an infinite set. Automata are used for proofs about computability. Language theory is a branch of mathematics concerned with describing languages as a set of operations over an alphabet. It is closely linked with theory, as automata are used to generate and recognize formal languages. Because automata are used as models for computation, formal languages are the preferred mode of specification for any problem that must be computed.
Theory of computation
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An artistic representation of a Turing machine. Turing machines are frequently used as theoretical models for computing.
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Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are words used for ordering are "ordinal numbers". These chains of extensions make the natural numbers canonically embedded in the other number systems. Properties such as divisibility and the distribution of prime numbers, are studied in number theory. Problems such as partitioning and enumerations, are studied in combinatorics. The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested by striking out a mark and removing an object from the set. The major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, all the powers of 10 up to over 1 million. A later advance was the development of the idea that 0 can be considered as a number, with its own numeral. This usage did not spread beyond Mesoamerica. The use of a 0 in modern times originated with the Indian mathematician Brahmagupta in 628. The systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all.
Natural number
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The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
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Natural numbers can be used for counting (one apple, two apples, three apples, …)
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Integer
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An integer is a number that can be written without a fractional component. For example, 21, − 2048 are integers, while 9.75, 5 1⁄2, √ 2 are not. The set of integers consists of the natural numbers, also called their additive inverses. This is often denoted by a boldface Z or bold standing for the German word Zahlen. ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the algebraic integers that are also rational numbers. However, with the inclusion of the negative natural numbers, and, importantly, 0, Z is also closed under subtraction. This universal property, namely to be an initial object in the category of rings, characterizes the ring Z. Z is not closed under division, since the quotient of two integers, need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the basic properties of multiplication for any integers a, c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z.
Integer
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Algebraic structure → Group theory Group theory
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Fermat's Last Theorem
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The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity. The successful proof was released by Andrew Wiles, formally published by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. His claim was discovered some 30 years later, after his death. This claim, which came to be known as Fermat's Last Theorem, stood unsolved in mathematics for the following three and a half centuries. The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in number theory, over time Fermat's Last Theorem gained prominence as an unsolved problem in mathematics. With the special case n = 4 proved, it suffices to prove the theorem for exponents n that are prime numbers. In the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. The solution came in a roundabout manner, from a completely different area of mathematics. Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two completely different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, as the modularity theorem, it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was widely considered completely inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermat's Last Theorem. This potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermat's Last Theorem that depended on the modularity theorem.
Fermat's Last Theorem
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The 1670 edition of Diophantus ' Arithmetica includes Fermat's commentary, particularly his "Last Theorem" (Observatio Domini Petri de Fermat).
Fermat's Last Theorem
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Problem II.8 in the 1621 edition of the Arithmetica of Diophantus. On the right is the margin that was too small to contain Fermat's alleged proof of his "last theorem".
Fermat's Last Theorem
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British mathematician Andrew Wiles
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Goldbach's conjecture
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Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. The conjecture remains unproven despite considerable effort. A Goldbach number is a positive integer that can be expressed as the sum of two odd primes. The expression of a given even number as a sum of two primes is called a Goldbach partition of that number. He then proposed a second conjecture in the margin of his letter: Every integer greater than 2 can be written as the sum of three primes. He considered 1 to be a prime number, a convention subsequently abandoned. This did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is: Every integer greater than 5 can be written as the sum of three primes. Goldbach's third version is the form in which the conjecture is usually expressed today. "binary" Goldbach conjecture, to distinguish it from a weaker corollary. While the weak Goldbach conjecture appears to have been finally proved in 2013, the strong conjecture has remained unsolved. For small values of n, the strong Goldbach conjecture can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n 105. One record from this search is that 3325581707333960528 is the smallest number that has no Goldbach partition with a prime below 9781.
Goldbach's conjecture
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Letter from Goldbach to Euler dated on 7. June 1742 (Latin-German).
Goldbach's conjecture
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Subset
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A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. The relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. We may also partially order P by defining A ≤ B ⟺ B ⊆ A. When quantified, A ⊆ B is represented as: ∀x. So for example, for these authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊃ to indicate proper subset and superset, respectively, instead of ⊊ and ⊋. This usage makes ⊂ analogous to the inequality symbols ≤ and <. The set A = is a proper subset of B =, thus A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, D ⊊ E is not true. Any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given X. It is always a proper subset of any set except itself. The set of rational numbers is a proper subset of the set of real numbers.
Subset
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Euler diagram showing A is a proper subset of B and conversely B is a proper superset of A
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Rational number
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Since q may be equal to 1, every integer is a rational number. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true just for base 10, but also for any other integer base. A real number, not rational is called irrational. Irrational numbers include √ 2, π, φ. The decimal expansion of an irrational number continues without repeating. Since the set of real numbers is uncountable, almost all real numbers are irrational. In abstract algebra, the rational numbers together with certain operations of multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. The algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed by completion using Cauchy sequences, Dedekind cuts, or infinite decimals. Zero divided by any other integer equals zero; therefore, zero is a rational number. The term rational to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun "rational number".
Rational number
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A diagram showing a representation of the equivalent classes of pairs of integers
161.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, simple fraction consists of a non-zero denominator, displayed below that line. Denominators are also used in fractions that are not common, including mixed numerals. The picture to the right illustrates 3 4 or ¾ of a cake. Fractional numbers can also be written by using negative exponents. An integer such as the number 7 can be thought of as having an implicit denominator of one: 7 equals 7/1. Other uses for fractions are to represent ratios and to represent division. Thus 3/4 is also used to represent 3 ÷ 4. The test for a number being a rational number is that it can be written in that form. Informally, they may be distinguished by placement alone but in formal contexts they are always separated by a fraction bar. The bar may be diagonal. These marks are respectively known as the fraction slash. The denominators of English fractions are generally expressed as ordinal numbers, in the plural if the numerator is not one. When the numerator is one, it may be omitted.
Fraction (mathematics)
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
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Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced by Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the fraction 4/3, all the irrational numbers, such as √ 2. Included within the irrationals are the transcendental numbers, such as π. Complex numbers include real numbers. These descriptions of the real numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions are thus equivalent. Around 500 BC, the Greek mathematicians led by Pythagoras realized the need in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of "number" and "magnitude" into a more general idea of real numbers. In the 17th century, Descartes introduced the term "real" to describe roots of a polynomial, distinguishing them from "imaginary" ones. In the 19th centuries, there was much work on irrational and transcendental numbers. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, Ferdinand von Lindemann, showed that π is transcendental. Lindemann's proof has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the entire set of real numbers without having defined them cleanly.
Real number
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A symbol of the set of real numbers (ℝ)
163.
Continuous function
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In mathematics, a continuous function is, roughly speaking, a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the core concepts of topology, treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. Especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article. As an example, consider the h, which describes the height of a growing flower at t. This function is continuous. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of variable quantities, his definition of continuity closely parallels the infinitesimal definition used today. All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. Such a point is called a discontinuity.
Continuous function
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Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.
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Complex number
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In this expression, b is the imaginary part of the complex number. A + bi can be identified with the point in the complex plane. As as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation = − 9 has no real solution, since the square of a real number can not be negative. Complex numbers provide a solution to this problem. According to the fundamental theorem of algebra, all polynomial equations with complex coefficients in a single variable have a solution in complex numbers. For example, 3.5 + 2i is a complex number. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part. For example, Re = − 3.5 Im = 2. Hence, in imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is sometimes known as the Cartesian form of z. A can be regarded as a complex number a + 0i whose imaginary part is 0.
Complex number
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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i 2 = −1.
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Quaternion
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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, therefore also a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H, or in blackboard bold by H. It can also be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Quaternion algebra was introduced by Hamilton in 1843. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. However, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. As he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. This letter was later published in the London, Edinburgh, Dublin Philosophical Magazine and Journal of Science, vol. xxv, pp 489–95.
Quaternion
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Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says: Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i 2 = j 2 = k 2 = ijk = −1 & cut it on a stone of this bridge
Quaternion
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Graphical representation of quaternion units product as 90°-rotation in 4D-space, ij = k, ji = − k, ij = − ji
166.
Octonion
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There are three normed division algebras over the reals: the quaternions H. The octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, namely they are alternative. Octonions are not well known as the complex numbers, which are much more widely used. Despite this, they are related among them the exceptional Lie groups. Additionally, octonions have applications in fields such as logic. The octonions were discovered in 1843 by John T. Graves, inspired by his friend W. R. Hamilton's discovery of quaternions. The octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery. Hamilton invented the word associative so that he could say that octonions were not associative. The octonions can be thought of as octets of real numbers. Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions. 7.
Octonion
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A mnemonic for the products of the unit octonions.
167.
Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, it is not the same sort of number as real numbers. Georg Cantor formalized many ideas related during the late 19th and 20th centuries. In the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from a Greek philosopher who lived in Miletus. He used the apeiron which means limitless. Aristotle called him the inventor of the dialectic. He is described as "immeasurably subtle and profound". However, recent readings of the Archimedes Palimpsest have hinted that Archimedes at least had an intuition about actual infinite quantities. The mathematical text Surya Prajnapti classifies all numbers into three sets: innumerable, infinite. On both ontological grounds, a distinction was made between rigidly bounded and loosely bounded infinities. European mathematicians started using infinite numbers in a systematic fashion in the 17th century.
Infinity
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Infinity represented in screenshot form
168.
Cardinal number
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In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The cardinal numbers describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, if there is a one-to-one correspondence between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. There is a transfinite sequence of cardinal numbers: 0, 1, 2, 3... n... ℵ 0, ℵ 1, ℵ 2... ℵ α.... This sequence starts with the natural numbers including zero, which are followed by the aleph numbers. The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs. Cardinality is studied as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, mathematical analysis. In theory, the cardinal numbers form a skeleton of the category of sets.
Cardinal number
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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.
169.
Aleph number
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They are named after the symbol used to denote them, the Hebrew letter aleph. The cardinality of the natural numbers is ℵ 0, the next larger cardinality is aleph-one ℵ 1, then ℵ 2 and so on. Continuing in this manner, it is possible to define a cardinal number ℵ α for every ordinal number α, as described below. The concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity commonly found in algebra and calculus. ℵ 0 is the cardinality of the set of all natural numbers, is an infinite cardinal. The set of all finite ordinals, called ω or ω0, has cardinality ℵ 0. A set has cardinality ℵ 0 if and only if it is countably infinite, there is a bijection between it and the natural numbers. These infinite ordinals: ω, ω+1, ω·2, ω2, ωω and ε0 are among the countably infinite sets. For example, the sequence of all positive odd integers followed by all positive even integers is an ordering of the set of positive integers. If the axiom of countable choice holds, then ℵ 0 is smaller than any other infinite cardinal. ℵ 1 is the cardinality of the set of all countable ordinal numbers, called ω1 or Ω. This ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 is distinct from ℵ 0. The definition of ℵ 1 implies that no cardinal number is between ℵ 0 and ℵ 1.
Aleph number
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Aleph-naught, the smallest infinite cardinal number
170.
Function (mathematics)
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An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f. In this example, if the input is 3, we may write f = 9. Likewise, if the input is 3, then the output is also 9, we may write f = 9. The input variable are sometimes referred to as the argument of the function. Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse as a solution of a equation. In the example above, f = x2, we have the ordered pair. More commonly the word "range" is used to mean, specifically the set of outputs. The image of this function is the set of non-negative real numbers. In analogy with arithmetic, it is possible to define addition, division of functions, in those cases where the output is a number.
Function (mathematics)
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A function f takes an input x, and returns a single output f (x). One metaphor describes the function as a "machine" or " black box " that for each input returns a corresponding output.
171.
Operation (mathematics)
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In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation. An operation of 0-ary operation is a constant. The mixed product is an example of ternary operation. Infinitary operations are sometimes considered. In this context, the usual operations, of finite arity are also called finitary operations. There are two common types of operations: binary. Unary operations involve only one value, such as trigonometric functions. Binary operations, on the other hand, include addition, subtraction, multiplication, division, exponentiation. Operations can involve mathematical objects other than numbers. The logical values false can be combined using logic operations, such as and, or, not. Vectors can be subtracted. Rotations can be combined performing the first rotation and then the second. Operations on sets include the unary operation of complementation. Operations on functions include convolution.
Operation (mathematics)
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– plus (addition) – minus (subtraction) – times (multiplication) – obelus (division)
172.
Abstraction
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In a type–token distinction, a type is more abstract than its tokens. Abstraction in its secondary use is a material process, discussed in the themes below. Its development is likely to have been closely connected with the development of human language, which appears to both involve and facilitate abstract thinking. Abstraction involves induction of ideas or the synthesis of particular facts into one general theory about something. It is the opposite of specification, the breaking-down of a general abstraction into concrete facts. Thales believed that everything in the universe comes from one main substance, water. He specified from a general idea, "everything is water", to the specific forms of water such as ice, snow, rivers. This conceptual scheme emphasizes the inherent equality of both constituent and abstract data, thus avoiding problems arising from the distinction between "abstract" and "concrete". In this sense the process of abstraction entails the identification of similarities between objects, the process of associating these objects with an abstraction. For example, picture 1 below illustrates the concrete relationship "Cat sits on Mat". For example, graph 1 below expresses the abstraction "agent sits on location". This conceptual scheme entails no specific hierarchical taxonomy, only a progressive exclusion of detail. Things that do not exist at any particular place and time are often considered abstract. By contrast, members, of such an abstract thing might exist in times. Those abstract things are then said to be multiply instantiated, in the sense of picture 1, picture 2, etc. shown below.
Abstraction
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Cat on Mat (picture 1)
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Group (mathematics)
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The operation satisfies four conditions called the group axioms, namely closure, invertibility. It allows entities beyond to be handled while retaining their essential structural aspects. The ubiquity of groups in numerous areas outside mathematics makes a central organizing principle of contemporary mathematics. Groups share a fundamental kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into better-understandable pieces, such as subgroups, simple groups. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric theory, which studies finitely generated groups as geometric objects, has become a particularly active area in theory. The following properties of integer addition serve as a model for the abstract group axioms given in the definition below. For a + b is also an integer. That is, addition of integers always yields an integer. This property is known as closure under addition. For c, + c = a +.
Group (mathematics)
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A periodic wallpaper pattern gives rise to a wallpaper group.
Group (mathematics)
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The manipulations of this Rubik's Cube form the Rubik's Cube group.
174.
Ring (mathematics)
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In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, functions. The conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, Noether. Afterward, they also proved to be useful in other branches of mathematics such as geometry and mathematical analysis. A ring is an group with a second operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the second operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object. As a result, commutative theory, commonly known as commutative algebra, is a key topic in theory. Its development has been greatly influenced by ideas occurring naturally in algebraic geometry. The most familiar example of a ring is the set of all integers, Z, consisting of the numbers. . . , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5.
Ring (mathematics)
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Richard Dedekind, one of the founders of ring theory.
Ring (mathematics)
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Chapter IX of David Hilbert 's Die Theorie der algebraischen Zahlkörper. The chapter title is Die Zahlringe des Körpers, literally "the number rings of the field". The word "ring" is the contraction of "Zahlring".
175.
Abstract algebra
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In algebra, a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, algebras. The term abstract algebra was coined in the 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, called variety of groups. As in other parts of mathematics, concrete examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many -- perhaps most -- of these problems were in some way related to the theory of algebraic equations. Numerous textbooks in abstract algebra then proceed to establish their properties. This creates a false impression that in algebra axioms had then served as a motivation and as a basis of further study. The true order of historical development was exactly the opposite. For example, the hypercomplex numbers of the nineteenth century challenged comprehension. An archetypical example of this progressive synthesis can be seen in the history of theory. There were several threads in the early development of theory, in modern language loosely corresponding to number theory, theory of equations, geometry.
Abstract algebra
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The permutations of Rubik's Cube form a group, a fundamental concept within abstract algebra.
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Compass and straightedge constructions
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The idealized ruler, known as a straightedge, is assumed to be infinite in length, has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. More formally, the only permissible constructions are those granted by Euclid's first three postulates. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone. The ancient Greek mathematicians first conceived compass-and-straightedge constructions, a number of ancient problems in plane geometry impose this restriction. The ancient Greeks developed many constructions, but in some cases were unable to do so. Gauss showed that some polygons are constructible but that most are not. Some of the most famous straightedge-and-compass problems were proven impossible by Pierre Wantzel in 1837, using the mathematical theory of fields. In spite of existing proofs of impossibility, some persist in trying to solve these problems. Circles can only be drawn starting from two given points: the centre and a point on the circle. The compass may or may not collapse when it's not drawing a circle. The straightedge is infinitely long, but it has no markings on it and has only one straight edge, unlike ordinary rulers. It can only be used to draw a line segment between two points or to extend an existing segment. The modern compass generally does not collapse and several modern constructions use this feature. It would appear that the modern compass is a "more powerful" instrument than the ancient collapsing compass.
Compass and straightedge constructions
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A compass
Compass and straightedge constructions
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Creating a regular hexagon with a ruler and compass
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Galois theory
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In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, in some sense, simpler and better understood. Originally, Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions. Further abstraction of Galois theory is achieved by the theory of Galois connections. Further, it gives a conceptually clear, often practical, means of telling when some particular equation of higher degree can be solved in that manner. Galois theory also gives a clear insight into questions concerning problems in compass and straightedge construction. It gives an elegant characterisation of the ratios of lengths that can be constructed with this method. This was first formalized by the 16th-century French mathematician François Viète, in Viète's formulas, for the case of positive real roots. He was the first who discovered the rules for summing the powers of the roots of any equation. See Discriminant:Nature of the roots for details. This solution was then rediscovered independently in 1535 by Niccolò Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano then extended this to numerous other cases, using similar arguments; see more details at Cardano's method. After the discovery of Ferro's work, he felt that Tartaglia's method was no longer secret, thus he published his solution in his 1545 Ars Magna. His student Lodovico Ferrari solved the quartic polynomial; his solution was also included in Ars Magna.
Galois theory
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Évariste Galois (1811–1832)
178.
Linear algebra
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Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, subspaces, but is also concerned with properties common to all vector spaces. The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models. The study of linear algebra first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, initially listed as an advancement in geodesy. The study of algebra first emerged in the mid-1800s.
Linear algebra
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The three-dimensional Euclidean space R 3 is a vector space, and lines and planes passing through the origin are vector subspaces in R 3.
179.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. There are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a space. In the same vein, but in a more geometric sense, vectors representing displacements in three-dimensional space also form vector spaces. Infinite-dimensional vector spaces arise naturally as function spaces, whose vectors are functions. These vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of continuity. Among these topologies, those that are defined by inner product are more commonly used, as having a notion of distance between two vectors. This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis. Vector spaces are applied throughout mathematics, science and engineering. Furthermore, vector spaces furnish an coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading in geometry and abstract algebra. This is used in physics to describe velocities. Given any two such arrows, w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too.
Vector space
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Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2 w.
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Vector (geometric)
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In mathematics, physics, engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added according to algebra. A vector is what is needed to "carry" the A to the B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planet rotation around the Sun. The direction refers to B. Associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a space. Physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their direction can still be represented by the direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Vector-like objects that transform in a similar way under changes of the coordinate system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years. About a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence. Working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane and thus erected the first space of vectors in the plane.
Vector (geometric)
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This article is about the vectors mainly used in physics and engineering to represent directed quantities. For mathematical vectors in general, see Vector (mathematics and physics).
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Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, later shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle. In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative and algebraic combinatorics. Graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem.
Combinatorics
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An example of change ringing (with six bells), one of the earliest nontrivial results in Graph Theory.
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Graph theory
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In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of points which are connected by edges, arcs, or lines. Graphs are one of the prime objects of study in discrete mathematics. Refer for basic definitions in theory. Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. To avoid ambiguity, this type of graph may be described precisely as undirected and simple. Other senses of graph stem from different conceptions of the edge set. In one more generalized notion, V is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices. Many authors call pseudograph. All of these variants and others are described more fully below. The vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge. The order of a graph is |V|, its number of vertices.
Graph theory
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A drawing of a graph.
183.
Order theory
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Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Orders are everywhere in mathematics and related fields like computer science. This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems in general. Other familiar examples of orderings are the alphabetical order of words in a dictionary and the genealogical property of lineal descent within a group of people. The notion of order is very general, extending beyond contexts that have an immediate, intuitive feel of sequence or relative quantity. In other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the subset relation, e.g. "Pediatricians are physicians," and "Circles are merely special-case ellipses." However, many other orders do not. Order theory captures the intuition of orders that arises from such examples in a general setting. This is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes much sense, because one can derive numerous theorems in the general setting, without focusing on the details of any particular order.
Order theory
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Hasse diagram of the set of all divisors of 60, partially ordered by divisibility
184.
Euclidean geometry
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Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in deducing other propositions from these. The Elements begins with geometry, still taught as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms to propositions without the use of coordinates. This is in contrast to analytic geometry, which uses coordinates. The Elements is mainly a systematization of earlier knowledge of geometry. There are 13 total books in the Elements: I -- IV and VI discuss geometry. Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as irrational numbers are introduced. The infinitude of prime numbers is proved.
Euclidean geometry
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Detail from Raphael 's The School of Athens featuring a Greek mathematician – perhaps representing Euclid or Archimedes – using a compass to draw a geometric construction.
Euclidean geometry
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A surveyor uses a level
Euclidean geometry
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Sphere packing applies to a stack of oranges.
Euclidean geometry
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Geometry is used in art and architecture.
185.
Trigonometry
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Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. Trigonometry is also the foundation of surveying. Trigonometry is most simply associated with planar right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in schools, either as a separate course or as a part of a precalculus course. Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. The ancient Nubians used a similar method. In the 2nd AD, the Greco-Egyptian Ptolemy printed trigonometric tables in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. The modern sine convention is first attested in the Surya Siddhanta, its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata. These Indian works were expanded by Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, were applying them to problems in spherical geometry.
Trigonometry
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Hipparchus, credited with compiling the first trigonometric table, is known as "the father of trigonometry".
Trigonometry
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All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
Trigonometry
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Sextants are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.
186.
Non-Euclidean geometries
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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. The essential difference between the metric geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ. In elliptic geometry the lines "curve toward" each other and intersect. The debate that eventually led to the discovery of the non-Euclidean geometries began soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions and sought to prove all the other results in the work. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates: 1. To draw a straight line from any point to any point. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4.
Non-Euclidean geometries
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On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.
Non-Euclidean geometries
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Behavior of lines with a common perpendicular in each of the three types of geometry
187.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. In particular, the curvature of spacetime is directly related to the momentum of whatever matter and radiation are present. The relation is specified by a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational time delay. The predictions of general relativity have been confirmed to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. Einstein's theory has astrophysical implications. General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics LIGO. In addition, general relativity is the basis of cosmological models of a consistently expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his relativistic framework. The Einstein field equations are very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory. But early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. The objects known today as black holes. In 1917, Einstein applied his theory as a whole initiating the field of relativistic cosmology.
General relativity
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A simulated black hole of 10 solar masses within the Milky Way, seen from a distance of 600 kilometers.
General relativity
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Albert Einstein developed the theories of special and general relativity. Picture from 1921.
General relativity
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Einstein cross: four images of the same astronomical object, produced by a gravitational lens
General relativity
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Artist's impression of the space-borne gravitational wave detector LISA
188.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. Important topological properties include connectedness and compactness. Topology developed through analysis of concepts such as transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. By the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all other branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology and homotopy groups. Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds. A particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler.
Topology
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Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
189.
Analytic geometry
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In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry. Usually the Cartesian system is applied to manipulate equations for planes, straight lines, squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean Euclidean space. The numerical output, however, might also be a shape. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. He further developed relations between the corresponding ordinates that are equivalent to rhetorical equations of curves. Curves were not determined by equations. Coordinates, equations were subsidiary notions applied to a specific geometric situation. Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. The alternative term used for analytic geometry, is named after Descartes. This work, written in its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 did Descartes's masterpiece receive due recognition.
Analytic geometry
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Cartesian coordinates
190.
Differential geometry
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Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques characteristic of this field. Differential geometry developed to the mathematical analysis of surfaces. These unanswered questions indicated greater, hidden relationships and symmetries in nature, which the standard methods of analysis could not address. Initially applied to the Euclidean space, further explorations led to non-Euclidean space, metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a symmetric bilinear form defined on the tangent space at each point. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.
Differential geometry
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A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.
191.
Algebraic geometry
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Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the points at infinity. More advanced questions involve relations between the curves given by different equations. Algebraic geometry has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. In the 20th century, algebraic geometry split into several subareas. The study of the real points of an algebraic variety is the subject of algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties. With the rise of the computers, a algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for finding the properties of explicitly given algebraic varieties. This means that a point of such a scheme may be either a subvariety. This approach also enables a unification of classical algebraic geometry mainly concerned with complex points, of algebraic number theory. Wiles's proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach.
Algebraic geometry
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This Togliatti surface is an algebraic surface of degree five. The picture represents a portion of its real locus.
192.
Discrete geometry
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Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, so forth. László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry. A polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, so on in higher dimensions. Some theories further generalize abstract polytopes. A packing is an arrangement of non-overlapping spheres within a containing space. The space is usually three-dimensional Euclidean space. However, packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space or to non-Euclidean spaces such as hyperbolic space. A tessellation of a flat surface is the tiling of a plane using called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions. Topics in this area include: Cauchy's theorem Flexible polyhedra Incidence structures generalize planes as can be seen from their axiomatic definitions. The finite structures are sometimes called finite geometries. Formally, an structure is a triple C =.
Discrete geometry
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A collection of circles and the corresponding unit disk graph
193.
Convex optimization
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Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum. The convexity of f makes the powerful tools of convex analysis applicable. With recent improvements in theory, minimization is nearly as straightforward as linear programming. Many optimization problems can be reformulated as convex minimization problems. For example, the problem of maximizing a concave function f can be re-formulated equivalently as a problem of minimizing the function -f, convex. F is a function defined on n. The following statements are true about the convex minimization problem: if a local minimum exists, then it is a global minimum. The set of all minima is convex. for each strictly convex function, if the function has a minimum, then the minimum is unique. Standard form is the usual and most intuitive form of describing a convex minimization problem. In practice, the terms "linear" and "affine" are often used interchangeably. Note that every constraint h = 0 can be equivalently replaced by a pair of inequality constraints ≤ 0 and h ≤ 0. Therefore, for theoretical purposes, equality constraints are redundant; however, it can be beneficial to treat them specially in practice. Following from this fact, it is easy to understand why h i = 0 has to be affine as opposed to merely being convex. If h i is convex, h i ≤ 0 is convex, but − h i ≤ 0 is concave.
Convex optimization
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Unconstrained nonlinear: Methods calling …
194.
Fiber bundles
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In mathematics, particularly topology, a fiber bundle is a space, locally a product space, but globally may have a different topological structure. The π, called the submersion of the bundle, is regarded as part of the structure of the bundle. The E is known as the total space of the fiber bundle, F the fiber. In the trivial case, the π is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include Klein bottle, well as nontrivial covering spaces. A bundle map from the base space itself to E is called a section of E. Fiber bundles became their own object of study in the period 1935-1940. The first general definition appeared in the works of Hassler Whitney. The B is called the base space of the bundle, F the fiber. The π is called the map. We shall assume in what follows that the B is connected. The set of all is called a local trivialization of the bundle. Thus for any p in B, the preimage π−1 is homeomorphic to F and is called the fiber over p. Every fiber bundle π: E → B is an open map, since projections of products are open maps.
Fiber bundles
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A cylindrical hairbrush showing the intuition behind the term "fiber bundle". This hairbrush is like a fiber bundle in which the base space is a cylinder and the fibers (bristles) are line segments. The mapping π: E → B would take a point on any bristle and map it to its root on the cylinder.
195.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights. Two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not. Manifolds naturally arise as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate.
Manifold
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The surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
Manifold
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The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface.
196.
Polynomial
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In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, non-negative integer exponents. An example of a polynomial of a indeterminate x is x2 − 4x + 7. An example in three variables is x3 2xyz2 − yz + 1. Polynomials appear in a wide variety of areas of science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, algebraic geometry. The polynomial joins two diverse roots: the Greek poly, meaning "many," and the Latin nomen, or name. It was derived by replacing the Latin root bi - with the Greek poly -. The polynomial was first used in the 17th century. The x occurring in a polynomial is commonly called either an indeterminate. When the polynomial is considered as an expression, x is a fixed symbol which does not have any value. It is thus more correct to call an "indeterminate". However, when one considers the function defined by the polynomial, then x is therefore called a "variable". Many authors use these two words interchangeably. It is a common convention to use uppercase letters for the variables of the associated function. However one may use it over any domain where multiplication are defined.
Polynomial
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The graph of a polynomial function of degree 3
197.
Topological groups
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A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, one may talk about continuous functions, because of the topology. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. Here G × G is viewed as a topological space with the product topology. The reasons, some equivalent conditions, are discussed below. In any case, any topological group can be made Hausdorff by taking an appropriate canonical quotient. Note that the axioms are given in terms of the maps, hence are categorical definitions. A homomorphism of topological groups means a continuous group homomorphism G → H. An isomorphism of topological groups is a isomorphism, also a homeomorphism of the topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism. Topological groups, together with their homomorphisms, form a category. Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups.
Topological groups
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Algebraic structure → Group theory Group theory
198.
Lie group
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Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in the thesis of Lie's student Arthur Tresse. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Lie groups play an enormous role in modern geometry, on different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate group that leaves certain geometric properties invariant. This idea later led to the notion of a G-structure, where G is a group of "local" symmetries of a manifold. The presence of continuous symmetries expressed via a Lie action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are studied in representation theory. This insight opened new possibilities by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The × 2 real invertible matrices form a group under multiplication, denoted by GL or by GL2: GL =. This is a four-dimensional noncompact real group. This group is disconnected; it has two connected components corresponding to the negative values of the determinant. The rotation matrices form a subgroup of GL, denoted by SO. It is a group in its own right: specifically, a one-dimensional compact connected Lie group, diffeomorphic to the circle.
Lie group
199.
Point-set topology
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In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation including differential topology, geometric topology, algebraic topology. Another name for general topology is point-set topology. The fundamental concepts in point-set topology are connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words'nearby','arbitrarily small', and'far apart' can all be made precise by using open sets. If we change the definition of ` open set', we change what connected sets are. Each choice of definition for'open set' is called a topology. A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, many of the most common topological spaces are metric spaces. General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics. Let X be a set and let τ be a family of subsets of X.
Point-set topology
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The Topologist's sine curve, a useful example in point-set topology. It is connected but not path-connected.
200.
Set-theoretic topology
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In mathematics, set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of Zermelo–Fraenkel set theory. In the mathematical field of general topology, a Dowker space is a topological space, T4 but not countably paracompact. The conjecture was not resolved until M.E. Rudin constructed one in 1971. Rudin's counterexample is generally not well-behaved. Zoltán Balogh gave the first ZFC construction of a small example, more well-behaved than Rudin's. Using PCF theory, S. Shelah constructed a subspace of Rudin's Dowker space of cardinality ℵ ω + 1, also Dowker. A famous problem is a question in general topology, the subject of intense research. The answer to the normal Moore question was eventually proved to be independent of ZFC. Cardinal functions are widely used as a tool for describing various topological properties. Below are some examples. Perhaps the simplest cardinal invariants of a topological X are its cardinality and the cardinality of its topology, denoted respectively by | X | and o. The w of a topological space X is the smallest possible cardinality of a base for X. When w ≤ ℵ 0 the space X is said to be second countable. The π -weight of a space X is the smallest cardinality of a π -base for X.
Set-theoretic topology
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The space of integers has cardinality, while the real numbers has cardinality. The topologies of both spaces have cardinality. These are examples of cardinal functions, a topic in set-theoretic topology.
201.
Algebraic topology
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Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about holes, of a topological space. In algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space that near each point resembles Euclidean space. Typically, results in algebraic topology focus on non-differentiable aspects of manifolds; for example Poincaré duality. Knot theory is the study of mathematical knots.
Algebraic topology
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A torus, one of the most frequently studied objects in algebraic topology
202.
Axiomatic set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of theory was initiated in the 1870s. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, theory is a branch of mathematics in its own right, with an active community. Mathematical topics typically evolve among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began on theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of theory led to the article "Mengenlehre" contributed in 1898 to Klein's encyclopedia. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox.
Axiomatic set theory
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Georg Cantor
Axiomatic set theory
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A Venn diagram illustrating the intersection of two sets.
203.
Homotopy theory
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A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. The two versions coincide by setting ht = H. It is not sufficient to require each map ht to be continuous. The animation, looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3. It pauses, then shows the image as t varies back from 1 to 0, pauses, repeats this cycle. Continuous functions f and g are said to be homotopic if and only if there is a homotopy H taking f to g as described above. Being homotopic is an equivalence relation on the set of all continuous functions from X to Y. The maps f and g are called homotopy equivalences in this case. As another example, the Möbius strip and an untwisted strip are homotopy equivalent but not homeomorphic. Spaces that are homotopy equivalent to a point are called contractible. Intuitively, X and Y are homotopy equivalent if they can be transformed by expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, R2 − is homotopy equivalent to the unit circle S1. A function f is said to be null-homotopic if it is homotopic to a constant function.
Homotopy theory
Homotopy theory
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The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible homotopy.
204.
Morse theory
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"Morse function" redirects here. In another context, a "Morse function" can also mean an anharmonic oscillator: see Morse potential. In topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical function on a manifold will reflect the topology quite directly. Morse theory allows one to obtain substantial information about their homology. Before Morse, James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics. These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Consider, for purposes of illustration, a mountainous landscape M. Each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. These are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines passes. Saddle points are points where the surrounding landscape curves up in one direction and down in the other. Imagine flooding this landscape with water.
Morse theory
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A saddle point
205.
Four color theorem
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Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May, "those that do usually require only three. The history of mapmaking do not mention the four-color property." A number of false counterexamples have appeared since the first statement of the four color theorem in 1852. In 1975 Gardner revisited the topic by publishing a map said to be a counter-example in his infamous April fool's column of April 1975. The four theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the major theorem to be proved using a computer. Appel and Haken used a special-purpose program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of analysis. Appel and Haken yet do not contain, one of these 1,936 maps. This contradiction means that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. Since then the proof has gained wider acceptance, although doubts remain. Additionally, in 2005, the theorem was proved by Georges Gonthier with general theorem proving software.
Four color theorem
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Example of a four-colored map
206.
Kepler conjecture
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The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. The density of these arrangements is around 74.05%. In 1998 Thomas Hales, following an approach suggested by Fejes Tóth, announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of individual cases using complex computer calculations. Now Kepler conjecture is accepted as a theorem. Imagine filling a large container with equal-sized spheres. The density of the arrangement is equal to the collective volume of the spheres divided by the volume of the container. Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. However, a higher density can be achieved by carefully arranging the spheres as follows. The conjecture was first stated by Johannes Kepler in his paper'On the six-cornered snowflake'. He had started to study arrangements of spheres as a result of his correspondence in 1606. Harriot went on to develop an early version of atomic theory. This meant that any arrangement that disproved the Kepler conjecture would have to be an irregular one. But eliminating all possible irregular this is what made the Kepler conjecture so hard to prove. After Gauss, no further progress was made towards proving the Kepler conjecture in the nineteenth century.
Kepler conjecture
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One of the diagrams from Strena Seu de Nive Sexangula, illustrating the Kepler conjecture
Kepler conjecture
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Diagrams of cubic close packing (left) and hexagonal close packing (right).
207.
Fractal
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A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge. Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set. Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from geometric figures because of the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, two to the power of three. But if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power, not necessarily an integer. It usually exceeds the fractal's topological dimension. As mathematical equations, fractals are usually nowhere differentiable. The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. There is some disagreement amongst authorities about how the concept of a fractal should be formally defined.
Fractal
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Mandelbrot set: Self-similarity illustrated by image enlargements. This panel, no magnification.
Fractal
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Figure 5. Self-similar branching pattern modeled in silico using L-systems principles
208.
Measure theory
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In this sense, a measure is a generalization of the concepts of length, area, volume. For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word – specifically, 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to subsets of a set X. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Indeed, their existence is a non-trivial consequence of the axiom of choice. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system. Let X be a set and Σ a σ-algebra over X. Null empty set: μ = 0. The pair is called a measurable space, the members of Σ are called measurable sets. A triple is called a measure space. A probability measure is a measure with total measure one – i.e. μ = 1. A space is a space with a measure.
Measure theory
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Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0.
209.
Real analysis
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Real analysis is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. There are several ways of defining the real system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a ordered field. These constructions are described in more detail in the main article. The real numbers have several lattice-theoretic properties that are absent in the complex numbers. Most importantly, the real numbers form an ordered field, in which multiplication preserve positivity. Moreover, the real numbers have the least upper bound property. However, while the results in real analysis are stated for real numbers, many of these results can be generalized to mathematical objects. Also, mathematicians consider imaginary parts of complex sequences, or by pointwise evaluation of operator sequences. In real analysis a sequence is a function from a subset of the natural numbers to the real numbers. In other words, a sequence is a map f: N → R. We might just write an: N → R. A limit is the value that sequence "approaches" as the input or index approaches some value. Limits are used to define continuity, derivatives, integrals. There are several ways to make this intuition rigorous.
Real analysis
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The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.
210.
Complex analysis
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. Complex analysis is particularly concerned with analytic functions of complex variables. Because the separate imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics. Complex analysis is one of the classical branches in mathematics, with roots in the 19th century and prior. Important mathematicians associated with complex analysis include many more in the 20th century. Complex analysis, in particular the theory of conformal mappings, is also used throughout analytic number theory. In modern times, it has become very popular through the pictures of fractals produced by iterating holomorphic functions. Another important application of complex analysis is in theory which studies conformal invariants in quantum field theory. A complex function is one in which the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain and range are subsets of the complex plane. The basic concepts of complex analysis are often introduced by extending the real functions into the complex domain. Holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. Although superficially similar to the derivative of a real function, the behavior of complex derivatives and differentiable functions is significantly different. Consequently, complex differentiability has much stronger consequences than usual differentiability. For instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not.
Complex analysis
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Plot of the function f (x) = (x 2 − 1)(x − 2 − i) 2 / (x 2 + 2 + 2 i). The hue represents the function argument, while the brightness represents the magnitude.
Complex analysis
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The Mandelbrot set, a fractal.
211.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, the equation defines a relationship between the two. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, biology. In pure mathematics, differential equations mostly concerned with their solutions -- the set of functions that satisfy the equation. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence by Newton and Leibniz. Jacob Bernoulli proposed the Bernoulli equation in 1695. In 1746, within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. Lagrange sent the solution to Euler. Both further applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fourier's proposal of his equation for conductive diffusion of heat. This partial equation is now taught to every student of mathematical physics. In classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one to express these variables dynamically as a equation for the unknown position of the body as a function of time.
Differential equation
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
212.
Dynamical system
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In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. The rule of the dynamical system is a function that describes what future states follow from the current state. The concept of a dynamical system has its origins in Newtonian mechanics. To determine the state for all future times requires iterating many times -- each advancing time a small step. The procedure is referred to as solving the system or integrating the system. Before the advent of computers, finding an orbit could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, most dynamical systems are too complicated to be understood in terms of individual trajectories. The approximations used bring into question the relevance of numerical solutions. To address several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of initial conditions for which the trajectories would be equivalent. The operation for comparing orbits to establish their equivalence changes with the different notions of stability. The type of trajectory may be more important than one particular trajectory. Some trajectories may be periodic, whereas others may wander through different states of the system. Applications often require maintaining the system within one class.
Dynamical system
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The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system.
213.
Chaos theory
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Small differences in initial conditions yield widely diverging outcomes for dynamical systems, rendering long-term prediction of their behavior impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The approximate present does not approximately determine the future. Chaotic behavior exists in natural systems, such as weather and climate. It also occurs spontaneously in some systems such as road traffic. This behavior can be studied through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications including meteorology, sociology, physics, environmental science, computer science, engineering, economics, biology, ecology, philosophy. Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then'appear' to become random. Some examples of Lyapunov times are: about 1 millisecond; weather systems, a few days; the solar system, 50 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction can not be made over three times the Lyapunov time.
Chaos theory
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The Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.
Chaos theory
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A plot of Lorenz attractor for values r = 28, σ = 10, b = 8/3
Chaos theory
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Turbulence in the tip vortex from an airplane wing. Studies of the critical point beyond which a system creates turbulence were important for chaos theory, analyzed for example by the Soviet physicist Lev Landau, who developed the Landau-Hopf theory of turbulence. David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory.
Chaos theory
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A conus textile shell, similar in appearance to Rule 30, a cellular automaton with chaotic behaviour.
214.
Professional
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A professional is a member of a profession or any person who earns their living from a specified professional activity. In addition, most professionals are subject to strict codes of conduct, enshrining rigorous moral obligations. Professional standards for a particular field are typically agreed upon and maintained through widely recognized professional associations, such as the IEEE. Some definitions of "professional" limit this term to those professions that serve the general good of society. In narrow usage, not all expertise is considered a profession. Although sometimes incorrectly referred to as professions, occupations such as skilled construction and work are more generally thought of as trades or crafts. The completion of an apprenticeship is generally associated with trades such as carpenter, electrician, mason, painter, plumber and other similar occupations. A related distinction would be that a professional does mainly mental work, as opposed to engaging in physical work. Although professional training appears to be ideologically neutral, it may be biased towards those with a formal education. His evidence is both quantitative, including professional examinations, industry statistics and personal accounts of trainees and professionals. A theoretical dispute arises from the observation that established professions are subject to strict codes of conduct. With a reputation to uphold, trusted workers of a society who have a specific trade are considered professionals. Ironically, the usage of the word ` profess' declined to the 1950s just as the term ` professional' was gaining popularity from 1900-2010. Centre for the Study of Professions Organizational culture Professional boundaries Professional sports
Professional
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Doctors in many Western countries take the Hippocratic Oath upon entering the profession, as a symbol of their commitment to upholding a number of ethical and moral standards.
215.
Probability theory
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Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events. Two mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature at atomic scales described in quantum mechanics. In the 19th century, Pierre Laplace completed what is today considered the classic interpretation. Initially, its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, presented his axiom system for probability theory in 1933. Most introductions to theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, more. Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment.
Probability theory
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The normal distribution, a continuous probability distribution.
Probability theory
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The Poisson distribution, a discrete probability distribution.
216.
Design of experiments
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The change in the predictor is generally hypothesized to result in a change in the second variable, hence called the outcome variable. Main concerns in experimental design include the establishment of replicability. Related concerns include achieving appropriate levels of statistical power and sensitivity. Correctly designed experiments advance knowledge in the natural and social sciences and engineering. Other applications include making. In 1747, while serving as surgeon on HMS Salisbury, James Lind carried out a systematic clinical trial to compare remedies for scurvy. This systematic clinical trial constitutes a type of DOE. Lind selected 12 men from the ship, all suffering from scurvy. Lind limited his subjects to men who "were as similar as I could have them,", he provided strict entry requirements to reduce extraneous variation. He divided them into six pairs, giving each pair different supplements to their basic diet for two weeks. The treatments were all remedies, proposed: A quart of cider every day. Twenty five gutts of vitriol three times a day upon an empty stomach. One half-pint of seawater every day. A mixture of garlic, mustard, horseradish in a lump the size of a nutmeg. Two spoonfuls of vinegar three times a day.
Design of experiments
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Design of experiments with full factorial design (left), response surface with second-degree polynomial (right)
Design of experiments
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Ronald Fisher
Design of experiments
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For the book, see The Design of Experiments.
217.
Model selection
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Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can also involve the design of experiments such that the data collected is well-suited to the problem of selection. Given candidate models of explanatory power, the simplest model is most likely to be the best choice. Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, Sir David Cox has said, "How translation from subject-matter problem to statistical model is done is often the most critical part of an analysis". In its most basic forms, selection is one of the fundamental tasks of scientific inquiry. Determining the principle that explains a series of observations is often linked directly to a mathematical model predicting those observations. For example, when Galileo performed his inclined plane experiments, he demonstrated that the motion of the balls fitted the parabola predicted by his model. Of the countless number of possible processes that could have produced the data, how can one even begin to choose the best model? The mathematical approach commonly taken decides among a set of candidate models; this set must be chosen by the researcher. Often simple models such as polynomials are used, at least initially. Once the set of candidate models has been chosen, the statistical analysis allows us to select the best of these models. What is meant by best is controversial. A good model technique will balance goodness of fit with simplicity.
Model selection
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The scientific observation cycle.
218.
Scientific method
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The scientific method is a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry is commonly based on measurable evidence subject to specific principles of reasoning. Experiments need to be designed to test hypotheses. The most important part of the scientific method is the experiment. The scientific method is a continuous process, which usually begins with observations about the natural world. The best hypotheses lead to predictions that can be tested including making further observations about nature. In general, the strongest tests of hypotheses come from carefully replicated experiments that gather empirical data. Depending on how well the tests match the predictions, the original hypothesis may require refinement, alteration, even rejection. If a particular hypothesis becomes well supported a general theory may be developed. Although procedures vary to another, identifiable features are frequently shared in common between them. The overall process of the scientific method involves making conjectures, then carrying out experiments based on those predictions. A hypothesis is a conjecture, based on knowledge obtained while formulating the question. It might be broad. Scientists then test hypotheses by conducting experiments. The purpose of an experiment is to determine whether observations conflict with the predictions derived from a hypothesis.
Scientific method
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Johannes Kepler (1571–1630). "Kepler shows his keen logical sense in detailing the whole process by which he finally arrived at the true orbit. This is the greatest piece of Retroductive reasoning ever performed." – C. S. Peirce, c. 1896, on Kepler's reasoning through explanatory hypotheses
Scientific method
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Ibn al-Haytham (Alhazen), 965–1039 Iraq. A polymath, considered by some to be the father of modern scientific methodology, due to his emphasis on experimental data and reproducibility of its results.
Scientific method
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According to Morris Kline, "Modern science owes its present flourishing state to a new scientific method which was fashioned almost entirely by Galileo Galilei " (1564−1642). Dudley Shapere takes a more measured view of Galileo's contribution.
Scientific method
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Flying gallop falsified; see image below.
219.
Statistical hypothesis testing
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A statistical hypothesis is a hypothesis, testable on the basis of observing a process, modeled via a set of random variables. A statistical hypothesis test is a method of statistical inference. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. Hypothesis tests are used in determining what outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance. The most common selection techniques are based on either Akaike information criterion or Bayes factor. Statistical hypothesis testing is sometimes called confirmatory data analysis. It can be contrasted with exploratory data analysis, which may not have pre-specified hypotheses. Statistical hypothesis testing is a key technique of both frequentist inference and Bayesian inference, although the two types of inference have notable differences. Statistical hypothesis tests define a procedure that controls the probability of incorrectly deciding that a default position is incorrect. The procedure is based on how likely it would be for a set of observations to occur if the null hypothesis were true. This contrasts with possible techniques of theory in which the null and alternative hypothesis are treated on a more equal basis. One naïve Bayesian approach to hypothesis testing is to base decisions on the posterior probability, but this fails when comparing point and continuous hypotheses. A number of other approaches to reaching a decision based on data are available via optimal decisions, some of which have desirable properties. Hypothesis testing, though, is a dominant approach to data analysis in many fields of science. Such considerations can be used for the purpose of sample size determination prior to the collection of data.
Statistical hypothesis testing
–
A likely originator of the "hybrid" method of hypothesis testing, as well as the use of "nil" null hypotheses, is E.F. Lindquist in his statistics textbook: Lindquist, E.F. (1940) Statistical Analysis In Educational Research. Boston: Houghton Mifflin.
220.
Statistical decision theory
–
Decision theory is the study of the reasoning underlying an agent's choices. Theory is an interdisciplinary topic, studied by economists, statisticians, psychologists, political and social scientists, philosophers. The practical application of this prescriptive approach is aimed at finding tools, methodologies and software to help people make better decisions. In contrast, positive or descriptive theory is concerned with describing observed behaviors under the assumption that the decision-making agents are behaving under some consistent rules. The predictions about behaviour that positive decision theory produces allow for further tests of the kind of decision-making that occurs in practice. There is a thriving dialogue with experimental economics, which uses field experiments to evaluate and inform theory. The area of choice under uncertainty represents the heart of theory. In his solution, he computes expected utility rather than expected financial value. The phrase "theory" itself was used in 1950 by E. L. Lehmann. At this time, Morgenstern theory of expected utility proved that expected utility maximization followed from basic postulates about rational behavior. The work of Daniel Ellsberg showed that human behavior has systematic and sometimes important departures from expected-utility maximization. The theory of Daniel Kahneman and Amos Tversky renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. Pascal's Wager is a classic example of a choice under uncertainty. Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realised at different points in time. What is the optimal thing to do?
Statistical decision theory
–
Daniel Kahneman
221.
Risk
–
Risk is the potential of gaining or losing something of value. Values can be lost when taking risk resulting from a given action or inaction, foreseen or unforeseen. Risk can also be defined as the intentional interaction with uncertainty. Uncertainty is a potential, uncontrollable outcome; risk is a consequence of action taken in spite of uncertainty. Perception is the subjective judgment people make about the severity and probability of a risk, may vary person to person. Some are much riskier than others. The Oxford English Dictionary cites the earliest use of the word in English as the spelling as risk from 1655. It defines risk as: the possibility of loss, other adverse or unwelcome circumstance; a chance or situation involving such a possibility. Risk is an uncertain condition that, if it occurs, has an effect on at least one objective.. . The probability of happening multiplied by the resulting cost or benefit if it does. Finance: The possibility that an actual return on an investment will be lower than the expected return. Insurance: A situation where the probability of a variable is known but when a mode of occurrence or the actual value of the occurrence is not. A risk is not an uncertainty, a hazard. Securities trading: The probability of a loss or drop in value.
Risk
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Firefighters at work
222.
Statistical method
–
Statistics is the study of the collection, analysis, interpretation, presentation, organization of data. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of collection in terms of the design of surveys and experiments. Statistician Sir Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed to each other". When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation. Inferences on mathematical statistics are made under the framework of theory, which deals with the analysis of random phenomena. Working from a null hypothesis, two basic forms of error are recognized: Type errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are also subject to error. Other types of errors can also be important. Specific techniques have been developed to address these problems. Statistics continues to be an area of active research, for example on the problem of how to analyze Big data. Statistics is a mathematical body of science that pertains as a branch of mathematics.
Statistical method
–
Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistical method
–
More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Statistical method
–
Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistical method
–
Karl Pearson, a founder of mathematical statistics.
223.
Hypothesis testing
–
A statistical hypothesis is a hypothesis, testable on the basis of observing a process, modeled via a set of random variables. A statistical hypothesis test is a method of statistical inference. Commonly, two statistical data sets are compared, or a data set obtained by sampling is compared against a synthetic data set from an idealized model. Hypothesis tests are used in determining what outcomes of a study would lead to a rejection of the null hypothesis for a pre-specified level of significance. The most common selection techniques are based on either Akaike information criterion or Bayes factor. Statistical hypothesis testing is sometimes called confirmatory data analysis. It can be contrasted with exploratory data analysis, which may not have pre-specified hypotheses. Statistical hypothesis testing is a key technique of both frequentist inference and Bayesian inference, although the two types of inference have notable differences. Statistical hypothesis tests define a procedure that controls the probability of incorrectly deciding that a default position is incorrect. The procedure is based on how likely it would be for a set of observations to occur if the null hypothesis were true. This contrasts with possible techniques of theory in which the null and alternative hypothesis are treated on a more equal basis. One naïve Bayesian approach to hypothesis testing is to base decisions on the posterior probability, but this fails when comparing point and continuous hypotheses. A number of other approaches to reaching a decision based on data are available via optimal decisions, some of which have desirable properties. Hypothesis testing, though, is a dominant approach to data analysis in many fields of science. Such considerations can be used for the purpose of sample size determination prior to the collection of data.
Hypothesis testing
–
A likely originator of the "hybrid" method of hypothesis testing, as well as the use of "nil" null hypotheses, is E.F. Lindquist in his statistics textbook: Lindquist, E.F. (1940) Statistical Analysis In Educational Research. Boston: Houghton Mifflin.
224.
Mathematical statistics
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Mathematical techniques which are used for this include mathematical analysis, linear algebra, measure-theoretic probability theory. Statistical science is concerned with the planning of surveys using random sampling. The initial analysis of the data from properly randomized studies often follows the study protocol. Of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis. Data analysis is divided into: descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties. Mathematical statistics has been inspired by and has extended many options in applied statistics. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures. A distribution can either be multivariate. Important and commonly encountered univariate probability distributions include the normal distribution. The normal distribution is a commonly encountered distribution. Inferential statistics are used to test hypotheses and make estimations using sample data. Whereas descriptive statistics describe a sample, inferential statistics infer predictions about a larger population that the sample represents. For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling. Data about a random process is obtained during a finite period of time.
Mathematical statistics
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Illustration of linear regression on a data set. Regression analysis is an important part of mathematical statistics.
225.
Cost
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In business, the cost may be one of acquisition, in which case the amount of money expended to acquire it is counted as cost. In this case, money is the input, gone in order to acquire the thing. Usually, the price also includes a mark-up for profit over the cost of production. Hence cost is the metric used in the standard paradigm applied to economic processes. Costs are often further described based on their applicability. It is the amount recorded in bookkeeping records as an expense or asset cost basis. It represents opportunities forgone. In theoretical economics, cost used without qualification often means cost. When a transaction takes place, it typically involves external costs. Private costs are the costs that the buyer of a service pays the seller. This can also be described as the costs internal to the firm's function. External costs, in contrast, are the costs that people other than the buyer are forced to pay as a result of the transaction. The bearers of such costs can be either particular individuals or society at large. Note that external costs are often both problematic to quantify for comparison with monetary values. Social costs are the sum of external costs.
Cost
–
Accounting
226.
Mathematical optimization
–
The generalization of optimization theory and techniques to other formulations comprises a large area of applied mathematics. Such a formulation is called a mathematical problem. Theoretical problems may be modeled in this general framework. The A of f is called the choice set, while the elements of A are called candidate solutions or feasible solutions. A feasible solution that minimizes the objective function is called an optimal solution. In mathematics, conventional optimization problems are usually stated in terms of minimization. Generally, unless both the feasible region are convex in a problem, there may be several local minima. Local maxima are defined similarly. While a local minimum is at least as good as any nearby points, a global minimum is at least as good as every feasible point. Optimization problems are often expressed with special notation. Here are some examples. The minimum value in this case is 1, occurring at x = 0. Similarly, the notation max x ∈ R 2 x asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined". This represents the value of the argument x in the interval.
Mathematical optimization
–
Graph of a paraboloid given by f(x, y) = −(x ² + y ²) + 4. The global maximum at (0, 0, 4) is indicated by a red dot.
227.
Decision science
–
Decision theory is the study of the reasoning underlying an agent's choices. Theory is an interdisciplinary topic, studied by economists, statisticians, psychologists, political and social scientists, philosophers. The practical application of this prescriptive approach is aimed at finding tools, methodologies and software to help people make better decisions. In contrast, positive or descriptive theory is concerned with describing observed behaviors under the assumption that the decision-making agents are behaving under some consistent rules. The predictions about behaviour that positive decision theory produces allow for further tests of the kind of decision-making that occurs in practice. There is a thriving dialogue with experimental economics, which uses field experiments to evaluate and inform theory. The area of choice under uncertainty represents the heart of theory. In his solution, he computes expected utility rather than expected financial value. The phrase "theory" itself was used in 1950 by E. L. Lehmann. At this time, Morgenstern theory of expected utility proved that expected utility maximization followed from basic postulates about rational behavior. The work of Daniel Ellsberg showed that human behavior has systematic and sometimes important departures from expected-utility maximization. The theory of Daniel Kahneman and Amos Tversky renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. Pascal's Wager is a classic example of a choice under uncertainty. Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realised at different points in time. What is the optimal thing to do?
Decision science
–
Daniel Kahneman
228.
Control theory
–
To do this a controller is designed, which monitors the output and compares it with the reference. Some topics studied in theory are observability. Extensive use is usually made of a diagrammatic style known as the block diagram. As the general theory of feedback systems, control theory is useful wherever feedback occurs. A few examples are in physiology, electronics, climate modeling, machine design, ecosystems, navigation, -- production theory. Control systems may be thought of as having four functions: correct. These four functions are completed by five elements: detector, transducer, final element. The function is completed by the transmitter. In practical applications these three elements are typically contained in one unit. A standard example of a measuring unit is a resistance thermometer. Older controller units have been mechanical, as in a centrifugal governor or a carburetor. The correct function is completed with a final control element. The final element changes an output in the system that affects the manipulated or controlled variable. Fundamentally, there are two types of control loop; open loop control, closed loop control. In open control, the action from the controller is independent of the "output".
Control theory
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Centrifugal governor in a Boulton & Watt engine of 1788
229.
Mathematical economics
–
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. An advantage claimed for the approach is its allowing formulation of theoretical relationships with rigor, simplicity. Mathematics allows economists to form testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make positive claims about controversial or contentious subjects that would be impossible without mathematics. This rapid systematizing of economics alarmed critics of the discipline well as some noted economists. The use of mathematics in the service of economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics. Petty's use of numerical data would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars. The mathematization of economics began in the 19th century. Most of the economic analysis of the time was what would later be called classical economics. Calculus was not used. Thünen's model of farmland use represents the first example of marginal analysis. He also mined empirical data in order to attempt to support his generalizations. In comparison to his contemporaries, Thünen built economic tools, rather than applying previous tools to new problems.
Mathematical economics
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Equilibrium quantities as a solution to two reaction functions in Cournot duopoly. Each reaction function is expressed as a linear equation dependent upon quantity demanded.
230.
Mathematical problem
–
A mathematical problem is a problem, amenable to being represented, analyzed, possibly solved, with the methods of mathematics. It can also be a problem referring to the nature such as Russell's Paradox. Informal "real-world" mathematical problems are questions related to a concrete setting, such as "Adam gives John three. How many has he left?". Known as word problems, they are used in education to teach students to connect real-world situations to the abstract language of mathematics. In general, to use mathematics for solving a real-world problem, the first step is to construct a mathematical model of the problem. After the problem has been solved in the world of mathematics, the solution must be translated back into the context of the original problem. Mathematical problems arise in all fields of mathematics. While mathematicians usually study them by doing so results may be obtained that find application outside the realm of mathematics. Theoretical physics remains, a rich source of inspiration. Also provably unsolvable are so-called undecidable problems, such as the halting problem for Turing machines. Some difficult abstract problems that have been solved relatively recently are the four-colour theorem, Fermat's Last Theorem, the Poincaré conjecture.. . The same issue was faced by Sylvestre Lacroix almost two centuries earlier... it is necessary to vary the questions that students might communicate with each other. Though they may fail the exam, they might pass later.
Mathematical problem
–
'Suppose you walk past a barber's shop one day, and see a sign that says: "Do you shave yourself? If not, please come in and I'll shave you! I shave anyone who does not shave himself, and no one else". So the question is: "Who shaves the barber?"' —the barber paradox
231.
Numerical analysis
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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is extremely important, in construction. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of 2, modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. Hedge funds use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use numerical programs for actuarial analysis.
Numerical analysis
–
Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296...
Numerical analysis
–
Direct method
Numerical analysis
232.
Analysis (mathematics)
–
Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, analytic functions. These theories are usually studied in the context of complex functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. The Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of mathematical analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones. In the 18th century, Euler introduced the notion of mathematical function. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals.
Analysis (mathematics)
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A strange attractor arising from a differential equation. Differential equations are an important area of mathematical analysis with many applications to science and engineering.
233.
Approximation theory
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In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, with quantitatively characterizing the errors introduced thereby. Note that what is meant by simpler will depend on the application. This is typically done with rational approximations. This is accomplished by narrowing the domain over which the polynomial has to approximate the function. Narrowing the domain can often be done for the function being approximated. Modern mathematical libraries often use a low-degree polynomial for each segment. Once the degree of the polynomial are chosen, the polynomial itself is chosen in such a way as to minimize the worst-case error. It is seen that an Nth-degree polynomial can interpolate N+1 points in a curve. Such a polynomial is always optimal. These occur rarely in practice. For example, the graphs shown to the right show the error in approximating exp for N = 4. The red curves, for the optimal polynomial, are level, they oscillate between + ε and ε exactly. Note that, in each case, the number of extrema is N+2, 6. Two of the extrema are at the left and right edges of the graphs. The red graph to the right shows what this function might look like for N = 4.
Approximation theory
–
Error between optimal polynomial and log(x) (red), and Chebyshev approximation and log(x) (blue) over the interval [2, 4]. Vertical divisions are 10 −5. Maximum error for the optimal polynomial is 6.07 x 10 −5.
234.
Discretization
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In mathematics, discretization concerns the process of transferring continuous functions, models, equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Processing on a digital computer requires another process called quantization. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable. Euler–Maruyama method Zero-order hold Discretization is also related to discrete mathematics, is an important component of granular computing. In this context, discretization may also refer to modification of granularity, as when discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand. Discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing. It can, however, be computed by first constructing a matrix, computing the exponential of it: F = T G = e F =. Now we want to discretise the above expression. We assume that u is constant during each timestep. Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps e A T ≈ I + A T. Each of them have different stability properties.
Discretization
–
A solution to a discretized partial differential equation, obtained with the finite element method.
235.
Algorithm
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In mathematics and computer science, an algorithm is a self-contained step-by-step set of operations to be performed. Algorithms perform calculation, data processing, and/or automated reasoning tasks. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem. In English, it was first used in about 1230 and then by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five. An informal definition could be "a set of rules that precisely defines a sequence of operations." Which would include all computer programs, including programs that do not perform numeric calculations. Generally, a program is only an algorithm if it stops eventually. An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules.
Algorithm
–
Alan Turing's statue at Bletchley Park.
Algorithm
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Logical NAND algorithm implemented electronically in 7400 chip
236.
Mathematical physics
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Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems. These roughly correspond to historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics. Moreover, they have provided basic ideas in geometry. The theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, aerodynamics. It has connections to molecular physics. Quantum information theory is another subspecialty. The special and general theories of relativity require a rather different type of mathematics. This was theory, which played an important role in both quantum field theory and geometry. This was, however, gradually supplemented by functional analysis in the mathematical description of cosmological well as quantum field theory phenomena. In this area both homological theory are important nowadays.
Mathematical physics
–
An example of mathematical physics: solutions of Schrödinger's equation for quantum harmonic oscillators (left) with their amplitudes (right).
237.
Fluid dynamics
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In physics, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics. Some of its principles are even used in engineering, where traffic is treated as crowd dynamics. Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected like hydrodynamic stability, both of which can also be applied to gases. The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of energy. These are modified in general relativity. They are expressed using the Reynolds Transport Theorem. In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of discrete molecules is ignored. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve. Some of the simplifications allow appropriate fluid dynamics problems to be solved in closed form.
Fluid dynamics
238.
Mathematical biology
–
Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications. The field is sometimes called mathematical biology or biomathematics to stress the mathematical side, or theoretical biology to stress the biological side. Mathematical biology aims at modeling of biological processes, using techniques and tools of applied mathematics. It has both practical applications in biomedical and biotechnology research. This requires precise mathematical models. Mathematical biology employs many components of mathematics, has contributed to the development of new techniques. Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the field. Ecology and evolutionary biology have traditionally been the dominant fields of mathematical biology. Evolutionary biology has been the subject of extensive mathematical theorizing. The traditional approach in this area, which includes complications from genetics, is population genetics. When infinitesimal effects at a large number of gene loci are considered, together with the assumption of linkage equilibrium or quasi-linkage equilibrium, one derives quantitative genetics. Ronald Fisher made fundamental advances in statistics, such as analysis of variance, via his work on quantitative genetics. Another important branch of population genetics that led to the extensive development of coalescent theory is phylogenetics. Many population genetics models assume that population sizes are constant. Variable population sizes, often in the absence of genetic variation, are treated by the field of population dynamics.
Mathematical biology
–
Contents
239.
Fields Medal
–
The Fields Medal is sometimes viewed as the highest honor a mathematician can receive. The Abel Prize have often been described as the mathematician's "Nobel Prize". The Fields Medal differs from the Abel in view of the restriction mentioned above. The prize comes with a monetary award, which since 2006 has been C$15,000. The colloquial name is in honour of Canadian mathematician John Charles Fields. Fields was instrumental in establishing the award, funding the monetary component. Its purpose is to give support to younger mathematical researchers who have made major contributions. However, in contrast to the Nobel Prize, the Fields Medal is awarded only every four years. This is similar to restrictions applicable to the Clark Medal in economics. The monetary award is much lower than the 8,000,000 Swedish kronor given with each Nobel prize as of 2014. Other major awards such as the Abel Prize and the Chern Medal, have larger monetary prizes, comparable to the Nobel. In 1954, Jean-Pierre Serre became the youngest winner of the Fields Medal, at 27. He still retains this distinction. In 1966, Alexander Grothendieck boycotted the ICM, held in Moscow, to protest military actions taking place in Eastern Europe. Founder and director of the Institut des Hautes Études Scientifiques attended and accepted Grothendieck's Fields Medal on his behalf.
Fields Medal
–
The obverse of the Fields Medal
Fields Medal
–
The reverse of the Fields Medal
240.
Nobel Prize
–
The will of the Swedish inventor Alfred Nobel established the prizes in 1895. The prizes in Chemistry, Literature, Peace, Physics, Physiology or Medicine were first awarded in 1901. Medals made before 1980 were struck in 23 gold, later from 18 green gold plated with a 24 carat gold coating. Between 2015, the Prize in Economic Sciences were awarded 573 times to 900 people and organisations. With some receiving the Nobel Prize more than once, this makes a total of 23 organisations, 870 individuals—of whom 48 were women. The prize ceremonies take place annually in Stockholm, Sweden. Laureate, receives a gold medal, a sum of money, decided by the Nobel Foundation. The Nobel Prize is widely regarded as the most prestigious award available in the fields of medicine, physics, chemistry, economics. The prize is not awarded posthumously; however, if a person is awarded a prize and dies before receiving it, the prize may still be presented. Though the average number of laureates per prize increased substantially during the 20th century, a prize may not be shared among more than three people. Alfred Nobel was born on 21 October 1833 in Stockholm, Sweden, into a family of engineers. He was inventor. In 1894, Nobel purchased the Bofors mill, which he made into a major armaments manufacturer. Nobel also invented ballistite. This invention was a precursor to military explosives, especially the British smokeless powder cordite.
Nobel Prize
–
Alfred Nobel had the unpleasant surprise of reading his own obituary, which was titled The merchant of death is dead, in a French newspaper.
Nobel Prize
–
The Nobel Prize
Nobel Prize
–
Alfred Nobel's will stated that 94% of his total assets should be used to establish the Nobel Prizes.
Nobel Prize
–
Wilhelm Röntgen received the first Physics Prize for his discovery of X-rays.
241.
Abel Prize
–
The Abel Prize /ˈɑːbəl/ is a Norwegian prize awarded annually by the Government of Norway to one or more outstanding mathematicians. It comes with a monetary award of million Norwegian kroner. The board has also established an Abel symposium, administered by the Norwegian Mathematical Society. The prize was first proposed to be part of the 1902 celebration of 100th anniversary of Abel's birth. In August 2001, the Norwegian government announced that the prize would be awarded beginning in the two-hundredth anniversary of Abel's birth. The first actual Abel Prize was only awarded in 2003. A series presenting Abel Prize laureates and their research was commenced in 2010. The first two volumes cover 2008 -- 2012 respectively. The committee is currently headed by John Rognes. The European Mathematical Society nominate members of the Abel Committee. The Norwegian Government gave an initial funding of NOK 200 million in 2001. The funding is controlled by the Board, which consists of members elected by the Norwegian Academy of Science and Letters. Self-nomination is not allowed. The nominee must be alive; however, if the awardee dies after being declared as the winner, the prize is awarded posthumously. The Abel Laureate is decided based on the recommendation of the Abel Committee.
Abel Prize
–
The prize is awarded in the atrium of the Domus Media building of the University of Oslo Faculty of Law, where the Nobel Peace Prize was formerly awarded
Abel Prize
–
Abel Prize
242.
Riemann hypothesis
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It was proposed by Bernhard Riemann, after whom it is named. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics. The Riemann ζ is whose values are also complex. It has zeros at the even integers;, ζ = 0 when s is one of 6.... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called non-trivial zeros. There are several nontechnical books on the Riemann hypothesis, such as Derbyshire, Rockmore, du Sautoy. Mazur & Stein give mathematical introductions, while Karatsuba & Voronin are advanced monographs. The convergence of the Euler product shows that ζ has no zeros in this region, as none of the factors have zeros. The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to give it a definition, valid for all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows.
Riemann hypothesis
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The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.
243.
Lists of mathematics topics
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This article itemizes the various lists of mathematics topics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. The purpose of this list is not similar to that of the Mathematics Subject Classification formulated by the American Mathematical Society. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. These lists include topics typically taught in secondary education or in the first year of university. As a rough guide this list is divided into pure and applied sections although in reality these branches are overlapping and intertwined. Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group. Analysis evolved from calculus. Geometry is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension. Outline of combinatorics List of theory topics Glossary of Logic is the foundation which underlies mathematical logic and the rest of mathematics.
Lists of mathematics topics
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Ray tracing is a process based on computational mathematics.
Lists of mathematics topics
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Fourier series approximation of square wave in five steps.
244.
Mathematics and art
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Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as textiles. This article focuses, however, on mathematics in the visual arts. Mathematics and art have a historical relationship. Popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, in his paintings. The engraver Albrecht Dürer made many references in his work Melencolia I. In modern times, the graphic artist M. C. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as linear perspective, mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates stellated polyhedra, originally as models for teaching. Mathematical concepts such as recursion and logical paradox can be seen in paintings by Rene Magritte and in engravings by M. C. Escher.
Mathematics and art
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Mathematics in art: Albrecht Dürer 's copper plate engraving Melencolia I, 1514. Mathematical references include a compass for geometry, a magic square and a truncated rhombohedron, while measurement is indicated by the scales and hourglass.
Mathematics and art
Mathematics and art
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Roman copy in marble of Doryphoros, originally a bronze by Polykleitos
Mathematics and art
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Brunelleschi 's experiment with linear perspective
245.
Mathematics education
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In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research. This article describes some of recent controversies. Elementary mathematics was part of the system in most ancient civilisations, including Ancient Greece, the Roman empire, ancient Egypt. In most cases, a formal education was only available to male children with a sufficiently high caste. In Plato's division of the liberal arts into the quadrivium, the quadrivium included the mathematical fields of geometry. This structure was continued in the structure of classical education, developed in medieval Europe. Teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as money-lenders could expect to learn such practical mathematics as was relevant to their profession. The first textbooks to be written in English and French were published beginning with The Grounde of Artes in 1540. However, there are different writings on mathematics and methodology that date back to 1800 BCE. These were mostly located in Mesopotamia where the Sumerians were practicing multiplication and division. There are also artifacts demonstrating their own methodology for solving equations like the quadratic equation. The more famous Rhind Papyrus has been dated to approximately 1650 BCE but it is thought to be a copy of an even older scroll. This papyrus was essentially an early textbook for Egyptian students. In the Renaissance, the academic status of mathematics declined, because it was strongly associated with trade and commerce.
Mathematics education
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A mathematics lecture at Aalto University School of Science and Technology
Mathematics education
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Illustration at the beginning of a 14th-century translation of Euclid's Elements.
Mathematics education
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Games can motivate students to improve skills that are usually learned by rote. In "Number Bingo," players roll 3 dice, then perform basic mathematical operations on those numbers to get a new number, which they cover on the board trying to cover 4 squares in a row. This game was played at a "Discovery Day" organized by Big Brother Mouse in Laos.
246.
Relationship between mathematics and physics
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Before giving a mathematical proof for the formula for the volume of a sphere, Archimedes used physical reasoning to discover the solution. From the seventeenth century, this continued in the following centuries. During this period there was little distinction between mathematics; as an example, Newton regarded geometry as a branch of mechanics. As time progressed, increasingly sophisticated mathematics started to be used in physics. The current situation is that the mathematical knowledge used in physics is becoming increasingly sophisticated, in the case of superstring theory. —Albert Einstein, in Geometry and Experience. Clearly delineate physics: For some results or discoveries, it is difficult to say to which area they belong: to the mathematics or to physics. What is the geometry of physical space? What is the origin of the axioms of mathematics? How does the already existing mathematics influence in the creation and development of physical theories? Is arithmetic a priori or synthetic? What is essentially different between doing a physical experiment to see the result and making a mathematical calculation to see the result? Do Gödel's incompleteness theorems imply that physical theories will always be incomplete? Is math invented or discovered? In recent times the two disciplines have most often been taught separately, despite all the interrelations between mathematics.
Relationship between mathematics and physics
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A cycloidal pendulum is isochronous, a fact discovered and proved by Christiaan Huygens under certain mathematical assumptions.
247.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each variation of a book. For example, an e-book, a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned after 1 January 2007, 10 digits long if assigned before 2007. The method of assigning an ISBN varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated based upon the 9-digit Standard Book Numbering created in 1966. The 10-digit ISBN format was published in 1970 as international standard ISO 2108. The International Standard Serial Number, identifies periodical publications such as magazines; and the International Standard Music Number covers for musical scores. The ISBN configuration of recognition was generated in 1967 in the United Kingdom by Emery Koltay. The 10-digit ISBN format was published as international standard ISO 2108. The United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978. An SBN may be converted by prefixing the digit "0". This can be converted to ISBN 0-340-01381-8; the digit does not need to be re-calculated. Since 1 ISBNs have contained 13 digits, a format, compatible with "Bookland" European Article Number EAN-13s.
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
248.
Science (journal)
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It was first published in 1880, has a print subscriber base of around 130,000. Because online access serve a larger audience, its estimated readership is 570,400 people. Unlike most scientific journals, which focus on a specific field, its rival Nature cover the full range of scientific disciplines. According to the Journal Citation Reports, Science's 2015 factor was 34.661. Although it is the journal of the AAAS, membership in the AAAS is not required to publish in Science. Papers are accepted from authors around the world. Fewer than 7% of articles submitted are accepted for publication. Science is based with a second office in Cambridge, England. Science was founded with financial support from Thomas Edison and later from Alexander Graham Bell. However, the journal ended publication in March 1882. Entomologist Samuel H. Scudder had some success while covering the meetings of prominent American scientific societies, including the AAAS. However, by 1894, Science was sold to psychologist James McKeen Cattell for $500. After Cattell died in 1944, the ownership of the journal was transferred to the AAAS. After Cattell's death in 1944, the journal lacked a consistent presence until Graham DuShane became editor in 1956. Under DuShane's leadership, Science absorbed The Scientific Monthly, thus increasing the journal's circulation by over 60 % from 38,000 to more than 61,000.
Science (journal)
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Issue from February–June 1883
Science (journal)
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Science
249.
Keith Devlin
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Keith J. Devlin is a British mathematician and popular science writer. Since 1987 he has lived in the United States. He has American-British citizenship. He is a commentator on National Public Radio's Weekend Edition Saturday, where he is known as "The Math Guy." As of 2012, he is the author of over 80 research articles. Several of his books are aimed at an audience of the general public. Springer. 1984. ISBN 3-540-13258-9. Logic and Information. Cambridge University Press. 1991. ISBN 0-521-49971-2. The Joy of Sets: Fundamentals of Contemporary Set Theory. Springer.
Keith Devlin
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Keith Devlin (2011)
250.
BBC Radio 4
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It replaced the BBC Home Service in 1967. The controller is Gwyneth Williams; and the station is part of BBC Radio and the BBC Radio department. The station is broadcast at Broadcasting House, London. It is also available on the Internet. It also won a Peabody Award in 2002 for File On 4: Export Controls. Costing # million, it is the BBC's most expensive national radio network and is considered by many to be its flagship. In 2010 Gwyneth Williams replaced Mark Damazer as Radio 4 controller. Damazer became Master of Oxford. Sport are the only fields that largely fall outside the station's remit. Documentaries related to various forms of both popular and classical music, the long-running music-based Desert Island Discs. The BBC Home Service was broadcast between 1939 and 1967. It was broadcast on medium wave with a network of VHF FM transmitters being added from 1955. Radio 4 replaced it on 30 September 1967, when the BBC renamed many of its domestic radio stations, to the challenge of offshore radio. Plus programme variations for parts of England not served by BBC Local Radio stations. Roundabout East Anglia came in 1980 when local radio services were introduced to East Anglia with the launch of BBC Radio Norfolk.
BBC Radio 4
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Logo of Radio 4 until 2007
BBC Radio 4
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BBC Radio 4
251.
PubMed Identifier
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PubMed is a free search engine accessing primarily the MEDLINE database of references and abstracts on life sciences and biomedical topics. The United States National Library of Medicine at the National Institutes of Health maintains the database as part of the Entrez system of information retrieval. From 1971 to 1997, MEDLINE online access to the MEDLARS Online computerized database had been primarily through institutional facilities, such as university libraries. PubMed, first released in January 1996, ushered in the era of private, free, home- and office-based MEDLINE searching. Information about the journals indexed in MEDLINE and available through PubMed is found in the NLM Catalog. As of the same date, 13.1 million of PubMed's records are listed with their abstracts, 14.2 million articles have links to full-text. In 2016, NLM changed the indexing system so that publishers will be able to directly correct typos and errors in PubMed indexed articles. Simple searches on PubMed can be carried out by entering key aspects of a subject into PubMed's search window. When a journal article is indexed, numerous article parameters are extracted and stored as structured information. Such parameters are: Article Type, publication history. Publication type parameter enables many special features. Since July 2005, the MEDLINE process puts those in a field called Secondary Identifier. The secondary field is to store accession numbers to various databases of clinical trial IDs. For clinical trials, PubMed extracts trial IDs for the two largest trial registries: ClinicalTrials.gov and the International Standard Randomized Controlled Trial Number Register. A reference, judged particularly relevant can be marked and "related articles" can be identified.
PubMed Identifier
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PubMed
252.
Online Etymology Dictionary
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The Online Etymology Dictionary is a free online dictionary that describes the origins of English-language words. Douglas Harper compiled the etymology dictionary to record the evolution of more than 30,000 words, including technical terms. The core body of its etymology information stems from Ernest Weekley's An Etymological Dictionary of Modern English. Other sources include the Middle English Dictionary and the Barnhart Dictionary of Etymology. Harper works as a Copy editor/Page designer for LNP Media Group. As of June 2015, there were nearly 50,000 entries in the dictionary. It is cited as a source for explaining the evolution of words. Official website
Online Etymology Dictionary
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Online Etymology Dictionary
Online Etymology Dictionary
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Screenshot of etymonline.com
253.
Henry Liddell
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Lewis Carroll wrote Alice's Adventures in Wonderland for Henry Liddell's daughter Alice. Liddell received his education at Charterhouse and Christ Church, Oxford. He was ordained in 1838. Liddell was Headmaster of Westminster School from 1846 to 1855. It immediately became the standard Greek-English dictionary, with the 8th edition published in 1897. As Headmaster of Westminster Liddell enjoyed a period of great success, followed in the school. In 1855 he accepted the deanery of Christ Church, Oxford. In the same year he brought out his History of Ancient Rome and took a very active part in the first Oxford University Commission. Aristocratic mien were for many years associated with all, characteristic of Oxford life. Before then the school was housed within Christ Church itself. In July 1846, Liddell married Miss Lorina Reeve, with whom he had several children, including Alice Liddell of Lewis Carroll fame. In 1891, owing to advancing years, he resigned the deanery. The last years of his life were spent at Ascot, where he died on 18 January 1898. Two roads in Carroll Crescent honour the relationship between Henry Liddell and Lewis Carroll. Liddell was an Oxford “character” in later years.
Henry Liddell
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Henry Liddell, in an 1891 portrait by Sir Hubert von Herkomer.
Henry Liddell
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Caricature of Rev. Henry Liddell by 'Ape' from Vanity Fair (1875).
254.
Ohio State University
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The Ohio State University, commonly referred to as Ohio State or OSU, is a public university in Columbus, Ohio. Hayes, in 1878 the Ohio General Assembly passed a law changing the name to "The Ohio State University". It has since grown into the third largest campus in the United States. Along with its main campus in Columbus, Ohio State also operates a regional system with regional campuses in Lima, Mansfield, Marion, Newark, Wooster. Ohio State athletic teams are known as the Ohio State Buckeyes. Athletes from Ohio State have won 100 Olympic medals. The university is a member of the Big Ten Conference for the majority of sports. The Ohio State men's ice program competes in the Big Ten Conference, while its women's hockey program competes in the Western Collegiate Hockey Association. In addition, the OSU men's volleyball team is a member of the Midwestern Intercollegiate Volleyball Association. OSU is one of only fourteen universities in the nation that plays Division I FBS football and Division I hockey. As of August 2015, a total of 714,512 degrees had been awarded since the university was founded. Former students have gone on to prominent careers in government, business, science, medicine, education, sports, entertainment. Hayes. The school was originally situated within a community located on the northern edge of Columbus. The university opened its doors on September 17, 1873.
Ohio State University
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The East Atrium at the William Oxley Thompson Memorial Library
Ohio State University
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The Ohio State University
Ohio State University
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Detail of the Wexner Center
Ohio State University
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Scott Laboratory, housing the Mechanical and Aerospace Engineering department. This facility is a joint effort between BHDP Architecture and Polshek Partnership Architects.
255.
Oxford English Dictionary
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The Oxford English Dictionary is a descriptive dictionary of the English language, published by the Oxford University Press. The second edition came in 20 volumes, published in 1989. In 1933, The Oxford English Dictionary fully replaced the former name in all occurrences in its reprinting as twelve volumes with a one-volume supplement. More supplements came until 1989 when the second edition was published. Since 2000, a third edition of the dictionary has been underway, approximately a third of, now complete. The electronic version of the dictionary was made available in 1988. As of April 2014 was receiving over two million hits per month. As a historical dictionary, the Oxford English Dictionary explains words by showing their development rather than merely their present-day usages. Therefore, it shows definitions in the order that the sense of the word began being used, including word meanings which are no longer used. The format of the OED's entries has influenced numerous historical lexicography projects. This influenced later volumes of other lexicographical works. As of 30 the Oxford English Dictionary contained approximately 301,100 main entries. The dictionary's latest, complete print edition was printed in 20 volumes, comprising 291,500 entries in 21,730 pages. The longest entry in the OED2 was for the verb set, which required 60,000 words to describe some 430 senses. Despite its impressive size, the OED is the earliest exhaustive dictionary of a language.
Oxford English Dictionary
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Seven of the twenty volumes of the printed version of the second edition of the OED
Oxford English Dictionary
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Frederick Furnivall, 1825–1910
Oxford English Dictionary
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James Murray in the Scriptorium at Banbury Road
Oxford English Dictionary
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The 78 Banbury Road, Oxford, house, erstwhile residence of James Murray, Editor of the Oxford English Dictionary
256.
Florian Cajori
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Florian Cajori was a Swiss-American historian of mathematics. Florian Cajori immigrated at the age of sixteen. He received both master's degrees from the University of Wisconsin -- Madison. He taught for a few years before being appointed as professor of applied mathematics there in 1887. He was then driven north by tuberculosis. While in Colorado, he received his doctorate in 1894. A History of Mathematics was the first popular presentation of the history of mathematics in the United States. He remained until his death in 1930. Cajori did no mathematical research unrelated to the history of mathematics. In addition to his numerous books, he also contributed highly recognized and popular historical articles to the American Mathematical Monthly. He died before it was completed. The work was finished by R.T.Crawford of Berkeley, California. 1893: A History of Mathematics, Macmillan & Company. 1898: A History of Elementary Mathematics, Macmillan. 1909: A History of the Logarithmic Slide Rule and Allied Instruments The Engineering News Publishing Company.
Florian Cajori
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Florian Cajori at Colorado College
257.
JSTOR
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JSTOR is a digital library founded in 1995. Originally containing back issues of academic journals, it now also includes books and primary sources, current issues of journals. It provides full-text searches of almost 2,000 journals. President of Princeton University from 1972 to 1988, founded JSTOR. Most libraries found it prohibitively expensive in terms of space to maintain a comprehensive collection of journals. By digitizing many journal titles, JSTOR allowed libraries to outsource the storage of journals with the confidence that they would remain available long-term. Full-text search ability improved access dramatically. Bowen initially considered using CD-ROMs for distribution. JSTOR originally encompassed ten economics and history journals. It became a fully searchable index accessible from any ordinary web browser. Special software was put in place to make graphs clear and readable. With the success of this limited project, then-president of JSTOR, wanted to expand the number of participating journals. The work of adding these volumes to JSTOR was completed by December 2000. The Andrew W. Mellon Foundation funded JSTOR initially. Until January 2009 JSTOR operated in New York City and in Ann Arbor, Michigan.
JSTOR
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The JSTOR front page
258.
Oxford University Press
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Oxford University Press is the largest university press in the world, the second oldest after Cambridge University Press. They are headed to the delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University has used a similar system to oversee OUP since the 17th century. The university grew into a major printer of Bibles, prayer books, scholarly works. OUP took on the project that expanded to meet the ever-rising costs of the work. Moves into international markets led to OUP opening its own offices outside the United Kingdom, beginning with New York City in 1896. By contracting out binding operations, the modern OUP publishes some 6,000 new titles around the world each year. OUP was first exempted from United Kingdom corporation tax in 1978. The Oxford University Press Museum is located on Oxford. Visits are led by a member of the archive staff. Displays include a 19th-century printing press, the printing and history of the Oxford Almanack, Alice in Wonderland and the Oxford English Dictionary. The first printer associated with Oxford University was Theoderic Rood. An edition of Rufinus's Expositio in symbolum apostolorum, was printed by another, anonymous, printer. Famously, this was mis-dated in Roman numerals as "1468", thus apparently pre-dating Caxton. Rood's printing included John Ankywyll's Compendium totius grammaticae, which set new standards for teaching of Latin grammar.
Oxford University Press
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Oxford University Press on Walton Street.
Oxford University Press
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2008 conference booth
259.
Diophantus
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These texts deal with solving algebraic equations. This led to tremendous advances in number theory, the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality. Diophantus was the Greek mathematician who recognized fractions as numbers; thus he allowed rational numbers for the solutions. In modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation. Little is known about the life of Diophantus. He lived to 298. Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems states:'Here lies Diophantus,' the wonder behold. The dear child of sage After attaining half the measure of his father's life fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' However, the accuracy of the information cannot be independently confirmed. The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations.
Diophantus
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Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac.
Diophantus
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Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's Last Theorem.
260.
Franciscus Vieta
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He served as a privy councillor to both Henry III and Henry IV. Viete was born in present-day Vendée. His grandfather was a merchant from La Rochelle. Etienne Viète, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of parliament during the ascendancy of the Catholic League of France. Viete in 1558 studied law at Poitiers, graduating as a Bachelor of Laws in 1559. He began his career as an attorney in his native town. The same year, in the commune of Mouchamps in present-day Vendée, Vieta became the tutor of Catherine de Parthenay, Soubise's twelve-year-old daughter. He wrote for her numerous treatise on astronomy, geography and trigonometry, some of which have survived. John V de Parthenay presented him to King Charles IX of France. Following the death of Jean V de Parthenay-Soubise in 1566, his biography. In 1571, he continued to visit his student Catherine. He regularly lived in Fontenay-le-Comte, where he took on some municipal functions. He wrote new mathematical research by night or during periods of leisure. He was known to dwell for up to three days, his elbow on the desk, feeding himself without changing position.
Franciscus Vieta
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François Viète, French mathematician
Franciscus Vieta
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Opera, 1646
261.
False proof
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In mathematics, certain kinds of mistaken proof are often exhibited, sometimes collected, as illustrations of a concept of mathematical fallacy. For example, the validity fails may be a division by zero, hidden by algebraic notation. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Beyond pedagogy, the resolution of a fallacy can lead into a subject. An ancient lost book of false proofs, is attributed to Euclid. Mathematical fallacies exist in many branches of mathematics. Well-known fallacies also calculus. Examples exist of correct results derived by incorrect lines of reasoning. However true the conclusion, is mathematically invalid and is commonly known as a howler. Consider for instance the calculation: 16 64 = 16 / 6 / 4 = 1 4. Although the conclusion 64 = 1 4 is correct, there is a fallacious, invalid cancellation in the middle step. Another classical example of a howler is proving the Cayley-Hamilton theorem by simply substituting the scalar variables of the characteristic polynomial by the matrix. Bogus proofs, derivations constructed to produce a correct result in spite of incorrect logic or operations were termed howlers by Maxwell. Outside the field of mathematics the term "howler" has various meanings, generally less specific. The division-by-zero fallacy has many variants.
False proof
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Contents
262.
C.R. Rao
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Calyampudi Radhakrishna Rao, FRS known as C R Rao is an Indian-born, naturalized American, mathematician and statistician. He is currently emeritus at Penn State University and Research Professor at the University at Buffalo. Rao was awarded the US National Medal of Science in 2002. The Times of India listed Rao as one of the top 10 Indian scientists of all time. Rao is also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on raising South Asian cardiovascular awareness. C. R. Rao was born into a Telugu family in Hadagali, Bellary, Karnataka, India. He received an M.Sc. in mathematics from Andhra University and an M.A. in statistics from Calcutta University in 1943. He was in the world to hold a master's degree in Statistics. Among his best-known discoveries are the Cramér–Rao bound and the Rao–Blackwell theorem both related to the quality of estimators. Other areas he worked in include multivariate analysis, differential geometry. His other contributions include the Fisher -- Rao Theorem, orthogonal arrays. He has published over 400 journal publications. He is a member of eight National Academies in India, the United Kingdom, Italy. Rao was awarded the United States National Medal of that nation's highest award for lifetime achievement in fields of scientific research, in June 2002. He has been the President of the International Statistical Institute, the International Biometric Society.
C.R. Rao
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Prof. Rao at Indian Statistical Institute, Chennai in April 2012
263.
Yadolah Dodge
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Yadolah Dodge is an Iranian and Swiss statistician. His major contributions are in the theory of design of experiments, simulation and regression. He spent his early years in Abadan, Iran. He obtained his Post Licentiate in Engineering in Agriculture in 1966 with distinction. He got his PhD in 1974. In the 1980s he became professor of statistics at the University of Neuchâtel, Switzerland, where as of 2011 he is a emeritus. He is 25 books, many of which have appeared in multiple editions. Arthanari, T.S. and Y Dodge, Y. Mathematical Programming in Statistics, Wiley, 1993. ISBN 0-471-59212-9 Birkes, D. and Dodge, Y. Alternative Methods of Regression, Wiley, 1993, ISBN 0-471-56881-3 Dodge, Y. Analysis of Experiments with Missing Data, John Wiley & Sons, 1985, ISBN 0471887366. Dodge, Y. The Oxford Dictionary of Statistical Terms, Oxford University Press, Oxford, 2003. ISBN 0-19-850994-4 Dodge, Y. and Rousson, V. Analyse de regression appliquée, Dunod, 1999. ISBN 0-19-850994-4 Dodge, Y. and Melfi, G. Premiers pas en simulation, Paris, 2008. ISBN 978-2-287-79493-3
Yadolah Dodge
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Yadolah Dodge
264.
Frank Kelly (mathematician)
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Francis Patrick "Frank" Kelly, CBE, FRS is Professor of the Mathematics of Systems at the Statistical Laboratory, University of Cambridge. He served to 2016. Kelly's research interests are in random networks and optimisation, especially in very large-scale systems such as telecommunication or transportation networks. He has also worked to congestion control and fair resource allocation in the internet. From 2003 to 2006 he served for Transport. Kelly was elected a Fellow of the Royal Society in 1989. In December 2006 he was elected 37th Master of Cambridge. He was appointed Commander of the Order of the British Empire in the 2013 New Year Honours to mathematical science. 2015 David Crighton Medal of the London Mathematical Society and Institute of Mathematics and its Applications Kelly, F. P.. Probability, statistics and optimisation: A Tribute to Peter Whittle. Chichester: John Wiley & Sons. Frank Kelly's homepage, retrieved 8 December 2006 Biography from Frank Kelly's website, retrieved 8 December 2006 Who's Who
Frank Kelly (mathematician)
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Professor Frank Kelly at the EPFL, 15 October 2007
265.
Richard Courant
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Richard Courant was a German American mathematician. He is best known by the general public for the book What is Mathematics?, co-written with Herbert Robbins. Courant was born in the Prussian Province of Silesia. His parents were Martha Courant née Freund of Oels. Edith Stein was Richard's cousin on the paternal side. During his youth his parents moved often, including to Glatz, then to Berlin. He stayed in Breslau and then continued his studies at the University of Zürich and the University of Göttingen. He obtained his doctorate there in 1910. He was wounded shortly after enlisting and therefore dismissed from the military. He continued his research in Göttingen and then engaged a two-year period as professor of mathematics. There he founded the Mathematical Institute, which he headed as director until 1933. Courant left Germany in 1933, earlier than Jewish escapees. After one year at Cambridge, Courant accepted a professorship at New York University in New York City. There he founded an institute in applied mathematics. The Courant Institute of Mathematical Sciences is now one of the most respected research centers in applied mathematics.
Richard Courant
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Richard Courant
266.
Herbert Robbins
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Herbert Ellis Robbins was one of the most prominent American mathematicians and statisticians of the 20th century. He did research in topology, measure theory, a variety of other fields. He was the co-author, with Richard Courant, of What is Mathematics?, a popularization, still in print. The Robbins lemma, used in empirical Bayes methods, is named after him. Robbins algebras are named after him because of a conjecture that he posed concerning Boolean algebras. The Robbins theorem, in theory, is also named after him, as is the Whitney -- Robbins synthesis, a tool he introduced to prove this theorem. Robbins was born in Pennsylvania. As an undergraduate, Robbins attended Harvard University, where Marston Morse influenced him to become interested in mathematics. Robbins was an instructor at New York University from 1939 to 1941. In 1953, he became a professor of mathematical statistics at Columbia University. He was then a professor at Rutgers University until his retirement in 1997. He has 567 descendants listed at the Mathematics Genealogy Project. In 1955, Robbins introduced empirical Bayes methods on Mathematical Statistics and Probability. These policies were simplified with MN Katehakis. Books by Herbert Robbins What is Mathematics?
Herbert Robbins
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Herbert Robbins visiting Purdue in 1966
267.
Clarendon Press
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Oxford University Press is the largest university press in the world, the second oldest after Cambridge University Press. They are headed to the delegates, who serves as its major representative on other university bodies. Oxford University has used a similar system to oversee OUP since the 17th century. The university grew into a major printer of Bibles, scholarly works. OUP took on the project that became the Oxford English Dictionary in the late 19th century, expanded to meet the ever-rising costs of the work. Moves into international markets led to OUP opening its own offices outside the United Kingdom, beginning with New York City in 1896. By contracting out binding operations, the modern OUP publishes some 6,000 new titles around each year. OUP was first exempted from United Kingdom tax in 1978. The Oxford University Press Museum is located on Great Clarendon Street, Oxford. Visits are led by a member of the staff. Displays include the printing and history of the Oxford Almanack, Alice in Wonderland and the Oxford English Dictionary. The first printer associated with Oxford University was Theoderic Rood. An edition of Rufinus's Expositio in apostolorum, was printed by another, anonymous, printer. Famously, this was mis-dated in Roman numerals as "1468", thus apparently pre-dating Caxton. Rood's printing included John Ankywyll's Compendium totius grammaticae, which set new standards for teaching of Latin grammar.
Clarendon Press
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Oxford University Press on Walton Street.
Clarendon Press
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Oxford University Press
Clarendon Press
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2008 conference booth
268.
Charles Sanders Peirce
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Charles Sanders Peirce was an American philosopher, logician, mathematician, scientist, sometimes known as "the father of pragmatism." He was employed as a scientist for 30 years. He is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, semiotics, for his founding of pragmatism. An innovator in mathematics, statistics, philosophy, research methodology, various sciences, Peirce considered himself, first and foremost, a logician. Logic for him encompassed much of that, now called epistemology and philosophy of science. In 1934, the philosopher Paul Weiss called Peirce "America's greatest logician." Webster's Biographical Dictionary said in 1943 that Peirce was "now regarded as greatest logician of his time." Keith Devlin similarly referred to Peirce as one of the greatest philosophers ever. Peirce was born at 3 Phillips Place in Cambridge, Massachusetts. At age 12, Charles read his older brother's copy of Richard Whately's Elements of Logic, then the English-language text on the subject. So began his lifelong fascination with logic and reasoning. At Harvard, he began lifelong friendships with Francis Ellingwood Abbot, William James. Charles William Eliot, formed an unfavorable opinion of Peirce. This opinion proved fateful, because Eliot, while President of Harvard 1869–1909—a period encompassing nearly all of Peirce's working life—repeatedly vetoed Harvard's employing Peirce in any capacity. Peirce suffered from his late teens onward from a nervous condition then known as "facial neuralgia", which would today be diagnosed as trigeminal neuralgia.
Charles Sanders Peirce
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Charles Sanders Peirce
Charles Sanders Peirce
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Peirce's birthplace. Now part of Lesley University 's Graduate School of Arts and Social Sciences
Charles Sanders Peirce
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Juliette and Charles by a well at their home Arisbe in 1907
Charles Sanders Peirce
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Arisbe in 2011
269.
Wolfgang Sartorius von Waltershausen
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Wolfgang Sartorius Freiherr von Waltershausen was a German geologist. Waltershausen was educated at the university in that city. There he devoted his attention to mineralogy. Waltershausen was named after Johann Wolfgang von Goethe, close friends with his parents. Georg, was a writer, lecturer and professor of economics and history at Göttingen. Georg Sartorius is best known in his role of popularizer of Adam Smith's Wealth of Nations. August, was a well known economist who studied the American economy, had at least one of his books translated into English. During a tour in 1834-1835 Waltershausen carried out a series of magnetic observations in various parts of Europe. The chief result of this undertaking was his great Atlas des Ätna, in which he distinguished the lava streams formed during the later centuries. Meanwhile, he was held this post for about thirty years, until his death. He died at Göttingen. In 1880, Arnold von Lasaulx edited Waltershausen published the book Der Aetna. Waltershausen was also the author of Gauss zum Gedächtnis, in 1856. This biography, published upon the death of Carl Friedrich Gauss, is viewed as Gauss's biography as Gauss wished it to be told. It is also the source of the most famous mathematical quotes: Mathematics is the queen of the sciences.
Wolfgang Sartorius von Waltershausen
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Wolfgang Sartorius von Waltershausen by August Kestner
Wolfgang Sartorius von Waltershausen
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Front page of Der Aetna
Wolfgang Sartorius von Waltershausen
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Waltershausen Glacier
Wolfgang Sartorius von Waltershausen
270.
Reuben Hersh
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Reuben Hersh is an American mathematician and academic, best known for his writings on the nature, practice, social impact of mathematics. This work complements mainstream philosophy of mathematics. After receiving a B.A. in English literature from Harvard University in 1946, Hersh spent a decade working as a machinist. After losing his right thumb when working with a band saw he decided to study mathematics at the Courant Institute of Mathematical Sciences. In 1962, he was awarded a Ph.D. in mathematics from New York University; his advisor was P.D. Lax. He has been affiliated with the University of New Mexico since 1964, where he is now emeritus. Hersh has written a number of technical articles on partial differential equations, probability, linear operator equations. He is 12 articles in the Mathematical Intelligencer. Hersh is best known as the coauthor with Philip J. Davis of The Mathematical Experience, which won a National Book Award in Science. Hersh advocates what he calls a "humanist" philosophy of mathematics, opposed to its rivals nominalism/fictionalism/formalism. Its reality is social-cultural-historical, located in the shared thoughts of those who learn it, teach it, create it. His article "The Kingdom of Math is Within You" explains how mathematicians' proofs compel agreement, even when they are inadequate as formal logic. He sympathizes with the perspectives on mathematics of Imre Lakatos and Where Mathematics Comes From, Rafael Nunez, Basic Books. 1981, Hersh and Philip Davis.
Reuben Hersh
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Reuben Hersh
271.
Jan Gullberg
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Jan Gullberg was a Swedish surgeon and anaesthesiologist, but became known as a writer on popular science and medical topics. He is best known as the author of Mathematics: From the Birth of Numbers, published by W. W. Norton in 1997. Gullberg was trained as a surgeon in Sweden. He qualified in medicine in 1964. He practised in the United States, as well as in Sweden. Gullberg saw himself as a doctor rather than a writer. He died at the hospital where he was working. He was twice married: first to Anne-Marie Hallin, with whom he had three children; and Ann, with whom he adopted two sons. Gullberg's second book, Mathematics: From the Birth of Numbers, took ten years to write, consuming all of Gullberg's spare time. It proved a major success; its first edition of 17,000 copies was virtually sold out within six months. ... It is a wonderful read. I take it with me everywhere I go." Allen says the book has "special charm", providing "excellent quotes and quips" throughout. His favourite chapter is "Cornerstones of Mathematics" which he believes should appeal both to "old hands".
Jan Gullberg
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Jan Gullberg
272.
W. W. Norton & Company
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W. W. Norton & Company is an American publishing company based in New York City. It has been owned wholly by its employees since the early 1960s. The roots of the company date back to 1923, when William Warder Norton founded the firm with his wife Mary Norton, became its first president. Storer D. Lunt took over in 1945 after Norton's death, was succeeded by George Brockway, Donald S. Lamm, now W. Drake McFeely. W. W. Norton & Company is an employee-owned publisher in the United States, which publishes fiction, nonfiction, professional books. Norton Anthologies Norton Critical Editions Oxford World's Classics Verso Book's Radical Thinkers Official website Making the Cut - Chronicle of Higher Education
W. W. Norton & Company
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500 Fifth Avenue
W. W. Norton & Company
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W. W. Norton & Company
273.
Hazewinkel, Michiel
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Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. After graduation Hazewinkel started his academic career in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was also Professor at the Universitaire Instelling Antwerpen, were Marcel van de Vel was his PhD student. In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical moduli for linear, constant, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978.
Hazewinkel, Michiel
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Michiel Hazewinkel, 1987
274.
Encyclopaedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available on CD-ROM. The presentation is technical in nature. The encyclopedia was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains three-dimensional objects. A dynamic version of the encyclopedia is now available as a public wiki online. This new wiki is a collaboration between the European Mathematical Society. All entries will be monitored by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989.
Encyclopaedia of Mathematics
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Encyclopedia of Mathematics snap shot
Encyclopaedia of Mathematics
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A complete set of Encyclopedia of Mathematics at a university library.
275.
Kluwer Academic Publishers
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Springer Science+Business Media or Springer is a global publishing company that publishes books, e-books and peer-reviewed journals in science, technical and medical publishing. Springer also hosts a number including SpringerImages. Book publications include book series; more than 168,000 titles are available as e-books in 24 subject collections. Springer has major offices in Berlin, Heidelberg, Dordrecht, New York City. On January 15, 2015, Holtzbrinck Publishing Group / Nature Publishing Group and Springer Science+Business Media announced a merger. In 1964, Springer expanded its business internationally, opening an office in New York City. Offices in Tokyo, Paris, Milan, Hong Kong, Delhi soon followed. The academic BertelsmannSpringer was formed after Bertelsmann bought a majority stake in 1999. The British investment groups Cinven and Candover bought BertelsmannSpringer from Bertelsmann in 2003. They merged the company in 2004 with the Dutch publisher Kluwer Academic Publishers which they bought from Wolters Kluwer in 2002, to form Springer Science+Business Media. Springer acquired the open-access publisher BioMed Central in October 2008 for an undisclosed amount. In 2009, Cinven and Candover sold Springer to two private equity firms, EQT Partners and Government of Singapore Investment Corporation. The closing of the sale was confirmed in February 2010 after the competition authorities in the USA and in Europe approved the transfer. In 2011, Springer acquired Pharma Marketing and Publishing Services from Wolters Kluwer. In 2013, the private firm BC Partners acquired a majority stake in Springer for $4.4 billion.
Kluwer Academic Publishers
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Springer Science+Business Media
276.
Wikiversity
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Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, resulting activities. Wikiversity's beta phase officially began on August 15, 2006, with the English language Wikiversity. Two proposals were made. The first project proposal was not approved and the second, modified proposal, was approved. The launch of Wikiversity was announced at Wikimania 2006 as:.... We're also going to be hosting and fostering research into how these kinds of things can be used more effectively. Wikiversity is a center for the creation of and use of free learning materials, the provision of learning activities. Wikiversity is one of many wikis used in educational contexts, as well as many initiatives that are creating free and open educational resources. The primary goals for Wikiversity are to: host a range of multilingual learning materials/resources, for all age groups in all languages. Host scholarly/learning projects and communities that support these materials. The Wikiversity e-Learning model places emphasis on "learning groups" and "learning by doing". Wikiversity's motto and slogan is "set learning free", indicating that groups/communities of Wikiversity participants will engage in learning projects. Learning is facilitated on projects that are detailed, results reported by editing Wikiversity pages. Wikiversity learning projects include collections of wiki webpages concerned with the exploration of a particular topic. Wikiversity participants are encouraged to express their learning goals, the Wikiversity community collaborates to develop learning activities and projects to accommodate those goals.
Wikiversity
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Wikiversity
277.
BBC
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The British Broadcasting Corporation is a British public service broadcaster. The BBC operates under its Agreement with the Secretary of State for Culture, Media and Sport. Britain's first public broadcast from the Marconi factory in Chelmsford took place in June 1920. It was featured the famous Australian Soprano Dame Nellie Melba. The broadcast caught the people's imagination and marked a turning point in the British public's attitude to radio. However, this public enthusiasm was not shared in official circles where such broadcasts were held to interfere with important civil communications. A Scottish Calvinist, was appointed its General Manager in December 1922 a few weeks after the company made its first official broadcast. The company was to be financed by a royalty on the sale of BBC wireless receiving sets from approved manufacturers. To this day, the BBC aims to follow the Reithian directive to "inform, entertain". The financial arrangements soon proved inadequate. Set sales were disappointing as amateurs made listeners bought rival unlicensed sets. By mid-1923, the Postmaster-General commissioned a review of broadcasting by the Sykes Committee. This was to be followed by a simple 10 shillings licence fee with no royalty once the wireless manufactures protection expired. The BBC's broadcasting monopoly was made explicit for the duration of its current licence, as was the prohibition on advertising. The BBC was also required to source all news from external wire services.
BBC
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BBC Television Centre at White City, West London, which opened in 1960 and closed in 2013
BBC
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BBC Pacific Quay in Glasgow, which was opened in 2007
BBC
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BBC New Broadcasting House, London which came into use during 2012–13.
BBC
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The headquarters of the BBC at Broadcasting House in Portland Place, London, England. This section of the building is called 'Old Broadcasting House'.
278.
University of Cambridge
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The University of Cambridge is a collegiate public research university in Cambridge, England. The university grew out of an association of scholars who left the University of Oxford after a dispute with the townspeople. The two ancient universities are often referred to jointly as "Oxbridge". Cambridge is formed over 100 academic departments organised into six schools. A department of the university, is the world's oldest publishing house and the second-largest university press in the world. The university also operates eight scientific museums, including the Fitzwilliam Museum, a botanic garden. Cambridge's libraries hold a total of around million books, eight million of which are in Cambridge University Library, a legal deposit library. In the year ended July 2015, the university had a total income of # 1.64 billion, of which # 398 million was from research grants and contracts. Colleges have a combined endowment of around # 5.89 billion, the largest of any university outside the United States. The university is closely linked with the development of the high-tech cluster known as "Silicon Fen". Cambridge is consistently ranked as one of the world's best universities. The university has educated notable alumni, including eminent mathematicians, scientists, politicians, lawyers, philosophers, writers, actors, foreign Heads of State. Two Chief Scientists of the U.S. Air Force and ten Fields medalists have been affiliated with Cambridge as students, faculty, staff or alumni. By the 12th century, the Cambridge region already had a scholarly and ecclesiastical reputation, due to monks from the nearby bishopric church of Ely. Most scholars moved to cities such as Paris, Reading, Cambridge.
University of Cambridge
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Emmanuel College Chapel
University of Cambridge
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University of Cambridge coat of arms
University of Cambridge
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Sir Isaac Newton was a student of the University of Cambridge
University of Cambridge
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Trinity Lane in the snow, with King's College Chapel (centre), Clare College Chapel (right), and the Old Schools (left)
279.
Wiki
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A wiki is a website that provides collaborative modification of its content and structure directly from the web browser. In a typical wiki, text is written using a simplified language, often edited with the help of a rich-text editor. A wiki is run using wiki software, otherwise known as a wiki engine. There are dozens of different wiki engines in both standalone and part of other software, such as bug tracking systems. Some wiki engines are open source, whereas others are proprietary. Some permit control over different functions; for example, editing rights may permit changing, removing material. Others may permit access without enforcing control. Other rules may also be imposed to organize content. Wikipedia is not a single wiki but rather a collection of hundreds of one for each language. The English-language Wikipedia has the largest collection of articles; as of September 2016, it had over million articles. The developer of the first wiki software, WikiWikiWeb, originally described it as "the simplest online database that could possibly work". "Wiki" is a Hawaiian word meaning "quick". Wiki promotes meaningful topic associations between different pages by showing whether an intended target page exists or not. A wiki is not a carefully crafted site designed for casual visitors. Instead, it seeks to involve the typical visitor/user in an ongoing process of collaboration that constantly changes the website landscape.
Wiki
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Ward Cunningham, inventor of the wiki
Wiki
Wiki
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Wiki Wiki Shuttle at Honolulu International Airport
Wiki
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Homepage of Wikipedia
280.
CC BY SA
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A Creative Commons license is one of several public copyright licenses that enable the free distribution of an otherwise copyrighted work. A CC license is used when an author wants to give people the right to share, use, build upon a work that they have created. CC licensed music is available through several outlets such as SoundCloud, is available for use in video and music remixing. There are several types of CC licenses. The licenses differ by several combinations that condition the terms of distribution. They were initially released on December 16, 2002 by Creative Commons, a U.S. non-profit corporation founded in 2001. There have also been five versions of the suite of licenses, numbered 1.0 through 4.0. As of 2016, the 4.0 license suite is the most current. Work licensed under a Creative Commons license is governed by applicable copyright law. This allows Creative Commons licenses to be applied to all work falling under copyright, including: books, plays, movies, music, articles, photographs, blogs, websites. Creative Commons does not recommend the use of Creative Commons licenses for software. There are over 35,000 works that are available in hardcopy and have a registered ISBN number. Creative Commons splits these works into two categories, one of which encompasses self-published books. However, application of a Creative Commons license may not modify the rights allowed by fair use or fair dealing or exert restrictions which violate copyright exceptions. Furthermore, Creative Commons licenses are non-exclusive and non-revocable.
CC BY SA
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Creative Commons licenses are explained in many languages and used around the world, such as pictured here in Cambodia.
281.
TeX
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TeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. TeX is popular in academia, especially in mathematics, computer science, economics, engineering, physics, quantitative psychology. It has largely displaced the other favored formatting system, in many Unix installations, which use both for different purposes. It is also used for other typesetting tasks, especially in the form of LaTeX, ConTeXt, other template packages. The widely used MIME type for TeX is application/x-tex. Within the typesetting system, its name is stylized as TeX. This method, dating back to the 19th century, produced a "classic style" appreciated by Knuth. When Knuth received the galley proofs of the new book on 30 March 1977, he found them awful. Around that time, Knuth became interested in digital typography. The galley proofs gave him the final motivation to solve the problem at hand once and for all by designing his own typesetting system. On 13 he wrote a memo to himself describing the basic features of TeX.. He planned as it happened the language was not "frozen" until 1989, more than ten years later. Guy Steele happened to be during the summer of 1978, when Knuth was developing his first version of TeX.. When Steele returned to Massachusetts Institute of Technology that autumn, he rewrote TeX's input/output to run under the Incompatible Timesharing System system. The first version of TeX was written in the SAIL language to run on a PDP-10 under Stanford's WAITS operating system.
TeX
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A sample page produced using TeX with the LaTeX macros
282.
Elementary algebra
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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned outside the realm of real and complex numbers. Quantitative relationships in science and mathematics are expressed as algebraic equations. Algebraic notation describes how algebra is written. It has its own terminology. Letters represent constants. They are usually written in italics. Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation. and are applied to algebraic variables and terms. Multiplication symbols implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 × x 2 is written as 3 x 2, 2 × x × y may be written 2 x y. Usually terms with the highest power, are written on the left, for example, x 2 is written to the left of x. When a coefficient is one, it is usually omitted.
Elementary algebra
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A typical algebra problem.
Elementary algebra
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Two-dimensional plot (magenta curve) of the algebraic equation
283.
Dynamical systems theory
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Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is called discrete dynamical systems. Some situations may also be modeled by mixed operators, such as differential-difference equations. Much of modern research is focused on the study of chaotic systems. This field of study is also called the mathematical theory of dynamical systems. Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems. Some of these fixed points are attractive, meaning that if the system starts out in a nearby state, it converges towards the fixed point. Similarly, one is interested in periodic points, states of the system that repeat after several timesteps. Periodic points can also be attractive. Sharkovskii's theorem is an interesting statement about the number of periodic points of a discrete system. Even simple nonlinear dynamical systems often exhibit seemingly random behavior, called chaos. The branch of dynamical systems that deals with the clean investigation of chaos is called theory. The concept of dynamical systems theory has its origins in Newtonian mechanics. Some excellent presentations of dynamic theory include.
Dynamical systems theory
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The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to chaos theory.
284.
Finite geometry
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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains many points. A geometry based on the graphics displayed on a screen, where the pixels are considered to be the points, would be a finite geometry. Finite geometries can also be defined purely axiomatically. However, dimension two has projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries. The following remarks apply only to finite planes. There are two main kinds of finite geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Finite projective plane geometry may be described by fairly simple axioms. There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not trivial, while the first two specify the nature of the geometry. The simplest plane contains only four points; it is called the affine plane of order 2. Since no three are collinear, any pair of points so this plane contains six lines.
Finite geometry
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Finite affine plane of order 2, containing 4 points and 6 lines. Lines of the same color are "parallel".
285.
Representation theory
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The algebraic objects amenable to such a description include groups, Lie algebras. Theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject, well understood. A feature of theory is its pervasiveness in mathematics. There are two sides to this. The second aspect is the diversity of approaches to theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, topology. The success of theory has led to numerous generalizations. One of the most general is in theory. Let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard n-dimensional space of column vectors over the real or complex numbers respectively. In this case, the idea of theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers. There are three main sorts of algebraic objects for which this can be done: groups, Lie algebras. There are two ways to say what a representation is. The first uses the idea of an action, generalizing the way that matrices act by matrix multiplication. First, for any g in G, the φ: V → V v ↦ Φ is linear.
Representation theory
286.
Discrete mathematics
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Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included by what is excluded: related notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research. Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well. In curricula, "Discrete Mathematics" appeared in the 1980s, initially as a computer science course; its contents were somewhat haphazard at the time. Some discrete mathematics textbooks have appeared well. At this level, discrete mathematics is sometimes seen as a preparatory course, not unlike precalculus in this respect. The Fulkerson Prize is awarded for outstanding papers in discrete mathematics. The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field.
Discrete mathematics
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Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms.
287.
Integrated Authority File
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The Integrated Authority File or GND is an international authority file for the organisation of personal names, subject headings and corporate bodies from catalogues. It is used mainly increasingly also by archives and museums. The GND is managed with various regional library networks in German-speaking Europe and other partners. The GND falls under the Creative Commons Zero license. The GND specification provides a hierarchy of high-level sub-classes, useful in library classification, an approach to unambiguous identification of single elements. It also comprises an ontology intended for knowledge representation in the semantic web, available in the RDF format.
Integrated Authority File
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GND screenshot
288.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established for the purpose of assisting members of the National Diet of Japan in researching matters of public policy. The library is similar in scope to the United States Library of Congress. The National Diet Library consists of several other branch libraries throughout Japan. Its need for information was "correspondingly small." The original Diet libraries "never developed either the services which might have made them vital adjuncts of genuinely responsible legislative activity." Until Japan's defeat, moreover, the executive had controlled all political documents, depriving the Diet of access to vital information. In 1946, each house of the Diet formed its own National Diet Library Standing Committee. Hani envisioned the new body as "both a ` citadel of popular sovereignty," and the means of realizing a "peaceful revolution." The National Diet Library opened with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori. The philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL became the only national library in Japan. At this time the collection gained an additional million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, adjacent to the National Diet.
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo
289.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, change. There is a range of views among philosophers as to the exact scope and definition of mathematics. Mathematicians use them to formulate new conjectures. Mathematicians resolve the falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of logic, mathematics developed from counting, calculation, measurement, the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Galileo Galilei said, "The universe can not become familiar with the characters in which it is written. Without these, one is wandering about in a dark labyrinth." Carl Friedrich Gauss referred as "the Queen of the Sciences". Benjamin Peirce called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules.
Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem
Mathematics
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Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
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Carl Friedrich Gauss, known as the prince of mathematicians