1.
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. The tips of the caliper are adjusted to fit across the points to be measured, the caliper is then removed and it is used in many fields such as mechanical engineering, metalworking, forestry, woodworking, science and medicine. A plurale tantum sense of the word calipers coexists in natural usage with the regular noun sense of caliper, also existing colloquially but not in formal usage is referring to a vernier caliper as a vernier or a pair of verniers. In imprecise colloquial usage, some extend this even to dial calipers. In machine-shop usage, the caliper is often used in contradistinction to micrometer. In this usage, caliper implies only the factor of the vernier or dial 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 bronze caliper, dating from 9 AD, was used for minute measurements during the Chinese Xin dynasty, the caliper had an inscription stating that it was made on a gui-you day at new moon of the first month of the first year of the Shijian guo period. The calipers included a slot and pin and graduated in inches, the modern vernier caliper, reading to thousandths of an inch, was invented by American Joseph R. Brown in 1851. It was the first practical tool for exact measurements that could be sold at a price within the reach of ordinary machinists, the inside calipers are used to measure the internal size of an object. The upper caliper in the image requires manual adjustment prior to fitting, fine setting of this caliper type is performed by tapping the caliper legs lightly on a handy surface until they will almost pass over the object. A light push against the resistance of the pivot screw then spreads the legs to the correct dimension and provides the required. The lower caliper in the image has a screw that permits it to be carefully adjusted without removal of the tool from the workpiece. Outside calipers are used to measure the size of an object. The same observations and technique apply to this type of caliper, with some understanding of their limitations and usage, these instruments can provide a high degree of accuracy and repeatability. They are especially useful when measuring over very large distances, consider if the calipers are used to measure a large diameter pipe, a vernier caliper does not have the depth capacity to straddle this large diameter while at the same time reach the outermost points of the pipes diameter
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
2.
Number
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A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1,2,3, a notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are used for labels, for ordering. In common usage, number may refer to a symbol, a word, calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, the same term may also refer to number theory, the study of the properties of numbers. Besides their practical uses, numbers have cultural significance throughout the world, for example, in Western society the number 13 is regarded as unlucky, and a million may signify a lot. Though it is now regarded as pseudoscience, numerology, the belief in a significance of numbers, permeated ancient. Numerology heavily influenced the development of Greek mathematics, stimulating the investigation of problems in number theory which are still of interest today. During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Numbers should be distinguished from numerals, the used to represent numbers. Boyer showed that Egyptians created the first ciphered numeral system, Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets. The number five can be represented by digit 5 or by the Roman numeral Ⅴ, notations used to represent numbers are discussed in the article numeral systems. The Roman numerals require extra symbols for larger numbers, different types of numbers have many different uses. Numbers can be classified into sets, called number systems, such as the natural numbers, the same number can be written in many different ways. For different methods of expressing numbers with symbols, such as the Roman numerals, each of these number systems may be considered as a proper subset of the next one. This is expressed, symbolically, by writing N ⊂ Z ⊂ Q ⊂ R ⊂ C, the most familiar numbers are the natural numbers,1,2,3, and so on. Traditionally, the sequence of numbers started with 1 However, in the 19th century, set theorists. Today, different mathematicians use the term to both sets, including 0 or not
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.
3.
Mathematician
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A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, quantity, structure, space, models, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and 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 works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked. Science and mathematics in the Islamic world during the Middle Ages followed various models and 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 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 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. ”Thus, seminars, overall, science became the focus of universities in the 19th and 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and then proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
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
4.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
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.
5.
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 relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and 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 much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and 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, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
Logic
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Aristotle, 384–322 BCE.
6.
Counting
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Counting is the action of finding the number of elements of a finite set of objects. The related term 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, 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 of social and economic data such as number of members, prey animals, property. The development of counting led to the development of mathematical notation, numeral systems, counting can occur in a variety of forms. Counting can be verbal, that is, speaking every number out loud to keep track of progress and 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 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, counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations, finger-counting uses unary notation, and is thus limited to counting 10. Older finger counting used the four fingers and the three bones in each finger to count to the number twelve, other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary, it is possible to keep a count up to 1023 =210 −1. Various devices can also be used to facilitate counting, such as tally counters. Inclusive counting is usually encountered when dealing with time in the Romance languages, in exclusive counting languages such as English, when counting 8 days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. When counting inclusively, the Sunday will be day 1 and therefore the following Sunday will be the eighth day, for example, the French phrase for fortnight is quinzaine, and similar words are present in Greek, Spanish and Portuguese. In contrast, the English word fortnight itself derives from a fourteen-night, as the archaic sennight does from a seven-night, learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a very first step into mathematics. However, some cultures in Amazonia and the Australian Outback do not count, many children at just 2 years of age have some skill in reciting the count list. They can also answer questions of ordinality for small numbers, e. g and they can even be skilled at pointing to each object in a set and reciting the words one after another
Counting
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Counting using tally marks at Hanakapiai Beach
7.
Shape
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A shape is the form of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material composition. Psychologists have theorized that humans mentally break down images into simple geometric shapes called geons, examples of geons include cones and spheres. Some simple shapes can be put into broad categories, for instance, polygons are classified according to their number of edges as triangles, quadrilaterals, pentagons, etc. Each of these is divided into categories, triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, among the most common 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 exactly or even approximately, thus, we say that the shape of a manhole cover is a disk, because it is approximately the same geometric object as an actual geometric disk. Similarity, Two objects are similar if one can be transformed into the other by a scaling, together with a sequence of rotations, translations. Isotopy, Two objects are isotopic if one can be transformed into the other by a sequence of deformations that do not tear the object or put holes in it. 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. For instance, the b and d are a reflection of each other, and hence they are congruent and similar. Sometimes, only the outline or 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 determine whether or not two objects have the shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same. Simple shapes can often be classified into basic objects such as a point, a line, a curve, a plane. However, most shapes occurring in the world are complex. Some, such as plant structures and coastlines, may be so complicated as to defy traditional mathematical description – in which case they may be analyzed by differential geometry, or as fractals. In geometry, two subsets of a Euclidean space have the shape if one can be transformed to the other by a combination of translations, rotations. In other words, the shape of a set of points is all the information that is invariant to translations, rotations
Shape
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A variety of polygonal shapes.
8.
Motion (physics)
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In physics, motion is a change in position of an object over time. Motion is described in terms of displacement, distance, velocity, acceleration, time, motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame. If the position of a body is not changing with respect to a frame of reference. An objects motion cannot change unless it is acted upon by a force, momentum is a quantity which is used for measuring motion of an object. As there is no frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving, more generally, motion is a concept that applies to objects, bodies, and matter particles, to radiation, radiation fields and radiation particles, and to space, its curvature and space-time. One can also speak of motion of shapes and boundaries, so, the term motion in general signifies a continuous change in the configuration of a physical system. For example, one can talk about motion of a wave or about motion of a quantum particle, in physics, motion is described through two sets of apparently contradictory laws of mechanics. Motions of all large scale and familiar objects in the universe are described by classical mechanics, whereas the motion of very small atomic and sub-atomic objects is described by quantum mechanics. It produces very accurate results within these domains, and is one of the oldest and largest in science, engineering, classical mechanics is fundamentally based on Newtons laws of motion. These laws describe the relationship between the acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica and his three laws are, A body either is at rest or moves with constant velocity, until and unless an outer force is applied to it. An object will travel in one direction only until an outer force changes its direction, whenever one body exerts a force F onto a second body, the second body exerts the force −F on the first body. F and −F are equal in magnitude and opposite in sense, so, the body which exerts F will go backwards. Newtons three laws of motion, along with his Newtons law of motion, which were the first to provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of bodies and motion of objects on earth. Classical mechanics was later enhanced by Albert Einsteins special relativity. Motion of objects with a velocity, approaching the speed of light
Motion (physics)
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Motion involves a change in position, such as in this perspective of rapidly leaving Yongsan Station.
9.
History of Mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind 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 refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
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.
10.
Greek mathematics
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Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, 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 study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. 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 possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, 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. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first 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, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
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.
11.
Euclid's Elements
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Euclids 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, propositions, the books cover Euclidean geometry and the ancient Greek version of elementary number theory. Elements is the second-oldest extant Greek mathematical treatise after Autolycus On the Moving Sphere and it has proven instrumental in the development of logic and modern science. According to Proclus, the element was used to describe a theorem that is all-pervading. The word element in the Greek language is the same as letter and this suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Euclids Elements has been referred to as the most successful and influential textbook ever written, for centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclids Elements was required of all students. Not until the 20th century, by which time its content was taught through other school textbooks. Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, the Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions, papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although known to, for instance, Cicero, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century. The Arabs received the Elements from the Byzantines around 760, this version was translated into Arabic under Harun al Rashid circa 800, the Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century. Although known in Byzantium, the Elements was lost to Western Europe until about 1120, the first printed edition appeared in 1482, and since then it has been translated into many languages and published in about a thousand different editions. Theons Greek edition was recovered in 1533, in 1570, John Dee provided a widely respected Mathematical Preface, along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library, the manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text, ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text, also of importance are the scholia, or annotations to the text. These additions, which distinguished themselves from the main text. The Elements is still considered a masterpiece in the application of logic to mathematics, in historical context, it has proven enormously influential in many areas of science
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
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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.
12.
Giuseppe Peano
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Giuseppe Peano was an Italian mathematician. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, the standard axiomatization of the natural numbers is named the Peano axioms in his honor. As part of effort, he made key contributions to the modern rigorous. He spent most of his career teaching mathematics at the University of Turin, Peano was born and raised on a farm at Spinetta, a hamlet now belonging to Cuneo, Piedmont, Italy. Due to Genocchis poor health, Peano took over the teaching of calculus course within two years and his first major work, 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 union and intersection of sets appeared for the first time. In 1887, Peano married Carola Crosio, the daughter of the Turin-based painter Luigi Crosio, in 1886, he began teaching concurrently at the Royal Military Academy, and was promoted to Professor First Class in 1889. The next year, the University of Turin also granted him his full professorship, Peanos famous space-filling curve appeared in 1890 as a counterexample. He used it to show that a continuous curve cannot always be enclosed in a small region. This was an example of what came to be known as a fractal. In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891, in 1891 Peano started the Formulario Project. It was to be an Encyclopedia of Mathematics, containing all known formulae, in 1897, the first International Congress of Mathematicians was held in Zürich. Peano was a key participant, presenting a paper on mathematical logic and he also started to become increasingly occupied with Formulario to the detriment of his other work. In 1898 he presented a note to the Academy about binary numeration and he also became so frustrated with publishing delays that he purchased a printing press. Paris was the venue for the Second International Congress of Mathematicians in 1900, the conference was preceded by the First International Conference of Philosophy where Peano was a member of the patronage committee. He presented a paper which posed the question of correctly formed definitions in mathematics and this became one of Peanos main philosophical interests for the rest of his life. At the conference Peano met Bertrand Russell and gave him a copy of Formulario, Russell was so struck by Peanos innovative logical symbols that he left the conference and returned home to study Peanos text. Peanos students Mario Pieri and Alessandro Padoa had papers presented at the congress also
Giuseppe Peano
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Giuseppe Peano
Giuseppe Peano
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Aritmetica generale e algebra elementare, 1902
Giuseppe Peano
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Giuseppe Peano and his wife Carola Crosio in 1887
13.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, 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 as a Privatdozent from 1886 to 1895, in 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. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
David Hilbert
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David Hilbert (1912)
David Hilbert
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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
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Hilbert's tomb: Wir müssen wissen Wir werden wissen
14.
Truth
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Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth may also often be used in modern contexts to refer to an idea of truth to self, the commonly understood opposite of truth is falsehood, which, correspondingly, can also take on a logical, factual, or ethical meaning. The concept of truth is discussed and debated in several contexts, including philosophy, art, Some philosophers view the concept of truth as basic, and unable to be explained in any terms that are more easily understood than the concept of truth itself. Commonly, truth is viewed as the correspondence of language or thought to an independent reality, other philosophers take this common meaning to be secondary and derivative. On this view, the conception of truth as correctness is a derivation from the concepts original essence. Various theories and views of truth continue to be debated among scholars, philosophers, language and words are a means by which humans convey information to one another and the method used to determine what is a truth is termed a criterion of truth. The English word truth is derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, like troth, it is a -th nominalisation of the adjective true. Old Norse trú, faith, word of honour, religious faith, thus, truth involves both the quality of faithfulness, fidelity, loyalty, sincerity, veracity, and that of agreement with fact or reality, in Anglo-Saxon expressed by sōþ. All Germanic languages besides English have introduced a distinction between truth fidelity and 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, trust, pact. Romance languages use terms following the Latin veritas, while the Greek aletheia, Russian pravda, each presents perspectives that are widely shared by published scholars. However, the theories are not universally accepted. More recently developed deflationary or minimalist theories of truth have emerged as competitors to the substantive theories. Minimalist reasoning centres around the notion that the application of a term like true to a statement does not assert anything significant about it, for instance, anything about its nature. Minimalist reasoning realises truth as a label utilised in general discourse to express agreement, to stress claims, correspondence theories emphasise that true beliefs and true statements correspond to the actual state of affairs. This type of theory stresses a relationship between thoughts or statements on one hand, and things or objects on the other and it is a traditional model tracing its origins to ancient Greek philosophers such as Socrates, Plato, and Aristotle. This class of theories holds that the truth or the falsity of a representation is determined in principle entirely by how it relates to things, Aquinas also restated the theory as, A judgment is said to be true when it conforms to the external reality. Many modern theorists have stated that this ideal cannot be achieved without analysing additional factors, for example, language plays a role in that all languages have words to represent concepts that are virtually undefined in other languages
Truth
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Time Saving Truth from Falsehood and Envy, François Lemoyne, 1737
Truth
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Truth, holding a mirror and a serpent (1896). Olin Levi Warner, Library of Congress Thomas Jefferson Building, Washington, D.C.
Truth
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An angel carrying the banner of "Truth", Roslin, Midlothian
Truth
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Walter Seymour Allward 's Veritas (Truth) outside Supreme Court of Canada, Ottawa, Ontario Canada
15.
Galileo Galilei
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Galileo Galilei was an Italian polymath, astronomer, physicist, engineer, philosopher, and mathematician. He played a role in the scientific revolution of the seventeenth century. Galileo also worked in applied science and technology, inventing an improved military compass, Galileos 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 a stellar parallax. He was tried by the Inquisition, found vehemently suspect of heresy and he spent the rest of his life under house arrest. He has been called the father of observational astronomy, the father of modern physics, the father of scientific method, and the father of science. Galileo was born in Pisa, Italy, on 15 February 1564, the first of six children of Vincenzo Galilei, a famous lutenist, composer, and music theorist, and Giulia, three of Galileos five siblings survived infancy. The youngest, Michelangelo, also became a noted lutenist and composer although he contributed to financial burdens during Galileos young adulthood, Michelangelo was unable to contribute his fair share of their fathers promised dowries to their brothers-in-law, who would later attempt to seek legal remedies for payments due. Michelangelo would also occasionally have to borrow funds from Galileo to support his musical endeavours and these financial burdens may have contributed to Galileos early fire to develop inventions that would bring him additional income. When Galileo Galilei was eight, his family moved to Florence and he then was educated in the Vallombrosa Abbey, about 30 km southeast of Florence. Galileo Bonaiuti was buried in the church, the Basilica of Santa Croce in Florence. It was common for mid-sixteenth century Tuscan families to name the eldest son after the parents surname, hence, Galileo Galilei was not necessarily named after his ancestor Galileo Bonaiuti. The Italian male given name Galileo derives from the Latin Galilaeus, meaning of Galilee, the biblical roots of Galileos name and surname were to become the subject of a famous pun. In 1614, during the Galileo affair, one of Galileos opponents, in it he made a point of quoting Acts 1,11, Ye men of Galilee, why stand ye gazing up into heaven. Despite being a genuinely pious Roman Catholic, Galileo fathered three children out of wedlock with Marina Gamba and they had two daughters, Virginia and Livia, and a son, Vincenzo. Their only worthy alternative was the religious life, both girls were accepted by the convent of San Matteo in Arcetri and remained there for the rest of their lives. Virginia took the name Maria Celeste upon entering the convent and she died on 2 April 1634, and is buried with Galileo at the Basilica of Santa Croce, Florence. Livia took the name Sister Arcangela and was ill for most of her life, Vincenzo was later legitimised as the legal heir of Galileo and married Sestilia Bocchineri
Galileo Galilei
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Portrait of Galileo Galilei by Giusto Sustermans
Galileo Galilei
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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
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Galileo Galilei. Portrait by Leoni
Galileo Galilei
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Cristiano Banti 's 1857 painting Galileo facing the Roman Inquisition
16.
Albert Einstein
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Albert Einstein was a German-born theoretical physicist. He developed the theory of relativity, one of the two pillars of modern physics, Einsteins work is also known for its influence on the philosophy of science. Einstein is best known in popular culture for his mass–energy equivalence formula E = mc2, near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led him to develop his theory of relativity during his time at the Swiss Patent Office in Bern. Briefly before, he aquired the Swiss citizenship in 1901, which he kept for his whole life and he continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the properties of light which laid the foundation of the photon theory of light. In 1917, Einstein applied the theory of relativity to model the large-scale structure of the universe. He was visiting the United States when Adolf Hitler came to power in 1933 and, being Jewish, did not go back to Germany and he settled in the United States, becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project, Einstein supported defending the Allied forces, but generally denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, Einstein signed the Russell–Einstein Manifesto, Einstein was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 300 scientific papers along with over 150 non-scientific works, on 5 December 2014, universities and archives announced the release of Einsteins papers, comprising more than 30,000 unique documents. Einsteins intellectual achievements and originality have made the word Einstein synonymous with genius, Albert Einstein was born in Ulm, in the Kingdom of Württemberg in the German Empire, on 14 March 1879. His parents were Hermann Einstein, a salesman and engineer, the Einsteins were non-observant Ashkenazi Jews, and Albert attended a Catholic elementary school in Munich from the age of 5 for three years. At the age of 8, he was transferred to the Luitpold Gymnasium, the loss forced the sale of the Munich factory. In search of business, the Einstein family moved to Italy, first to Milan, when the family moved to Pavia, Einstein stayed in Munich to finish his studies at the Luitpold Gymnasium. His father intended for him to electrical engineering, but Einstein clashed with authorities and resented the schools regimen. He later wrote that the spirit of learning and creative thought was lost in strict rote learning, at the end of December 1894, he travelled to Italy to join his family in Pavia, convincing the school to let him go by using a doctors note. During his time in Italy he wrote an essay with the title On the Investigation of the State of the Ether in a Magnetic Field
Albert Einstein
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Albert Einstein in 1921
Albert Einstein
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Einstein at the age of 3 in 1882
Albert Einstein
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Albert Einstein in 1893 (age 14)
Albert Einstein
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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)
17.
Natural science
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Natural science is a branch of science concerned with the description, prediction, and understanding of natural phenomena, based on observational and empirical evidence. Mechanisms such as review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two branches, life science and physical science. Physical science is subdivided into branches, including physics, space science, chemistry and 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, galileo, Descartes, Francis Bacon, and Newton debated the benefits of using approaches which were more mathematical and more experimental in a methodical way. Still, philosophical perspectives, conjectures, and presuppositions, often overlooked, systematic data collection, including discovery science, succeeded natural history, which emerged in the 16th century by describing and classifying plants, animals, minerals, and so on. Today, natural history suggests observational descriptions aimed at popular audiences, philosophers of science have suggested a number of criteria, including Karl Poppers controversial falsifiability criterion, to help them differentiate scientific endeavors from non-scientific ones. Validity, accuracy, and quality control, such as peer review and 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, biology is concerned with the characteristics, classification and behaviors of organisms, as well as how species were formed and their interactions with each other and the environment. The biological fields of botany, zoology, and medicine date back to periods of civilization. However, it was not until the 19th century that became a unified science. Once scientists discovered commonalities between all living things, it was decided they were best studied as a whole, modern biology is divided into subdisciplines by the type of organism and by the scale being studied. Molecular biology is the study of the chemistry of life, while cellular biology is the examination of the cell. At a higher level, anatomy and physiology looks at the internal structures, constituting the scientific study of matter at the atomic and molecular scale, chemistry deals primarily with collections of atoms, such as gases, molecules, crystals, and metals. The composition, statistical properties, transformations and reactions of these materials are studied, chemistry also involves understanding the properties and interactions of individual atoms and molecules for use in larger-scale applications. Most chemical processes can be studied directly in a laboratory, using a series of techniques for manipulating materials, chemistry is often called the central science because of its role in connecting the other natural sciences. Early experiments in chemistry had their roots in the system of Alchemy, the science of chemistry began to develop with the work of Robert Boyle, the discoverer of gas, and Antoine Lavoisier, who developed the theory of the Conservation of mass. The success of science led to a complementary chemical industry that now plays a significant role in the world economy
Natural science
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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
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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
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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
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Isaac Newton is widely regarded as one of the most influential scientists of all time.
18.
Medicine
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Medicine is the science and practice of the diagnosis, treatment, and prevention of disease. The word medicine is derived from Latin medicus, meaning a physician, Medicine encompasses a variety of health care practices evolved to maintain and restore health by the prevention and treatment of illness. Medicine has existed for thousands of years, during most of which it was an art frequently having connections to the religious and philosophical beliefs of local culture. For example, a man would apply herbs and say prayers for healing, or an ancient philosopher. In recent centuries, since the advent of modern science, most medicine has become a combination of art, while stitching technique for sutures is an art learned through practice, the knowledge of what happens at the cellular and molecular level in the tissues being stitched arises through science. Prescientific forms of medicine are now known as medicine and folk medicine. They remain commonly used with or instead of medicine and are thus called alternative medicine. For example, evidence on the effectiveness of acupuncture is variable and inconsistent for any condition, in contrast, treatments outside the bounds of safety and efficacy are termed quackery. Medical availability and clinical practice varies across the world due to differences in culture. In modern clinical practice, physicians personally assess patients in order to diagnose, treat, the doctor-patient relationship typically begins an interaction with an examination of the patients medical history and medical record, followed by a medical interview and a physical examination. Basic diagnostic medical devices are typically used, after examination for signs and 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 an important part of the relationship and the development of trust. The medical encounter is then documented in the record, which is a legal document in many jurisdictions. Follow-ups may be shorter but follow the general procedure. The diagnosis and treatment may take only a few minutes or a few weeks depending upon the complexity of the issue, the components of the medical interview and encounter are, Chief complaint, the reason for the current medical visit. They are in the patients own words and are recorded along with the duration of each one, also called chief concern or presenting complaint. History of present illness, the order of events of symptoms. Distinguishable from history of illness, often called past medical history
Medicine
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Early Medicine Bottles
Medicine
Medicine
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The Doctor, by Sir Luke Fildes (1891)
Medicine
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The Hospital of Santa Maria della Scala, fresco by Domenico di Bartolo, 1441–1442
19.
Applied mathematics
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Applied mathematics is a branch of mathematics that deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Thus, applied mathematics is a combination of science and specialized knowledge. The term applied mathematics also describes the professional specialty in which work on practical problems by formulating and studying mathematical models. The activity of applied mathematics is thus connected with research in pure mathematics. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations, approximation theory, 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 made use of applied mathematics. Today, the applied mathematics is used in a broader sense. It includes the areas noted above as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of mathematics are now important in applications. There is no consensus as to what the various branches of applied mathematics are, such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Many mathematicians distinguish between applied mathematics, which is concerned with methods, and the applications of mathematics within science. Mathematicians such as Poincaré and Arnold deny the existence of applied mathematics, similarly, non-mathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to industrial problems is also called industrial mathematics. Historically, mathematics was most important in the sciences and engineering. Academic institutions are not consistent in the way they group and label courses, programs, at some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and Mathematics. It is very common for Statistics departments to be separated at schools with graduate programs, many applied mathematics programs consist of primarily cross-listed courses and jointly appointed faculty in departments representing applications. Some Ph. D. programs in applied mathematics require little or no coursework outside of mathematics, in some respects this difference reflects the distinction between application of mathematics and applied mathematics. Research universities dividing their mathematics department into pure and applied sections include MIT, brigham Young University also has an Applied and Computational Emphasis, a program that allows student to graduate with a Mathematics degree, with an emphasis in Applied Math
Applied mathematics
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Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming.
20.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, 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 data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and 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 probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
Statistics
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Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics
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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
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Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistics
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Karl Pearson, a founder of mathematical statistics.
21.
Pure mathematics
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Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. 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, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, 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, 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 mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central
Pure mathematics
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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.
22.
History of mathematics
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Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322, the Rhind 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 refined the methods and expanded the subject matter of mathematics, Chinese mathematics made early contributions, including a place value system. Islamic mathematics, in turn, developed and expanded the known to these civilizations. Many Greek and Arabic texts on mathematics were then translated into Latin, from ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Italy in the 16th century, new mathematical developments, the origins of mathematical thought lie in the concepts of number, magnitude, and form. Modern studies of cognition have shown that these concepts are not unique to humans. Such concepts would have part of everyday life in hunter-gatherer societies. The idea of the number concept evolving gradually over time is supported by the existence of languages which preserve the distinction between one, two, and many, but not of numbers larger than two. Prehistoric artifacts discovered in Africa, dated 20,000 years old or more suggest early attempts to quantify time. The Ishango bone, found near the headwaters of the Nile river, may be more than 20,000 years old, common interpretations are that the Ishango bone shows either the earliest known demonstration of sequences of prime numbers or a six-month lunar calendar. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10, predynastic Egyptians of the 5th millennium BC pictorially represented geometric designs. All of the above are disputed however, and the currently oldest undisputed mathematical documents are from Babylonian, Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia from the days of the early Sumerians through the Hellenistic period almost to the dawn of Christianity. The majority of Babylonian mathematical work comes from two widely separated periods, The first few hundred years of the second millennium BC, and it is named Babylonian mathematics due to the central role of Babylon as a place of study. Later under the Arab Empire, Mesopotamia, especially Baghdad, once again became an important center of study for Islamic mathematics, in contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and they developed a complex system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period
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.
23.
Pythagoras
–
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and the putative founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written centuries after he lived. He was born on the island of Samos, and travelled, visiting Egypt and Greece, around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild. In 520 BC, he returned to Samos, Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a mathematician and scientist and 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 colleagues, some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato. Burkert states that Aristoxenus and Dicaearchus are the most important accounts, Aristotle had written a separate work On the Pythagoreans, which is no longer extant. However, the Protrepticus possibly contains parts of On the Pythagoreans and his disciples Dicaearchus, Aristoxenus, and Heraclides Ponticus had written on the same subject. These writers, late as they are, were among the best sources from whom Porphyry and Iamblichus drew, while adding some legendary accounts. Herodotus, Isocrates, and other writers agree that Pythagoras was the son of Mnesarchus and born on the Greek island of Samos. His father is said to have been a gem-engraver or a wealthy merchant, a late source gives his mothers name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, around 530 BC he arrived in the Greek colony of Croton in what was then Magna Graecia. There he founded his own school the members of which he engaged to a disciplined. He furthermore aquired some political influence, on Greeks and non-Greeks of the region, following a conflict with the neighbouring colony of Sybaris, internal discord drove most of the Pythagoreans out of Croton. Pythagoras left the city before the outbreak of civil unrest and moved to Metapontum, after his death, his house was transformed into a sanctuary of Demeter, out of veneration for the philosopher, by the local population. In ancient sources there was disagreement and inconsistency about the late life of Pythagoras. His tomb was shown at Metapontum in the time of Cicero, according to Walter Burkert, Most obvious is the contradiction between Aristoxenus and Dicaearchus, regarding the catastrophe that overwhelmed the Pythagorean society
Pythagoras
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Bust of Pythagoras of Samos in the Capitoline Museums, Rome.
Pythagoras
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Bust of Pythagoras, Vatican
Pythagoras
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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
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Croton on the southern coast of Magna Graecia (Southern Italy), to which Pythagoras ventured after feeling overburdened in Samos.
24.
Tally sticks
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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, historical reference is made by Pliny the Elder about the best wood to use for tallies, and by Marco Polo who mentions the use of the tally in China. Tallies have been used for purposes such as messaging and scheduling. Principally, there are two different kinds of sticks, the single tally and the split tally. A common form of the kind of primitive counting device is seen in various kinds of prayer beads. A number of artefacts have been conjectured to be tally sticks. It is a dark brown length of bone, the fibula of a baboon and 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 baboons 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, 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 tally stick was an elongated piece of bone, ivory, wood, or stone which is 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. Herodotus reported the use of a knotted cord by Darius I of Persia, the split tally was a technique which became common in medieval Europe, which was constantly short of money and predominantly illiterate, in order to record bilateral exchange and debts. A stick was marked with a system of notches and then split lengthwise and this way the two halves both record the same notches and each party to the transaction received one half of the marked stick as proof. Later this technique was refined in ways and became virtually tamper proof. One of the refinements was to make the two halves of the stick of different lengths, the longer part was called stock and was given to the party which had advanced money to the receiver. The shorter portion of the stick was called foil and was given to the party which had received the funds or goods, using this technique each of the parties had an identifiable record of the transaction. If one party tried to change the value of his half of the tally stick by adding more notches. The split tally was accepted as proof in medieval courts
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
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Entrance gates to the UK National Archives, Kew, from Ruskin Avenue. The notched vertical elements were inspired by medieval tally sticks.
25.
Babylonia
–
Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia. A small Amorite-ruled state emerged in 1894 BC, which contained at this time the city of Babylon. Babylon greatly expanded during the reign of Hammurabi in the first half of the 18th century BC, 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 rivalry with its older fellow Akkadian-speaking state of Assyria in northern Mesopotamia and it retained the Sumerian language for religious use, but by the time Babylon was founded, this was no longer a spoken language, having been wholly subsumed by Akkadian. The earliest mention of the city of Babylon can be found in a tablet from the reign of Sargon of Akkad. During the 3rd millennium BC, a cultural symbiosis occurred between Sumerian and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian and vice versa is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the millennium as a sprachbund. Traditionally, the religious center of all Mesopotamia was the city of Nippur. The empire eventually disintegrated due to decline, climate change and civil war. Sumer rose up again with the Third Dynasty of Ur in the late 22nd century BC and they also seem to have gained ascendancy over most of the territory of the Akkadian kings of Assyria in northern Mesopotamia for a time. The states of the south were unable to stem the Amorite advance, King Ilu-shuma of the Old Assyrian Empire in a known inscription describes his exploits to the south as follows, The freedom of the Akkadians and their children I established. I established their freedom from the border of the marshes and Ur and Nippur, Awal, past scholars originally extrapolated from this text that it means he defeated the invading Amorites to the south, but there is no explicit record of that. More recently, the text has been taken to mean that Asshur supplied the south with copper from Anatolia and these policies were continued by his successors Erishum I and Ikunum. During the first centuries of what is called the Amorite period and his reign was concerned with establishing statehood amongst a sea of other minor city states and kingdoms in the region. However Sumuabum appears never to have bothered to give himself the title of King of Babylon, suggesting that Babylon itself was only a minor town or city. He was followed by Sumu-la-El, Sabium, Apil-Sin, each of whom ruled in the same manner as Sumuabum. Sin-Muballit was the first of these Amorite rulers to be regarded officially as a king of Babylon, the Elamites occupied huge swathes of southern Mesopotamia, and the early Amorite rulers were largely held in vassalage to Elam
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
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Geography
26.
Arithmetic
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Arithmetic is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations between them—addition, subtraction, multiplication and division. Arithmetic is an part of number theory, and number theory is considered to be one of the top-level divisions of modern mathematics, along with algebra, geometry. The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are still used to refer to a wider part of number theory. The earliest written records indicate the Egyptians and Babylonians used all the elementary arithmetic operations as early as 2000 BC and these artifacts do not always reveal the specific process used for solving problems, but the characteristics of the particular numeral system strongly influence the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numerals, 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 board or the Roman abacus to obtain the results. Early number systems that included positional notation were not decimal, including the system for Babylonian numerals. Because of this concept, the ability to reuse the same digits for different values contributed to simpler. The continuous historical development of modern arithmetic starts with the Hellenistic civilization of ancient Greece, 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 numbers, Greek numerals were used by Archimedes, Diophantus and others in a positional notation not very different from ours. Because the ancient Greeks lacked a symbol for zero, they used three separate sets of symbols, one set for the units place, one for the tens place, and one for the hundreds. Then for the place they would reuse the symbols for the units place. Their addition algorithm was identical to ours, and their multiplication algorithm was very slightly different. Their long division algorithm was the same, and the square root algorithm that was taught in school was known to Archimedes. He preferred it to Heros method of successive approximation because, once computed, a digit doesnt change, and the square roots of perfect squares, such as 7485696, terminate immediately as 2736. For numbers with a part, such as 546.934. The ancient Chinese used a positional notation. Because they also lacked a symbol for zero, they had one set of symbols for the place
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.
27.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and 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 as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
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.
28.
Taxation
–
A tax is a financial charge or other levy imposed upon a taxpayer by a state or the functional equivalent of a state to fund various public expenditures. A failure to pay, or evasion of or resistance to taxation, is punishable by law. Taxes consist of direct or indirect taxes and may be paid in money or as its labour equivalent, the legal definition and the economic definition of taxes differ in that economists do not regard many transfers to governments as taxes. For example, some transfers to the sector are comparable to prices. Examples include tuition at public universities and fees for utilities provided by local governments, governments also obtain resources by creating money and coins, through voluntary gifts, by imposing penalties, by borrowing, and by confiscating wealth. In modern taxation systems, governments levy taxes in money, but in-kind and corvée taxation are characteristic of traditional or pre-capitalist states, the method of taxation and the government expenditure of taxes raised is often highly debated in politics and economics. Tax collection is performed by a government agency such as the Canada Revenue Agency, when taxes are not fully paid, the state may impose civil penalties or criminal penalties on the non-paying entity or individual. The levying of taxes aims to raise revenue to fund governing and/or to alter prices in order to affect demand, States and their functional equivalents throughout history have used money provided by taxation to carry out many functions. A governments ability to raise taxes is called its fiscal capacity, when expenditures exceed tax revenue, a government accumulates debt. A portion of taxes may be used to service past debts, governments also use taxes to fund welfare and public services. These services can include education systems, pensions for the elderly, unemployment benefits, energy, water and waste management systems are also common public utilities. A tax effectively changes relative prices of products and they have therefore sought to identify the kind of tax system that would minimize this distortion. Governments use different kinds of taxes and vary the tax rates, historically, taxes on the poor supported the nobility, modern social-security systems aim to support the poor, the disabled, or the retired by taxes on those who are still working. A states tax system often reflects its communal values and the values of those in current political power. To create a system of taxation, a state must make choices regarding the distribution of the tax burden—who will pay taxes and how much they will pay—and how the taxes collected will be spent. In democratic nations where the public elects those in charge of establishing or administering the tax system, in countries where the public does not have a significant amount of influence over the system of taxation, that system may reflect more closely the values of those in power. All large businesses incur administrative costs in the process of delivering revenue collected from customers to the suppliers of the goods or services being purchased. Taxation is no different, the resource collected from the public through taxation is always greater than the amount which can be used by the government, the difference is called the compliance cost and includes the labour cost and other expenses incurred in complying with tax laws and rules
Taxation
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Pieter Brueghel the Younger, The tax collector's office, 1640
Taxation
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Taxation
Taxation
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Egyptian peasants seized for non-payment of taxes. (Pyramid Age)
29.
Astronomy
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Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry, in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, galaxies, and comets, while the phenomena include supernovae explosions, gamma ray bursts, more generally, all astronomical phenomena that originate outside Earths atmosphere are within the purview of astronomy. A related but distinct subject, 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 the Babylonians, Greeks, Indians, Egyptians, Nubians, Iranians, Chinese, during the 20th century, the field of professional astronomy split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain the results and observations being used to confirm theoretical results. Astronomy is one of the few sciences where amateurs can play an active role, especially in the discovery. Amateur astronomers have made and contributed to many important astronomical discoveries, Astronomy means law of the stars. Astronomy should not be confused with astrology, the system which claims that human affairs are correlated with the positions of celestial objects. Although the two share a common origin, they are now entirely distinct. Generally, either the term astronomy or astrophysics may be used to refer to this subject, however, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Few fields, such as astrometry, are purely astronomy rather than also astrophysics, some titles of the leading scientific journals in this field includeThe Astronomical Journal, The Astrophysical Journal and Astronomy and Astrophysics. In early times, astronomy only comprised the observation and predictions of the motions of objects visible to the naked eye, in some locations, early cultures assembled massive artifacts that possibly had some astronomical purpose. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye, most of early astronomy actually consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon, the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the model of the Universe, or the Ptolemaic system. The Babylonians discovered that lunar eclipses recurred in a cycle known as a saros
Astronomy
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A star -forming region in the Large Magellanic Cloud, an irregular galaxy.
Astronomy
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A giant Hubble mosaic of the Crab Nebula, a supernova remnant
Astronomy
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19th century Sydney Observatory, Australia (1873)
Astronomy
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19th century Quito Astronomical Observatory is located 12 minutes south of the Equator in Quito, Ecuador.
30.
Painting
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Painting is the practice of applying paint, pigment, color or other medium to a solid surface. The medium is commonly applied to the base with a brush, but other implements, such as knives, sponges, Painting is a mode of creative expression, and the forms are numerous. Drawing, gesture, composition, narration, or abstraction, among other aesthetic modes, may serve to manifest the expressive, Paintings can be naturalistic and representational, photographic, abstract, narrative, symbolistic, emotive, or political in nature. A portion of the history of painting in both Eastern and Western art is dominated by motifs and ideas. In art, the term painting describes both the act and the result of the action, the term painting is also used outside of art as a common trade among craftsmen and builders. What enables painting is the perception and representation of intensity, every point in space has different intensity, which can be represented in painting by black and white and all the gray shades between. In practice, painters can articulate shapes by juxtaposing surfaces of different intensity, thus, the basic means of painting are distinct from ideological means, such as geometrical figures, various points of view and organization, and symbols. In technical drawing, thickness of line is ideal, demarcating ideal outlines of an object within a perceptual frame different from the one used by painters. Color and tone are the essence of painting as pitch and rhythm are the essence of music, color is highly subjective, but has observable psychological effects, although these can differ from one culture to the next. Black is associated with mourning in the West, but in the East, some painters, theoreticians, writers and scientists, including Goethe, Kandinsky, and Newton, have written their own color theory. Moreover, the use of language is only an abstraction for a color equivalent, the word red, for example, can cover a wide range of variations from the pure red of the visible spectrum of light. There is not a register of different colors in the way that there is agreement on different notes in music. For a painter, color is not simply divided into basic, painters deal practically with pigments, so blue for a painter can be any of the blues, phthalocyanine blue, Prussian blue, indigo, cobalt, ultramarine, and so on. Psychological and symbolical meanings of color are not, strictly speaking, colors only add to the potential, derived context of meanings, and because of this, the perception of a painting is highly subjective. The analogy with music is quite clear—sound in music is analogous to light in painting, shades to dynamics and these elements do not necessarily form a melody of themselves, rather, they can add different contexts to it. Modern artists have extended the practice of painting considerably to include, as one example, collage, some modern painters incorporate different materials such as sand, cement, straw or wood for their texture. Examples of this are the works of Jean Dubuffet and Anselm Kiefer, there is a growing community of artists who use computers to paint color onto a digital canvas using programs such as Adobe Photoshop, Corel Painter, and many others. These images can be printed onto traditional canvas if required, rhythm is important in painting as it is in music
Painting
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The Mona Lisa, by Leonardo da Vinci, is one of the most recognizable paintings in the world.
Painting
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Chen Hongshou (1598–1652), Leaf album painting (Ming Dynasty)
Painting
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Circus Sideshow (French: Parade de cirque), Georges Seurat, 1887–88
Painting
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Cave painting of aurochs, (French: Bos primigenius primigenius), Lascaux, France, prehistoric art
31.
Weaving
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Weaving is a method of textile production in which two distinct sets of yarns or threads are interlaced at right angles to form a fabric or cloth. Other methods are knitting, felting, and braiding or plaiting, the longitudinal threads are called the warp and 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 loom, a device that holds the warp threads in place while filling threads are woven through them. A fabric band which meets this definition of cloth can also be using other methods, including tablet weaving, back-strap. The way the warp and filling threads interlace with each other is called the weave, the majority of woven products are created with one of three basic weaves, plain weave, satin weave, or twill. Woven cloth can be plain, or can be woven in decorative or artistic design, in general, weaving involves using a loom to interlace two sets of threads at right angles to each other, the warp which runs longitudinally and the weft that crosses it. One warp thread is called an end and one weft thread is called a pick, the warp threads are held taut and in parallel to each other, 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. The warp is divided into two overlapping groups, or lines that run in two planes, one another, so the shuttle can be passed between them in a straight motion. Then, the group is lowered by the loom mechanism. Repeating these actions form a fabric mesh but without beating-up, the 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, the threads of the warp extend in parallel order from the warp-beam to the front of the loom where they are attached to the cloth-roll. Each thread or group of threads of the passes through an opening in a heddle. The warp threads are separated by the heddles into two or more groups, each controlled and automatically drawn up and down by the motion of the heddles, where a complex design is required, the healds are raised by harness cords attached to a Jacquard machine. Every time the harness moves up or down, an opening is made between the threads of warp, through which the pick is inserted, traditionally the weft thread is inserted by a shuttle. On a conventional loom, the thread is carried on a pirn. A handloom weaver could propel the shuttle by throwing it from side to side with the aid of a picking stick, the picking΅ on a power loom is done by rapidly hitting the shuttle from each side using an overpick or underpick mechanism controlled by cams 80-250 times a minute
Weaving
–
Warp and weft in plain weaving
Weaving
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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
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An Indian weaver preparing his warp on a pegged loom (another type of hand loom)
32.
Babylonian mathematics
–
Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited, in respect of time they fall in two distinct groups, one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content there is any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for 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, the majority of recovered clay tablets date from 1800 to 1600 BCE, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC7289 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 second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages. Babylonian mathematics is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, 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, the Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a highly composite number, having factors of 1,2,3,4,5,6,10,12,15,20,30,60. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, the ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises, the earliest traces of the Babylonian numerals also date back to this period. Most clay tablets that describe Babylonian mathematics belong to the Old Babylonian, some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The Babylonians used pre-calculated tables to assist with arithmetic, for example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulae a b =2 − a 2 − b 22 a b =2 −24 to simplify multiplication, the Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a b = a ×1 b together with a table of reciprocals
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...
33.
Subtraction
–
Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign, for example, in the picture on the right, there are 5 −2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. It is anticommutative, meaning that changing the order changes the sign of the answer and 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, starting with the subtraction of integers and generalizing up through the real numbers, 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 system, starting with single digits. Subtraction is written using the minus sign − between the terms, that is, in infix notation, the result is expressed with an equals sign. This is most common in accounting, formally, the number being subtracted is known as the subtrahend, while the number it is subtracted from is the minuend. All of this terminology derives from Latin, subtraction is an English word derived from the Latin verb subtrahere, which is in turn a compound of sub from under and trahere to pull, thus to subtract is to draw from below, take away. Using the gerundive suffix -nd results in subtrahend, thing to be subtracted, likewise from minuere to reduce or diminish, one gets minuend, thing to be diminished. Imagine a line segment of length b with the left end labeled a, starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition, a + b = c, from c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction, c − b = a, now, a line segment labeled with the numbers 1,2, and 3. From position 3, it takes no steps to the left to stay at 3 and it takes 2 steps to the left to get to position 1, so 3 −2 =1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3, to represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number, from 3, it takes 3 steps to the left to get to 0, so 3 −3 =0. But 3 −4 is still invalid since it leaves the line
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
34.
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 product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. 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, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
Multiplication
–
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).
35.
Writing
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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 complement to speech or spoken language, Writing is not a language but a form of technology that developed as tools developed with human society. Within a language system, writing relies on many of the structures as speech, such as vocabulary, grammar and semantics. The result of writing is called text, and the recipient of text is called a reader. Motivations for writing include publication, storytelling, correspondence and diary, Writing has been instrumental in keeping history, 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, codifying laws, in both ancient Egypt and 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, laws and it made the growth of states larger than the old city states possible. It made a continuous historical consciousness possible, the command of the priest or king and 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, alphabetic, another category, ideographic, has never been developed sufficiently to represent language. A sixth category, pictographic, is insufficient to represent language on its own, a logogram is a written character which represents a word or morpheme. A vast number of logograms are needed to write Chinese characters, cuneiform, and Mayan, where a glyph may stand for a morpheme, many logograms have an ideographic component. For example, in Mayan, the glyph for fin, pronounced ka, was used to represent the syllable ka whenever the pronunciation of a logogram needed to be indicated. In Chinese, about 90% of characters are compounds of an element called a radical with an existing character to indicate the pronunciation. However, such phonetic elements complement the elements, rather than vice versa. A syllabary is a set of symbols that represent syllables. A glyph in a syllabary typically represents a consonant followed by a vowel, or just a vowel alone, phonetically related syllables are not so indicated in the script. For instance, the syllable ka may look nothing like the syllable ki, syllabaries are best suited to languages with a relatively simple syllable structure, such as Japanese. Other languages that use syllabic writing include the Linear B script for Mycenaean Greek, Cherokee, Ndjuka, an English-based creole language of Surinam, most logographic systems have a strong syllabic component
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
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Narmer Palette, with the two serpopards representing unification of Upper and Lower Egypt, 3000 B. C.
36.
Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, 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 notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the 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. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
Numeral system
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Numeral systems
37.
Ancient Egypt
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Ancient Egypt was a civilization of ancient Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. It is one of six civilizations to arise independently, Egyptian civilization followed prehistoric Egypt and coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh Narmer. In the aftermath of Alexander the Greats death, one of his generals, Ptolemy Soter and this Greek Ptolemaic Kingdom ruled Egypt until 30 BC, when, under Cleopatra, it fell to the Roman Empire and became a Roman province. The success of ancient 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 a more dense population, and social development and culture. Its art and architecture were widely copied, and its antiquities carried off to far corners of the world and its monumental ruins have inspired the imaginations of travelers and writers for centuries. The Nile has been the lifeline of its region for much of human history, nomadic modern human hunter-gatherers began living in the Nile valley through the end of the Middle Pleistocene some 120,000 years ago. By the late Paleolithic period, the climate of Northern Africa became increasingly hot and dry. In Predynastic and Early Dynastic times, the Egyptian climate was less arid than it is today. Large regions of Egypt were covered in treed savanna and traversed by herds of grazing ungulates, foliage and fauna were far more prolific in all environs and the Nile region supported large populations of waterfowl. Hunting would have been common for Egyptians, and this is also the period when many animals were first domesticated. The largest of these cultures in upper Egypt was the Badari, which probably originated in the Western Desert, it was known for its high quality ceramics, stone tools. The Badari was followed by the Amratian and Gerzeh cultures, which brought a number of technological improvements, as early as the Naqada I Period, predynastic Egyptians imported obsidian from Ethiopia, used to shape blades and other objects from flakes. In Naqada II times, early evidence exists of contact with the Near East, particularly Canaan, establishing a power center at Hierakonpolis, and later at Abydos, Naqada III leaders expanded their control of Egypt northwards along the Nile. They also traded with Nubia to the south, the oases of the desert to the west. Royal Nubian burials at Qustul produced artifacts bearing the oldest-known examples of Egyptian dynastic symbols, such as the crown of Egypt. They also developed a ceramic glaze known as faience, which was used well into the Roman Period to decorate cups, amulets, and figurines. During the last predynastic phase, the Naqada culture began using written symbols that eventually were developed into a system of hieroglyphs for writing the ancient Egyptian language. The Early Dynastic Period was approximately contemporary to the early Sumerian-Akkadian civilisation of Mesopotamia, the third-century BC Egyptian priest Manetho grouped the long line of pharaohs from Menes to his own time into 30 dynasties, a system still used today
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.
38.
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, during the Middle Kingdom period, Osiris became the most important deity in popular religion. The period comprises two phases, the 11th Dynasty, which ruled from Thebes and the 12th Dynasty onwards which was centered on el-Lisht, after the collapse of the Old Kingdom, Egypt entered a period of weak Pharaonic power and decentralization called the First Intermediate Period. Towards the end of period, two rival dynasties, known in Egyptology as the Tenth and Eleventh, fought for power over the entire country. The Theban 11th Dynasty only ruled southern Egypt from the first cataract to the Tenth Nome of Upper Egypt, to the north, Lower Egypt was ruled by the rival 10th Dynasty from Herakleopolis. The struggle was to be concluded by Mentuhotep II, who ascended the Theban throne in 2055 B. C, during Mentuhotep IIs fourteenth regnal year, he took advantage of a revolt in the Thinite Nome to launch an attack on Herakleopolis, which met little resistance. After toppling the last rulers of the 10th Dynasty, Mentuhotep began consolidating his power over all Egypt, for this reason, Mentuhotep II is regarded as the founder of the Middle Kingdom. Mentuhotep II commanded military campaigns south as far as the Second Cataract in Nubia and he also restored Egyptian hegemony over the Sinai region, which had been lost to Egypt since the end of the Old Kingdom. He also sent the first expedition to Punt during the Middle Kingdom, by means of ships constructed at the end of Wadi Hammamat, Mentuhotep III was succeeded by Mentuhotep IV, whose name significantly is omitted from all ancient Egyptian king lists. The Turin Papyrus claims that after Mentuhotep III came seven kingless years, despite this absence, his reign is attested from a few inscriptions in Wadi Hammamat that record expeditions to the Red Sea coast and to quarry stone for the royal monuments. The leader of expedition was his vizier Amenemhat, who is widely assumed to be the future pharaoh Amenemhet I. Mentuhotep IVs absence from the king lists has prompted the theory that Amenemhet I usurped his throne, while there are no contemporary accounts of this struggle, certain circumstantial evidence may point to the existence of a civil war at the end of the 11th dynasty. Inscriptions left by one Nehry, the Haty-a of Hermopolis, suggest that he was attacked at a place called Shedyet-sha by the forces of the reigning king, but his forces prevailed. Khnumhotep I, an official under Amenemhet I, claims to have participated in a flotilla of 20 ships to pacify Upper Egypt, donald Redford has suggested these events should be interpreted as evidence of open war between two dynastic claimants. What is certain is that, however he came to power, from the 12th dynasty onwards, pharaohs often kept well-trained standing armies, which included Nubian contingents. These formed the basis of larger forces which were raised for defence against invasion, however, the Middle Kingdom was basically defensive in its military strategy, with fortifications built at the First Cataract of the Nile, in the Delta and across the Sinai Isthmus. Early in his reign, Amenemhet I was compelled to campaign in the Delta region, in addition, he strengthened defenses between Egypt and Asia, building the Walls of the Ruler in the East Delta region. Perhaps in response to this perpetual unrest, Amenemhat I built a new capital for Egypt in the north, known as Amenemhet Itj Tawy, or Amenemhet, the location of this capital is unknown, but is presumably near the citys necropolis, the present-day el-Lisht
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
39.
Ancient Greeks
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Ancient Greece was a civilization belonging to a period of Greek history from the Greek Dark Ages of the 12th-9th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and this was followed by the period of Classical Greece, an era that began with the Greco-Persian Wars, lasting from the 5th to 4th centuries BC. Due to the conquests by Alexander the Great of Macedonia, Hellenistic civilization flourished from Central Asia to the end of the Mediterranean Sea. Classical Greek culture, especially philosophy, had a influence on ancient Rome. For this reason Classical Greece is generally considered to be the culture which provided the foundation of modern Western culture and is considered the cradle of Western civilization. Classical Antiquity in the Mediterranean region is considered to have begun in the 8th century BC. Classical Antiquity in Greece is preceded by the Greek Dark Ages and this period is succeeded, around the 8th century BC, by the Orientalizing Period during which a strong influence of Syro-Hittite, Jewish, Assyrian, Phoenician and Egyptian cultures becomes apparent. The end of the Dark Ages is also dated to 776 BC. The Archaic period gives way to the Classical period around 500 BC, Ancient Periods Astronomical year numbering Dates are approximate, consult particular article for details The history of Greece during Classical Antiquity may be subdivided into five major periods. The earliest of these is the Archaic period, in which artists made larger free-standing sculptures in stiff, 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. It was followed by the Classical period, characterized by a style which was considered by observers to be exemplary, i. e. classical, as shown in the Parthenon. 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, most of these authors were either Athenian or pro-Athenian, which is why far more is known about the history and politics of Athens than those of many other cities. Their scope is limited by a focus on political, military and diplomatic history, ignoring economic. In the 8th century BC, Greece began to emerge from the Dark Ages which followed the fall of the Mycenaean civilization, literacy had been lost and Mycenaean script forgotten, but the Greeks adopted the Phoenician alphabet, modifying it to create the Greek alphabet. The Lelantine War is the earliest documented war of the ancient Greek period and it was fought between the important poleis of Chalcis and Eretria over the fertile Lelantine plain of Euboea. Both cities seem to have suffered a decline as result of the long war, a mercantile class arose in the first half of the 7th century BC, shown by the introduction of coinage in about 680 BC
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
40.
Muhammad ibn Musa al-Khwarizmi
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Muḥammad ibn Mūsā al-Khwārizmī, formerly Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwārizmīs The Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and he is often considered one of the fathers of algebra. He revised Ptolemys Geography and wrote on astronomy and astrology, some words reflect the importance of al-Khwārizmīs 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, few details of al-Khwārizmīs 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 epithet al-Qutrubbulli could indicate he might instead have come from Qutrubbul and this would not be worth mentioning if a series of errors concerning the personality of al-Khwārizmī, occasionally even the origins of his knowledge, had not been made. Recently, G. J. Toomer. with naive confidence constructed an entire fantasy on the error which cannot be denied the merit of amusing the reader. Regarding al-Khwārizmīs religion, Toomer writes, Another epithet given to him by al-Ṭabarī, al-Majūsī, Ibn al-Nadīms 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, douglas Morton Dunlop suggests that it may have been possible that Muḥammad ibn Mūsā al-Khwārizmī was in fact the same person as Muḥammad ibn Mūsā ibn Shākir, the eldest of the three Banū Mūsā. Al-Khwārizmīs contributions to mathematics, geography, astronomy, and cartography established the basis for innovation in algebra, on the Calculation with Hindu Numerals written about 825, was principally responsible for spreading the Hindu–Arabic numeral system throughout the Middle East and Europe. It was translated into Latin as Algoritmi de numero Indorum, al-Khwārizmī, rendered as Algoritmi, led to the term algorithm. Some of his work was based on Persian and Babylonian astronomy, Indian numbers, al-Khwārizmī systematized and corrected Ptolemys data for Africa and the Middle East. Another major book was Kitab surat al-ard, presenting the coordinates of places based on those in the Geography of Ptolemy but with improved values for the Mediterranean Sea, Asia and he also wrote on mechanical devices like the astrolabe and sundial. He assisted a project to determine the circumference of the Earth and in making a map for al-Mamun. When, in the 12th century, his works spread to Europe through Latin translations, the Compendious Book on Calculation by Completion and Balancing is a mathematical book written approximately 830 CE. The term algebra is derived from the name of one of the operations with equations described in this book
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.
41.
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 engineering and medicine, may also be considered to be applied sciences. However, during the Islamic Golden Age foundations for the method were laid by Ibn al-Haytham in his Book of Optics. In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws, over the course of the 19th century, the word science became increasingly associated with the scientific method itself as a disciplined way to study the natural world. It was during this time that scientific disciplines such as biology, chemistry, Science in a broad sense existed before the modern era and in many historical civilizations. Modern science is distinct in its approach and successful in its results, 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 each other, for example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thought. This is shown by the construction of calendars, techniques for making poisonous plants edible. For this reason, it is claimed these men were the first philosophers in the strict sense and they were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen by scientists as a more appropriate interest for lower class artisans. A clear-cut distinction between formal and empirical science was made by the pre-Socratic philosopher Parmenides, although his work Peri Physeos is a poem, it may be viewed as an epistemological essay on method in natural science. Parmenides ἐὸν may refer to a system or calculus which can describe nature more precisely than natural languages. Physis may be identical to ἐὸν and he criticized the older type of study of physics as too purely speculative and lacking in self-criticism. He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of things had been the realm of mythology and tradition, however. Aristotle later created a less controversial systematic programme of Socratic philosophy which was teleological and he rejected many of the conclusions of earlier scientists. For example, in his physics, the sun goes around the earth, each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, while the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being, they did not argue for any other types of applied science
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)
42.
Pythagoreanism
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Later revivals of Pythagorean doctrines led to what is now called Neopythagoreanism or Neoplatonism. Pythagorean ideas exercised an influence on Aristotle, and Plato. According to tradition, pythagoreanism developed at some point into two schools of thought, the mathēmatikoi and the akousmatikoi. There is the inner and outer circle John Burnet noted Lastly, we have one admitted instance of a philosophic guild, that of the Pythagoreans. And it will be found that the hypothesis, if it is to be called by that name, of a regular organisation of scientific activity will alone explain all the facts. The development of doctrine in the hands of Thales, Anaximander, according to Iamblichus in The life of Pythagoras, by Thomas Taylor There were also two forms of philosophy, for the two genera of those that pursued it, the Acusmatici and the Mathematici. The latter are acknowledged to be Pythagoreans by the rest but the Mathematici do not admit that the Acusmatici derived their instructions from Pythagoras, memory was the most valued faculty. All these auditions were of three kinds, some signifying what a thing is, others what it especially is, others what ought or ought not to be done. By musical sounds alone unaccompanied with words they healed the passions of the soul and certain diseases, enchanting in reality and it is probable that from hence this name epode, i. e. enchantment, came to be generally used. Each of these he corrected through the rule of virtue, attempering them through appropriate melodies, therefore its function is none of what are called ‘parts of virtue’, for it is better than all of them and the end produced is always better than the knowledge that produces it. Nor is every virtue of the soul in that way a function, nor is success, for if it is to be productive, different ones will produce different things, as the skill of building produces a house. However, intelligence is a part of virtue and of success, according to historians like Thomas Gale, Thomas Taler, or 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 historians from the Stanford Encyclopedia of Philosophy, Philolaus and Eurytus are identified by Aristoxenus as teachers of the last generation of Pythagoreans, a Echecrates is mentioned by Aristoxenus as a student of Philolaus and Eurytus. The mathēmatikoi were supposed to have extended and developed the more mathematical, the mathēmatikoi did think that the akousmatikoi were Pythagorean, but felt that their own group was more representative of Pythagoras. Commentary from Sir William Smith, Dictionary of Greek and Roman Biography, Aristotle states the fundamental maxim of the Pythagoreans in various forms. According to Philolaus, number is the dominant and self-produced bond of the continuance of things. But number has two forms, the even and the odd, and a third, resulting from the mixture of the two, the even-odd and this third species is one itself, for it is both even and odd
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)
43.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and 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, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and 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-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
Physics
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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
44.
Group theory
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In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra, linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is central to public key cryptography. The first class of groups to undergo a systematic study was permutation groups, given any set X and a collection G of bijections of X into itself that is closed under compositions and inverses, G is a group acting on X. If X consists of n elements and G consists of all permutations, G is the symmetric group Sn, in general, an early construction due to Cayley exhibited any group as a permutation 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. For example, in this way one proves that for n ≥5 and this fact plays a key role in the impossibility of solving a general algebraic equation of degree n ≥5 in radicals. The next important class of groups is given by matrix groups, here G is a set consisting of invertible matrices of given order n over a field K that is closed under the products and inverses. Such a group acts on the vector space Kn by linear transformations. In the case of groups, X is a set, for matrix groups. The concept of a group is closely related with the concept of a symmetry group. The theory of groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, the groups themselves may be discrete or continuous. Most groups considered in the first stage of the development of group theory were concrete, having been realized through numbers, permutations, or matrices. It was not until the nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold. A typical way of specifying an abstract group is through a presentation by generators and relations, a significant source of abstract groups is given by the construction of a factor group, or quotient group, G/H, of a group G by a normal subgroup H. Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory
Group theory
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Water molecule with symmetry axis
Group theory
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The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups.
45.
Bertrand Russell
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Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life he considered himself a liberal, a socialist, and a pacifist and he was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British revolt against idealism and he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th centurys 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, he advocated preventive nuclear war, before the opportunity provided by the monopoly is gone. He went to prison for his pacifism during World War I, in 1950 Russell was awarded the Nobel Prize in Literature in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire and his parents, Viscount and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wifes affair with their childrens tutor, both were early advocates of birth control at a time when this was considered scandalous. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russells secular godfather, Mill died the year after Russells birth, but his writings had a great effect on Russells life. His paternal grandfather, the Earl Russell, had asked twice by Queen Victoria to form a government. The Russells had been prominent in England for several centuries before this, coming to power, Lady Amberley was the daughter of Lord and Lady Stanley of Alderley. Russell often feared the ridicule of his grandmother, one of the campaigners for education of women. Russell had two siblings, brother Frank, and sister Rachel, in June 1874 Russells mother died of diphtheria, followed shortly by Rachels death. In January 1876, his father died of bronchitis following a period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents and his grandfather, former Prime Minister Earl Russell, died in 1878, and was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russells childhood, the countess was from a Scottish Presbyterian family, and successfully petitioned the Court of Chancery to set aside a provision in Amberleys will requiring the children to be raised as agnostics. Her favourite Bible verse, Thou shalt not follow a multitude to do evil, the atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, and formality, Frank reacted to this with open rebellion, but the young Bertrand learned to hide his feelings
Bertrand Russell
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Russell as a four year-old
Bertrand Russell
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Bertrand Russell
Bertrand Russell
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Childhood home, Pembroke Lodge
Bertrand Russell
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Russell in 1907
46.
Scholasticism
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It originated as an outgrowth of and a departure from Christian monastic schools at the earliest European universities. The Scholastic thought is known for rigorous conceptual analysis and the careful drawing of distinctions. Because of its emphasis on rigorous dialectical method, scholasticism was eventually applied to other fields of study. Some of the figures of scholasticism include Anselm of Canterbury, Peter Abelard, Alexander of Hales, Albertus Magnus, Duns Scotus, William of Ockham, Bonaventure. Important work in the tradition has been carried on well past Aquinass time, for instance by Francisco Suárez and Luis de Molina. The terms scholastic and scholasticism derive from the Latin word scholasticus and the latter from the Greek σχολαστικός, forerunners of Christian scholasticism were Islamic Ilm al-Kalām, literally science of discourse, and Jewish philosophy, especially Jewish Kalam. The first significant renewal of learning in the West came with the Carolingian Renaissance of the Early Middle Ages, charlemagne, advised by Peter of Pisa and Alcuin of York, attracted the scholars of England and Ireland. By decree in AD787, he established schools in every abbey in his empire and these schools, from which the name scholasticism is derived, became centers of medieval learning. During this period, knowledge of Ancient Greek had vanished in the west except in Ireland, 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 the most significant Irish intellectual of the early monastic period and an outstanding philosopher in terms of originality. He had considerable familiarity with the Greek language and translated works into Latin, affording access to the Cappadocian Fathers. The other three founders of scholasticism were the 11th-century scholars Peter Abelard, Archbishop Lanfranc of Canterbury and Archbishop Anselm of Canterbury and this period saw the beginning of the rediscovery of many Greek works which had been lost to the Latin West. As early as the 10th century, scholars in Spain had begun to gather translated texts and, in the half of that century. After a successful burst of Reconquista in the 12th century, Spain opened even further for Christian scholars, as these Europeans encountered Islamic philosophy, they opened a wealth of Arab knowledge of mathematics and astronomy. Scholars such as Adelard of Bath traveled to Spain and Sicily, translating works on astronomy and mathematics, at the same time, Anselm of Laon systematized the production of the gloss on Scripture, followed by the rise to prominence of dialectic in the work of Abelard. Peter Lombard produced a collection of Sentences, or opinions of the Church Fathers and other authorities The 13th, the early 13th century witnessed the culmination of the recovery of Greek philosophy. Schools of translation grew up in Italy and Sicily, and eventually in the rest of Europe, powerful Norman kings gathered men of knowledge from Italy and other areas into their courts as a sign of their prestige. His work formed the basis of the commentaries that followed
Scholasticism
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14th-century image of a university lecture
Scholasticism
47.
Organon
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The Organon is the standard collection of Aristotles six works on logic. The name Organon was given by Aristotles followers, the Peripatetics and they are as follows, The order of the works is not chronological but 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 by Andronicus of Rhodes around 40 BC. The Categories introduces Aristotles 10-fold classification of that exists, substance, quantity, quality, relation, place, time, situation, condition, action. On Interpretation introduces Aristotles conception of proposition and judgment, and the relations between affirmative, negative, universal, and particular propositions. Aristotle discusses the square of opposition or square of Apuleius in Chapter 7, Chapter 9 deals with the problem of future contingents. The Prior Analytics introduces his syllogistic method, argues for its correctness, the Posterior Analytics deals with demonstration, definition, and scientific knowledge. The Topics treats issues in constructing valid arguments, and inference that is probable and it is in this treatise that Aristotle mentions the Predicables, later discussed by Porphyry and the scholastic logicians. The Sophistical Refutations gives a treatment of logical fallacies, and provides a key link to Aristotles work on rhetoric, the Organon was used in the school founded by Aristotle at the Lyceum, and some parts of the works seem to be a scheme of a lecture on logic. So much so that after Aristotles death, his publishers collected these works, following the collapse of the Western Roman Empire in the fifth century, much of Aristotles work was lost in the Latin West. The Categories and On Interpretation are the only significant logical works that were available in the early Middle Ages and these had been translated into Latin by Boethius. The other logical works were not available in Western Christendom until translated to Latin in the 12th century, however, the original Greek texts had been preserved in the Greek-speaking lands of the Eastern Roman Empire. In the mid-twelfth century, James of Venice translated into Latin the Posterior Analytics from Greek manuscripts found in Constantinople. The books of Aristotle were available in the early Arab Empire, all the major scholastic philosophers wrote commentaries on the Organon. Aquinas, Ockham and Scotus wrote commentaries on On Interpretation, Ockham and Scotus wrote commentaries on the Categories and Sophistical Refutations. Grosseteste wrote a commentary on the Posterior Analytics. During this period, while the logic certainly was based on that of Aristotle, there was a tendency in this period to regard the logical systems of the day to be complete, which in turn no doubt stifled innovation in this area. However Francis Bacon published his Novum Organum as an attack in 1620
Organon
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Aristotelianism
48.
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 vast and eclectic field, composed of branches and subdisciplines. However, despite the broad scope of biology, there are certain unifying concepts within it that consolidate it into single, coherent field. In general, biology recognizes the cell as the unit of life, genes as the basic unit of heredity. It is also understood today that all organisms survive by consuming and transforming energy and by regulating their internal environment to maintain a stable, the term 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, the first German use, Biologie, was in a 1771 translation of Linnaeus work. In 1797, Theodor Georg August Roose used the term in the preface of a book, karl Friedrich Burdach used the term in 1800 in a more restricted sense of the study of human beings from a morphological, physiological and psychological perspective. The science that concerns itself with these objects we will indicate by the biology or the doctrine of life. Although modern biology is a recent development, sciences related to. Natural philosophy was studied as early as the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent, however, the origins of modern biology and its approach to the study of nature are most often traced back to ancient Greece. While the formal study of medicine back to Hippocrates, it was Aristotle who contributed most extensively to the development of biology. Especially important are his History of Animals and other works where he showed naturalist leanings, and later more empirical works that focused on biological causation and the diversity of life. Aristotles successor at the Lyceum, Theophrastus, wrote a series of books on botany that survived as the most important contribution of antiquity to the plant sciences, even into the Middle Ages. Scholars of the medieval Islamic world who wrote on biology included al-Jahiz, Al-Dīnawarī, who wrote on botany, biology began to quickly develop and grow with Anton van Leeuwenhoeks dramatic improvement of the microscope. It was then that scholars discovered spermatozoa, bacteria, infusoria, investigations by Jan Swammerdam led to new interest in entomology and helped to develop the basic techniques of microscopic dissection and staining. Advances in microscopy also had a impact on biological thinking. In the early 19th century, a number of biologists pointed to the importance of the cell. Thanks to the work of Robert Remak and Rudolf Virchow, however, meanwhile, taxonomy and classification became the focus of natural historians
Biology
Biology
Biology
Biology
49.
Chemistry
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Chemistry is a branch of physical science that studies the composition, structure, properties and change of matter. Chemistry is sometimes called the science because it bridges other natural sciences, including physics. For the differences between chemistry and physics see comparison of chemistry and physics, the history of chemistry can be traced to alchemy, which had been practiced for several millennia in various parts of the world. The word chemistry comes from alchemy, which referred to a set of practices that encompassed elements of chemistry, metallurgy, philosophy, astrology, astronomy, mysticism. An alchemist was called a chemist in popular speech, and later the suffix -ry was added to this to describe the art of the chemist as chemistry, the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία and this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, which is in turn derived from the word Chemi or Kimi, which is the ancient name of Egypt in Egyptian. 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, 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, early civilizations, such as the Egyptians Babylonians, Indians amassed practical knowledge concerning the arts of metallurgy, pottery and dyes, but didnt develop a systematic theory. Greek atomism dates back to 440 BC, arising in works by such as Democritus and Epicurus. In 50 BC, the Roman philosopher Lucretius expanded upon the theory in his book De rerum natura, unlike modern concepts of science, Greek atomism was purely philosophical in nature, with little concern for empirical observations and no concern for chemical experiments. Work, particularly the development of distillation, continued in the early Byzantine period with the most famous practitioner being the 4th century Greek-Egyptian Zosimos of Panopolis. He formulated Boyles law, rejected the four elements and proposed a mechanistic alternative of atoms. Before his work, though, many important discoveries had been made, the Scottish chemist Joseph Black and the Dutchman J. B. English scientist John Dalton proposed the theory of atoms, that all substances are composed of indivisible atoms of matter. Davy discovered nine new elements including the alkali metals by extracting them from their oxides with electric current, british William Prout first proposed ordering all the elements by their atomic weight as all atoms had a weight that was an exact multiple of the atomic weight of hydrogen. The inert gases, later called the noble gases were discovered by William Ramsay in collaboration with Lord Rayleigh at the end of the century, thereby filling in the basic structure of the table. Organic chemistry was developed by Justus von Liebig and others, following Friedrich Wöhlers synthesis of urea which proved that organisms were, in theory
Chemistry
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Solutions of substances in reagent bottles, including ammonium hydroxide and nitric acid, illuminated in different colors
Chemistry
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Democritus ' atomist philosophy was later adopted by Epicurus (341–270 BCE).
Chemistry
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Antoine-Laurent de Lavoisier is considered the "Father of Modern Chemistry".
Chemistry
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Laboratory, Institute of Biochemistry, University of Cologne.
50.
Hypothesis
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A hypothesis is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the scientific theories. Even though the hypothesis and theory are often used synonymously. 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 ancient Greek ὑπόθεσις word hupothesis, in Platos Meno, Socrates dissects virtue with a method used by mathematicians, that of investigating from a hypothesis. In this sense, hypothesis refers to an idea or to a convenient mathematical approach that simplifies cumbersome calculations. In common usage in the 21st century, a hypothesis refers to an 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 hypothesis may become part of a theory or occasionally may grow to become a theory itself. Normally, scientific hypotheses have the form of a mathematical model, in entrepreneurial science, a hypothesis is used to formulate provisional ideas within a business setting. The formulated hypothesis is then evaluated where either the hypothesis is proven to be true or false through a verifiability- or falsifiability-oriented Experiment, any useful hypothesis will enable predictions by reasoning. It might predict the outcome of an experiment in a setting or the observation of a phenomenon in nature. The prediction may also invoke statistics and only talk about probabilities, other philosophers of science have rejected the criterion of falsifiability or supplemented it with other criteria, such as verifiability or coherence. The scientific method involves experimentation, to test the ability of some hypothesis to adequately answer the question under investigation. In contrast, unfettered observation is not as likely to raise unexplained issues or open questions in science, a thought experiment might also be used to test the hypothesis as well. In framing a hypothesis, the investigator must not currently know the outcome of a test or that it remains reasonably under continuing investigation, only in such cases does the experiment, test or study potentially increase the probability of showing the truth of a hypothesis
Hypothesis
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Andreas Cellarius hypothesis, demonstrating the planetary motions in eccentric and epicyclical orbits
51.
Imre Lakatos
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Lakatos was born Imre Lipschitz to a Jewish family in Debrecen, Hungary in 1922. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944, in March 1944 the Germans invaded Hungary and Lakatos along with Éva Révész, his then-girlfriend and subsequent wife, formed soon after that event a Marxist resistance group. In May of that year, the group was joined by Éva Izsák, Lakatos, considering that there was a risk that she would be captured and forced to betray them, decided that her duty to the group was to commit suicide. Subsequently, a member of the group took her to Debrecen, during the occupation, Lakatos avoided Nazi persecution of Jews by changing his name to Imre Molnár. His mother and grandmother died in Auschwitz and he changed his surname once again to Lakatos in honor of Géza Lakatos. After the war, from 1947 he worked as an official in the Hungarian ministry of education. He also continued his education with a PhD at Debrecen University awarded in 1948 and he also studied at the Moscow State University under the supervision of Sofya Yanovskaya in 1949. When he returned, however, he found himself on the side of internal arguments within the Hungarian communist party and was imprisoned on charges of revisionism from 1950 to 1953. More of Lakatos activities in Hungary after World War II have recently become known, after his release, Lakatos returned to academic life, doing mathematical research and translating George Pólyas How to Solve It into Hungarian. Still nominally a communist, his views had shifted markedly. After the Soviet Union invaded Hungary in November 1956, Lakatos fled to Vienna and he received a doctorate in philosophy in 1961 from the University of Cambridge, his thesis advisor was R. B. The book Proofs and Refutations, The Logic of Mathematical Discovery, in 1960 he was appointed to a position in the London School of Economics, where he wrote on the philosophy of mathematics and the philosophy of science. The LSE philosophy of science department at that time included Karl Popper, Joseph Agassi and it was Agassi who first introduced Lakatos to Popper under the rubric of his applying a fallibilist methodology of conjectures and refutations to mathematics in his Cambridge PhD thesis. With co-editor Alan Musgrave, he edited the often cited Criticism and the Growth of Knowledge, published in 1970, the 1965 Colloquium included well-known speakers delivering papers in response to Thomas Kuhns The Structure of Scientific Revolutions. Lakatos remained at the London School of Economics until his death in 1974 of a heart attack at the age of just 51. The Lakatos Award was set up by the school in his memory and his last LSE lectures in scientific method in Lent Term 1973 along with parts of his correspondence with his friend and critic Paul Feyerabend have been published in For and Against Method. Lakatos philosophy of mathematics was inspired by both Hegels and Marxs dialectic, by Karl Poppers theory of knowledge, and by the work of mathematician George Pólya. The 1976 book Proofs and Refutations is based on the first three chapters of his four chapter 1961 doctoral thesis Essays in the logic of mathematical discovery
Imre Lakatos
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Imre Lakatos, c. 1960s
52.
Falsificationism
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Falsifiability or refutability of a statement, hypothesis, or theory is the inherent possibility that it can be proven false. A statement is called 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 shown to be false, thus, the term falsifiability is sometimes synonymous to testability. Some statements, such as It will be raining here in one years, are falsifiable in principle. The concern with falsifiability gained attention by way of philosopher of science Karl Poppers scientific epistemology falsificationism, the classical view of the philosophy of science is that it is the goal of science to prove hypotheses like All swans are white or to induce them from observational data. Popper argued that this would require the inference of a rule from a number of individual cases. However, if one finds one single swan that is not white, 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. A key observation of falsificationism is thus that a criterion of demarcation is needed to distinguish those statements that 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 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 —, for example, while all men are mortal is unfalsifiable, it is a logical consequence of the falsifiable theory that every man dies before he reaches the age of 150 years. Similarly, the ancient metaphysical and unfalsifiable idea of the existence of atoms has led to corresponding falsifiable modern theories, Popper invented the notion of metaphysical research programs to name such unfalsifiable ideas. Criticizability, in contrast to falsifiability, and thus rationality, may be comprehensive, though this claim is controversial, even proponents of Poppers philosophy. In work beginning in the 1930s, Popper gave falsifiability a renewed emphasis as a criterion of empirical statements in science, Popper noticed that two types of statements are of particular value to scientists. The first are statements of observations, such as there is a white swan, logicians call these statements singular existential statements, since they assert the existence of some particular thing. They are equivalent to a predicate calculus statement of the form, There exists an x such that x is a swan, the second are statements that categorize all instances of something, such as all swans are white. They are usually parsed in the form, For all x, if x is a swan, Scientific laws are commonly supposed to be of this type
Falsificationism
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Are all swans white?
53.
Intuition (knowledge)
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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 the Latin verb intueri translated as consider or from late middle English word intuit, 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, belief, and meaning in philosophical discussion. In the East intuition is mostly intertwined with religion and spirituality, in Hinduism various attempts have been made to interpret the Vedic and other esoteric texts. However with knowledge by identity which we currently only gives the awareness of human existence can be extended further to outside of ourselves resulting in intuitive knowledge. He finds that this process seems to be a decent, is actual a circle of progress. As a lower faculty is being pushed to take up as much from a way of working. Advaita vedanta takes intuition to be an experience through which one can come in contact with, in Zen Buddhism various techniques have been developed to help develop ones intuitive capability, such as kó-an – the resolving of which leads to states of minor enlightenment. In parts of Zen Buddhism intuition is deemed a state between the Universal mind and ones individual, discriminating mind. In Islam there are scholars with varied interpretation of intuition. While Ibn Sīnā finds the ability of having intuition as a prophetic capacity terms it as a knowledge obtained without intentionally acquiring it and he finds regular knowledge is based on imitation while intuitive knowledge as based on intellectual certitude. In the West, intuition does not appear as a field of study. In his book Republic he tries to define intuition as a capacity of human reason to comprehend the true nature of reality. In his discussion with Meno & Phaedo, he describes intuition as a pre-existing knowledge residing in the soul of eternity, and he provides an example of mathematical truths, and posits that they are not arrived at by reason. He argues that these truths are accessed using an already present in a dormant form. This concept by Plato is also referred to as anamnesis. The study was continued by his followers. In his book Meditations on first philosophy, Descartes refers to an intuition as a pre-existing knowledge gained through rational reasoning or discovering truth through contemplation and this definition is commonly referred to as rational intuition
Intuition (knowledge)
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A phrenological mapping of the brain – phrenology was among the first attempts to correlate mental functions with specific parts of the brain
Intuition (knowledge)
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Papirus Oxyrhynchus, with fragment of Plato's Republic
Intuition (knowledge)
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Girl with a Book by José Ferraz de Almeida Júnior
54.
Experiment
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An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated, experiments vary greatly in goal and scale, but always rely on repeatable procedure and logical analysis of the results. There also exists natural experimental studies, a child may carry out basic experiments to understand gravity, while teams of scientists may take years of systematic investigation to advance their understanding of a phenomenon. Experiments and other types of activities are very important to student learning in the science classroom. Experiments can raise test scores and help a student become more engaged and interested in the material they are learning, experiments can vary from personal and informal natural comparisons, to highly controlled. Uses of experiments vary considerably between the natural and human sciences, experiments typically include controls, which are designed to minimize the effects of variables other than the single independent variable. This increases the reliability of the results, often through a comparison between control measurements and the other measurements, scientific controls are a part of the scientific method. Ideally, all variables in an experiment are controlled and none are uncontrolled, in such an experiment, if all controls work as expected, it is possible to conclude that the experiment works as intended, and that results are due to the effect of the tested variable. In the scientific method, an experiment is a procedure that arbitrates between competing models or hypotheses. Researchers also use experimentation to test existing theories or new hypotheses to support or disprove them, an experiment usually tests a hypothesis, which is an expectation about how a particular process or phenomenon works. However, an experiment may also aim to answer a question, without a specific expectation about what the experiment reveals. If an experiment is conducted, the results usually either support or disprove the hypothesis. According to some philosophies of science, an experiment can never prove a hypothesis, on the other hand, an experiment that provides a counterexample can disprove a theory or hypothesis. An experiment must also control the possible confounding factors—any factors that would mar the accuracy or repeatability of the experiment or the ability to interpret the results, confounding is commonly eliminated through scientific controls and/or, in randomized experiments, through random assignment. In engineering and the sciences, experiments are a primary component of the scientific method. They are used to test theories and hypotheses about how physical processes work under particular conditions, typically, experiments in these fields focus on replication of identical procedures in hopes of producing identical results in each replication. In medicine and the sciences, the prevalence of experimental research varies widely across disciplines. In contrast to norms in the sciences, the focus is typically on the average treatment effect or another test statistic produced by the experiment
Experiment
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Even very young children perform rudimentary experiments to learn about the world and how things work.
Experiment
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Original map by John Snow showing the clusters of cholera cases in the London epidemic of 1854
55.
Liberal arts
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Grammar, logic, and rhetoric were the core liberal arts, while arithmetic, geometry, the theory of music, and astronomy also played a part in education. In modern times, liberal arts education term that can be interpreted in different ways and it can refer to academic subjects such as literature, philosophy, mathematics, and social and physical sciences, or it can also refer to overall studies in a liberal arts degree program. For example, Harvard University offers a Bachelor of Arts degree, for both interpretations, the term generally refers to matters not relating to the professional, vocational, or technical curriculum. The four scientific artes – music, arithmetic, geometry and astronomy – were known from the time of Boethius onwards as the Quadrivium. After the 9th century, the three arts of the humanities – grammar, logic, and rhetoric – were classed as well as the Trivium. It was in that 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 parts of the Trivium. In the Renaissance, the Italian humanists and their Northern counterparts, despite in many respects continuing the traditions of the Middle Ages, the ideal of a liberal arts, or humanistic education grounded in classical languages and literature, persisted until the middle of the twentieth century. Some subsections of the arts are in the trivium—the verbal arts, grammar, logic, and rhetoric, and in the quadrivium—the numerical arts, arithmetic, geometry, music. 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, today, a number of other areas of specialization exist, such as gymnasiums specializing in economics, technology or domestic sciences. In some countries, there is a notion of progymnasium, which is equivalent to beginning classes of the full gymnasium, here, the prefix pro is equivalent to pre. In the United States, liberal arts colleges are schools emphasizing undergraduate study in the liberal arts, in most parts of Europe, liberal arts education is deeply rooted. In Germany, Austria and countries influenced by their education system, the term is not to be mixed up with some modern educational concepts that use a similar wording. Educational institutions that see themselves in that tradition are often a Gymnasium and they aim at providing their pupils with comprehensive education in order to form personality with regard to a pupils own humanity as well as his/her innate intellectual skills. Going back to the tradition of the liberal arts in Europe, education in the above sense was freed from scholastic thinking. In particular, Wilhelm von Humboldt played a key role in that regard, universities encourage students to do so and offer respective opportunities, but do not make such activities part of the universitys curriculum
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
56.
University
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A university is an institution of higher education and research which grants academic degrees in various academic disciplines. Universities typically provide undergraduate education and postgraduate education, the word university is derived from the Latin universitas magistrorum et scholarium, which roughly means community of teachers and scholars. Universities were created in Italy and 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 documentary evidence of this comes from early in the life of the first university, the University of Bologna adopted an academic charter, the Constitutio Habita, in 1158 or 1155, which guaranteed the right of a traveling scholar to unhindered passage in the interests of education. Today this is claimed as the origin of academic freedom and this is now widely recognised internationally - on 18 September 1988,430 university rectors signed the Magna Charta Universitatum, marking the 900th anniversary of Bolognas foundation. The number of universities signing the Magna Charta Universitatum continues to grow, the university is generally regarded as a formal institution that has its origin in the Medieval Christian setting. The earliest universities were developed under the aegis of the Latin Church by papal bull as studia generalia and 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 founded by Kings or municipal administrations. In the early period, most new universities were founded from pre-existing schools. Many historians state that universities and cathedral schools were a continuation of the interest in learning promoted by monasteries, the first universities in Europe with a form of corporate/guild structure were the University of Bologna, the University of Paris, and the University of Oxford. The students had all the power … and dominated the masters, princes and leaders of city governments perceived the potential benefits of having a scholarly expertise develop with the ability to address difficult problems and achieve desired ends. The emergence of humanism was essential to understanding of the possible utility of universities as well as the revival of interest in knowledge gained from ancient Greek texts. The rediscovery of Aristotles works–more than 3000 pages of it would eventually be translated–fuelled a spirit of inquiry into natural processes that had begun to emerge in the 12th century. Some scholars believe that these represented one of the most important document discoveries in Western intellectual history. Richard Dales, for instance, calls the discovery of Aristotles works a turning point in the history of Western thought and this became the primary mission of lecturers, and the expectation of students. The university culture developed differently in northern Europe than it did in the south, Latin was the language of the university, used for all texts, lectures, disputations and examinations. Professors lectured on the books of Aristotle for logic, natural philosophy, and metaphysics, while Hippocrates, Galen, outside of these commonalities, great differences separated north and south, primarily in subject matter
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.
57.
Philosophy of mathematics
–
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics, the logical and structural nature of mathematics itself makes this study both broad and 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 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 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 play a role in mathematics. What are the objectives of mathematical inquiry, what gives mathematics its hold on experience. What are the human traits behind mathematics, what is the source and nature of mathematical truth. What is the relationship between the world of mathematics and the material universe. The origin of mathematics is subject to argument, whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics, there are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Greek philosophy on mathematics was strongly influenced by their study of geometry, for example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude, therefore,3, for example, represented a certain multitude of units, and was thus not truly a number. At another point, an argument was made that 2 was not a number. These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the root of two
Philosophy of mathematics
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David Hilbert
58.
Mathematical beauty
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Mathematical beauty describes the notion that some mathematicians may derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful, Mathematicians describe mathematics as an art form or, at a minimum, as a creative activity. Comparisons are often made music and poetry. The true spirit of delight, the exaltation, the sense of being more than Man, Paul Erdős expressed his views on the ineffability of mathematics when he said, Why are numbers beautiful. Its like asking why is Beethovens Ninth Symphony beautiful, if you dont see why, someone cant tell you. If they arent 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 additional assumptions or previous results, a proof that is unusually succinct. A proof that derives a result in a surprising way A proof that is based on new, a method of proof that can be easily generalized to solve a family of similar problems. In the search for an elegant proof, mathematicians often look for different independent ways to prove a result—the first proof that is found may not be the best. The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, another theorem that has been proved in many different ways is the theorem of quadratic reciprocity—Carl Friedrich Gauss alone published eight different proofs of this theorem. Some mathematicians see beauty in mathematical results that establish connections between two areas of mathematics that at first sight appear to be unrelated and these results are often described as deep. While it is difficult to find agreement on whether a result is deep. One is Eulers identity, e i π +1 =0 and this is a special case of Eulers formula, which the physicist Richard Feynman called our jewel and the most remarkable formula in mathematics. Other examples of deep results include unexpected insights into mathematical structures, for example, Gausss Theorema Egregium is a deep theorem which relates a local phenomenon to a global phenomenon in a surprising way. In particular, the area of a triangle on a surface is proportional to the excess of the triangle. Another example is the theorem of calculus. The opposite of deep is trivial, sometimes, however, a statement of a theorem can be original enough to be considered deep, even though its proof is fairly obvious. In his A Mathematicians Apology, Hardy suggests that a proof or result possesses inevitability, unexpectedness
Mathematical beauty
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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
59.
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, cultural and technological systems that are in operation in any country or internationally, commerce is derived from the Latin commercium, from cum, together, and merx, merchandise. Some commentators trace the origins of commerce to the start of transaction in prehistoric times. Apart from traditional self-sufficiency, trading became a facility of prehistoric people. Historian Peter Watson and Ramesh Manickam dates the history of commerce from circa 150,000 years ago. In historic times, the introduction of currency as a money, facilitated a wider exchange of goods. Numismatists have collections of these tokens, which include coins from some Ancient World large-scale societies. 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. But he/she cannot make an equivalent number of pots to equal this service done for him/her, because if the builder could build the house. During the Middle Ages, commerce developed in Europe by trading luxury goods at trade fairs, wealth became converted into movable wealth or capital. Banking systems developed where money on account was transferred across national boundaries, hand to hand markets became a feature of town life, and were regulated by town authorities. Today commerce includes as a subset a complex system of companies which try to maximize their profits by offering products and services to the market at the lowest production cost
Commerce
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The caduceus has been used today as the symbol of commerce with which Mercury has traditionally been associated.
60.
Architecture
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Architecture is both the process and the product of planning, designing, and constructing buildings and other physical structures. Architectural works, in the form of buildings, are often perceived as cultural symbols. Historical civilizations are often identified with their surviving architectural achievements, Architecture can mean, A general term to describe buildings and other 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. A unifying or coherent form or structure Knowledge of art, science, technology, the design activity of the architect, from the macro-level to the micro-level. The practice of the architect, where architecture means offering or rendering services in connection with the design and construction of buildings. The earliest surviving work on the subject of architecture is De architectura. According to Vitruvius, a building should satisfy the three principles of firmitas, utilitas, venustas, commonly known by the original translation – firmness, commodity. An equivalent in modern English would be, Durability – a building should stand up robustly, utility – it should be suitable for the purposes for which it is used. Beauty – it should be aesthetically pleasing, according to Vitruvius, the architect should strive to fulfill each of these three attributes as well as possible. Leon Battista Alberti, who elaborates on the ideas of Vitruvius in his treatise, De Re Aedificatoria, saw beauty primarily as a matter of proportion, for Alberti, the rules of proportion were those that governed the idealised human figure, the Golden mean. The most important aspect of beauty was, therefore, an inherent part of an object, rather than something applied superficially, Gothic architecture, Pugin believed, was the only true Christian form of architecture. The 19th-century English art critic, John Ruskin, in his Seven Lamps of Architecture, Architecture was the art which so disposes and adorns the edifices raised by men. That the sight of them contributes to his health, power. For Ruskin, the aesthetic was of overriding significance and his work goes on to state that a building is not truly a work of architecture unless it is in some way adorned. For Ruskin, a well-constructed, well-proportioned, functional building needed string courses or rustication, but suddenly you touch my heart, you do me good. I am happy and I say, This is beautiful, le Corbusiers contemporary Ludwig Mies van der Rohe said Architecture starts when you carefully put two bricks together. The notable 19th-century architect of skyscrapers, Louis Sullivan, promoted an overriding precept to architectural design, function came to be seen as encompassing all criteria of the use, perception and enjoyment of a building, not only practical but also aesthetic, psychological and cultural
Architecture
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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
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Section of Brunelleschi 's dome drawn by the architect Cigoli (c. 1600)
Architecture
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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
61.
Physicist
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A physicist is a scientist who has specialized knowledge in the field of physics, the exploration of the interactions of matter and energy across the physical universe. 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, and 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 study and practice of physics is based on a ladder of discoveries. Many mathematical and physical ideas used today found their earliest expression in ancient Greek culture and Asian culture, the bulk of physics education can be said to flow from the scientific revolution in Europe, starting with the work of Galileo and Kepler in the early 1600s. New knowledge in the early 21st century includes an increase in understanding physical 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 Masters degree like MSc, MPhil, MPhys or MSci. In a research oriented level, students tend to specialize in a particular field, Physics students also need training in mathematics, and also in computer science and programming. For being employed as a physicist a doctoral background may be required for certain positions, undergraduate students like BSc Mechanical Engineering, BSc Electrical and Computer Engineering, BSc Applied Physics. etc. With physics orientation are chosen as research assistants with faculty members, 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 academic institutions, laboratories, and 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. As per the American Institute for Physics, some 20% of new physics Ph. D. s holds jobs in engineering development programs, while 14% turn to computer software, a majority of physicists employed apply their skills and training to interdisciplinary sectors. For industry or self-employment. and also in science and programming. Hence a majority of Physics bachelors degree holders are employed in the private sector, other fields are academia, government and military service, nonprofit entities, labs and teaching
Physicist
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Albert Einstein, physicist who developed the theory of general relativity.
62.
Eugene Wigner
–
Eugene Paul E. P. Wigner, was a Hungarian-American theoretical physicist, engineer and mathematician. Wigner and Hermann Weyl were responsible for introducing group theory into physics, along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigners theorem is a cornerstone in the formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus, in 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States. Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first, during the Manhattan Project, he led a team whose task was to design nuclear reactors to convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical, Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. In later life, he became more philosophical, and published The Unreasonable Effectiveness of Mathematics in the Natural Sciences, 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. He had a sister, Bertha, known as Biri, and a younger sister Margit, known as Manci. He was home schooled by a teacher until the age of 9. During this period, Wigner developed an interest in mathematical problems, at the age of 11, Wigner contracted what his doctors believed to be tuberculosis. His parents sent him to live for six weeks in a sanatorium in the Austrian mountains, Wigners family was Jewish, but not religiously observant, and his Bar Mitzvah was a secular one. From 1915 through 1919, he studied at the grammar school called Fasori Evangélikus Gimnázium. Religious education was compulsory, and he attended classes in Judaism taught by a rabbi, a fellow student was János von Neumann, who was a year behind Wigner. They both benefited from the instruction of the mathematics teacher László Rátz. In 1919, to escape the Béla Kun communist regime, the Wigner family briefly fled to Austria, partly as a reaction to the prominence of Jews in the Kun regime, the family converted to Lutheranism. Wigner explained later in his life that his decision to convert to Lutheranism was not at heart a religious decision. On religious views, Wigner was an atheist, after graduating from the secondary school in 1920, Wigner enrolled at the Budapest University of Technical Sciences, known as the Műegyetem. He was not happy with the courses on offer, and in 1921 enrolled at the Technische Hochschule Berlin and he also attended the Wednesday afternoon colloquia of the German Physical Society
Eugene Wigner
–
Eugene Wigner
Eugene Wigner
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Signature
Eugene Wigner
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Werner Heisenberg and Eugene Wigner (1928)
Eugene Wigner
–
Wigner receiving the Medal for Merit for his work on the Manhattan Project from Robert P. Patterson (left), March 5, 1946
63.
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 operational analysis is used in the British military, as an intrinsic part of capability development, management. In particular, operational analysis forms part of the Combined Operational Effectiveness and Investment Appraisals and it is often considered to be a sub-field of applied mathematics. The terms management science and decision science are used as synonyms. Operation research is concerned with determining the maximum or minimum of some real-world objective. Originating in military efforts 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 computational and statistical nature of most of these fields, OR also has ties to computer science. In the decades after the two wars, the techniques were more widely applied to problems in business, industry. Early work in research was carried out by individuals such as Charles Babbage. Percy Bridgman brought operational research to bear on problems in physics in the 1920s, modern operational research originated at the Bawdsey Research Station in the UK in 1937 and was the result of an initiative of the stations superintendent, A. P. Rowe. Rowe conceived the idea as a means to analyse and improve the working of the UKs early warning radar system, initially, he analysed the operating of the radar equipment and its communication networks, expanding later to include the operating personnels behaviour. This revealed unappreciated limitations of the CH network and allowed action to be taken. Scientists in the United Kingdom including Patrick Blackett, Cecil Gordon, Solly Zuckerman, other names for it included operational analysis and quantitative management. During the Second World War close to 1,000 men and women in Britain were engaged in operational research, about 200 operational research scientists worked for the British Army. Patrick Blackett worked for different organizations during the war. In 1941, Blackett moved from the RAE to the Navy, after first working with RAF Coastal Command, in 1941, blacketts team at Coastal Commands Operational Research Section included two future Nobel prize winners and many other people who went on to be pre-eminent in their fields. They undertook a number of analyses that aided the war effort. Convoys travel at the speed of the slowest member, so small convoys can travel faster and it was also argued that small convoys would be harder for German U-boats to detect
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
64.
Computer science
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Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, 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 information theorist, for, among other reasons and 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 card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base
Computer science
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Ada Lovelace is credited with writing the first algorithm intended for processing on a computer.
Computer science
Computer science
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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
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Digital logic
65.
Proof (mathematics)
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
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.
66.
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 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a 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 of forms, 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 by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
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.
67.
A Mathematician's Apology
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A Mathematicians Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some content. In the books title, Hardy uses the word apology in the sense of a justification or defence. Hardy felt the need to justify his lifes work in mathematics at this time mainly for two reasons, firstly, at age 62, Hardy felt the approach of old age and the decline of his mathematical creativity and skills. By devoting time to writing the Apology, Hardy was admitting that his own time as a mathematician was finished. In his foreword to the 1967 edition of the book, C. P. Snow describes the Apology as a lament for creative powers that used to be. In Hardys words, Exposition, criticism, appreciation, is work for second-rate minds and it is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to new theorems, to add to mathematics. Secondly, at the start of the World War II, Hardy, Hardy was an atheist, and makes his justification not to God but to his fellow man. One of the themes of the book is the beauty that mathematics possesses. For Hardy, the most beautiful mathematics was that which had no applications in the outside world and, in particular. He justifies the pursuit of mathematics with the argument that its very uselessness on the whole meant that it could not be misused to cause harm. On the other hand, Hardy denigrates much of the mathematics as either being trivial, ugly, or dull, and contrasts it with real mathematics. Hardy expounds by commenting about a phrase attributed to Carl Friedrich Gauss that Mathematics is the queen of the sciences, if an application of number theory were to be found, then certainly no one would try to dethrone the queen of mathematics because of that. What Gauss meant, according to Hardy, is that the concepts that constitute number theory are deeper. This view reflects Hardys increasing depression at the wane of his own mathematical powers, for Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one. Hardys opinions were influenced by the academic culture of the universities of Cambridge. Some of Hardys examples seem unfortunate in retrospect, for example, he writes, No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years
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
68.
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. These conditions are very loose, and they allow enormous flexibility in the choice of open sets, in the two extremes, every set can be open, or 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 intervals of the real line. The notion of an open set provides a way to speak of nearness of points in a topological space. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of importance in point-set topology. Intuitively, an open set provides a method to distinguish two points, for example, if about one point in a topological space there exists an open set not containing another point, the two points are referred to as topologically distinguishable. In this manner, one may speak of 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 the set of all real numbers, one has the natural Euclidean metric, that is, a function which measures the distance between two real numbers, d = |x - y|. Therefore, given a number, one can speak of the set of all points close to that real number. In essence, points within ε of x approximate x to an accuracy of degree ε, note that ε >0 always but as ε becomes smaller and smaller, one obtains points that approximate x to a higher and higher degree of accuracy. For example, if x =0 and ε =1, the points within ε of x are precisely the points of the interval, that is, however, with ε =0.5, the points within ε of x are precisely the points of. Clearly, these points approximate x to a degree of accuracy compared to when ε =1. The previous discussion shows, for the case x =0, in particular, sets of the form give us 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, thus, we find that in some sense, every real number is distance 0 away from 0. It may help in case to think of the measure as being a binary condition, all things in R are equally close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a neighborhood basis, in fact, one may generalize these notions to an arbitrary set, rather than just the real numbers. In this case, given a point of that set, one may define a collection of sets around x, of course, this collection would have to satisfy certain properties for otherwise we may not have a well-defined method to measure distance
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.
69.
Homeomorphism
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In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word homeomorphism comes from the Greek words ὅμοιος = similar and μορφή = shape, roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. A function f, X → Y between two spaces and is called a homeomorphism if it has the following properties, f is a bijection, f is continuous. A function with 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 a 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 unit 2-disc D2 and the square in R2 are homeomorphic. An example of a mapping from the square to the disc is, in polar coordinates. The graph of a 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, a chart of a manifold is an homeomorphism between an open subset of the manifold and an open subset of a Euclidean space. The stereographic projection is a homeomorphism between the sphere in R3 with a single point removed and the set of all points in R2. If G is a group, its inversion map x ↦ x −1 is a homeomorphism. Also, for any x ∈ G, the left translation y ↦ x y, the right translation y ↦ y x, rm and Rn are not homeomorphic for m ≠ n. The Euclidean real line is not homeomorphic to the circle as a subspace of R2, since the unit circle is compact as a subspace of Euclidean R2. The third requirement, that f −1 be continuous, is essential, consider for instance the function f, [0, 2π) → S1 defined by f =
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.
70.
Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two 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 the antiderivative. In this case, it is called an integral and is written. 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 procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
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.
71.
Abacus
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The exact origin of the abacus is still unknown. Today, abaci are often constructed as a frame with beads sliding on wires. 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 Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank. Alternatively, without reference to ancient texts on etymology, it has suggested that it means a square tablet strewn with dust. Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all, Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq, dust. The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci in use, the user of an abacus is called an abacist. The period 2700–2300 BC saw the first appearance of the Sumerian abacus, 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 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 century 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, pre-set with small counters in wood or metal for mathematical calculations and this Greek abacus saw use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world. A tablet found on the Greek island Salamis in 1846 AD, dates back to 300 BC and it is a slab of white marble 149 cm long,75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. Below these lines is a space with a horizontal crack dividing it. Also from this frame the Darius Vase was unearthed in 1851. It was covered with pictures including a holding a wax tablet in one hand while manipulating counters on a table with the other. The earliest known documentation of the Chinese abacus dates to the 2nd century BC. The Chinese abacus, known as the suanpan, is typically 20 cm tall and it usually has more than seven rods. There are two beads on each rod in the deck and five beads each in the bottom for both decimal and hexadecimal computation
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
72.
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, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, 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 Cavalieris 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 Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of 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. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
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.
73.
Uncertainty
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Uncertainty is a situation which involves imperfect and/or unknown information. However, uncertainty is an expression without a straightforward description. It applies to predictions of events, to physical measurements that are already made. 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 effect or significant loss. Measurement of risk A set of measured uncertainties where some possible outcomes are losses, and the magnitudes of those losses – this also includes loss functions over continuous variables. It will appear that a measurable uncertainty, or risk proper, 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, if there is a major, costly, outdoor event planned for tomorrow then there is a risk since there is a 10% chance of rain, and rain would be undesirable. Furthermore, if this is an event and $100,000 would be lost if it rains. These situations can be even more realistic by quantifying light rain vs. heavy rain. 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 and 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 insurance company, for example, would compute an EOL as a minimum for any insurance coverage, then add onto that other operating costs and profit. Since many people are willing to buy insurance for many reasons, quantitative uses of the terms uncertainty and risk are fairly consistent from fields such as probability theory, actuarial science, and information theory. Some also create new terms without substantially changing the definitions of uncertainty or risk, for example, surprisal is a variation on uncertainty sometimes used in information theory. But outside of the more mathematical uses of the term, usage may vary widely, in cognitive psychology, uncertainty can be real, or just a matter of perception, such as expectations, threats, etc. Vagueness or ambiguity are sometimes described as second order uncertainty, where there is uncertainty even about the definitions of uncertain states or outcomes, the difference here is that this uncertainty is about the human definitions and concepts, not an objective fact of nature. It is usually modelled by some variation on Zadehs fuzzy logic and it has been argued that ambiguity, however, is always avoidable while uncertainty is not necessarily avoidable. Uncertainty may be purely a consequence of a lack of knowledge of obtainable facts and that is, there may be uncertainty about whether a new rocket design will work, but this uncertainty can be removed with further analysis and experimentation
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.
74.
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. Riemann surfaces can be thought of as deformed versions of the plane, locally near every point they look like patches of the complex plane. For example, they can look like a sphere or a torus or several sheets glued together, the main point of Riemann surfaces is that holomorphic functions may be defined between them. Every Riemann surface is a real analytic manifold, but it contains more structure which is needed for the unambiguous definition of holomorphic functions. A two-dimensional real manifold can be turned into a Riemann surface if, so the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and projective plane do not. Geometrical facts about Riemann surfaces are as nice as possible, and they provide the intuition and motivation for generalizations to other curves. The Riemann–Roch theorem is an example of this influence. There are several equivalent definitions of a Riemann surface, a Riemann surface X is a complex manifold of complex dimension one. This means that X is a Hausdorff topological space endowed with an atlas, the map carrying the structure of the complex plane to the Riemann surface is called a chart. Additionally, the maps between two overlapping charts are required to be holomorphic. A Riemann surface is a manifold of dimension two – a two-sided surface – together with a conformal structure. Again, manifold means that locally at any point x of X, the supplement Riemann signifies that X is endowed with an additional structure which allows angle measurement on the manifold, namely an equivalence class of so-called Riemannian metrics. Two such metrics are considered equivalent if the angles they measure are the same, choosing an equivalence class of metrics on X is the additional datum of the conformal structure. A complex structure gives rise to a structure by choosing the standard Euclidean metric given on the complex plane. Showing that a structure determines a complex structure is more difficult. The complex plane C is the most basic Riemann surface, the map f = z defines a chart for C, and is an atlas for C. The map g = z* also defines a chart on C and is an atlas for C, the charts f and g are not compatible, so this endows C with two distinct Riemann surface structures. In fact, given a Riemann surface X and its atlas A, the conjugate atlas B = is never compatible with A, in an analogous fashion, every non-empty open subset of the complex plane can be viewed as a Riemann surface in a natural way
Riemann surface
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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.
75.
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 two properties, the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, several terms used in category theory, including the term morphism, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself, Category theory has practical applications in programming language theory, in particular for the study of monads in functional programming. Categories represent abstraction of other mathematical concepts, many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate, a basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The arrows of category theory are 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 more abstract concepts are represented by objects. The most important property of the arrows is that they can be composed, in other words, linear algebra can also be expressed in terms of categories of matrices. A systematic study of category theory allows us to prove general results about any of these types of mathematical structures from the axioms of a category. The class Grp of groups consists of all objects having a group structure, one can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique, in the case of groups, the morphisms are the group homomorphisms. The study of group homomorphisms then provides a tool for studying properties of groups. Not all categories arise as structure preserving functions, however, the example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories, a category is itself a type of mathematical structure, so we can look for processes which preserve this structure in some sense, such a process is called a functor. Diagram chasing is a method of arguing with abstract arrows joined in diagrams. Functors are represented by arrows between categories, subject to specific defining commutativity conditions, functors can define categorical diagrams and sequences
Category theory
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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.)
76.
Theoretical computer science
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It is not easy to circumscribe the theoretical areas precisely. Work in this field is often distinguished by its emphasis on mathematical technique, despite this broad scope, the theory people in computer science 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 theory and application. This means that the theory people regularly use experimental science done in less-theoretical areas such as software system research. It also means there is more cooperation than mutually exclusive competition between theory and application. These developments have led to the study of logic and computability. Information theory was added to the field with a 1948 mathematical theory of communication 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 neural networks, in 1971, Stephen Cook and, working independently, Leonid Levin, proved that there exist practically relevant problems that are NP-complete – a landmark result in computational complexity theory. With the development of mechanics in the beginning of the 20th century came the concept that mathematical operations could be performed on an entire particle wavefunction. In other words, one could compute functions on multiple states simultaneously, modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed. An algorithm is a procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. The transition from one state to the next is not necessarily deterministic, some algorithms, known as randomized algorithms, a data structure is a particular way of organizing data in a computer so that it can be used efficiently. Different kinds of structures are suited to different kinds of applications. For example, databases use B-tree indexes for small percentages of data retrieval and compilers, data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms, some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Storing and retrieving can be carried out on data stored in main memory and in secondary memory. A problem is regarded as inherently difficult if its solution requires significant resources, the theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage
Theoretical computer science
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An artistic representation of a Turing machine. Turing machines are used to model general computing devices.
77.
Turing machine
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Despite the models simplicity, given any computer algorithm, a Turing machine can be constructed that is capable of simulating that algorithms logic. The machine operates on an infinite memory tape divided into discrete cells, the machine positions its head over a cell and reads the symbol there. The Turing machine was invented in 1936 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, a Turing machine is a general example of a CPU that controls all data manipulation done by a computer, with the canonical machine using sequential memory to store data. More specifically, it is a capable of enumerating some arbitrary subset of valid strings of an alphabet. 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 and this is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing. The Turing machine is capable of processing an unrestricted grammar, which implies that it is capable of robustly evaluating first-order logic in an infinite number of ways. This is famously demonstrated through lambda calculus, a Turing machine that is able to simulate any other Turing machine is called a universal Turing machine. The thesis states that Turing machines indeed capture the notion of effective methods in logic and mathematics. Studying their abstract properties yields many insights into computer science and complexity theory, at any moment there is one symbol in the machine, it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, however, the tape can be moved back and 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. On this tape are symbols, which the machine can read and write, one at a time, in the original article, Turing imagines not a mechanism, but a person whom he calls the computer, who executes these deterministic mechanical rules slavishly. If δ is not defined on the current state and the current tape symbol, Q0 ∈ Q is the initial state F ⊆ Q is the set of final or accepting states. The initial tape contents is said to be accepted by M if it eventually halts in a state from F, Anything that operates according to these specifications is a Turing machine. The 7-tuple for the 3-state busy beaver looks like this, Q = Γ = b =0 Σ = q 0 = A F = δ = see state-table below Initially all tape cells are marked with 0. In the words of van Emde Boas, p.6, The set-theoretical object provides only partial information on how the machine will behave and what its computations will look like. For instance, There will need to be many decisions on what the symbols actually look like, and a failproof way of reading and writing symbols indefinitely
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
78.
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 lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy, no information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information, the process of reducing the size of a data file is referred to as data compression. In the context of data transmission, it is called coding in opposition to channel coding. Compression is useful because it reduces resources required to store and transmit data, computational resources are consumed in the compression process and, usually, in the reversal of the process. Data compression is subject to a space–time complexity trade-off, Lossless data 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, for example, an image may have areas of color that do not change over several pixels, instead of coding red pixel, red pixel. The data may be encoded as 279 red pixels and this is a basic example of run-length encoding, there are many schemes to reduce file size by eliminating redundancy. The Lempel–Ziv compression methods are among the most popular algorithms for lossless storage, DEFLATE is a variation on LZ optimized for decompression speed and compression ratio, but compression can be slow. DEFLATE is used in PKZIP, Gzip, and PNG, LZW is used in GIF images. LZ methods use a table-based compression model where table entries are substituted for repeated strings of data, for most LZ methods, this table is generated dynamically from earlier data in the input. The table itself is often Huffman encoded, current LZ-based coding schemes that perform well are Brotli and LZX. LZX is used in Microsofts CAB format, the best modern lossless compressors use probabilistic models, such as prediction by partial matching. The Burrows–Wheeler transform can also be viewed as a form of statistical modelling. The basic task of grammar-based codes is constructing a context-free grammar deriving a single string, sequitur and Re-Pair are practical grammar compression algorithms for which software is publicly available. In a further refinement of the use of probabilistic modelling. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of a machine to produce a string of encoded bits from a series of input data symbols
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
79.
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 cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, 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 for equality, excess or shortage, by striking out a mark, the first 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 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first 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, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
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, …)
80.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z 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 integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. 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 properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
Integer
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Algebraic structure → Group theory Group theory
81.
Fermat's Last Theorem
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In number theory, Fermats Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n =1 and n =2 have been known to have many solutions since antiquity. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, 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. The Pythagorean equation, x2 + y2 = z2, has an number of positive integer solutions for x, y, and z. Around 1637, Fermat wrote in the margin of a book that the general equation an + bn = cn had no solutions in positive integers. Although he claimed to have a proof of his conjecture, Fermat left no details of his proof. His claim was discovered some 30 years later, after his death and this claim, which came to be known as Fermats 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, and over time Fermats Last Theorem gained prominence as an unsolved problem in mathematics. With the special case n =4 proved, it suffices to prove the theorem for n that are prime numbers. Over the next two centuries, the conjecture was proved for only the primes 3,5, and 7, in the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and as the modularity theorem, it stood on its own and 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 Fermats Last Theorem and this potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermats Last Theorem that depended on the modularity theorem. On hearing this, English mathematician Andrew Wiles, who had a fascination with Fermats Last Theorem. In 1993, after six years working secretly on the problem, Wiless paper was massive in size and scope. A flaw was discovered in one part of his paper during peer review and required a further year and collaboration with a past student, Richard Taylor. As a result, the proof in 1995 was accompanied by a second smaller joint paper to that effect
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
82.
Goldbach's conjecture
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Goldbachs 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 has been shown to hold up through 4 ×1018, but remains unproven despite considerable effort. A Goldbach number is an integer that can be expressed as the sum of two odd primes. The expression of an even number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some numbers,6 =3 +38 =3 +510 =3 +7 =5 +512 =7 +5. 100 =3 +97 =11 +89 =17 +83 =29 +71 =41 +59 =47 +53. He then proposed a second conjecture in the margin of his letter and he considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time, a modern version of Goldbachs marginal conjecture is, Every integer greater than 5 can be written as the sum of three primes. In the letter dated 30 June 1742, Euler stated, Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, Goldbachs third version is the form in which the conjecture is usually expressed today. It is also known as the strong, even, or binary Goldbach conjecture, 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. With the advent of computers, many more values of n have been checked, one record from this search is that 3325581707333960528 is the smallest number that has no Goldbach partition with a prime below 9781. A very crude version of the heuristic argument is as follows. The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime. Thus if n is an even integer and m is a number between 3 and n/2, then one might expect the probability of m and n − m simultaneously being prime to be 1 /. Since this quantity goes to infinity as n increases, we expect that every even integer has not just one representation as the sum of two primes, but in fact has very many such representations. This heuristic argument is somewhat inaccurate, because it assumes that the events of m and n − m being prime are statistically independent of each other. For instance, if m is odd then n − m is odd, and if m is even, then n − m is even
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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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is 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 set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
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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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, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, 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 the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
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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Continuous function
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In mathematics, a continuous function is 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 concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, 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 function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
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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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a 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, 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 space or equivalently as the quotient of two vectors. Quaternions are generally represented in the form, a + bi + cj + dk where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units. In practical applications, they can be used other methods, such as Euler angles and rotation matrices, or as an alternative to them. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, 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 and 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. The unit quaternions can be thought of as a choice of a structure on the 3-sphere S3 that gives the group Spin. 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. Hamilton knew that the numbers could be interpreted as points in a plane. Points in space can be represented by their coordinates, which are triples of numbers, 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. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i2 = j2 = k2 = ijk = −1, into the stone of Brougham Bridge as he paused on it. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves and this letter was later published in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. xxv, pp 489–95. In the letter, Hamilton states, And here there dawned on me the notion that we must admit, in some sense, an electric circuit seemed to close, and a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, Hamiltons treatment is more geometric than the modern approach, which emphasizes quaternions algebraic properties
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
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Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the 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, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system
Infinity
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Infinity represented in screenshot form
89.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and 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 smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
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.
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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 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, series, matrices, the conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether, rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry. A ring is a group with a second binary operation that is associative, is distributive over the abelian group operation. By extension from the integers, the group operation is called addition. Whether a ring is commutative or not has profound implications on its behavior as an abstract object, as a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has greatly influenced by problems and ideas occurring naturally in algebraic number theory. The most familiar example of a ring is the set of all integers, Z, −5, −4, −3, −2, −1,0,1,2,3,4,5. The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings, a ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms 1. R is a group under addition, meaning that, + c = a + for all a, b, c in R. a + b = b + a for all a, b in R. There is an element 0 in R such that a +0 = a for all a in R, for each a in R there exists −a in R such that a + =0. R is a monoid under multiplication, meaning that, · c = a · for all a, b, c in R. There is an element 1 in R such that a ·1 = a and 1 · a = a for all a in R.3. Multiplication is distributive with respect to addition, a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many follow a alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless stated, a ring is assumed to have such an identity
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".
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Vector (geometric)
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In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the 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, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
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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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 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. The first order often discussed in primary school is the order on the natural numbers e. g.2 is less than 3,10 is greater than 5. This intuitive concept can be extended to orders on sets of numbers, such as the integers. 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, in other contexts orders may capture notions of containment or specialization. Abstractly, this type of order amounts to the relation, e. g. Pediatricians are physicians. However, many other orders do not and those orders like the subset-of relation for which there exist incomparable elements are called partial orders, orders for which every pair of elements is comparable are total orders. Order theory captures the intuition of orders that arises from such examples in a general setting and this is achieved by specifying properties that a relation ≤ must have to be a mathematical order. This more abstract approach makes sense, because one can derive numerous theorems in the general setting. These insights can then be transferred to many less abstract applications. Driven by the wide usage of orders, numerous special kinds of ordered sets have been defined. In addition, order theory does not restrict itself to the classes of ordering relations. A simple example of an order theoretic property for functions comes from analysis where monotone functions are frequently found and this section introduces ordered sets by building upon the concepts of set theory, arithmetic, and binary relations. Suppose that P is a set and that ≤ is a relation on P, a set with a partial order on it is called a partially ordered set, poset, or just an ordered set if the intended meaning is clear. By checking these properties, one sees that the well-known orders on natural numbers, integers, rational numbers
Order theory
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Hasse diagram of the set of all divisors of 60, partially ordered by divisibility
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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, when the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the geometries is the nature of parallel lines. In hyperbolic geometry, by contrast, there are many lines through A not intersecting ℓ, while in elliptic geometry. 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 almost as soon as Euclids work Elements was written. In the Elements, Euclid began with a number of assumptions. 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 Euclids other postulates,1. To draw a line from any point to any point. To produce a straight line continuously in a straight line. To describe a circle with any centre and distance and that all right angles are equal to one another. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate. Many attempted to find a proof by contradiction, including Ibn al-Haytham, Omar Khayyám, Nasīr al-Dīn al-Tūsī and these theorems along with their alternative postulates, such as Playfairs axiom, played an important role in the later development of non-Euclidean geometry. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Another example is al-Tusis son, Sadr al-Din, who wrote a book on the subject in 1298, based on al-Tusis later thoughts and he essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements. His work was published in Rome in 1594 and was studied by European geometers and he finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, in 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral and he quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle
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
94.
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. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
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
95.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to 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 and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
Topology
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Möbius strips, which have only one surface and one edge, are a kind of object studied in topology.
96.
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 finite or discrete sets of geometric objects, such as points, lines, planes, circles, spheres, polygons. The subject focuses on the properties of these objects, such as how they intersect one another. Although polyhedra and tessellations had been studied for years by people such as Kepler and Cauchy. 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, and so on in higher dimensions, some theories further generalize the idea to include such objects as unbounded polytopes, and abstract polytopes. A sphere packing is an arrangement of non-overlapping spheres within a containing space, the spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, a tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions, topics in this area include, Cauchys theorem Flexible polyhedra Incidence structures generalize planes as can be seen from their axiomatic definitions. Incidence structures also generalize the higher-dimensional analogs and the structures are sometimes called finite geometries. Formally, a structure is a triple C =. Where P is a set of points, L is a set of lines, the elements of I are called flags. If ∈ I, we say that point p lies on line l, a geometric graph is a graph in which the vertices or edges are associated with geometric objects. Examples include Euclidean graphs, the 1-skeleton of a polyhedron or polytope, intersection graphs, simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology, lovászs proof used the Borsuk-Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has used in the study of fair division problems. Topics in this include, Sperners lemma Regular maps A discrete group is a group G equipped with the discrete topology
Discrete geometry
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A collection of circles and the corresponding unit disk graph
97.
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, the convexity of f makes the powerful tools of convex analysis applicable. With recent improvements in computing and in theory, convex 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, which is convex. The general form of a problem is to find some x ∗ ∈ X such that f = min, for some feasible set X ⊂ R n. The optimization problem is called an optimization problem if X is a convex set. The following statements are true about the convex minimization problem, if a local minimum exists, 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. Such constraints can be expressed in the h i = a i T x + b i. A convex minimization problem is thus written as x f s u b j e c t t o g i ≤0, i =1, …, m h i =0, i =1, …, p. Note that every equality constraint h =0 can be replaced by a pair of inequality constraints h ≤0 and − h ≤0. Therefore, for theoretical purposes, equality constraints are redundant, however, 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, therefore, the only way for h i =0 to be convex is for h i to be affine. Then the domain X is, X =, the Lagrangian function for the problem is L = λ0 f + λ1 g 1 + ⋯ + λ m g m. If there exists a strictly feasible point, that is, a point z satisfying g 1, …, g m <0, dual subgradient methods are subgradient methods applied to a dual problem. The drift-plus-penalty method is similar to the dual subgradient method, problems with convex level sets can be efficiently minimized, in theory. Yurii Nesterov proved that quasi-convex minimization problems could be solved efficiently, however, such theoretically efficient methods use divergent-series stepsize rules, which were first developed for classical subgradient methods. Solving even close-to-convex but non-convex problems can be computationally intractable, minimizing a unimodal function is intractable, regardless of the smoothness of the function, according to results of Ivanov
Convex optimization
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Unconstrained nonlinear: Methods calling …
98.
Fiber bundles
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In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. The map π, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle, the space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the trivial case, E is just B × F, and this is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, Fiber bundles such as the tangent bundle of a manifold and more general vector bundles play an important role in differential geometry and differential topology, as do principal bundles. Mappings between total spaces of bundles that commute with the projection maps are known as bundle maps. A bundle map from the 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. Whitney came to the definition of a fiber bundle from his study of a more particular notion of a sphere bundle. A fiber bundle is a structure, where E, B, the space B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map and we shall assume in what follows that the base space B is connected. That is, the diagram should commute, where proj1, U × F → U is the natural projection and φ. The set of all is called a 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. Therefore B carries the quotient topology determined by the map π, a fiber bundle is often denoted that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space. A smooth fiber bundle is a bundle in the category of smooth manifolds. That is, E, B, and F are required to be smooth manifolds, let E = B × F and let π, E → B be the projection onto the first factor. Then E is a fiber bundle over B, here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle, any fiber bundle over a contractible CW-complex is trivial
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.
99.
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, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is x2 − 4x +7, an example in three variables is x3 + 2xyz2 − yz +1. Polynomials appear in a variety of areas of mathematics and science. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra, the word polynomial joins two diverse roots, the Greek poly, meaning many, and the Latin nomen, or name. It was derived from the binomial by replacing the Latin root bi- with the Greek poly-. The word polynomial was first used in the 17th century, the x occurring in a polynomial is commonly called either a variable or an indeterminate. When the polynomial is considered as an expression, x is a symbol which does not have any value. It is thus correct to call it an indeterminate. However, when one considers the function defined by the polynomial, then x represents the argument of the function, many authors use these two words interchangeably. It is a convention to use uppercase letters for the indeterminates. However one may use it over any domain where addition and multiplication are defined, in particular, when a is the indeterminate x, then the image of x by this function is the polynomial P itself. This equality allows writing let P be a polynomial as a shorthand for let P be a polynomial in the indeterminate x. A polynomial is an expression that can be built from constants, the word indeterminate means that x represents no particular value, although any value may be substituted for it. The mapping that associates the result of substitution to the substituted value is a function. This can be expressed concisely by using summation notation, ∑ k =0 n a k x k That is. Each term consists of the product of a number—called the coefficient of the term—and a finite number of indeterminates, because x = x1, the degree of an indeterminate without a written exponent is one. A term and a polynomial with no indeterminates are called, respectively, a constant term, the degree of a constant term and of a nonzero constant polynomial is 0. The degree of the polynomial,0, is generally treated as not defined
Polynomial
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The graph of a polynomial function of degree 3
100.
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, Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. A topological group, G, is a space which is also a group such that the group operations of product, G × G → G, ↦ x y and taking inverses. Here G × G is viewed as a space with the product topology. Although not part of this definition, many require that the topology on G be Hausdorff. The reasons, and 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, a homomorphism of topological groups means a continuous group homomorphism G → H. An isomorphism of groups is a group isomorphism which is also a homeomorphism of the underlying 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 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 made into a topological group by considering it with the discrete topology. In this sense, the theory of topological groups subsumes that of ordinary groups, the real numbers, R with the usual topology form a topological group under addition. More generally, Euclidean n-space Rn is a group under addition. Some other examples of topological groups are the circle group S1. The classical groups are important examples of topological groups. Another classical group is the orthogonal group O, the group of all maps from Rn to itself that preserve the length of all vectors. The orthogonal group is compact as a topological space, much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related group O ⋉ Rn of isometries of Rn
Topological groups
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Algebraic structure → Group theory Group theory
101.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
Lie group
102.
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, 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 group is again a free group. Below are some of the areas studied in algebraic topology, In mathematics. The first and simplest homotopy group is the group, which records information about loops in a space. Intuitively, homotopy groups record information about the shape, or holes. In homology theory and algebraic topology, cohomology is a term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the study of cochains, cocycles. Cohomology can be viewed as a method of assigning algebraic invariants to a 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 space that near each point resembles Euclidean space. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds, knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, in precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. A simplicial complex is a space of a certain kind, constructed by gluing together points, line segments, triangles. Simplicial complexes should not be confused with the abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a complex is an abstract simplicial complex. A CW complex is a type of space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, an older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones
Algebraic topology
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A torus, one of the most frequently studied objects in algebraic topology
103.
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 set theory was initiated by Georg Cantor, 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, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions 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. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors 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 set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
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.
104.
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 difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. If we think of the parameter of H as time then H describes a continuous deformation of f into g, at time 0 we have the function f. We can also think of the second parameter as a control that allows us to smoothly transition from f to g as the slider moves from 0 to 1. The two versions coincide by setting ht = H and it is not sufficient to require each map ht to be continuous. The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3. The animation shows the image of ht as a function of the parameter t and it pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle. Continuous functions f and g are said to be homotopic if, 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, every homeomorphism is a homotopy equivalence, but the converse is not true, for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent. As another example, the Möbius strip and an untwisted strip are homotopy equivalent, spaces that are homotopy equivalent to a point are called contractible. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations, the first example of a homotopy equivalence is R n with a point, denoted R n ≃. There is an equivalence between S1 and R2 −. More generally, R n − ≃ S n −1, any fiber bundle π, E → B with fibers F b homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since π, R n − → S n −1 is a bundle with fiber R >0. Every vector bundle is a bundle with a fiber homotopy equivalent to a point. For any 0 ≤ k < n, R n − R k ≃ S n − k −1 by writing R n as R k × R n − k, a function f is said to be null-homotopic if it is homotopic to a constant function. For example, a map f from the unit circle S1 to any space X is null-homotopic precisely when it can be extended to a map from the unit disk D2 to X that agrees with f on the boundary
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.
105.
Morse theory
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In another context, a Morse function can also mean an anharmonic oscillator, see Morse potential. In mathematics, specifically in topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds, before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics and these techniques were used in Raoul Botts 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. If f is the function M → R sending each point to its elevation, each connected component of a contour line is either a point, a simple closed curve, or a closed curve with a double point. Contour lines may also have points of order, but these are unstable. Double points in contour lines occur at points, or passes. Saddle points are points where the surrounding landscape curves up in one direction, imagine flooding this landscape with water. Then, the covered by water when the water reaches an elevation of a is f−1(−∞, a]. Consider how the topology of this changes as the water rises. It appears, intuitively, that it does not change except when a passes the height of a point, that is. In other words, it does not change except when the water either starts filling a basin, covers a saddle, to each of these three types of critical points – basins, passes, and peaks – one associates a number called the index. Intuitively speaking, the index of a point b is the number of independent directions around b in which f decreases. Therefore, the indices of basins, passes, and peaks are 0,1, rigorously, the index of a critical point is the dimension of the negative-definite submatrix of the hessian matrix calculated at that point. In case of smooth maps, the matrix turns out to be a symmetric matrix. Starting from the bottom of the torus, let p, q, r, and s be the four points of index 0,1,1
Morse theory
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A saddle point
106.
Four color theorem
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Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. 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, “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. Martin Gardner wrote an account of what was known at the time about the four color theorem in his September 1960 Mathematical Games column in Scientific American magazine. In 1975 Gardner revisited the topic by publishing a map said to be a counter-example in his infamous April fools hoax column of April 1975, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer, Appel and Hakens approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a 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 exist because any must contain, yet do not contain and 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 proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. To dispel remaining doubt about the Appel–Haken proof, a proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour. Additionally, in 2005, the theorem was proved by Georges Gonthier with general purpose theorem proving software, the intuitive statement of the four color theorem, i. e. First, all corners, points that belong to three or more countries, must be ignored. In addition, bizarre maps can require more than four colors, second, for the purpose of the theorem, every country has to be a connected region, or contiguous. In the real world, this is not true, because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map, In this map and this map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. A similar construction also applies if a color is used for all bodies of water. For maps in which more than one country may have multiple disconnected regions, a simpler statement of the theorem uses graph theory
Four color theorem
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Example of a four-colored map
107.
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 expanding symmetry or 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 other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals
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
108.
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. The theorems of real analysis rely intimately upon the structure of the number line. The real number system consists of a set, together with two operations and an order, and is, formally speaking, an ordered quadruple consisting of these objects, there are several ways of formalizing the definition of the real number system. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. These constructions are described in detail in the main article. In addition to these notions, the real numbers, equipped with the absolute value function as a metric. Many important theorems in real analysis remain valid when they are restated as statements involving metric spaces and these theorems are frequently topological in nature, and placing them in the more abstract setting of metric spaces may lead to proofs that are shorter, more natural, or more elegant. The real numbers have several important lattice-theoretic properties that are absent in the complex numbers, most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the numbers is total. These order-theoretic properties lead to a number of important results in analysis, such as the monotone convergence theorem, the intermediate value theorem. However, while the results in analysis are stated for real numbers. In particular, many ideas in analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces. Also, mathematicians consider real and imaginary parts of complex sequences, a sequence is a function whose domain is a countable, totally ordered set, usually taken to be the natural numbers or whole numbers. Occasionally, it is convenient to consider bidirectional sequences indexed by the set of all integers. Of interest in analysis, a real-valued sequence, here indexed by the natural numbers, is a map a, N → R, n ↦ a n. Each a = a n is referred to as a term of the sequence, a sequence that tends to a limit is said to be convergent, otherwise it is divergent
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.
109.
Design of experiments
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The design of experiments is the design of any task that aims to describe or explain the variation of information under conditions that are hypothesized to reflect the variation. In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, 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 design include the establishment of validity, reliability. Related concerns include achieving appropriate levels of power and sensitivity. Correctly designed experiments advance knowledge in the natural and social sciences, other applications include marketing and policy making. In 1747, while serving as surgeon on HMS Salisbury, James Lind carried out a 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 and he divided them into six pairs, giving each pair different supplements to their basic diet for two weeks. The treatments were all remedies that had 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, and horseradish in a lump the size of a nutmeg, two spoonfuls of vinegar three times a day. Two oranges and one every day. The citrus treatment stopped after six days when they ran out of fruit, apart from that, only group one showed some effect of its treatment. The remainder of the crew served as a control. Charles S. Peirce randomly assigned volunteers to a blinded, repeated-measures design to evaluate their ability to discriminate weights, peirces experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1800s. Charles S. Peirce also contributed the first English-language publication on a design for regression models in 1876. A pioneering optimal design for regression was suggested by Gergonne in 1815. In 1918 Kirstine Smith published optimal designs for polynomials of degree six, herman Chernoff wrote an overview of optimal sequential designs, while adaptive designs have been surveyed by S. Zacks
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.
110.
Statistical decision theory
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Decision theory is the study of the reasoning underlying an agents choices. Decision theory is a topic, studied by economists, statisticians, psychologists, political and social scientists. Empirical applications of this theory are usually done with the help of statistical and econometric methods, especially via the so-called choice models. Estimation of such models is usually done via parametric, semi-parametric and non-parametric maximum likelihood methods, the practical application of this prescriptive approach is called decision analysis, and is aimed at finding tools, methodologies and software to help people make better decisions. In contrast, positive or descriptive decision theory is concerned with describing observed behaviors under the assumption that the agents are behaving under some consistent rules. The prescriptions or 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 laboratory and field experiments to evaluate and inform theory. The area of choice under uncertainty represents the heart of decision theory and he gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter. In his solution, he defines a utility function and computes expected utility rather than expected financial value, the phrase decision theory itself was used in 1950 by E. L. Lehmann. At this time, von Neumann and Morgenstern theory of expected utility proved that expected utility maximization followed from basic postulates about rational behavior, the work of Maurice Allais and Daniel Ellsberg showed that human behavior has systematic and sometimes important departures from expected-utility maximization. The prospect theory of Daniel Kahneman and Amos Tversky renewed the study of economic behavior with less emphasis on rationality presuppositions. Pascals Wager is an 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. The answer depends partly on factors such as the rates of interest and inflation, the persons life expectancy. Some decisions are difficult because of the need to take into account how people in the situation will respond to the decision that is taken. The analysis of such decisions is more often treated under the label of game theory, rather than decision theory. From the standpoint of game theory most of the treated in decision theory are one-player games. Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, one example is the model of economic growth and resource usage developed by the Club of Rome to help politicians make real-life decisions in complex situations. Decisions are also affected by whether options are framed together or separately, one example of common and incorrect thought process is the gamblers fallacy, or believing that a random event is affected by previous random events
Statistical decision theory
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Daniel Kahneman
111.
Risk
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Risk is the potential of gaining or losing something of value. Values can be gained or lost when taking risk resulting from an action or inaction. Risk can also be defined as the interaction with uncertainty. Uncertainty is a potential, unpredictable, and uncontrollable outcome, risk is a consequence of action taken in spite of uncertainty, Risk perception is the subjective judgment people make about the severity and probability of a risk, and may vary person to person. Any human endeavor carries some risk, but some are much riskier than others, the Oxford English Dictionary cites the earliest use of the word in English as of 1621, and the spelling as risk from 1655. It defines risk as, the possibility of loss, injury, or other adverse or unwelcome circumstance, Risk is an uncertain event or condition that, if it occurs, has an effect on at least one objective. The probability of something 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, a risk is not an uncertainty, a peril, or a hazard. Securities trading, The probability of a loss or drop in value, non-systematic risk is any risk that isnt market-related. Also called non-market risk, extra-market risk or diversifiable risk, workplace, Product of the consequence and probability of a hazardous event or phenomenon. For example, the risk of developing cancer is estimated as the probability of developing cancer over a lifetime as a result of exposure to potential carcinogens. The International Organization for Standardization publication ISO31000 / ISO Guide 73,2002 definition of risk is the effect of uncertainty on objectives, in this definition, uncertainties include events and uncertainties caused by ambiguity or a lack of information. It also includes both negative and positive impacts on objectives, very different approaches to risk management are taken in different fields, e. g. Risk is the unwanted subset of a set of uncertain outcomes. Risk can be seen as relating to the probability of future events. For example, according to analysis of information risk, risk is. In computer science this definition is used by The Open Group, OHSAS defines risk as the combination of the probability of a hazard resulting in an adverse event, and the severity of the event. In information security risk is defined as the potential that a threat will exploit vulnerabilities of an asset or group of assets. Financial risk is defined as the unpredictable variability or volatility of returns
Risk
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Firefighters at work
112.
Mathematical optimization
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In mathematics, computer science and operations research, mathematical optimization, also spelled mathematical optimisation, is the selection of a best element from some set of available alternatives. The generalization of optimization theory and techniques to other formulations comprises an area of applied mathematics. Such a formulation is called a problem or a mathematical programming problem. Many real-world and theoretical problems may be modeled in this general framework, typically, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the space or the choice set. The function f is called, variously, a function, a loss function or cost function, a utility function or fitness function, or, in certain fields. 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 function and the feasible region are convex in a minimization problem, there may be several local minima. 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. In a convex problem, if there is a minimum that is interior, it is also the global minimum. Optimization problems are often expressed with special notation, consider the following notation, min x ∈ R This denotes the minimum value of the objective function x 2 +1, when choosing x from the set of real numbers R. The minimum value in case is 1, occurring at x =0. Similarly, the notation max x ∈ R2 x asks for the value of the objective function 2x. In this case, there is no such maximum as the function is unbounded. This represents the value of the argument x in the interval, John Wiley & Sons, Ltd. pp. xxviii+489. (2008 Second ed. in French, Programmation mathématique, Théorie et algorithmes, Editions Tec & Doc, Paris,2008. Nemhauser, G. L. Rinnooy Kan, A. H. G. Todd, handbooks in Operations Research and Management Science. Amsterdam, North-Holland Publishing Co. pp. xiv+709, J. E. Dennis, Jr. and Robert B
Mathematical optimization
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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.
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Analysis (mathematics)
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, 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 Cavalieris 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 Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of 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. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
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.
114.
Discretization
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In mathematics, discretization concerns the process of transferring continuous functions, models, and equations into discrete counterparts. This process is carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the case of discretization in which the number of discrete classes is 2. Discretization is also related to mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, 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 not always distinguished from quantization in any clearly defined way. The two terms share a semantic field, the same is true of discretization error and quantization error. Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold, 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, the discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G, Q d = T. Now we want to discretise the above expression and we assume that u is constant during each timestep. Exact discretization may sometimes be due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate model, based on that for small timesteps e A T ≈ I + A T. The approximate solution then becomes, x ≈ x + T B u Other possible approximations are e A T ≈ −1 and e A T ≈ −1, each of them have different stability properties. The last one is known as the transform, or Tustin transform. In statistics and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features and this can be useful when creating probability mass functions. Discrete space Time-scale calculus Discrete event simulation Stochastic simulation Finite volume method for unsteady flow Discrete time, introduction to random signals and applied Kalman filtering. Philadelphia, PA, USA, Saunders College Publishing, computing integrals involving the matrix exponential. Digital control and estimation, a unified approach
Discretization
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A solution to a discretized partial differential equation, obtained with the finite element method.
115.
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, there are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics, both formulations are embodied in analytical mechanics. These approaches and ideas can be and, in fact, have extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory. Moreover, they have provided several examples and basic ideas in differential geometry, the theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics. The theory of atomic spectra developed almost concurrently with the fields of linear algebra. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic, Quantum information theory is another subspecialty. The special and general theories of relativity require a different type of mathematics. This was group theory, which played an important role in quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the description of cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays, statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics and it is related with the more mathematical ergodic theory. There are increasing interactions between combinatorics and physics, in statistical physics. The usage of the mathematical physics is sometimes idiosyncratic. Certain parts of mathematics that arose from the development of physics are not, in fact, considered parts of mathematical physics. The term mathematical physics is sometimes used to research aimed at studying and solving problems inspired by physics or thought experiments within a mathematically rigorous framework
Mathematical physics
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An example of mathematical physics: solutions of Schrödinger's equation for quantum harmonic oscillators (left) with their amplitudes (right).
116.
Fluid dynamics
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In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids. It has several subdisciplines, including aerodynamics and hydrodynamics, before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, the foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on mechanics and are modified in quantum mechanics. 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 molecules is ignored. The unsimplified equations do not have a general 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 some simple fluid dynamics problems to be solved in closed form. Three conservation laws are used to solve fluid dynamics problems, the conservation laws may be applied to a region of the flow called a control volume. A control volume is a volume in space through which fluid is assumed to flow. The integral formulations of the laws are used to describe the change of mass, momentum. Mass continuity, The rate of change of fluid mass inside a control volume must be equal to the net rate of flow into the volume. Mass flow into the system is accounted as positive, and since the vector to the surface is opposite the sense of flow into the system the term is negated. The first term on the right is the net rate at which momentum is convected into the volume, the second term on the right is the force due to pressure on the volumes surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, the third term on the right is the net acceleration of the mass within the volume due to any body forces. Surface forces, such as forces, are represented by F surf. The following is the form of the momentum conservation equation
Fluid dynamics
117.
Abel Prize
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The Abel Prize /ˈɑːbəl/ is a Norwegian prize awarded annually by the Government of Norway to one or more outstanding mathematicians. It comes with an award of 6 million Norwegian kroner. The award ceremony takes place in the Atrium of the University of Oslo Faculty of Law, the prize 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 Abels birth, king Oscar II was willing to finance a mathematics prize in 1902, and the mathematicians Ludwig Sylow and Carl Størmer drew up statutes and rules for the proposed prize. However, Lies influence waned after his death, and the dissolution of the union between Sweden and Norway in 1905 ended the first attempt to create the Abel Prize. After interest in the concept of the prize had risen in 2001, a group was formed to develop a proposal. In August 2001, the Norwegian government announced that the prize would be awarded beginning in 2002, atle Selberg received an honorary Abel Prize in 2002, but the first actual Abel Prize was only awarded in 2003. A book series presenting Abel Prize laureates and their research was commenced in 2010, the first two volumes cover the years 2003–2007 and 2008–2012 respectively. The Norwegian Academy of Science and Letters declares the winner of the Abel Prize each March after recommendation by the Abel Committee, the committee is currently headed by John Rognes. The International Mathematical Union and the European Mathematical Society nominate members of the Abel Committee, the Norwegian Government gave the prize an initial funding of NOK200 million in 2001. The funding is controlled by the Board, which consists of members elected by the Norwegian Academy of Science, anyone may submit a nomination, but self-nomination is not allowed. The nominee must be alive, however, if the awardee dies after being declared as the winner, the Abel Laureate is decided by the Norwegian Academy of Science and Letters based on the recommendation of the Abel Committee. List of prizes, medals, and awards – mathematics Official website Official website of the Abel Symposium Barile, Margherita and Weisstein, cS1 maint, Multiple names, authors list
Abel Prize
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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
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Abel Prize
118.
Riemann hypothesis
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In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. 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 zeta function ζ is a function whose argument s may be any complex number other than 1 and it has zeros at the negative even integers, that is, ζ =0 when s is one of −2, −4, −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, the Riemann hypothesis is concerned with the locations of these non-trivial zeros, and states that, The real part of every non-trivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the non-trivial zeros lie on the line consisting of the complex numbers 1/2 + i t. There are several books on the Riemann hypothesis, such as Derbyshire, Rockmore. The books Edwards, Patterson, Borwein et al. and Mazur & Stein give mathematical introductions, while Titchmarsh, Ivić, furthermore, the book Open Problems in Mathematics, edited by John Forbes Nash Jr. and Michael Th. Rassias, features an essay on the Riemann hypothesis by Alain Connes. The convergence of the Euler product shows that ζ has no zeros in this region, 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 continue the function to give it a definition that is valid for all complex s. This can be done by expressing it in terms of the Dirichlet eta function as follows. If the real part of s is greater than one, then the function satisfies ζ = ∑ n =1 ∞ n +1 n s =11 s −12 s +13 s − ⋯. However, the series on the right converges not just when the part of s is greater than one. Thus, this alternative series extends the function from Re >1 to the larger domain Re >0. The zeta function can be extended to these values, as well, by taking limits, in the strip 0 < Re <1 the zeta function satisfies the functional equation ζ =2 s π s −1 sin Γ ζ. If s is an even integer then ζ =0 because the factor sin vanishes
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.
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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 only to a few. The template to the right links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a 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 and 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, 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 structure is studied, for instance. Geometry is initially the study of figures like circles and cubes. Topology developed from geometry, it looks at those properties that do not change even when the figures are deformed by stretching and bending, outline of combinatorics List of graph theory topics Glossary of graph theory Logic is the foundation which underlies mathematical logic and the rest of mathematics. It tries to formalize valid reasoning, in particular, it attempts to define what constitutes a proof. One of the concepts in number theory is that of the prime number. In a dynamical system, a fixed rule describes the dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, historically, information theory was developed to find fundamental limits on compressing and reliably communicating data. Signal processing is the analysis, interpretation, and manipulation of signals, signals of interest include sound, images, biological signals such as ECG, radar signals, and many others. Processing of such signals includes filtering, storage and reconstruction, separation of information from noise, compression, the related field of mathematical statistics develops statistical theory with mathematics. Statistics, the science concerned with collecting and analyzing data, is an autonomous discipline and it has applications in a variety of fields, including economics, evolutionary biology, political science, social psychology and military strategy. The following pages list the integrals of many different functions
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.
120.
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 music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts, Mathematics and art have a long historical relationship. Artists have used mathematics since the 5th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions based on the ratio 1, persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De Divina Proportione, illustrated with woodcuts by Leonardo da Vinci, another Italian painter, Piero della Francesca, developed Euclids ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics 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 evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jaali pierced stone screens, and widespread muqarnas vaulting. Mathematics has directly influenced art with conceptual tools such as perspective, the analysis of symmetry, and mathematical objects such as polyhedra. Magnus Wenninger creates colourful 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. Computer art often makes use of including the Mandelbrot set. Polykleitos the elder was a Greek sculptor from the school of Argos, and his works and statues consisted mainly of bronze and were of athletes. While his sculptures may not be as famous as those of Phidias, Polykleitos uses the distal phalanx of the little finger as the basic module for determining the proportions of the human body. Next, he takes the length and multiplies that by √2 to get the length of the palm from the base of the finger to the ulna. This geometric series of measurements progresses until Polykleitos has formed the arm, chest, body, the influence of the Canon of Polykleitos is immense in Classical Greek, Roman, and Renaissance sculpture, many sculptors following Polykleitoss prescription. While none of Polykleitoss original works survive, Roman copies demonstrate his ideal of physical perfection, some scholars argue that Pythagorean thought influenced the Canon of Polykleitos. In classical times, rather than making distant figures smaller with linear perspective, painters sized objects, in the Middle Ages, some artists used reverse perspective for special emphasis. The Muslim mathematician Alhazen described a theory of optics in his Book of Optics in 1021, the Renaissance saw a rebirth of Classical Greek and Roman culture and ideas, among them the study of mathematics to understand nature and the arts
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
121.
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 the history, influences and recent controversies, elementary mathematics was part of the education system in most ancient civilisations, including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most cases, an education was only available to male children with a sufficiently high status. In Platos division of the arts into the trivium and the quadrivium. This structure was continued in the structure of education that was developed in medieval Europe. Teaching of geometry was almost universally based on Euclids Elements, apprentices to trades such as masons, merchants and money-lenders could expect to learn such practical mathematics as was relevant to their profession. The first mathematics textbooks to be written in English and French were published by Robert Recorde, however, there are many different writings on mathematics and mathematics 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. After the Sumerians some of the most famous ancient works on come from Egypt in the form of the Rhind Mathematical Papyrus. The more famous Rhind Papyrus has been dated to approximately 1650 BCE and this papyrus was essentially an early textbook for Egyptian students. In the Renaissance, the status of mathematics declined, because it was strongly associated with trade. Although it continued to be taught in European universities, it was seen as subservient to the study of Natural, Metaphysical and Moral Philosophy, however, it was uncommon for mathematics to be taught outside of the universities. Isaac Newton, for example, received no formal mathematics teaching until he joined Trinity College, Cambridge in 1661, in the 18th and 19th centuries, the industrial revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money and carry out simple arithmetic, within the new public education systems, mathematics became a central part of the curriculum from an early age. By the twentieth century, mathematics was part of the curriculum in all developed countries. During the twentieth century, mathematics education was established as an independent field of research. S. A, had generated more than 4000 articles after 1920, so in 1941 William L. Schaaf published a classified index, sorting them into their various subjects. While previous approach focused on working with specialized problems in arithmetic, at different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. The teaching of heuristics and other problem-solving strategies to solve non-routine problems, the method or methods used in any particular context are largely determined by the objectives that the relevant educational system is trying to achieve
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.
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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 edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and 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 in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, 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 to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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
123.
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, Devlin earned a BSc in Mathematics at Kings College London in 1968, and a PhD in Mathematics at the University of Bristol in 1971 under the supervision of Frederick Rowbottom. He is a commentator on National Public Radios Weekend Edition Saturday, as of 2012, he is the author of 34 books and over 80 research articles. Several of his books are aimed at an audience of the general public, the Joy of Sets, Fundamentals of Contemporary Set Theory. Goodbye, Descartes, the End of Logic and the Search for a New Cosmology of the Mind, john Wiley & Sons, Inc.1997. The Language of Mathematics, Making the Invisible Visible, the Math Gene, How Mathematical Thinking Evolved and Why Numbers Are Like Gossip. The Millennium Problems, the Seven Greatest Unsolved Mathematical Puzzles of Our Time, the Math Instinct, Why Youre a Mathematical Genius. The Numbers Behind NUMB3RS, Solving Crime with Mathematics, with coauthor Gary Lorden The Unfinished Game, Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern. The Man of Numbers, Fibonaccis Arithmetic Revolution, Mathematics Education for a New Era, Video Games as a Medium for Learning. Official website including his curriculum vitae Devlins Angle — column at the Mathematical Association of America
Keith Devlin
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Keith Devlin (2011)
124.
Henry Liddell
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Lewis Carroll wrote Alices Adventures in Wonderland for Henry Liddells daughter Alice. Liddell received his education at Charterhouse and Christ Church, Oxford and he gained a double first degree in 1833, then became a college tutor, and was ordained in 1838. Liddell was Headmaster of Westminster School from 1846 to 1855, meanwhile, his life work, the great lexicon, which he and Robert Scott began as early as 1834, had made good progress, and the first edition of Liddell and Scotts Lexicon appeared in 1843. 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 by trouble due to the outbreak of fever and cholera 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. His tall figure, fine presence and aristocratic mien were for years associated with all that was characteristic of Oxford life. In 1859 Liddell welcomed the then Prince of Wales when he matriculated at Christ Church, while he was Dean of Christ Church, he arranged for the building of a new choir school and classrooms for the staff and pupils of Christ Church Cathedral School on its present site. 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 conjunction with Sir Henry Acland, Liddell did much to encourage the study of art at Oxford, 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 Ascot, Liddell Way and Carroll Crescent honour the relationship between Henry Liddell and Lewis Carroll. Liddell was an Oxford character in later years and he figures in contemporary undergraduate doggerel, I am the Dean, this Mrs Liddell. She plays first, I, second fiddle and she is the Broad, I am the High – We are the University. The Victorian journalist, George W. E and his father was Henry Liddell, Rector of Easington, the younger son of Sir Henry Liddell, 5th Baronet and the former Elizabeth Steele. His fathers elder brother, Sir Thomas Liddell, 6th Baronet, was raised to the Peerage as Baron Ravensworth in 1821 and his mother was the former Charlotte Lyon, a daughter of Thomas Lyon and the former Mary Wren. On 2 July 1846, Henry married Lorina Reeve and they were parents of ten children, Edward Harry Liddell. Alice Pleasance Liddell, for whom the story of the childrens classic Alices Adventures in Wonderland was originally told, rhoda Caroline Anne Liddell, she was invested as an Officer, Order of the British Empire in 1920. Albert Edward Arthur Liddell, he died in infancy, violet Constance Liddell, she was invested as a Member, Order of the British Empire in 1920. Sir Frederick Francis Liddell, First Parliamentary Counsel and Ecclesiastical Commissioner, lionel Charles Liddell, he was British Consul to Lyons and Copenhagen
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).
125.
Ohio State University
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The Ohio State University, commonly referred to as Ohio State or OSU, is a large, primarily residential, public university in Columbus, Ohio. Founded in 1870 as a land-grant university and ninth university in Ohio with the Morrill Act of 1862, Hayes, and 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 university campus in the United States, along with its main campus in Columbus, Ohio State also operates a regional campus system with regional campuses in Lima, Mansfield, Marion, Newark, and Wooster. Ohio State athletic teams compete in Division I of the NCAA and 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 mens ice hockey program competes in the Big Ten Conference, while its womens hockey program competes in the Western Collegiate Hockey Association. In addition, the OSU mens volleyball team is a member of the Midwestern Intercollegiate Volleyball Association, OSU is one of only 14 universities that plays Division I FBS football and Division I ice hockey. As of August 2015, the university had awarded a total of 714,512 degrees, alumni and former students have gone on to prominent careers in government, business, science, medicine, education, sports, and entertainment. Championed by the Republican stalwart Governor Rutherford B, Hayes, the Ohio State University was founded in 1870 as a land-grant university under the Morrill Act of 1862 as the Ohio Agricultural and Mechanical College. The school was originally within a community on the northern edge of Columbus. The university opened its doors to 24 students on September 17,1873, in 1878, the first class of six men graduated. The first woman graduated the following year, also in 1878, in light of its expanded focus, the Ohio legislature changed the name to the now-familiar The Ohio State University, with The as part of its official name. Ohio State began accepting students in the 1880s, and in 1891. It would later acquire colleges of medicine, dentistry, optometry, veterinary medicine, commerce, in 1916, Ohio State was elected into membership in the Association of American Universities. Michael V. Drake, former chancellor of the University of California, Irvine, in an attack against the campus on November 28,2016, an unrelated fluorine leak was called in for Watts Hall, resulting in the evacuation of the building to an outside courtyard. As firetrucks began to depart, Abdul Razak Ali Artan drove into the crowd, then emerged, the attack was stopped in under two minutes by OSU Police Officer Alan Horujko, who witnessed the attack after responding to the reported gas leak, and who shot and killed Artan. The universitys Buckeye Alert system was triggered and the campus was placed on lockdown, Ten were transported to local hospitals and one suspect was killed according to multiple sources. Local law enforcement and the FBI launched an investigation, according to authorities, Artan was inspired by terrorist propaganda from the Islamic State and radical Muslim cleric Anwar al-Awlaki. Ohio States 1, 764-acre main campus is about 2.5 miles north of the citys downtown, the historical center of campus is the Oval, quad of about 11 acres
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.
126.
Diophantus
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Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and 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 and this term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers, thus he allowed positive rational numbers for the coefficients, 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 in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 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 states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. 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, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found
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.
127.
False proof
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In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a quality of the mathematical fallacy, as typically presented, it leads not only to an absurd result. Therefore, these fallacies, for reasons, usually take the form of spurious proofs of obvious contradictions. The latter applies normally to a form of argument that is not a rule of logic. Beyond pedagogy, the resolution of a fallacy can lead to insights into a subject. Pseudaria, an ancient lost book of false proofs, is attributed to Euclid, mathematical fallacies exist in many branches of mathematics. Well-known fallacies also exist in elementary Euclidean geometry and calculus, examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion, is invalid and is commonly known as a howler. Consider for instance the calculation,1664 =16 /6 /4 =14, although the conclusion 1664 =14 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, calculations, or 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, the division-by-zero fallacy has many variants. The following example uses division by zero to prove that 2 =1, the fallacy is in line 5, the progression from line 4 to line 5 involves division by a − b, which is zero since a equals b. Since division by zero is undefined, the argument is invalid, many functions do not have a unique inverse. For instance squaring a number gives a value, but there are two possible square roots of a positive number. The square root of the square of −2 is 2, calculus as the mathematical study of infinitesimal change and limits can lead to mathematical fallacies if the properties of integrals and differentials are ignored. For instance, a use of integration by parts can be used to give a false proof that 0 =1. The problem is that antiderivatives are only defined up to a constant, the error really comes to light when we introduce arbitrary integration limits a and b
False proof
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Contents
128.
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 professor emeritus at Penn State University and Research Professor at the University at Buffalo, Rao has been honoured by numerous colloquia, honorary degrees, and festschrifts and 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 disease awareness. C. R. Rao was born in Hadagali, Bellary, Karnataka and he received an M. Sc. in mathematics from Andhra University and an M. A. in statistics from Calcutta University in 1943. He was among the first few people in the world to hold a 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, estimation theory, and differential geometry. His other contributions include the Fisher–Rao Theorem, Rao distance, and he is the author of 14 books and has published over 400 journal publications. Rao has received 38 honorary doctoral degrees from universities in 19 countries around the world and numerous awards and medals for his contributions to statistics and he is a member of eight National Academies in India, the United Kingdom, the United States, and Italy. Rao was awarded the United States National Medal of Science, that nations highest award for achievement in fields of scientific research. The latest addition to his collection of awards is the India Science Award for 2010 and he has been the President of the International Statistical Institute, Institute of Mathematical Statistics, and the International Biometric Society. The Journal of Quantitative Economics published an issue in Raos honour in 1991. Dr Rao is a distinguished scientist and a highly eminent statistician of our time. He is a teacher and has guided the research work of numerous students in all areas of statistics. His early work had influenced the course of statistical research during the last four decades. One of the purposes of special issue is to recognize Dr Raos own contributions to econometrics. Bush, on June 12,2002, honoured him with the National Medal of Science, also honorary doctorates from a number of universities and institutes around the world. The Pennsylvania State University has established C. R, the road from IIIT Hyderabad passing along Central University of Hyderabad crossroads to Alind Factory, Lingampally is named as Prof. C. R. Rao Road. Testing Point Null Hypothesis of a Normal Mean and the Truth, Data Mining Using Neural Networks, A Guide for Statisticians
C.R. Rao
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Prof. Rao at Indian Statistical Institute, Chennai in April 2012
129.
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 as Master of Christs College, Cambridge from 2006 to 2016, Kellys research interests are in random processes, networks and optimisation, especially in very large-scale systems such as telecommunication or transportation networks. He has also worked on the theory of pricing to congestion control. From 2003 to 2006 he served as Chief Scientific Advisor to the United Kingdom Department for Transport, Kelly was elected a Fellow of the Royal Society in 1989. In December 2006 he was elected 37th Master of Christs College, Cambridge and he was appointed Commander of the Order of the British Empire in the 2013 New Year Honours for services to mathematical science. Sc. Central to all third-generation cellular networks,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. Frank Kellys homepage, retrieved 8 December 2006 Biography from Frank Kellys website, retrieved 8 December 2006 Whos Who
Frank Kelly (mathematician)
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Professor Frank Kelly at the EPFL, 15 October 2007
130.
Herbert Robbins
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Herbert Ellis Robbins was an American mathematician and statistician. He did research in topology, measure theory, statistics, and he was the co-author, with Richard Courant, of What is Mathematics. A popularization that is 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 graph theory, is also named after him, as is the Whitney–Robbins synthesis, a tool he introduced to prove this theorem. Robbins was born in New Castle, Pennsylvania, as an undergraduate, Robbins attended Harvard University, where Marston Morse influenced him to become interested in mathematics. Robbins received a doctorate from Harvard in 1938 under the supervision of Hassler Whitney and was an instructor at New York University from 1939 to 1941, in 1953, he became a professor of mathematical statistics at Columbia University. He retired from full-time activity at Columbia in 1985 and was then a professor at Rutgers University until his retirement in 1997 and he has 567 descendants listed at the Mathematics Genealogy Project. In 1955, Robbins introduced empirical Bayes methods at the Third Berkeley Symposium on Mathematical Statistics, Robbins was also one of the inventors of the first stochastic approximation algorithm, the Robbins-Monro method, and worked on the theory of power-one tests and optimal stopping. These policies were simplified in the 1995 paper Sequential choice from several populations and he was a member of the National Academy of Sciences and the American Academy of Arts and Sciences and was past president of the Institute of Mathematical Statistics. Books by Herbert Robbins What is Mathematics, an Elementary Approach to Ideas and Methods, with Richard Courant, London, Oxford University Press,1941. Great Expectations, The Theory of Optimal Stopping, with Y. S. Chow, introduction to Statistics, with John Van Ryzin, Science Research Associates,1975. Articles A theorem on graphs with an application to a problem on traffic control, American Mathematical Monthly, the central limit theorem for dependent random variables, with Wassily Hoeffding, Duke Mathematical Journal, vol. A stochastic approximation method, with Sutton Monro, Annals of Mathematical Statistics, some aspects of the sequential design of experiments, in Bulletin of the American Mathematical Society, vol. Two-stage procedures for estimating the difference between means, with Ghurye, SG, Biometrika,41, 146-152,1954, the strong law of large numbers when the first moment does not exist, with C. Derman, in the Proceedings of the National Academy of Sciences of the United States of America, an empirical Bayes approach to statistics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Jerzy Neyman, ed. vol. 1, Berkeley, California, University of California Press,1956, on the asymptotic theory of fixed-width sequential confidence intervals for the mean, with Chow, Y. S. The Annals of Mathematical Statistics,36, 457-462,1965, Statistical methods related to the law of the iterated logarithm, The Annals of Mathematical Statistics,41, 1397–1409,1970. Optimal stopping, The American Mathematical Monthly,77, 333-343,1970, a convergence theorem for nonnegative almost supermartingales and some applications, with David Siegmund, Optimizing methods in statistics, 233–257,1971
Herbert Robbins
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Herbert Robbins visiting Purdue in 1966
131.
Charles Sanders Peirce
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Charles Sanders Peirce was an American philosopher, logician, mathematician, and scientist who is sometimes known as the father of pragmatism. He was educated as a chemist and employed as a scientist for 30 years, today he is appreciated largely for his contributions to logic, mathematics, philosophy, scientific methodology, and semiotics, and for his founding of pragmatism. An innovator in mathematics, statistics, philosophy, research methodology, and various sciences, Peirce considered himself, first and foremost and he made major contributions to logic, but logic for him encompassed much of that which is now called epistemology and philosophy of science. As early as 1886 he saw that logical operations could be carried out by electrical switching circuits, in 1934, the philosopher Paul Weiss called Peirce the most original and versatile of American philosophers and Americas greatest logician. Websters Biographical Dictionary said in 1943 that Peirce was now regarded as the most original thinker, keith Devlin similarly referred to Peirce as one of the greatest philosophers ever. Peirce was born at 3 Phillips Place in Cambridge, Massachusetts and he was the son of Sarah Hunt Mills and Benjamin Peirce, himself a professor of astronomy and mathematics at Harvard University and perhaps the first serious research mathematician in America. At age 12, Charles read his older brothers copy of Richard Whatelys Elements of Logic, so began his lifelong fascination with logic and reasoning. At Harvard, he began lifelong friendships with Francis Ellingwood Abbot, Chauncey Wright, one of his Harvard instructors, 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 Peirces working life—repeatedly vetoed Harvards employing Peirce in any capacity. Peirce suffered from his late teens onward from a condition then known as facial neuralgia. Its consequences may have led to the isolation which made his lifes later years so tragic. That employment exempted Peirce from having to part in the Civil War, it would have been very awkward for him to do so. At the Survey, he worked mainly in geodesy and gravimetry and he was elected a resident fellow of the American Academy of Arts and Sciences in January 1867. From 1869 to 1872, he was employed as an Assistant in Harvards astronomical observatory, doing important work on determining the brightness of stars, on April 20,1877 he was elected a member of the National Academy of Sciences. Also in 1877, he proposed measuring the meter as so many wavelengths of light of a certain frequency, during the 1880s, Peirces indifference to bureaucratic detail waxed while his Survey works quality and timeliness waned. Peirce took years to write reports that he should have completed in months, meanwhile, he wrote entries, ultimately thousands during 1883–1909, on philosophy, logic, science, and other subjects for the encyclopedic Century Dictionary. In 1885, an investigation by the Allison Commission exonerated Peirce, in 1891, Peirce resigned from the Coast Survey at Superintendent Thomas Corwin Mendenhalls request. He never again held regular employment, in 1879, Peirce was appointed Lecturer in logic at Johns Hopkins University, which had strong departments in a number of areas that interested him, such as philosophy, psychology, and mathematics
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
132.
Wolfgang Sartorius von Waltershausen
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Wolfgang Sartorius Freiherr von Waltershausen was a German geologist. Waltershausen was born at Göttingen and educated at the university in that city, there he devoted his attention to physical and natural science, and in particular to mineralogy. Waltershausen was named after Johann Wolfgang von Goethe, who was friends with his parents. Waltershausens father, Georg, was a writer, lecturer and professor of economics, Georg Sartorius is best known in his role of translator and popularizer of Adam Smiths Wealth of Nations. Waltershausens son, August, was a well known economist who studied the American economy, 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, meanwhile, he was appointed professor of mineralogy and geology at Göttingen, and held this post for about thirty years, until his death. In 1880, Arnold von Lasaulx edited Waltershausen notes and 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 Gausss biography as Gauss wished it to be told and it is also the source of one of the most famous mathematical quotes, Mathematics is the queen of the sciences. The mineral Sartorite as well as the Waltershausen Glacier in Northeast Greenland73°52′N 24°20′W were named in his honour
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
133.
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 outside Sweden as the author of Mathematics, From the Birth of Numbers, Gullberg grew up and was trained as a surgeon in Sweden. He qualified in medicine at the University of Lund in 1964 and he practised as a surgeon in Saudi Arabia, Norway and Virginia Mason Hospital, Seattle in the United States, as well as in Sweden. Gullberg saw himself as a rather than a writer. His first book, on science, won the Swedish Medical Societys Jubilee Prize in 1980 and he died of a stroke in Nordfjordeid, Norway 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, Gullbergs second book, Mathematics, From the Birth of Numbers, took ten years to write, consuming all of Gullbergs spare time. It proved a success, its first edition of 17,000 copies was virtually sold out within six months. I take it with me everywhere I go, allen says the book has special charm, making innovative use of the margin and providing excellent quotes and quips throughout. His favourite chapter is Cornerstones of Mathematics which he believes should appeal both to beginners and old hands and he admires the efficient Babylonian method of finding square roots, using division and averaging. He learns from Gullberg how to multiply and divide using an abacus, allen is delighted by the chapter on combinatorics, with its approach to graph theory and magic squares, complete with 1740 map of the seven bridges of Königsberg. And he loved the chapter on probability and he records that he finds its introductory accounts useful for engineers who use maths only occasionally, and suggests how the book could be used for undergraduate students. He concludes by describing the book as gigantic, in every sense and was 10 years in the making, and calls it a giant leap forward for mathematics and all those who love it. The book was reviewed in Scientific American, but more reservedly in New Scientist. Kevin Kelly comments that the book is an able to provide answers on obscure mathematical concepts, in his view The book has wit and humor. Gullberg commented At the start no real mathematician would accept my book, and perhaps it was a bit crazy of me to write a book on mathematics, as it would be for a mathematician to write a book on surgery. Vätska Gas Energi – Kemi och Fysik med tillämpningar i vätskebalans-, blodgas- och näringslära Kiruna
Jan Gullberg
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Jan Gullberg
134.
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 company is known for its Norton Anthologies and its texts in the Norton Critical Editions series, the latter of which are frequently assigned in university literature courses. The roots of the date back to 1923, when William Warder Norton founded the firm with his wife Mary Norton. Storer D. Lunt took over in 1945 after Nortons death, and was succeeded by George Brockway, Donald S. Lamm, and now W. Drake McFeely. W. W. Norton & Company is a publisher in the United States, which publishes fiction, nonfiction, poetry, college textbooks, cookbooks, art books. Norton Anthologies Norton Critical Editions Oxford Worlds Classics Verso Books 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
135.
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 career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, and 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. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra, Analysis, in 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited books, and numerous articles. With Michel Demazure and Pierre Gabriel, on invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems. Moduli and canonical forms for linear dynamical systems II, The topological case, on Lie algebras and finite dimensional filtering. Stochastics, a journal of probability and stochastic processes 7. 1–2. Nonexistence of finite-dimensional filters for conditional statistics of the sensor problem. Systems & control letters 3.6, 331–340, the algebra of quasi-symmetric functions is free over the integers
Hazewinkel, Michiel
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Michiel Hazewinkel, 1987
136.
Kluwer Academic Publishers
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Springer also hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages. Book publications include major works, textbooks, monographs and book series. Springer has major offices in Berlin, Heidelberg, Dordrecht, on 15 January 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, and Delhi soon followed. The academic publishing company BertelsmannSpringer was formed after Bertelsmann bought a majority stake in Springer-Verlag 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, 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 London-based private equity firm BC Partners acquired a majority stake in Springer from EQT and GIC for $4.4 billion. In 2014, it was revealed that Springer had published 16 fake papers in its journals that had been computer-generated using SCIgen, Springer subsequently removed all the papers from these journals. IEEE had also done the thing by removing more than 100 fake papers from its conference proceedings. In 2015, Springer retracted 64 of the papers it had published after it was found that they had gone through a fraudulent peer review process, Springer provides its electronic book and journal content on its SpringerLink site, which launched in 1996. SpringerProtocols is home to a collection of protocols, recipes which provide step-by-step instructions for conducting experiments in research labs, SpringerImages was launched in 2008 and offers a collection of currently 1.8 million images spanning science, technology, and medicine. SpringerMaterials was launched in 2009 and is a platform for accessing the Landolt-Börnstein database of research and information on materials, authorMapper is a free online tool for visualizing scientific research that enables document discovery based on author locations and geographic maps. The tool helps users explore patterns in scientific research, identify trends, discover collaborative relationships. While open-access publishing typically requires the author to pay a fee for copyright retention, for example, a national institution in Poland allows authors to publish in open-access journals without incurring any personal cost - but using public funds. Springer is a member of the Open Access Scholarly Publishers Association, the Academic Publishing Industry, A Story of Merger and Acquisition – via Northern Illinois University
Kluwer Academic Publishers
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Springer Science+Business Media
137.
Wikiversity
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Wikiversity is a Wikimedia Foundation project that supports learning communities, their learning materials, and resulting activities. It differs from more structured projects such as Wikipedia in that it offers a series of tutorials, or courses, for the fostering of learning. Wikiversitys beta phase began on August 15,2006, with the English language Wikiversity. The first project proposal was not approved and the second, modified proposal, was approved, the launch of Wikiversity was announced at Wikimania 2006 as, Wikiversity is a center for the creation of and use of free learning materials, and the provision of learning activities. Wikiversity is one of many used in educational contexts, as well as many initiatives that are creating free. The primary priorities and goals for Wikiversity are to, Create and host a range of free-content, multilingual learning materials/resources, host scholarly/learning projects and communities that support these materials. The Wikiversity e-Learning model places emphasis on learning groups and learning by doing, wikiversitys motto and slogan is set learning free, indicating that groups/communities of Wikiversity participants will engage in learning projects. Learning is facilitated through collaboration on projects that are detailed, outlined, summarized or 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, and the Wikiversity community collaborates to develop learning activities, the Wikiversity e-Learning activities give learners the opportunity to build knowledge. Students have to be aware in order to be able to correct their classmates. By doing this, students develop their reflection skills, secondly, they enable students to be autonomous deciding what to write or edit, also when and how to do it. Students are able to free resort to any mean of support, at the same time, it fosters the Cognitive development engaging students to collaborate between them. However, as the project is still in its early stages, Learning resources are developed by an individual or groups, either on their own initiative, or as part of a learning project. Texts useful to others are hosted at Wikibooks for update and maintenance, Learning groups with interests in each subject area create a web of resources that form the basis of discussions and activities at Wikiversity. Learning resources can be used by educators outside of Wikiversity for their own purposes, under the terms of the GFDL, such research content may lack any peer review. WikiJournal is a project that provides quality control by having expert peer review of all included content and this activity started with WikiJournal of Medicine in 2014. WikiJournal of Medicine can also review and publish Wikipedia content. For newly established specific language Wikiversities to move out of the initial beta phase
Wikiversity
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Wikiversity
138.
BBC
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The British Broadcasting Corporation is a British public service broadcaster headquartered at Broadcasting House in London, England. The total number of staff is 35,402 when part-time, flexible, the BBC is established under a Royal Charter and operates under its Agreement with the Secretary of State for Culture, Media and Sport. The fee is set by the British Government, agreed by Parliament, and used to fund the BBCs radio, TV, britains first live public broadcast from the Marconi factory in Chelmsford took place in June 1920. It was sponsored by the Daily Mails Lord Northcliffe and featured the famous Australian Soprano Dame Nellie Melba, the Melba broadcast caught the peoples imagination and marked a turning point in the British publics attitude to radio. However, this public enthusiasm was not shared in official circles where such broadcasts were held to interfere with important military and civil communications. By late 1920, pressure from these quarters and uneasiness among the staff of the licensing authority, the General Post Office, was sufficient to lead to a ban on further Chelmsford broadcasts. But by 1922, the GPO had received nearly 100 broadcast licence requests, John Reith, 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, educate and entertain. The financial arrangements soon proved inadequate, set sales were disappointing as amateurs made their own receivers and listeners bought rival unlicensed sets. By mid-1923, discussions between the GPO and the BBC had become deadlocked and the Postmaster-General commissioned a review of broadcasting by the Sykes Committee and this was to be followed by a simple 10 shillings licence fee with no royalty once the wireless manufactures protection expired. The BBCs broadcasting monopoly was made explicit for the duration of its current broadcast licence, the BBC was also banned from presenting news bulletins before 19.00, and required to source all news from external wire services. Mid-1925 found the future of broadcasting under further consideration, this time by the Crawford committee, by now the BBC under Reiths leadership had forged a consensus favouring a continuation of the unified broadcasting service, but more money was still required to finance rapid expansion. Wireless manufacturers were anxious to exit the loss making consortium with Reith keen that the BBC be seen as a service rather than a commercial enterprise. The recommendations of the Crawford Committee were published in March the following year and were still under consideration by the GPO when the 1926 general strike broke out in May. The strike temporarily interrupted newspaper production and with restrictions on news bulletins waived the BBC suddenly became the source of news for the duration of the crisis. The crisis placed the BBC in a delicate position, the Government was divided on how to handle the BBC but ended up trusting Reith, whose opposition to the strike mirrored the PMs own. Thus the BBC was granted sufficient leeway to pursue the Governments objectives largely in a manner of its own choosing, supporters of the strike nicknamed the BBC the BFC for British Falsehood Company. Reith personally announced the end of the strike which he marked by reciting from Blakes Jerusalem signifying that England had been saved, Reith argued that trust gained by authentic impartial news could then be used
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'.
139.
University of Cambridge
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The University of Cambridge is a collegiate public research university in Cambridge, England, often regarded as one of the most prestigious universities in the world. Founded in 1209 and given royal status by King Henry III in 1231, Cambridge is the second-oldest university in the English-speaking world. 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 share many common features and are often referred to jointly as Oxbridge. Cambridge is formed from a variety of institutions which include 31 constituent colleges, Cambridge University Press, a department of the university, is the worlds oldest publishing house and the second-largest university press in the world. The university also operates eight cultural and scientific museums, including the Fitzwilliam Museum, Cambridges libraries hold a total of around 15 million books, eight million of which are in Cambridge University Library, a legal deposit library. In the year ended 31 July 2015, the university had an income of £1.64 billion. The central university and colleges have an endowment of around £5.89 billion. The university is linked with the development of the high-tech business cluster known as Silicon Fen. It is a member of associations and forms part of the golden triangle of leading English universities and Cambridge University Health Partners. As of 2017, Cambridge is ranked the fourth best university by three ranking tables and no other institution in the world ranks in the top 10 for as many subjects. Cambridge is consistently ranked as the top university in the United Kingdom, the university has educated many notable alumni, including eminent mathematicians, scientists, politicians, lawyers, philosophers, writers, actors, and foreign Heads of State. Ninety-five Nobel laureates, fifteen British prime ministers and ten Fields medalists have been affiliated with Cambridge as students, faculty, by the late 12th century, the Cambridge region already had a scholarly and ecclesiastical reputation, due to monks from the nearby bishopric church of Ely. The University of Oxford went into suspension in protest, and most scholars moved to such as Paris, Reading. After the University of Oxford reformed several years later, enough remained in Cambridge to form the nucleus of the new university. A bull in 1233 from Pope Gregory IX gave graduates from Cambridge the right to teach everywhere in Christendom, the colleges at the University of Cambridge were originally an incidental feature of the system. No college is as old as the university itself, the colleges were endowed fellowships of scholars. There were also institutions without endowments, called hostels, the hostels were gradually absorbed by the colleges over the centuries, but they have left some indicators of their time, such as the name of Garret Hostel Lane. Hugh Balsham, Bishop of Ely, founded Peterhouse, Cambridges first college, the most recently established college is Robinson, built in the late 1970s
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)
140.
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, there are several types of CC licenses. The licenses differ by several combinations that condition the terms of distribution and 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, as of 2016, the 4.0 license suite is the most current. In October 2014 the Open Knowledge Foundation approved the Creative Commons CC BY, CC BY-SA, 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, 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. Any work or copies of the work obtained under a Creative Commons license may continue to be used under that license, in the case of works protected by multiple Creative Common licenses, the user may choose either. The CC licenses all grant the rights, such as the right to distribute the copyrighted work worldwide for non-commercial purposes. For software, Creative Commons includes three free licenses created by other institutions, the BSD License, the GNU LGPL, mixing and matching these conditions produces sixteen possible combinations, of which eleven are valid Creative Commons licenses and five are not. Of the five invalid combinations, four include both the nd and sa clauses, which are exclusive, and one includes none of the clauses. Of the eleven valid combinations, the five that lack the by clause have been retired because 98% of licensors requested attribution, to address this issue, Creative Commons asked its affiliates to translate the various licenses to reflect local laws in a process called porting. As of July 2011, Creative Commons licenses have been ported to over 50 jurisdictions worldwide, the latest version 4.0 of the Creative Commons licenses, released on November 25,2013, are generic licenses that are applicable to most jurisdictions and do not usually require ports. No new ports have implemented in version 4.0 of the license. Version 4.0 discourages using ported versions and instead acts as a single global license, since 2004, all current licenses require attribution of the original author, the BY component. The attribution must be given to the best of ability using the information available, generally this implies the following, Include any copyright notices. Cite the authors name, screen name, or user ID, if the work is being published on the Internet, it is nice to link that name to the persons profile page, if such a page exists
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.
141.
Elementary algebra
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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values and this use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real, the use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations, algebraic notation describes how algebra is written. It follows certain rules and conventions, and has its own terminology, a term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. By convention, letters at the beginning of the alphabet are used to 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 are usually omitted, and 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, 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, likewise when the exponent is one. When the exponent is zero, the result is always 1, however 00, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters, for example, exponents are usually formatted using superscripts, e. g. x 2. In plain text, and in the TeX mark-up language, the symbol ^ represents exponents. In programming languages such as Ada, Fortran, Perl, Python and Ruby, many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example,3 x is written 3*x. Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general numbers and this is useful for several reasons. Variables may represent numbers whose values are not yet known, for example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as C = P +20. Variables allow one to describe general problems, without specifying the values of the quantities that are involved, for example, it can be stated specifically that 5 minutes is equivalent to 60 ×5 =300 seconds
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
142.
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 called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems. Dynamical systems theory and chaos theory deal with the qualitative behavior of dynamical systems. Or Does the long-term behavior of the system depend on its initial condition, an important goal is to describe the fixed points, or steady states of a given dynamical system, these are values of the variable that dont change over time. Some of these points are attractive, meaning that if the system starts out in a nearby state. Similarly, one is interested in points, states of the system that repeat after several timesteps. Periodic points can also be attractive, sharkovskiis theorem is an interesting statement about the number of periodic points of a one-dimensional discrete dynamical system. Even simple nonlinear dynamical systems often exhibit seemingly random behavior that has been called chaos, the branch of dynamical systems that deals with the clean definition and investigation of chaos is called chaos theory. The concept of systems theory has its origins in Newtonian mechanics. Before the advent of fast computing machines, solving a system required sophisticated mathematical techniques. Some excellent presentations of mathematical dynamic system theory include, and, the dynamical system concept is a mathematical formalization for any fixed rule that describes the time dependence of a points position in its ambient space. Examples include the models that describe the swinging of a clock pendulum, the flow of water in a pipe. A dynamical system has a state determined by a collection of real numbers, small changes in the state of the system correspond to small changes in the numbers. The numbers are also the coordinates of a geometrical space—a manifold, the evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule may be deterministic or stochastic and it argues that differential equations are more suited to modelling cognition than more traditional computer models. In mathematics, a system is a system that is not linear—i. e
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.
143.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. 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 in June 1948 in the present-day State Guest-House 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 merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
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