The Ancient Greek language includes the forms of Greek used in Ancient Greece and the ancient world from around the 9th century BCE to the 6th century CE. It is roughly divided into the Archaic period, Classical period, Hellenistic period, it is succeeded by medieval Greek. Koine is regarded as a separate historical stage of its own, although in its earliest form it resembled Attic Greek and in its latest form it approaches Medieval Greek. Prior to the Koine period, Greek of the classic and earlier periods included several regional dialects. Ancient Greek was the language of Homer and of fifth-century Athenian historians and philosophers, it has contributed many words to English vocabulary and has been a standard subject of study in educational institutions of the Western world since the Renaissance. This article contains information about the Epic and Classical periods of the language. Ancient Greek was a pluricentric language, divided into many dialects; the main dialect groups are Attic and Ionic, Aeolic and Doric, many of them with several subdivisions.
Some dialects are found in standardized literary forms used in literature, while others are attested only in inscriptions. There are several historical forms. Homeric Greek is a literary form of Archaic Greek used in the epic poems, the "Iliad" and "Odyssey", in poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic and other Classical-era dialects; the origins, early form and development of the Hellenic language family are not well understood because of a lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period, they differ in some of the detail. The only attested dialect from this period is Mycenaean Greek, but its relationship to the historical dialects and the historical circumstances of the times imply that the overall groups existed in some form. Scholars assume that major Ancient Greek period dialect groups developed not than 1120 BCE, at the time of the Dorian invasion—and that their first appearances as precise alphabetic writing began in the 8th century BCE.
The invasion would not be "Dorian" unless the invaders had some cultural relationship to the historical Dorians. The invasion is known to have displaced population to the Attic-Ionic regions, who regarded themselves as descendants of the population displaced by or contending with the Dorians; the Greeks of this period believed there were three major divisions of all Greek people—Dorians and Ionians, each with their own defining and distinctive dialects. Allowing for their oversight of Arcadian, an obscure mountain dialect, Cypriot, far from the center of Greek scholarship, this division of people and language is quite similar to the results of modern archaeological-linguistic investigation. One standard formulation for the dialects is: West vs. non-west Greek is the strongest marked and earliest division, with non-west in subsets of Ionic-Attic and Aeolic vs. Arcadocypriot, or Aeolic and Arcado-Cypriot vs. Ionic-Attic. Non-west is called East Greek. Arcadocypriot descended more from the Mycenaean Greek of the Bronze Age.
Boeotian had come under a strong Northwest Greek influence, can in some respects be considered a transitional dialect. Thessalian had come under Northwest Greek influence, though to a lesser degree. Pamphylian Greek, spoken in a small area on the southwestern coast of Anatolia and little preserved in inscriptions, may be either a fifth major dialect group, or it is Mycenaean Greek overlaid by Doric, with a non-Greek native influence. Most of the dialect sub-groups listed above had further subdivisions equivalent to a city-state and its surrounding territory, or to an island. Doric notably had several intermediate divisions as well, into Island Doric, Southern Peloponnesus Doric, Northern Peloponnesus Doric; the Lesbian dialect was Aeolic Greek. All the groups were represented by colonies beyond Greece proper as well, these colonies developed local characteristics under the influence of settlers or neighbors speaking different Greek dialects; the dialects outside the Ionic group are known from inscriptions, notable exceptions being: fragments of the works of the poet Sappho from the island of Lesbos, in Aeolian, the poems of the Boeotian poet Pindar and other lyric poets in Doric.
After the conquests of Alexander the Great in the late 4th century BCE, a new international dialect known as Koine or Common Greek developed based on Attic Greek, but with influence from other dialects. This dialect replaced most of the older dialects, although Doric dialect has survived in the Tsakonian language, spoken in the region of modern Sparta. Doric has passed down its aorist terminations into most verbs of Demotic Greek. By about the 6th century CE, the Koine had metamorphosized into Medieval Greek. Ancient Macedonian was an Indo-European language at least related to Greek, but its exact relationship is unclear because of insufficient data: a dialect of Greek; the Macedonian dialect (or l
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Doron Zeilberger is an Israeli mathematician, known for his work in combinatorics. He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym. with the thesis "New Approaches and Results in the Theory of Discrete Analytic Functions." He is a Board of Governors Professor of Mathematics at Rutgers University. Zeilberger has made contributions to combinatorics, hypergeometric identities, q-series. Zeilberger gave the first proof of the alternating sign matrix conjecture, noteworthy not only for its mathematical content, but for the fact that Zeilberger recruited nearly a hundred volunteer checkers to "pre-referee" the paper. In 2011, together with Manuel Kauers and Christoph Koutschan, Zeilberger proved the q-TSPP conjecture, independently stated in 1983 by George Andrews and David P. Robbins. Zeilberger is an ultrafinitist, he is known for crediting his computer "Shalosh B. Ekhad" as a co-author, for his provocative opinions. Zeilberger received a Lester R. Ford Award in 1990.
Together with Herbert Wilf, Zeilberger was awarded the American Mathematical Society's Leroy P. Steele Prize for Seminal Contributions to Research in 1998 for their development of WZ theory, which has revolutionized the field of hypergeometric summation. In 2004, Zeilberger was awarded the Euler Medal. In 2016 he received, together with Manuel Kauers and Christoph Koutschan, the David P. Robbins Prize of the American Mathematical Society. Zeilberger was a member of the inaugural 2013 class of fellows of the American Mathematical Society. MacMahon Master theorem Wilf–Zeilberger pair Doron Zeilberger's homepage Biography from ScienceWorld Weisstein, Eric W. "Zeilberger's Algorithm". MathWorld. Weisstein, Eric W. "Wilf-Zeilberger Pair". MathWorld. Weisstein, Eric W. "Alternating Sign Matrix Conjecture". MathWorld. Weisstein, Eric W. "Refined Alternating Sign Matrix Conjecture". MathWorld. Weisstein, Eric W. "Zeilberger-Bressoud Theorem". MathWorld. From A = B to Z = 60, a conference in honor of Doron Zeilberger's 60th birthday, 27 and 28 May 2010
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Urysohn's lemma is used to construct continuous functions with various properties on normal spaces, it is applicable since all metric spaces and all compact Hausdorff spaces are normal. The lemma is generalized by the Tietze extension theorem; the lemma is named after the mathematician Pavel Samuilovich Urysohn. Two subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhoods U of A and V of B that are disjoint. In particular A and B are disjoint. Two plain subsets A and B are said to be separated by a function if there exists a continuous function f from X into the unit interval such that f = 0 for all a in A and f = 1 for all b in B. Any such function is called a Urysohn function for A and B. In particular A and B are disjoint, it follows that if two subsets A and B are separated by a function so are their closures.
It follows that if two subsets A and B are separated by a function A and B are separated by neighbourhoods. A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function; the sets A and B need not be separated by f, i.e. we do not, in general cannot, require that f ≠ 0 and ≠ 1 for x outside of A and B. The spaces in which this property holds are the normal spaces. Urysohn's lemma has led to the formulation of other topological properties such as the'Tychonoff property' and'completely Hausdorff spaces'. For example, a corollary of the lemma is; the procedure is an straightforward application of the definition of normality, beginning with two disjoint closed sets. The clever part of the proof is the indexing the open sets thus constructed by dyadic fractions. For every dyadic fraction r ∈, we are going to construct an open subset U of X such that: U contains A and is disjoint from B for all r for r < s, the closure of U is contained in U.
Once we have these sets, we define f = 1. Using the fact that the dyadic rationals are dense, it is not too hard to show that f is continuous and has the property f ⊆ and f ⊆. In order to construct the sets U, we do a little bit more: we construct sets U and V such that A ⊆ U and B ⊆ V for all r U and V are open and disjoint for all r for r < s, V is contained in the complement of U and the complement of V is contained in U. Since the complement of V is closed and contains U, the latter condition implies condition from above; this construction proceeds by mathematical induction. First define U = X \ B and V = X \ A. Since X is normal, we can find two disjoint open sets V which contain A and B, respectively. Now assume that n≥1 and the sets U and V have been constructed for k = 1...2n-1. Since X is normal, for any a ∈, we can find two disjoint open sets which contain X \ V and X \ U, respectively. Call these two open sets U and V, verify the above three conditions; the Mizar project has formalized and automatically checked a proof of Urysohn's lemma in the URYSOHN3 file.
Cutoff function Willard, Stephen. General Topology. Dover Publications. ISBN 0-486-43479-6. Hazewinkel, Michiel, ed. "Urysohn lemma", Encyclopedia of Mathematics, Springer Science+Business Media B. V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 "proof of Urysohn's lemma". PlanetMath. Mizar system proof: http://mizar.org/version/current/html/urysohn3.html#T20
In mathematics, a theorem is a statement, proven on the basis of established statements, such as other theorems, accepted statements, such as axioms. A theorem is a logical consequence of the axioms; the proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, experimental. Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called premises. In light of the interpretation of proof as justification of truth, the conclusion is viewed as a necessary consequence of the hypotheses, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.
Although they can be written in a symbolic form, for example, within the propositional calculus, theorems are expressed in a natural language such as English. The same is true of proofs, which are expressed as logically organized and worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, from which a formal symbolic proof can in principle be constructed; such arguments are easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but explains in some way why it is true. In some cases, a picture alone may be sufficient to prove a theorem; because theorems lie at the core of mathematics, they are central to its aesthetics. Theorems are described as being "trivial", or "difficult", or "deep", or "beautiful"; these subjective judgments vary not only from person to person, but with time: for example, as a proof is simplified or better understood, a theorem, once difficult may become trivial.
On the other hand, a deep theorem may be stated but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a well-known example of such a theorem. Logically, many theorems are of the form of an indicative conditional: if A B; such a theorem does not assert B, only that B is a necessary consequence of A. In this case A is called B the conclusion; the theorem "If n is an natural number n/2 is a natural number" is a typical example in which the hypothesis is "n is an natural number" and the conclusion is "n/2 is a natural number". To be proved, a theorem must be expressible as a formal statement. Theorems are expressed in natural language rather than in a symbolic form, with the intention that the reader can produce a formal statement from the informal one, it is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses are called axioms or postulates.
The field of mathematics known as proof theory studies formal languages and the structure of proofs. Some theorems are "trivial", in the sense that they follow from definitions and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example is Fermat's Last Theorem, there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas. Other theorems have a known proof that cannot be written down; the most prominent examples are the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search, verified by a computer program. Many mathematicians did not accept this form of proof, but it has become more accepted.
The mathematician Doron Zeilberger has gone so far as to claim that these are the only nontrivial results that mathematicians have proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. To establish a mathematical statement as a theorem, a proof is required, that is, a line of reasoning from axioms in the system to the given statement must be demonstrated. However, the proof is considered as separate from the theorem statement. Although more than one proof may be known for a single theorem, only one proof is required to establish the status of a statement as a theorem; the Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs. Theorems in mathematics and theories in science are fundamentally different in their epistemology. A scientific theory cannot be proved.