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Spartans
Sparta (Doric Greek: Σπάρτα, ''Spártā''; Attic Greek: wikt:Σπάρτη, Σπάρτη, ''Spártē'') was a prominent city-state in Laconia, in ancient Greece. In antiquity, the city-state was known as Lacedaemon (, ), while the name Sparta referred to its main settlement on the banks of the Eurotas (river), Eurotas River in Laconia, in south-eastern Peloponnese. Around 650 BC, it rose to become the dominant Military, military land-power in ancient Greece. Given its military pre-eminence, Sparta was recognized as the leading force of the unified Greek military during the Greco-Persian Wars, in rivalry with the rising naval power of Classical Athens, Athens. Sparta was the principal enemy of Athens during the Peloponnesian War (431–404 BC), from which it emerged victorious after the Battle of Aegospotami. The decisive Battle of Leuctra in 371 BC ended the Spartan hegemony, although the city-state maintained its Independence, political independence until its for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn–Newman Constant
The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zeros of a certain function ''H''(''λ'', ''z''), where ''λ'' is a real parameter and ''z'' is a complex variable. More precisely, :H(\lambda, z):=\int_^ e^ \Phi(u) \cos (z u) d u, where \Phi is the super-exponentially decaying function :\Phi(u) = \sum_^ (2\pi^2n^4e^ - 3 \pi n^2 e^ ) e^ and Λ is the unique real number with the property that ''H'' has only real zeros if and only if ''λ'' ≥ Λ. The constant is closely connected with Riemann's hypothesis concerning the zeros of the Riemann zeta-function: since the Riemann hypothesis is equivalent to the claim that all the zeroes of ''H''(0, ''z'') are real, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0. (announcement post) Brad Rodgers and Terence Tao proved that Λ < 0 cannot be true, so |
Eigendecomposition Of A Matrix
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Fundamental theory of matrix eigenvectors and eigenvalues A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form :\mathbf \mathbf = \lambda \mathbf for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues : p\left(\lambda\right) = \d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as well as consonants. In Archaic and early Classical times, the Greek alphabet existed in many local variants, but, by the end of the 4th century BCE, the Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard and it is this version that is still used for Greek writing today. The uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , /ς, , , , , , . The Greek alphabet is the ancestor of the Latin and Cyrillic scripts. Like Latin and Cyrillic, Greek originally had only a single form of each letter; it developed the letter case distinction between uppercase and lowercase in parallel with Latin during the modern era. Sound values and conventional transcriptions for some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epichoric Alphabets
Many local variants of the Greek alphabet were employed in ancient Greece during the archaic and early classical periods, until around 400 BC, when they were replaced by the classical 24-letter alphabet that is the standard today. All forms of the Greek alphabet were originally based on the shared inventory of the 22 symbols of the Phoenician alphabet, with the exception of the letter Samekh, whose Greek counterpart Xi () was used only in a sub-group of Greek alphabets, and with the common addition of Upsilon () for the vowel . The local, so-called ''epichoric'', alphabets differed in many ways: in the use of the consonant symbols , and ; in the use of the innovative long vowel letters ( and ), in the absence or presence of Η in its original consonant function (); in the use or non-use of certain archaic letters ( = , = , = ); and in many details of the individual shapes of each letter. The system now familiar as the standard 24-letter Greek alphabet was originally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Particle
The lambda baryons (Λ) are a family of subatomic hadron particles containing one up quark, one down quark, and a third quark from a higher flavour generation, in a combination where the quantum wave function changes sign upon the flavour of any two quarks being swapped (thus slightly different from a neutral sigma baryon, ). They are thus baryons, with total isospin of 0, and have either neutral electric charge or the elementary charge +1. Overview The lambda baryon was first discovered in October 1950, by V. D. Hopper and S. Biswas of the University of Melbourne, as a neutral V particle with a proton as a decay product, thus correctly distinguishing it as a baryon, rather than a meson, i.e. different in kind from the K meson discovered in 1947 by Rochester and Butler; they were produced by cosmic rays and detected in photographic emulsions flown in a balloon at . Though the particle was expected to live for , it actually survived for . The property that caused it to l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
El (Cyrillic)
El (Л л; italics: ) is a letter of the Cyrillic script. El commonly represents the alveolar lateral approximant . In Slavic languages it may be either palatalized or slightly velarized; see below. Allography In some typefaces the Cyrillic letter El has a grapheme which may be confused with the Cyrillic letter Pe (Пп) Note that Pe has a straight left leg, without the hook. An alternative form of El (Ʌ ʌ) is more common in Russian, Ukrainian, Belarusian, Bulgarian, Macedonian, and Serbian. History The Cyrillic letter El was derived from the Greek letter lambda (Λ λ). In the Early Cyrillic alphabet its name was (''ljudije''), meaning "people". In the Cyrillic numeral system, Л had a value of 30. Pronunciation As used in the alphabets of various languages, El represents the following sounds: * alveolar lateral approximant , like the pronunciation of in "lip" * palatalized alveolar lateral approximant * velarized alveolar lateral approximant , like the pronun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Mangoldt Function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mangoldt function, denoted by , is defined as :\Lambda(n) = \begin \log p & \textn=p^k \text p \text k \ge 1, \\ 0 & \text \end The values of for the first nine positive integers (i.e. natural numbers) are :0 , \log 2 , \log 3 , \log 2 , \log 5 , 0 , \log 7 , \log 2 , \log 3, which is related to . Properties The von Mangoldt function satisfies the identityApostol (1976) p.32Tenenbaum (1995) p.30 :\log(n) = \sum_ \Lambda(d). The sum is taken over all integers that divide . This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to . For example, consider the case . Then :\begin \sum_ \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann's Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(''s'') is a function whose argument ''s'' may be any complex num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal numbers, however, modern Greece uses Arabic numerals. History The Minoan and Mycenaean civilizations' Linear A and Linear B alphabets used a different system, called Aegean numerals, which included number-only symbols for powers of ten: = 1, = 10, = 100, = 1000, and = 10000. Attic numerals comprised another system that came into use perhaps in the 7th century BCE. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran = 1, = 5, = 10, = 100, = 1,000, and =&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamed
Lamedh or Lamed is the twelfth Letter (alphabet), letter of the Semitic abjads, including Hebrew alphabet, Hebrew Lāmed , Aramaic alphabet, Aramaic Lāmadh , Syriac alphabet, Syriac Lāmaḏ ܠ, Arabic alphabet, Arabic , and Phoenician alphabet, Phoenician Lāmed . Its sound value is . The Phoenician letter gave rise to the Greek alphabet, Greek Lambda (Λ), Latin alphabet, Latin L, and Cyrillic script, Cyrillic El (Cyrillic), El (Л). Origin The letter is usually considered to have originated from the representation of a goad, i.e. a cattle prod, or a shepherd's crook, i.e. a crosier, pastoral staff. Hebrew Lamed Hebrew spelling: Pronunciation Lamed transcribes as an alveolar lateral approximant . Significance Lamed in gematria represents the number 30. With the letter Waw (letter), Vav it refers to the Lamed Vav Tzadikim, Lamedvavniks, the 36 righteous people who save the world from destruction. As an abbreviation, it can stand for litre. Also, a sign on a car with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Greek
Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic period (), and the Classical period (). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, 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 primarily contains information about the Epic and Classical periods of the language. From the Hellenistic period (), Ancient Greek was followed by Koine Greek, which is regarded as a separate historical stage, although its earliest form closely resembles Attic Greek and its latest form approaches Medieval Greek. There were several regional dialects of Ancient Greek, of which Attic Greek developed into Koine. Dial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
