|
Spartans
Sparta 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 in the valley of Evrotas river in Laconia, in southeastern Peloponnese. Around 650 BC, it rose to become the dominant military land-power in ancient Greece. 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 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 against Thebes in 371 BC ended the Spartan hegemony, although the city-state maintained its political independence until its forced integration into the Achaean League in 192 BC. The city nevertheless recovered much autonomy after the Roman conquest of Greece in 146 BC and prospered during the Roman Emp ... [...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 BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as well as consonants. In Archaic Greece, Archaic and early Classical Greece, Classical times, the Greek alphabet existed in Archaic Greek alphabets, many local variants, but, by the end of the 4th century BC, the Ionia, Ionic-based Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard throughout the Greek-speaking world and is the version that is still used for Greek writing today. The letter case, uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , , , , , , , The Greek alphabet is the ancestor of several scripts, such as the Latin script, Latin, Gothic alphabet, Gothic, Coptic script, Coptic, and Cyrillic scripts. Throughout antiquity, Greek had only a single uppercas ... [...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 Greece, archaic and Classical Greece, 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 (letter), Xi () was used only in a subgroup 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 standa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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) = ... [...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 Root of a function, zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important List of unsolved problems in mathematics, 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 Hilbert's problems, twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to 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' ... [...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 liv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
El (Cyrillic)
El (Л л or Ʌ ʌ; 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. 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. Forms El has two forms: one form resembles Greek capital Lambda (Ʌ ʌ), and the other form resembles the Hebrew letter ת (Л л). In some typeface A typeface (or font family) is a design of Letter (alphabet), letters, Numerical digit, numbers and other symbols, to be used in printing or for electronic display. Most typefaces include variations in size (e.g., 24 point), weight (e.g., light, ...s the Cyrillic letter El has a grapheme which may be confused with the Cyrillic letter Pe (П п). Note that Pe has a straight left leg, without the hoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn–Newman Constant
The de Bruijn–Newman constant, denoted by \Lambda and named after Nicolaas Govert de Bruijn and Charles Michael Newman, is a mathematical constant defined via the zeros of a certain function H(\lambda,z), where \lambda is a real parameter and z is a complex variable. More precisely, :H(\lambda, z):=\int_^ e^ \Phi(u) \cos (z u) \, du, where \Phi is the super-exponentially decaying function :\Phi(u) = \sum_^ (2\pi^2n^4e^-3\pi n^2 e^ ) e^ and \Lambda is the unique real number with the property that H has only real zeros if and only if \lambda\geq \Lambda. The constant is closely connected with Riemann hypothesis. Indeed, the Riemann hypothesis is equivalent to the conjecture that \Lambda\leq 0. (announcement post) Brad Rodgers and Terence Tao proved that \Lambda\geq 0, so the Riemann hypothesis is equivalent to \Lambda=0. A simplified proof of the Rodgers–Tao result was later given by Alexander Dobner. History De Bruijn showed in 1950 that H has only real zeros ... [...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] |
Lamed
Lamedh or lamed is the twelfth letter of the Semitic abjads, including Hebrew ''lāmeḏ'' , Aramaic ''lāmaḏ'' 𐡋, Syriac ''lāmaḏ'' ܠ, Arabic ''lām'' , and Phoenician ''lāmd'' 𐤋. Its sound value is . It is also related to the Ancient North Arabian 𐪁, South Arabian , and Ge'ez . The Phoenician letter gave rise to the Greek Lambda (Λ), Latin L, and Cyrillic El (Л). Origin The letter is usually considered to have originated from the representation of an ox-goad, i.e. a cattle prod, or a shepherd's crook, i.e. a pastoral staff. In Proto-Semitic a goad was called *''lamed''-. Arabic lām The letter is named لام . Orthography Its form depends on its position in the word: Grammatical functions has functions as a grammatical particle when used as a prefix: * Prepositional () * of ownership () * of association () * of purpose () * of absolute negation () * Imperative () * of affirmative emphasis () (, ) is essentially a prepos ... [...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 or Homeric 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, which are the best-attested periods and considered most typical of Ancient Greek. From the Hellenistic period (), Ancient Greek was followed by Koine Greek, which is regarded as a separate historical stage, though its earliest form closely resembles Attic Greek, and its latest form approaches Medieval Greek, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
