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Xi (letter)
Xi is the 14th letter of the Greek alphabet (uppercase Ξ, lowercase ξ; el, ξι), representing the voiceless consonant cluster . It is pronounced in Modern Greek, and generally or in English. In the system of Greek numerals, it has a value of 60. Xi was derived from the Phoenician letter samekh . Xi is distinct from the letter chi, which gave its form to the Latin letter X. Greek Both in classical Ancient Greek and in Modern Greek, the letter Ξ represents the consonant cluster /ks/. In some archaic local variants of the Greek alphabet, this letter was missing. Instead, especially in the dialects of most of the Greek mainland and Euboea, the cluster /ks/ was represented by Χ (which in classical Greek is chi, used for ). Because this variant of the Greek alphabet was used in Magna Graecia (the Greek colonies in Sicily and the southern part of the Italian peninsula), the Latin alphabet borrowed Χ rather than Ξ as the Latin letter that represented the /ks/ cluster ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triple Bar
The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an equals sign sign with a third line. The triple bar character in Unicode is code point .. The closely related code point is the same symbol with a slash through it, indicating the negation of its mathematical meaning. In LaTeX mathematical formulas, the code \equiv produces the triple bar symbol and \not\equiv produces the negated triple bar symbol as output. Uses Mathematics and philosophy In logic, it is used with two different but related meanings. It can refer to the if and only if connective, also called material equivalence. This is a binary operation whose value is true when its two arguments have the same value as each other. Alternatively, in some texts ⇔ is used with this meaning, while ≡ is used for the higher-level metalogical notion of logical equivalence, according to which two formulas are logically equivalent when all models give the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sicily
(man) it, Siciliana (woman) , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = Ethnicity , demographics1_footnotes = , demographics1_title1 = Sicilian , demographics1_info1 = 98% , demographics1_title2 = , demographics1_info2 = , demographics1_title3 = , demographics1_info3 = , timezone1 = CET , utc_offset1 = +1 , timezone1_DST = CEST , utc_offset1_DST = +2 , postal_code_type = , postal_code = , area_code_type = ISO 3166 code , area_code = IT-82 , blank_name_sec1 = GDP (nominal) , blank_info_sec1 = €89.2 billion (2018) , blank1_name_sec1 = GDP per capita , blank1_info_sec1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Canonical Ensemble
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system. The thermodynamic variables of the grand canonical ensemble are chemical potential (symbol: ) and absolute temperature (symbol: . The ensemble is also dependent on mechanical variables such as volume (symbol: which influence the nature of the system's internal states. This ensemble is therefore sometimes called the ensemble, as each of these three quantities ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (statistical Mechanics)
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless. Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xi Baryon
The Xi baryons or ''cascade particles'' are a family of subatomic hadron particles which have the symbol and may have an electric charge () of +2 , +1 , 0, or −1 , where is the elementary charge. Like all conventional baryons, particles contain three quarks. baryons, in particular, contain either one up or one down quark and two other, more massive quarks. The two more massive quarks are any two of strange, charm, or bottom (doubles allowed). For notation, the assumption is that the two heavy quarks in the are both strange; subscripts "c" and "b" are added for each even heavier charm or bottom quark that replaces one of the two presumed strange quarks. They are historically called the ''cascade particles'' because of their unstable state; they are typically observed to decay rapidly into lighter particles, through a chain of decays (cascading decays). The first discovery of a charged Xi baryon was in cosmic ray experiments by the Manchester group in 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Xi Function
In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. Definition Riemann's original lower-case "xi"-function, \xi was renamed with an upper-case ~\Xi~ ( Greek letter "Xi") by Edmund Landau. Landau's lower-case ~\xi~ ("xi") is defined as :\xi(s) = \frac s(s-1) \pi^ \Gamma\left(\frac\right) \zeta(s) for s \in \mathbb. Here \zeta(s) denotes the Riemann zeta function and \Gamma(s) is the Gamma function. The functional equation (or reflection formula) for Landau's ~\xi~ is :\xi(1-s) = \xi(s)~. Riemann's original function, rebaptised upper-case ~\Xi~ by Landau, satisfies :\Xi(z) = \xi \left(\tfrac + z i \right), and obeys the functional equation :\Xi(-z) = \Xi(z)~. Both functions are entire and purely real for real arguments. Values The general form for positive even integers is :\xi(2n) = (-1)^\fracB_2^\pi^(2n-1) where ''Bn'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are Multiple (mathematics), integer multiples of one another, as are the frequencies of the Harmonic series (music), harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harish-Chandra's Ξ Function
In mathematical harmonic analysis, Harish-Chandra's ''Ξ'' function is a special spherical function on a semisimple Lie group, studied by . Harish-Chandra used it to define Harish-Chandra's Schwartz space. gives a detailed description of the properties of ''Ξ''. Definition :\Xi(g)=\int_Ka(kg)^\rho dk, where *''K'' is a maximal compact subgroup of a semisimple Lie group with Iwasawa decomposition ''G''=''NAK'' *''g'' is an element of ''G'' *''ρ'' is a Weyl vector In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the c ... *''a''(''g'') is the element ''a'' in the Iwasawa decomposition ''g''=''nak'' References * * {{DEFAULTSORT:Harish-Chandra's Xi function Harmonic analysis Representation theory Special functions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ksi (Cyrillic Letter)
Ksi (Ѯ, ѯ) is a letter of the early Cyrillic alphabet, derived from the Greek letter Xi (Ξ, ξ). It was mainly used in Greek loanwords, especially words relating to the Church. Ksi was eliminated from the Russian alphabet along with psi, omega, and yus in the Civil Script of 1708 ( Peter the Great's ''Grazhdanka''), and has also been dropped from other secular languages. It was briefly restored in 1710 and ultimately removed in 1735. While it was no longer used in typographic fonts, it continued to be used by the church, and since clergy actively participated in civil censuses, Ksi can be found in multiple handwritten civil texts all the way until the early 1800s. In the Civil Script during Peter the Great's time, ksi was also written similarly to an izhitsa Izhitsa or Izhica (Ѵ, ѵ; italics: ; OCS: Ѷжица, Russian: Ижица, Ukrainian: Іжиця) is a letter of the early Cyrillic alphabet and several later alphabets, usually the last in the row. It origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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