翻訳と辞書 Words near each other ・ Q'umir Qucha (Bolivia) ・ Q'umir Qucha (disambiguation) ・ Q'umirqucha (Camanti) ・ Q'umirqucha (Canchis) ・ Q'umirqucha (Checacupe) ・ Q'umirqucha (Ocongate) ・ Q'umirqucha (Q'umir Qucha) ・ Q'umirqucha (Yanaq Qusqu K'uchu) ・ Q'uq'umatz ・ Q'urawiri ・ Q'uwa Laki ・ Q'Viva! The Chosen ・ Q*bert ・ Q+/Papias Hypothesis ・ Q+A ・ Q-analog ・ Q-analysis ・ Q-ball ・ Q-based narrowing ・ Q-Be ・ Q-Bessel polynomials ・ Q-Bond ・ Q-Bus ・ Q-carbon ・ Q-Cells ・ Q-CERT ・ Q-Charlier polynomials ・ Q-Chastic EP ・ Q-Chem ・ Q-Collection Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Q-analog ： ウィキペディア英語版 Q-analog Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a ''q''-analog of a theorem, identity or expression is a generalization involving a new parameter ''q'' that returns the original theorem, identity or expression in the limit as (this limit is often formal, as q is often discrete-valued). Typically, mathematicians are interested in ''q''-analogues that arise naturally, rather than in arbitrarily contriving ''q''-analogues of known results. The earliest ''q''-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.〔Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538〕 ''q''-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic dynamical systems. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket) and the modular group in particular. The connection passes through hyperbolic geometry and ergodic theory, where the elliptic integrals and modular forms play a prominent role; the ''q''-series themselves are closely related to elliptic integrals. ''q''-analogs also appear in the study of quantum groups and in ''q''-deformed superalgebras. The connection here is similar, in that much of string theory is set in the language of Riemann surfaces, resulting in connections to elliptic curves, which in turn relate to ''q''-series. There are two main groups of ''q''-analogs, the "classical" ''q''-analogs, with beginnings in the work of Leonhard Euler and extended by F. H. Jackson〔F. H. Jackson (1908), "On q-functions and a certain difference operator", ''Trans. Roy. Soc. Edin.'', 46 253–281.〕 and others. =="Classical" ''q''-theory== Classical ''q''-theory begins with the ''q''-analogs of the nonnegative integers.〔 The equality :$\backslash lim\_\backslash frac=n$ suggests that we define the ''q''-analog of ''n'', also known as the ''q''-bracket or ''q''-number of ''n'', to be :$()\_q=\backslash frac\; =\; 1\; +\; q\; +\; q^2\; +\; \backslash ldots\; +\; q^.$ By itself, the choice of this particular ''q''-analog among the many possible options is unmotivated. However, it appears naturally in several contexts. For example, having decided to use ()''q'' as the ''q''-analog of ''n'', one may define the ''q''-analog of the factorial, known as the ''q''-factorial, by : $$ \begin \big()_q! & =()_q \cdot ()_q \cdots ()_q \cdot ()_q \\() & =\frac \cdot \frac \cdots \frac \cdot \frac \\() & =1\cdot (1+q)\cdots (1+q+\cdots + q^) \cdot (1+q+\cdots + q^). \end This ''q''-analog appears naturally in several contexts. Notably, while ''n''! counts the number of permutations of length ''n'', ()''q''! counts permutations while keeping track of the number of inversions. That is, if inv(''w'') denotes the number of inversions of the permutation ''w'' and ''S''''n'' denotes the set of permutations of length ''n'', we have :$\backslash sum\_\; q^\; =\; ()\_q\; !\; .$ In particular, one recovers the usual factorial by taking the limit as $q\backslash rightarrow\; 1$. The ''q''-factorial also has a concise definition in terms of the ''q''-Pochhammer symbol, a basic building-block of all ''q''-theories: :$()\_q!=\backslash frac.$ From the ''q''-factorials, one can move on to define the ''q''-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomial coefficients: :$$ \binom_q = \frac. The ''q''-exponential is defined as: :$e\_q^x\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$ ''q''-trigonometric functions, along with a ''q''-Fourier transform have been defined in this context. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Q-analog」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.