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Q-derivative
In mathematics, in the area of combinatorics and quantum calculus, the ''q''-derivative, or Jackson derivative, is a ''q''-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's ''q''-integration. For other forms of q-derivative, see . Definition The ''q''-derivative of a function ''f''(''x'') is defined as :\left(\frac\right)_q f(x)=\frac. It is also often written as D_qf(x). The ''q''-derivative is also known as the Jackson derivative. Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator :D_q= \frac ~ \frac ~, which goes to the plain derivative, D_q \to \frac as q \to 1. It is manifestly linear, :\displaystyle D_q (f(x)+g(x)) = D_q f(x) + D_q g(x)~. It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms :\displaystyle D_q (f(x)g(x)) = g(x)D_q f(x) + f(qx)D_q g(x) = g(qx)D_q f(x) + f(x)D_q g(x). Similarly, it satisfies a quotient r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative (generalizations)
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet derivative defines the derivative for general normed vector spaces V, W. Briefly, a function f : U \to W, where U is an open subset of V, is called ''Fréchet differentiable'' at x \in U if there exists a bounded linear operator A:V\to W such that \lim_ \frac = 0. Functions are defined as being differentiable in some open neighbourhood of x, rather than at individual points, as not doing so tends to lead to many pathological counterexamples. The Fréchet derivative is quite similar to the formula for the derivative found in elementary one-variable calculus, \lim_\frac = A, and simply moves ''A'' to the left hand side. However, the Fréchet derivative ''A'' denotes the function t \mapsto f'(x) \cdot t. In multivariable calculus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-exponential
The term ''q''-exponential occurs in two contexts. The q-exponential distribution, based on the Tsallis q-exponential is discussed in elsewhere. In combinatorial mathematics, a ''q''-exponential is a ''q''-analog of the exponential function, namely the eigenfunction of a ''q''-derivative. There are many ''q''-derivatives, for example, the classical ''q''-derivative, the Askey–Wilson operator, etc. Therefore, unlike the classical exponentials, ''q''-exponentials are not unique. For example, e_q(z) is the ''q''-exponential corresponding to the classical ''q''-derivative while \mathcal_q(z) are eigenfunctions of the Askey–Wilson operators. The ''q''-exponential is also known as the quantum dilogarithm. Definition The ''q''-exponential e_q(z) is defined as :e_q(z)= \sum_^\infty \frac = \sum_^\infty \frac = \sum_^\infty z^n\frac where _q is the ''q''-factorial and :(q;q)_n=(1-q^n)(1-q^)\cdots (1-q) is the ''q''-Pochhammer symbol. That this is the ''q''-analog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-difference Polynomial
In combinatorial mathematics, the ''q''-difference polynomials or ''q''-harmonic polynomials are a polynomial sequence defined in terms of the ''q''-derivative. They are a generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence. Definition The q-difference polynomials satisfy the relation :\left(\frac \right)_q p_n(z) = \frac = \frac p_(z)= qp_(z) where the derivative symbol on the left is the q-derivative. In the limit of q\to 1, this becomes the definition of the Appell polynomials: :\fracp_n(z) = np_(z). Generating function The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely :A(w)e_q(zw) = \sum_^\infty \frac w^n where e_q(t) is the q-exponential: :e_q(t)=\sum_^\infty \frac= \sum_^\infty \frac. Here, q! is the q-factorial and :(q;q)_n=(1-q^n)(1-q^)\cdots (1-q) is the q-Pochhammer symbol In the mathematical field of combinatorics, the ''q''-P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Calculus
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. The two types of calculus in quantum calculus are ''q''-calculus and ''h''-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit, the original object is returned. In ''q''-calculus, the limit as ''q'' tends to 1 is taken of the ''q''-analog. Likewise, in ''h''-calculus, the limit as h tends to 0 is taken of the ''h''-analog. The parameters q and h can be related by the formula q = e^h. Differentiation The ''q''-differential and ''h''-differential are defined as: :d_q(f(x)) = f(qx) - f(x) and :d_h(f(x)) = f(x + h) - f(x), respectively. The ''q''-derivative and ''h''-derivative are then defined as :D_q(f(x)) = \frac = \frac and :D_h(f(x)) = \frac = \frac respectively. By taking the limit as q \rightarrow 1 of the ''q''-derivative or as h \rightarrow 0 of the ''h''-derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tsallis Entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann–Gibbs entropy. It is proportional to the expectation of the q-logarithm of a distribution. History The concept was introduced in 1988 by Constantino Tsallis as a basis for generalizing the standard statistical mechanics and is identical in form to Havrda–Charvát structural α-entropy, introduced in 1967 within information theory. Definition Given a discrete set of probabilities \ with the condition \sum_i p_i=1, and q any real number, the Tsallis entropy is defined as :S_q() = k \cdot \frac \left( 1 - \sum_i p_i^q \right), where q is a real parameter sometimes called ''entropic-index'' and k a positive constant. In the limit as q \to 1, the usual Boltzmann–Gibbs entropy is recovered, namely :S_\text = S_1(p) = -k \sum_i p_i \ln p_i , where one identifies k with the Boltzmann constant k_B. For continuous probability distributions, we define the entropy as :S_q = \left( 1 - \int (p(x)) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Calculus
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. The two types of calculus in quantum calculus are ''q''-calculus and ''h''-calculus. The goal of both types is to find "analogs" of mathematical objects, where, after taking a certain limit, the original object is returned. In ''q''-calculus, the limit as ''q'' tends to 1 is taken of the ''q''-analog. Likewise, in ''h''-calculus, the limit as h tends to 0 is taken of the ''h''-analog. The parameters q and h can be related by the formula q = e^h. Differentiation The ''q''-differential and ''h''-differential are defined as: :d_q(f(x)) = f(qx) - f(x) and :d_h(f(x)) = f(x + h) - f(x), respectively. The ''q''-derivative and ''h''-derivative are then defined as :D_q(f(x)) = \frac = \frac and :D_h(f(x)) = \frac = \frac respectively. By taking the limit as q \rightarrow 1 of the ''q''-derivative or as h \rightarrow 0 of the ''h''-derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Q-bracket
In the mathematical field of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product (a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^), with (a;q)_0 = 1. It is a ''q''-analog of the Pochhammer symbol (x)_n = x(x+1)\dots(x+n-1), in the sense that \lim_ \frac = (x)_n. The ''q''-Pochhammer symbol is a major building block in the construction of ''q''-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series. Unlike the ordinary Pochhammer symbol, the ''q''-Pochhammer symbol can be extended to an infinite product: (a;q)_\infty = \prod_^ (1-aq^k). This is an analytic function of ''q'' in the interior of the unit disk, and can also be considered as a formal power series in ''q''. The special case \phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k) is known as Euler's function, and is important in combinatorics, number t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
