|
Pascal's Inversion Formula
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. Definition The binomial transform, , of a sequence, , is the sequence defined by s_n = \sum_^n (-1)^k \binom a_k. Formally, one may write s_n = (Ta)_n = \sum_^n T_ a_k for the transformation, where is an infinite-dimensional operator with matrix elements . The transform is an involution, that is, TT = 1 or, using index notation, \sum_^\infty T_ T_ = \delta_ where \delta_ is the Kronecker delta. The original series can be regained by a_n=\sum_^n (-1)^k \binom s_k. The binomial transform of a sequence is just the -th forward differences of the sequence, with odd differences carrying a negative sign, namely: \begin s_0 &= a_0 \\ s_1 &= - (\Delta a)_0 = -a_1+a_0 \\ s_2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analytic
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, ''holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which deals with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Factorial And Binomial Topics
{{Short description, none This is a list of factorial and binomial topics in mathematics. See also binomial (other). * Abel's binomial theorem *Alternating factorial *Antichain *Beta function * Bhargava factorial *Binomial coefficient **Pascal's triangle *Binomial distribution *Binomial proportion confidence interval * Binomial-QMF ( Daubechies wavelet filters) *Binomial series *Binomial theorem *Binomial transform *Binomial type * Carlson's theorem *Catalan number ** Fuss–Catalan number *Central binomial coefficient *Combination *Combinatorial number system *De Polignac's formula *Difference operator * Difference polynomials *Digamma function * Egorychev method * Erdős–Ko–Rado theorem *Euler–Mascheroni constant *Faà di Bruno's formula *Factorial * Factorial moment *Factorial number system *Factorial prime * Factoriangular number *Gamma distribution *Gamma function *Gaussian binomial coefficient * Gould's sequence * Hyperfactorial *Hypergeometric distribution * Hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Liouville Integral
In mathematics, the Riemann–Liouville integral associates with a real function f: \mathbb \rightarrow \mathbb another function of the same kind for each value of the parameter . The integral is a manner of generalization of the repeated antiderivative of in the sense that for positive integer values of , is an iterated antiderivative of of order . The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions. It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential. Motivation The Riemann-Liouville integral is motivated from Cauchy formula for repeated integration. For a function continuous on the interval the Cauchy formula for -fold repeated integration states that I^n f(x) = f^(x) = \frac \int_a^x\left(x-t\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial QMF
A binomial QMF – properly an orthonormal binomial quadrature mirror filter – is an orthogonal wavelet developed in 1990. The binomial QMF bank with perfect reconstruction (PR) was designed by Ali Akansu, and published in 1990, using the family of binomial polynomials for subband decomposition of discrete-time signals. Akansu and his fellow authors also showed that these binomial-QMF filters are identical to the wavelet filters designed independently by Ingrid Daubechies from compactly supported orthonormal wavelet transform perspective in 1988 ( Daubechies wavelet). It was an extension of Akansu's prior work on Binomial coefficient and Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ... wherein he developed the Modified Hermite Transformation (MHT) in 1987. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Summation
In the mathematics of convergent and divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ..., Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ''a''''n'', if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series. Euler summation can be generalized into a family of methods denoted (E, ''q''), where ''q'' ≥ 0. The (E, 1) sum is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for ''q'' > 0 they are incomparable with Abel su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Transform
In combinatorial mathematics, the Stirling transform of a sequence of numbers is the sequence given by :b_n=\sum_^n \left\ a_k, where \left\ is the Stirling number of the second kind, which is the number of partitions of a set of size n into k parts. This is a linear sequence transformation. The inverse transform is :a_n=\sum_^n (-1)^ \leftrightb_k, where (-1)^ \leftright/math> is a signed Stirling number of the first kind, where the unsigned \leftright/math> can be defined as the number of permutations on n elements with k cycles. Berstein and Sloane (cited below) state "If ''a''''n'' is the number of objects in some class with points labeled 1, 2, ..., ''n'' (with all labels distinct, i.e. ordinary labeled structures), then ''b''''n'' is the number of objects with points labeled 1, 2, ..., ''n'' (with repetitions allowed)." If :f(x) = \sum_^\infty x^n is a formal power series, and :g(x) = \sum_^\infty x^n with ''a''''n'' and ''b''''n'' as above, then :g(x) = f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transform
Moebius, Mœbius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Friedrich Möbius (art historian) (1928–2024), German art historian and architectural historian * Theodor Möbius (1821–1890), German philologist, son of August Ferdinand * Karl Möbius (1825–1908), German zoologist and ecologist * Paul Julius Möbius (1853–1907), German neurologist, grandson of August Ferdinand * Dieter Moebius (1944–2015), Swiss-born German musician * Mark Mobius (born 1936), emerging markets investments pioneer * Jean Giraud (1938–2012), French comics artist who used the pseudonym Mœbius Fictional characters * Mobius M. Mobius, a character in Marvel Comics * Mobius, also known as the Anti-Monitor, a supervillain in DC Comics * Johann Wilhelm Möbius, a character in the play '' The Physicists'' * Moebius, the main antagonistic faction in the video game ''Xenoblade Chronicles 3'' * Mobius, or Dr. Ignatio Mobius, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hankel Matrix
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example, \qquad\begin a & b & c & d & e \\ b & c & d & e & f \\ c & d & e & f & g \\ d & e & f & g & h \\ e & f & g & h & i \\ \end. More generally, a Hankel matrix is any n \times n matrix A of the form A = \begin a_0 & a_1 & a_2 & \ldots & a_ \\ a_1 & a_2 & & &\vdots \\ a_2 & & & & a_ \\ \vdots & & & a_ & a_ \\ a_ & \ldots & a_ & a_ & a_ \end. In terms of the components, if the i,j element of A is denoted with A_, and assuming i \le j, then we have A_ = A_ for all k = 0,...,j-i. Properties * Any Hankel matrix is symmetric. * Let J_n be the n \times n exchange matrix. If H is an m \times n Hankel matrix, then H = T J_n where T is an m \times n Toeplitz matrix. ** If T is real symmetric, then H = T J_n will have the same eigenvalues as T up to sign. * The Hilbert matrix is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Series
A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly denoted \Delta, is the operator that maps a function to the function \Delta /math> defined by \Delta x) = f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term "finite difference" is often used as an abbreviation of "finite difference approximation of derivatives". Finite differences were introduced by Brook Taylor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Operator
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag operator''. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator. Definition Functions of a real variable The shift operator (where ) takes a fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hankel Transform Of A Series
In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a rectangular matrix in which each ascending skew-diagonal from left to right is constant. For example, \qquad\begin a & b & c & d & e \\ b & c & d & e & f \\ c & d & e & f & g \\ d & e & f & g & h \\ e & f & g & h & i \\ \end. More generally, a Hankel matrix is any n \times n matrix A of the form A = \begin a_0 & a_1 & a_2 & \ldots & a_ \\ a_1 & a_2 & & &\vdots \\ a_2 & & & & a_ \\ \vdots & & & a_ & a_ \\ a_ & \ldots & a_ & a_ & a_ \end. In terms of the components, if the i,j element of A is denoted with A_, and assuming i \le j, then we have A_ = A_ for all k = 0,...,j-i. Properties * Any Hankel matrix is symmetric. * Let J_n be the n \times n exchange matrix. If H is an m \times n Hankel matrix, then H = T J_n where T is an m \times n Toeplitz matrix. ** If T is real symmetric, then H = T J_n will have the same eigenvalues as T up to sign. * The Hilbert matrix is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
