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Triangular Array
In mathematics and computing, a triangular array of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index. That is, the ''i''th row contains only ''i'' elements. Examples Notable particular examples include these: *The Bell triangle, whose numbers count the Partition of a set, partitions of a set in which a given element is the largest singleton (mathematics), singleton * Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched * Euler's triangle, which counts permutations with a given number of ascents * Floyd's triangle, whose entries are all of the integers in order * Hosoya's triangle, based on the Fibonacci numbers * Lozanić's triangle, used in the mathematics of chemical compounds * Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings * Pascal's triangle, whose entries are the binomial coefficients Triangular arrays of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula ''n''2 − 1 for the ''n''th term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the ''n''th perfect number. Examples Integer sequences that have their own name include: *Abundant numbers *Baum–Sweet sequence *Bell numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boustrophedon Transform
In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-like) manner—as opposed to a "Raster Scan" sawtooth-like manner. Definition The boustrophedon transform is a numerical, sequence-generating transformation, which is determined by an "addition" operation. Generally speaking, given a sequence: (a_0, a_1, a_2, \ldots), the boustrophedon transform yields another sequence: (b_0, b_1, b_2, \ldots), where b_0 is likely defined equivalent to a_0. The entirety of the transformation itself can be visualized (or imagined) as being constructed by filling-out the triangle as shown in Figure 1. Boustrophedon Triangle To fill-out the numerical Isosceles triangle (Figure 1), you start with the input sequence, (a_0, a_1, a_2, \ldots), and place one value (from the input sequence) per row, using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definite Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Romberg's Method
In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. The method is named after Werner Romberg (1909–2003), who published the method in 1955. Method Using h_n = \frac, the method can be inductively defined by \begin R(0,0) &= h_1 (f(a) + f(b)) \\ R(n,0) &= \tfrac R(n-1,0) + h_n \sum_^ f(a + (2k-1)h_n) \\ R(n,m) &= R(n,m-1) + \tfrac (R(n,m-1) - R(n-1,m-1)) \\ &= \frac ( 4^m R( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have ''optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; Rives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Context-free Grammar
In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules are of the form :A\ \to\ \alpha with A a ''single'' nonterminal symbol, and \alpha a string of terminals and/or nonterminals (\alpha can be empty). A formal grammar is "context-free" if its production rules can be applied regardless of the context of a nonterminal. No matter which symbols surround it, the single nonterminal on the left hand side can always be replaced by the right hand side. This is what distinguishes it from a context-sensitive grammar. A formal grammar is essentially a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the first rule in the picture, :\langle\text\rangle \to \langle\text\rangle = \langle\text\rangle ; replaces \langle\text\rangle with \langle\text\rangle = \langle\text\rangle ;. There can be multiple replacement rules for a given nonterminal symbol. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CYK Algorithm
In computer science, the Cocke–Younger–Kasami algorithm (alternatively called CYK, or CKY) is a parsing algorithm for context-free grammars published by Itiroo Sakai in 1961. The algorithm is named after some of its rediscoverers: John Cocke, Daniel Younger, Tadao Kasami, and Jacob T. Schwartz. It employs bottom-up parsing and dynamic programming. The standard version of CYK operates only on context-free grammars given in Chomsky normal form (CNF). However any context-free grammar may be transformed (after convention) to a CNF grammar expressing the same language . The importance of the CYK algorithm stems from its high efficiency in certain situations. Using big ''O'' notation, the worst case running time of CYK is \mathcal\left( n^3 \cdot \left, G \ \right), where n is the length of the parsed string and \left, G \ is the size of the CNF grammar G . This makes it one of the most efficient parsing algorithms in terms of worst-case asymptotic complexity, although other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a Heuristic (computer science), heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are zero. Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix ''L'' and an upper triangular matrix ''U'' if and only if all its leading principal minors are non-zero. Description A matrix of the form :L = \begin \ell_ & & & & 0 \\ \ell_ & \ell_ & & & \\ \ell_ & \ell_ & \ddots & & \\ \vdots & \vdots & \ddots & \ddots & \\ \ell_ & \ell_ & \ldots & \ell_ & \ell_ \end is called a lower triangular matrix or left triangular matrix, and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bell Polynomials
In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula. Definitions Exponential Bell polynomials The ''partial'' or ''incomplete'' exponential Bell polynomials are a triangular array of polynomials given by :B_(x_1,x_2,\dots,x_) = \sum \left(\right)^\left(\right)^\cdots\left(\right)^, where the sum is taken over all sequences ''j''1, ''j''2, ''j''3, ..., ''j''''n''−''k''+1 of non-negative integers such that these two conditions are satisfied: :j_1 + j_2 + \cdots + j_ = k, :j_1 + 2 j_2 + 3 j_3 + \cdots + (n-k+1)j_ = n. The sum :B_n(x_1,\dots,x_n)=\sum_^n B_(x_1,x_2,\dots,x_) is called the ''n''th ''complete exponential Bell polynomial''. Ordinary Bell polynomials Likewise, the partial ''ordinary'' Bell polynomial is defined by :\hat_(x_1,x_2,\ldots,x_) = \sum \frac x_1^ x_2^ \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
