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MacMahon's Master Theorem
In mathematics, MacMahon's master theorem (MMT) is a result in enumerative combinatorics and linear algebra. It was discovered by Percy MacMahon and proved in his monograph ''Combinatory analysis'' (1916). It is often used to derive binomial identities, most notably Dixon's identity. Background In the monograph, MacMahon found so many applications of his result, he called it "a master theorem in the Theory of Permutations." He explained the title as follows: "a Master Theorem from the masterly and rapid fashion in which it deals with various questions otherwise troublesome to solve." The result was re-derived (with attribution) a number of times, most notably by I. J. Good who derived it from his multilinear generalization of the Lagrange inversion theorem. MMT was also popularized by Carlitz who found an exponential power series version. In 1962, Good found a short proof of Dixon's identity from MMT. In 1969, Cartier and Foata found a new proof of MMT by combining a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumerative Combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are '' closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koszul Algebra
In abstract algebra, a Koszul algebra R is a graded k-algebra over which the ground field k has a linear minimal graded free resolution, ''i.e.'', there exists an exact sequence: :\cdots \rightarrow (R(-i))^ \rightarrow \cdots \rightarrow (R(-2))^ \rightarrow (R(-1))^ \rightarrow R \rightarrow k \rightarrow 0. for some nonnegative integers b_i. Here R(-j) is the graded algebra R with grading shifted up by j, ''i.e.'' R(-j)_i = R_, and the exponent b_i refers to the b_i-fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, ''e.g'', R = k ,y(xy) . The concept is named after the French mathematician Jean-Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying and the coefficient of each term is a specific positive integer depending on and . For example, for , (x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. The coefficient in each term is known as the binomial coefficient or (the two have the same value). These coefficients for varying and can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where gives the number of different combinations (i.e. subsets) of elements that can be chosen from an -element set. Therefore is usually pronounced as " choose ". Statement According to the theorem, the expansion of any nonnegative integer power of the binomial is a sum of the form (x+y)^n = x^n y^0 + x^ y^1 + x^ y^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derangement
In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points. The number of derangements of a set of size is known as the subfactorial of or the derangement number or de Montmort number (after Pierre Remond de Montmort). Notations for subfactorials in common use include , , , or . For , the subfactorial equals the nearest integer to , where denotes the factorial of and is Euler's number. The problem of counting derangements was first considered by Pierre Raymond de Montmort in his '' Essay d'analyse sur les jeux de hazard'' in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time. Example Suppose that a professor gave a test to 4 students – A, B, C, and D – and wants to let them grade each other's tests. Of course, no student should grade their own test. How many ways could the prof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac14 + \tfrac18 + \cdots is a geometric series with common ratio , which converges to the sum of . Each term in a geometric series is the geometric mean of the term before it and the term after it, in the same way that each term of an arithmetic series is the arithmetic mean of its neighbors. While Ancient Greek philosophy, Greek philosopher Zeno's paradoxes about time and motion (5th century BCE) have been interpreted as involving geometric series, such series were formally studied and applied a century or two later by Greek mathematics, Greek mathematicians, for example used by Archimedes to Quadrature of the Parabola, calculate the area inside a parabola (3rd century BCE). Today, geometric series are used in mathematical finance, calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 4 \end\right/math>. An identity matrix of any size, or any multiple of it is a diagonal matrix called a ''scalar matrix'', for example, \left begin 0.5 & 0 \\ 0 & 0.5 \end\right/math>. In geometry, a diagonal matrix may be used as a '' scaling matrix'', since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with columns and rows is diagonal if \forall i,j \in \, i \ne j \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coefficient
In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless quantity, number without units, in which case it is known as a numerical factor. It may also be a constant (mathematics), constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any mathematical expression, expression (including Variable (mathematics), variables such as , and ). When the combination of variables and constants is not necessarily involved in a product (mathematics), product, it may be called a ''parameter''. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. A , also known as constant term or simply constant, is a quantity either implicitly attach ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-particle System
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Terminology ''Microscopic'' here implies that quantum mechanics has to be used to provide an accurate description of the system. ''Many'' can be anywhere from three to infinity (in the case of a practically infinite, homogeneous or periodic system, such as a crystal), although three- and four-body systems can be treated by specific means (respectively the Faddeev and Faddeev–Yakubovsky equations) and are thus sometimes separately classified as few-body systems. Explanation of the problem In general terms, while the underlying physical laws that govern the motion of each individual particle may (or may not) be simple, the study of the collection of particles can be extremely complex. In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction (geometry), direction and a magnitude, and both are conserved. Bicycle and motorcycle dynamics, Bicycles and motorcycles, flying discs, Rifling, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a point particle is classically represented as a pseudovector , the cross product of the particle's position vector (relative to some origin) and its mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
