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Complete Homogeneous Symmetric Polynomial
In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials. Definition The complete homogeneous symmetric polynomial of degree in variables , written for , is the sum of all monomials of total degree in the variables. Formally, :h_k (X_1, X_2, \dots,X_n) = \sum_ X_ X_ \cdots X_. The formula can also be written as: :h_k (X_1, X_2, \dots,X_n) = \sum_ X_^ X_^ \cdots X_^. Indeed, is just the multiplicity of in the sequence . The first few of these polynomials are :\begin h_0 (X_1, X_2, \dots,X_n) &= 1, \\ 0pxh_1 (X_1, X_2, \dots,X_n) &= \sum_ X_j, \\ h_2 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k, \\ h_3 (X_1, X_2, \dots,X_n) &= \sum_ X_j X_k X_l. \end Thus, for each nonnegative integer , there exists exactly one complete homogeneous symmetric polynomi ... [...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] |
Plethystic Exponential
In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions, as a concise relation between the generating series for elementary, complete and power sums homogeneous symmetric polynomials in many variables. Its name comes from the operation called plethysm, defined in the context of so-called lambda rings. In combinatorics, the plethystic exponential is a generating function for many well studied sequences of integers, polynomials or power series, such as the number of integer partitions. It is also an important technique in the enumerative combinatorics of unlabelled graphs, and many other combinatorial objects. In geometry and topology, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariant of its symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension (vector Space)
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is wiktionary:finite, finite, and if its dimension is infinity, infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as [V : F], read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any Field (mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Identities
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial ''P'' in one variable, they allow expressing the sums of the ''k''-th powers of all roots of ''P'' (counted with their multiplicity) in terms of the coefficients of ''P'', without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity. Mathematical statement Formulation in terms of symmetric polynomials Let ''x''1, ..., ''x''''n'' be variables, denote for ''k'' ≥ 1 by ''p''''k''(''x''1, ..., ''x''''n'') the ''k''-th p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial Symmetric Polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every ''symmetric'' polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Number
In mathematics, Stirling numbers arise in a variety of Analysis (mathematics), analytic and combinatorics, combinatorial problems. They are named after James Stirling (mathematician), James Stirling, who introduced them in a purely algebraic setting in his book ''Methodus differentialis'' (1730). They were rediscovered and given a combinatorial meaning by Masanobu Saka in his 1782 ''Sanpō-Gakkai'' ''(The Sea of Learning on Mathematics)''. Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendence Basis
In mathematics, a transcendental extension L/K is a field extension such that there exists an element in the field L that is transcendental element, transcendental over the field K; that is, an element that is not a root of any univariate polynomial with coefficients in K. In other words, a transcendental extension is a field extension that is not Algebraic extension, algebraic. For example, \mathbb and \mathbb are both transcendental extensions of \mathbb. A transcendence basis of a field extension L/K (or a transcendence basis of L over K) is a maximal algebraically independent subset of L over K. Transcendence bases share many properties with basis (linear algebra), bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is nonzero. Transcendental extensions are widely used in alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator (mathematics)
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. List of generators A list of examples of generating sets follow. * Generating set or spanning set of a vector space: a set that spans the vector space * Generating set of a group: a subset of a group that is not contained in any sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Function
In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f\left(x_2,x_1\right) for all x_1 and x_2 such that \left(x_1,x_2\right) and \left(x_2,x_1\right) are in the domain of f. The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials. A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k-tensors on a vector space V is isomorphic to the space of homogeneous polynomials of degree k on V. Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry. Symmetrization Given any function f in n variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
