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Symmetrization
In mathematics, symmetrization is a process that converts any function in n variables to a symmetric function in n variables. Similarly, antisymmetrization converts any function in n variables into an antisymmetric function. Two variables Let S be a set and A be an additive abelian group. A map \alpha : S \times S \to A is called a if \alpha(s,t) = \alpha(t,s) \quad \text s, t \in S. It is called an if instead \alpha(s,t) = - \alpha(t,s) \quad \text s, t \in S. The of a map \alpha : S \times S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x). Similarly, the or of a map \alpha : S \times S \to A is the map (x,y) \mapsto \alpha(x,y) - \alpha(y,x). The sum of the symmetrization and the antisymmetrization of a map \alpha is 2 \alpha. Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function. The symmetrization of a symmetric map is its dou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Subrepresentation
In representation theory, a subrepresentation of a representation (\pi, V) of a group ''G'' is a representation (\pi, _W, W) such that ''W'' is a vector subspace of ''V'' and \pi, _W(g) = \pi(g), _W. A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations. If (\pi, V) is a representation of ''G'', then there is the trivial subrepresentation: :V^G = \. If f: V \to W is an equivariant map In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ... between two representations, then its kernel is a subrepresentation of V and its image is a subrepresentation of W. References * Representation theory {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bootstrapping (statistics)
Bootstrapping is a procedure for estimating the distribution of an estimator by resampling (often with replacement) one's data or a model estimated from the data. Bootstrapping assigns measures of accuracy ( bias, variance, confidence intervals, prediction error, etc.) to sample estimates.software This technique allows estimation of the sampling distribution of almost any statistic using random sampling methods. Bootstrapping estimates the properties of an estimand (such as its ) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distributi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Symmetric Polynomials
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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Representation Theory Of The Symmetric Group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. The symmetric group S''n'' has order ''n''!. Its conjugacy classes are labeled by partitions of ''n''. Therefore according to the representation theory of a finite group, the number of inequivalent irreducible representations, over the complex numbers, is equal to the number of partitions of ''n''. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representations by the same set that parametrizes conjugacy classes, namely by partitions of ''n'' or equivalently Young diagrams of size ''n''. Each such irreducible representation can in fact be realized over the integers (every permutation acting by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Even Permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of ''X'' is fixed, the parity (oddness or evenness) of a permutation \sigma of ''X'' can be defined as the parity of the number of inversions for ''σ'', i.e., of pairs of elements ''x'', ''y'' of ''X'' such that and . The sign, signature, or signum of a permutation ''σ'' is denoted sgn(''σ'') and defined as +1 if ''σ'' is even and −1 if ''σ'' is odd. The signature defines the alternating character of the symmetric group S''n''. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol (''ε''''σ''), which is defined for all maps from ''X'' to ''X'', and has value zero for non-bijective maps. The sign of a permutation can be explicitly expressed as : where ''N''('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
