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Separability Criteria
Separability may refer to: Mathematics * Separable algebra, a generalization to associative algebras of the notion of a separable field extension * Separable differential equation, in which separation of variables is achieved by various means * Separable extension, in field theory, an algebraic field extension * Separable filter, a product of two or more simple filters in image processing * Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals * Separable partial differential equation, a class of equations that can be broken down into differential equations in fewer independent variables * Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation * Separable polynomial, a polynomial whose number of distinct roots is equal to its degree * Separable sigma algebra, a separable space in measure theory * Separable space, a topological space that contains a countable, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Algebra
In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension. Definition and first properties A homomorphism of (unital, but not necessarily commutative) rings : K \to A is called ''separable'' if the multiplication map : \begin \mu :& A \otimes_K A &\to& A \\ & a \otimes b &\mapsto & ab \end admits a section : \sigma: A \to A \otimes_K A that is a homomorphism of ''A''-''A''-bimodules. If the ring K is commutative and K \to A maps K into the center of A, we call A a ''separable algebra over'' K. It is useful to describe separability in terms of the element : p := \sigma(1) = \sum a_i \otimes b_i \in A \otimes_K A The reason is that a section ''σ'' is determined by this element. The condition that ''σ'' is a section of ''μ'' is equivalent to : \sum a_i b_i = 1 and the condition that σ is a homomorphism of ''A''-''A''-bimodules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Of Variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary differential equation, ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) A differential equation for the unknown f(x) is separable if it can be written in the form :\frac f(x) = g(x)h(f(x)) where g and h are given functions. This is perhaps more transparent when written using y = f(x) as: :\frac=g(x)h(y). So now as long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, where the two variables ''x'' and ''y'' have been separated. Note ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Extension
In field theory (mathematics), field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial (field theory), minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated zero of a function, roots in any extension field).Isaacs, p. 281 There is also a more general definition that applies when is not necessarily algebraic over . An extension that is not separable is said to be ''inseparable''. Every algebraic extension of a field (mathematics), field of characteristic (algebra)#Case of fields, characteristic zero is separable, and every algebraic extension of a finite field is separable.Isaacs, Theorem 18.11, p. 281 It follows that most extensions that are considered in mathematics are separable. Nevertheless, the concept of separability is important, as the existence of inseparable extensions is the main ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Filter
A separable filter in image processing can be written as product of two more simple filters. Typically a 2-dimensional convolution operation is separated into two 1-dimensional filters. This reduces the computational costs on an N\times M image with a m\times n filter from \mathcal(M\cdot N\cdot m\cdot n) down to \mathcal(M\cdot N\cdot (m + n)). Examples 1. A two-dimensional smoothing filter: : \frac \begin 1 \\ 1 \\ 1 \end * \frac \begin 1 & 1 & 1 \end = \frac \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end 2. Another two-dimensional smoothing filter with stronger weight in the middle: : \frac \begin 1 \\ 2 \\ 1 \end * \frac \begin 1 & 2 & 1 \end = \frac \begin 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end 3. The Sobel operator, used commonly for edge detection: : \begin 1 \\ 2 \\ 1 \end * \begin 1 & 0 & -1 \end = \begin 1 & 0 & -1 \\ 2 & 0 & -2 \\ 1 & 0 & -1 \end This works also for the Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Partial Differential Equation
A separable partial differential equation can be broken into a set of equations of lower dimensionality (fewer independent variables) by a method of separation of variables. It generally relies upon the problem having some special form or symmetry. In this way, the partial differential equation (PDE) can be solved by solving a set of simpler PDEs, or even ordinary differential equations (ODEs) if the problem can be broken down into one-dimensional equations. The most common form of separation of variables is simple separation of variables. A solution is obtained by assuming a solution of the form given by a product of functions of each individual coordinate. There is a special form of separation of variables called R-separation of variables which is accomplished by writing the solution as a particular fixed function of the coordinates multiplied by a product of functions of each individual coordinate. Laplace's equation on ^n is an example of a partial differential equation that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Permutation
In combinatorics, combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by Direct sum of permutations, direct sums and Skew sum of permutations, skew sums. Separable permutations may be characterized by the forbidden permutation patterns 2413 and 3142;; , Theorem 2.2.36, p. p.58. they are also the permutations whose permutation graphs are cographs and the permutations that Order dimension, realize the series-parallel partial orders. It is possible to test in polynomial time whether a given separable permutation is a pattern in a larger permutation, or to find the longest common subpattern of two separable permutations. Definition and characterization define a separable permutation to be a permutation that has a ''separating tree'': a rooted binary tree in which the elements of the permutation appear (in permutation order) at the leaves of the tree, and in which the descendants of each tree node form a contiguou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Polynomial
In mathematics, a polynomial ''P''(''X'') over a given field ''K'' is separable if its roots are distinct in an algebraic closure of ''K'', that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If ''K'' is a perfect field then the two concepts coincide. In general, ''P''(''X'') is separable if and only if it is square-free over any field that contains ''K'', which holds if and only if ''P''(''X'') is coprime to its formal derivative ''D'' ''P''(''X''). Older definition In an older definition, ''P''(''X'') was considered separable if each of its irreducible factors in ''K'' 'X''is separable in the modern definition. In this definition, separability depended on the field ''K''; for example, any polynomial over a perfect field would have been considered separable. This definition, although it can be convenient for Galois theory, is no longer in use. Separable field extensions Separable pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Sigma Algebra
Separability may refer to: Mathematics * Separable algebra, a generalization to associative algebras of the notion of a separable field extension * Separable differential equation, in which separation of variables is achieved by various means * Separable extension, in field theory, an algebraic field extension * Separable filter, a product of two or more simple filters in image processing * Separable ordinary differential equation, a class of equations that can be separated into a pair of integrals * Separable partial differential equation, a class of equations that can be broken down into differential equations in fewer independent variables * Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation * Separable polynomial, a polynomial whose number of distinct roots is equal to its degree * Separable sigma algebra, a separable space in measure theory * Separable space, a topological space that contains a countable, dens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x_n )_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Separability
In Euclidean geometry, linear separability is a property of two sets of points. This is most easily visualized in two dimensions (the Euclidean plane) by thinking of one set of points as being colored blue and the other set of points as being colored red. These two sets are ''linearly separable'' if there exists at least one line in the plane with all of the blue points on one side of the line and all the red points on the other side. This idea immediately generalizes to higher-dimensional Euclidean spaces if the line is replaced by a hyperplane. The problem of determining if a pair of sets is linearly separable and finding a separating hyperplane if they are, arises in several areas. In statistics and machine learning, classifying certain types of data is a problem for which good algorithms exist that are based on this concept. Mathematical definition Let X_ and X_ be two sets of points in an ''n''-dimensional Euclidean space. Then X_ and X_ are ''linearly separable'' if ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursively Inseparable Sets
In computability theory, two disjoint sets of natural numbers are called computably inseparable or recursively inseparable if they cannot be "separated" with a computable set.Monk 1976, p. 100 These sets arise in the study of computability theory itself, particularly in relation to \Pi^0_1 classes. Computably inseparable sets also arise in the study of Gödel's incompleteness theorem. Definition The natural numbers are the set \mathbb = \. Given disjoint subsets A and B of \mathbb, a separating set C is a subset of \mathbb such that A \subseteq C and B \cap C = \emptyset (or equivalently, A \subseteq C and B \subseteq C', where C' = \mathbb \setminus C denotes the complement of C). For example, A itself is a separating set for the pair, as is B'. If a pair of disjoint sets A and B has no computable separating set, then the two sets are computably inseparable. Examples If A is a non-computable set, then A and its complement are computably inseparable. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
