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Reduced Free Group
Reduction, reduced, or reduce may refer to: Science and technology Chemistry * Reduction (chemistry), part of a reduction-oxidation (redox) reaction in which atoms have their oxidation state changed. ** Organic redox reaction, a redox reaction that takes place with organic compounds ** Ore reduction: see smelting Computing and algorithms * Reduction (complexity), a transformation of one problem into another problem * Reduction (recursion theory), given sets A and B of natural numbers, is it possible to effectively convert a method for deciding membership in B into a method for deciding membership in A? * Bit Rate Reduction, an audio compression method * Data reduction, simplifying data in order to facilitate analysis * Graph reduction, an efficient version of non-strict evaluation * L-reduction, a transformation of optimization problems which keeps the approximability features * Partial order reduction, a technique for reducing the size of the state-space to be searched by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction (chemistry)
Redox (reduction–oxidation, , ) is a type of chemical reaction in which the oxidation states of substrate (chemistry), substrate change. Oxidation is the loss of Electron, electrons or an increase in the oxidation state, while reduction is the gain of electrons or a decrease in the oxidation state. There are two classes of redox reactions: * ''Electron-transfer'' – Only one (usually) electron flows from the reducing agent to the oxidant. This type of redox reaction is often discussed in terms of redox couples and electrode potentials. * ''Atom transfer'' – An atom transfers from one substrate to another. For example, in the rusting of iron, the oxidation state of iron atoms increases as the iron converts to an oxide, and simultaneously the oxidation state of oxygen decreases as it accepts electrons released by the iron. Although oxidation reactions are commonly associated with the formation of oxides, other chemical species can serve the same function. In hydrogen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Reduction
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was introduced by the mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics. Lambda calculus consists of constructing § lambda terms and performing § reduction operations on them. In the simplest form of lambda calculus, terms are built using only the following rules: * x – variable, a character or string representing a parameter or mathematical/logical value. * (\lambda x.M) – abstraction, function definition (M is a lambda term). The variable x becomes bound in the expression. * (M\ N) – application, applying a function M to an argument N. M and N are lambda terms. The reduction operations include: * (\lambda x.M \rightarrow(\la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Word
In group theory, a word is any written product of group elements and their inverses. For example, if ''x'', ''y'' and ''z'' are elements of a group ''G'', then ''xy'', ''z''−1''xzz'' and ''y''−1''zxx''−1''yz''−1 are words in the set . Two different words may evaluate to the same value in ''G'', or even in every group. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory. Definitions Let ''G'' be a group, and let ''S'' be a subset of ''G''. A word in ''S'' is any expression of the form :s_1^ s_2^ \cdots s_n^ where ''s''1,...,''sn'' are elements of ''S'', called generators, and each ''εi'' is ±1. The number ''n'' is known as the length of the word. Each word in ''S'' represents an element of ''G'', namely the product of the expression. By convention, the unique Uniqueness of identity element and inverses identity element can be represented by the empty word, which is the unique word ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Row Echelon Form
In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices. Specifically, a matrix is in row echelon form if * All rows consisting of only zeroes are at the bottom. * The leading entry (that is the left-most nonzero entry) of every nonzero row is to the right the leading entry of every row above. Some texts add the condition that the leading coefficient must be 1 while others regard this as ''reduced'' row echelon form. These two conditions imply that all entries in a column below a leadin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Ring
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring ''R'' form an ideal of ''R'', called the nilradical of ''R''; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring ''R/I'' is reduced if and only if ''I'' is a radical ideal. Let ''D'' be the set of all zero-divisors in a reduced ring ''R''. Then ''D'' is the union of all minimal prime ideals. Over a Noetherian ring ''R'', we say a finitely generated module ''M'' has locally constant rank if \mathfrak \mapsto \operatorname_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Residue System
In mathematics, a subset ''R'' of the integers is called a reduced residue system modulo ''n'' if: #gcd(''r'', ''n'') = 1 for each ''r'' in ''R'', #''R'' contains φ(''n'') elements, #no two elements of ''R'' are congruent modulo ''n''. Here φ denotes Euler's totient function. A reduced residue system modulo ''n'' can be formed from a complete residue system modulo ''n'' by removing all integers not relatively prime to ''n''. For example, a complete residue system modulo 12 is . The so-called totatives 1, 5, 7 and 11 are the only integers in this set which are relatively prime to 12, and so the corresponding reduced residue system modulo 12 is . The cardinality of this set can be calculated with the totient function: φ(12) = 4. Some other reduced residue systems modulo 12 are: * * * * Facts *If is a reduced residue system modulo ''n'' with ''n'' > 2, then \sum r_i \equiv 0\!\!\!\!\mod n. *Every number in a reduced residue system modulo ''n'' is a generator for the additive g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Product
In model theory, a branch of mathematical logic, and in algebra, the reduced product is a construction that generalizes both direct product and ultraproduct. Let be a family of structures of the same signature σ indexed by a set ''I'', and let ''U'' be a filter on ''I''. The domain of the reduced product is the quotient of the Cartesian product :\prod_ S_i by a certain equivalence relation ~: two elements (''ai'') and (''bi'') of the Cartesian product are equivalent if :\left\\in U If ''U'' only contains ''I'' as an element, the equivalence relation is trivial, and the reduced product is just the original Cartesian product. If ''U'' is an ultrafilter, the reduced product is an ultraproduct. Operations from σ are interpreted on the reduced product by applying the operation pointwise. Relations are interpreted by :R((a^1_i)/,\dots,(a^n_i)/) \iff \\in U. For example, if each structure is a vector space In mathematics and physics, a vector space (also called a li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If ''P'' is a single-point space, then with the usual definitions the integral homology group :''H''0(''P'') is isomorphic to \mathbb (an infinite cyclic group), while for ''i'' ≥ 1 we have :''H''''i''(''P'') = . More generally if ''X'' is a simplicial complex or finite CW complex, then the group ''H''0(''X'') is the free abelian group with the connected components of ''X'' as generators. The reduced homology should replace this group, of rank ''r'' say, by one of rank ''r'' − 1. Otherwise the homology groups should remain unchanged. An ''ad hoc'' way to do this is to think of a 0-th homology class not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Form
In statistics, and particularly in econometrics, the reduced form of a system of equations is the result of solving the system for the endogenous variables. This gives the latter as functions of the exogenous variables, if any. In econometrics, the equations of a structural form model are estimated in their theoretically given form, while an alternative approach to estimation is to first solve the theoretical equations for the endogenous variables to obtain reduced form equations, and then to estimate the reduced form equations. Let ''Y'' be the vector of the variables to be explained (endogeneous variables) by a statistical model and ''X'' be the vector of explanatory (exogeneous) variables. In addition let \varepsilon be a vector of error terms. Then the general expression of a structural form is f(Y, X, \varepsilon) = 0 , where ''f'' is a function, possibly from vectors to vectors in the case of a multiple-equation model. The reduced form of this model is given by Y = g(X, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction System
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Of The Structure Group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(''n'')-structure defines a Riemannian metric, and for the special linear group an SL(''n'',R)-structure is the same as a volume form. For the trivial group, an -structure consists of an absolute parallelism of the manifold. Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal G-bundle over a group G "comes from" a subgroup H of G. This is called reduction of the structure group (to H). Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are ''G''-structures with an additional integrability c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Of Order
Reduction of order is a technique in mathematics for solving second-order linear ordinary differential equations. It is employed when one solution y_1(x) is known and a second linearly independent solution y_2(x) is desired. The method also applies to ''n''-th order equations. In this case the ansatz will yield an (''n''−1)-th order equation for v. Second-order linear ordinary differential equations An example Consider the general, homogeneous, second-order linear constant coefficient ordinary differential equation. (ODE) : a y''(x) + b y'(x) + c y(x) = 0, where a, b, c are real non-zero coefficients. Two linearly independent solutions for this ODE can be straightforwardly found using characteristic equations except for the case when the discriminant, b^2 - 4 a c, vanishes. In this case, : a y''(x) + b y'(x) + \frac y(x) = 0, from which only one solution, :y_1(x) = e^, can be found using its characteristic equation. The method of reduction of order is used to obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
