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Reduction (Sweden), Reductions
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 searche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction (chemistry)
Redox ( , , reduction–oxidation or oxidation–reduction) is a type of chemical reaction in which the oxidation states of the reactants change. Oxidation is the loss of electrons or an increase in the oxidation state, while reduction is the gain of electrons or a decrease in the oxidation state. The oxidation and reduction processes occur simultaneously in the chemical reaction. There are two classes of redox reactions: * Electron transfer, Electron-transfer – Only one (usually) electron flows from the atom, ion, or molecule being oxidized to the atom, ion, or molecule that is reduced. This type of redox reaction is often discussed in terms of redox couples and electrode potentials. * Atom transfer – An atom transfers from one Substrate (chemistry), 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 r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable. Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Linear approaches can be further divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance Reduction
In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main variance reduction methods are * common random numbers * antithetic variates * control variates * importance sampling * stratified sampling * moment matching * conditional Monte Carlo * and quasi random variables (in Quasi-Monte Carlo method) For simulation with black-box models subset simulation and line sampling can also be used. Under these headings are a variety of specialized techniques; for example, parti ... [...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 wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Row Echelon Form
In linear algebra, a matrix is in row echelon form if it can be obtained as the result of Gaussian elimination. Every matrix can be put in row echelon form by applying a sequence of elementary row operations. The term ''echelon'' comes from the French ''échelon'' ("level" or step of a ladder), and refers to the fact that the nonzero entries of a matrix in row echelon form look like an inverted staircase. For square matrices, an upper triangular matrix with nonzero entries on the diagonal is in row echelon form, and a matrix in row echelon form is (weakly) upper triangular. Thus, the row echelon form can be viewed as a generalization of upper triangular form for rectangular matrices. A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to 1 and is the only nonzero entry of its column. The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementar ... [...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 \mathcal_R denote nilradical of a commutative ring R. There is a functor R \mapsto R/\mathcal_R of the category of commutative rings \text into the category of reduced rings \text and it is left adjoint to the inclusion functor I of \text into ... [...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 *Every number in a reduced residue system modulo ''n'' is a generator for the additive group of integers modulo ''n''. *A reduced residue system modulo ''n'' is a group under multipli ... [...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 nonempty family of structures of the same signature σ indexed by a set ''I'', and let ''U'' be a proper 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 direct 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 cal ... [...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 no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Form
In statistics, and particularly in econometrics, the reduced form of a simultaneous equations model, 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 model (econometrics), structural form model are estimation theory, 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 ... [...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 ... [...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 integrabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
