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Basis Of Membrane Potential2-en
Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items * Basis point, 0.01%, often used in the context of interest rates * Basis swap, an interest rate swap * Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation * Tax basis, cost of an asset Securities markets and trading strategies * Basis trading, a trading strategy consisting of the purchase of a security and the sale of a similar security Fixed income markets: * Treasury basis trade, a leveraged arbitrage strategy exploiting price differences between Treasury securities and futures contracts * Index arbitrage, a strategy that exploits price differences between a stock index and its futures contract Commodities and physical assets: * Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjusted Basis
In tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other basis of property, reduced by depreciation deductions and increased by capital expenditures. Example: Michael buys a lot for $100,000. He then erects a retail facility for $600,000, then depreciates the improvements for tax purposes at the rate of $15,000 per year. After three years his adjusted tax basis is $655,000 = $100,000 + $600,000 - (3 x $15,000). Adjusted basis is one of two variables in the formula used to compute gains and losses when determining gross income for tax purposes. The Amount Realized – Adjusted Basis tells the amount of Realized Gain (if positive) or Realized Loss (if negative). Statutory definition Section 1012 of the Internal Revenue Code defines “basis” as a taxpayer's cost in acquiring property, except as provided in Sections 1001–1092. There is also an exceptio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Function
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points). Examples Monomial basis for ''Cω'' The monomial basis for the vector space of analytic functions is given by \. This basis is used in Taylor series, amongst others. Monomial basis for polynomials The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n for some n \in \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greedoid
In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Hassler Whitney, Whitney in 1935 to study planar graphs and was later used by Jack Edmonds, Edmonds to characterize a class of optimization problems that can be solved by greedy algorithms. Around 1980, Bernhard Korte, Korte and László Lovász, Lovász introduced the greedoid to further generalize this characterization of greedy algorithms; hence the name greedoid. Besides mathematical optimization, greedoids have also been connected to graph theory, language theory, order theory, and other areas of mathematics. Definitions A set system is a collection of subsets of a ground set (i.e. is a subset of the power set of ). When considering a greedoid, a member of is called a feasible set. When considering a matroid, a feasible set is also known as an ''independent set''. An accessible set system is a set system in which every nonempty feasible set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Change Of Basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector on one basis is, in general, different from the coordinate vector that represents on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis. Such a conversion results from the ''change-of-basis formula'' which expresses the coordinates relative to one basis in terms of coordinates relative to the other basis. Using matrices, this formula can be written :\mathbf x_\mathrm = A \,\mathbf x_\mathrm, where "old" and "new" refer respectively to the initially defined basis and the other basis, \mathbf x_\mathrm and \mathbf x_\mathrm are the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base (topology)
In mathematics, a base (or basis; : bases) for the topology of a topological space is a family \mathcal of open subsets of such that every open set of the topology is equal to the union of some sub-family of \mathcal. For example, the set of all open intervals in the real number line \R is a basis for the Euclidean topology on \R because every open interval is an open set, and also every open subset of \R can be written as a union of some family of open intervals. Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called , are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier. Not all families of subsets of a set X form a base for a topology on X. Under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group (mathematics), group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their Inverse element, inverses. In other words, if S is a subset of a group G, then \langle S\rangle, the ''subgroup generated by S'', is the smallest subgroup of G containing every element of S, which is equal to the intersection over all subgroups containing the elements of S; equivalently, \langle S\rangle is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If G=\langle S\rangle, then we say that S ''generates'' G, and the elements in S are called ''generators'' or ''group generators''. If S is the empty set, then \langle S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Basis Theorem
In mathematics Hilbert's basis theorem asserts that every ideal (ring theory), ideal of a polynomial ring over a field (mathematics), field has a finite generating set of an ideal, generating set (a finite ''basis'' in Hilbert's terminology). In modern algebra, ring (mathematics), rings whose ideals have this property are called Noetherian rings. Every field, and the ring of integers are Noetherian rings. So, the theorem can be generalized and restated as: ''every polynomial ring over a Noetherian ring is also Noetherian''. The theorem was stated and proved by David Hilbert in 1890 in his seminal article on invariant theory, where he solved several problems on invariants. In this article, he proved also two other fundamental theorems on polynomials, the Nullstellensatz (zero-locus theorem) and the syzygy theorem (theorem on relations). These three theorems were the starting point of the interpretation of algebraic geometry in terms of commutative algebra. In particular, the basis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gröbner Basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K _1,\ldots,x_n/math> over a field K. A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced by Bruno Buchberger in his 1965 Ph.D. thesis, which also included an algorithm to compute them ( Buchberger's alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Of A Matroid
In mathematics, a basis of a matroid is a maximal independent set of the matroid—that is, an independent set that is not contained in any other independent set. Examples As an example, consider the matroid over the ground-set R2 (the vectors in the two-dimensional Euclidean plane), with the following independent sets: It has two bases, which are the sets , . These are the only independent sets that are maximal under inclusion. The basis has a specialized name in several specialized kinds of matroids: * In a graphic matroid, where the independent sets are the forests, the bases are called the '' spanning forests'' of the graph. * In a transversal matroid, where the independent sets are endpoints of matchings in a given bipartite graph, the bases are called ''transversals''. * In a linear matroid, where the independent sets are the linearly-independent sets of vectors in a given vector-space, the bases are just called ''bases'' of the vector space. Hence, the concept ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (universal Algebra)
In universal algebra, a basis is a structure inside of some (universal) algebras, which are called free algebras. It generates all algebra elements from its own elements by the algebra operations in an independent manner. It also represents the endomorphisms of an algebra by certain indexings of algebra elements, which can correspond to the usual matrices when the free algebra is a vector space. Definitions A basis (or reference frame) of a (universal) algebra is a function b that takes some algebra elements as values b(i) and satisfies either one of the following two equivalent conditions. Here, the set of all b(i) is called the basis set, whereas several authors call it the "basis". The set I of its arguments i is called the dimension set. Any function, with all its arguments in the whole I, that takes algebra elements as values (even outside the basis set) will be denoted by m. Then, b will be an m. Outer condition This condition will define bases by the set L of the I- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schauder Basis
In mathematics, a Schauder basis or countable basis is similar to the usual ( Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927, although such bases were discussed earlier. For example, the Haar basis was given in 1909, and Georg Faber discussed in 1910 a basis for continuous functions on an interval, sometimes called a Faber–Schauder system.Faber, Georg (1910), "Über die Orthogonalfunktionen des Herrn Haar", ''Deutsche Math.-Ver'' (in German) 19: 104–112. ; http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN37721857X ; http://resolver.sub.uni-goettingen.de/purl?GDZPPN002122553 Definitions Let ''V'' denote a topological vector space over the field ''F' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
