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Dependence Relation
In mathematics, a dependence relation is a binary relation which generalizes the relation of linear dependence. Let X be a set. A (binary) relation \triangleleft between an element a of X and a subset S of X is called a ''dependence relation'', written a \triangleleft S, if it satisfies the following properties: * if a \in S, then a \triangleleft S; * if a \triangleleft S, then there is a finite subset S_0 of S, such that a \triangleleft S_0; * if T is a subset of X such that b \in S implies b \triangleleft T, then a \triangleleft S implies a \triangleleft T; * if a \triangleleft S but a \not\!\triangleleft S-\lbrace b \rbrace for some b \in S, then b \triangleleft (S-\lbrace b \rbrace)\cup\lbrace a \rbrace. Given a ''dependence relation'' \triangleleft on X, a subset S of X is said to be ''independent'' if a \not\!\triangleleft S - \lbrace a \rbrace for all a \in S. If S \subseteq T, then S is said to ''span'' T if t \triangleleft S for every t \in T. S is said to be a ''bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dependency Relation
In computer science, in particular in concurrency theory, a dependency relation is a binary relation on a finite domain \Sigma, symmetric, and reflexive; i.e. a finite tolerance relation. That is, it is a finite set of ordered pairs D, such that * If (a,b)\in D then (b,a) \in D (symmetric) * If a \in \Sigma, then (a,a) \in D (reflexive) In general, dependency relations are not transitive; thus, they generalize the notion of an equivalence relation by discarding transitivity. \Sigma is also called the alphabet on which D is defined. The independency induced by D is the binary relation I :I = (\Sigma \times \Sigma) \setminus D That is, the independency is the set of all ordered pairs that are not in D. The independency relation is symmetric and irreflexive. Conversely, given any symmetric and irreflexive relation I on a finite alphabet, the relation :D = (\Sigma \times \Sigma) \setminus I is a dependency relation. The pair (\Sigma, D) is called the concurrent alphabe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called ''linear manifolds'' for emphasizing that there are also manifolds. is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to distinguish it from other types of subspaces. Definition If ''V'' is a vector space over a field ''K'' and if ''W'' is a subset of ''V'', then ''W'' is a linear subspace of ''V'' if under the operations of ''V'', ''W'' is a vector space over ''K''. Equivalently, a nonempty subset ''W'' is a subspace of ''V'' if, whenever are elements of ''W'' and are elements of ''K'', it follows that is in ''W''. As a corollary, all vector spaces are equipped with at least two ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Dependence
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically independent over K if and only if \alpha is transcendental over K. In general, all the elements of an algebraically independent set S over K are by necessity transcendental over K, and over all of the field extensions over K generated by the remaining elements of S. Example The two real numbers \sqrt and 2\pi+1 are each transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, each of the two singleton sets \ and \ are algebraically independent over the field \mathbb of rational numbers. However, the set \ is ''not'' algebraically independent over the rational numbers, because the nontrivial polynomial :P(x,y)=2x^2-y+1 is zero when x=\sqrt and y=2\pi+1. Algebraic independence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Element
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic over are called transcendental over . These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is , being the field of complex numbers and being the field of rational numbers). Examples * The square root of 2 is algebraic over , since it is the root of the polynomial whose coefficients are rational. * Pi is transcendental over but algebraic over the field of real numbers : it is the root of , whose coefficients (1 and −) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses , not .) Properties The following conditions are equivalent for an element a of L: * a is algebraic over K, * the field extension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ''F''. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry. Subfield A subfield K of a field L is a subset K\subseteq L that is a field with respect to the field operations inherited from L. Equivalently, a subfield is a subset that contains 1, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of K. As , the latter definition implies K and L have the same zero eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Independence
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes the zero vector. This implies that at least one of the scalars is nonzero, say a_1\ne 0, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Equivalence
In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on the notation being used. However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related. Logical equivalences In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these. General logical equivalences Logical equivalences involving conditional statements :#p \implies q \equiv \neg p \vee q :#p \implies q \equiv \neg q \implies \neg p :#p \vee q \equiv \neg p \implies q :#p \wedge q \equiv \neg (p \implies \neg q) :#\neg (p \implies q) \equiv p \wedge \neg q :#(p \implies q) \wedge (p \implie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. 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 fields of rational functions, algebraic function fields, algebraic number fields, and ''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 elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
