|
Rank Factorization
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy of the Catholic Church * Imperial, royal and noble ranks * Military rank * Police rank Unofficial ranks * Social class * Social position * Social status Either * Seniority Mathematics * Rank (differential topology) * Rank (graph theory) * Rank (linear algebra), the dimension of the vector space generated (or spanned) by a matrix's columns * Rank (set theory) * Rank (type theory) * Rank of an abelian group, the cardinality of a maximal linearly independent subset * Rank of a free module * Rank of a greedoid, the maximal size of a feasible set * Rank of a group, the smallest cardinality of a generating set for the group * Rank of a Lie group – see Cartan subgroup * Rank of a matroid, the maximal size of an independent set * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hierarchy
A hierarchy (from Ancient Greek, Greek: , from , 'president of sacred rites') is an arrangement of items (objects, names, values, categories, etc.) that are represented as being "above", "below", or "at the same level as" one another. Hierarchy is an important concept in a wide variety of fields, such as architecture, philosophy, design, mathematics, computer science, organizational theory, systems theory, systematic biology, and the social sciences (especially political science). A hierarchy can link entities either directly or indirectly, and either vertically or diagonally. The only direct links in a hierarchy, insofar as they are hierarchical, are to one's immediate superior or to one of one's subordinates, although a system that is largely hierarchical can also incorporate alternative hierarchies. Hierarchical links can extend "vertically" upwards or downwards via multiple links in the same direction, following a path (graph theory), path. All parts of the hierarchy that are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank (graph Theory)
In graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ..., a branch of mathematics, the rank of an undirected graph has two unrelated definitions. Let equal the number of vertices of the graph. * In the matrix theory of graphs the rank of an undirected graph is defined as the rank of its adjacency matrix. :Analogously, the nullity of the graph is the nullity of its adjacency matrix, which equals . * In the matroid theory of graphs the rank of an undirected graph is defined as the number , where is the number of connected components of the graph. Equivalently, the rank of a graph is the rank of the oriented incidence matrix associated with the graph.. See in particular the discussion on p. 218. :Analogously, the nullity of the graph is the nullity of its o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Statistics
In statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ..., ranking is the data transformation (statistics), data transformation in which number, numerical or Ordinal scale, ordinal values are replaced by their rank when the data are sorted. For example, the ranks of the numerical data 3.4, 5.1, 2.6, 7.3 are 2, 3, 1, 4. As another example, the ordinal data hot, cold, warm would be replaced by 3, 1, 2. In these examples, the ranks are assigned to values in ascending order, although descending ranks can also be used. Ranks are related to the indexed list of order statistics, which consists of the original dataset rearranged into ascending order. Use for testing Some kinds of statistical tests employ calculations based on ranks. Examples include: * Friedman test * Kruskal– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of A Partition
In number theory and combinatorics, the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ... in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatoric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, 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 Axiomatic system, 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 simple matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terms used in 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 Cryptomorphism, equivalent ways to define a (finite) matroid. Independent sets In terms of independence, a finite matroid M is a pair (E, \mathcal), where E is a finite set (called the ''gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Subgroup
In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G over a (not necessarily algebraically closed) field k is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If k is algebraically closed, they are all conjugate to each other. Notice that in the context of algebraic groups a ''torus'' is an algebraic group T such that the base extension T_ (where \bar is the algebraic closure of k) is isomorphic to the product of a finite number of copies of the \mathbf_m=\mathbf_1. Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...s. If G is reductive (in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of A Group
In the mathematical subject of group theory, the rank of a group ''G'', denoted rank(''G''), can refer to the smallest cardinality of a generating set for ''G'', that is : \operatorname(G)=\min\. If ''G'' is a finitely generated group, then the rank of ''G'' is a non-negative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for ''p''-groups, the rank of the group ''P'' is the dimension of the vector space ''P''/Φ(''P''), where Φ(''P'') is the Frattini subgroup. The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as affine groups. To distinguish these different definitions, one sometimes calls this rank the subgroup rank. Explicitly, the subgroup rank of a group ''G'' is the maximum of the ranks of its subgroups: : \operatorname(G)=\max_ \m ... [...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] |
Free Module
In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set and ring , there is a free -module with basis , which is called the ''free module on'' or ''module of formal'' -''linear combinations'' of the elements of . A free abelian group is precisely a free module over the ring \Z of integers. Definition For a ring R and an R- module M, the set E\subseteq M is a basis for M if: * E is a generating set for M; that is to say, every element of M is a finite sum of elements of E multiplied by coefficients in R; and * E is linearly independent: for every set \\subset E of distinct elements, r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M implies that r_1 = r_2 = \cdots = r_n = 0_R (where 0_M is the zero element of M and 0_R is the zer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of An Abelian Group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset. The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If ''A'' is torsion-free then it embeds into a vector space over the rational numbers of dimension rank ''A''. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved. The term rank has a different meaning in the context of elementary abelian groups. Definition A subset of an abelian group ''A'' is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if : \sum_\alpha n_\alpha a_\alpha = 0, \quad n_\alpha\in\mathbb, where all but finitely man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
