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Coxeter Group
In mathematics, a Coxeter
Coxeter
group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter
Coxeter
groups are finite, and not all can be described in terms of symmetries and Euclidean reflections
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Edge (graph Theory)
This is a glossary of graph theory terms. Graph theory
Graph theory
is the study of graphs, systems of nodes or vertices connected in pairs by edges.Contents: !$@ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also ReferencesSymbols[edit]Square brackets [ ] G[S] is the induced subgraph of a graph G for vertex subset S. Prime symbol ' The prime symbol is often used to modify notation for graph invariants so that it applies to the line graph instead of the given graph. For instance, α(G) is the independence number of a graph; α′(G) is the matching number of the graph, which equals the independence number of its line graph
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Commutative Operation
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations such as the multiplication and addition of numbers are commutative, was for many years implicitly assumed
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Connected Component (graph Theory)
In graph theory, a connected component (or just component) of an undirected graph is a subgraph in which any two vertices are connected to each other by paths, and which is connected to no additional vertices in the supergraph. For example, the graph shown in the illustration has three connected components. A vertex with no incident edges is itself a connected component. A graph that is itself connected has exactly one connected component, consisting of the whole graph.Contents1 An equivalence relation 2 The number of connected components 3 Algorithms 4 See also 5 References 6 External linksAn equivalence relation[edit] An alternative way to define connected components involves the equivalence classes of an equivalence relation that is defined on the vertices of the graph. In an undirected graph, a vertex v is reachable from a vertex u if there is a path from u to v
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Disjoint Union
In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in. Or slightly different from this, the disjoint union of a family of subsets is the usual union of the subsets which are pairwise disjoint – disjoint sets means they have no element in common. Note that these two concepts are different but strongly related. Moreover, it seems that they are essentially identical to each other in category theory
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Dot Product
In mathematics, the dot product or scalar product[note 1] is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates
Cartesian coordinates
of two vectors is widely used and often called inner product (or rarely projection product); see also inner product space. Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces
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Eigenvalues
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v
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Vertex (graph Theory)
In mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices
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Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses. In other words, if S is a subset of a group G, then 〈S〉, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, 〈S〉 is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.(Notice: here inverses only need for the case of infinite group, while for finite group, inverse can be expressed as a power of the element itself, and no need here.) If G = 〈S〉, then we say that S generates G, and the elements in S are called generators or group generators
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Conjugate Elements
In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial pK,α(x) of α over K
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Quotient Group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory. In a quotient of a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written G / N, where G is the original group and N is the normal subgroup. (This is pronounced "G mod N", where "mod" is short for modulo.) Much of the importance of quotient groups is derived from their relation to homomorphisms
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Linear Group
In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is an abstract group that is isomorphic to a matrix group over a field K, in other words, admitting a faithful, finite-dimensional representation over K. Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups form an interesting and tractable class
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Hyperplane
In geometry a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces
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Linear Representation
Representation theory
Representation theory
is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.[1] In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras
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Exceptional Object
Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions that do not fit into any series. These are known as exceptional objects. Frequently these exceptional objects play a further and important role in the subject
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