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Coxeter Group In mathematics , a COXETER GROUP, named after H. S. M. Coxeter 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 groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced (Coxeter 1934 ) as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 (Coxeter 1935 ) [...More...]  "Coxeter Group" on: Wikipedia Yahoo 

Mathematics MATHEMATICS (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity , structure , space , and change . There are many views among mathematicians and philosophers as to the exact scope and definition of mathematics . Mathematicians seek out patterns 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. The research required to solve mathematical problems can take years or even centuries of sustained inquiry [...More...]  "Mathematics" on: Wikipedia Yahoo 

Dot Product In mathematics , the DOT PRODUCT or SCALAR PRODUCT is an algebraic operation that takes two equallength sequences of numbers (usually coordinate vectors ) and returns a single number. In Euclidean geometry , the dot product of the 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 [...More...]  "Dot Product" on: Wikipedia Yahoo 

Eigenvalues In linear algebra , an EIGENVECTOR or CHARACTERISTIC VECTOR of a linear transformation is a nonzero vector whose direction does not change 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. This condition can be written as the equation T ( v ) = v , {displaystyle T(mathbf {v} )=lambda mathbf {v} ,} where λ is a scalar in the field F, known as the EIGENVALUE, CHARACTERISTIC VALUE, or CHARACTERISTIC ROOT associated with the eigenvector V [...More...]  "Eigenvalues" on: Wikipedia Yahoo 

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 . That is, both are realizations of the coproduct of category of sets [...More...]  "Disjoint Union" on: Wikipedia Yahoo 

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. CONTENTS * 1 An equivalence relation * 2 The number of connected components * 3 Algorithms * 4 See also * 5 References * 6 External links AN EQUIVALENCE RELATIONAn 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 [...More...]  "Connected Component (graph Theory)" on: Wikipedia Yahoo 

Symmetric Matrix In linear algebra , a SYMMETRIC MATRIX is a square matrix that is equal to its transpose . Formally, matrix A is symmetric if A = A T . {displaystyle A=A^{mathrm {T} }.} Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal . So if the entries are written as A = (aij), then aij = aji, for all indices i and j. The following 3 × 3 matrix is symmetric: . {displaystyle {begin{bmatrix}1&7&3\7&4&5\3&5 width:16.331ex; height:9.176ex;" alt="{begin{bmatrix}1&7&3\7&4&5\3&5"> Mat n = Sym n Skew n , {displaystyle {mbox{Mat}}_{n}={mbox{Sym}}_{n}oplus {mbox{Skew}}_{n},} where ⊕ denotes the direct sum . Let X ∈ Matn then X = 1 2 ( X + X T ) + 1 2 ( X X T ) [...More...]  "Symmetric Matrix" on: Wikipedia Yahoo 

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. Thus, this property was not named until the 19th century, when mathematics started to become formalized [...More...]  "Commutative Operation" on: Wikipedia Yahoo 

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 [...More...]  "Vertex (graph Theory)" on: Wikipedia Yahoo 

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 : * !$@ * A * B * C * D * E * F * G * H * I * J * K * L * M * N * O * P * Q * R * S * T * U * V * W * X * Y * Z * See also * References !$@ [] G 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. Similarly, χ(G) is the chromatic number of a graph; χ ′(G) is the chromatic index of the graph, which equals the chromatic number of its line graph [...More...]  "Edge (graph Theory)" on: Wikipedia Yahoo 

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. Furthermore, the exceptional objects in one branch of mathematics are often related to the exceptional objects in others. A related phenomenon is exceptional isomorphism , when two series are in general different, but agree for some small values [...More...]  "Exceptional Object" on: Wikipedia Yahoo 

Crystallographic Restriction Theorem The CRYSTALLOGRAPHIC RESTRICTION THEOREM in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2fold, 3fold, 4fold, and 6fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5fold; these were not discovered until 1982 by Dan Shechtman . Crystals are modeled as discrete lattices , generated by a list of independent finite translations (Coxeter 1989 ). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group (alternatively, the point is the only system allowing for infinite rotational symmetry). The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups [...More...]  "Crystallographic Restriction Theorem" on: Wikipedia Yahoo 

Order (group Theory) In group theory , a branch of mathematics , the term order is used in two unrelated senses: * The ORDER of a group is its cardinality , i.e., the number of elements in its set . Also, the ORDER, sometimes PERIOD, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order. * The ordering relation of a partially or totally ordered group .This article is about the first sense of order. The order of a group G is denoted by ord(G) or G and the order of an element a is denoted by ord(a) or a. CONTENTS * 1 Example * 2 Order and structure * 3 Counting by order of elements * 4 In relation to homomorphisms * 5 Class equation * 6 Open questions * 7 See also * 8 References EXAMPLEEXAMPLE. The symmetric group S3 has the following multiplication table [...More...]  "Order (group Theory)" on: Wikipedia Yahoo 

Linear Representation 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. 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 . The most prominent of these (and historically the first) is the representation theory of groups , in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication. Representation theory Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra , a subject that is well understood [...More...]  "Linear Representation" on: Wikipedia Yahoo 

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 , finitedimensional 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. Examples of groups that are not linear include groups which are "too big" (for example, the group of permutations of an infinite set), or which exhibit some pathological behaviour (for example finitely generated infinite torsion groups) [...More...]  "Linear Group" on: Wikipedia Yahoo 

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. If G = 〈S〉, 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 〈S〉 is the trivial group {e}, since we consider the empty product to be the identity. When there is only a single element x in S, 〈S〉 is usually written as 〈x〉 [...More...]  "Generating Set Of A Group" on: Wikipedia Yahoo 