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Simplex In geometry , a SIMPLEX (plural: SIMPLEXES or SIMPLICES) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions . Specifically, a KSIMPLEX is a kdimensional polytope which is the convex hull of its k + 1 vertices . More formally, suppose the k + 1 points u 0 , , u k R k {displaystyle u_{0},dots ,u_{k}in mathbb {R} ^{k}} are affinely independent , which means u 1 u 0 , , u k u 0 {displaystyle u_{1}u_{0},dots ,u_{k}u_{0}} are linearly independent . Then, the simplex determined by them is the set of points C = { 0 u 0 + + k u k i = 0 k i = 1 and i 0 for all i } {displaystyle C=left{theta _{0}u_{0}+dots +theta _{k}u_{k}~{bigg }~sum _{i=0}^{k}theta _{i}=1{mbox{ and }}theta _{i}geq 0{mbox{ for all }}iright}} . For example, a 2simplex is a triangle, a 3simplex is a tetrahedron, and a 4simplex is a 5cell [...More...]  "Simplex" on: Wikipedia Yahoo 

Finite Set In mathematics , a FINITE SET is a set that has a finite number of elements . Informally, a finite set is a set which one could in principle count and finish counting. For example, { 2 , 4 , 6 , 8 , 10 } {displaystyle {2,4,6,8,10}} is a finite set with five elements. The number of elements of a finite set is a natural number (a nonnegative integer ) and is called the cardinality of the set. A set that is not finite is called INFINITE . For example, the set of all positive integers is infinite: { 1 , 2 , 3 , } . {displaystyle {1,2,3,ldots }.} Finite sets are particularly important in combinatorics , the mathematical study of counting . Many arguments involving finite sets rely on the pigeonhole principle , which states that there cannot exist an injective function from a larger finite set to a smaller finite set [...More...]  "Finite Set" on: Wikipedia Yahoo 

Simplicial Homology In algebraic topology , SIMPLICIAL HOMOLOGY formalizes the idea of the number of holes of a given dimension in a simplicial complex . This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are nsimplices , the ndimensional analogs of triangles. This includes a point (0simplex), a line segment (1simplex), a triangle (2simplex) and a tetrahedron (3simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex ). Such a homeomorphism is referred to as a triangulation of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead ) [...More...]  "Simplicial Homology" on: Wikipedia Yahoo 

Abstract Simplicial Complex In mathematics , an ABSTRACT SIMPLICIAL COMPLEX is a purely combinatorial description of the geometric notion of a simplicial complex , consisting of a family of nonempty finite sets closed under the operation of taking nonempty subsets . In the context of matroids and greedoids , abstract simplicial complexes are also called INDEPENDENCE SYSTEMS. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring ; this sets up a powerful relation between combinatorics and commutative algebra . CONTENTS * 1 Definitions * 2 Geometric realization * 3 Examples * 4 Enumeration * 5 See also * 6 References DEFINITIONSA family Δ of nonempty finite subsets of a set S is an ABSTRACT SIMPLICIAL COMPLEX if, for every set X in Δ, and every nonempty subset Y ⊂ X, Y also belongs to Δ [...More...]  "Abstract Simplicial Complex" on: Wikipedia Yahoo 

Combinatorics COMBINATORICS is a branch of mathematics concerning the study of finite or countable discrete structures . Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics ), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization ), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics ). Combinatorial problems arise in many areas of pure mathematics , notably in algebra , probability theory , topology , and geometry , and combinatorics also has many applications in mathematical optimization , computer science , ergodic theory and statistical physics [...More...]  "Combinatorics" on: Wikipedia Yahoo 

Topology In mathematics , TOPOLOGY (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets , that satisfy certain properties, turning the given set into what is known as a topological space . Important topological properties include connectedness and compactness . Topology Topology developed as a field of study out of geometry and set theory , through analysis of concepts such as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz Gottfried Leibniz , who in the 17th century envisioned the geometria situs (GreekLatin for "geometry of place") and analysis situs (GreekLatin for "picking apart of place") [...More...]  "Topology" on: Wikipedia Yahoo 

Coxeter HAROLD SCOTT MACDONALD "DONALD" COXETER, FRS, FRSC, CC (February 9, 1907 – March 31, 2003) was a Britishborn Canadian geometer. Coxeter is regarded as one of the greatest geometers of the 20th century. He was born in London London but spent most of his adult life in Canada Canada . He was always called Donald, from his third name MacDonald. CONTENTS * 1 Biography * 2 Awards * 3 Works * 4 See also * 5 References * 6 Further reading * 7 External links BIOGRAPHYIn his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Mathematics and Music" in the Canadian Music Journal. He worked for 60 years at the University of Toronto University of Toronto and published twelve books . He was most noted for his work on regular polytopes and higherdimensional geometries [...More...]  "Coxeter" on: Wikipedia Yahoo 

Hypercubic Honeycomb In geometry , a HYPERCUBIC HONEYCOMB is a family of regular honeycombs (tessellations ) in ndimensions with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter Coxeter group Rn (or B~n1) for n>=3. The tessellation is constructed from 4 nhypercubes per ridge . The vertex figure is a crosspolytope {3...3,4}. The hypercubic honeycombs are selfdual . Coxeter Coxeter named this family as δn+1 for an ndimensional honeycomb. WYTHOFF CONSTRUCTION CLASSES BY DIMENSIONThere are two general forms of the hypercube honeycombs, the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard . A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lowerdimensional elements [...More...]  "Hypercubic Honeycomb" on: Wikipedia Yahoo 

Isosceles Triangle In geometry , an ISOSCELES TRIANGLE is a triangle that has two sides of equal length. Sometimes it is specified as having two and only two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case . By the isosceles triangle theorem , the two angles opposite the equal sides are themselves equal, while if the third side is different then the third angle is different. By the Steiner–Lehmus theorem , every triangle with two angle bisectors of equal length is isosceles [...More...]  "Isosceles Triangle" on: Wikipedia Yahoo 

Power Of Two In mathematics , a POWER OF TWO means a number of the form 2n where n is an integer , i.e. the result of exponentiation with number two as the base and integer n as the exponent . In a context where only integers are considered, n is restricted to nonnegative values, so we have 1, 2, and 2 multiplied by itself a certain number of times. Because two is the base of the binary numeral system , powers of two are common in computer science . Written in binary, a power of two always has the form 100…000 or 0.00…001, just like a power of ten in the decimal system [...More...]  "Power Of Two" on: Wikipedia Yahoo 

Edge (geometry) In geometry , an EDGE is a particular type of line segment joining two vertices in a polygon , polyhedron , or higherdimensional polytope . In a polygon, an edge is a line segment on the boundary, and is often called a SIDE. In a polyhedron or more generally a polytope, an edge is a line segment where two faces meet. A segment joining two vertices while passing through the interior or exterior is not an edge but instead is called a diagonal . CONTENTS * 1 Relation to edges in graphs * 2 Number of edges in a polyhedron * 3 Incidences with other faces * 4 Alternative terminology * 5 See also * 6 References * 7 External links RELATION TO EDGES IN GRAPHSIn graph theory , an edge is an abstract object connecting two graph vertices , unlike polygon and polyhedron edges which have a concrete geometric representation as a line segment [...More...]  "Edge (geometry)" on: Wikipedia Yahoo 

Tetrahedron Number A TETRAHEDRAL NUMBER, or TRIANGULAR PYRAMIDAL NUMBER, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron . The nth tetrahedral number is the sum of the first n triangular numbers [...More...]  "Tetrahedron Number" on: Wikipedia Yahoo 

Triangle Number A TRIANGULAR NUMBER or TRIANGLE NUMBER counts objects arranged in an equilateral triangle , as in the diagram on the right. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers (sequence A000217 in the OEIS ), starting at the 0th triangular number , is 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666.. [...More...]  "Triangle Number" on: Wikipedia Yahoo 

Convex Set In convex geometry , a CONVEX SET is a subset of an affine space that is closed under convex combinations . More specifically, in a Euclidean space Euclidean space , a CONVEX REGION is a region where, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region. For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve . The intersection of all convex sets containing a given subset A of Euclidean space Euclidean space is called the convex hull of A. It is the smallest convex set containing A. A convex function is a realvalued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set [...More...]  "Convex Set" on: Wikipedia Yahoo 

Binomial Coefficient In mathematics , any of the positive integers that occurs as a coefficient in the binomial theorem is a BINOMIAL COEFFICIENT. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) {displaystyle {tbinom {n}{k}}} . It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, which is equal to n ! k ! ( n k ) ! {displaystyle {tfrac {n!}{k!(nk)!}}} . Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal\'s triangle . The binomial coefficients occur in many areas of mathematics, especially in the field of combinatorics [...More...]  "Binomial Coefficient" on: Wikipedia Yahoo 

Linearly Independent In the theory of vector spaces , a set of vectors is said to be LINEARLY DEPENDENT if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be LINEARLY INDEPENDENT. These concepts are central to the definition of dimension . A vector space can be of finitedimension or infinitedimension depending on the number of linearly independent basis 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 a basis for a vector space [...More...]  "Linearly Independent" on: Wikipedia Yahoo 