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Power Of Three
In mathematics, a power of three is a number of the form where is an integer, that is, the result of exponentiation with number 3, three as the Base (exponentiation), base and integer as the exponent. The first seven non-negative powers of three are: 1, 3, 9, 27 (number), 27, 81 (number), 81, 243 (number), 243, 729 (number), 729, etc. (sequence oeis:A000244, A000244 in OEIS) Applications The powers of three give the place values in the ternary numeral system. Graph theory In graph theory, powers of three appear in the Moon–Moser bound on the number of maximal independent sets of an -vertex graph (discrete mathematics), graph, and in the time analysis of the Bron–Kerbosch algorithm for finding these sets. Several important strongly regular graphs also have a number of vertices that is a power of three, including the Brouwer–Haemers graph (81 vertices), Berlekamp–van Lint–Seidel graph (243 vertices), and Games graph (729 vertices). Enumerative combinatoric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bron–Kerbosch Algorithm
In computer science, the Bron–Kerbosch algorithm is an enumeration algorithm for finding all maximal cliques in an undirected graph. That is, it lists all subsets of vertices with the two properties that each pair of vertices in one of the listed subsets is connected by an edge, and no listed subset can have any additional vertices added to it while preserving its complete connectivity. The Bron–Kerbosch algorithm was designed by Dutch scientists Coenraad Bron and Joep Kerbosch, who published its description in 1973. Although other algorithms for solving the clique problem have running times that are, in theory, better on inputs that have few maximal independent sets, the Bron–Kerbosch algorithm and subsequent improvements to it are frequently reported as being more efficient in practice than the alternatives. It is well-known and widely used in application areas of graph algorithms such as computational chemistry. A contemporaneous algorithm of , although presented i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kalai's 3^d Conjecture
In geometry, more specifically in polytope theory, Kalai's 3''d'' conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989.. It states that every ''d''-dimensional centrally symmetric polytope has at least 3''d'' nonempty faces (including the polytope itself as a face but not including the empty set). Examples In two dimensions, the simplest centrally symmetric convex polygons are the parallelograms, which have four vertices, four edges, and one polygon: . A cube is centrally symmetric, and has 8 vertices, 12 edges, 6 square sides, and 1 solid: . Another three-dimensional convex polyhedron, the regular octahedron, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid: . In higher dimensions, the hypercube , 1sup>''d'' has exactly 3''d'' faces, each of which can be determined by specifying, for each of the ''d'' coordinate axes, whether the face projects onto that axis o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degree (angle), degrees, or Pi, /2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called square (algebra), squaring. Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art. The formula for the area of a square forms the basis of the calculation of area and motivates the search for methods for s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empty Set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called ''non-empty''. In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). Notation Common notations for the empty set include "", "\emptyset", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø () in the Danish orthography, Danish and Norwegian orthography, Norwegian a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hanner Polytope
In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Swedish mathematician Olof Hanner, who introduced them in 1956.. Construction The Hanner polytopes are constructed recursively by the following rules:. *A line segment is a one-dimensional Hanner polytope. *The Cartesian product of every two Hanner polytopes is another Hanner polytope, whose dimension is the sum of the dimensions of the two given polytopes. *The dual of a Hanner polytope is another Hanner polytope of the same dimension. They are exactly the polytopes that can be constructed using only these rules: that is, every Hanner polytope can be formed from line segments by a sequence of product and dual operations. Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products and direct sums, the dual of the Cartesian products. This direct sum operation comb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Combinatorics
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas. Mathematicians in this area study the combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ... of polytopes; for instance, they seek inequality (mathematics), inequalities that describe the relations between the numbers of vertex (geometry), vertices, edge (geometry), edges, and faces of higher dimensions in arbitrary polytopes or in certain important subclasses of polytopes, and study other combinatorial properties of polytopes such as their Connectivity (graph theory), connectivity and dia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Set
In mathematics, a signed set is a set of elements together with an assignment of a sign (positive or negative) to each element of the set. Representation Signed sets may be represented mathematically as an ordered pair of disjoint sets, one set for their positive elements and another for their negative elements. Alternatively, they may be represented as a Boolean function, a function whose domain is the underlying unsigned set (possibly specified explicitly as a separate part of the representation) and whose range is a two-element set representing the signs. Signed sets may also be called \Z_2- graded sets. Application Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a hypercube. If the hypercube consists of all points in Euclidean space of a given dimension whose Cartesian coordinates are in the interval 1,+1/math>, then a signed subset of the coordinate axes can be used to specify the points whose coordinates with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enumerative Combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are '' closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. In 2020, most of the editorial board of ''JCTA'' resigned to form a new, |
Games Graph
In graph theory, the Games graph is the largest known locally linear strongly regular graph. Its parameters as a strongly regular graph are (729,112,1,20). This means that it has 729 vertices, and 40824 edges (112 per vertex). Each edge is in a unique triangle (it is a locally linear graph) and each non-adjacent pair of vertices have exactly 20 shared neighbors. It is named after Richard A. Games, who suggested its construction in an unpublished communication and wrote about related constructions. Construction The construction of this graph involves the 56-point cap set in PG(5,3). This is a subset of points with no three in line in the five-dimensional projective geometry over a three-element field, and is unique up to symmetry. The six-dimensional projective geometry, PG(6,3), can be partitioned into a six-dimensional affine space AG(6,3) and a copy of PG(5,3), which forms the set of points at infinity with respect to the affine space. The Games graph has as its vertices the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
