|
Toida's Conjecture
In combinatorial mathematics, Toida's conjecture, due to Shunichi Toida in 1977, is a refinement of the disproven Ádám's conjecture from 1967. Statement Both conjectures concern circulant graphs. These are graphs defined from a positive integer n and a set S of positive integers. Their vertices can be identified with the numbers from 0 to n-1, and two vertices i and j are connected by an edge whenever their difference modulo n belongs to set S. Every symmetry of the cyclic group of addition modulo n gives rise to a symmetry of the n-vertex circulant graphs, and Ádám conjectured (incorrectly) that these are the only symmetries of the circulant graphs. However, the known counterexamples to Ádám's conjecture involve sets S in which some elements share non-trivial divisors with n. Toida's conjecture states that, when every member of S is relatively prime to n, then the only symmetries of the circulant graph for n and S are symmetries coming from the underlying cyclic group. Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shunichi Toida
Shun'ichi or Shunichi (written: or ) is a masculine Japanese given name. Notable people with the name include: *, Japanese baseball player and manager *, Japanese academic *, Japanese footballer *, Japanese sprint canoeist *, Japanese engineer *, Japanese film director *, Japanese diplomat *, Japanese volleyball player, announcer and television personality *, Japanese footballer *Shun'ichi Kuryu Shun'ichi Kuryu (born 6 December 1958) is a Japanese police bureaucrat who served as Deputy Chief Cabinet Secretary in the First and Second Kishida Cabinet The Second Kishida Cabinet is the 101st Cabinet of Japan and was formed by Fumio Kish ... (born 1958), Japanese bureaucrat *, Japanese diplomat *, Japanese musician and voice actor *, Japanese politician *, Japanese film director and screenwriter *, Japanese footballer *, Japanese baseball player *, Japanese mixed martial artist *, Japanese politician *, Japanese footballer *, Japanese composer *, Japanese mixed martial artist, k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ádám's Conjecture
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings. Equivalent definitions Circulant graphs can be described in several equivalent ways:. *The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices. *The graph has an adjacency matrix that is a circulant matrix. *The vertices of the graph can be numbered from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. *The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing. *The graph is a Cayley graph of a cyc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circulant Graph
In graph theory, a circulant graph is an undirected graph acted on by a cyclic group of symmetries which takes any vertex to any other vertex. It is sometimes called a cyclic graph, but this term has other meanings. Equivalent definitions Circulant graphs can be described in several equivalent ways:. *The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has a graph automorphism, which is a cyclic permutation of its vertices. *The graph has an adjacency matrix that is a circulant matrix. *The vertices of the graph can be numbered from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. *The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing. *The graph is a Cayley graph of a cyclic group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a ''generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Algebra
In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. Green first published in 1980. The name "Schur algebra" is due to Green. In the modular case (over infinite fields of positive characteristic) Schur algebras were used by Gordon James and Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent. Schur algebras were used by Friedlander and Suslin to prove finite generation of cohomology of finite group schemes. Construction The Schur algebra S_k(n, r) can be defined for any commutative ring k and integers n, r \geq 0. Consider the algebra k _/math> of polynomials (with coefficients in k) in n^2 commuting v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joy Morris
Joy Morris (born 1970) is a Canadian mathematician whose research involves group theory, graph theory, and the connections between the two through Cayley graphs. She is also interested in mathematics education, is the author of two open-access undergraduate mathematics textbooks, and oversees a program in which university mathematics education students provide a drop-in mathematics tutoring service for parents of middle school students. She is a professor of mathematics at the University of Lethbridge. Education and career Morris is originally from Toronto, Ontario. Both her parents had doctorates; she was the youngest of their four children, another of whom also earned a Ph.D.. She was educated through various alternative-education and gifted-student programs in the Toronto public school system. She graduated from Trent University in 1992 with a double major in mathematics and English, and with fourth-year honours in mathematics earned in part through a summer research project wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Finite Simple Groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six or twenty-seven exceptions, called sporadic. The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
