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Problems In Latin Squares
In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the ''Loops (Prague)'' conferences and the ''Milehigh (Denver)'' conferences. Open problems Bounds on maximal number of transversals in a Latin square A ''transversal'' in a Latin square of order ''n'' is a set ''S'' of ''n'' cells such that every row and every column contains exactly one cell of ''S'', and such that the symbols in ''S'' form . Let ''T''(''n'') be the maximum number of transversals in a Latin square of order ''n''. Estimate ''T''(''n''). *''Proposed:'' by Ian Wanless at Loops '03, Prague 2003 *''Comments:'' Wanless, McKay and McLeod have bounds of the form ''c''''n'' exp(''c'' ''n'' log ''n''). A related question is to estimate the number of transversals in the Cayley tables of cyclic groups of odd order. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow Matching
In the mathematical discipline of graph theory, a rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Definition Given an edge-colored graph , a rainbow matching in is a set of pairwise non-adjacent edges, that is, no two edges share a common vertex, such that all the edges in the set have distinct colors. A maximum rainbow matching is a rainbow matching that contains the largest possible number of edges. History Rainbow matchings are of particular interest given their connection to transversals of Latin squares. Denote by the complete bipartite graph on vertices. Every proper -edge coloring of corresponds to a Latin square of order . A rainbow matching then corresponds to a transversal of the Latin square, meaning a selection of positions, one in each row and each column, containing distinct entries. This connection between transversals of Latin squares and rainbow matchings in has inspired additional interest in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Problems In Loop Theory And Quasigroup Theory
In mathematics, especially abstract algebra, loop (algebra), loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the ''Loops (Prague)'' conferences and the ''Mile High (Denver)'' conferences. Open problems (Moufang loops) Abelian by cyclic groups resulting in Moufang loops Let ''L'' be a Moufang loop with normal Abelian group, abelian subgroup (associative subloop) ''M'' of odd order such that ''L''/''M'' is a cyclic group of order bigger than 3. (i) Is ''L'' a Group (mathematics), group? (ii) If the orders of ''M'' and ''L''/''M'' are relatively prime, is L a group? *''Proposed:'' by Michael Kinyon, based on (Chein and Rajah, 2000) *''Comments:'' The assumption that ''L''/''M'' has order bigger than 3 is important, as there is a (commutative) Moufang loop ''L'' of order 81 with norma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist. The presumed difficulty of this problem is important for the algorithms used in cryptography such as RSA public-key encryption and the RSA digital signature. Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing. In 2019, Fabrice Boudot, Pierrick Gaudry, Aurore Guillevic, Nadia Heninger, Emmanuel Thomé and Paul Zimmermann factored a 240-digit (795-bit) number (RSA-240) utilizing approximately 900 core-years of computing power. The researchers estimated that a 1024-bit RSA mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-order
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades. These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization. *A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd. *A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even. The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics, coding theory (see even codes), among others. Definitions The ancient Greek terms "even-times-even" ( grc, ἀρτιάκι ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication Table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9. History In pre-modern time The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moufang Loop (born 1966), German ambient techno musician
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Moufang is the family name of the following people: *Christoph Moufang (1817–1890), a Roman Catholic cleric *Ruth Moufang (1905–1977), a German mathematician, after whom several concepts in mathematics are named: ** Moufang–Lie algebra ** Moufang loop ** Moufang polygon ** Moufang plane *David Moufang David Moufang (born 1966, in Heidelberg, West Germany) is a German ambient techno musician. He records with his partner, Jonas Grossmann as Deep Space Network project and his solo releases as Move D.Profileat Allmusic guide His other projects inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order (group Theory)
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group operation is denoted as a multiplication, the order of an element of a group, is thus the smallest positive integer such that , where denotes the identity element of the group, and denotes the product of copies of . If no such exists, the order of is infinite. The order of a group is denoted by or , and the order of an element is denoted by or , instead of \operatorname(\langle a\rangle), where the brackets denote the generated group. Lagrange's theorem states that for any subgroup of a finite group , the order of the subgroup divides the order of the group; that is, is a divisor of . In particular, the order of any element is a divisor of . Example The symmetric group S3 has th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latin Squares
In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin square is The name "Latin square" was inspired by mathematical papers by Leonhard Euler (1707–1783), who used Latin characters as symbols, but any set of symbols can be used: in the above example, the alphabetic sequence A, B, C can be replaced by the integer sequence 1, 2, 3. Euler began the general theory of Latin squares. History The Korean mathematician Choi Seok-jeong was the first to publish an example of Latin squares of order nine, in order to construct a magic square in 1700, predating Leonhard Euler by 67 years. Reduced form A Latin square is said to be ''reduced'' (also, ''normalized'' or ''in standard form'') if both its first row and its first column are in their natural order. For example, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
