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Modular Invariant Of A Group
In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914 by . Dickson invariant When ''G'' is the finite general linear group GL''n''(F''q'') over the finite field F''q'' of order a prime power ''q'' acting on the ring F''q'' 'X''1, ...,''X''''n''in the natural way, found a complete set of invariants as follows. Write 'e''1, ..., ''e''''n''for the determinant of the matrix whose entries are ''X'', where ''e''1, ..., ''e''''n'' are non-negative integers. For example, the Moore determinant ,1,2of order 3 is :\begin x_1 & x_1^q & x_1^\\x_2 & x_2^q & x_2^\\x_3 & x_3^q & x_3^ \end Then under the action of an element ''g'' of GL''n''(F''q'') these determinants are all multiplied by det(''g''), so they are all invariants of SL''n''(F''q'') and the ratios 'e''1, ...,''e''''n''thinsp;/ , 1,&nb ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * '' Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sanderson's Theorem
Mildred Leonora Sanderson (May 12, 1889 – October 10, 1914) was an American mathematician, best known for her mathematical theorem concerning modular invariants. Life Sanderson was born in Waltham, Massachusetts, in 1889 and was the valedictorian of her class at the Waltham High School. She graduated from Mount Holyoke College in 1910, winning Senior Honors in Mathematics. She obtained her Ph.D degree from the University of Chicago in 1913, publishing the thesis in which she set forth her mathematical theorem. She was Leonard Eugene Dickson's first female doctoral student. After completing her Ph.D., Sanderson briefly taught at the University of Wisconsin before her untimely death in 1914 due to tuberculosis Tuberculosis (TB) is an infectious disease usually caused by '' Mycobacterium tuberculosis'' (MTB) bacteria. Tuberculosis generally affects the lungs, but it can also affect other parts of the body. Most infections show no symptoms, .... Sanderson's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vandermonde Determinant
In algebra, the Vandermonde polynomial of an ordered set of ''n'' variables X_1,\dots, X_n, named after Alexandre-Théophile Vandermonde, is the polynomial: :V_n = \prod_ (X_j-X_i). (Some sources use the opposite order (X_i-X_j), which changes the sign \binom times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.) It is also called the Vandermonde determinant, as it is the determinant of the Vandermonde matrix. The value depends on the order of the terms: it is an alternating polynomial, not a symmetric polynomial. Alternating The defining property of the Vandermonde polynomial is that it is ''alternating'' in the entries, meaning that permuting the X_i by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is the basic alternating polynomial, as will be made precise below. It thus depends on the order, and is zero if two entries are equa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore Determinant Over A Finite Field
In linear algebra, a Moore matrix, introduced by , is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism applied to its columns (beginning with the zeroth power of the Frobenius automorphism in the first column), so it is an ''m'' × ''n'' matrix M=\begin \alpha_1 & \alpha_1^q & \dots & \alpha_1^\\ \alpha_2 & \alpha_2^q & \dots & \alpha_2^\\ \alpha_3 & \alpha_3^q & \dots & \alpha_3^\\ \vdots & \vdots & \ddots &\vdots \\ \alpha_m & \alpha_m^q & \dots & \alpha_m^\\ \end or M_ = \alpha_i^ for all indices ''i'' and ''j''. (Some authors use the transpose of the above matrix.) The Moore determinant of a square Moore matrix (so ''m'' = ''n'') can be expressed as: \det(V) = \prod_ \left( c_1\alpha_1 + \cdots + c_n\alpha_n \right), where c runs over a complete set of direction vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for 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 symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2) has a primitive root; thus the multiplicative group of integers modulo ''p''''n'' (i.e. the group of units of the ring Z/''p''''n''Z) i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
