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Total Algebra
In abstract algebra, the total algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring. Suppose that ''S'' is a monoid with the property that, for all s\in S, there exist only finitely many ordered pairs (t,u)\in S\times S for which tu=s. Let ''R'' be a ring. Then the total algebra of ''S'' over ''R'' is the set R^S of all functions \alpha:S\to R with the addition law given by the (pointwise) operation: :(\alpha+\beta)(s)=\alpha(s)+\beta(s) and with the multiplication law given by: :(\alpha\cdot\beta)(s) = \sum_\alpha(t)\beta(u). The sum on the right-hand side has finite support, and so is well-defined in ''R''. These operations turn R^S into a ring. There is an embedding of ''R'' into R^S, given by the constant functions, which turns R^S into an ''R''-algebra. An example is the ring of formal power series, where the monoid ''S'' is the natural numbers. The product is then the Cauchy product In mathematics, more specifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being . Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid. Transition monoids and syntactic monoids are used in describing finite-state machines. Trace monoids and history monoids provide a foundation for process calculi and concurrent computing. In theoretical computer science, the study of monoids is fundamental for automata theory (Krohn–Rhodes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Monoid Ring
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Definition Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' over ''R'', denoted ''R'' 'G''or ''RG'', is the set of formal sums \sum_ r_g g, where r_g \in R for each g \in G and ''r''''g'' = 0 for all but finitely many ''g'', equipped with coefficient-wise addition, and the multiplication in which the elements of ''R'' commute with the elements of ''G''. More formally, ''R'' 'G''is the free ''R''-module on the set ''G'', endowed with ''R''-linear multiplication defined on the base elements by ''g·h'' := ''gh'', where the left-hand side is understood as the multiplication in ''R'' 'G''and the right-hand side is understood in ''G''. Alternatively, one can identify the element g \in R with the function ''eg'' that maps ''g'' to 1 and every other element of ''G'' to 0. This way, ''R'' 'G''is identifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Series (mathematics)
In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greece, Ancient Greeks, the idea that a potential infinity, potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the Quadrature of the Parabola, quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Natural Numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cauchy Product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section. Cauchy product of two infinite series Let \sum_^\infty a_i and \sum_^\infty b_j be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows: :\left(\sum_^\infty a_i\right) \cdot \left(\sum_^\infty b_j\right) = \sum_^\infty c_k where c_k=\sum_^k a_l b_. Cauchy product of two power series Consider the following two power series :\sum_^\infty a_i x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in mathematical analysis, analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the ''Éléments de mathématique'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups, and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Cartan co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |