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Completely Additive Function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the function applied to ''a'' and ''b'':Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207online/ref> f(a b) = f(a) + f(b). Completely additive An additive function ''f''(''n'') is said to be completely additive if f(a b) = f(a) + f(b) holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If ''f'' is a completely additive function then ''f''(1) = 0. Every completely additive function is additive, but not vice versa. Examples Examples of arithmetic functions which are completely additive are: * The restriction of the Logarithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
