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Lyapunov Vector-measure Theorem
In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Definitions and first consequences Given a field of sets (\Omega, \mathcal F) and a Banach space X, a finitely additive vector measure (or measure, for short) is a function \mu:\mathcal \to X such that for any two disjoint sets A and B in \mathcal one has \mu(A\cup B) =\mu(A) + \mu (B). A vector measure \mu is called countably additive if for any sequence (A_i)_^ of disjoint sets in \mathcal F such that their union is in \mathcal F it holds that \mu = \sum_^\mu(A_i) with the series on the right-hand side convergent in the norm of the Banach space X. It can be proved that an additive vector measure \mu is countably additive if and only if for any sequence (A_i)_^ as above one has where \, \cdot\, is the norm on X. Countably additive vector ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
