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Set Function
In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R and \pm \infty. A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning. Definitions If \mathcal is a family of sets over \Omega (meaning that \mathcal \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a is a function \mu with domain \mathcal and codomain \infty, \infty/math> or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The domain of a set function may have any number properties; the commonly encountered properties and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integral, integration theory, and can be generalized to assume signed measure, negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation (measure Theory)
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval 'a'', ''b''⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation ''x'' ↦ ''f''(''x''), for ''x'' ∈ 'a'', ''b'' Functions whose total variation is finite are called '' functions of bounded variation''. Historical note The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper . He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons. Definitions Total variation for functions of one real variabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unconditional Convergence
In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditional convergence, conditionally convergent if it converges but different orderings do not all converge to that same value. Unconditional convergence is equivalent to absolute convergence in Dimension (vector space), finite-dimensional vector spaces, but is a weaker property in infinite dimensions. Definition Let X be a topological vector space. Let I be an index set and x_i \in X for all i \in I. The series \textstyle \sum_ x_i is called unconditionally convergent to x \in X, if * the indexing set I_0 := \left\ is countable, and * for every permutation (bijection) \sigma : I_0 \to I_0 of I_0 = \left\_^\infty the following relation holds: \sum_^\infty x_ = x. Alternative definition Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma-additive Set Function
In mathematics, an additive set function is a function \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of Set (mathematics), sets is the set of all element (set theory), elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of Zero, zero () sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the List of mathematical symbols, table of mathematical symbols. Binary union The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, : A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairwise Disjoint
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (set theory), intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. Generalizations This definition of disjoint sets can be extended to family of sets, families of sets and to indexed family, indexed families of sets. By definition, a collection of sets is called a ''family of sets'' (such as the power set, for example). In some sources this is a set of sets, while other sources allow it to be a multiset of sets, with some sets repeated. An \left(A_i\right)_, is by definition a set-valued Function (mathematics), function (that is, it is a function that assigns a set A_i to every ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Additive Set Function
In mathematics, an additive set function is a function \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of ''k'' disjoint sets (where ''k'' is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an ''infinite'' number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, \mu\left(\bigcup_^\infty A_n\right) = \sum_^\infty \mu(A_n). Additivity and sigma-additivity are particularly important properties of measures. They are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physics. It was found that different atoms and different elementary particle, elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple Mass in special relativity, definitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure (mathematics), measure of the body's inertia, meaning the resistance to acceleration (change of velocity) when a net force is applied. The object's mass also determines the Force, strength of its gravitational attraction to other bodies. The SI base unit of mass is the kilogram (kg). In physics, mass is Mass versus weight, not the same as weight, even though mass is often determined by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war period. It was badly damaged during World War II (1939–45). In the first thirty years after the war the shipyard again experienced a boom and employed up to 3,000 workers making oil tankers, and then liquid natural gas tankers. Demand dropped off in the 1970s and 1980s. In 1972 the shipyard became Chantiers de France-Dunkerque, and in 1983 merged with others yards to become part of Chantiers du Nord et de la Mediterranee, or Normed. The shipyard closed in 1987. Foundation (1898–99) The Ateliers et Chantiers de France (ACF) company was officially founded on 6 July 1898 by a consortium of six shipping brokers, the Dunkirk chamber of commerce and the state. The state asked that the shipyard be able to build steamships and also four-masted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seminormed Space
In mathematics, particularly in functional analysis, a seminorm is like a norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Definition Let X be a vector space over either the real numbers \R or the complex numbers \Complex. A real-valued function p : X \to \R is called a if it satisfies the following two conditions: # Subadditivity/Triangle inequality: p(x + y) \leq p(x) + p(y) for all x, y \in X. # Absolute homogeneity: p(s x) =, s, p(x) for all x \in X and all scalars s. These two conditions imply that p(0) = 0If z \in X denotes the zero vector in X while 0 denote the zero scalar, then absolute homogeneity implies that p(z) = p(0 z) = , 0, p(z) = 0 p(z) = 0. \blacksquare and that ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
