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Subadditive Set Function
In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions. Definition Let \Omega be a set and f \colon 2^ \rightarrow \mathbb be a set function, where 2^\Omega denotes the power set of \Omega. The function ''f'' is ''subadditive'' if for each subset S and T of \Omega, we have f(S) + f(T) \geq f(S \cup T). Note that by substitution of T=S into the defining equation, it follows that f(S) \ge 0 for all . Examples of subadditive functions Every non-negative submodular set function is subadditive (the family of non-negative submodular functions is strictly contained in the family of subadditive functions). The function that counts the number of sets required to cover a given set is subadditive. Let T_1, \dotsc, T_m \subseteq \Omega su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Subadditivity
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions. Definitions A subadditive function is a function f \colon A \to B, having a domain ''A'' and an ordered codomain ''B'' that are both closed under addition, with the following property: \forall x, y \in A, f(x+y)\leq f(x)+f(y). An example is the square root function, having the non-negative real numbers as domain and codomain: since \forall x, y \geq 0 we have: \sqrt\leq \sqrt+\sqrt. A sequence \left \_ is called subadditive if it satisfies the inequality a_\leq a_n+a_m for all ''m'' and ''n''. This is a special case of subadditive function, if a sequ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principle Of Sigma-subadditivity
A principle may relate to a fundamental truth or proposition that serves as the foundation for a system of beliefs or behavior or a chain of reasoning. They provide a guide for behavior or evaluation. A principle can make values explicit, so they are expressed in the form of rules and standards. Principles unpack the values underlying them more concretely so that the values can be more easily operationalized in policy statements and actions. In law, higher order, overarching principles establish rules to be followed, modified by sentencing guidelines relating to context and proportionality. In science and nature, a principle may define the essential characteristics of the system, or reflect the system's designed purpose. The effective operation would be impossible if any one of the principles was to be ignored. A system may be explicitly based on and implemented from a document of principles as was done in IBM's 360/370 ''Principles of Operation''. It is important to differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit ( diminishing returns). The natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of \Omega, which satisfies one of the following equivalent condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Cover
The set cover problem is a classical question in combinatorics, computer science, operations research, and Computational complexity theory, complexity theory. Given a Set (mathematics), set of elements (henceforth referred to as the Universe (mathematics), universe, specifying all possible elements under consideration) and a collection, referred to as , of a given subsets whose union (set theory), union equals the universe, the set cover problem is to identify a smallest sub-collection of whose union equals the universe. For example, consider the universe, and the collection of sets In this example, is equal to 4, as there are four subsets that comprise this collection. The union of is equal to . However, we can cover all elements with only two sets: , see picture, but not with only one set. Therefore, the solution to the set cover problem for this and has size 2. More formally, given a universe \mathcal and a family \mathcal of subsets of \mathcal, a set cover is a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum
In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (mathematics), range (the ''local'' or ''relative'' extrema) or on the entire domain of a function, domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set (mathematics), set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued Function (mathematics), function ''f'' defined on a Domain of a function, domain ''X'' has a global (or absolute) m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z- module homomorphism. Since an abelian group is a \Z- module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superadditive
In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f. Similarly, a sequence a_1, a_2, \ldots is called superadditive if it satisfies the inequality a_ \geq a_n + a_m for all m and n. The term "superadditive" is also applied to functions from a boolean algebra to the real numbers where P(X \lor Y) \geq P(X) + P(Y), such as lower probabilities. Examples of superadditive functions * The map f(x) = x^2 is a superadditive function for nonnegative real numbers because f(x + y) = (x + y)^2 = x^2 + y^2 + 2 x y = f(x) + f(y) + 2 x y \ge f(x) + f(y). * The determinant is superadditive for nonnegative Hermitian matrix, that is, if A, B \in \text_n(\Complex) are nonnegative Hermitian then \det(A + B) \geq \det(A) + \det(B). This follows from the Minkowski determinant theorem, which more generally states that \det(\cdot)^ is superadditive (equivalently, concave) for nonnegative Hermitian matrices of size n: If A, B \in \text_n(\Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodular Set Function
In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit ( diminishing returns). The natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set, a submodular function is a set function f:2^\rightarrow \mathbb, where 2^\Omega denotes the power set of \Omega, which satisfies one of the following equivalent condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
