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Separation Theorem (other)
Separation theorem may refer to several theorems in different scientific fields. Economics * Fisher separation theorem (corporation theory) - asserts that the objective of a corporation will be the maximization of its present value, regardless of the preferences of its shareholders. *Mutual fund separation theorem (portfolio theory) states that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Mathematics * Gabbay's separation theorem (mathematical logic and computer science) states that any arbitrary temporal logic formula can be rewritten in a logically equivalent "past → future" form. *Planar separator theorem (graph theory) states that any planar graph can be split into smaller pieces by removing a small number of vertices. * Lusin's separation theorem (descriptive set theory) states t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fisher Separation Theorem
In economics, the Fisher separation theorem asserts that the primary objective of a corporation will be the maximization of its present value, regardless of the preferences of its shareholders. The theorem therefore separates management's "productive opportunities" from the entrepreneur's "market opportunities". It was proposed by—and is named after—the economist Irving Fisher. The theorem has its "clearest and most famous expositionin the ''Theory of Interest'' (1930); particularly in the "second approximation to the theory of interest"II:VI. {{Ref improve section, date=January 2011 The Fisher separation theorem states that: * the firm's Corporate_finance#The_investment_decision, investment decision is independent of the consumption preferences of the owner; * the investment decision is independent of the financing decision. * the value of a capital project (investment) is independent of the mix of methods – equity, debt, and/or cash – used to finance the project. Fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Fund Separation Theorem
In portfolio theory, a mutual fund separation theorem, mutual fund theorem, or separation theorem is a theorem stating that, under certain conditions, any investor's optimal portfolio can be constructed by holding each of certain mutual funds in appropriate ratios, where the number of mutual funds is smaller than the number of individual assets in the portfolio. Here a mutual fund refers to any specified benchmark portfolio of the available assets. There are two advantages of having a mutual fund theorem. First, if the relevant conditions are met, it may be easier (or lower in transactions costs) for an investor to purchase a smaller number of mutual funds than to purchase a larger number of assets individually. Second, from a theoretical and empirical standpoint, if it can be assumed that the relevant conditions are indeed satisfied, then implications for the functioning of asset markets can be derived and tested. Portfolio separation in mean-variance analysis Portfolios can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabbay's Separation Theorem
In mathematical logic and computer science, Gabbay's separation theorem, named after Dov Gabbay, states that any arbitrary temporal logic formula can be rewritten in a logically equivalent Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ... "past → future" form. I.e. the future becomes what must be satisfied. This form can be used as execution rules; a MetateM program is a set of such rules.. References Artificial intelligence Theorems Temporal logic {{Comp-sci-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Separator Theorem
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of vertices from an -vertex graph (where the invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most vertices. A weaker form of the separator theorem with vertices in the separator instead of was originally proven by , and the form with the tight asymptotic bound on the separator size was first proven by . Since their work, the separator theorem has been reproven in several different ways, the constant in the term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs. Repeated application of the separator theorem produces a separator hierarchy which may take the form of either a tree decomposition or a branch-decomposition of the graph. Separator hierarc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lusin's Separation Theorem
In descriptive set theory and mathematical logic, Lusin's separation theorem states that if ''A'' and ''B'' are disjoint analytic subsets of Polish space, then there is a Borel set ''C'' in the space such that ''A'' ⊆ ''C'' and ''B'' ∩ ''C'' = ∅.. It is named after Nikolai Luzin Nikolai Nikolaevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlaɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 January 1950) was a Soviet/Ru ..., who proved it in 1927.. The theorem can be generalized to show that for each sequence (''A''''n'') of disjoint analytic sets there is a sequence (''B''''n'') of disjoint Borel sets such that ''A''''n'' ⊆ ''B''''n'' for each ''n''. An immediate consequence is Suslin's theorem, which states that if a set and its complement are both analytic, then the set is Borel. Notes References * ( for the Euro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Separation Theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ''n''-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces. A related result is the supporting hyperplane theorem. In the context of support-vector machines, the ''optimally separating hyperplane'' or ''maximum-margin hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Separator
A geometric separator is a line (or another shape) that partitions a collection of geometric shapes into two subsets, such that proportion of shapes in each subset is bounded, and the number of shapes that do not belong to any subset (i.e. the shapes intersected by the separator itself) is small. When a geometric separator exists, it can be used for building divide-and-conquer algorithms for solving various problems in computational geometry. Separators that are lines General question In 1979, Helge Tverberg raised the following question. For two positive integers ''k'', ''l'', what is the smallest number ''n''(''k'',''l'') such that, for any family of pairwise-disjoint convex objects in the plane, there exists a straight line that has at least ''k'' objects on one side and at least ''l'' on the other side? The following results are known. * Obviously, ''n''(1,1)=1. * Hope and Katchalski proved that ''n''(''k'',1) ≤ 12(''k''-1) for all ''k'' ≥ 2. * Villanger prove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
