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Subanalytic Set
In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real power series to be positive there). Subanalytic sets still have a reasonable local description in terms of submanifolds. Formal definitions A subset ''V'' of a given Euclidean space ''E'' is semianalytic if each point has a neighbourhood ''U'' in ''E'' such that the intersection of ''V'' and ''U'' lies in the Boolean algebra of sets generated by subsets defined by inequalities ''f'' > 0, where f is a real analytic function. There is no Tarski–Seidenberg theorem for semianalytic sets, and projections of semianalytic sets are in general not semianalytic. A subset ''V'' of ''E'' is a subanalytic set if for each point there exists a relatively compact In mathematics, a relatively compact subspace (or relatively compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Real Analytic Geometry
Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) * ''Real'' (Belinda Carlisle album) (1993) * ''Real'' (Gorgon City EP) (2013) * ''Real'' (IU EP) (2010) * ''Real'' (Ivy Queen album) (2004) * ''Real'' (Mika Nakashima album) (2013) * ''Real'' (Ednita Nazario album) (2007) * ''Real'' (Jodie Resther album), a 2000 album by Jodie Resther * ''Real'' (Michael Sweet album) (1995) * ''Real'' (The Word Alive album) (2014) * ''Real'', a 2002 album by Israel Houghton recording as Israel & New Breed Songs * "Real" (Goo Goo Dolls song) (2008) * "Real" (Gorgon City song) (2013) * "Real" (Plumb song) (2004) * "Real" (Vivid song) (2012) * "Real" (James Wesley song) (2010) * "Real", a song by Kendrick Lamar from '' Good Kid, M.A.A.D City'' * "Real", a song by NF from ''Therapy Session'' * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (paral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class ''C''''r'' for a fixed , and all morphisms are differentiable of class ''C''''r''. Immersed submanifolds An immersed submanifold of a manifold ''M'' is the image ''S'' of an immersion map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset ''S'' together with a topology and differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Real Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its doma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Tarski–Seidenberg Theorem
In mathematics, the Tarski–Seidenberg theorem states that a set in (''n'' + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto ''n''-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–Seidenberg projection property—is named after Alfred Tarski and Abraham Seidenberg. It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectives (''or''), (''and''), (''not'') and quantifiers (''for all''), (''exists'') is equivalent to a similar formula without quantifiers. An important consequence is the decidability of the theory of real-closed fields. Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Relatively Compact
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Locally Finite Collection
In the mathematical field of topology, local finiteness is a property of collections of subsets of a topological space. It is fundamental in the study of paracompactness and topological dimension. A collection of subsets of a topological space X is said to be locally finite if each point in the space has a neighbourhood that intersects only finitely many of the sets in the collection. Note that the term locally finite has different meanings in other mathematical fields. Examples and properties A finite collection of subsets of a topological space is locally finite. Infinite collections can also be locally finite: for example, the collection of all subsets of \mathbb of the form (n, n+2) for an integer n. A countable collection of subsets need not be locally finite, as shown by the collection of all subsets of \mathbb of the form (-n, n) for a natural number ''n''. If a collection of sets is locally finite, the collection of all closures of these sets is also locally ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Semialgebraic Set
In mathematics, a semialgebraic set is a subset ''S'' of ''Rn'' for some real closed field ''R'' (for example ''R'' could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n) = 0) and inequalities (of the form Q(x_1,...,x_n) > 0), or any finite union of such sets. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Properties Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another such (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |