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O-minimality
In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) that is totally ordered by < is called an o-minimal structure if and only if every definable subset ''X'' ⊆ ''M'' (with parameters taken from ''M'') is a finite union of intervals and points. O-minimality can be regarded as a weak form of . A structure ''M'' is o-minimal if and only if every |
Weakly O-minimal Structure
In model theory, a weakly o-minimal structure is a model-theoretic structure whose definable sets in the domain are just finite unions of convex sets. Definition A linearly ordered structure, ''M'', with language ''L'' including an ordering relation <, is called weakly o-minimal if every parametrically definable subset of ''M'' is a finite union of convex (definable) subsets. A is weakly o-minimal if all its models are weakly o-minimal. Note that, in contrast to o-minimality, it is possible for a theory to have models that are weakly o-minimal and to have other models that are not weakly o-minimal. Difference from o-minimality In an o-minimal structure the definable sets in are finite unions of points ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Definable Set
In mathematical logic, a definable set is an ''n''-ary relation on the domain of a structure whose elements satisfy some formula in the first-order language of that structure. A set can be defined with or without parameters, which are elements of the domain that can be referenced in the formula defining the relation. Definition Let \mathcal be a first-order language, \mathcal an \mathcal-structure with domain M, X a fixed subset of M, and m a natural number. Then: * A set A\subseteq M^m is ''definable in \mathcal with parameters from X'' if and only if there exists a formula \varphi _1,\ldots,x_m,y_1,\ldots,y_n/math> and elements b_1,\ldots,b_n\in X such that for all a_1,\ldots,a_m\in M, :(a_1,\ldots,a_m)\in A if and only if \mathcal\models\varphi _1,\ldots,a_m,b_1,\ldots,b_n :The bracket notation here indicates the semantic evaluation of the free variables in the formula. * A set ''A is definable in \mathcal without parameters'' if it is definable in \mathcal with parameters ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Algebraic Geometry
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets. Terminology Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semialgebraic Set
In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic 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. Definition Let \mathbb be a real closed field (For example \mathbb could be the field of real numbers \mathbb). A subset S of \mathbb^n is a ''semialgebraic set'' if it is a finite union of sets defined by polynomial equalities of the form \ and of sets defined by polynomial inequalities of the form \. 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–Seide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Łojasiewicz Inequality
In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : ''U'' → R be a real analytic function on an open set ''U'' in R''n'', and let ''Z'' be the zero locus of ƒ. Assume that ''Z'' is not empty. Then for any compact set ''K'' in ''U'', there exist positive constants α and ''C'' such that, for all ''x'' in ''K'' :\operatorname(x,Z)^\alpha \le C, f(x), . Here, \alpha can be small. The following form of this inequality is often seen in more analytic contexts: with the same assumptions on ''f'', for every ''p'' ∈ ''U'' there is a possibly smaller open neighborhood ''W'' of ''p'' and constants θ ∈ (0,1) and ''c'' > 0 such that :, f(x)-f(p), ^\theta\le c, \nabla f(x), . Polyak inequality A special case of the Łojasiewicz inequality, due to , is comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stratification (mathematics)
Stratification has several usages in mathematics. In mathematical logic In mathematical logic, stratification is any consistent assignment of numbers to predicate symbols guaranteeing that a unique formal interpretation of a logical theory exists. Specifically, we say that a set of clauses of the form Q_1 \wedge \dots \wedge Q_n \wedge \neg Q_ \wedge \dots \wedge \neg Q_ \rightarrow P is stratified if and only if there is a stratification assignment S that fulfills the following conditions: # If a predicate P is positively derived from a predicate Q (i.e., P is the head of a rule, and Q occurs positively in the body of the same rule), then the stratification number of P must be greater than or equal to the stratification number of Q, in short S(P) \geq S(Q). # If a predicate P is derived from a negated predicate Q (i.e., P is the head of a rule, and Q occurs negatively in the body of the same rule), then the stratification number of P must be greater than the stratification numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. 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 + \cdots in which the coefficients a_0, a_1, \dots a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersion (mathematics), immersions, characteristic classes and, geometric integration theory. Biography Life Hassler Whitney was born on March 23, 1907, in New York City, where his father, Edward Baldwin Whitney, was the First District New York Supreme Court judge. His mother, A. Josepha Newcomb Whitney, was an artist and political activist. He was the paternal nephew of Connecticut Governor and Chief Justice Simeon E. Baldwin, his paternal grandfather was William Dwight Whitney, professor of Ancient Languages at Yale University, linguist and Sanskrit scholar. Whitney was the great-grandson of Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman. His maternal grandparents were astronomer and mathematician Simon Newcomb (183 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Algebraic Geometry
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them (in particular real polynomial mappings). Semialgebraic geometry is the study of semialgebraic sets, i.e. real-number solutions to algebraic inequalities with-real number coefficients, and mappings between them. The most natural mappings between semialgebraic sets are semialgebraic mappings, i.e., mappings whose graphs are semialgebraic sets. Terminology Nowadays the words 'semialgebraic geometry' and 'real algebraic geometry' are used as synonyms, because real algebraic sets cannot be studied seriously without the use of semialgebraic sets. For example, a projection of a real algebraic set along a coordinate axis need not be a real algebraic set, but it is always a semialgebraic set: this is the Tarski–Seidenberg theorem. Related fields are o-minimal theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semialgebraic Set
In mathematics, a basic semialgebraic set is a set defined by polynomial equalities and polynomial inequalities, and a semialgebraic set is a finite union of basic semialgebraic 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. Definition Let \mathbb be a real closed field (For example \mathbb could be the field of real numbers \mathbb). A subset S of \mathbb^n is a ''semialgebraic set'' if it is a finite union of sets defined by polynomial equalities of the form \ and of sets defined by polynomial inequalities of the form \. 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–Seide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
