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Wilkie's Theorem
In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties. Formulations In terms of model theory, Wilkie's theorem deals with the language ''L''exp = (+, −, ·, ''m''. Gabrielov's theorem states that any formula in this language is equivalent to an existential one, as above. Hence the theory of the real ordered field with restricted analytic functions is model complete. Intermediate results Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential function. As an intermediate result Wilkie asked when the complement of a sub-analytic set could be defined using the same analytic functions that described the original set. It turns out the required functions are the Pfaffian function In mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pfaffian Function
In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff. Basic definition Some functions, when differentiated, give a result which can be written in terms of the original function. Perhaps the simplest example is the exponential function, ''f''(''x'') = ''e''''x''. If we differentiate this function we get ''ex'' again, that is :f^\prime(x) = f(x). Another example of a function like this is the reciprocal function, ''g''(''x'') = 1/''x''. If we differentiate this function we will see that :g^\prime(x) = -g(x)^2. Other functions may not have the above property, but their derivative may be written in terms of functions like those above. For example, if we take the function ''h''(''x'') = ''e''''x'' log(''x'') then we see :h^\prime(x) = e^x\log x+x^ ... [...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 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 domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 semianalytic set ''X'' in a Euclidean space ''F'' of dimension at least ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Polynomial
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function. Definition In fields An exponential polynomial generally has both a variable ''x'' and some kind of exponential function ''E''(''x''). In the complex numbers there is already a canonical exponential function, the function that maps ''x'' to '' e''''x''. In this setting the term exponential polynomial is often used to mean polynomials of the form ''P''(''x'', ''e''''x'') where ''P'' ∈ C 'x'', ''y''is a polynomial in two variables. There is nothing particularly special about C here; exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of ''e''''x'' above. Similarly, there is no reason to have one variable, and an exponential polynomial in ''n'' variables would be of the form ''P''(''x''1, ..., ''x''''n'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...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/math> :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 paramet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, Aerospace engineering, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including Algebraic geometry, algebraic, Differential geometry, differential, Discrete geometry, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Complete
In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula. This notion was introduced by Abraham Robinson. Model companion and model completion A companion of a theory ''T'' is a theory ''T''* such that every model of ''T'' can be embedded in a model of ''T''* and vice versa. A model companion of a theory ''T'' is a companion of ''T'' that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if ''T'' is an \aleph_0-categorical theory, then it always has a model companion. A model completion for a theory ''T'' is a model companion ''T''* such that for any model ''M'' of ''T'', the theory of ''T''* together with the diagram of ''M'' is complete. Roughly speaking, this means every model of ''T'' is embeddable in a model of ''T''* in a uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^ = e^x e^y \text x,y\in\mathbb, which, along with the definition e = \exp(1), shows that e^n=\underbrace_ for positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier Elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a question "When is there an x such that \ldots?", and the statement without quantifiers can be viewed as the answer to that question. One way of classifying formulas is by the amount of quantification. Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula \alpha, there exists another formula \alpha_ without quantifiers that is equivalent to it ( modulo this theory). Examples An example from high school mathematics says that a single-variable quadratic polynomial has a real root if and only if its discriminant is non-negative: :: \exists x\in\mathbb. (a\neq 0 \wedge ax^2+bx+c=0)\ \ \Longleftrightarrow\ \ a\neq 0 \wedge b^2-4ac\geq 0 He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
