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Exponential Field
In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation. Definition A field is an algebraic structure composed of a set of elements, ''F'', two binary operations, addition (+) such that ''F'' forms an abelian group with identity 0''F'' and multiplication (·), such that ''F'' excluding 0''F'' forms an abelian group under multiplication with identity 1''F'', and such that multiplication is distributive over addition, that is for any elements ''a'', ''b'', ''c'' in ''F'', one has . If there is also a function ''E'' that maps ''F'' into ''F'', and such that for every ''a'' and ''b'' in ''F'' one has :\begin&E(a+b)=E(a)\cdot E(b),\\&E(0_F)=1_F \end then ''F'' is called an exponential field, and the function ''E'' is called an exponential function on ''F''. Thus an exponential function on a field is a homomorphism between the additive group of ''F'' and its multiplicative group. Trivial exponential f ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit (ring Theory)
In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More generally, any root of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Exponential Field
In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers. Definition An exponential E on an ordered field K is a strictly increasing isomorphism of the additive group of K onto the multiplicative group of positive elements of K. The ordered field K\, together with the additional function E\, is called an ordered exponential field. Examples * The canonical example for an ordered exponential field is the ordered field of real numbers R with any function of the form a^x where a is a real number greater than 1. One such function is the usual exponential function, that is . The ordered field R equipped with this function gives the ordered real exponential field, denoted by . It was proved in the 1990s that Rexp is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that Rexp i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tarski's Exponential Function Problem
In model theory, Tarski's exponential function problem asks whether the theory of the real numbers together with the exponential function is decidable. Alfred Tarski had previously shown that the theory of the real numbers (without the exponential function) is decidable. The problem The ordered real field R is a structure over the language of ordered rings ''L''or = (+,·,−, ''η''−1. Workaround Recently there are attempts at handling the theory of the real numbers with functions such as the exponential function or sine by relaxing decidability to the weaker notion of quasi-decidability. A theory is quasi-decidable if there is a procedure that decides satisfiability but that may run forever for inputs that are not robust in a certain, well-defined sense. Such a procedure exists for systems of equations in variables . References * * *{{citation, first1=Peter, last1=Franek, first2=Stefan, last2=Ratschan, first3=Piotr, last3=Zgliczynski, chapter=S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Tarski
Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician and mathematician. A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983. Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he changed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decidability (logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
O-minimal
In mathematical logic, and more specifically in model theory, an infinite structure (''M'',<,...) which 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 formula with one free variable and parameters in ''M'' is equivalent to a quantifier-free formula involving only the ordering, als ... [...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] |
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] |
Model Complete Theory
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] |
Boris Zilber
Boris Zilber (russian: Борис Иосифович Зильбер, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Sciences) from the Novosibirsk State University in 1975 under the supervision of Mikhail Taitslin and his habilitation (Doctor of Sciences) from the Saint Petersburg State University in 1986. He received the Senior Berwick Prize (2004) and the Pólya Prize (2015) from the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical .... He also gave the Tarski Lectures in 2002. References External links Prof. Zilber's homepage 20th-century British mathematicians 21st-century Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
