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Exponential Field
In mathematics, an exponential field is a Field (mathematics), field with a further unary operation that is a homomorphism from the field's additive group to its multiplicative group. This generalizes the usual idea of exponentiation on the real numbers, where the base is a chosen positive real number. 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 (mathematics), 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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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_=(+,-,\tfrac. References {{Reflist Model theory Unsolved problems in mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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, type theory, and analytic philosophy. Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school, 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.#FefA, Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
O-minimal
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 quantifier elimination. A structure ''M'' is o-minimal if and only if every with one [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
