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Supernatural Number
In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory. A supernatural number \omega is a formal product: : \omega = \prod_p p^, where p runs over all prime numbers, and each n_p is zero, a natural number or infinity. Sometimes v_p(\omega) is used instead of n_p. If no n_p = \infty and there are only a finite number of non-zero n_p then we recover the positive integers. Slightly less intuitively, if all n_p are \infty, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide \omega "infinitely often," by taking that prime's corresponding exponent to be the symbol \infty. There is no natural way to add supernatural numbers, but they can be multiplied, with \prod_p p^\cdot\prod_p p^=\prod_p p^. Si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinitz Numbers Svg
Steinitz may refer to: * Steinitz, Germany Steinitz is a village and a former municipality in the district Altmarkkreis Salzwedel, in Saxony-Anhalt, Germany. Since January 2011, it is part of the town Salzwedel. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profinite Integer
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) :\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p where :\varprojlim \mathbb/n\mathbb indicates the profinite completion of \mathbb, the index p runs over all prime numbers, and \mathbb_p is the ring of ''p''-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable example of a profinite group. Construction The profinite integers \widehat can be constructed as the set of sequences \upsilon of residues represented as : \upsilon = (\upsilon_1 \bmod 1, ~ \upsilon_2 \bmod 2, ~ \upsilon_3 \bmod 3, ~ \ldots) such that m \ , \ n \implies \upsilon_m \equiv \upsilon_n \bmod m. Pointwise addition and multiplication make it a commutative ring. The ring of integers embeds into the ring of profinite integers by the canonical injection: :\eta: \mathbb \ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Hyperfinite Algebra
In mathematics, particularly in the theory of C*-algebras, a uniformly hyperfinite, or UHF, algebra is a C*-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras. Definition A UHF C*-algebra is the direct limit of an inductive system where each ''An'' is a finite-dimensional full matrix algebra and each ''φn'' : ''An'' → ''A''''n''+1 is a unital embedding. Suppressing the connecting maps, one can write :A = \overline . Classification If :A_n \simeq M_ (\mathbb C), then ''rk''''n'' = ''k''''n'' + 1 for some integer ''r'' and :\phi_n (a) = a \otimes I_r, where ''Ir'' is the identity in the ''r'' × ''r'' matrices. The sequence ...''kn'', ''k''''n'' + 1, ''k''''n'' + 2... determines a formal product :\delta(A) = \prod_p p^ where each ''p'' is prime and ''tp'' = sup , possibly zero or infinite. The formal product ''δ''(''A'') is said to be the supernatural number correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field extension that is not algebraic, is said to be transcendental, and must contain transcendental elements, that is, elements that are not algebraic. The algebraic extensions of the field \Q of the rational numbers are called algebraic number fields and are the main objects of study of algebraic number theory. Another example of a common algebraic extension is the extension \Complex/\R of the real numbers by the complex numbers. Some properties All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group Theory
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause is a clause whose main verb is non-finite See also * Infinite (other) Infin ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profinite Group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the larges ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
