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S-unit
In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units. Definition Let ''K'' be a number field with ring of integers ''R''. Let ''S'' be a finite set of prime ideals of ''R''. An element ''x'' of ''K'' is an ''S''-unit if the principal fractional ideal (''x'') is a product of primes in ''S'' (to positive or negative powers). For the ring of rational integers Z one may take ''S'' to be a finite set of prime numbers and define an ''S''-unit to be a rational number whose numerator and denominator are divisible only by the primes in ''S''. Properties The ''S''-units form a multiplicative group containing the units of ''R''. Dirichlet's unit theorem holds for ''S''-units: the group of ''S''-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to ''r'' + ''s'', where ''r' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S Meter
An S meter (signal strength meter) is an indicator often provided on communications receivers, such as amateur radio or shortwave broadcast receivers. The scale markings are derived from a system of reporting signal strength from S1 to S9 as part of the R-S-T system. The term S unit refers to the amount of signal strength required to move an S meter indication from one marking to the next. Technical description Analogue S meters are actually sensitive microammeters, with a full scale deflection of 50 to 100 μA. In AM receivers, the S meter can be connected to the main detector or use a separate detector at the final IF stage. This is the preferred method for CW and SSB receivers. Another approach in the days of electronic tubes (valves) was to connect the S meter to the screen grid circuit of the final IF amplifier tube. A third option is to connect the S meter to the AGC line through a suitable level conversion circuit. In FM receivers, the S meter circuit must be connec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Unit Theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a positive real number that determines how "dense" the units are. The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to where is the ''number of real embeddings'' and the ''number of conjugate pairs of complex embeddings'' of . This characterisation of and is based on the idea that there will be as many ways to embed in the complex number field as the degree n = : \mathbb/math>; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that Note that if is Galois over \mathbb then either or . Other ways of determining and are * use the primitive element theorem to write K = \mathbb(\alpha), and the ... [...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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of An Abelian Group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset. The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If ''A'' is torsion-free then it embeds into a vector space over the rational numbers of dimension rank ''A''. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved. The term rank has a different meaning in the context of elementary abelian groups. Definition A subset of an abelian group ''A'' is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if : \sum_\alpha n_\alpha a_\alpha = 0, \quad n_\alpha\in\mathbb, where all but finitely many ... [...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 international ... [...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 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Providence, RI
Providence is the capital and most populous city of the U.S. state of Rhode Island. One of the oldest cities in New England, it was founded in 1636 by Roger Williams, a Reformed Baptist theologian and religious exile from the Massachusetts Bay Colony. He named the area in honor of "God's merciful Providence" which he believed was responsible for revealing such a haven for him and his followers. The city developed as a busy port as it is situated at the mouth of the Providence River in Providence County, at the head of Narragansett Bay. Providence was one of the first cities in the country to industrialize and became noted for its textile manufacturing and subsequent machine tool, jewelry, and silverware industries. Today, the city of Providence is home to eight hospitals and eight institutions of higher learning which have shifted the city's economy into service industries, though it still retains some manufacturing activity. At the 2020 census, Providence had a populatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics. The first version of SageMath was released on 24 February 2005 as free and open-source software under the terms of the GNU General Public License version 2, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB". The originator and leader of the SageMath project, William Stein, was a mathematician at the University of Washington. SageMath uses a syntax resembling Python's, supporting procedural, functional and object-oriented constructs. Development Stein realized when designing Sage that there were many open-source mathematics software packages already written in different languages, namely C, C++, Common Lisp, Fortran and Python. Rather th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superelliptic Curve
In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form :y^m = f(x), where m \geq 2 is an integer and ''f'' is a polynomial of degree d\geq 3 with coefficients in a field k; more precisely, it is the smooth projective curve whose function field defined by this equation. The case m=2 and d=3 is an ''elliptic curve'', the case m=2 and d\ge 5 is a ''hyperelliptic curve'', and the case m=3 and d\geq 4 is an example of a ''trigonal curve''. Some authors impose additional restrictions, for example, that the integer m should not be divisible by the characteristic of k, that the polynomial f should be square free, that the integers ''m'' and ''d'' should be coprime, or some combination of these. The Diophantine problem of finding integer points on a superelliptic curve can be solved by a method similar to one used for the resolution of hyperelliptic equations: a Siegel identity is used to reduce to a Thue equation. Definition More generally, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel's Theorem
In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve ''C'' of genus ''g'' defined over a number field ''K'', presented in affine space in a given coordinate system, there are only finitely many points on ''C'' with coordinates in the ring of integers ''O'' of ''K'', provided ''g'' > 0. The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations. For ''g'' > 1 it was superseded by Faltings's theorem in 1983. History In 1929, Siegel proved the theorem by combining a version of the Thue–Siegel–Roth theorem, from diophantine approximation, with the Mordell–Weil theorem from diophantine geometry (required in Weil's version, to apply to the Jacobian variety of ''C''). In 2002, Umberto Zannier and Pietro Corvaja gave a new proof by using a new method based on the subspace theorem.Corvaja, P. and Zannier, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
