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S-units
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] |
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 unit ... [...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] |
Finitely Generated Abelian Group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n_s. In this case, we say that the set \ is a ''generating set'' of G or that x_1,\dots, x_s ''generate'' G. Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. Examples * The integers, \left(\mathbb,+\right), are a finitely generated abelian group. * The integers modulo n, \left(\mathbb/n\mathbb,+\right), are a finite (hence finitely generated) abelian group. * Any direct sum of finitely many finitely generated abelian groups is again a finitely generated abelian group. * Every lattice forms a finitely generated free abelian group. There are no other examples (up to isomorphism). In particular, the group \left(\mathbb,+\right) of rational numbers is not finitely generated ... [...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 internationally, o ... [...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 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 population o ... [...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 tha ... [...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 In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ... of degree d\geq 3 with coefficients in a field k; more precisely, it is the smooth scheme, smooth projective curve whose Algebraic function field, 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 (algebra), characteristic of k, that the polynomial f should be square free, that the integers ''m'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
