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Tropical Geometry
In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x\oplus y=\min\, : x\otimes y=x+y. So for example, the classical polynomial x^3+xy+y^4 would become \min\. Such polynomials and their solutions have important applications in optimization problems, for example the problem of optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear meshes, and in which numbers belong to the tropical semiring instead of a field. Because classical and tropical geometry are closely related, results and methods can be converted between them. Algebraic varieties can be mapped to a tropical counterpart and, since this process still retains some geometric information about the original variety, it can be used to help prove and generalize classical results from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tropical
The tropics are the regions of Earth surrounding the equator, where the sun may shine directly overhead. This contrasts with the temperate or polar regions of Earth, where the Sun can never be directly overhead. This is because of Earth's axial tilt; the width of the tropics (in latitude) is twice the tilt. The tropics are also referred to as the tropical zone and the torrid zone (see geographical zone). Due to the overhead sun, the tropics receive the most solar energy over the course of the year, and consequently have the highest temperatures on the planet. Even when not directly overhead, the sun is still close to overhead throughout the year, therefore the tropics also have the lowest seasonal variation on the planet; "winter" and "summer" lose their temperature contrast. Instead, seasons are more commonly divided by precipitation variations than by temperature variations. The tropics maintain wide diversity of local climates, such as rain forests, monsoons, sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. Definition The Laurent series for a complex function f(z) about an arbitrary point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where the coefficients a_n are defined by a contour integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic ( analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Valuation
In number theory, the valuation or -adic order of an integer is the exponent of the highest power of the prime number that divides . It is denoted \nu_p(n). Equivalently, \nu_p(n) is the exponent to which p appears in the prime factorization of n. The P-adic number, -adic valuation is a Valuation (algebra), valuation and gives rise to an analogue of the usual absolute value. Whereas the Complete metric space, completion of the rational numbers with respect to the usual absolute value results in the real numbers \mathbb, the completion of the rational numbers with respect to the p-adic absolute value results in the p-adic number, numbers \mathbb_p. Definition and properties Let be a prime number. Integers The -adic valuation of an integer n is defined to be : \nu_p(n)= \begin \mathrm\ & \text n \neq 0\\ \infty & \text n=0, \end where \mathbb_0 denotes the set of natural numbers (including zero) and m \mid n denotes divisor, divisibility of n by m. In particular, \nu_p is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valued Field
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : that satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (algebra)
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of the size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : that satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as group (mathematics), groups and ring (mathematics), rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an Additive identity, (often denoted as 0) and an identity with respect to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggestin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lecture Notes In Computer Science
''Lecture Notes in Computer Science'' is a series of computer science books published by Springer Science+Business Media since 1973. Overview The series contains proceedings, post-proceedings, monographs, and Festschrifts. In addition, tutorials, state-of-the-art surveys, and "hot topics" are increasingly being included. The series is indexed by DBLP. See also *'' Monographiae Biologicae'', another monograph series published by Springer Science+Business Media *'' Lecture Notes in Physics'' *'' Lecture Notes in Mathematics'' *'' Electronic Workshops in Computing'', published by the British Computer Society image:Maurice Vincent Wilkes 1980 (3).jpg, Sir Maurice Wilkes served as the first President of BCS in 1957. The British Computer Society (BCS), branded BCS, The Chartered Institute for IT, since 2009, is a professional body and a learned ... References External links * Academic journals established in 1973 Computer science books Series of non-fiction books ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominique Perrin
Dominique Pierre Perrin (b. 1946) is a French mathematician and theoretical computer scientist known for his contributions to coding theory and to combinatorics on words. He is a professor of the University of Marne-la-Vallée and currently serves as the President of ESIEE Paris. Biography Perrin earned his PhD from Paris 7 University in 1975. In his early career, he was a CNRS researcher (1970–1977) and taught at the University of Chile (1972–1973). Later, he worked as a professor at the University of Rouen (1977–1983), Paris 7 University (1983–1993), and École Polytechnique (1982–2002). Since 1993, Perrin is a professor at the University of Marne-la-Vallée, and since 2004, he is the President of ESIEE Paris. Perrin is a member of Academia Europaea since 1989. Scientific contributions Perrin has been a member of the Lothaire group of mathematicians that developed the foundations of combinatorics on words. He has co-authored three scientific monographs: "Theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Éric Pin
Jean-Éric Pin is a French mathematician and theoretical computer scientist known for his contributions to the algebraic automata theory and semigroup theory. He is a CNRS research director. Biography Pin earned his undergraduate degree from ENS Cachan in 1976 and his doctorate (Doctorat d'état) from the Pierre and Marie Curie University in 1981. Since 1988 he has been a CNRS research director at Paris Diderot University. In the years 1992–2006 he was a professor at École Polytechnique. Pin is a member of the Academia Europaea (2011) and an EATCS fellow (2014). In 2018, Pin became the first recipient of the Salomaa Prize in Automata Theory, Formal Languages, and Related Topics. Notable Work Pin is the author of the prominent textbook ''Varieties of Formal Languages'' on automata theory and formal language theory In logic, mathematics, computer science, and linguistics, a formal language is a set of string (computer science), strings whose symbols are taken from a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
