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Gröbner Fan
In symbolic computation, computer algebra, the Gröbner fan of an Ideal (ring theory), ideal in the Polynomial ring, ring of polynomials is a concept in the theory of Gröbner basis, Gröbner bases. It is defined to be a Polyhedral complex#Fans, fan consisting of cones that correspond to different monomial orders on that ideal. The concept was introduced by Teo Mora, Mora and Lorenzo Robbiano, Robbiano in 1988. The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison. Gröbner fan is a base for the nowadays active field of tropical geometry. One implementation of the Gröbner fan is called Gfan, based on an article of Fukuda, et al. which is included in some computer algebra systems such as SINGULAR computer algebra system, Singular, Macaulay 2, Macaulay2, and CoCoA. See also * Gröbner basis * Tropical geometry References Computer algebra Algebraic geometry Commutative algebra {{commutative-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Computation
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 + 2xy + 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 classica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the languag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 + 2xy + 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 classica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CoCoA
Cocoa may refer to: Chocolate * Chocolate * ''Theobroma cacao'', the cocoa tree * Cocoa bean, seed of ''Theobroma cacao'' * Chocolate liquor, or cocoa liquor, pure, liquid chocolate extracted from the cocoa bean, including both cocoa butter and cocoa solids * Cocoa butter, a pale yellow, edible fat extracted from the cocoa bean * Cocoa solids or cocoa powder * Hot chocolate, also called hot cocoa * Cocoa Mountain, a gourmet chocolate enterprise in Durness in the North West Highlands of Scotland * Cocoa Processing Company, a Ghanaian company Computing * Cocoa (API), an API and programming environment for macOS * Cocoa Touch, an API and programming environment for iOS, iPadOS, watchOS and tvOS * CoCoA, a computer algebra system * COCOA (digital humanities), an early FORTRAN program for generating concordances and word counts from natural language texts * Stagecast Creator, formerly Cocoa, a language developed by Apple to teach programming to children Given name * Cocoa Brown ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macaulay 2
Macaulay2 is a free computer algebra system created by Daniel Grayson (from the University of Illinois at Urbana–Champaign) and Michael Stillman (from Cornell University) for computation in commutative algebra and algebraic geometry. Overview Macaulay2 is built around fast implementations of algorithms useful for computation in commutative algebra and algebraic geometry. This core functionality includes arithmetic on rings, modules, and matrices, as well as algorithms for Gröbner bases, free resolutions, Hilbert series, determinants and Pfaffians, factoring, and similar. In addition, the system has been extended by a large number of packages. Nearly 200 packages are included in the distribution of Macaulay2 as of 2019, and notable package authors include Craig Huneke and Frank-Olaf Schreyer. The Journal of Software for Algebra and Geometry has published numerous packages and programs for Macaulay2. Macaulay2 has an interactive command-line interface used from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SINGULAR Computer Algebra System
Singular (typeset ) is a computer algebra system for polynomial computations with special emphasis on the needs of commutative and non-commutative algebra, algebraic geometry, and singularity theory. Singular has been released under the terms of GNU General Public License. Problems in non-commutative algebra can be tackled with the Singular offspring Plural. Singular is developed under the direction of Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann, who head Singular's core development team within the Department of Mathematics of the Technische Universität Kaiserslautern. In the DFG Priority Program 1489, interfaces to GAP, Polymake and Gfan are being developed in order to cover recently established areas of mathematics involving convex and algebraic geometry, such as toric and tropical geometry. See also *Comparison of computer algebra systems The following tables provide a comparison of computer algebra systems (CAS). A CAS is a package compri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of Computation
''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than five years are available electronically free of charge. Abstracting and indexing The journal is abstracted and indexed in Mathematical Reviews, Zentralblatt MATH, Science Citation Index, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.417. References External links * Delayed open access journals English-language journals Mathematics journals Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorenzo Robbiano
Lorenzo may refer to: People * Lorenzo (name) Places Peru * San Lorenzo Island (Peru), sometimes referred to as the island of Lorenzo United States * Lorenzo, Illinois * Lorenzo, Texas * San Lorenzo, California, formerly Lorenzo * Lorenzo State Historic Site, house in New York State listed on the National Register of Historic Places Art, entertainment, and media ;Films and television * ''Lorenzo'' (film), an animated short film * ''Lorenzo's Oil'', a film based on a true story about a boy suffering from Adrenoleukodystrophy and his parents' journey to find a treatment. * ''Lorenzo's Time'', a 2012 Philippine TV series that aired on ABS-CBN ;Music *Lorenzo (rapper), French rapper * "Lorenzo", a 1996 song by Phil Collins Other uses * List of storms named Lorenzo * Lorenzo patient record systems, a type of electronic health record in the United Kingdom See also * San Lorenzo (other) * De Lorenzo * di Lorenzo * Lorenzen (other) Lorenzen may refer to People * Lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teo Mora
Ferdinando 'Teo' Mora is an Italian mathematician, and since 1990 until 2019 a professor of algebra at the University of Genoa. Life and work Mora's degree is in mathematics from the University of Genoa in 1974. Mora's publications span forty years; his notable contributions in computer algebra are the tangent cone algorithm and its extension of Buchberger theory of Gröbner bases and related algorithm earlier to non-commutative polynomial rings and more recently to effective rings; less significantThe result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison. the notion of Gröbner fan; marginal, with respect to the other authors, his contribution to the FGLM algorithm. Mora is on the managing-editorial-board of the journal '' AAECC'' published by Springer, and was also formerly an editor of the ''Bulletin of the Iranian Mathematical Society''. He is the author of the tetralogy ''Solving Polynomial Equation Systems'': ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial Order
In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all ( monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * If u \leq v and w is any other monomial, then uw \leq vw. Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of ''being'' a Gröbner basis is always relative to a specific monomial order. Definition, details and variations Besides respecting multiplication, monomial orders are often required to be well-orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders. In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions: # The order is a total ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
