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Numerical Algebraic Geometry
Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations. Homotopy continuation The primary computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated solutions (points) of one are continued to the other. This is a specialization of the more general method of numerical continuation. Let z represent the variables of the system. By abuse of notation, and to facilitate the spectrum of ambient spaces over which one can solve the system, we do not use vector notation for z. Similarly for the polynomial systems f and g. Current canonical notation calls the start system g, and the target system, i.e., the system to solve, f. A very common homotopy, the straight-line homotopy, between f and g is H(z,t) = (1- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Mathematics
Computational mathematics is the study of the interaction between mathematics and calculations done by a computer.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006. Retrieved April 2007. A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants. Areas of computational mathematics Computational mathematics emerged as a distinct part of applied mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MiNuS
The plus sign () and the minus sign () are mathematical symbols used to denote positive and negative functions, respectively. In addition, the symbol represents the operation of addition, which results in a sum, while the symbol represents subtraction, resulting in a difference. Their use has been extended to many other meanings, more or less analogous. and are Latin terms meaning 'more' and 'less', respectively. The forms and are used in many countries around the world. Other designs include for plus and for minus. History Though the signs now seem as familiar as the alphabet or the Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembles a pair of legs walking in the direction in which the text was written ( Egyptian could be written either from right to left or left to right), with the reverse sign indicating subtraction: Nicole Oresme's manuscripts from the 14th century show what may be one of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macaulay2
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 the te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Certification
Numerical certification is the process of verifying the correctness of a candidate solution to a system of equations. In (numerical) computational mathematics, such as numerical algebraic geometry, candidate solutions are computed algorithmically, but there is the possibility that errors have corrupted the candidates. For instance, in addition to the inexactness of input data and candidate solutions, numerical errors or errors in the discretization of the problem may result in corrupted candidate solutions. The goal of numerical certification is to provide a certificate which proves which of these candidates are, indeed, approximate solutions. Methods for certification can be divided into two flavors: ''a priori'' certification and ''a posteriori'' certification. ''A posteriori'' certification confirms the correctness of the final answers (regardless of how they are generated), while ''a priori'' certification confirms the correctness of each step of a specific computation. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be Irreducible component, irreducible, which means that it is not the Union (set theory), union of two smaller Set (mathematics), sets that are Closed set, closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem Of Bertini
In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0. Statement for hyperplane sections of smooth varieties Let ''X'' be a smooth quasi-projective variety over an algebraically closed field, embedded in a projective space \mathbf P^n. Let , H, denote the complete system of hyperplane divisors in \mathbf P^n. Recall that it is the dual space (\mathbf P^n)^ of \mathbf P^n and is isomorphic to \mathbf P^n. The theorem of Bertini states that the set of hyperplanes not containing ''X'' and with smooth intersection with ''X'' contains an open dense subset of the total system of divisors , H, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular points, inflection points and points at infinity. More advanced questions involve the topology of the curve and the relationship ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predictor–corrector Method
In numerical analysis, predictor–corrector methods belong to a class of algorithms designed to integrate ordinary differential equationsto find an unknown function that satisfies a given differential equation. All such algorithms proceed in two steps: # The initial, "prediction" step, starts from a function fitted to the function-values and derivative-values at a preceding set of points to extrapolate ("anticipate") this function's value at a subsequent, new point. # The next, "corrector" step refines the initial approximation by using the ''predicted'' value of the function and ''another method'' to interpolate that unknown function's value at the same subsequent point. Predictor–corrector methods for solving ODEs When considering the numerical solution of ordinary differential equations (ODEs), a predictor–corrector method typically uses an explicit method for the predictor step and an implicit method for the corrector step. Example: Euler method with the trapezoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Start System
Start can refer to multiple topics: * Takeoff, the phase of flight where an aircraft transitions from moving along the ground to flying through the air * Starting lineup in sports * Starts use in sport race * Standing start, and rolling start, in an auto race Acronyms * National Consortium for the Study of Terrorism and Responses to Terrorism * Strategic Arms Reduction Treaties, a series of arms reduction treaties between the US and USSR ** START I (1991) ** START II (1993) ** START III (1997), never signed into effect ** New START (2010), initiated to continue the effects of previous START treaties ** "START" (''The Americans''), a 2018 episode and the series finale of the period spy thriller ''The Americans'' * Simple triage and rapid treatment * Small Tight Aspect Ratio Tokamak * Spanish Technical Aid Response Team * Stanislaus Regional Transit, predecessor to the Stanislaus Regional Transit Authority Books and publications * ''STart'' (magazine), an Atari ST computer mag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
