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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 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-t) f(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Mathematics
Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.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 ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. 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: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Polynomial Equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field . A ''solution'' of a polynomial system is a set of values for the s which belong to some algebraically closed field extension of , and make all equations true. When is the field of rational numbers, is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of , which is isomorphic to a subfield of the complex numbers. This article is about the methods for solving, that is, finding all solutions or describing them. As these methods are designed for being implemented in a computer, emphasis is given on fields in which computation (including equality testing) is easy and efficient, that is the field of rational numbers and finite fields. Searching for solutions that belong to a specific set is a problem which is generally much more difficult, and is outs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Continuation
Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, :F(\mathbf u,\lambda) = 0. The ''parameter'' \lambda is usually a real scalar, and the ''solution'' \mathbf u an ''n''-vector. For a fixed ''parameter value'' \lambda, F(\ast,\lambda) maps Euclidean n-space into itself. Often the original mapping F is from a Banach space into itself, and the Euclidean n-space is a finite-dimensional Banach space. A steady state, or fixed point, of a parameterized family of flows or maps are of this form, and by discretizing trajectories of a flow or iterating a map, periodic orbits and heteroclinic orbits can also be posed as a solution of F=0. Other forms In some nonlinear systems, parameters are explicit. In others they are implicit, and the system of nonlinear equations is written :F(\mathbf u) = 0 where \mathbf u is an ''n''-vector, and its image F(\mathbf u) is an ''n-1'' vector. This formulation, without an e ... [...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 *Standing start, and rolling start, in an auto race Acronyms *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", 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'' (newspaper), a daily tabloid published in Serbia * ''STart'' (magazine), an Atari ST publication *Start, by Susan Long (journalist) *''Start'', by Terry Virgo Places *Start, ... [...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 trapezoida ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. 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 , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...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] |
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 set of solutions of a system of polynomial equations over the 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, which means that it is not the union of two smaller sets that are 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 monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
