Wolfram Research Mathworld, Repository hosted at UIUC
(13.662 recursos)
Wolfram Research's MathWorld is a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers. It has been assembled by internet encyclopedist Eric W.ÊWeisstein with assistance from the mathematics and internet communities. Like the discipline of mathematics, this site is continuously updated to include new material and incorporate new discoveries. This is a free service for the mathematical community provided by Wolfram Research, makers of Mathematica, with additional support from the National Science Foundation.
Mostrando recursos 1 - 20 de 549
1.
Adams' Method -- from MathWorld - Weisstein, Eric W.
Adams' method is a numerical method for solving linear first-order ordinary differential equations of the form {dy\over dx}=f(x,y). Let h=x_{n+1}-x_n be the step interval, and consider the Maclaurin series of y about x_n, y_{n+1}=y_n+\left({dy\over dx}\right)_n(x-x_n)+{1\over 2}\left({d^2y\over dx^2}\right)_n(x-x_n)^2+\dots \left({dy\over dx}\right)_{n+1} = \left({dy\over dx}\right)_n+\left({d^2y\over dx^2}\right)_n(x-x_n)^2+\dots. Here, the derivatives of y are given by the backward...
2.
Aitken Interpolation -- from MathWorld - Weisstein, Eric W.
An algorithm similar to Neville's algorithm for constructing the Lagrange interpolating polynomial. Let f(x\vert x_0, x_1, \dots, x_k) be the unique polynomial of kth polynomial order coinciding with f(x) at x_0, ..., x_k. Then \begin{eqnarray*} f(x\vert x_0, x_1) &=& {1\over x_1-x_0}\left\vert\begin{array... ...2-x\\ f(x\vert x_0, x_1, x_3) & \!\!x_3-x\end{array}\right\vert. \end{eqnarray*} See also: Lagrange Interpolating Polynomial
3.
Allometric -- from MathWorld - Weisstein, Eric W.
Mathematical growth in which one population grows at a rate proportional to the power of another population.
4.
Alpha-Test -- from MathWorld - Weisstein, Eric W.
For some constant \alpha_0, \alpha(f,z)<\alpha_0 implies that z is an approximate zero of f, where \alpha(f,z) = {\left\vert{f(z)}\right\vert\over \left\vert{f... ...ert{f^{(k)}(z)\over k!f'(z)}\right\vert^{1/(k-1)}. Smale (1986) found a constant \alpha\approx 0.130707 for the test, and this value was subsequently improved to \alpha_0=3-2\sqrt{2}\approx 0.171573 by Wang and Han (1989), and further improved by Wang and Zhao (1995; Petkovic et al. 1997, p. 2). See also: Approximate Zero,...
5.
Alpha -- from MathWorld - Weisstein, Eric W.
Alpha is the name for the first letter in the Greek alphabet: \alpha. In finance, alpha is a financial measure giving the difference between a fund's actual return and its expected level of performance, given its level of risk (as measured by beta). A positive alpha indicates that a fund has performed better than expected based on its beta, whereas a negative alpha indicates poorer performance. See also: Alpha Function, Alpha-Test, Alpha Value, Beta, Sharpe Ratio
7.
Angel Problem -- from MathWorld - Weisstein, Eric W.
In a game proposed by J. H. Conway, a devil chases an angel on an infinite chessboard. At each move, the devil can eliminate one of the squares, and the angel can make a leap in any direction, covering a distance of at most n squares. Here, n is a positive integer previously fixed, and is called the "power" of the angel. The devil's aim is to trap the angel on an island surrounded by a hole of width at least n. Can the angel indefinitely escape the devil, if his power is sufficiently...
8.
Anonymous -- from MathWorld - Weisstein, Eric W.
A term in social choice theory meaning invariance of a result under permutation of voters. See also: Dual Voting, Monotonic Voting
10.
Apollonian Gasket -- from MathWorld - Weisstein, Eric W.
Consider three mutually tangent circles, and draw their inner Soddy circles. Then draw the inner Soddy circles of this circle with each pair of the original three, and continue iteratively. The steps in the process are illustrated above (Trott 2002). An animation illustrating the construction of the gasket is shown above. The points which are never inside a circle form a set of measure 0 having fractal dimension approximately 1.3058 (Mandelbrot 1983, p. 172). The Apollonian gasket corresponds...
11.
Approximate Zero -- from MathWorld - Weisstein, Eric W.
An initial point that provides safe convergence of Newton's method (Smale 1981; Petkovic et al. 1997, p. 1). See also: Alpha-Test, Newton's Method, Point Estimation Theory
12.
Approximation Theory -- from MathWorld - Weisstein, Eric W.
The mathematical study of how given quantities can be approximated by other (usually simpler) ones under appropriate conditions. Approximation theory also studies the size and properties of the error introduced by approximation. Approximations are often obtained by power series expansions in which the higher order terms are dropped. See also: Lagrange Remainder
13.
Arnold Tongue -- from MathWorld - Weisstein, Eric W.
Consider the circle map. If K is nonzero, then the motion is periodic in some finite region surrounding each rational \Omega. This execution of periodic motion in response to an irrational forcing is known as mode locking. If a plot is made of K versus \Omega with the regions of periodic mode-locked parameter space plotted around rational \Omega values (the map winding numbers), then the regions are seen to widen upward from 0 at K = 0 to some finite width at K = 1. The region surrounding each...
14.
Arrow's Paradox -- from MathWorld - Weisstein, Eric W.
Perfect democratic voting is, not just in practice but in principle, impossible. See also: Social Choice Theory, Voting
15.
Asymptotic Equipartition Property -- from MathWorld - Weisstein, Eric W.
A theorem from information theory that is a simple consequence of the weak law of large numbers. It states that if a set of values X_1, X_2, ..., X_n is drawn independently from a random variable X distributed according to P(x), then the joint probability P(X_1,\dots,X_n) satisfies -{1\over n} \log_2 P(X_1,X_2,\dots,X_n)\rightarrow H(X), where H(X) is the entropy of the random variable X. See also: Entropy
16.
Attractor -- from MathWorld - Weisstein, Eric W.
An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction. This restriction is necessary since a dynamical system may have multiple attractors, each with its own basin of attraction. Conservative...
17.
Auction -- from MathWorld - Weisstein, Eric W.
A type of sale in which members of a group of buyers offer ever increasing amounts. The bidder making the last bid (for which no higher bid is subsequently made within a specified time limit: "going once, going twice, sold") must then purchase the item in question at this price. Variants of simple bidding are also possible, as in a Vickrey auction. See also: Vickrey Auction
18.
Average Power -- from MathWorld - Weisstein, Eric W.
The average power of a complex signal f(t) as a function of time t is defined as \left\langle{f^2(t)}\right\rangle{} = \lim_{T\to\infty} {1\o... ...2T} \int_{-T}^T \left\vert{f(t)}\right\vert^2\,dt, where \left\vert{z}\right\vert is the complex modulus (Papoulis 1962, p. 240). See also: Autocorrelation
19.
B-Spline -- from MathWorld - Weisstein, Eric W.
A B-spline is a generalization of the Bézier curve. Let a vector known as the knot vector be defined \mathbf{T}=\{t_0, t_1, \dots, t_m\}, where T is a nondecreasing sequence with t_i\in[0,1], and define control points \mathbf{P}_{0}, ..., \mathbf{P}_{n}. Define the degree as p\equiv m-n-1. The "knots" t_{p+1}, ..., t_{m-p-1} are called internal knots. Define the basis functions as N_{i,0}(t) = \left\{\begin{array}{ll} 1 & \mbox{if t_i\leq t
20.
Backward Difference -- from MathWorld - Weisstein, Eric W.
The backward difference is a finite difference defined by \nabla_p \equiv \nabla f_p\equiv f_p-f_{p-1}. Higher order differences are obtained by repeated operations of the backward difference operator, so \nabla^2_p = \nabla(\nabla p)=\nabla(f_p-f_{p-1})=\nabla f_p-\nabla f_{p-1} = (f_p-f_{p-1})-(f_{p-1}-f_{p-2}) = f_p-2f_{p-1}+f_{p-2} In general, \nabla^k_p \equiv \nabla^k f_p \equiv \sum_{m=0}^k (-1)^m {k\choose m} f_{p-m}, where {k\choose m} is a binomial coefficient. Newton's backward...
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