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.
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A-Integrable -- from MathWorld - Weisstein, Eric W.
A generalization of the Lebesgue integral. A measurable function f(x) is called A-integrable over the closed interval [a, b] if m\{x: \left\vert{f(x)}\right\vert>n\}=O(n^{-1}), where m is the Lebesgue measure, and I=\lim_{n\to\infty} \int_a^b [f(x)]_n\,dx exists, where [f(x)]_n=\cases{ f(x) & if \left\vert{f(x)}\right\vert\leq n\cr 0 & if \left\vert{f(x)}\right\vert>n.\cr}
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AND -- from MathWorld - Weisstein, Eric W.
A connective in logic which yields true if all conditions are true, and false if any condition is false. A AND B is denoted A\land B (Mendelson 1997, p. 12), A\&B, A\cap B (Simpson 1987, p. 538), A\cdot B, A.B (Carnap 1958, p. 7), or simply AB (Simpson 1987, p. 538). The way to distinguish the similar symbols \land (AND) and \lor (OR) is to note that the symbol for AND is oriented in the same direction as the capital letter `A." The AND operation is implemented in Mathematica as And[A,...
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ANOVA -- from MathWorld - Weisstein, Eric W.
"Analysis of Variance." A statistical test for heterogeneity of means by analysis of group variances. ANOVA is implemented as ANOVA[data] in the Mathematica 4.2 standard add-on package Statistics`ANOVA` (which can be loaded using the command <
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Abel Prize -- from MathWorld - Weisstein, Eric W.
The Abel prize is a new mathematics prize of the Norwegian Academy of Science and Letters, dedicated to the memory of Niels Henrik Abel (1802-1829) on the occasion of the bicentenary of his birth. It is modeled after the Nobel Prize, and developed from a proposal by the mathematics department at the University of Oslo in fulfillment of a request formulated by the Norwegian mathematician Sophus Lie towards the end of the 19th century. The Abel Prize will be awarded annually beginning in the year...
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Abel Transform -- from MathWorld - Weisstein, Eric W.
The following integral transform relationship, known as the Abel transform, exists between two functions f(x) and g(t) for 0<\alpha<1, f(x) = \int_0^x {g(t)\,dt\over(x-t)^\alpha} g(t) = -{\sin(\pi\alpha)\over\pi} {d\over dt}\int_0^t {f(x)\,dx\over(x-t)^{1-\alpha}} = -{\sin(\pi\alpha)\over\pi}\left[{\int_0^t{df\over dx} {dx\over(t-x)^{1-\alpha}} + {f(0)\over t^{1-\alpha}}}\right]. The Abel transform is used in calculating the radial mass distribution of galaxies (Binney and Tremaine 1987)...
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Abelian Category -- from MathWorld - Weisstein, Eric W.
An Abelian category is a category for which the constructions and techniques of homological algebra are available. The basic examples of such categories are the category of Abelian groups and, more generally, the category of modules over a ring. Abelian categories are widely used in algebra, algebraic geometry, and topology. Many of the same constructions that are found in categories of modules, such as kernels, exact sequences, and commutative diagrams are available in Abelian categories. A...
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Abelian Differential -- from MathWorld - Weisstein, Eric W.
An Abelian differential is an analytic or meromorphic differential on a compact or closed Riemann surface.
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Abelian Function -- from MathWorld - Weisstein, Eric W.
An inverse function of an Abelian integral. Abelian functions have two variables and four periods, and can be defined by \Theta\left({v,\tau; \matrix{q'\cr q\cr}}\right)= \sum_{\lam... ...pi iv(\lambda+q')+\pi i\tau(\lambda+q')^2+2\pi iq(\lambda+q')} Baker (1907, p. 21). Abelian functions are a generalization of elliptic functions, and are also called hyperelliptic functions. Any Abelian function can be expressed as a ratio of homogenous polynomials of the Riemann theta function (Igusa 1972,...
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Abelian Integral -- from MathWorld - Weisstein, Eric W.
An integral of the form \int_0^x {dt\over{\sqrt{R(t)}}}, where R(t) is a polynomial of degree >4. They are also called hyperelliptic integrals. See also: Abelian Function, Elliptic Integral
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Abelian Theorem -- from MathWorld - Weisstein, Eric W.
A theorem which asserts that if a sequence or function behaves regularly, then some average of it behaves regularly. For example, A(x)\sim x implies A_1(x)=\int_0^x A(t)\,dt\sim {{1\over 2}}x^2 for any A(x). The converse is false, but can be made into a correct Tauberian theorem if A(x) is subjected to an appropriate additional condition (Hardy 1999, p. 46). See also: Tauberian Theorem
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Abel's Convergence Theorem -- from MathWorld - Weisstein, Eric W.
Given a Taylor series f(z) = \sum_{n=0}^\infty C_n z^n = \sum_{n=0}^\infty C_n r^n e^{in \theta}, where the complex number z has been written in the polar form z = re^{i\theta}, examine the real and imaginary parts u(r,\theta) = \sum_{n=0}^\infty C_n r^n \cos(n\theta) v(r,\theta) = \sum_{n=0}^\infty C_n r^n \sin(n\theta). Abel's theorem states that, if u(1,\theta) and v(1,\theta) are convergent, then u(1,\theta)+iv(1,\theta) = \lim_{r\to 1} f(re^{i\theta}). Stated in words, Abel's theorem...
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Abel's Differential Equation -- from MathWorld - Weisstein, Eric W.
The Abel equation of the first kind is given by y'=f_0(x)+f_1(x)y+f_2(x)y^2+f_3(x)y^3+\dots (Murphy 1960, p. 23; Zwillinger 1997, p. 120), and the Abel equation of the second kind by [g_0(x)+g_1(x)y]y'=f_0(x)+f_1(x)y+f_2(x)y^2+f_3(x)y^3 (Murphy 1960, p. 25; Zwillinger 1997, p. 120).
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Abel's Differential Equation Identity -- from MathWorld - Weisstein, Eric W.
Given a homogeneous linear second-order ordinary differential equation, y"+P(x)y'+Q(x)y=0, call the two linearly independent solutions y_1(x) and y_2(x). Then y"_1+P(x)y'_1+Q(x)y_1=0 y"_2+P(x)y'_2+Q(x)y_2=0. Now, take y_1\times (3) minus y_2\times (2), y_1[y"_2+P(x)y'_2+Q(x)y_2]-y_2[y"_1+P(x)y'_1+Q(x)y_1]=0 (y_1y"_2-y_2y"_1)+P(y_1y'_2-y'_1y_2)+Q(y_1y_2-y_1y_2)=0 (y_1y"_2-y_2y"_1)+P(y_1y'_2-y'_1y_2)=0. Now, use the definition of the Wronskian...
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Abel's Duplication Formula -- from MathWorld - Weisstein, Eric W.
The duplication formula for Rogers L-function follows from Abel's functional equation and is given by {{1\over 2}}L(x^2)=L(x)-L\left({x\over 1+x}\right). See also: Abel's Functional Equation, Dilogarithm
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Abel's Functional Equation -- from MathWorld - Weisstein, Eric W.
Let L(x) denote the Rogers L-function defined in terms of the usual dilogarithm by L(x) = {6\over\pi^2}[\mathop{\rm Li}\nolimits _2(x)+{{1\over 2}}\ln x\ln(1-x)] = {6\over\pi^2}\left[{\sum_{n=1}^\infty {x^n\over n^2}+{{1\over 2}}\ln x\ln(1-x)}\right], then L(x) satisfies the functional equation L(x)+L(y)=L(xy)+L\left({x(1-y)\over 1-xy}\right)+L\left({y(1-x)\over 1-xy}\right). Abel's duplication formula follows from this identity. See also: Abel's Duplication Formula, Dilogarithm,...
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Abel's Inequality -- from MathWorld - Weisstein, Eric W.
Let \{f_n\} and \{a_n\} be sequences with f_n \geq f_{n+1} > 0 for n = 1, 2, ..., then \left\vert{\sum_{n=1}^m a_n f_n}\right\vert \leq A f_1, where A = \max\{\left\vert{a_1}\right\vert,\left\vert{a_1+a_2}\rig... ...,\dots,\left\vert{a_1+a_2+\dots+a_m}\right\vert\}.
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