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 1.776
1.
(-1,0,1)-Matrix -- from MathWorld - Weisstein, Eric W.
A (-1,0,1) matrix is a matrix whose elements consist only of the numbers -1, 0, or 1. The number of distinct (-1,0,1)-n\times n matrices (counting row and column permutations, the transpose, and multiplication by -1 as equivalent) having 2n different row and column sums for n = 2, 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (Kleber). For example, the 2\times 2 matrix is given by \left[{\matrix{-1 & -1\cr \phantom{-}0 & \phantom{-}1\cr}}\right]. To get the total number from these counts...
2.
(-1,1)-Matrix -- from MathWorld - Weisstein, Eric W.
A (-1,1) matrix is a matrix whose elements consist only of the numbers -1 or 1. For an n\times n(-1,0,1)-matrix, the largest possible determinants (Hadamard's maximum determinant problem) for n = 1, 2, ... are 1, 2, 4, 16, 48, 160, ... (Sloane's A003433; Ehrlich and Zeller 1962, Ehrlich 1964). The numbers of distinct n\times n(-1,1)-matrices having the largest possible determinant are 1, 4, 96, 384, .... See also: Hadamard Matrix, Integer Matrix
3.
(0,1)-Matrix -- from MathWorld - Weisstein, Eric W.
A (0,1)-integer matrix, i.e., a matrix each of whose elements is 0 or 1, also called a binary matrix. The number of m\times n binary matrices is 2^{mn}, so the number of square n\times n binary matrices is 2^{n^2} which, for n = 1, 2, ..., gives 2, 16, 512, 65536, 33554432, ... (Sloane's A002416). The numbers of positive definite n\times n(0,1)-matrices for n = 1, 2, ... are 1, 3, 25, 543, 29281, ... (Sloane's A003024). Weisstein's conjecture proposed that positive eigenvalued (0,1)-matrices...
4.
Abel-Regularized Sum -- from MathWorld - Weisstein, Eric W.
See also: Hadamard-Regularized Sum, Hölder-Regularized Sum, Regularization, Regularized Sum, Zeta-Regularized Sum
5.
Abelian -- from MathWorld - Weisstein, Eric W.
A group or other algebraic object is said to be Abelian is the law of commutativity always holds. If an algebraic object is not Abelian, it is said to be non-Abelian. See also: Abelian Category, Abelian Differential, Abelian Function, Abelian Group, Abelian Integral, Abelian Variety, Commutative, Non-Abelian
6.
Abelian Extension -- from MathWorld - Weisstein, Eric W.
If F is an algebraic Galois extension of K such that the Galois group of the extension is Abelian, then F is said to be an Abelian extension of K. For example, \mathbb{Q}(\sqrt{2}\,) =\{a+b\sqrt{2}\,\} is the field of rational numbers with the square root of two adjoined, a extension field degree-two extension of \mathbb{Q}. Its Galois group has two elements, the nontrivial element sending \sqrt{2} to -\sqrt{2}, and is Abelian. By contrast, the degree-six extension F=\mathbb{Q}(2^{1/3},...
7.
Abelian Group -- from MathWorld - Weisstein, Eric W.
A group for which the elements commute (i.e., AB = BA for all elements A and B) is called an Abelian group. All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator. No general formula is known for giving the number of nonisomorphic finite groups of a given group order. ...
8.
Abelian Variety -- from MathWorld - Weisstein, Eric W.
An Abelian variety is an algebraic group which is a complete algebraic variety. An Abelian variety of dimension 1 is an elliptic curve. See also: Albanese Variety, Algebraic Variety, Variety
9.
Abelianization -- from MathWorld - Weisstein, Eric W.
In general, groups are not Abelian. However, there is always a group homomorphism h:G \rightarrow G' to an Abelian group, and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the commutator subgroup [G, G], which is the unique smallest normal subgroup of G such that the quotient group G'=G/[G,G] is Abelian. Roughly speaking, in any expression, every product becomes commutative after Abelianization. As a consequence, some previously unequal...
10.
Abel's Curve Theorem -- from MathWorld - Weisstein, Eric W.
The sum of the values of an integral of the "first" or "second" sort \int_{x_0,y_0}^{x_1,y_1} {P\,dx\over Q}+\dots+\int_{x_0,y_0}^{x_N,y_N} {P\,dx\over Q}=F(z) and {P(x_1,y_1)\over Q(x_1,y_1)} {dx_1\over dz}+\dots+{P(x_N,y_N)\over Q(x_N,y_N)}{dx_N\over dz}={dF\over dz}, from a fixed point to the points of intersection with a curve depending rationally upon any number of parameters is a rational function of those parameters.
11.
Abel's Impossibility Theorem -- from MathWorld - Weisstein, Eric W.
In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions. This was also shown by Ruffini in 1813 (Wells 1986, p. 59). See also: Cubic Equation, Galois's Theorem, Polynomial, Quadratic Equation, Quartic Equation, Quintic Equation
12.
Abel's Irreducibility Theorem -- from MathWorld - Weisstein, Eric W.
If one root of the equation f(x)=0, which is irreducible over a field K, is also a root of the equation F(x)=0 in K, then all the roots of the irreducible equation f(x)=0 are roots of F(x)=0. Equivalently, F(x) can be divided by f(x) without a remainder, F(x)=f(x)F_1(x), where F_1(x) is also a polynomial over K. See also: Abel's Lemma, Kronecker's Polynomial Theorem, Schönemann's Theorem
13.
Abel's Lemma -- from MathWorld - Weisstein, Eric W.
The pure equation x^p=C of prime degree p is irreducible over a field when C is a number of the field but not the pth power of an element of the field. Jeffreys and Jeffreys (1988) use the term "Abel's lemma" for another lemma related to Abel's uniform convergence test. See also: Abel's Irreducibility Theorem, Gauss's Polynomial Theorem, Kronecker's Polynomial Theorem, Schönemann's Theorem
14.
Abhyankar's Conjecture -- from MathWorld - Weisstein, Eric W.
For a finite group G, let p(G) be the subgroup generated by all the Sylow p-subgroups of G. If X is a projective curve in characteristic p > 0, and if x_0, ..., x_t are points of X (for t > 0), then a necessary and sufficient condition that G occur as the Galois group of a finite covering Y of X, branched only at the points x_0, ..., x_t, is that the quotient group G/p(G) has 2g+t generators. Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the...
15.
Absorption Law -- from MathWorld - Weisstein, Eric W.
The law appearing in the definition of Boolean algebras and lattice which states that a\land (a\lor b)=a\lor (a\land b)=a for binary operators \lor and \land (which most commonly are logical OR and logical AND). The two parts of the absorption law are sometimes called the "absorption identities" (Grätzer 1971, p. 5). See also: Boolean Algebra, Lattice
16.
Abstract Group -- from MathWorld - Weisstein, Eric W.
An abstract group is a group characterized only by its abstract properties and not by the particular representations chosen for elements. For example, there are two distinct abstract groups on four elements: the viergruppe C_2\times C_2 and the cyclic group C4. A number of particular examples of the abstract group C_4 are the point groups C_4 (unfortunately, the symbols for the point groups C_n are the same as those for the abstract cyclic groups C_n to which they are isomorphic) and S_4. See...
17.
Abstract Vector Space -- from MathWorld - Weisstein, Eric W.
An abstract vector space of dimension n over a field k is the set of all formal expressions a_1v_1+a_2v_2+\dots+a_nv_n, where \{v_1,v_2,\dots, v_n\} is a given set of n objects (called a basis) and (a_1,a_2,\dots, a_n) is any n-tuple of elements of k. Two such expressions can be added together by summing their coefficients, (a_1v_1+a_2v_2+\dots +a_nv_n)+(b_1v_1+b_2v_2+\cdots +b_nv_n)=(a_1+b_1)v_1+(a_2+b_2)v_2+\dots+ (a_n+b_n)v_n. (2) This addition is a commutative group operation, since...
18.
Acnode -- from MathWorld - Weisstein, Eric W.
Another name for an isolated point. See also: Crunode, Spinode, Tacnode
19.
Action -- from MathWorld - Weisstein, Eric W.
Let M(X) denote the group of all invertible maps X\to X and let G be any group. A homomorphism \theta:G\to M(X) is called an action of G on X. Therefore, \theta satisfies 1. For each g\in G, \theta(g) is a map X\to X:x\mapsto \theta(g)x, 2. \theta(gh)x=\theta(g)(\theta(h)x), 3. \theta(e)x=x, where e is the group identity in G, 4. \theta(g^{-1})x=\theta(g)^{-1}x. See also: Cascade, Flow, Semidirect Product, Semiflow
20.
Additive Group -- from MathWorld - Weisstein, Eric W.
A group where the operation is called addition and is denoted +. In an additive group, the identity element is called zero, and the inverse of the element a is denoted -a (minus a). The symbols and terminology are borrowed from the additive groups of numbers: the ring of integers \mathbb{Z}, the field of rational numbers \mathbb{Q}, the field of real numbers \mathbb{R}, and the field of complex numbers \mathbb{C} are all additive groups. In general, every ring and every field is an additive...
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