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 1.420
1.
15 Puzzle -- from MathWorld - Weisstein, Eric W.
The "15 puzzle" is a sliding square puzzle introduced by Sam Loyd in 1878. It consists of 15 squares numbered from 1 to 15 which are placed in a 4\times 4 box leaving one position out of the 16 empty. The goal is to reposition the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements, this rearrangement is possible, but for others, it is not. To address the solubility of a given initial...
2.
2x mod 1 Map -- from MathWorld - Weisstein, Eric W.
Let x_0 be a rational number in the closed interval [0,1], and generate a sequence using the map x_{n+1}\equiv 2x_n\ ({\rm mod}\ 1). Then the number of periodic map orbits of period p (for p prime) is given by N_p={2^p-2\over p} (i.e, the number of period-p repeating bit strings, modulo shifts). Since a typical map orbit visits each point with equal probability, the natural invariant is given by \rho(x)=1. See also: Logistic Map, Tent Map
3.
36 Officer Problem -- from MathWorld - Weisstein, Eric W.
How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6\times 6 array such that no row or column duplicates a rank or a regiment? The answer is that no such arrangement is possible. See also: Euler's Graeco-Roman Squares Conjecture, Latin Square, Trigonometry Angles Pi/3, Trigonometry Angles Pi/6
5.
Abel Polynomial -- from MathWorld - Weisstein, Eric W.
A polynomial A_n(x;a) given by the associated Sheffer sequence with f(t)=te^{at}, given by A_n(x;a)=x(x-an)^{n-1}. The generating function is \sum_{k=0}^\infty {A_k(x;a)\over k!}t^k=e^{x W(at)/a}, where W(x) is the Lambert W-function. The associated binomial identity is (x+y)(x+y-an)^{n-1}=\sum_{k=0}^n{n\choose k}xy(x-ak)^{k-1}[y-a(n-k)]^{n-k-1}, where {n\choose k} is a binomial coefficient, a formula originally due to Abel (Riordan 1979, p. 18; Roman 1984, pp. 30 and 73). The first few...
6.
Abel's Binomial Theorem -- from MathWorld - Weisstein, Eric W.
The identity \sum_{y=0}^m {m\choose y}(w-y)^{m-y-1}(z+y)^y=w^{-1}(z+w+m)^m (Bhatnagar 1995, p. 51). There are a host of other such binomial identities. See also: Binomial Identity, q-Abel's Theorem
7.
Actuarial Polynomial -- from MathWorld - Weisstein, Eric W.
The polynomials a_n^{(\beta)}(x) given by the Sheffer sequence with g(t) = (1-t)^{-\beta} f(t) = \ln(1-t), giving generating function \sum_{k=0}^\infty {a_n^{(\beta)}\over k!}t^k=e^{x(1-e^t)+\beta t}. The Sheffer identity is a_n^{(\beta)}(x+y)=\sum_{k=0}^n{n\choose k}a_k^{(\beta)}(y)\phi_{n-k}(-x), where \phi_n(x) is an exponential polynomial. The actuarial polynomials are given in terms of the exponential polynomials \phi_n(x) by a_n^{(\beta)}(x) = (1-t)^\beta\phi_n(-x) = \sum_{k=0}^n...
8.
Acyclic Digraph -- from MathWorld - Weisstein, Eric W.
An acyclic digraph is a directed graph containing no directed cycles, also known as a directed acyclic graph or a "DAG." Every acyclic digraph has at least one node of outdegree 0. The numbers of acyclic digraphs on n = 1, 2, ... vertices are 1, 2, 6, 31, 302, 5984, ... (Sloane's A003087). The numbers of labeled acyclic digraphs on n = 1, 2, ... nodes are 1, 3, 25, 543, 29281, ... (Sloane's A003024). E. W. Weisstein (July 10, 2003) conjectured that positive eigenvalued (0,1)-matrices...
9.
Additive Cellular Automaton -- from MathWorld - Weisstein, Eric W.
An additive cellular automaton is a cellular automaton whose rule is compatible with an addition of states. Typically, this addition is derived from modular arithmetic. Additive rules allow the evolution for different initial conditions to be computed independently, then the results combined by simply adding. The results for arbitrary starting conditions can therefore be computed very efficiently by convolving the evolution of a single cell with an appropriate convolution kernel (which, in the...
10.
Adjacency List -- from MathWorld - Weisstein, Eric W.
The adjacency list representation of a graph consists of n lists one for each vertex v_i, 1\leq i\leq n, which gives the vertices to which v_i is adjacent. The adjacency lists of a graph g may be computed using ToAdjacencyLists[g] in the Mathematica add-on package DiscreteMath`Combinatorica` (which can be loaded with the command <
11.
Adjacency Relation -- from MathWorld - Weisstein, Eric W.
The set E of graph edges of a graph (V, E), being a set of unordered pairs of elements of V, constitutes a relation on V. Formally, an adjacency relation is any relation which is irreflexive and symmetric. See also: Irreflexive, Relation, Symmetric
12.
Adjacent Vertices -- from MathWorld - Weisstein, Eric W.
In a graph G, two graph vertices are adjacent if they are joined by an graph edge. See also: Graph, Graph Edge, Graph Vertex
13.
Admissible -- from MathWorld - Weisstein, Eric W.
A string or word is said to be admissible if that word appears in a given sequence. For example, in the sequence aabaabaabaabaab\dots, a, aa, baab are all admissible, but bb is inadmissible. See also: Block Growth
14.
Affine Plane -- from MathWorld - Weisstein, Eric W.
A two-dimensional affine geometry constructed over a finite field. For a field F of size n, the affine plane consists of the set of points which are ordered pairs of elements in F and a set of lines which are themselves a set of points. Adding a point at infinity and line at infinity allows a projective plane to be constructed from an affine plane. An affine plane of order n is a block design of the form (n^2, n, 1). An affine plane of order n exists iff a projective plane of order n exists. ...
15.
Algebraic Combinatorics -- from MathWorld - Weisstein, Eric W.
The use of techniques from algebra, topology, and geometry in the solution of combinatorial problems, or the use of combinatorial methods to attack problems in these areas (Billera et al. 1999, p. ix). See also: Combinatorics
16.
Algebraic Connectivity -- from MathWorld - Weisstein, Eric W.
The second smallest eigenvalue of the Laplacian matrix of a graph G. This eigenvalue is greater than 0 iff G is a connected graph. See also: Connected Graph, Fiedler Vector, Laplacian Matrix
17.
Algebraic Language -- from MathWorld - Weisstein, Eric W.
Let X be an alphabet (i.e., a finite and nonempty set), and call its member letters. A word on X is a finite sequence of letters a_1\dots a_n, where a_1,\dots,a_n\in X. Denote the empty word by e, and the set of all words in X by X^*. Define the concatenation (also called product) of a word u=a_1\dots a_n with a word v=b_1\dots b_m as uv=a_1\dots a_nb_1\dots b_m. In general, concatenation is not commutative. Use the notation \left\vert{u}\right\vert _a to mean the number of letters a in the...
18.
All-Pairs Shortest Path -- from MathWorld - Weisstein, Eric W.
The shortest distance between any pair of vertices in the shortest-path spanning tree, as long as the path giving the shortest path does not pass through the root of the spanning tree (Skiena 1990, p. 228). The problem can be solved using n applications of Dijkstra's algorithm or Floyd's algorithm. The latter also works in the case of a weighted graph where the edges have negative weights. See also: Floyd's Algorithm, Dijkstra's Algorithm, Graph Geodesic
20.
Alphabet -- from MathWorld - Weisstein, Eric W.
A set (usually of letters) from which a subset is drawn. A sequence of letters is called a word, and a set of words is called a code. See also: Code, String, Word
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