Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples.| mathworld.wolfram.com
The m×n knight graph is a graph on mn vertices in which each vertex represents a square in an m×n chessboard, and each edge corresponds to a legal move by a knight (which may only make moves which simultaneously shift one square along one axis and two along the other). It is therefore a (1,2)-leaper graph, as well as the Euclidean distance-sqrt(5) graph. n×n knight graphs abstracted from the chessboard are illustrated above for n=3, ..., 6. The 1×1 knight graph is the...| mathworld.wolfram.com
An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum. Due to the lack of specificity in the above definition, different branches...| mathworld.wolfram.com
The triangular number T_n is a figurate number that can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one. This is illustrated above for T_1=1, T_2=3, .... The triangular numbers are therefore 1, 1+2, 1+2+3, 1+2+3+4, ..., so for n=1, 2, ..., the first few are 1, 3, 6, 10, 15, 21, ... (OEIS A000217). More formally, a triangular number is a number obtained by adding...| mathworld.wolfram.com
The Heaviside step function is a mathematical function denoted H(x), or sometimes theta(x) or u(x) (Abramowitz and Stegun 1972, p. 1020), and also known as the "unit step function." The term "Heaviside step function" and its symbol can represent either a piecewise constant function or a generalized function. When defined as a piecewise constant function, the Heaviside step function is given by H(x)={0 x<0; 1/2 x=0; 1 x>0 (1) (Abramowitz and Stegun 1972,...| mathworld.wolfram.com
A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function f(x) is a point x_0 such that f(x_0)=x_0. (1) The fixed point of a function f starting from an initial value x can be computed in the Wolfram Language using FixedPoint[f, x]. Similarly, to get a list of the values obtained by iterating the function until a fixed point is reached, the command FixedPointList[f, x] can be used. The...| mathworld.wolfram.com
An nth-rank tensor in m-dimensional space is a mathematical object that has n indices and m^n components and obeys certain transformation rules. Each index of a tensor ranges over the number of dimensions of space. However, the dimension of the space is largely irrelevant in most tensor equations (with the notable exception of the contracted Kronecker delta). Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly...| mathworld.wolfram.com
The natural logarithm lnx is the logarithm having base e, where e=2.718281828.... (1) This function can be defined lnx=int_1^x(dt)/t (2) for x>0. This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola y=1/x, the x-axis, and the vertical lines x=1 and x=e is 1. In other words, int_1^e(dx)/x=lne=1. (3) The notation lnx is used in physics and engineering to denote the natural logarithm, while mathematicians...| mathworld.wolfram.com