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Following the work of Fuchs in classifying first-order ordinary differential equations, Painlevé studied second-order ordinary differential equation of the form ...

The parabolic cylinder differential equation is the second-order ordinary differential equation y^('')+(nu+1/2-1/4z^2)y=0 (1) whose solution is given by ...

A geodesic on a paraboloid x = sqrt(u)cosv (1) y = sqrt(u)sinv (2) z = u (3) has differential parameters defined by P = ...

A patch (also called a local surface) is a differentiable mapping x:U->R^n, where U is an open subset of R^2. More generally, if A is any subset of R^2, then a map x:A->R^n ...

The Pell polynomials P(x) are the W-polynomials generated by the Lucas polynomial sequence using the generator p(x)=2x, q(x)=1. This gives recursive equations for P(x) from ...

The term perimeter refers either to the curve constituting the boundary of a lamina or else to the length of this boundary. The perimeter of a circle is called the ...

A function f(x) is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period p if f(x)=f(x+np) for ...

The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the ...

An operator which describes the time evolution of densities in phase space. The operator can be defined by rho_(n+1)=L^~rho_n, where rho_n are the natural invariants after ...

Erdős and Rényi (1960) showed that for many monotone-increasing properties of random graphs, graphs of a size slightly less than a certain threshold are very unlikely to have ...