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d_n=p_(n+1)-p_n. (1) The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEIS A001223). Rankin has shown that d_n>(clnnlnlnnlnlnlnlnn)/((lnlnlnn)^2) ...

Consider the sum (1) where the x_js are nonnegative and the denominators are positive. Shapiro (1954) asked if f_n(x_1,x_2,...,x_n)>=1/2n (2) for all n. It turns out ...

A requirement necessary for a given statement or theorem to hold. Also called a criterion.

If f(x,y) is an analytic function in a neighborhood of the point (x_0,y_0) (i.e., it can be expanded in a series of nonnegative integer powers of (x-x_0) and (y-y_0)), find a ...

For any system that seeks to minimize a function, only seven different local forms of catastrophe "typically" occur for four or fewer variables: 1. Fold catastrophe, 2. Cusp ...

The following conditions are equivalent for a conservative vector field on a particular domain D: 1. For any oriented simple closed curve C, the line integral ∮_CF·ds=0. 2. ...

The one-dimensional wave equation is given by (partial^2psi)/(partialx^2)=1/(v^2)(partial^2psi)/(partialt^2). (1) In order to specify a wave, the equation is subject to ...

A partial differential equation of second-order, i.e., one of the form Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, (1) is called parabolic if the matrix Z=[A B; B C] (2) ...

A polynomial equation whose roots all have negative real parts. For a real quadratic equation z^2+Bz+C=0, the stability conditions are B,C>0. For a real cubic equation ...

To find the motion of a rectangular membrane with sides of length L_x and L_y (in the absence of gravity), use the two-dimensional wave equation ...