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If n=1,2 (mod 4), and the squarefree part of n is divisible by a prime p=3 (mod 4), then no difference set of order n exists. Equivalently, if a projective plane of order n ...

A catastrophe which can occur for four control factors and one behavior axis. The butterfly catastrophe is the universal unfolding of the singularity f(x)=x^6 of codimension ...

The surface of revolution given by the parametric equations x(u,v) = cosusin(2v) (1) y(u,v) = sinusin(2v) (2) z(u,v) = sinv (3) for u in [0,2pi) and v in [-pi/2,pi/2]. It is ...

Define I_n=(-1)^nint_0^infty(lnz)^ne^(-z)dz, (1) then I_n=(-1)^nGamma^((n))(1), (2) where Gamma^((n))(z) is the nth derivative of the gamma function. Particular values ...

The exponential factorial is defined by the recurrence relation a_n=n^(a_(n-1)), (1) where a_0=1. The first few terms are therefore a_1 = 1 (2) a_2 = 2^1=2 (3) a_3 = ...

Define F(1)=1 and S(1)=2 and write F(n)=F(n-1)+S(n-1), where the sequence {S(n)} consists of those integers not already contained in {F(n)}. For example, F(2)=F(1)+S(1)=3, so ...

The kiss surface is the quintic surface of revolution given by the equation x^2+y^2=(1-z)z^4 (1) that is closely related to the ding-dong surface. It is so named because the ...

Given a parabola with parametric equations x = at^2 (1) y = 2at, (2) the negative pedal curve for a pedal point (x_0,0) has equation x_n = (at^2[a(3t^2+4)-x_0])/(at^2+x_0) ...

A triangle with rows containing the numbers {1,2,...,n} that begins with 1, ends with n, and such that the sum of each two consecutive entries being a prime. Rows 2 to 6 are ...

A figurate number corresponding to a configuration of points which form a pyramid with r-sided regular polygon bases can be thought of as a generalized pyramidal number, and ...