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While the Catalan numbers are the number of p-good paths from (n,n) to (0,0) which do not cross the diagonal line, the super Catalan numbers count the number of lattice paths ...

A transpose of a doubly indexed object is the object obtained by replacing all elements a_(ij) with a_(ji). For a second-tensor rank tensor a_(ij), the tensor transpose is ...

The point on a line segment dividing it into two segments of equal length. The midpoint of a line segment is easy to locate by first constructing a lens using circular arcs, ...

Two cones placed base-to-base. The bicone with base radius r and half-height h has surface area and volume S = 2pirsqrt(r^2+h^2) (1) V = 2/3pir^2h. (2) The centroid is at the ...

A pair of positive integers (a_1,a_2) such that the equations a_1+a_2x=sigma(a_1)=sigma(a_2)(x+1) (1) have a positive integer solution x, where sigma(n) is the divisor ...

Let there be N_i observations of the ith phenomenon, where i=1, ..., p and N = sumN_i (1) y^__i = 1/(N_i)sum_(alpha)y_(ialpha) (2) y^_ = 1/Nsum_(i)sum_(alpha)y_(ialpha). (3) ...

A set function mu is said to possess countable subadditivity if, given any countable disjoint collection of sets {E_k}_(k=1)^n on which mu is defined, mu( union ...

If x_1/n_1 and x_2/n_2 are the observed proportions from standard normally distributed samples with proportion of success theta, then the probability that ...

A normal distribution with mean 0, P(x)=h/(sqrt(pi))e^(-h^2x^2). (1) The characteristic function is phi(t)=e^(-t^2/(4h^2)). (2) The mean, variance, skewness, and kurtosis ...

A factorization algorithm which works by expressing N as a quadratic form in two different ways. Then N=a^2+b^2=c^2+d^2, (1) so a^2-c^2=d^2-b^2 (2) (a-c)(a+c)=(d-b)(d+b). (3) ...