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The Barnes-Wall lattice is a d-dimensional lattice that exists when d is a power of 2. It is implemented in the Wolfram Language as LatticeData[{"BarnesWall", n}]. Special ...

The Cayley-Menger determinant is a determinant that gives the volume of a simplex in j dimensions. If S is a j-simplex in R^n with vertices v_1,...,v_(j+1) and B=(beta_(ik)) ...

A n×n matrix A is an orthogonal matrix if AA^(T)=I, (1) where A^(T) is the transpose of A and I is the identity matrix. In particular, an orthogonal matrix is always ...

The integer sequence defined by the recurrence relation P(n)=P(n-2)+P(n-3) (1) with the initial conditions P(0)=P(1)=P(2)=1. This is the same recurrence relation as for the ...

A generalization of the factorial and double factorial, n! = n(n-1)(n-2)...2·1 (1) n!! = n(n-2)(n-4)... (2) n!!! = n(n-3)(n-6)..., (3) etc., where the products run through ...

A rational function P(x)/Q(x) can be rewritten using what is known as partial fraction decomposition. This procedure often allows integration to be performed on each term ...

Let sopfr(n) be the sum of prime factors (with repetition) of a number n. For example, 20=2^2·5, so sopfr(20)=2+2+5=9. Then sopfr(n) for n=1, 2, ... is given by 0, 2, 3, 4, ...

For positive numbers a and b with a!=b, (a+b)/2>(b-a)/(lnb-lna)>sqrt(ab).

The intersection of an ellipse centered at the origin and semiaxes of lengths a and b oriented along the Cartesian axes with a line passing through the origin and point ...

A fractal based on iterating the map F(x)=ax+(2(1-a)x^2)/(1+x^2) (1) according to x_(n+1) = by_n+F(x_n) (2) x_(y+1) = -x_n+F(x_(n+1)). (3) The plots above show 10^4 ...