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The mutual information between two discrete random variables X and Y is defined to be I(X;Y)=sum_(x in X)sum_(y in Y)P(x,y)log_2((P(x,y))/(P(x)P(y))) (1) bits. Additional ...

The series which arises in the binomial theorem for negative integer -n, (x+a)^(-n) = sum_(k=0)^(infty)(-n; k)x^ka^(-n-k) (1) = sum_(k=0)^(infty)(-1)^k(n+k-1; k)x^ka^(-n-k) ...

The evolute of the nephroid given by x = 1/2[3cost-cos(3t)] (1) y = 1/2[3sint-sin(3t)] (2) is given by x = cos^3t (3) y = 1/4[3sint+sin(3t)], (4) which is another nephroid.

Two functions f(x) and g(x) are orthogonal over the interval a<=x<=b with weighting function w(x) if <f(x)|g(x)>=int_a^bf(x)g(x)w(x)dx=0. (1) If, in addition, ...

The quintic surface given by the equation x^2+y^3+z^5=1 was considered by S. Plouffe (pers. comm., Dec. 21, 1998) and here dubbed the peninsula surface. It has been dubbed ...

Polyrects are polyforms obtained from a rectangular grid, illustrated above. The numbers of polyrects with n=1, 2, ... components are 1, 2, 3, 9, 21, 68, 208, ... (OEIS ...

Polyrhombs are polyforms obtained from a rhombic grid, illustrated above. The numbers of polyrhombs with n=1, 2, ... components are 1, 1, 3, 7, 20, 62, 204, ... (OEIS ...

The derivative identity d/(dx)[f(x)g(x)] = lim_(h->0)(f(x+h)g(x+h)-f(x)g(x))/h (1) = (2) = lim_(h->0)[f(x+h)(g(x+h)-g(x))/h+g(x)(f(x+h)-f(x))/h] (3) = f(x)g^'(x)+g(x)f^'(x), ...

A proper factor of a positive integer n is a factor of n other than 1 or n (Derbyshire 2004, p. 32). For example, 2 and 3 are positive proper factors of 6, but 1 and 6 are ...

A proposition is a mathematical statement such as "3 is greater than 4," "an infinite set exists," or "7 is prime." An axiom is a proposition that is assumed to be true. With ...