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The Legendre transform of a sequence {c_k} is the sequence {a_k} with terms given by a_n = sum_(k=0)^(n)(n; k)(n+k; k)c_k (1) = sum_(k=0)^(n)(2k; k)(n+k; n-k)c_k, (2) where ...

Legendre's formula counts the number of positive integers less than or equal to a number x which are not divisible by any of the first a primes, (1) where |_x_| is the floor ...

The line integral of a vector field F(x) on a curve sigma is defined by int_(sigma)F·ds=int_a^bF(sigma(t))·sigma^'(t)dt, (1) where a·b denotes a dot product. In Cartesian ...

A linear congruence equation ax=b (mod m) (1) is solvable iff the congruence b=0 (mod d) (2) with d=GCD(a,m) is the greatest common divisor is solvable. Let one solution to ...

In calculus, geometry, and plotting contexts, the term "linear function" means a function whose graph is a straight line, i.e., a polynomial function of degree 0 or 1. A ...

The logarithmic capacity of a compact set E in the complex plane is given by gamma(E)=e^(-V(E)), (1) where V(E)=inf_(nu)int_(E×E)ln1/(|u-v|)dnu(u)dnu(v), (2) and nu runs over ...

For a polynomial P(x_1,x_2,...,x_k), the Mahler measure of P is defined by (1) Using Jensen's formula, it can be shown that for P(x)=aproduct_(i=1)^(n)(x-alpha_i), ...

For triangles in the plane, AD·BE·CF=BD·CE·AF. (1) For spherical triangles, sinAD·sinBE·sinCF=sinBD·sinCE·sinAF. (2) This can be generalized to n-gons P=[V_1,...,V_n], where ...

The metabigyrate rhombicosidodecahedron is a convex equilateral solid obtained by rotating two of the pentagonal cupolas of a small rhombicosidodecahedron located in the ...

1. Find a complete system of invariants, or 2. Decide when two metrics differ only by a coordinate transformation. The most common statement of the problem is, "Given metrics ...