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A second-order ordinary differential equation arising in the study of stellar interiors, also called the polytropic differential equations. It is given by ...

Let s=1/(sqrt(2pi))[Gamma(1/4)]^2=5.2441151086... (1) (OEIS A064853) be the arc length of a lemniscate with a=1. Then the lemniscate constant is the quantity L = 1/2s (2) = ...

A quasi-cubic graph is a quasi-regular graph, i.e., a graph such that degree of every vertex is the same delta except for a single vertex whose degree is Delta=delta+1 ...

A canonical brick is a 1×2×4 cuboid.

Euler (1772ab) conjectured that there are no positive integer solutions to the quartic Diophantine equation A^4=B^4+C^4+D^4. This conjecture was disproved by Elkies (1988), ...

Two matrices A and B are said to be equal iff a_(ij)=b_(ij) (1) for all i,j. Therefore, [1 2; 3 4]=[1 2; 3 4], (2) while [1 2; 3 4]!=[0 2; 3 4]. (3)

The following conditions are equivalent for a conservative vector field on a particular domain D: 1. For any oriented simple closed curve C, the line integral ∮_CF·ds=0. 2. ...

A list of five properties of a topological space X expressing how rich the "population" of open sets is. More precisely, each of them tells us how tightly a closed subset can ...

The Goh-Schmutz constant is defined by the integrals C = int_0^infty(ln(1+t))/(e^t-1)dt (1) = int_0^inftyln[1-ln(1-e^(-t))]dt (2) = ...

If x takes only nonnegative values, then P(x>=a)<=(<x>)/a. (1) To prove the theorem, write <x> = int_0^inftyxP(x)dx (2) = int_0^axP(x)dx+int_a^inftyxP(x)dx. (3) Since P(x) is ...