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The area of a rational right triangle cannot be a square number. This statement is equivalent to "a congruum cannot be a square number."

The abscissas of the N-point Gaussian quadrature formula are precisely the roots of the orthogonal polynomial for the same interval and weighting function.

On a Riemannian manifold, there is a unique connection which is torsion-free and compatible with the metric. This connection is called the Levi-Civita connection.

If a function phi is harmonic in a sphere, then the value of phi at the center of the sphere is the arithmetic mean of its value on the surface.

The index of a vector field with finitely many zeros on a compact, oriented manifold is the same as the Euler characteristic of the manifold.

Consider (1) If the probability distribution is governed by a Markov process, then P_3(y_1,t_1;y_2,t_2|y_3,t_3) = P_2(y_2,t_2|y_3,t_3) (2) = P_2(y_2|y_3,t_3-t_2). (3) ...

Let f(z) be a transcendental meromorphic function, and let D_1, D_2, ..., D_5 be five simply connected domains in C with disjoint closures (Ahlfors 1932). Then there exists j ...

A triangle and its polar triangle with respect to a conic are perspective.

A (presumably autobiographical) character in one of astrophysicist Fred Hoyle's novels opined the following. "I figure that if to be totally known and totally loved is worth ...

States that for a nondissipative Hamiltonian system, phase space density (the area between phase space contours) is constant. This requires that, given a small time increment ...