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A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

Let Omega be an open, bounded, and connected subset of R^d for some d and let dx denote d-dimensional Lebesgue measure on R^d. In functional analysis, the Friedrichs ...

Let Omega be an open, bounded, and connected subset of R^d for some d and let dx denote d-dimensional Lebesgue measure on R^d. In functional analysis, the Poincaré inequality ...

Borwein et al. (2004, pp. 4 and 44) term the expression of the integrals I_1 = int_0^1x^xdx (1) = 0.783430510... (2) I_2 = int_0^1(dx)/(x^x) (3) = 1.291285997... (4) (OEIS ...

Ergodic theory can be described as the statistical and qualitative behavior of measurable group and semigroup actions on measure spaces. The group is most commonly N, R, R-+, ...

Borsuk conjectured that it is possible to cut an n-dimensional shape of generalized diameter 1 into n+1 pieces each with diameter smaller than the original. It is true for ...

The Darboux integral, also called a Darboux-Stieltjes integral, is a variant of the Stieltjes integral that is defined as a common value for the lower and upper Darboux ...

The hyperbolic cotangent is defined as cothz=(e^z+e^(-z))/(e^z-e^(-z))=(e^(2z)+1)/(e^(2z)-1). (1) The notation cthz is sometimes also used (Gradshteyn and Ryzhik 2000, p. ...

Jackson's theorem is a statement about the error E_n(f) of the best uniform approximation to a real function f(x) on [-1,1] by real polynomials of degree at most n. Let f(x) ...

A theorem, also known as Bachet's conjecture, which Bachet inferred from a lack of a necessary condition being stated by Diophantus. It states that every positive integer can ...