Description: This article discusses an abstract index theorem on non-compact Riemannian manifolds. Abstract: We prove an abstract index theorem for essentially self-adjoint Fredholm supersymmetric first-order elliptic differential operators on Hermitian vector bundles over complete oriented Riemannian manifolds. According to our main result the supersymmetric L2-index of such an operator can be expressed as the sum of a "local contribution" (the familiar Atiyah-Singer index form, suitably restricted to and integrated over a finite region) and a "boundary contribution" (which depends only on the restriction of the operator at large distances). This is done by splicing together local parametrices and Green's operators defined "at infinity". The result yields (in fact is equivalent to) a generalisation of the relative index theorem of Gromov and Lawson.

Description: This article addresses, from a historical perspective, Newton's sums of like powers of natural numbers.

Description: This paper proposes a new direct proof of the fact that L^1 verifies the strong maximum principle, i.e., any analytic map from the complex unit disk into L^1, constant in norm, must be constant.

Description: The '5 lei coin' problem of Titeica is generalized to circles of arbitrary radii.

Description: This paper studies the relationship between the Stieltjes sums and the elementary symmetric sums.