The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the Rayleigh quotient ‖u‖2 H1(Ω) /‖u‖2 L2(∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed volume. First, we find a formula for the first variation of the ...

where 5 > 0, a is a constant, and F\(p) is analytic and bounded in the half plane Re(j>) > c. We assume that this condition is satisfied. Whether this condition is also sufficient for the convergence of the w-point quadrature formula to the true value of f(t) in (1), when n tends to infinity, has not been determined. The author makes use here of the fact that the convergence occurs whenever F(p...

Consider an insurer who is allowed to make risk-free and risky investments. The price process of the investment portfolio is described as a geometric Lévy process. We study the tail probability of the stochastic present value of future aggregate claims. When the claim-size distribution is of Pareto type, we obtain a simple asymptotic formula which holds uniformly for all time horizons. The same...

The celebrated formula of Schll aai relates the variation of the dihedral angles of a smooth family of polyhedra in a space form and the variation of volume. We give a smooth analogue of this classical formula { our result relates the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. The argument is d...

An analysis is presented of the standardized Wahlund's variances (f) in gene frequencies of the ABO, Rh and MNS blood group systems among 19 villages of the Atsera isolate of the upper Markham Valley, Papua New Guinea. In the past, there has been some disagreement over the relative importance of population structure and natural selection in the determination of these variances. The Lewontin-Kra...

The main contribution of this paper is to propose a closed expression for the Ramanujan constant alternating series, based on Euler–Boole summation formula. Such an not present in literature. We also highlight only choice parameter formula proposed by Hardy series positive terms, so value obtained as agrees with other methods divergent series. Additionally, we derive closed-formula chosen, unde...