We show a general formula of the one loop effective potential of the 5D SU(N) gauge theory compactified on an orbifold, S/Z2. The formula shows the case when there are fundamental, (anti-)symmetric tensor and adjoint representational bulk fields. Our calculation method is also applicable when there are bulk fields belonging to higher dimensional representations. The supersymmetric version of th...

For a planar directed graph G, Postnikov’s boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit formula for Postnikov’s map by expressing each Plücker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the nonplanar set...

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic func...

In this paper, a hybrid approach for solving the Laplace equation in general threedimensional (3-D) domains is presented. The approach is based on a local method for the Dirichletto-Neumann (DtN) mapping of a Laplace equation by combining a deterministic (local) boundary integral equation (BIE) method and the probabilistic Feynman–Kac formula for solutions of elliptic partial differential equat...

Asymptotic symmetry of the Euclidean 3D gravity with torsion is described by two independent Virasoro algebras with different central charges. Elements of this boundary conformal structure are combined with Cardy’s formula to calculate the black hole entropy.

We investigate a class of over-determined parabolic problems involving a non-constant boundary condition. The Weinstein’s technique known for the elliptic problems is extended to the parabolic one by means of auxiliary functions and Green classical formula.

The aim of this study is to prove Lyapunov-type inequalities for a quasilinear elliptic equation in [Formula: see text]. Also the lower bound for the first positive eigenvalue of the boundary value problem is obtained.

We prove a variant of Krein’s resolvent formula for self-adjoint extensions given by arbitrary boundary conditions. A parametrization of all such extensions is suggested with the help of two bounded operators instead of multivalued operators and selfadjoint linear relations.

We present an algorithm for computing the direction of pitchfork bifurcation for two-point boundary value problems. The formula is rather involved, but its computational evaluation is quite feasible. As an application, we obtain a multiplicity result.

For a family of Dirac operators acting on Hermitian Cli ord modules over the odd dimensional compact manifolds with boundary which are the bres of a bration with compact base we compute the Chern character of the index in K of the base Although we assume a product decomposition near the boundary we make no assumptions on invertibility of the bound ary family and instead obtain a family of self ...