Representation of the Green’s function of the exterior Neumann problem for the Laplace operator

We represent the Green’s function of the classical Neumann problem for the exterior of the unit ball of arbitrary dimension. We show that the Green’s function can be expressed through elementary functions. The explicit form of the function is written out.     

Fast algorithms for implementation of the Green’s function

This paper is devoted to new fast algorithms for implementation of the Green’s function for the Helmholtz operator in high-frequency regions in periodic and helical structures.     

Stability of radial symmetry for a Monge-Ampère overdetermined problem

Recently the symmetry of solutions to overdetermined problems has been established for the class of Hessian operators, including the Monge-Ampère operator. In this paper we prove that the radial symmetry of the domain and of the solution to an overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.      

Bounds for canonical Green’s function at cusps

In this article, we derive bounds for the canonical Green’s function defined on a noncompact hyperbolic Riemann surface, when evaluated at two inequivalent cusps.     

Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter \(\lambda >0\), where \(m\in (1/2,1/2+\varepsilon )\) with \(\varepsilon >0\) small, and f is superlinear or sublinear at \(\infty \) with no sign-conditions at 0 assumed.

     

Symmetry in the Green’s Function for Birth-death Chains

Methodology and Computing in Applied Probability - A symmetric relation in the probabilistic Green’s function for birth-death chains is explored. Two proofs are given, each of which makes use...     

On the oscillation property of Green’s function of a fourth-order discontinuous boundary-value problem

The paper deals with conditions under which the Green function of a multipoint boundary-value problem for fourth-order equations describing small strains of a rod fastened to a solid elastic basement and additionally fixed by “concentrated” elastic supports at separate points has the oscillation property. It is shown that the condition that the Green function is positive is necessary and sufficient for the Green function to have the oscillation property.     

On Karatsuba’s problem concerning the divisor function

We study an asymptotic behavior of the sum ${\sum_{n \leqslant x} \frac{\tau(n)}{\tau(n+a)}}$ . Here τ(n) denote the number of divisors of n and ${a\,\geqslant\,1}$ is a fixed integer.     

The two-time Green’s function and the diagram technique

Based on constructing the equations of motion for the two-time Green’s functions, we discuss calculating the dynamical spin susceptibility and correlation functions in the Heisenberg model. Using a Mori-type projection, we derive an exact Dyson equation with the self-energy operator in the form of a multiparticle Green’s function. Calculating the self-energy operator in the mode-coupling approximation in the ferromagnetic phase, we reproduce the results of the temperature diagram technique, including the correct formula for low-temperature magnetization. We also consider calculating the spin fluctuation spectrum in the paramagnetic phase in the framework of the method of equations of motion for the relaxation function.     

Farsighted Stability in an n-Person Prisoner’s Dilemma

Farsighted stability is examined in an n-person prisoner’s dilemma. It is shown that any individually rational and Pareto efficient outcome is a farsighted stable set and no other farsighted stable set exists; it is further shown that the largest consistent set consists of all individually rational outcomes.