We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on Z4) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability P2n ω (0,0) after 2n steps is at most C(ω)n−2 logn, but the best lower bound till now has been C(ω)n−2. Here we will show t...

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebys...

Let g be a finite-dimensional semisimple Lie algebra and (· , ·) its Killing form, σ an elliptic automorphism of g, and a a σ-invariant reductive subalgebra of g, such that the restriction of the form (· , ·) to a is non-degenerate. Let L̂(g, σ) and L̂(a, σ) be the associated twisted affine Lie algebras and F σ(p) the σ-twisted Clifford module over L̂(a, σ), associated to the orthocomplement p of ...

In this article, a systematic approach is proposed to calculate the torsional rigidity and stress of a circular bar containing multiple circular inclusions. To fully capture the circular geometries, the kernel function is expanded to the degenerate form and the boundary density is expressed into Fourier series. The approach is seen as a semi-analytical manner since error purely attributes to th...

We consider a class of non-trivial perturbations A of the degenerate OrnsteinUhlenbeck operator in R . In fact we perturb both the diffusion and the drift part of the operator (say Q and B) allowing the diffusion part to be unbounded in R . Assuming that the kernel of the matrix Q(x) is invariant with respect to x ∈ R and the Kalman rank condition is satisfied at any x ∈ R by the same m < N , a...

We consider partial differential operators H = − div(C∇) in divergence form on R with a positive-semidefinite, symmetric, matrix C of real L∞-coefficients. First, we prove that one can define H as a selfadjoint operator on L2(R ) such that the corresponding semigroup extends as a positive, contraction semigroup to all the Lp-spaces. Secondly, we establish that H is strongly elliptic if and only...

We obtain a vanishing theorem for the kernel of a Dirac operator on a Clifford module twisted by a sufficiently large power of a line bundle, whose curvature is non-degenerate at any point of the base manifold. In particular, if the base manifold is almost complex, we prove a vanishing theorem for the kernel of a spinc Dirac operator twisted by a line bundle with curvature of a mixed sign. In t...

for the polynomial planar vector fields with a hyperbolic or nilpotent critical point at the origin, the monodromy problem has been solved, but for the strongly degenerate critical points this problem is still open. when the critical point is monodromic, the stability problem or the center- focus problem is an open problem too. in this paper we will consider the polynomial planar vector fields ...