We study the existence of weak solutions to (E) (−∆)u+g(u) = ν in a bounded regular domain Ω in R (N ≥ 2) which vanish in R \Ω, where (−∆) denotes the fractional Laplacian with α ∈ (0, 1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for pr...

A random walk on a countable group $G$ acting metric space $X$ gives characteristic called the drift which depends only transition probability measure $\mu$ of walk. The is `translation distance' In this paper, we prove that varies continuously with measures, under assumption distance and horofunctions are expressed by certain ratios. As an example, consider mapping class $\mathrm{MCG}(S)$ Teic...

ABSTRACT : Quaternion Dirac equation has been obtained from the square root of Klein-Gordon equation in compact and consistent way. Dirac matrices are described as quaternion valued and the Dirac Hamiltonian is considered as Hermitian with real eigenvalues of energy. Dirac spinors and free particle energy solution has been obtained in terms of one component, two-component and four-component Dir...

In this article, we survey the gluing formulae of the spectral invariants the ζ-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator. After these spectral invariants had been originally introduced by Ray-Singer [30] and Atiyah-Patodi-Singer [1] respectively, these invariants have been studied by many people in many different parts of mathematics and ...

We present a matrix technique to obtain the spectrum and the analytical index of some elliptic operators defined on compact Riemannian manifolds. The method uses matrix representations of the derivative which yield exact values for the derivative of a trigono-metric polynomial. These matrices can be used to find the exact spectrum of an elliptic operator in particular cases and in general, to g...

Abstract. We first give an elementary proof of a result relating the eigenvalues of the Dirac operator to Branson’s Q-curvature on 4-dimensional spin compact manifolds. In the case of n-dimensional closed compact (spin) manifolds we then use the conformal covariance of the Dirac, Yamabe and Branson-Paneitz operators to compare appropriate powers of their first eigenvalues. Equality cases are al...

The honeycomb lattice of graphene is characterized by linear dispersion and pseudospin chirality of fermions on the Dirac cones. If lattice anisotropy is introduced, the Dirac cones stay intact but move in reciprocal space. Dirac point movement can lead to a topological transition from semimetal to semiconductor when two inequivalent Dirac points merge, an idea that has attracted significant re...

The electromagnetic scattering of a spin-0 charged particle off a fixed center is calculated in first-order quantum perturbation theory. This implies evaluating the square of a ‘Dirac delta-function,’ an operation that is not defined in Schwartz distribution theory, and which in elementary text-books is dealt with according to ‘Fermi’s golden rule.’ In this paper these conventional calculations...

A novelWeyl-Heisenberg algebra in Clifford-spaces is constructed that is based on a matrix-valued H extension of Planck’s constant. As a result of this modified Weyl-Heisenberg algebra one will no longer be able to measure, simultaneously, the pairs of variables (x, px); (x, py); (x, pz); (y, px), ... with absolute precision. New Klein-Gordon and Dirac wave equations and dispersion relations in...