The transmission problem is a system of two second-order elliptic equations unknowns equipped with the Cauchy data on boundary. After four decades research motivated by scattering theory, spectral properties this are now known to depend type contrast between coefficients near Previously, we established discreteness eigenvalues for large class anisotropic which related celebrated complementing c...

Let S(RG) be the normed p-unit group in a group ring RG, formed by an abelian group G and a commutative ring R with identity of prime characteristic p. All unexplained symbols and letters as well as the terminology and definitions from the abelian group theory (including the topological ones) can be found in the classical book monographs [7]. For a background material in that direction, we refe...

The behaviour of test fields near a compact Cauchy horizon is investigated. It is shown that solutions of nonlinear wave equations on Taub spacetime with generic initial data cannot be continued smoothly to both extensions of the spacetime through the Cauchy horizon. This is proved using an energy method. Similar results are obtained for the spacetimes of Moncrief containing a compact Cauchy ho...

1. Free holomorphic functions and Hausdorff derivations 2. Cauchy, Liouville, and Schwartz type results for free holomorphic functions 3. Algebras of free holomorphic functions 4. Free analytic functional calculus and noncommutative Cauchy transforms 5. Weierstrass and Montel theorems for free holomorphic functions 6. Free pluriharmonic functions and noncommutative Poisson transforms 7. Hardy s...

The corresponding topology is called norm-topology. The norm-topology naturally gives rise to the concept of convergence and continuity. A sequence txnun“1 Ă X is said to converge to x in the norm-metric, or strongly, if }xn ́ x} Ñ 0 as nÑ 8. A function f : X Ñ Y , where Y is a topological space, is continuous if fpxnq Ñ fpxq whenever xn Ñ x, or equivalently, if f ́1pUq is open for any open U Ă Y...

We study the Cauchy problem for the Aw-Rascle-Zhang model for traffic flow with a flux constraint at x = 0. More precisely we consider the Riemann solver, conserving the number of cars at x = 0 but not the generalized momentum, introduced in [9] for the problem with flux constrained. For such a Riemann solver, we prove existence of a solution for the Cauchy problem. The proof is based on the wa...

Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous diffus...

D.V. Chudnovsky and G.V. Chudnovsky [CH] introduced a generalization of the FrobeniusStickelberger determinantal identity involving elliptic functions that generalize the Cauchy determinant. The purpose of this note is to provide a simple essentially non-analytic proof of this evaluation. This method of proof is inspired by D. Zeilberger’s creative application in [Z1]. AMS Subject Classificatio...