The subclasses N (∞) κ of the classes N κ of generalized Nevanlinna functions appear in the context of Pontryagin space models, where they correspond to model relations having a particular spectral behaviour. Applications are found, for instance, in the investigation of differential expressions with singular coefficients. We study representations of N (∞) κ-functions as Cauchy-type integrals in...

In this paper we study different conditions whose presence is required for A. the admissibility and stability of large shocks present in solutions of a strictly hyperbolic n × n system of conservation laws in one space dimension ut + f(u)x = 0, B. the solvability and L well posedness of the Cauchy problem for the above equation, near solutions containing large and stable, but noninteracting sho...

We study the splitting scheme associated with the linear stochastic Cauchy problem dU(t) = AU(t) dt+ dW (t), t ∈ [0, T ], U(0) = x, Here A is the generator of an analytic C0-semigroup S = {S(t)}t>0 on a Banach space E and W = {W (t)}t>0 is a Brownian motion with values in a fractional domain space Eβ associated with A. We prove that if α, β, γ, θ > 0 are such that γ + θ < 1 and (α − β + θ)+ + γ...

We compute the spectra of the Tanaka type Laplacians = ∂̄ Q∂̄Q + ∂̄Q∂̄ ∗ Q and △ = ∂ ∗ Q∂Q+ ∂Q∂ ∗ Q on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S in Cn. We prove that Szegö map is a unitary operator from a subspace of (p, q− 1)-forms on the sphere defined by the operators △ and the normal vector field onto the space of L-harmonic (p, q)-forms on th...

Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analyt...

and Applied Analysis 3 Moreover, Bourgin 15 and Găvruţa 16 have considered the stability problem with unbounded Cauchy differences see also 17–27 . On the other hand, J. M. Rassias 28–33 considered the Cauchy difference controlled by a product of different powers of norm. However, there was a singular case; for this singularity a counterexample was given by Găvruţa 34 . This stability phenomeno...

We consider a Cauchy problem for a scalar conservation law in one space dimension with a compactly supported initial datum,

We prove that the space-time developments of generic solutions of the vacuum constraint Einstein equations do not possess any global or local Killing vectors, when Cauchy data are prescribed on an asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean curvature close to a constant, or for CMC asymptotically hyperbolic initial data sets. More generally, we show that non-exi...

This paper considers totally bounded quasi-uniformities and quasi-proximities for frames and shows that for a given quasi-proximity ⊳ on a frame L there is a totally bounded quasi-uniformity on L that is the coarsest quasi-uniformity, and the only totally bounded quasi-uniformity, that determines ⊳. The constructions due to B. Banaschewski and A. Pultr of the Cauchy spectrum ψL and the compacti...