A fixed point theorem of Jungck in $$b_v(s)$$-metric spaces

In this paper, we prove the common fixed point theorem of Jungck in \(b_v(s)\)-metric spaces. As a corollary, a common fixed point theorem in the rectangular b-metric space and the well known common fixed point theorem in b-metric spaces are obtained.     

Asymptotic quadratic convergence of the parallel block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices

The proof of the asymptotic quadratic convergence is provided for the parallel two-sided block-Jacobi EVD algorithm with dynamic ordering for Hermitian matrices. The discussion covers the case of well-separated eigenvalues as well as clusters of eigenvalues. Having p processors, each parallel iteration step consists of zeroing 2p off-diagonal blocks chosen by dynamic ordering with the aim to maximize the decrease of the off-diagonal Frobenius norm. Numerical experiments illustrate and confirm the developed theory.     

Statistical causality,martingale problems and local uniqueness

The paper considers a statistical concept of causality in continuous time in the filtered probability spaces which is based on the Granger’s definition of causality. The given causality concept is then applied to the solution of the martingale problem (associated with the stochastic differential equation driven with semimartingales). More precisely, we show that the given causality concept is closely connected to the concept of extremality of measures for the solutions of the martingale problem, for the stopped martingale problem and for the local martingale problem. We also show the equivalence between some models of causality and local uniqueness (for the solutions of the martingale problem).     

On the scattering problem for infinitely many fermions in dimensions d ≥ 3 at positive temperature

In this paper, we study the dynamics of a system of infinitely many fermions in dimensions $d\ge 3$ near thermal equilibrium and prove scattering in the case of small perturbation around equilibrium in a certain generalized Sobolev space of density operators. This work is a continuation of our previous paper [11], and extends the important recent result of M. Lewin and J. Sabin in [19] of a similar type for dimension $d=2$. In the work at hand, we establish new, improved Strichartz estimates that allow us to control the case $d\ge 3$.     

On graded nil clean rings

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.     

The unsaturated flow in porous media with dynamic capillary pressure

In this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two-phase flow in porous media with dynamic capillary pressure-saturation relationship where the relaxation parameter depends on the saturation. Following the approach given in [13] the existence of a weak solution is proved using Galerkin approximation and regularization techniques. A priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test-functions, motivated by the fact that considered PDE allows a natural generalization of the classical Kullback entropy. Finally, a special care was given in obtaining an estimate of the mixed-derivative term by combining the information from the capillary pressure with the obtained a priori estimates on the saturation.     

Complex backflow dynamics in nematic liquid crystals

Received December 31, 2001 / Published online May 21, 2002     

Propagation of Reactions in Inhomogeneous Media

Consider reaction‐diffusion equation u _t =Δ u + f (x,u ) with and general inhomogeneous ignition reaction f ≥ 0 vanishing at u = 0,1. Typical solutions 0 ≤ u ≤ 1 transition from 0 to 1 as time progresses, and we study them in the region where this transition occurs. Under fairly general qualitative hypotheses on f we show that in dimensions d ≤ 3, the Hausdorff distance of the superlevel sets {u ≥ ε } and {u ≥ 1‐ε} remains uniformly bounded in time for each ε ? (0,1). Thus, u remains uniformly in time close to the characteristic function of in the sense of Hausdorff distance of superlevel sets. We also show that each {u ≥ ε} expands with average speed (over any long enough time interval) between the two spreading speeds corresponding to any x ‐independent lower and upper bounds on f . On the other hand, these results turn out to be false in dimensions d ≥ 4, at least without further quantitative hypotheses on f . The proof for d ≤ 3 is based on showing that as the solution propagates, small values of u cannot escape far ahead of values close to 1. The proof for d ≥ 4 is via construction of a counterexample for which this fails. Such results were before known for d =1 but are new for general non‐periodic media in dimensions d ≥ 2 (some are also new for homogeneous and periodic media). They extend in a somewhat weaker sense to monostable, bistable, and mixed reaction types, as well as to transitions between general equilibria of the PDE and to solutions not necessarily satisfying . © 2016 Wiley Periodicals, Inc.     

Stability,periodicity, and symmetries of certain second‐order fractional difference equation with quadratic terms via KAM theory

By using the Kolmogorov–Arnold–Moser theory, we investigate the stability of the equilibrium solution of the difference equation where A ,B ,D > 0,u _?1,u ₀>0. We also use the symmetries to find effectively the periodic solutions with feasible periods. Copyright © 2016 John Wiley & Sons, Ltd.     

Bounds for Shannon and Zipf‐Mandelbrot entropies

Shannon and Zipf‐Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well‐know Jensen inequality and obtain different bounds for Shannon and Zipf‐Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf‐Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf‐Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf‐Mandelbrot entropy.