Transition probabilities of Lévy‐type processes: Parametrix construction

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.     

Global existence and time decay estimate of solutions to the Keller–Segel system

We consider the chemotaxis‐Navier–Stokes system 1.1-1.4 (Keller–Segel system) in the whole space, which describes the motion of oxygen‐driven bacteria, eukaryotes, in a fluid. We proved the global existence and time decay estimate of solutions to the Cauchy problem 1.1-1.2 in with the small initial data. Moreover, when the fluid motion is described by the Stokes equations, we established the global weak solutions to 1.3-1.4 in with the potential function ? is small and the initial density n₀(x) has finite mass.     

Partial widths and branching ratios for the emitted electron resulting from interatomic Coulombic decay in quantum wells heterostructure

Interatomic coulomb decay (ICD) is a decay process relying on the Coulombic interaction between neighbouring atoms, molecules or nanostructures. Due to this process, an electron is emitted into the continuum. We study the ICD process in a system of the double quantum well heterostructure and investigate how we can manipulate the structure's parameters such that a better detection of the ICD's emitted electron is achieved. For this purpose, we calculated the partial widths (PWs) and branching ratios (BRs) of the ICD's emitted electron to the left and right asymptotes of the heterostructure; these will give an estimation of the detection current. We manipulated the structure's parameters and took into account the repulsion from the electron in the ground state located in the left well. By introducing two small barriers in the vicinity of the right QW, we observed a BR three times larger than in the structure without the barriers. We also investigate the effect of repulsion due to the second electron. This work gives a better understanding of the dynamics of the scattered ICD's electron, and realisation of better design rules for future experimental observation of ICD in nanostructures.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Weak attractors for the dissipative fractional Heisenberg equation

In this short paper, we consider the long time behaviors of the fractional Heisenberg equation and the existence of a global weak attractor is proved for the shift dynamics in the path space. The key ingredient is some new type of commutator structure introduced in this paper, which seems indispensable in proving the compactness of the dynamics. The technique introduced in this paper may also be useful to other fractional order partial differential equations.     

Two- and three-dimensional restricted geometry case of luminescence quenching

We present general analytical expression for two and three-dimensional cases of static energy transfer kinetics in doped nanoparticles (of round, spherical and cylindrical shape). A series of numerical experiments has been performed using Monte-Carlo simulation. The analytical expressions have shown very good coincidence with the computer simulation.