Branching Brownian Motion with catalytic branching at the origin

We consider a branching Brownian motion in which binary fission takes place only when particles are at the origin at a rate \beta > 0 on the local time scale. We obtain results regarding the asymptotic behaviour of the number of particles above \lambda t at time t, for \lambda > 0. As a corollary, we establish the almost sure asymptotic speed of the rightmost particle. We also prove a Strong Law of Large Numbers for this catalytic branching Brownian motion.Comment: 25 page

Fluctuations of the rightmost particle in the catalytic branching Brownian motion

In this article we establish the magnitude of fluctuations of the extreme particle in the model of binary branching Brownian motion with a single catalytic point at the origin.Comment: 28 page

Ancient economic and social concepts in the Genoese Gasaria Region

The main theme of the article is the consideration of issues related to the presence of the Genoese Republic in the Northern Black Sea region in the 13th - 15th centuries. In the region, a whole system of towns and settlements was established by the Genoese, which was transformed into a separate state entity. The boundaries of this state entity covered the territory from the delta of the Danube in the west to the Caucasus coast in the east, as well as the entire coast of the Crimean Peninsula. For the state entity, built by the Genoese in the region of the Northern Black Sea Coast, the name Genoese Gazaria is proposed. Therefore, all the material evidence of the Genoese presence on the territory of the previously owned state of Golden Horde and, especially, of trade settlements in the Northern Black Sea region in the 13th and 15th centuries fall under this definition. The main difference between the Genoese Gazaria and Genoese Romania is the issue of the initial ownership of the land in the places of settlement of trading factories (settlements). In the case of Genoese Romania, these lands belonged to Byzantium and were granted to the Genoese by Byzantine emperors. The Genoese Gazaria emerged under similar circumstances, but on the lands of the Mongolian empire, these lands were given to them by khans of the Golden Horde.peer-reviewe

Limiting distribution of the rightmost particle in catalytic branching Brownian motion

We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate βδ0(·), where δ0(·) is the Dirac delta function and β is some positive constant. We show that the distribution of the rightmost particle centred about β/2 t converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [10] for the degenerate case of catalytic branching.</p

Branching Lévy Processes with Inhomogeneous Breeding Potentials

The object of study in this thesis is a number of different models of branching Levy processes in inhomogeneous breeding potential. We employ some widely-used spine techniques to investigate various features of these models for their subsequent comparison. The thesis is divided into 5 chapters. In the first chapter we introduce the general framework for branching Markov processes within which we are going to present all our results. In the second chapter we consider a branching Brownian motion in the potential β|·|p, β> 0, p ≥0. We give a new proof of the result about the critical value of p for the explosion time of the population. The main advantage of the new proof is that it can be easily generalised to other models. The third chapter is devoted to continuous-time branching random walks in the potential β|·|p, β> 0, p ≥0. We give results about the explosion time and the right most particle behaviour comparing them with the known results for the branching Brownian motion. In the fourth chapter we look at general branching Levy processes in the potential β|·|p, β> 0, p ≥0. Subject to certain assumptions we prove some results about the explosion time and the rightmost particle. We exhibit how the corresponding results for the branching Brownian motion and and the branching random walk fit into the general structure. The last chapter considers a branching Brownian motion with branching taking place at the origin on the local time scale. We present some results about the population dynamics and the right most particle behaviour. We also prove the Strong Law of Large Numbers for this model.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Limiting distribution of the rightmost particle in catalytic branching Brownian motion

We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $\beta \delta_0(\cdot)$, where $\delta_0(\cdot)$ is the Dirac delta function and $\beta$ is some positive constant. We show that the distribution of the rightmost particle centred about $\frac{\beta}{2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [6] for the degenerate case of catalytic branching.Comment: 12 page

First Principle Evaluation of Photocatalytic Suitability for TiO2-Based Nanotubes

Water splitting under the influence of solar light on semiconducting electrodes Immersed in aqueous electrolyte is a potentially clean and renewable source for hydrogen fuel production. Its efficiency depends on relative position of the band gap edges (the visible light interval between infrared and ultraviolet (UV) ranges of electromagnetic spectrum corresponds to gap widths 1.5–2.8 eV) accompanied by a proper band alignment relative to both reduction (H+/H2) and oxidation (O2/H2O) potentials (−4.44 eV and −5.67 eV on energy scale for vacuum, respectively) which must be positioned inside the band gap. Its width for TiO2 anatase-structured bulk is experimentally found to be 3.2 eV, which corresponds to photocatalytic activity under UV light possessing only ~1% efficiency of sunlight energy conversion. Noticeable growth of this efficiency can be achieved by by adjusting the band gap edges for titania bulk through nanoscale transformation of its morphology to anatase-type nanotubes (NTs) (formed by folding of (001) or (101) nanothin TiO2 sheets consisting of 9 or 6 atomic layers and possessing either (n,0) or (−n,n) chiralities, respectively) accompanied by partial substitution of pristine atoms by CO, FeTi, NO and SO single dopants as well as NO+SO codopants. In the latter case, the band gap can be reduced down to 2.2 eV while the efficiency is achieved up to ~15%. The energy differences between the edges of band gap (VB and CB), the highest occupied and lowest unoccupied impurity levels inside the band gap (HOIL and LUIL, respectively) induced in doped NTs, while preserving the proper disposition of these levels relatively to the redox potentials, so that εVB<εHOIL<εO2/H2O<εH+/H2<εLUIL<εCB, thus reducing the photon energy required for dissociation of H2O molecule. In this chapter, we analyze applicability of large-scale first principle calculations on the doped single-wall titania NTs of different morphologies with the aim of establishment of their suitability for photocatalytic water splitting