Time-dependent gradient perturbations of fractional Laplacian

We construct a fundamental solution of the equation {partial_t - Delta^{alpha/2} - b(cdot, cdot) cdotnabla_{x} = 0} for TeX{alpha in (1, 2)} and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds

Overcoming State Support for School Consolidation: How Schools in the Empire State react.

Since 1958, the New York State Education Department has officially promoted the policy of consolidating small, rural schools. This policy is delineated in the Master Plan for School Reorganization, and while the centralization of most one-room rural schools has been successful, the state has been less successful in the consolidation of smaller, centralized rural school districts. This paper examines some of the efforts made by those smaller, centralized rural schools to overcome the outside pressures that have emerged within the process of state-backed consolidation. Based on findings in literature explored and data collected concerning consolidation, it is clear that New York State’s policy of supporting consolidations has not proven effective. Schools that are allowed creative freedom are examples for rural schools across the nation

Transnational tourist destination management: a case study of the Baltic sea region

The issue of tourist destination management has been widely reflected in scientific literature. However, transnational destinations, i.e. those that are located on the territory of several countries, have not been given enough attention. The development of transnational destinations can occur provided there is close cooperation among key stakeholders. In the Baltic Sea Region (BSR) conditions for effective cooperation were created at the end of the 20th century when integration processes intensified. Currently, the BSR is the first macro-region of the European Union to implement a transnational strategy, in which tourism is recognised as a development priority. Creating an effective tourist destination management system has become a key challenge in this process. This article aims to identify and critically analyse the concept of transnational tourist destination management. Another goal is to identify the changing determinants of tourism management in the BSR and to explore relations between key stakeholders. To this end, we conducted a review of literature on destination management. We carried out a critical analysis of secondary sources (documents, information materials, the Internet resources) and supplemented the review with our insights from participatory observation. We also obtained additional information from interviews. Three models of destination management were identified: marketing-oriented, planning-oriented, and governance-oriented. Our research shows that the evolution of destination management is characterised by a gradual increase in the participation of public entities. This is related to the marginalisation of the tourism industry, which resulted in a systematic decrease in state funding for maintaining destination management structures. Acquiring the European Union funds from the European Territorial Cooperation Programs has become a partial but not entirely effective solution to this problem

On Harnack inequality for α-stable Ornstein–Uhlenbeck processes

We consider the α-stable Ornstein–Uhlenbeck process in {mathbb{R}}^d with the generator TeXL = Delta^{alpha/2} - lambda x cdotnabla_x . We show that if 2 > α ≥ 1 or α < 1 = d the Harnack inequality holds. For α < 1 < d we construct a counterexample that shows that the Harnack inequality does not hold

On Combinatorics of Schrödinger Perturbations

We give a tight upper bound for Schrödinger-type perturbations of integral kernels

Shape-changing Collisions of Coupled Bright Solitons in Birefringent Optical Fibers

Wecritically review the recent progress in understanding soliton propagation in birefringent optical fibers.By constructing the most general bright two-soliton solution of the integrable coupled nonlinear Schroedinger equation (Manakov model) we point out that solitons in birefringent fibers can in general change their shape after interaction due to a change in the intensity distribution among the modes even though the total energy is conserved. However, the standard shape-preserving collision (elastic collision) property of the (1+1)-dimensional solitons is recovered when restrictions are imposed on some of the soliton parameters. As a consequence the following further properties can be deduced using this shape-changing collision. (i) The exciting possibility of switching of solitons between orthogonally polarized modes of the birefringent fiber exists. (ii) When additional effects due to periodic rotation of birefringence axes are considered, the shape changing collision can be used as a switch to suppress or to enhance the periodic intensity exchange between the orthogonally polarized modes. (iii) For ultra short optical soliton pulse propagation in non-Kerr media, from the governing equation an integrable system of coupled nonlinear Schroedinger equation with cubic-quintic terms is identified. It admits a nonlocal Poisson bracket structure. (iv) If we take the higher-order terms in the coupled nonlinear Schroedinger equation into account then their effect on the shape-changing collision of solitons, during optical pulse propagation, can be studied by using a direct perturbational approach.Comment: 14 pages, ROMP31, 4 EPS figure

Analysis of Ornstein-Uhlenbeck and Laguerre Stochastic Processes

The objective of these lectures is to present Ornstein-Uhlenbeck and related stochastic processes to a wide mathematical audience with a modest preparation in stochastic analysis. The aim of the first part of the lectures (Chapter 1) is to discuss the Ornstein-Uhlenbeck and the Squared Radial Ornstein-Uhlenbeck stochastic diffusion processes, whose infinitesimal generators are, respectively, the Ornstein-Uhlenbeck operator and the Laguerre operator. In the second chapter of these lectures the Ornstein-Uhlenbeck processes governed by -stable rotationally invariant processes are studied. This corresponds to replacing the Laplacian by the fractionnary Laplacian -(-)^/2 in the Ornstein-Uhlenbeck generator L=12-x. More general drift terms b(x) are also considered at the end of Chapter 2.[2mm