On the construction of partial difference schemes II: discrete variables and Schwarzian lattices

In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut--Schwarz--Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut--Schwarz--Young theorem. A discussion on the numerical results is also presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.Comment: 14 pages, 4 figure

Discrete Reductive Perturbation Technique

We expand a partial difference equation (P$\Delta$E) on multiple lattices and obtain the P$\Delta$E which governs its far field behaviour. The perturbative--reductive approach is here performed on well known nonlinear P$\Delta$Es, both integrable and non integrable. We study the cases of the lattice modified Korteweg--de Vries (mKdV) equation, the Hietarinta equation, the lattice Volterra--Kac--Van Moerbeke (VKVM) equation and a non integrable lattice KdV equation. Such reductions allow us to obtain many new P$\Delta$Es of the nonlinear Schr\"odinger (NLS) type.Comment: 18 pages, 1 figure. submitted to Journal of Mathematical Physic

Lie group analysis of a generalized Krichever-Novikov differential-difference equation

The symmetry algebra of the differential--difference equation $$\dot u_n = [P(u_n)u_{n+1}u_{n-1} + Q(u_n)(u_{n+1}+u_{n-1})+ R(u_n)]/(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension 1 \le \mbox{dim}L \le 5. When $P$, $Q$ and $R$ are specific second order polynomials in $u_n$ (depending on 6 constants) this is the integrable discretization of the Krichever--Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies \mbox{dim}L=2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity $u_{n+N}=u_n$ and thus to reduce the differential--difference equation to a system of $N$ coupled ordinary three points difference equations

On a discrete Davey-Stewartson system

We propose a differential difference equation in ${\mathcal R}^1\times {\mathcal Z}^2$ and study it by Hirota's bilinear method. This equation has a singular continuum limit into a system which admits the reduction to the Davey-Stewartson equation. The solutions of this discrete DS system are characterized by Casorati and Grammian determinants. Based on the bilinear form of this discrete DS system, we construct the bilinear B\"{a}cklund transformation which enables us to obtain its Lax pair.Comment: 12 pages, 2 figure

Biomarkers of islet beta cell stress and death in type 1 diabetes

Recent work on the pathogenesis of type 1 diabetes has led to an evolving recognition of the heterogeneity of this disease, both with regards to clinical phenotype and responses to therapies to prevent or revert diabetes. This heterogeneity not only limits efforts to accurately predict clinical disease but also is reflected in differing responses to immunomodulatory therapeutics. Thus, there is a need for robust biomarkers of beta cell health, which could provide insight into pathophysiological differences in disease course, improve disease prediction, increase the understanding of therapeutic responses to immunomodulatory interventions and identify individuals most likely to benefit from these therapies. In this review, we outline current literature, limitations and future directions for promising circulating markers of beta cell stress and death in type 1 diabetes, including markers indicating abnormal prohormone processing, circulating RNAs and circulating DNAs

Lie discrete symmetries of lattice equations

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlev\'e I equation and of the Toda lattice equation

On Fourier integral transforms for qq-Fibonacci and qq-Lucas polynomials

We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform

The Kundu--Eckhaus equation and its discretizations

In this article we show that the complex Burgers and the Kundu--Eckhaus equations are related by a Miura transformation. We use this relation to discretize the Kundu--Eckhaus equation.Comment: 10 page

Response of young and adult birds to the same environmental variables and different spatial scales during post breeding period

Context: How do young birds achieve spatial knowledge about the environment during the initial stages of their life? They may follow adults, so gaining social information and learning; alternatively, young birds may acquire knowledge of the environment themselves by experiencing habitat and landscape features. If learning is at least partially independent of adults then young birds should respond to landscape composition at finer spatial scale than adults, who possess knowledge over a larger area. Objectives: We studied the responses of juvenile, immature and adult Caspian Gull Larus cachinnans to the same habitat and landscape variables, but at several spatial scales (ranging from 2.5 to 15\ua0km), during post-breeding period. Methods: We surveyed 61 fish ponds (foraging patches) in southern Poland and counted Caspian gulls. Results: Juvenile birds responded at finer spatial scales to the factors than did adults. Immature birds showed complicated, intermediate responses to spatial scale. The abundance of juvenile birds was mostly correlated with the landscape composition (positively with the cover of corridors and negatively with barriers). Adult abundance was positively related to foraging patch quality (fish stock), which clearly required previous spatial experience of the environment. The abundance of all age classes were moderately correlated with each other indicating that social behaviour may also contribute to the learning of the environment. Conclusions: This study shows that as birds mature, they respond differently to components of their environment at different spatial scales. This has considerable ecological consequences for their distribution across environments

Cytokines Interleukin-1β and Tumor Necrosis Factor-α Regulate Different Transcriptional and Alternative Splicing Networks in Primary β-Cells

OBJECTIVE: Cytokines contribute to pancreatic beta-cell death in type 1 diabetes. This effect is mediated by complex gene networks that remain to be characterized. We presently utilized array analysis to define the global expression pattern of genes, including spliced variants, modified by the cytokines interleukin (IL)-1beta + interferon (IFN)-gamma and tumor necrosis factor (TNF)-alpha + IFN-gamma in primary rat beta-cells. RESEARCH DESIGN AND METHODS: Fluorescence-activated cell sorter-purified rat beta-cells were exposed to IL-1beta + IFN-gamma or TNF-alpha + IFN-gamma for 6 or 24 h, and global gene expression was analyzed by microarray. Key results were confirmed by RT-PCR, and small-interfering RNAs were used to investigate the mechanistic role of novel and relevant transcription factors identified by pathway analysis. RESULTS Nearly 16,000 transcripts were detected as present in beta-cells, with temporal differences in the number of genes modulated by IL-1beta + IFNgamma or TNF-alpha + IFN-gamma. These cytokine combinations induced differential expression of inflammatory response genes, which is related to differential induction of IFN regulatory factor-7. Both treatments decreased the expression of genes involved in the maintenance of beta-cell phenotype and growth/regeneration. Cytokines induced hypoxia-inducible factor-alpha, which in this context has a proapoptotic role. Cytokines also modified the expression of >20 genes involved in RNA splicing, and exon array analysis showed cytokine-induced changes in alternative splicing of >50% of the cytokine-modified genes. CONCLUSIONS: The present study doubles the number of known genes expressed in primary beta-cells, modified or not by cytokines, and indicates the biological role for several novel cytokine-modified pathways in beta-cells. It also shows that cytokines modify alternative splicing in beta-cells, opening a new avenue of research for the field.Journal ArticleResearch Support, Non-U.S. Gov'tinfo:eu-repo/semantics/publishe