Exact solutions to the focusing nonlinear Schrodinger equation

A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrodinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates.Comment: 60 pages, 18 figure

A unified approach to Darboux transformations

We analyze a certain class of integral equations related to Marchenko equations and Gel'fand-Levitan equations associated with various systems of ordinary differential operators. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution. We show how this result provides a unified approach to Darboux transformations associated with various systems of ordinary differential operators. We illustrate our theory by deriving the Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a discrete eigenvalue is added to the spectrum.Comment: final version that will appear in Inverse Problem

Increased serum levels of sortilin are associated with depression and correlated with BDNF and VEGF

AbstractNeurotrophic factors have been investigated in relation to depression. The aim of the present study was to widen this focus to sortilin, a receptor involved in neurotrophic signalling. The serum sortilin level was investigated in 152 individuals with depression and 216 control individuals, and eight genetic markers located within the SORT1 gene were successfully analysed for association with depression. Genotyping was performed using the Sequenom MassARRAY platform. All the individuals returned a questionnaire and participated in a semi-structured diagnostic interview. Sortilin levels were measured by immunoassay, and potential determinants of the serum sortilin level were assessed by generalized linear models. Serum levels of brain-derived neurotrophic factor (BDNF) and vascular endothelial growth factor (VEGF) were measured in previous studies. We identified a significant increase of serum sortilin levels in depressed individuals compared with controls (P=0.0002) and significant positive correlation between serum sortilin levels and the corresponding levels of BDNF and VEGF. None of the genotyped SNPs were associated with depression. Additional analyses showed that the serum sortilin level was influenced by several other factors. Alcohol intake and body mass index, as well as depression, serum BDNF and serum VEGF were identified as predictors of serum sortilin levels in our final multivariate model. In conclusion, the results suggest a role of circulating sortilin in depression which may relate to altered activity of neurotrophic factors.</jats:p

Exact Solutions to the Sine-Gordon Equation

A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable $x$ and the temporal variable $t,$ and they are exponentially asymptotic to integer multiples of $2\pi$ as $x\to\pm\infty.$ The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of a Marchenko integral equation. By expressing the kernel of that Marchenko equation as a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.Comment: 43 page

High prevalence of bronchiectasis is linked to HTLV-1-associated inflammatory disease.

BACKGROUND: Human T-lymphotropic virus type 1 (HTLV-1), a retrovirus, is the causative agent of HTLV-1-associated myelopathy/tropical spastic paraparesis (HAM/TSP) and adult T-cell leukaemia/lymphoma (ATLL). The reported association with pulmonary disease such as bronchiectasis is less certain. METHODS: A retrospective case review of a HTLV-1 seropositive cohort attending a national referral centre. The cohort was categorised into HTLV-1 symptomatic patients (SPs) (ATLL, HAM/TSP, Strongyloidiasis and HTLV associated inflammatory disease (HAID)) and HTLV-1 asymptomatic carriers (ACs). The cohort was reviewed for diagnosis of bronchiectasis. RESULT: 34/246 ACs and 30/167 SPs had been investigated for respiratory symptoms by computer tomography (CT) with productive cough +/- recurrent chest infections the predominant indications. Bronchiectasis was diagnosed in one AC (1/246) and 13 SPs (2 HAID, 1 ATLL, 10 HAM/TSP) (13/167, RR 19.2 95 % CI 2.5-14.5, p = 0.004) with high resolution CT. In the multivariate analysis ethnicity (p = 0.02) and disease state (p < 0.001) were independent predictors for bronchiectasis. The relative risk of bronchiectasis in SPs was 19.2 (95 % CI 2.5-14.5, p = 0.004) and in HAM/TSP patients compared with all other categories 8.4 (95 % CI 2.7-26.1, p = 0.0002). Subjects not of African/Afro-Caribbean ethnicity had an increased prevalence of bronchiectasis (RR 3.45 95 % 1.2-9.7, p = 0.02). CONCLUSIONS: Bronchiectasis was common in the cohort (3.4 %). Risk factors were a prior diagnosis of HAM/TSP and ethnicity but not HTLV-1 viral load, age and gender. The spectrum of HTLV-associated disease should now include bronchiectasis and HTLV serology should be considered in patients with unexplained bronchiectasis

On two possible ways to recover ordinary thermodynamics from extended thermodynamics of polyatomic gases

We consider two possible ways, i.e. the Maxwellian iteration (MI) and the Chapman–Enskog method (CEM), to recover relativistic ordinary thermodynamics from relativistic extended thermodynamics of Polyatomic gases with N moments. Both of these methods give the Eckart equations which are the relativistic version of the Navier–Stokes and Fourier laws as a first iteration. However, these methods do not lead to the same expressions of the heat conductivity χ, the shear viscosity μ, and the bulk viscosity ν which appear as coefficients in the Eckart equations. In particular, we prove that the expressions of χ, μ, and ν obtained via the CEM do not depend on N, while those obtained through the MI depend on N. Moreover, we also prove that these two methods lead to the same results in the nonrelativistic limit

Genome-wide study of association and interaction with maternal cytomegalovirus infection suggests new schizophrenia loci.

Genetic and environmental components as well as their interaction contribute to the risk of schizophrenia, making it highly relevant to include environmental factors in genetic studies of schizophrenia. This study comprises genome-wide association (GWA) and follow-up analyses of all individuals born in Denmark since 1981 and diagnosed with schizophrenia as well as controls from the same birth cohort. Furthermore, we present the first genome-wide interaction survey of single nucleotide polymorphisms (SNPs) and maternal cytomegalovirus (CMV) infection. The GWA analysis included 888 cases and 882 controls, and the follow-up investigation of the top GWA results was performed in independent Danish (1396 cases and 1803 controls) and German-Dutch (1169 cases, 3714 controls) samples. The SNPs most strongly associated in the single-marker analysis of the combined Danish samples were rs4757144 in ARNTL (P=3.78 × 10(-6)) and rs8057927 in CDH13 (P=1.39 × 10(-5)). Both genes have previously been linked to schizophrenia or other psychiatric disorders. The strongest associated SNP in the combined analysis, including Danish and German-Dutch samples, was rs12922317 in RUNDC2A (P=9.04 × 10(-7)). A region-based analysis summarizing independent signals in segments of 100 kb identified a new region-based genome-wide significant locus overlapping the gene ZEB1 (P=7.0 × 10(-7)). This signal was replicated in the follow-up analysis (P=2.3 × 10(-2)). Significant interaction with maternal CMV infection was found for rs7902091 (P(SNP × CMV)=7.3 × 10(-7)) in CTNNA3, a gene not previously implicated in schizophrenia, stressing the importance of including environmental factors in genetic studies

The International-Trade Network: Gravity Equations and Topological Properties

This paper begins to explore the determinants of the topological properties of the international - trade network (ITN). We fit bilateral-trade flows using a standard gravity equation to build a "residual" ITN where trade-link weights are depurated from geographical distance, size, border effects, trade agreements, and so on. We then compare the topological properties of the original and residual ITNs. We find that the residual ITN displays, unlike the original one, marked signatures of a complex system, and is characterized by a very different topological architecture. Whereas the original ITN is geographically clustered and organized around a few large-sized hubs, the residual ITN displays many small-sized but trade-oriented countries that, independently of their geographical position, either play the role of local hubs or attract large and rich countries in relatively complex trade-interaction patterns

On the Inverse Scattering Method for Integrable PDEs on a Star Graph

© 2015, Springer-Verlag Berlin Heidelberg. We present a framework to solve the open problem of formulating the inverse scattering method (ISM) for an integrable PDE on a star-graph. The idea is to map the problem on the graph to a matrix initial-boundary value (IBV) problem and then to extend the unified method of Fokas to such a matrix IBV problem. The nonlinear Schrödinger equation is chosen to illustrate the method. The framework unifies all previously known examples which are recovered as particular cases. The case of general Robin conditions at the vertex is discussed: the notion of linearizable initial-boundary conditions is introduced. For such conditions, the method is shown to be as efficient as the ISM on the full-line