Towards an Inverse Scattering theory for non decaying potentials of the heat equation

The resolvent approach is applied to the spectral analysis of the heat equation with non decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe $N$ solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a DBAR-problem explicitly in terms of the corresponding objects associated to the original potential. Regularity conditions of the potential in the cases N=1 and N=2 are investigated in details. The singularities of the resolvent for the case N=1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.Comment: 22 pages, submitted to Inverse Problem

On the equivalence of different approaches for generating multisoliton solutions of the KPII equation

The unexpectedly rich structure of the multisoliton solutions of the KPII equation has been explored by using different approaches, running from dressing method to twisting transformations and to the tau-function formulation. All these approaches proved to be useful in order to display different properties of these solutions and their related Jost solutions. The aim of this paper is to establish the explicit formulae relating all these approaches. In addition some hidden invariance properties of these multisoliton solutions are discussed

The Gould-Hopper Polynomials in the Novikov-Veselov equation

We use the Gould-Hopper (GH) polynomials to investigate the Novikov-Veselov (NV) equation. The root dynamics of the $\sigma$-flow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particulr, when $N=3,4,5$, one can get the Gold-fish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.Comment: 22 pages, no figur

Topological Phenomena in the Real Periodic Sine-Gordon Theory

The set of real finite-gap Sine-Gordon solutions corresponding to a fixed spectral curve consists of several connected components. A simple explicit description of these components obtained by the authors recently is used to study the consequences of this property. In particular this description allows to calculate the topological charge of solutions (the averaging of the $x$-derivative of the potential) and to show that the averaging of other standard conservation laws is the same for all components.Comment: LaTeX, 18 pages, 3 figure

Building extended resolvent of heat operator via twisting transformations

Twisting transformations for the heat operator are introduced. They are used, at the same time, to superimpose a` la Darboux N solitons to a generic smooth, decaying at infinity, potential and to generate the corresponding Jost solutions. These twisting operators are also used to study the existence of the related extended resolvent. Existence and uniqueness of the extended resolvent in the case of $N$ solitons with N "ingoing" rays and one "outgoing" ray is studied in details.Comment: 15 pages, 2 figure

Technologically-Treated Polyclonal Affinity-Purified Antibodies to the Tumor Necrosis Factor-α, Brain Specific S-100 Protein and Histamine in Treatment of Functional Dyspepsia: Results of the Multicenter, Randomized, Double-Blind, Placebo-Controlled Trial

The aim of the study was to evaluate the efficacy and safety of Kolofort® (a complex medicine containing technologically processed forms of antibodies to S-100 protein, tumor necrosis factor-α and histamine) in the management of functional dyspepsia (FD) in outpatient clinical practice.Methods: Outpatients (N = 309) at the age of 18–45 in whom FD was diagnosed according to the Rome IV criteria were enrolled in a multicenter, double-blind, placebo-controlled, randomized clinical trial. Patients were randomized in two groups receiving Kolofort® or placebo 2 tablets tid for 8 weeks. The primary endpoint of the study was a change in the FD symptoms severity score according to the Gastrointestinal symptom score (GIS) at week 8. ITT and [PP] analysis were performed.Results: at week 8 the reduction in GIS sum score was observed in Kolofort® group and placebo group (by 7.2 ± 3.3 [7.2 ± 3.4] and 6.3 ± 4.6 [6.2 ± 4.5], respectively, p = 0.041 [0.039]). The proportion of cases with GIS score reduction by ≥4 was 88,1 % [88.6 %] and 79.1 % [79.6 %] in Kolofort® group and placebo group, respectively (p = 0.046 [p = 0.051]). None of the patients in Kolofort® group had progression of FD symptoms or required additional therapy. There were 29 adverse events (AEs) recorded in 25 patients including 16 cases in 13 (8.6 %) patients in Kolofort® group and 13 AEs in 12 (7.6 %) patients in placebo group.Conclusion: the clinical trial demonstrates the positive effect of Kolofort® in FD with a favorable safety profile

Fermionic approach to the evaluation of integrals of rational symmetric functions

We use the fermionic construction of two-matrix model partition functions to evaluate integrals over rational symmetric functions. This approach is complementary to the one used in the paper ``Integrals of Rational Symmetric Functions, Two-Matrix Models and Biorthogonal Polynomials'' \cite{paper2}, where these integrals were evaluated by a direct method.Comment: 34 page

Gradient catastrophe and flutter in vortex filament dynamics

Gradient catastrophe and flutter instability in the motion of vortex filament within the localized induction approximation are analyzed. It is shown that the origin if this phenomenon is in the gradient catastrophe for the dispersionless Da Rios system which describes motion of filament with slow varying curvature and torsion. Geometrically this catastrophe manifests as a rapid oscillation of a filament curve in a point that resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painlev\'e-I equation.Comment: 11 pages, 3 figures, typos corrected, references adde

Oxytocin Signaling in the Central Amygdala Modulates Emotion Discrimination in Mice

Recognition of other's emotions influences the way social animals interact and adapt to the environment. The neuropeptide oxytocin (OXT) has been implicated in different aspects of emotion processing. However, the role of endogenous OXT brain pathways in the social response to different emotional states in conspecifics remains elusive. Here, using a combination of anatomical, genetic, and chemogenetic approaches, we investigated the contribution of endogenous OXT signaling in the ability of mice to discriminate unfamiliar conspecifics based on their emotional states. We found that OXTergic projections from the paraventricular nucleus of the hypothalamus (PVN) to the central amygdala (CeA) are crucial for the discrimination of both positively and negatively valenced emotional states. In contrast, blocking PVN OXT release into the nucleus accumbens, prefrontal cortex, and hippocampal CA2 did not alter this emotion discrimination. Furthermore, silencing each of these PVN OXT pathways did not influence basic social interaction. These findings were further supported by the demonstration that virally mediated enhancement of OXT signaling within the CeA was sufficient to rescue emotion discrimination deficits in a genetic mouse model of cognitive liability. Our results indicate that CeA OXT signaling plays a key role in emotion discrimination both in physiological and pathological conditions. Is endogenous oxytocin implicated in emotion discrimination? Ferretti, Maltese et al. demonstrate that oxytocin signaling in the central amygdala plays a key role in the ability of mice to discriminate unfamiliar conspecifics based on their emotional state, both in physiological and genetically determined pathological conditions

Increase of precision of casting blocks by applying acoustical oscillations in gas-impulsive moulding

The present article shows the results of the theoretical and empirical research of the study of the impact made by acoustical oscillations on the working process of a gas-impulsive moulding device with the purpose of increasing the quality of the casting blocks’ seal and decreasing the roughness of the surface of a casting block. It has been shown that acoustical oscillations intensify the working process of the gas-impulsive moulding device. With the increase of frequency and power of oscillations the hardness of a casting block first sharply rises, but then gradually decreases. It has been found out that oscillations may help to adjust the precision of a casting block due to high sensitivity of a gas-impulsive process to the frequency of oscillations. The use of acoustical oscillations broadens technological opportunities of the moulding device