scholarly journals Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 7
Author(s):  
Stevan Pilipović ◽  
Nenad Teofanov ◽  
Filip Tomić
Keyword(s):  
Additional Parameter ◽  
Boundary Values ◽  
New Techniques ◽  
Gevrey Regularity ◽  
Essential Condition ◽  
Growth Properties ◽  
Associated Functions ◽  
Wave Front Sets ◽  
The Given ◽  
Logarithmic Type

Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, known as stability under ultradifferential operators in the classical ultradistribution theory, is replaced by a weaker condition, in which the growth properties are controlled by an additional parameter. For that reason, new techniques were used in the proofs. As an application, we discuss the corresponding wave front sets.

scholarly journals Beyond gevrey regularity: Superposition and propagation of singularities

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2763-2782 ◽  
Author(s):  
Stevan Pilipovic ◽  
Nenad Teofanov ◽  
Filip Tomic
Keyword(s):  
Wave Front ◽  
Regularity Condition ◽  
Differential Operators ◽  
Ultradifferentiable Functions ◽  
Gevrey Regularity ◽  
Gevrey Classes ◽  
Partial Differential Operators ◽  
Wave Front Sets ◽  
Propagation Of Singularities ◽  
Two Parameters

We propose the relaxation of Gevrey regularity condition by using sequences which depend on two parameters, and define spaces of ultradifferentiable functions which contain Gevrey classes. It is shown that such a space is closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients are less regular then Gevrey-type ultradifferentiable functions. To that aim we introduce appropriate wave front sets and prove a theorem on propagation of singularities. This extends related known results in the sense that assumptions on the regularity of the coefficients are weakened.

scholarly journals The implementation of the unified transform to the nonlinear Schrödinger equation with periodic initial conditions

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
B. Deconinck ◽  
A. S. Fokas ◽  
J. Lenells
Keyword(s):  
Evolution Equations ◽  
Initial Conditions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Values ◽  
Initial Value ◽  
Integrable Evolution ◽  
Initial Boundary Value ◽  
The Given

AbstractThe unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

scholarly journals Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 161
Author(s):  
Pongsakorn Sunthrayuth ◽  
Nuttapol Pakkaranang ◽  
Poom Kumam ◽  
Phatiphat Thounthong ◽  
Prasit Cholamjiak
Keyword(s):  
Banach Spaces ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonexpansive Mappings ◽  
Nonlinear Evolution Equations ◽  
Fredholm Integral Equations ◽  
Explicit Method ◽  
Convergence Theorems ◽  
New Techniques ◽  
The Given

In this paper, we introduce a generalized viscosity explicit method (GVEM) for nonexpansive mappings in the setting of Banach spaces and, under some new techniques and mild assumptions on the control conditions, prove some strong convergence theorems for the proposed method, which converge to a fixed point of the given mapping and a solution of the variational inequality. As applications, we apply our main results to show the existence of fixed points of strict pseudo-contractions and periodic solutions of nonlinear evolution equations and Fredholm integral equations. Finally, we give some numerical examples to illustrate the efficiency and implementation of our method.

scholarly journals Focusing on the Cell Type Specific Regulatory Actions of NLRX1

2021 ◽  
Vol 22 (3) ◽  
pp. 1316
Author(s):  
Tünde Fekete ◽  
Dóra Bencze ◽  
Eduárd Bíró ◽  
Szilvia Benkő ◽  
Kitti Pázmándi
Keyword(s):  
General Pattern ◽  
Signaling Cascades ◽  
Special Focus ◽  
Cell Type ◽  
Cellular Functions ◽  
Associated Functions ◽  
Cell Type Specific ◽  
Regulatory Effects ◽  
The Given ◽  
Regulatory Properties

Cells utilize a diverse repertoire of cell surface and intracellular receptors to detect exogenous or endogenous danger signals and even the changes of their microenvironment. However, some cytosolic NOD-like receptors (NLR), including NLRX1, serve more functions than just being general pattern recognition receptors. The dynamic translocation between the cytosol and the mitochondria allows NLRX1 to interact with many molecules and thereby to control multiple cellular functions. As a regulatory NLR, NLRX1 fine-tunes inflammatory signaling cascades, regulates mitochondria-associated functions, and controls metabolism, autophagy and cell death. Nevertheless, literature data are inconsistent and often contradictory regarding its effects on individual cellular functions. One plausible explanation might be that the regulatory effects of NLRX1 are highly cell type specific and the features of NLRX1 mediated regulation might be determined by the unique functional activity or metabolic profile of the given cell type. Here we review the cell type specific actions of NLRX1 with a special focus on cells of the immune system. NLRX1 has already emerged as a potential therapeutic target in numerous immune-related diseases, thus we aim to highlight which regulatory properties of NLRX1 are manifested in disease-associated dominant immune cells that presumably offer promising therapeutic solutions to treat these disorders.

scholarly journals Local competing in interpolation problems

2020 ◽  
Vol 11 (4) ◽  
pp. 99-122
Author(s):  
Sergej Vital'evich Znamenskij
Keyword(s):  
Metric Spaces ◽  
Rational Functions ◽  
Small Degree ◽  
Additional Parameter ◽  
General Terms ◽  
Interpolation Problems ◽  
Computational Simplicity ◽  
The One ◽  
The Given ◽  
Selection Of

A simple example illustrates the insufficiency of the known approaches to interpolation in the problem of recovering a function from a few given specific values that clearly convey the form. A local choice between polynomial and rational local interpolants, which minimizes the local interpolant’s errors at the nearest external nodes from one or different sides, complements the known approaches. It combines the extreme computational simplicity of local interpolants with the thorought selection of them. The principles of constructing the algorithm are formulated in general terms for mappings of metric spaces. They provide accurate (with rare exceptions) reconstruction of mappings that locally coincide with some of the given possible interpolants. In the one-dimensional case, the two-stage algorithm guarantees the continuity of the interpolant and accurately reconstructs polynomials of small degree, simple rational functions with a linear denominator, and broken lines of long links with knots at the ends when these requirements do not contradict each other. An additional parameter allows you to replace the exact restoration of polylines with the required smoothness of interpolation.

scholarly journals Determination of source term for the fractional Rayleigh–Stokes equation with random data

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Tran Thanh Binh ◽  
Dumitru Baleanu ◽  
Nguyen Hoang Luc ◽  
Nguyen-H Can
Keyword(s):  
Stokes Equation ◽  
Source Term ◽  
Convergence Rates ◽  
Mean Square ◽  
New Techniques ◽  
The Mean ◽  
The Given ◽  
Well Posed ◽  
Regularized Solution

AbstractIn this article, we consider the problem of finding a source term of a Rayleigh–Stokes equation. Our problem is not well-posed in the sense of Hadamard. The sought solution does not depend continuously on the given data. Using the truncation method and some new techniques on trigonometric estimators, we give the regularized solution. Moreover, the mean square error and convergence rates are established.

scholarly journals On a multi-channel queue with state dependent input flow and interruptions

1989 ◽  
Vol 2 (3) ◽  
pp. 199-204
Author(s):  
Jewgeni Dshalalow
Keyword(s):  
Queueing System ◽  
Input Flow ◽  
Regenerative Processes ◽  
Waiting Room ◽  
New Techniques ◽  
Queueing Process ◽  
State Dependent ◽  
Flow Activity ◽  
The Given ◽  
Explicit Relation

This paper deals with a multi-channel queueing system with a finite waiting room but without losses. The latter is achieved by a temporary interruption of the input flow activity until the waiting room is ready to place a new customer. In addition, the input flow on its “busy period” is non-recurrent: It is state dependent and may be controlled over relevant times of decision making. A similar model without interruptions (i.e. with losses) was earlier studied by the author, where in particular, major probability characteristics of the queueing process in equilibrium were obtained. Now the author derives a simple explicit relation between the two models allowing the given queue to inherit the results previously obtained. New techniques for semi-regenerative processes are used.

scholarly journals Extended Hodge theory for fibred cusp manifolds

2018 ◽  
Vol 10 (03) ◽  
pp. 531-562 ◽  
Author(s):  
E. Hunsicker
Keyword(s):  
Exact Sequence ◽  
Hodge Theory ◽  
Intersection Cohomology ◽  
Boundary Values ◽  
Cohomology Groups ◽  
Harmonic Forms ◽  
Relative Cohomology ◽  
Manifold With Boundary ◽  
The Given ◽  
Boundary Space

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted [Formula: see text] harmonic forms for a complete metric on the regular stratum with respect to some weight determined by the perversity. Extended weighted [Formula: see text] harmonic forms are harmonic forms that are almost in the given weighted [Formula: see text] space for the metric in question, but not quite. This result is akin to the representation of absolute and relative cohomology groups for a manifold with boundary by extended harmonic forms on the associated manifold with cylindrical ends. In analogy with that setting, in the unweighted [Formula: see text] case, the boundary values of the extended harmonic forms define a Lagrangian splitting of the boundary space in the long exact sequence relating upper and lower middle perversity intersection cohomology groups.

scholarly journals Null electromagnetic fields and relative Cauchy–Riemann embeddings

2013 ◽  
Vol 469 (2152) ◽  
pp. 20120583
Author(s):  
Jonathan Holland ◽  
George Sparling
Keyword(s):  
Electromagnetic Field ◽  
Minkowski Space ◽  
Electromagnetic Fields ◽  
New Techniques ◽  
Cr Embedding ◽  
Null Electromagnetic Field ◽  
Cauchy Riemann ◽  
The Given

This paper applies the notion of relative Cauchy–Riemann (CR) embeddings to study two related questions. First, it answers negatively the question posed by Penrose whether every shear-free null rotating congruence is analytic. Second, it proves that, given any shear-free null rotating congruence in Minkowski space, there exists a null electromagnetic field that is null with respect to the given congruence. In the course of answering these questions, we introduce some new techniques for studying null electromagnetic fields and shear-free congruences, in general, based on the notion of a relative CR embedding.

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