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The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.

In this short paper, the initial value problem for the Navier-Stokes equations with the Coriolis force is investigated. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. We prove that existence of uniform global large solutions to the Navier-Stokes equations with the Coriolis force for a class of special initial data. The results obtained in this paper are different from the previous 2 types of results.

In a paper from 2001 (Journal of the LMS), Diestel and Leader offered a proof that a connected graph has a normal spanning tree if and only if it does not contain a minor from two specific forbidden classes of graphs, all of cardinality $\aleph_1$. Unfortunately, their proof contains a gap, and their result is incorrect. In this paper, we construct a third type of obstruction: an $\aleph_1$-sized graph without a normal spanning tree that contains neither of the two types described by Diestel and Leader as a minor. Further, we show that any list of forbidden minors characterising the graphs with normal spanning trees must contain graphs of arbitrarily large cardinality., Comment: 9 pages. arXiv admin note: text overlap with arXiv:2005.02833

This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula., Comment: 48 pages. Reformulation of Theorem 1 for readability. Extra explanation for the mixed terms in section 2.2.3. Reference updates

Full-waveform inversion (FWI) is today a standard process for the inverse problem of seismic imaging. PDE-constrained optimization is used to determine unknown parameters in a wave equation that represent geophysical properties. The objective function measures the misfit between the observed data and the calculated synthetic data, and it has traditionally been the least-squares norm. In a sequence of papers, we introduced the Wasserstein metric from optimal transport as an alternative misfit function for mitigating the so-called cycle skipping, which is the trapping of the optimization process in local minima. In this paper, we first give a sharper theorem regarding the convexity of the Wasserstein metric as the objective function. We then focus on two new issues. One is the necessary normalization of turning seismic signals into probability measures such that the theory of optimal transport applies. The other, which is beyond cycle skipping, is the inversion for parameters below reflecting interfaces. For the first, we propose a class of normalizations and prove several favorable properties for this class. For the latter, we demonstrate that FWI using optimal transport can recover geophysical properties from domains where no seismic waves travel through. We finally illustrate these properties by the realistic application of imaging salt inclusions, which has been a significant challenge in exploration geophysics.

In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower box dimension at least $n-0.5$ and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper., 9 pages. arXiv admin note: text overlap with arXiv:1711.06533

This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages

We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math

It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.

We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which were already known for Haar null sets.

In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

[EN] In this paper, a novel technique is formed to obtain the solution of a fractional gas dynamics equation. Some reproducing kernel Hilbert spaces are defined. Reproducing kernel functions of these spaces have been found. Some numerical examples are shown to confirm the efficiency of the reproducing kernel Hilbert space method. The accurate pulchritude of the paper is arisen in its strong implementation of Caputo fractional order time derivative on the classical equations with the success of the highly accurate solutions by the series solutions. Reproducing kernel Hilbert space method is actually capable of reducing the size of the numerical work. Numerical results for different particular cases of the equations are given in the numerical section., This research was partially supported by Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

The paper deals with boundedness problems of integral operators in weighted grand Bochner-Lebesgue spaces. We will treat both cases: when a weight function appears as a multiplier in the definition of the norm, or when it defines the absolute continuous measure of integration. Along with the diagonal case we deal with the off-diagonal case. To get the appropriate result for the Hardy-Littlewood maximal operator we rely on the reasonable bound of the sharp constant in the Buckley type theorem which is also derived in the paper.

The first goal of this paper is to prove multiple asymptotic results for a time-discrete and space-continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite volume limit for every fixed asymptotic slope; concentration inequalities for free energy implying a bound on its fluctuation exponent; straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one-sided infinite paths with given endpoint and slope, and interpretation of these infinite-volume Gibbs measures as thermodynamic limits. Moreover, we prove that marginals of polymer measures with the same slope and different endpoints are asymptotic to each other. The second goal of the paper is to develop ergodic theory of the Burgers equation with positive viscosity and random kick forcing on the real line without any compactness assumptions. Namely, we prove a One Force -- One Solution principle, using the infinite volume polymer measures to construct a family of stationary global solutions for this system, and proving that each of those solutions is a one-point pullback attractor on the initial conditions with the same spatial average. This provides a natural extension of the same program realized for the inviscid Burgers equation with the help of action minimizers that can be viewed as zero temperature limits of polymer measures., Comment: 67 pages. This is an extension of the ergodic program for the Burgers equation in arXiv:1205.6721 and arXiv:1406.5660 to the positive viscosity case. Minor clarifications and additions to bibliography in this version

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$ ($N\in{\mathbb N}$), written as \[ \frac{\partial u}{\partial t} + (-\Delta+1)\beta(u) = g \quad \mbox{in}\ \Omega\times(0, T), \] which represents the porous media, the fast diffusion equations, etc., where $\beta$ is a single-valued maximal monotone function on $\mathbb{R}$, and $T>0$. Existence and uniqueness for (P) were directly proved under a growth condition for $\beta$ even though the Stefan problem was excluded from examples of (P). This paper completely removes the growth condition for $\beta$ by confirming Cauchy's criterion for solutions of the following approximate problem (P)$_{\varepsilon}$ with approximate parameter $\varepsilon>0$: \[ \frac{\partial u_{\varepsilon}}{\partial t} + (-\Delta+1)(\varepsilon(-\Delta+1)u_{\varepsilon} + \beta(u_{\varepsilon}) + \pi_{\varepsilon}(u_{\varepsilon})) = g \quad \mbox{in}\ \Omega\times(0, T), \] which is called the Cahn--Hilliard system, even if $\Omega \subset \mathbb{R}^N$ ($N \in \mathbb{N}$) is an unbounded domain. Moreover, it can be seen that the Stefan problem is covered in the framework of this paper.

A Borel set B⊂Rn is visible from x∈Rn, if the radial projection of B with base point x has positive Hn−1 measure. I prove that if dimB>n−1, then B is visible from every point x∈Rn∖E, where E is an exceptional set with dimension dimE⩽2(n−1)−dimB. This is the sharp bound for all n⩾2. Many parts of the proof were already contained in a recent previous paper by P. Mattila and the author, where a weaker bound for dimE was derived as a corollary from a certain slicing theorem. Here, no improvement to the slicing result is obtained; in brief, the main observation of the present paper is that the proof method gives the optimal result, when applied directly to the visibility problem.

This paper is focused on following time-harmonic Maxwell equation: ∇×(μ−1(x)∇×u)−ω2e(x)u=f(x,u),inΩ,ν×u=0,on∂Ω, where Ω⊂R3 is a bounded Lipschitz domain, ν:∂Ω→R3 is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as |u|→∞, we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor μ∈R3×3 and permittivity tensor e∈R3×3, ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.

In this paper, we consider the Cauchy problem of a fluid-particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro-micro decomposition.

This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space-time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time-asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support., Comment: This paper is an improved and extended version of arXiv:1507.00883. Moreover this second version contains supplementary information on wave packets to motivate our results and comments on the applicability of our method to the study of certain hyperbolic equations

Shannon and Zipf-Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well-know Jensen inequality and obtain different bounds for Shannon and Zipf-Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf-Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf-Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf-Mandelbrot entropy.

In this paper we consider the following question. When does there exist a square root of a probability measure supported on R+? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that μ=ν*ν (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non-trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7-atomic probability measures which don't have any square root. Our results have a connection to the following long-open problem in Operator Theory: characterize the subnormal operators having a square root.

The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plucker coordinates.

The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed c⩾c∗. Then we show that the traveling wave fronts with speed c>c∗ are exponentially asymptotically stable.

In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.

The Bogdanov-Takens bifurcations of a Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339-366,” Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.

In the paper, a necessary and sufficient criterion it provided such that any local optimal solution is also global in a not necessarily differentiable constrained optimization problem. This criterion is compared to others earlier appeared in the literature, which are sufficient but not necessary for a local optimal solution to be global. The importance of the established criterion is illustrated by suitable examples of nonconvex optimization problems presented in the paper.

This paper studies the dynamics of the generalized Lengyel-Epstein reaction-diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady-state solution.

In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we examine embeddings of alternating and symmetric groups into almost simple groups of exceptional type. In particular, we prove that if the alternating or symmetric group has degree equal to 5, or 8 or more, then it cannot appear as the maximal subgroup of any almost simple exceptional group of Lie type. Furthermore, in the remaining open cases of degrees 6 and 7 we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aiming to substantially improve the state of knowledge about the maximal subgroups of exceptional groups of Lie type.

Ren and Zeng (2013) introduced a new kind of q-Bernstein-Schurer operators and studied some approximation properties. Acu etal. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K-functional. Next, we introduce the bivariate case of q-Bernstein-Schurer-Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K-functional. Finally, we define the generalized Boolean sum operators of the q-Bernstein-Schurer-Kantorovich type and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright (c) 2017 John Wiley & Sons, Ltd.

This paper studies a higher dimensional generalization of Frieze's $\zeta(3)$-limit theorem in the Erd\"os-R\'enyi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erd\"os-R\'enyi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$., Comment: 24 pages

In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type k-fractional integral operators recently given by Mubeen et al. Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper we prove a sharp distortion property of the Cassinian metric under Mobius transformations of a punctured ball onto another punctured ball. The paper also deals with discussion on local convexity properties of the Cassinian metric balls in proper subdomains of Rn. Inclusion properties of the Cassinian metric balls with other hyperbolic-type metric balls are also investigated. In particular, several conjectures are also stated in response to sharpness of the inclusion relations.

This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line x > as follows: ut−lux−uxx=−β(uvx)x,x>0,t>0,λv−vxx=u,x>0,t>0,lu(0,t)+ux(0,t)=vx(0,t)=0,t>0,u(x,0)=u0(x),x>0, where l is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of l > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of l

This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

In this note we point out an error in [2]. We show how to repair the proof in dimension 5. The results are true in general as can easily be seen from recent work of Borman, Eliashberg and Murphy [1]. The proof of Lemma 3.4 in [2] is incorrect. Below we will describe the problem with the proof and then show how it can easily be repaired in dimension 5. We then observe that Lemma 3.4, and thus the main results of the paper, is true in all dimensions based on recent work of Borman, Eliashberg and Murphy [1]. However this approach does not give an explicit construction and hence goes against the sprit of the original paper and in addition all the results of [2] follow directly from [1]. Acknowledgement: We thank Yasha Eliashberg for pointing out the error in the proof of Lemma 3.4 in [2]. The first author was partially supported by a grant from the Simons Foundation (#342144) and NSF grant DMS-1309073. 1. Exact Lagrangians, Liouville flows, and the error in the proof of Lemma 3.4 We begin by recalling the statement of Lemma 3.4 from [2]. To state the lemma we first establish some notation (that is slightly different that what was used in [2]). Consider T 2 × [0, 1] with coordinates (θ, φ, r) and the contact structure ξi = kerαi, i = 1, 2, where αi = ki(r) dθ + li(r) dφ. Here we have k1(r) = cos π 2 r and l(r) = sin π 2 r, and for i = 2 we have k2 and l2 agreeing with k1 and l1 near r = 0 and 1, and the curve (k2(r), l2(r)) in R has 5π/2 winding about the origin. In particular notice that ξ2 is obtained from ξ1 by adding Giroux torsion. Lemma 3.4 from [2] now reads as follows. Lemma 1. Let W be a manifold with contact form λ, there is a contact structure on W × [0, 1] × ([0, 1]× T ) such that the following properties are satisfied: (1) near W×{0}× [0, 1]×T 2 and W× [0, 1]×{0, 1}×T 2 the contact structure is contactomorphic to λ+ eα1, and (2) near W × {1} × [0, 1]× T 2 the contact structure is contactomorphic to λ+ eα2. Here t is the coordinate on the first [0, 1] factor. See [2] for details on how the main constructions and theorems of the paper follow from this lemma. The strategy of the proof in [2] was: (1) To construct a contact structure on W×[0, 1]×T 3 that near W×{0}×T 3 is given by λ+eβ0 and near W ×{1}× T 3 is given by λ+ e × β1, where βi is the contact structure on T 3 with Giroux torsion i and we are thinking of T 3 as S×T 2 with the S-factors Legendrian curves. (2) Then cut W × [0, 1]×T 3 along W × [0, 1]× ({θ0, θ1}×T ) so that one of the resulting pieces is as described in the lemma. To try to arrange this let β = p1 dθ1+p1 dθ2 be the Liouville form on T ∗T 2 = R×T 2 with coordinates (p1, p2, θ1, θ2). Notice that α = λ + β is a contact form on W × T ∗T . We will see below that we can arrange the two items above that are needed for our proof if there is a radial vector field v in R centered at a point p whose flow expands dβ (that is, Lvdβ = dβ) and a Lagrangian torus T 2 in a small neighborhood of {q}×T 2 ⊂ T ∗T 2 that is exact with respect to ιvdβ that is isotopic to {q}×T 2 by an isotopy disjoint from {p} × T . One may easily arrange all of this except for either the last 2010 Mathematics Subject Classification. 57R17 (primary), and 53D35(secondary).

In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper examines the initial-value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow-up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper we study direct and inverse problems for discrete and continuous skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo-exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.

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In this paper, we introduce a general Bayesian abstract fuzzy economy model of product measurable spaces, and we prove the existence of Bayesian fuzzy equilibrium for this model. Our results extend and improve the corresponding recent results announced by Patriche and many authors from the literature. It captures the idea that the uncertainties characterize the individual feature of the decisions of the agents involved in different economic activities. In this paper, the uncertainties can be described by using random fuzzy mappings. Further attention is needed for the study of applications of the established result in the game theory and the fuzzy economic field.Copyright © 2015 John Wiley & Sons, Ltd.

The paper introduces a class of functions where the resolving operator for a system of Kolmogorov–Feller-type equations with a small parameter is well posed in forward and backward times. The introduced class of functions is invariant under the resolving operator if the solution is understood in the weak sense with an exponential weight. The paper continues the study of [6].

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In this paper we consider the system of the non-steady Navier–Stokes equations with mixed boundary conditions. We study the existence and uniqueness of a solution of this system. We define Banach spaces X and Y, respectively, to be the space of “possible” solutions of this problem and the space of its data. We define the operator and formulate our problem in terms of operator equations. Let and be the Frechet derivative of at . We prove that is one-to-one and onto Y. Consequently, suppose that the system is solvable with some given data (the initial velocity and the right hand side). Then there exists a unique solution of this system for data which are small perturbations of the previous ones. The next result proved in the Appendix of this paper is W2, 2-regularity of solutions of steady Stokes system with mixed boundary condition for sufficiently smooth data.