Search

In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers “Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production” by Tao et al. (2019) [2] and “Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation” by Frassu and Viglialoro (2021) [1] . These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

In this paper we investigate the massive Sine-Gordon model in the ultraviolet finite regime in thermal states over the two-dimensional Minkowski spacetime. We combine recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive quantum field theory over Euclidean spacetimes to construct the correlation functions of the equilibrium state of the Sine-Gordon theory in the adiabatic limit. First of all, the observables of the Sine-Gordon theory are seen as functionals over the free configurations and are obtained as a suitable combination of the S-matrices of the interaction Lagrangian restricted to compact spacetime regions over the free massive theory. These S-matrices are given as power series in the coupling constant with values in the algebra of fields over the free massive theory. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian signature, we prove that these power series converge when evaluated on a generic field configuration. The latter observation implies convergence in the strong operator topology in the GNS representations of the considered states. In the second part of the paper, adapting the cluster expansion technique to the Lorentzian case, we prove that the correlation functions of the interacting equilibrium state at finite temperature (KMS state) can be constructed also in the adiabatic limit, where the interaction Lagrangian is supported everywhere in space., Comment: 55 pages

The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application concerns non-autonomous evolution equations with a new initial-value and a periodic boundary condition for the time variable.

In this paper we extend some previous results on weak pseudo-bosons and on their related bi-coherent states. The role of {\em compatible} functions is discussed in details, and some examples are considered. The pseudo-bosonic ladder operators analysed in this paper generalize significantly those considered so far, and a class of new diagonalizable manifestly non self-adjoint Hamiltonians are deduced., Comment: in press in JMAA

This paper is devoted to an investigation of blow-up phenomena occurring in high-contrast fiber-reinforced composites. When the distance between perfect conductors or between the conductors and the matrix boundary tends to zero, the electric field may appear blow-up. The major objective of this paper is to give a precise description for the singular behavior of such a high concentration in the presence of C 1 , α -inclusions with extreme conductivities. Our results contain the boundary and interior asymptotics of the concentrated field in all dimensions. In particular, the blow-up factor for each dimension is accurately captured.

The purpose of this paper is to present the derivation and mathematical analysis of a new asymptotic model that describes the evolution of medium amplitude internal waves propagating between a flat rigid-lid and a highly variable topography. The smallness assumptions on the topographic variation parameter used in [Communications on Pure & Applied Analysis, 2015, 14 (6): 2203-2230] and in [Asymptotic Analysis, vol. 106, no. 2, pp. 61-98, 2018] are now relaxed and the results of the aforementioned papers are improved and generalized to the complex case of large topography variation. Limiting the flow to one-layer, we also emphasize our model's well-posedness in comparison to the original asymptotic model.

While there have been extensive studies regarding the theory of composition operators in standard Bergman spaces, there have not been many results pertaining to large Bergman spaces due to a lack of useful tools. In this paper, we give the characterizations of the compact differences of composition operators in Bergman spaces with the exponential type weight using a newly defined Riemannian distance. Furthermore, we give a sufficient condition for the question when two composition operators lie in the same component., This paper is the corrected version of the published paper

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] in Ω × ( 0 , T ) , v t = Δ v − v + u in Ω × ( 0 , T ) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0 . Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.

The known family of woven collections of frames have interesting properties and applications. Yet, they cannot be easily constructed. The present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction. The frames { φ i } i ∈ I and { ψ i } i ∈ I for a Hilbert space H are called partition-woven (or briefly P-woven), if there exists a nontrivial subset σ of I such that the family { g i } i ∈ I represented as the triple ( { φ i } , { ψ i } , σ ) and called a P-weaven frame, is a frame defined by g i = φ i if i ∈ σ and g i = ψ i , otherwise. The paper also studies the class of CP-woven collections of frames which is a subset of the P-woven collections of frames and satisfy a stronger condition. The set of all woven frames is a strict subset of the set of all CP-woven frames which is further a strict subset of the set of all P-woven frames. Several examples and counterexamples are studied to show whether or not a pair of frames is P-woven or CP-woven.

In this paper, we study the large time behavior of a chemotaxis model with nonlinear diffusion and consumption { u t = Δ u m − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , v t − Δ v = − v u , where m > 1 . In a previous paper [5] , we have proved the existence and uniform boundedness of global weak solutions for any nonnegative initial data and any m > 1 . In this work, we show that the weak solutions strongly converge to ( 1 , 0 ) in the large time limit.

In the paper, the initial observability of abstract second-order infinite-dimensional dynamical systems is considered. Two cases of observation operator are explored. First, the observation operator is assumed to be bounded. Then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation. Using the frequency-domain method it is proved, that initial observability of second-order system under discussion can be verified by the initial observability conditions for the corresponding simplified first-order system. General results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures. Some special cases are also discussed and practical comments and remarks are given. The paper extends earlier results on initial observability for a class of infinite-dimensional second-order abstract dynamical systems.

Let $A,B\subset\mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A, y\in B\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\{f_1(x)=\lambda x, f_2(x)=\lambda x+c-\lambda,f_3(x)=\lambda x+1-\lambda\}$, where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\cdot K=[0,1]$ if and only if $(1-\lambda)^2\leq c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in[0,1]$, $u=x\cdot y$, where $x,y\in K$., Comment: 12 pages

In this paper we study the priori error analysis for semidiscrete Galerkin finite element approximation of the impulsive optimal control problem governed by linear heat equation with control constraint. We first derive Pontryagin's maximum principle and construct the semidiscrete finite element approximation scheme for our impulsive optimal control problem. Then, the error estimates for optimal control u, the related state y and the adjoint state φ will be obtained. Finally, an efficient algorithm for our numerical experiment is designed, which verifies the theoretical results obtained in this paper.

In this paper we will study an option pricing problem in incomplete markets by an analytic point of view. The incompleteness is generated by the presence of a non- traded asset. The aim of this paper is to use the semigroup theory in order to prove existence and uniqueness of solutions to generalized Black-Scholes type equations that are non-linearly associated with the price of European claims written exclusively on non-traded assets. Then, we derive analytic expressions of the solutions. An approximate representation in terms of a generalized Feynman-Kac type formula is derived in cases where an explicit closed form solution is not available. Numerical examples are also given (see Appendix E ) where theoretical approximations and numerical tests reveal a remarkable agreement.

Whether or not weak solutions of classical Navier-Stokes equations decay to zero in L 2 as time tends to infinity was posed by Leray in his pioneering paper [12] , [13] and has been well solved (see [15] for instance). This paper examines the three dimensional Navier-Stokes equations with damping term | u | β − 1 u and proves that the weak solutions decay to zero in L 2 as time tends to infinity for any β ≥ 1 , by developing local-in-space estimate for weak solutions.

In this paper we consider a competitive economic equilibrium problem where preferences of consumers are expressed by means of a binary relation. The aim is to find a suitable quasi-variational inequality which characterizes the equilibria and, by using tools of variational theory, to study such equilibria. The novelty of this paper consists in the study of an economic equilibrium problem by a variational approach without the need of representing the consumer's preferences by a utility function.

In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C ( K ) must be nontrivial on a subspace isomorphic to c 0 . We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of l 1 and c 0 .

In our previous article, a particular function was stated to give an equality in a particular inequality, proving that inequality was sharp. In this corrigendum it is shown that the function given in the original paper does not have the stated property, but prove that another function does have the desired property, thus preserving the overall result from the original paper.

This paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems, whose unperturbed system has a center at the origin and may have invariant multiply lines. By Melnikov function method, this paper provides the upper and lower bounds for the Poincare cyclicity. In particular, the exact cyclicities of two specific cases are given using analyzing method, one of which has been discussed in the paper [7] .

Using an asymptotically additive sequence of continuous functions as a constrained condition, this paper studies the relations of several constrained ergodic averages for asymptotically additive potentials. Basic properties of constrained maximum ergodic averages are studied. In particular, if the dynamical systems satisfy the specification property, the maximal growth rate of an asymptotically additive potential on the level sets is equal to its constrained maximum ergodic averages and the maximal growth rates on the irregular sets is its unconstrained maximum ergodic averages. Finally, the applications for suspension flows are given in the end of the paper.

In this paper, we are concerned with an initial–boundary value problem for a class of non-Newtonian fluids. The viscosity term of momentum equation possesses singularity and nonlinearity, and the initial vacuum is allowed. The main feature which distinguishes this paper from other related works lies in the fact that we study the global strong solution of the compressible non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals, and show that there exists a unique global-in-time strong solution to the problem while the initial energy is small.

We give a criterion for the global boundedness of integral operators which are known to be locally bounded. As an application, we discuss the global $L^p$-boundedness for a class of Fourier integral operators. While the local $L^p$-boundedness of Fourier integral operators is known from the work of Seeger, Sogge and Stein, not so many results are available for the global boundedness on $L^p({\mathbb R}^n)$. We give several natural sufficient conditions for them., 13 pages. This paper is a substantially reworked version of our paper arXiv:1510.03807, where we now deduce the result of that paper on the global boundedness for Fourier integral operators from a much more general principle for integral operators

In this paper, we consider the following discrete p-Laplacian nonlinear parabolic equations with q-nonlocal reaction { u t ( x , t ) = Δ p , ω u ( x , t ) + λ ∑ y ∈ S | u ( y , t ) | q − 1 u ( y , t ) , ( x , t ) ∈ S × ( 0 , ∞ ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ S × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ≥ 0 , x ∈ S ‾ . Here, S is a network with boundary ∂S. The goal of this paper is to characterize completely the parameters p > 1 and q > 0 to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates for the blow-up solutions are derived.

The paper is devoted to the existence of weak solutions of set-valued stochastic differential equations. Such equations were considered in the authors paper [22] , where existence and uniqueness of strong solutions have been investigated. Now, apart from the existence of weak solutions, the weak compactness in distribution of the set of all weak solutions of set-valued stochastic differential equations, is considered.

It was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles., M.A.G.F. acknowledges the financial support of the Spanish MINECO through a Severo Ochoa FPI scholarship. The work of M.A.G.F. is supported in part by the ERC Starting Grant 633152 and the ICMAT-Severo Ochoa project SEV-2015-0554 . The research of D.G.U. has been supported in part by Spanish MINECO-FEDER Grants MTM2012-31714 and MTM2015-65888-C4-3 and by the ICMAT-Severo Ochoa project SEV-2015-0554 . The research of the third author (RM) was supported in part by NSERC grant RGPIN-228057-2009 . D.G.U. would like to thank Dalhousie University for their hospitality during his visit in the Spring semester of 2014 where many of the results in this paper were obtained.

The paper introduces some notions of variation using the concept of statistical convergence. In particular, the ideas of variation such as regular, translationally regular, O -regular and rapid are presented in statistical context. First, some results are established connecting some statistical variations with their non-statistical counterparts. Then, certain conditions are explored under which various statistical variations can occur. Finally, the paper examined some conditions to connect two different statistical variations.

In this paper, we mainly utilize proper orthogonal decomposition (POD) to reduce the order for the coefficient vector of the classical Crank–Nicolson finite spectral element (CCNFSE) method of the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions so that the reduced-order method maintains all the advantages of the CCNFSE method. Toward this end, we first establish a CCNFSE format with the second-order time accuracy for the two-dimensional (2D) non-stationary Boussinesq equations about vorticity-stream functions and analyze the existence, stability, and convergence of the CCNFSE solutions. And then, by POD, we establish a reduced-order extrapolated Crank–Nicolson finite spectral element (ROECNFSE) method and analyze the existence, stability, and convergence of the ROECNFSE solutions as well as offer the flowchart for seeking ROECNFSE solutions. Finally, we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the ROECNFSE method are further verified. Both theory and method in this paper is new and completely different from the existing reduced-order methods.

This paper is concerned with the asymptotic behavior of the solution for the generalized Hamilton–Jacobi equation u t + f ( D u ) = Δ u . It is known that one dimensional equation v t + C 0 v x 2 = v x x has a self-similar solution v ( x 1 + t ) if v + ≠ v − where v + = v ( t , + ∞ ) , v − = v ( t , − ∞ ) . This kind of solution is called planar diffusion wave in multi-dimension (M-D). In this paper, it is shown that under some smallness conditions, the solutions to the generalized Hamilton–Jacobi equation converge to the above diffusion wave v ( x 1 + t ) with C 0 = 1 2 ∂ 2 f ( ξ 1 , ξ 2 , ⋯ , ξ n ) ∂ ξ 1 2 | ξ 1 = 0 , as time tends to infinity. The convergence rate in time is also obtained.

In this paper, we will investigate the approximations of inverse moments for double-indexed weighted sums of random variables under sub-linear expectations. The asymptotic approximations of inverse moments and the convergence rate of approximations are established under the meaning of upper expectation E ˆ and the lower expectation e ˆ , respectively. Moreover, we carry out some simulations to verify the validity of our theoretical results. The results obtained in the paper improve and extend the corresponding ones of Yang et al. [24] from independent random variables in the classical probability space to END random variables in sub-linear expectation space.

The solution of a parabolic problem is expected to reproduce the basic qualitative properties of the original phenomenon, such as nonnegativity/nonpositivity preservation, maximum/minimum principles and maximum norm contractivity, without which the model might lead to unrealistic quantities in conflict with physical reality. This paper presents characterizations of qualitative properties for a general class of nonlinear parabolic systems. This is a continuation of the authors' research, published in a previous paper in this journal on scalar equations. Cooperativity of the systems plays an essential role in most of the results.

This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type { u t = ∇ ⋅ ( ∇ u m − u q − 1 ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R N ( N ≥ 3 ), m ≥ 1 , q ≥ 2 . As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q > m + 2 N ( [5] ). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case.

In this paper, we consider the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p-Laplacian of the type u t i − ( A α i ( ∇ u ) ) x α = f i ( x , t , u , ∇ u ) , i = 1 , … , N where the main part satisfies some ellipticity and f i satisfies certain growth conditions. We prove boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and a nonlinear iteration procedure of the Moser type in this paper.

In this paper, we present a scheme for solving two-dimensional miscible displacement problems using Eulerian–Lagrangian localized adjoint methods and mixed finite element methods. Since only the velocity and not the pressure appears explicitly in the concentration equation, an Eulerian–Lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation. To linearize and decouple the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this fully discrete problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid using Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O ( h 1 / 2 ) in this paper.

In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ = σ ( x ) and the inner pressure p = p ( x ) . Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20] . In this paper we study existence of nonradial solutions by using the bifurcation method. Let { γ k } k = 2 ∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k ⁎ ⩾ 2 such that in the two dimension case for any k ⩾ k ⁎ , γ k is a bifurcation point, and in the three dimension case for any even k ⩾ k ⁎ , γ k is also a bifurcation point.

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables).

In this paper, we determine the wave front sets of solutions to Schrodinger equations of a harmonic oscillator with sub-quadratic perturbation by using the representation of the Schrodinger evolution operator of a harmonic oscillator introduced in Kato, Kobayashi and Ito (2012) via the wave packet transform. In the previous paper (2011, 2014), the authors have studied the wave front sets for Schrodinger equations with a free particle and a harmonic oscillator and Schrodinger equations with sub-quadratic potential.

In this paper, we consider the model of the Landau-Lifshitz-Maxwell system coupled with spin accumulation. This system consists of the Landau-Lifshitz equation, Maxwell equation describing the magnetic field and a quasi-linear parabolic equation describing the process of spin accumulation. This paper establishes the existence of weak solutions ( M , S , H , E ) to the coupled system in three dimensions and shows the magnetization vector field M is Holder continuous except a closed set Σ, which has locally finite 3-dimensional parabolic Hausdorff measure. Finally, we deduce the Holder continuity of ∇M away from Σ in two special cases.

In this paper we study L p Marcinkiewicz-Zygmund type inequalities c 1 ∑ 1 ≤ j ≤ N w j | g ( x j ) | p ≤ ‖ g ‖ L p ( K ) p ≤ c 2 ∑ 1 ≤ j ≤ N w j | g ( x j ) | p for general exponential sums of the form g ( x ) = ∑ 1 ≤ j ≤ n a j e 〈 λ j , x 〉 , x , λ j ∈ R d , a j ∈ R . One of the main results of the paper asserts that when 1 ≤ p ∞ , K = [ a , b ] and the exponents satisfy relations λ j + 1 − λ j ≥ ϵ , 1 ≤ j ≤ n − 1 , max 1 ≤ j ≤ n ⁡ | λ j | ≤ Λ then Marcinkiewicz-Zygmund type inequalities hold for certain point sets of cardinality N ≤ c n ln 1 p + 1 ⁡ Λ ϵ . Since the dependence of cardinality on the parameters ϵ , Λ appears only in the logarithmic term, this bound is “almost” degree and separation independent. Moreover, the discrete meshes x j and weights w j = x j + 1 − x j are given explicitly and they are universal in the sense that they work for any exponential sum as above. This result will rely on some new degree independent Bernstein-Markov type inequality for exponential sums. Moreover we will extend our considerations to multivariate exponential sums. In addition, it will be shown that much stronger results hold for exponential sums with nonnegative coefficients.

In this paper, we study the ground state solutions to the following Choquard equation involving singular potential: − Δ u + V ( | x | ) u = ( I α ⁎ F ( u ) ) f ( u ) , x ∈ R N , where N ⩾ 3 , α ∈ ( 0 , N ) , I α is the Riesz potential, V is a singular potential with parameter θ ∈ ( max ⁡ { 0 , 4 − N } , 2 ) ∪ ( 2 , 2 N − 2 ) ∪ ( 2 N − 2 , ∞ ) , F is the primitive of f and f satisfies critical growth in sense of the Hardy-Littlewood-Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki-Lions type conditions, we divide this paper into three parts. By virtue of two different kinds of Lions-type theorem and Nehari manifold, some existence results are established.

The first part of the paper concerns with infinite symmetric block Jacobi matrices J with p × p -matrix entries. We present new conditions for general block Jacobi matrices to be self-adjoint and have discrete spectrum. In the second part of the paper a special classes of block Jacobi matrices J X , α are investigated. Our approach here substantially relies on a close connection between Jacobi matrices J X , α from this class and symmetric 2 p × 2 p Dirac operators D X , α with point interactions in L 2 ( ( a , b ) ; C 2 p ) established in our previous papers. In particular, their deficiency indices are related by n ± ( D X , α ) = n ± ( J X , α ) and under a simple additional assumption they are discrete only simultaneously. For block Jacobi matrices of this class we present several conditions ensuring equality n ± ( J X , α ) = k with any k ≤ p . It is worth mentioning that a connection between Dirac and Jacobi operators is employed here in both directions for the first time. In particular, to prove the equality n ± ( J X , α ) = p for J X , α it is first established for Dirac operator D X , α . We also find several conditions for matrix Schrodinger and Dirac operators with point interactions on finite or infinite intervals to have discrete spectrum.

Congruences for total number parts functions associated to ranks of partitions are recently studied in many papers. In particular, Chan, Osburn and the author proved congruences for N T ‾ ( b , k , n ) (resp. N T 2 ‾ ( b , k , n ) ) which denotes the total number of parts in the overpartitions of n with D-rank (resp. M 2 -rank) congruent to b modulo k. In this paper, we replace these congruences by equalities.

This paper studies the asymptotic behavior of the solution to the Cauchy problem of a semilinear wave equation with damping v t t + v t + f ( D v ) = Δ v , x ∈ R n , under some smallness conditions. By applying elementary energy method, we prove the solution of the above equation tends to the planar diffusion wave v ¯ ( x 1 1 + t ) time-asymptotically, where v ¯ ( x 1 1 + t ) is a self-similar solution of the one dimensional equation v ¯ t + C 0 v ¯ x 1 2 = v ¯ x 1 x 1 , v ¯ ( ± ∞ , t ) = v ± , v + ≠ v − , with C 0 = 1 2 ∂ 2 f ( ξ ) ∂ ξ 1 2 | ξ = 0 . In addition, this paper gives the L ∞ time decay rate, namely, ‖ v − v ¯ ‖ L ∞ = O ( 1 ) e 2 ( 1 + t ) − γ 4 , where γ = min ⁡ { 3 , n } .

The history of Bergman spaces goes back to the book [4] in the early fifties by S. Bergman, where the first systematic treatment of the subject was given, and since then there have been a lot of papers devoted to this area. Some standard works here are [5] , [8] , [15] , [28] and the references therein, which contain a broad summary and historical notes of the subject, that frees us from referring to missing details. In their recent works Gonzalez-Cervantes, Luna-Elizarraras and Shapiro [11] , [12] , laid the foundations for the generalization of the theory of Bergman spaces induced by Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields by taking advantage on the intimate connections between harmonic vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator (MT-operator for short). A deeper discussion of the last mentioned relation can be found in [1] . On the setting of general bounded domains in R 3 , we extend the aforementioned study in a very natural way to the case of an introduced v-MT-operator for v ∈ R 3 , proving several properties of induced Bergman spaces and some relative results about Stokes and Borel-Pompieu formulas for v-MT-hyperholomorphic functions, i.e., functions which belong to kernel of the v-MT-operator. In addition, we show that this v-MT operator satisfies a conformal co-variant property. Application of all the above allows to study of Bergman type spaces induced by v-Laplacian vector fields theory, which represents the main goal of this paper.

This paper gives some new results on multi-time first-order PDE constrained control optimization problem in the face of data uncertainty (MCOPU). We obtain the robust sufficient optimality conditions for (MCOPU). Further, we construct an unconstrained multi-time control optimization problem (MCOPU)ϱ corresponding to (MCOPU) via absolute value penalty function method. Then, we show that the robust optimal solution to the constrained problem and a robust minimizer to the unconstrained problem are equivalent under suitable hypotheses. Moreover, we give some non-trivial examples to validate the results established in this paper.

In this paper, we consider the global regularity for a n-dimensional generalized liquid crystal system with fractional diffusion term ζ 2 u for the velocity field, where ζ u ˆ ( ξ ) = | ξ | α g ( | ξ | ) u ˆ ( ξ ) and g 2 ( | ξ | ) ≤ C 0 log ⁡ ( e + | ξ | ) for α ≥ 1 + n 2 . This paper provides a global regularity result for the generalized liquid crystal equations, which extends the result given by Jin, Zhu and Jin to the case β = 0 for n = 2 (Jin et al. (2015) [5] ).

In this paper, we first propose a reaction–diffusion waterborne pathogen model with general incidence rate, incorporating both direct and indirect transmission pathways. Then, using the basic reproduction number we investigate the global dynamical behaviors of the continuous model. By nonstandard finite difference scheme, we derive a discrete counterpart of the continuous model. Then, the global properties of the discretized model are investigated. Finally, we conclude the paper by an example and numerical simulations.

In the paper [10] the L p -realization L p of the matrix Schrodinger operator L u = d i v ( Q ∇ u ) + V u was studied. The generation of a semigroup in L p ( R d , C m ) and characterization of the domain D ( L p ) has been established. In this paper we perturb the operator L p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on L p ( R d , C m ) with domain embedded in W 2 , p ( R d , C m ) . We also study the analyticity, compactness, positivity and ultracontractivity of the semigroup and prove Gaussian kernel estimates. Further kernel estimates and asymptotic behaviour of eigenvalues of the matrix Schrodinger operator are investigated.

In this paper we consider general mean-field backward doubly stochastic differential equations (BDSDEs), i.e., the generator of our mean-field BDSDEs depends not only on the solution but also on the law of the solution. The first part of the paper is devoted to the existence and the uniqueness of solutions for general mean-field BDSDEs under the Lipschitz conditions. We emphasize that in the mean-field BDSDEs we study, from Pardoux-Peng's pioneering work [17] the expected condition that the Lipschitz constant of the coefficient for the backward integral with respect to z is strictly less than 1 is considerably weakened for the Z-component of the law variable of the coefficient. We also establish higher order moment estimates and obtain a comparison theorem of one dimensional general mean-field BDSDEs. Furthermore, we study an existence theorem and a comparison theorem for general mean-field BDSDEs with a continuous coefficient f. Finally, we investigate uniqueness theorems for general mean-field BDSDEs with a uniformly continuous coefficient f.

In this paper, we study existence, uniqueness and asymptotic behavior of the Laplace equation with dynamical boundary conditions on regular non-cylindrical domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator and use form methods in a more general framework to accomplish our goal. A class of non-autonomous elliptic problems with dynamical boundary conditions on Lipschitz domains is also considered in this same context., Comment: This is a new version of a previous paper submitted here in Arxiv. It has substantial changes from the previous one. A whole section was rewritten and some small mistakes were corrected