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Abstract 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 <2. [ABSTRACT FROM AUTHOR]

Abstract: In (2012), Leonetti and Siepe [10] considered solutions to boundary value problems of some anisotropic elliptic equations of the type Under some suitable conditions, they obtained an integrability result, which shows that higher integrability of the boundary datum forces solutions to have higher integrability as well. In the present paper, we consider -obstacle problems of the nonhomogeneous anisotropic elliptic equations under some controllable growth and monotonicity conditions. We obtain an integrability result, which can be regarded as a generalization of the result due to Leonetti and Siepe. [Copyright &y& Elsevier]

Abstract: An error is pointed out in the paper [J.R. Kang, Y.C. Kwun, J.Y. Park, Controllability of second order differential inclusion in Banach spaces, J. Math. Anal. Appl. 285 (2003) 537-550]. By an additional condition the considered system is controllable. [Copyright &y& Elsevier]

Starting from a selfadjoint Schro¨dinger operator A=−d2/dx2+q with a gap G in its spectrum F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267–324] succeed in constructing another Schro¨dinger operator A˜=−d2/dx2+q˜ that is unitarily equivalent (and thus isospectral) to A. As the means they apply come from the Weyl–Titchmarsh theory the connections prove to be intricate, in particular the relation between A and A˜. We show that a central assertion in GST's paper rests substantially on factorizations of the form (A−μ)(A−ν)=B&ast;B, A˜B=BA, μ, ν being numbers in G and B an invertible 2nd order differential operator generated by corresponding eigensolutions of A. Hence A˜=UAU&ast; where U is the unitary operator B&z.sfnc;B&z.sfnc;−1. The operators B and U do not occur explicitly in F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267–324]. [Copyright &y& Elsevier]

Abstract: Using the scalar curvature of the product manifold and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M.T. Mustafa [H. Azad, M.T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl. 333 (2007) 1180–1188] to nonlinear Klein–Gordon equations on the two-dimensional sphere. [Copyright &y& Elsevier]

Abstract: Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant and for each there are cones with normal constant . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results. [Copyright &y& Elsevier]

Abstract: In this paper, we give some characterizations of the integral and Carleson type measure both associated with higher radial derivative for holomorphic functions of and spaces in the unit ball of . As applications, some sharp forms of the relations between Dirichlet type spaces and spaces are shown by means of higher radial derivative, Carleson type measure and constructive methods for the cases of determinism and randomization, respectively. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the polyharmonic system in , for , , with , , but not both equal to 1, where is the polyharmonic operator. Set , , for , we prove the nonexistence of positive solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we study a boundary feedback system of a class of nonuniform undamped Timoshenko beam with both ends free. We give some sufficient conditions and some necessary conditions for the system to have exponential stability. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method. [Copyright &y& Elsevier]

We prove oscillation and nonoscillation theorems for the second order linear differential equation (E) y″+q(t)y=0, where q(t)&ges;0 and locally integrable on [t&ast;,∞). These results are extensions of earlier results of Huang [J. Math. Anal. Appl. 210 (1997) 712–723]. Furthermore, we show that the oscillation criterion established for Eq. (E) can be extended to the delayed differential equation y″+q(t)y(σ(t))=0, where σ(t)&les;t and limt→∞σ(t)=∞. [Copyright &y& Elsevier]

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x″(t)+q(t)x(τ(t))γsgnx(τ(t))=f(t) in the linear (γ=1) and the superlinear (γ>1) cases. [Copyright &y& Elsevier]

Using the method of interval integral average, we establish some new oscillation criteria which improve the oscillation criteria due to James S.W. Wong. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to show that various classes of operators having a common collection of root spaces whose closed linear span equals the entire space have dense sets of common cyclic vectors. [Copyright &y& Elsevier]

Abstract: This paper deals with a parabolic system, cross-coupled via a nonlinear source and a nonlinear boundary flux. We get a necessary and sufficient condition for the existence of non-simultaneous blow-up. In particular, four different simultaneous blow-up rates are obtained in different regions of parameters, described by an introduced characteristic algebraic system. It is observed that different initial data may result in different simultaneous blow-up rates even in the same region of parameters. [Copyright &y& Elsevier]

Abstract: This paper studies the characteristics and structure of the weak surface of the production possibility set. We apply techniques and methods of transferring a polyhedral cone from its intersection form to its sum form, identify an intersection representation of the production possibility set. We give the structure theorem of weak surface of the production possibility set, which includes three complementary slackness conditions. We define the input weak efficiency and output weak efficiency for different DEA models according to the representation of the intersection form. It investigates the characteristics of the weak surfaces, and proves the structure theorems of input weak DEA efficiency and output weak DEA efficiency. The structure theorems establish weighted combination of inputs and outputs that are weak DEA efficient. Numerical examples are provided for illustration. [Copyright &y& Elsevier]

Abstract: W.A. Kirk [W.A. Kirk, Fixed points of asymptotic contractions, J. Math. Anal. Appl. 277 (2003) 645–650] defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the Boyd–Wong fixed point theorem. Subsequently, I.D. Arand̄elović [I.D. Arand̄elović, On a fixed point theorem of Kirk, J. Math. Anal. Appl. 301 (2005) 384–385] established somewhat more general version of Kirk''s result and he gave an elementary proof of it. However, our purpose is to show that there is an error in this proof and, moreover, Arand̄elović''s theorem is false. We also explain how to correct this result. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the following degenerate Kirchhoff type problem { − (b ∫ R N | ∇ u | 2 d x) Δ u + V (x) u = | u | 2 ⁎ − 2 u in R N , u ∈ D 1 , 2 (R N) , where b is a positive constant, V ∈ L N 2 (R N) is a given nonnegative function and 2 ⁎ is the critical exponent. Quite a few papers have been published about the degenerate Kirchhoff type problem with a critical exponent; moreover, this degenerate problem in R N (N ≥ 5) has never been considered so far. We obtain some sufficient conditions on the existence of bounded state solution for this degenerate problem. As to the cases where N ≥ 5 , it is the first time to consider the degenerate problem. [ABSTRACT FROM AUTHOR]

This paper aims to develop the theory of Ford spheres in line with the current theory for Ford circles laid out in a recent paper by S. Chaubey, A. Malik and A. Zaharescu. As a first step towards this goal, we establish an asymptotic estimate for the first moment M 1 , I 2 (S) = ∑ r s , r ′ s ′ ∈ G S c o n s e c 1 2 | s | 2 + 1 2 | s ′ | 2 , where the sum is taken over pairs of fractions associated with 'consecutive' Ford spheres of radius less than or equal to 1 2 S 2 . [ABSTRACT FROM AUTHOR]

Let A , B ⊂ R. Define A ⋅ B = { x ⋅ y : x ∈ A , y ∈ 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) = λ x , f 2 (x) = λ x + c − λ , f 3 (x) = λ x + 1 − λ } , where f 1 (I) ∩ f 2 (I) ≠ ∅ , (f 1 (I) ∪ f 2 (I)) ∩ f 3 (I) = ∅ , and I = [ 0 , 1 ] is the convex hull of K. The main result of this paper is K ⋅ K = [ 0 , 1 ] if and only if (1 − λ) 2 ≤ c. Equivalently, we give a necessary and sufficient condition such that for any u ∈ [ 0 , 1 ] , there exist some x , y ∈ K such that u = x ⋅ y. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Consider general Lebesgue spaces with variable exponent p (.) and the Hardy-Littlewood maximal operator M. There are known sufficient conditions for p (.) which guarantee the boundedness of M on these spaces. These conditions are divided into two categories. The first one controls a local behavior of p (.) and the second one gives sufficient conditions to p (.) at infinity. We put in this paper emphasis to properties of p (.) at infinity. Certain sufficient conditions to p (.) at infinity are known to guarantee the boundedness of the maximal operator on variable Lebesgue spaces. In this paper we find a weaker condition to p (.) which still preserves the boundedness of M. Moreover, it is known that there exist some functions p (.) which have no limit at infinity for which the maximal operator is bounded. We give here a wider class of such functions p (.) with no limit which nevertheless preserves the boundedness of M. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract We study properties of compactly supported, 4 parameter (ρ 12 , ρ 23 , ρ 13 , q) ∈ (− 1 , 1) × 4 family of continuous type 3 dimensional distributions, that have the property that for q → 1 − members of this family tend to members of some 3 dimensional Normal family of distribution. For q = 0 we deal with 3 dimensional generalization of Kesten–McKay distributions. In a very special case when ρ 12 ρ 13 ρ 23 = q all one dimensional marginals are identical, semicircle distributions. We find both all marginals, as well as all conditional distributions. Moreover, we find also families of polynomials that are orthogonalized by these one-dimensional margins and one-dimensional conditional distributions. Consequently, we find moments of both conditional and unconditional distributions of dimensions one and two. In particular, we show that all one-dimensional and two-dimensional conditional moments of, say, order n and are polynomials of the same order n in the conditioning random variables. Finding above mentioned orthogonal polynomials leads us to a probabilistic interpretation of these polynomials. Among them are the famous Askey–Wilson, Al-Salam–Chihara polynomials considered in the complex, but conjugate, parameters, as well as q-Hermite and Rogers polynomials. It seems that this paper is one of the first papers that give a probabilistic interpretation of Rogers (continuous q-ultraspherical) polynomials. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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 Poincaré 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]. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in z and z ‾ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behaviour of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in z and z ‾ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in z and z ‾. [ABSTRACT FROM AUTHOR]

Abstract The celebrated Schoenberg theorem establishes a relation between positive definite and conditionally positive definite functions. In this paper, we consider the classes of real-valued functions P(J) and CP(J), which are positive definite and respectively, conditionally positive definite, with respect to a given class of test functions J. For suitably chosen J , the classes P(J) and CP(J) contain classically positive definite (respectively, conditionally positive definite) functions, as well as functions which are singular at the origin. The main result of the paper is a generalization of Schoenberg's theorem to such function classes. [ABSTRACT FROM AUTHOR]

Abstract This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric domains and some cannot. The aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains. In particular, we show that a Toeplitz operator on the Bergman space A ν p is Fredholm if and only if all of its limit operators are invertible. Even more generally, we show that this is in fact true for all band-dominated operators, an algebra that contains the Toeplitz algebra. Moreover, we characterize compactness and explain how the Berezin transform comes into play. In particular, we show that a bounded linear operator is compact if and only if it is band-dominated and its Berezin transform vanishes at the boundary. For p = 2 "band-dominated" can be replaced by "contained in the Toeplitz algebra". [ABSTRACT FROM AUTHOR]

Abstract In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss a method of constructing separable representations of the C ⁎ -algebras associated to strongly connected row-finite k -graphs Λ. We begin by giving an alternative characterization of the Λ-semibranching function systems introduced in an earlier paper, with an eye towards constructing such representations that are faithful. Our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions. We present a variety of new examples relying on this characterization. We then use some of these methods and a direct limit procedure to identify a faithful separable representation for any row-finite source-free k -graph. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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. [ABSTRACT FROM AUTHOR]

Abstract 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 ε 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. [ABSTRACT FROM AUTHOR]