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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: 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]

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 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]

When studying non-symmetric nonlocal operators on R d : L f ( x ) = ∫ R d ( f ( x + z ) − f ( x ) − ∇ f ( x ) ⋅ z 1 { | z | ⩽ 1 } ) κ ( x , z ) | z | d + α d z , where 0 < α < 2 , d ⩾ 1 , and κ ( x , z ) is a function on R d × R d that is bounded between two positive constants, it is customary to assume that κ ( x , z ) is symmetric in z . In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption κ ( x , z ) = κ ( x , − z ) . In fact, we allow the kernel κ to be time-dependent and x → κ ( t , x , z ) to be only locally β -Hölder continuous with Hölder constant possibly growing at a polynomial rate in | z | . We also derive gradient estimate when β ∈ ( 0 ∨ ( 1 − α ) , 1 ) as well as fractional derivative estimate of order θ ∈ ( 0 , ( α + β ) ∧ 2 ) for the heat kernel. Moreover, when α ∈ ( 1 , 2 ) , drift perturbation of the time-dependent non-local operator L t with drift in Kato's class is also studied in this paper. As an application, when κ ( x , z ) = κ ( z ) does not depend on x , we show the boundedness of nonlocal Riesz's transformation: for any p > 2 d / ( d + α ) , ‖ L 1 / 2 f ‖ p ≍ ‖ Γ ( f ) 1 / 2 ‖ p , where Γ ( f ) : = 1 2 L ( f 2 ) − f L f is the carré du champ operator associated with L , and L 1 / 2 is the square root operator of L defined by using Bochner's subordination. Here ≍ means that both sides are comparable up to a constant multiple. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [ABSTRACT FROM AUTHOR]

This paper examines the Euler–Rodrigues formula in dual 3-space D 3 by analyzing its variations such as vectorial form, exponential map, point–line theory and quaternions which have some intrinsic relations. Contrary to the Euclidean case, dual rotation in dual 3-space corresponds to a screw motion in Euclidean 3-space. This paper begins by explaining dual motion in terms of the given dual axis and angle. It will then go on to express dual Euler–Rodrigues formula with algebraic methods. Furthermore, an application of dual Euler–Rodrigues formula to point–line geometry is accomplished and point–line displacement operator is obtained by dual Euler–Rodrigues formula. Finally, dual Euler–Rodrigues formula is presented with the help of dual Euler–Rodrigues parameters that is expressed as a dual quaternion. [ABSTRACT FROM AUTHOR]

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L 2 -spaces. However, in subsequent papers, we intend to extend the study to L p -spaces ( 1 < p < ∞ ) . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions to a class of p ( x ) -curl systems arising in electromagnetism. The results obtained in this paper extend several contributions concerning the p -curl operator and we focus on new existence results which are due to the presence of variable exponent. To our best knowledge, our results are new even in the semilinear case. [ABSTRACT FROM AUTHOR]

In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X . We also explore the left symmetry of Birkhoff–James orthogonality of linear operators defined on X . Using some of the related results proved in this paper, we finally prove that T ∈ L ( l p 2 ) ( p ≥ 2 , p ≠ ∞ ) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator. [ABSTRACT FROM AUTHOR]

In this paper we study a particular class of bilinear multipliers which are given by Littlewood-Paley decompositions. In the first part of the paper, we show that if Κ (ξ - η) is a bilinear multiplier for (p, q, r), 1≤ p, q≤ ∞ satisfying the Hölder's condition 1p+1q=1r and have support inside [0, 1), then its periodization Κ ♯ (ξ) =Φ j⊂ ZΚ (ξ - j) is also a bilinear multiplier for the same triplet (p, q, r). Further, we show that for a given triplet (p, q, r) of exponents outside the local L2-range, there exists sequence {Κ j}j⊂ Z of uniformly bounded bilinear multipliers so that the function σ (ξ) =Φ j⊂ ZΚ j (ξ) is not a bilinear multiplier for the triplet (p, q, r). In the second part, we describe several results for bilinear multipliers of the type m (ξ, η) which are similar to the first part in nature. In particular, we point out that the results described by P. Honzik (2014) [13] can be generalized to a more general setting. [ABSTRACT FROM AUTHOR]

We study the 1-d isentropic Euler equations with time-dependent damping { ∂ t ρ + ∂ x ( ρ u ) = 0 , ∂ t ( ρ u ) + ∂ x ( ρ u 2 ) + ∂ x p ( ρ ) = − μ ( 1 + t ) λ ρ u , ρ | t = 0 = 1 + ε ρ 0 ( x ) , u | t = 0 = ε u 0 ( x ) . In a previous paper [8] , we have proven that, when λ = 1 , μ > 2 , the 1-D Euler equations have global existence of small data solutions. However in this paper, we will show that, when the damping, with respect to time, decays faster or equal to 2 1 + t , the C 1 solution of the above system will blow up in finite time. More precisely, when λ = 1 , 0 ≤ μ ≤ 2 or λ > 1 , μ ≥ 0 , we will give a finite upper bound for the lifespan. Combining the results in this paper and [8] , we see that, when the damping decays with time like μ ( 1 + t ) λ , the critical exponents for λ , μ to separate the global existence and finite-time blow up of small data solutions are λ = 1 , μ = 2 . [ABSTRACT FROM AUTHOR]

In this paper we investigate a new class of bounded operators called weighted shifts on directed trees introduced recently in [11] . This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour and similarity properties in order to obtain cyclicity results. We also show that a weighted backward shift operator is cyclic if and only if there is at most one zero weight. [ABSTRACT FROM AUTHOR]

In this paper, the consistency of the nearest neighbor estimator of the density function based on widely orthant dependent (WOD, in short) samples is investigated. The convergence rate of strong consistency, the complete consistency, the uniformly complete consistency and uniformly strong consistency of the nearest neighbor estimator of the density function based on WOD samples are established. Our results established in the paper generalize or improve the corresponding ones for independent samples and some negatively dependent samples. [ABSTRACT FROM AUTHOR]

In this paper we employ a C-algebra approach to the study of Fredholm properties of differential and pseudodifferential operators on the half line. The algebra investigated in this paper has compact commutators, so the Gelfand theory applies to the quotient algebra, and we obtain an explicit description of the corresponding maximal ideal space and necessary and sufficient conditions for Fredholmness of operators in the algebra. [ABSTRACT FROM AUTHOR]

Let B denote the unit ball in RN with boundary S. For a non-negative C2 subharmonic function f on B and ∈S, we define the Lusin square area integral Sα(,f) by Sα(f)=[Γα(1-|x|)2-NΔf2(x)dx]12,where for α>1, Γα={x∈B:|x-|<α(1-|x|)} is the non-tangential approach region at ∈S, and Δ is the Laplacian in RN. In the paper we will prove the following: Let f be a non-negative subharmonic function such thatfpois subharmonic for somepo>0. Iffpp=sup0po, then for everyα>1,Sα(•,f)p≤Aα,pfpfor some constantAα,pindependent of f. The above result includes the known results for harmonic or holomorphic functions in the Hardy Hp spaces, as well as for a system F=(u1,...,uN) of conjugate harmonic functions for which it is known that |F|p=(Σuj2)p/2 is subharmonic for p≥(N-2)/(N-1),N≥3. We also consider analogues of the functions g and g\* of Littlewood-Paley, and introduce the function gλ\*, λ>1, defined bygλ\*(,f)=[B(1-|y|)Δf2(y)Kλ(y,)dy]12,whereKλ(y,)=(1-|y|)(λ-1)(N-1)|y-|λ(N-1). In the paper we prove that the inequality gλ\*(•,f)p≤Cpfp holds for all λ≥N/(N-1) when p≥2, and for λ>3-p whenever 1(2N-3)/(N-1). [ABSTRACT FROM AUTHOR]

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [11] proved that the C\*-algebra C\*(F∞) of the free group F∞ on countably many generators characterises relative weak injectivity for pairs of C\*-algebras by means of the maximal tensor product. One of the main results of this paper shows that C\*(F∞) also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products [9,10]. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for null curves in E13 to be W-slant helix in terms of their curvature functions. Mainly, throughout this paper relationships between the null W-slant helices and their pseudo spherical images are obtained. Furthermore, some illustrative examples of the null W-slant helices and their pseudo spherical indicatrices in E13 are demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we deal with the functional equation which is connected to Hermite quadrature rule. It is easy to note that particular cases of this equation generalize many well known functional equations connected to quadrature rules and mean value theorems. Thus the set of solutions is too complicated to be described completely and therefore we prove that (under some assumptions) all solutions of the above equation must be polynomials. We obtain the aforementioned result using a lemma proved by M. Sablik, however this lemma works only in case . Taking , we obtain the following equation which is also solved in the paper. [Copyright &y& Elsevier]

Abstract: This paper addresses periodic wavelet bi-frames associated with general expansive matrices. Periodization is an important method to obtain periodic wavelets from wavelets on . MEP and MOEP provide us with criteria for the construction of wavelet bi-frames on . Based on periodization techniques, MEP and MOEP, periodic wavelet bi-frames associated with the dyadic matrix have been constructed. However, the problem of constructing periodic wavelet bi-frames associated with general expansive matrices is still open. The geometry of a general expansive matrix is much more complicated than the dyadic matrix. In this paper, with the help of quasi-norms, MEP and MOEP we construct periodic wavelet bi-frames associated with general expansive matrices. [Copyright &y& Elsevier]

Abstract: In this paper some characterizations and properties of domains in with boundary accessible externally by some circular cones are given. It is shown that the boundary of such domains is also conically accessible from the interior. It is also characterized -accessible domains of , which are biholomorphically equivalent to the open Euclidean ball. The paper is also a continuation of a concept of V.V. Starkov (2011) [13] and refers to papers of D.A. Brannan and W.E. Kirwan (1969) [3], W. Ma and D. Minda (1991) [10], J. Stankiewicz (1966) [11], and T. Sugawa (2005) [15] connected with a planar case. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain existence of ground state solutions and ground state sign-changing solutions for the following Schrödinger equation where , , and are sign-changing functions satisfying very mild assumptions. The proofs rely on a new perturbed method which we design in this paper. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the long-time behaviour of the solutions of the damped wave equation on . This equation has travelling front solutions of the form . Gallay and Joly have proved in Gallay and Joly (2009) [7] that when the nonlinearity is of bistable type, if the initial data are sufficiently close to the profile of a front for large , the solution of the damped wave equation converges uniformly on to a travelling front as . In this paper, we establish a global stability result under more general assumptions on the function , which include in particular nonlinearities of combustion type. We impose, however, more restrictive conditions on the initial data in the region . We also apply our method to the case of a monostable pushed front. [Copyright &y& Elsevier]

Abstract: In this paper, the divergence theorem is proved by the Kurzweil–Henstock approach. The physical definition of the divergence of a vector field is used, instead of the usual definition used in mathematics papers. Sufficient conditions for the existence of the divergence of a vector field on -dimensional manifolds in are given. Concepts of strong differentiability are used in the sufficient conditions. [Copyright &y& Elsevier]

Abstract: This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations with Robin boundary conditions on an exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations. Our result in the general case and the special case is shorter than that of the initial–boundary value problem for one dimensional general quasilinear wave equations with Dirichlet boundary conditions. The results in this paper are not the trivial generalization of that in the case of Dirichlet boundary conditions. The lower bound estimates of life span of classical solutions to initial–boundary value problems are consistent with the actual physical meaning. The physical phenomenon also explains our results. [Copyright &y& Elsevier]

Abstract: The notation means the finite sum of all numbers as ranges over the integers that divide including and . The purpose of this paper is to define a theory of summability suggested by this type of sum and an associated theory of convergence. This paper studies five sequence spaces that are related to the arithmetical summability thus defined. [Copyright &y& Elsevier]

Abstract: This paper considers stochastic population dynamics driven by Lévy noise. The contributions of this paper lie in that: (a) Using the Khasminskii–Mao theorem, we show that the stochastic differential equation associated with our model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such a model. [Copyright &y& Elsevier]

Abstract: Let be i.i.d. random variables with partial sums , . The now classical Baum–Katz theorem provides necessary and sufficient moment conditions for the convergence of for fixed . An equally classical paper by Heyde in 1975 initiated what is now called precise asymptotics, namely asymptotics for the same sum (for the case and ) when, instead, . In this paper we extend a result due to Klesov (1994), in which he determined the convergence rate in Heydeʼs theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we study the stability of stationary solutions w for the Navier–Stokes flows in an exterior domain with zero velocity at infinity. With suitable assumptions of w, by the works of Chen (1993), Kozono–Ogawa (1994) and Borchers–Miyakawa (1995), if then one can obtain where is a solution of the Navier–Stokes equations with the initial condition . In this paper, we will prove that for any if belongs to then one has [Copyright &y& Elsevier]

Abstract: In this paper we prove for , where k is an integer in , the existence of an initial value ψ, odd with respect to the k first coordinates, and with , such that the resulting solution of is global. In the case and , it is known that the solution u with the initial value blows up in finite time if either sufficiently small or sufficiently large. The result in this paper extends a similar result of Cazenave, Dickstein, and Weissler in the case , i.e. with and . [Copyright &y& Elsevier]

Abstract: This paper deals with a generating function of the Jacobi polynomials that satisfies the following properties (I) and (II). (I) The generating function is the kernel of an integral operator that is unitary. (II) The image of the unitary operator is a reproducing kernel Hilbert space of analytic functions and the reproducing kernel is given as a special value of the generating function above. A generating function that satisfies (I) is given in Watanabe (1998) . The purpose of this paper is to give a generating function that satisfies (I) and (II). From a group theoretical point of view, a similar construction for zonal spherical functions is given in Watanabe (2006) . [Copyright &y& Elsevier]

Abstract: We investigate the problem of enhancing the stability of a coupled transport–diffusion system with Dirichlet actuation and Dirichlet measurement. In the recent paper [H. Sano, Neumann boundary control of a coupled transport–diffusion system with boundary observation, J. Math. Anal. Appl. 377 (2011) 807–816], we treated the stabilization problem for the case with Neumann actuation and Dirichlet measurement, where the variable transformation of the state is performed by using the fractional power of an unbounded operator. However, we cannot use the similar transformation for the case with Dirichlet actuation and Dirichlet measurement, since it brings an ill-posed expression of the system. So, we use an algebraic approach for the formulation of the system. In this paper, it is shown that a reduced-order model with a finite-dimensional state variable is controllable and observable. The fact enables us to construct a finite-dimensional stability-enhancing controller for the original infinite-dimensional system by using a residual mode filter (RMF) approach. The novelty of this paper is the structure that the controller contains the dynamics with respect to the control variable. As a result, the state vector of the resulting closed-loop system includes the control variable as its entry. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study sustainability of the pseudo-linear superposition principle, that is, of the nonlinear superposition principle (NLSP) observed in the pseudo-analysisʼ framework. The focus of this paper is on the so-called generated partial differential equation that is, for the purpose of this study, based on the stochastic Burgerʼs equation. [Copyright &y& Elsevier]

Abstract: Let be a metric measure space satisfying the upper doubling and the geometrically doubling conditions in the sense of T. Hytönen. In this paper, the authors prove that the boundedness of a Calderón–Zygmund operator T on is equivalent to either of the boundedness of T from the atomic Hardy space to or from to . To this end, the authors first establish an interpolation result that a sublinear operator which is bounded from to and from to for some is also bounded on for all . A main tool used in this paper is the Calderón–Zygmund decomposition in this setting established by B.T. Anh and X.T. Duong. [Copyright &y& Elsevier]

Abstract: In a recent paper, we proved that assuming some initial data lead to a singularity for the 3D Navier–Stokes equations, there are also initial data with the minimal -norm which produce a singularity and the set of such data is compact up to translations and the natural scaling of the equation. The purpose of this paper is to analyze a more general case where the set of initial data potentially leading to a singularity is on a sphere centered at non-zero initial data leading to a global solution. [Copyright &y& Elsevier]

Abstract: Stability in distribution of stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems. There are several papers which study this stability for stochastic differential equations with Markovian switching and stochastic differential delay equations with Markovian switching technically. In our paper, we are concerned with the general neutral stochastic functional differential equations with Markovian switching and we derive the sufficient conditions for stability in distribution. At the end of our paper, one example is established to illustrate the theory of our work. [Copyright &y& Elsevier]

Abstract: In the present paper we prove some representation theorems for t-Wright convex functions, as a consequence of a support theorem, which was proved by the author in earlier paper. [Copyright &y& Elsevier]

Abstract: This paper concerns the optimal stopping problem for discrete time multiparameter stochastic processes with the index set . In the classical optimal stopping problems, the comparisons between the expected reward of a player with complete foresight and the expected reward of a player using nonanticipating stop rules, known as prophet inequalities, have been studied by many authors. Ratio comparisons between these values in the case of multiparameter optimal stopping problems are studied by Krengel and Sucheston (1981) and Tanaka (2007, 2006) . In this paper an additive comparison in the case of finite stage multiparameter optimal stopping problems is given. [Copyright &y& Elsevier]

Abstract: In this paper, we revisit elliptic estimates invariant under domain expansion. We improve the invariant elliptic estimates in the previous paper [Y. Cho, T. Ozawa, Y. Shim, Calc. Var. PDE 34 (2009) 321–339] via gradient estimate and discuss an application to the Lamé system. [Copyright &y& Elsevier]

Abstract: This paper studies exponential stabilization of distributed semilinear systems. The paper (i) gives a constrained feedback control that ensure the exponential stabilizability and (ii) shows that this control is the unique solution of an appropriate minimization problem. Examples of hyperbolic partial equations are provided. [Copyright &y& Elsevier]

Abstract: In this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time, generalizing the classical result, which assumes Lipschitz-continuity. We start showing a general existence and uniqueness result for a general sort of hyperbolic equations. Then, we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones. We conclude the paper making an analysis of the corrector. Firstly, we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous. This result was known for non-time dependent coefficients. After, we show, with a counterexample, that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded. [Copyright &y& Elsevier]

Abstract: It is well known that the sequence of Bell numbers ( being the number of partitions of the set ) is the sequence of moments of a mean 1 Poisson random variable τ (a fact expressed in the Dobiński formula), and the shifted sequence is the sequence of moments of . In this paper, we generalize these results by showing that both and (where is the number of m-partitions of , as they are defined in the paper) are moment sequences of certain random variables. Moreover, such sequences also are sequences of falling factorial moments of related random variables. Similar results are obtained when is replaced by the number of ordered m-partitions of . In all cases, the respective random variables are constructed from sequences of independent standard Poisson processes. [Copyright &y& Elsevier]

Abstract: We study the Neumann boundary stabilization problem of a coupled transport-diffusion system in the case where the observation is done at the boundary. In the recent paper of Sano and Nakagiri [H. Sano, S. Nakagiri, Stabilization of a coupled transport-diffusion system with boundary input, J. Math. Anal. Appl. 363 (2010) 57–72], we treated the stabilization problem for the case with Neumann boundary control and distributed observation. The novelty of this paper is the formulation of the boundary observation equation in a Hilbert space. We have an interesting result of its being expressed by using an -bounded operator with . Moreover, it is shown that a reduced-order model with a finite-dimensional state variable is controllable and observable. This means that one can always construct a finite-dimensional stabilizing controller for the original infinite-dimensional system by using a residual mode filter (RMF) approach. [Copyright &y& Elsevier]

Abstract: In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439–3444]. [Copyright &y& Elsevier]

Abstract: This paper studies two widely used stochastic non-autonomous logistic models. For the first system, sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The critical number between weak persistence and extinction is obtained. For the second system, sufficient criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and stochastic permanence are established. The critical number between weak persistence in the mean and extinction is obtained. It should be pointed out that this research is systematical and complete. In fact, the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. By using the variant fountain theorem, we obtain a new criterion for guaranteeing that second-order Hamiltonian systems has infinitely many homoclinic solutions. Recent results from the literature are generalized and significantly improved. An example is also given in this paper to illustrate our main results. [Copyright &y& Elsevier]

Abstract We prove that all possible values of the deficiency indices r , s of a symmetric (formally self-adjoint) differential expression M with complex coefficients which satisfy the well known inequalities are realized. Subject to these inequalities, in 1974 Kogan and Rofe-Beketov showed that all values of r , s which differ by no more than 1 can be realized, in 1978, 1979 Gilbert showed that their difference can be arbitrarily large provided the order of M is large enough. In this paper we show that all values of r and s are realized. [ABSTRACT FROM AUTHOR]

Abstract We obtain a simple two-sided inequality for the ratio L ν (x) / L ν − 1 (x) in terms of the ratio I ν (x) / I ν − 1 (x) , where L ν (x) is the modified Struve function of the first kind and I ν (x) is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for I ν (x) / I ν − 1 (x) to immediately deduce bounds for L ν (x) / L ν − 1 (x). We note some consequences and obtain further bounds for L ν (x) / L ν − 1 (x) by adapting techniques used to bound the ratio I ν (x) / I ν − 1 (x). We apply these results to obtain new bounds for the condition numbers x L ν ′ (x) / L ν (x) , the ratio L ν (x) / L ν (y) and the modified Struve function L ν (x) itself. Amongst other results, we obtain two-sided inequalities for x L ν ′ (x) / L ν (x) and L ν (x) / L ν (y) that are given in terms of x I ν ′ (x) / I ν (x) and I ν (x) / I ν (y) , respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce BBI spaces (“big balls of itself”), which based on the notion of BPI spaces (“big pieces of itself”) used by David and Semmes to study self-similarity. We prove that the “self-similar” construction described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior. We apply this result to self-conformal sets satisfying the WSC and prove that positive Lebesgue measure implies nonempty interior for such sets. This generalizes Zerner's corresponding result for self-similar sets. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce some local versions of Bishop–Phelps–Bollobás type property for operators. That is, the function η which appears in their definitions depends not only on a given ε > 0 , but also on either a fixed norm-one operator T or a fixed norm-one vector x. We investigate those properties and show differences between local and uniform versions. [ABSTRACT FROM AUTHOR]