Linear statistics of matrix ensembles in classical background

Given a joint probability density function of N real random variables, , obtained from the eigenvector–eigenvalue decomposition of N × N random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely, . For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment‐generating function , where denotes expectation value over the orthogonal (β = 1) and symplectic (β = 4) ensembles, in the form one plus a Schwartz function, none vanishing over for the Gaussian ensembles and for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large N asymptotic of the linear statistics from suitably scaled F(·). Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many peak solutions for a biharmonic equation involving critical exponent

In this paper, we study the following biharmonic equation where , K(1) > 0,K′(1) > 0, B₁(0) is the unit ball in (N≥6). We show that the aforementioned problem has infinitely many peak solutions, whose energy can be made arbitrarily large. Copyright © 2016 John Wiley & Sons, Ltd.     

Regular orthogonal basis on Heisenberg group and application to function spaces

In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov–Morrey spaces. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinitely many solutions for a class of fractional Hamiltonian systems via critical point theory

In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems where , , and . The novelty of this paper is that, relaxing the conditions on the potential function W(t,x), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Ground state solution for differential equations with left and right fractional derivatives

In this work, we study the existence of positive solutions for a class of fractional differential equation given by (1) where . Using the mountain pass theorem and comparison argument, we prove that (1) at least has one nontrivial solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Non‐autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

Consider the non‐autonomous equations: where and also These are some non‐autonomous homogeneous rational difference equations of degree one. A reduction in order is consi‐dered. Convergence and monoton character of positive solutions are studied. Copyright © 2013 John Wiley & Sons, Ltd.     

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global‐in‐time strong solutions to the Cauchy problem of the three‐dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data with 1 < p < ∞ and further assume that with 1 < q≤p and , then the global‐in‐time strong solution (u,d) to the nematic liquid crystal flows admits the following temporal decay rate: Here, is a constant unit vector. The highlight of our argument is to show that the ‐norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.     

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy–Sobolev equation with critical growth: provided N > 6 + t, where and Ω is an open bounded domain in , which contains some points x₀ = (0,z₀). Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of solutions for an anisotropic elliptic problem with variable exponent and singularity

In this paper, we study the following anisotropic problem with singularity: where is a bounded domain with smooth boundary. Using the critical point theory, we obtain the existence of weak solutions for the problem under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

On Korn's first inequality for tangential or normal boundary conditions with explicit constants

We will prove that for piecewise C²‐concave domains in Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant . Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple solutions for a modified Kirchhoff‐type equation in RN

In this paper, we study the following modified Kirchhoff‐type equations of the form: where a > 0, b ≥ 0, and . Under appropriate assumptions on V (x) and h(x,u), some existence results for positive solutions, negative solutions, and sequence of high energy solutions are obtained via a perturbation method. Copyright © 2014 John Wiley & Sons, Ltd.     

Global dynamics of a SEIR model with information dependent vaccination and periodically varying transmission rate

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease‐free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo‐rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity

This paper deals with the parabolic–elliptic Keller–Segel system with signal‐dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smooth bounded domain , with initial data satisfying u₀ ≥ 0 and . The chemotactic sensitivity function χ(v) is assumed to satisfy The global existence of weak solutions in the special case is shown by Biler (Adv. Math. Sci. Appl. 1999; 9:347–359). Uniform boundedness and blow‐up of radial solutions are studied by Nagai and Senba (Adv. Math. Sci. Appl. 1998; 8:145–156). However, the global existence and uniform boundedness of classical nonradial solutions are left as an open problem. This paper gives an answer to the problem. Namely, it is shown that the system possesses a unique global classical solution that is uniformly bounded if , where γ > 0 is a constant depending on Ω and u₀. Copyright © 2014 John Wiley & Sons, Ltd.     

ψ‐Hyperholomorphic functions and a Kolosov–Muskhelishvili formula

Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector‐valued) monogenic, an anti‐monogenic and a ψ‐hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed virus infection model with Beddington–DeAngelis infection function and cytotoxic T‐lymphocyte immune response

In this paper, a class of virus infection model with Beddington–DeAngelis infection function and cytotoxic T‐lymphocyte immune response is investigated. Time delay in the immune response term is incorporated into the model. We show that the dynamics of the model are determined by the basic reproduction number and the immune response reproduction number . If , then the infection‐free equilibrium is globally asymptotically stable. If , then the immune‐free equilibrium is globally asymptotically stable. If , then the stability of the interior equilibrium is investigated. We conclude that Hopf bifurcation occurs as the time delay passes through a critical value. Numerical simulations are carried out to support our theoretical conclusion well. Copyright © 2015 John Wiley & Sons, Ltd.     

A Mehta–Ramanathan theorem for linear systems with basepoints

Let be a normal complex projective polarized variety and an H‐semistable sheaf on X. We prove that the restriction to a sufficiently positive general complete intersection curve passing through a prescribed finite set of points remains semistable, provided that at each , the variety X is smooth and the factors of a Jordan–Hölder filtration of are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.     

On the solutions of a max‐type difference equation system

In this paper, we study behavior of the solution of the following max‐type difference equation system: where , the parameter A is positive real number, and the initial values x₀,y₀ are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Multiple non semi‐trivial solutions of systems of Kirchhoff‐type equations with discontinuous nonlinearities in RN

In the present paper, we study the problem of multiple non semi‐trivial solutions for the following systems of Kirchhoff‐type equations with discontinuous nonlinearities (1.1) where F∈C¹(R^N×R⁺×R⁺,R),V∈C(R^N,R), and By establishing a new index theory, we obtain some multiple critical point theorems on product spaces, and as applications, three multiplicity results of non semi‐trivial solutions for (1.1) are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

On some length problems for univalent functions

Let be the class of functions which are analytic in the unit disk . Let C(r) be the closed curve that is the image of the circle |z|=r < 1 under the mapping w = f(z), L(r) the length of C(r), and let A(r) be the area enclosed by the curve C(r). In 1968 D. K. Thomas shown that if , f is starlike with respect to the origin, and for 0≤r < 1, A(r) < A, an absolute constant, then Later, in 1969 Nunokawa has shown that if f is convex univalent, then This paper is devoted to obtaining a related correspondence between f(z) and L(r) for the case when f is univalent. Copyright © 2016 John Wiley & Sons, Ltd.