The truncated Hankel operator is the compression to the model space of Hankel operator on the Hardy space. In this paper, the complex symmetry for a class of truncated Hankel operators is studied and the complete characterization is given. The obtained results show that, the complex symmetry of truncated Hankel operator may be related to the model space only, or to the model space and the symbol function of the operator both.

Quantum Bernoulli noises (QBN) are annihilation and creation operators acting on the space of square integrable Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time and can play an important role in describing the environment of an open quantum system. In this paper, we address a type of perturbations of the canonical unitary involutions associated with QBN. We analyze these perturbations from a perspective of spectral theory and obtain exactly their spectra, which coincide with their point spectra. We also discuss eigenvectors of these perturbations from an algebraic point of view and unveil the structures of the subspaces consisting of their eigenvectors. Finally, as application, we consider the abstract quantum walks driven by these perturbations and obtain infinitely many invariant probability distributions of these walks.

For $\beta>1$, let $T_\beta$ be the $\beta$-transformation defined on $[0, 1)$. We study the sets of points whose orbits of $T_\beta$ have uniform Diophantine approximation properties. Precisely, for two given positive functions $\psi_1, \ \psi_2:{\Bbb N}\rightarrow{\Bbb R}^+$, define ${\cal L}(\psi_1):=\left\{x\in[0, 1]:T_\beta^n x<\psi_1(n), \mbox{ for infinitely many $n\in{\Bbb N}$}\right\}, $ ${\cal U}(\psi_2):=\left\{x\in [0, 1]:\forall \ N\gg1, \ \exists \ n\in[0, N], \ s.t. \ T^n_\beta x<\psi_2(N)\right\}, $ where $\gg$ means large enough. We calculate the Hausdorff dimension of the set ${\cal L}(\psi_1)\cap{\cal U}(\psi_2)$. As a corollary, we obtain the Hausdorff dimension of the set ${\cal U}(\psi_2)$. Our work generalizes the results of ^[4] where only exponential functions $\psi_1, \ \psi_2$ were taken into consideration.

Let ${\cal A}$ be a unital $*$-algebra with the unit $I$ and let $\xi\in{\Bbb C}\setminus\{0\}.$ Assume that ${\cal A}$ contains a nontrivial projection $P$ which satisfies $X{\cal A} P=0$ implies $X=0$ and $X{\cal A}(I-P)=0$ implies $X=0.$ Then $\phi$ is a nonlinear $\xi$-$*$-Jordan-type derivations on ${\cal A}$ if and only if $\phi$ is an additive $*$-derivation and $\phi(\xi A)=\xi\phi(A)$ for all $A\in{\cal A}.$

Fractional quantum mechanics is a generalization of standard quantum mechanics, which is described by fractional Schrödinger equation with fractional Riesz derivative operator. In this paper, we consider a free particle moving in a two-dimensional infinite square well, By using Lévy path integral method, the wave function and energy eigenvalue of the two-dimensional infinite square well are obtained. Then the perturbation expansion method is used to study the two-dimensional infinite square well with $\delta$ function, and the corresponding energy-dependent Green's function is obtained.

In this paper, we study the Well-posedness of weak solutions for Cahn-Hilliard-Brinkman system with general nonlinear conditions, analyze the asymptotic behavior of the solutions, and obtain the existence of global attractors in ${H^s}(\Omega ){\kern 1pt} {\kern 1pt} (s = 1, 2, 3, 4)$ in virtue of asymptotic energy estimates.

Karman vortex street is a kind of periodic traveling wave solution. In this paper, the vortex method is used to study the existence of Karman vortex street for two-dimensional incompressible Euler equation. We construct a family of Karman vortex street type vortex patch solutions by using the variational method, and analyze the asymptotic behavior of the family of solutions. When the vortex strength parameters tend to infinity, the family of solutions constitute a desingularization of vortex street type point vortex pairs.

Based on the boundary layer function method, a class of singularly perturbed problems with integral boundary conditions in weakly nonlinear critical cases are studied. In the framework of this paper, we not only construct the asymptotic expansion of the solution of the original equation, but also prove the uniformly effective asymptotic expansion. At the same time, we give an example to illustrate our results, The comparison images of approximate solution and exact solution under different small parameters are drawn.

In this paper, we investigate the blow-up of the smooth solutions to the quantum Navier-Stokes-Landau-Lifshitz systems(QNSLL) in the domains $\Omega \subseteq \mathbb{R} ^n(n =1, 2)$. We prove that the smooth solutions to the QNSLL will blow up in finite time in the domains half-space $\mathbb{R} _+^n$, whole-space $\mathbb{R} ^n$ and ball. The paper also shows that the blow-up time of the smooth solutions in half-space or whole-space only depends on boundary conditions, while the the blow-up time of the smooth solutions in the ball depends on initial data and boundary conditions. In particular, the above conclusions are also valid for NSLL systems.

In this paper, we consider a class of one dimensional hyperbolic mean curvature flow, which is related to the Lagrangian parabolic mean curvature flow. First we derive the point Lie symmetries of this flow and obtain several ordinary differential equations. The existence of solutions is investigated. Finally, the global BV solutions, the lifespan for classical solutions and global existence of smooth solutions for this hyperbolic mean curvature flow are studied in detail.

This article concerns the following Klein-Gordon-Maxwell system $ \left\{ \begin{array}{*{35}{l}} -\Delta u+\lambda V(x)u-K(x)(2\omega +\phi )\phi u=f(x,u), & x\in {{\mathbb{R}}^{3}}, \\ \Delta \phi =K(x)(\omega +\phi ){{u}^{2}}, & x\in {{\mathbb{R}}^{3}}, \\\end{array} \right. $ where $ \omega> 0 $ is a constant and $ \lambda\geq1 $ is a parameter. When the nonlinearity satisfies asymptotically linear growth at infinity, the existence result of positive solutions for the system is obtained via variational methods. Our result completes some recent works concerning the existence of solutions of this system.

A nonlinear von Kármán equation with partial free boundary is considered. The equation is viewed as a mathematical model for suspension bridges with large deformation. The buckling loads, which carry a nonlocal effect into the model, are introduced. Uniqueness and multiplicity results are obtained by analyzing the critical points of the energy functionals.

We investigate the global structures and wave interactions of non-selfsimilar solutions for two dimensional non-homogeneous Burgers equation, where the initial data has three constant states, separated by two disjoint circles. We first get the expressions of solutions of shock waves and rarefaction waves starting from the initial discontinuity. Secondly, we discuss the interactions of these elementary waves and find some new phenomena that the time of the interaction of shock wave and rarefaction wave has no critical point at which the structures begin to change, which are different from the homogeneous case. Finally, we construct the global structures of the non-selfsimilar solutions and find the new asymptotic behavior that the diameter of the region of elementary waves is bounded.

In this paper, we study the Riemann problem for the Euler equations of compressible fluid flow with a composite source term. The source can cover a Coulomb-like friction or a damping or both. Different from the homogeneous system, Riemann solutions of the inhomogeneous system are non self-similar. Concentration and cavitation in the pressureless limit of solutions to the Riemann problem for the Euler equations of compressible fluid flow with a composite source term are investigated in detail as the adiabatic exponent tends to one. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a composite source term, and the intermediate density between the two shocks tends to a weighted $ \delta$-mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a composite source term, whose intermediate state between the two contact discontinuities is a vacuum state.

The long-time behavior of solution to Navier-Stokes equations with weak damping is studied in this paper. With some assumptions on the external force and initial datum, the global wellposedness and regularity of weak solution are proved by Galerkin method, and the existence and convergence of pullback attractors are established finally.

In this paper, we will study a class of fractional stochastic heat equation of the form $\frac{ \partial}{ \partial t} u(t, x)={\cal D}_\delta^\alpha u(t, x)+g( u(t, x)) \frac{\partial^2}{\partial t\partial x}w_\rho(t, x), \quad 0<t\leq T, \quad x\in{\Bbb R} , $ with $T>0$, where ${\cal D}_\delta^\alpha$ denotes a nonlocal fractional differential operator with $\alpha\in(1, 2]$ and $|\delta|\leq2-\alpha$, and $\frac{\partial^2}{\partial t\partial x}w_\rho(t, x)$ is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet $w_\rho(t, x)$, we prove the existence, uniqueness and Hölder regularity of the solution. Upper and lower moment bounds for the solution are also derived.

In this paper, we investigate the existence and nonexistence of traveling wave solution (TWS) for a reaction-diffusion dengue epidemic model with time delays. Firstly, by introducing an auxiliary system and combining with Schauder's fixed-point theorem, it is proved that when the basic reproduction number ${\cal R}_0>1$, $c>c_\ast$, the system admits a positive bounded monotone TWS. Secondly, when ${\cal R}_0>1$, $0<c<c_\ast$, by means of two-sided Laplace transform, the nonexistence of TWS is obtained. When ${\cal R}_0\leq1$, there is no TWS for any wave speed $c>0$ with the aid of comparison principle and contradictory arguments. Lastly, the effects of incubation period and individual diffusion on the threshold speed $c_\ast$ are studied theoretically and numerically. The conclusion shows that prolonging the length of incubation period or decreasing the individual diffusion will reduce the speed of disease transmission.

In this paper, we study the inverse problems for the determination of the shape of a human vocal tract, which consist in recovering the radius of a human vocal tract from either the absolute sound pressure at the lips at all positive frequencies or the poles of the sound pressure at the lips in the complex plane. Two uniqueness theorems are proved, and the corresponding reconstruction algorithms are provided.

It studies an induction heating model described by Maxwell's equations coupled with a heat equation. In the induction heating model, it assumes a nonlinear relation between the magnetic field and the magnetic induction field, and the electric conductivity is temperature dependent. It presents a fully discrete H-based finite element scheme in time and space and discusses its solvability. Moreover, it proves the fully discrete solution converges to a weak solution of the continuous problem. Finally, theoretical results are supported by some numerical experiments.

In this paper, we consider the long-term asymptotic behavior of a class of finite-dimensional optimal control problems with exponential weights. The main method of this article is to decouple the Hamilton system of optimal control problems, based on the Pontryagin maximum principle and the algebraic Riccati theory. The exponential turnpike properties of optimal trajectory and optimal control are established under two different conditions.

This paper studies the optimal investment and reinsurance strategies of a loss-averse insurer, considering inflation risk and minimum performance guarantee. Assume that the insurance surplus is correlated with the inflation-indexed bond process and the stock price process. The investment options of the insurer include inflation-indexed bond, stock and risk-free asset. Meanwhile, the insurer can purchase proportional reinsurance to diversify risks. Under the criterion of maximizing the expected S-shaped utility, the detailed expressions of the optimal investment and reinsurance strategies are derived by using the martingale method, and the influences of parameter variations on the investment and reinsurance strategies are analyzed by numerical simulations.