The paper focuses on the quasi-periodicity of meromorphic functions with $Ac$-shared functions. At the latter part of the paper, we give a uniqueness theorem for quasi-periodic functions. The proofs of the uniqueness theorem of additive quasi-periodic function are simplified by converting some equations into vectors and using MATLAB for some tedious procedures.

In this paper, we study the Hardy operator $H_{\mathbb{R}} $ defined on the space $ L^2(\mathbb{R})$. By introducing an orthogonal basis associated with Laguerre polynomials, we prove that the operator $ I - H_{\mathbb{R}} $ acts as an isometric shift on $ L^2(\mathbb{R})$. We further analyze the properties of its adjoint and establish that $ I - H_{\mathbb{R}}^*$ is also an isometric shift, satisfying $(I - H_{\mathbb{R}})(I - H_{\mathbb{R}}^*) = I,$ which implies that the operator norm of $ H_{\mathbb{R}}$ is 2. Moreover, we investigate the spectral structure of $H_{\mathbb{R}} $ in detail, proving that its spectrum lies on the circle in the complex plane centered at $ (1,0)$ with radius 1, and that $ H_{\mathbb{R}}$ has an empty point spectrum. Finally, we generalize the definition of the Hardy operator by considering a weighted version and demonstrate its application to solving certain first-order differential equations.

In this paper, the existence and uniqueness of periodic solutions for the second-order evolution equation with weak damping in a Hilbert space $H$

$$ u''(t)+2c u'(t)+Au(t)=f(t, u(t)),\quad t\in \mathbb{R} $$

are discussed, where $ A: D(A)\subset H\to H$ is a positive definite self-adjoint operator with a compact resolvent in $H$, $f: \mathbb{R}\times H\to H$ is continuous, $f(t, x)$ is $\omega$-periodic in $t$, and $c>0$ is the damping coefficient. By applying the semigroup theory of linear operators and fixed-point theorem, we obtain existence and uniqueness results of $\omega$-periodic weak solution

and classical solution of the equations.

In this paper, an inverse diffusion problem with the Caputo fractional derivative in time is considered. We prove that such a problem is ill-posed and apply the Landweber iteration method. The selection criteria for the number of iterations (regularization parameter) under both prior and posterior conditions are provided respectively, along with a rigorous mathematical proof of the convergence of the method. Finally, a numerical example is given to illustrate the effectiveness of this method.

This paper introduces a novel multi-step inertial regularization algorithm in Hilbert spaces, designed to solve the common solution of hierarchical variational inequality problems and quasi-non expansive mapping fixed-point problems. Under certain conditions, the strong convergence theorem for this iterative sequence generated by the algorithm is established. Finally, a numerical example is provided to demonstrate that this algorithm is more efficient than the existing ones.

The nonlocal positive flow short-pules equation is first proposed based on the nonlinear transverse oscillation of elastic beam under tension in physics. By the nonlinear steepest descent method, the long-time asymptotics of the solution of the Cauchy problem for the equation is discussed. Starting from the WKI-type Lax pair it satisfies, the corresponding basic Riemann-Hilbert problem and reconstruction formula for the solution are established. Through a series of deformations such as reorientation, extending, cuting and rescaling, the basic Riemann-Hilbert problem is transformed into the model Riemann-Hilbert problem that can be solved using parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the nonlocal positive flow short-pules equation is obtained.

This paper investigates a nonlocal Lotka-Volterra competition model with seasonal succession. The existence of a monostable traveling waves connecting two semi-trivial equilibria is proven by using the theory of monotone semiflows. Meanwhile, we obtain the rightward spreading speed, we establish the existence of the minimal wave speed for rightward traveling waves and its coincidence with the rightward spreading speed.

Current studies on predator-prey systems mainly consider single interaction mechanisms, which cannot fully characterize the complex ecological interactions between populations. Therefore, this paper establishes a cross-diffusion predator-prey system incorporating multiple interaction mechanisms including fear effects, saturation effects, intraspecific competition, and predator interference, based on the Beddington-DeAngelis functional response function and modified Leslie-Gower terms. For the non-diffusion system, we analyze the stability of positive equilibrium points and the Hopf bifurcation conditions induced by prey intraspecific competition. For the diffusion system, we derive the conditions for Turing instability and focus on investigating how various interaction mechanisms affect the formation and evolution of Turing patterns. The results demonstrate that changing the intensity of interaction mechanisms (such as fear effects) and cross-diffusion coefficients can lead to transitions between different Turing patterns, while different interaction mechanisms can also alter the system stability and the stabilization rate of Turing patterns to varying degrees. The results indicate that interactions between predators and prey, along with cross-diffusion, significantly impact the dynamical behavior of the system.

The periodic traveling waves for a class of delayed SIR system with nonlocal dispersal are considered. Firstly, the basic reproduction number $\Re_{0}$ is defined by the method of subalgebraic operators. Secondly, the existence of periodic traveling waves of the system when the basic reproduction number $\Re_{0}>1$ is established based on the non-compactness Kuratowski measure theory and the asymptotic fixed point theorem. Finally, the non-existence of periodic traveling waves when the basic reproduction number $\Re_{0}<1$ is investigated using the method of contradiction proof and the analysis technique.

In this paper, we consider the following second order variable coefficient weighted degenerate elliptic equation with singular potential constituted by Generalized Baouendi-Grushion vector fields

$$ - \sum\limits_{i,j = 1}^N {{X_j}({a_{ij}}(x,y){X_i}u)} + V(x,y)u = 0,$$

where

$$\lambda {\left| \eta \right|^2}w \leqslant \left\langle {A\eta,\eta } \right\rangle \leqslant {\lambda ^{ - 1}}{\left| \eta \right|^2}w,\ \ A = {({a_{ij}})_{N \times N}},$$

$w$ is a weight function related to quasi distance. By using the weighted Hardy inequality, Rellich type identity and the estimates of Dirichlet energy, we first get the monotonicity of the frequency function of the weak solutions. Then the doubling and unique continuation properties of the weak solutions are obtained.

This article concerns the following Klein-Gordon-Maxwell system

$$\begin{cases}-\Delta u+ V(x)u-(2\omega+\phi)\phi u=\lambda |u|^{s-2}u+ f(u), & x\in \mathbb{R}^{3},\\\Delta \phi=(\omega+\phi)u^2, & x\in \mathbb{R}^{3},\end{cases}$$

where $\omega> 0$ is a constant, $\lambda> 0$ is a real parameter, $s\geq6$. When $V, f$ satisfy suitable conditions and $\lambda$ is relatively small, existence of nontrivial solution can be proved via variational methods and Moser iteration. The result in this paper completes some recent works concerning research on solutions of this system.

In this paper, the initial boundary value problem of a three-dimensional compressible nematic liquid crystal flows is studied. The global existence and large time behavior of a strong solution on a bounded domain is obtained when the initial value is close to a equilibrium state in $H^2$ Sobolev space.

The three-dimensional (3D) compressible magnetohydrodynamic system is studied in a bounded rectangular domain with slip boundary condition for the velocity field and perfect conduction for the magnetic field. For the regular initial data with large energy, the global well-posedness of strong solutions to the initial-boundary-value problem of this system is obtained.

Soliton resonance and soliton molecules play a significant role in the study of optics, physics, and fluid mechanics. This paper systematically investigates the soliton resonance and soliton molecule properties of the Hirota equation in the context of nonlinear optical systems. First, based on the Lax pair, the $N$-fold Darboux transformation of the Hirota equation is constructed, from which the exact analytical expressions of the $N$-soliton solutions are derived. Subsequently, by adjusting the spectral parameters and initial phase parameters, novel soliton resonance and soliton molecule solutions of the Hirota equation are constructed, and their dynamical characteristics are analyzed. The research results indicate that localized soliton resonance can evolve into soliton molecules, with soliton molecules being the limiting form of localized soliton resonance. Furthermore, the study reveals that the energy carried by solitons exhibits a growth trend during the formation of soliton resonance and soliton molecules. It also demonstrates that the spatiotemporal evolution patterns of soliton molecules become increasingly complex as the soliton order increases. Meanwhile, the wave density of soliton molecules is positively correlated with the coefficients of higher-order dispersion and higher-order nonlinear terms. These findings provide new theoretical insights into the soliton dynamics of the Hirota equation and contribute to a better understanding of soliton interaction mechanisms in nonlinear optical systems.

In this paper, We discuss Riemann-Liouville fractional evolution differential systems in Banach spaces. Firstly, the existence of $C_{1-\alpha}$-mild solutions for the Riemann-Liouville fractional evolution differential equations are established in Banach spaces. Secondly, we make some general assumptions to guarantee the approximate controllability of the associated Riemann-Liouville fractional evolution systems is also formulated and proved. In the end, an example is given to illustrate our main results.

This article applies Riemann coordinate transformation and coordinate scaling to non-uniform transmission line equations, and establishes a class of hyperbolic PDE-PDE coupled systems with variable coefficients. Assuming that the control input voltage of the boundary condition remains constant, the proportional feedback boundary condition of the system is obtained. Then rewrite the closed-loop system into the form of abstract evolution equations, and use the semigroup method and equivalent norm theorem to obtain the dissipativity conditions of feedback control parameters, ensuring the dissipativity of system operators. Finally, spectral analysis was conducted on the system operators, and the asymptotic expression of the eigenvalues was derived using the matrix operator pencil method.

Algebraic degeneration in $L^2$ sense is studied for non-recurrent diffusion process on a semi-line. The sufficient and necessary conditions are presented under two boundary conditions. The similar conclusions are also proved on the real line. The conclusions are applied to two examples, yielding precise results.

Motivated by the paper of Vasudeva (Metrika, 2024, 87: 571-584), this paper studied the joint limiting distribution of the upper and the lower extreme order statistics with random sample size. Let $\{X_{n}, n\geq1\}$ be a sequence of random variables and $N(n)$ be a sequence of positive integer random variables. Under some conditions, we derive first the joint limiting distribution of $M_{N(n)} =\max\left \{X_{1}, X_{2}, \cdots, X_{N(n)} \right \}$ and $W_{N(n)} =\min\left \{ X_{1}, X_{2}, \cdots, X_{N(n)} \right \}$ and then we extended the result to the case of the upper and the lower extreme order statistics. The obtained results extended that of Vasudeva (Metrika, 2024, 87: 571-584).

The Ornstein-Uhlenbeck (O-U) process, as an important diffusion process, plays a significant role in fields such as statistics, finance, and physics. In this paper, the sharp large deviations of the maximum likelihood estimation for the O-U process with linear drift in the explosive cases are studied by change of measure, and a refined characterization of the tail probability is obtained. As an application, the large deviation principle is obtained. The results demonstrate that the maximum likelihood estimations of the O-U process with and without linear drift in the explosive cases have the same large deviations.

The Variable Dielectric Poisson-Nernst-Planck (VDPNP) equations are models that describe ion transport behavior in electrolyte solutions. These equations extend the classical Poisson-Nernst-Planck (PNP) equations by introducing a dependence of the dielectric constant on ion concentration, which allows for a better description of the complex dynamic processes in electrolyte solutions. To investigate the impact of the VDPNP equations on system interactions, this paper first solves the VDPNP model using the Gummel finite element method. Subsequently, numerical simulations of nanopore systems are performed. The simulation results show that, in mixed solutions, the traditional PNP equations cannot distinguish between sodium and potassium ions based on concentration, whereas the VDPNP models can clearly differentiate between sodium and potassium ions. Furthermore, the distinguishing effect becomes more pronounced as the surface charge increases.

The Landweber iterative regularization method is an effective approach for solving nonlinear ill-posed inverse problems. However, its convergence speed is often slow, which greatly limits its practical applications. An adaptive two-point gradient method with momentum (ATPGM) is proposed to accelerate the classic Landweber method. The main idea of the ATPGM is that the momentum is introduced into the gradient direction of the two-point gradient method, thus it can take full advantage of the previous gradients information, resulting in a highly fast convergence.The convergence and regularity of ATPGM are given. In numerical experiments, we test the numerical performance of ATPGM, based on one-dimensional and two-dimensional elliptic parameter identification problems. A comprehensive comparision between the Landweber iterative regularization method, the two-point gradient method (TPG) and ATPGM is also presented. The numerical results show that ATPGM performs little better in terms of iterations and running time. Numerical results also show that ATPGM is not sensitive to noise in terms of iterations.

In this paper, the uncertainty of the multidimensional European maximal call option is characterized by a combination of fuzziness and stochasticity. The pricing formulas for the correspond-ing European maximal call option are given by L$\acute{\rm e}$vy-It$\hat{\rm o}$ formula, when the logarithm of the jump amplitude is with fuzzy normal sum and fuzzy double exponential distribution. A clarity number is given by weighted probability averaging in the valuation of the fuzzy option. Finally, the effects of the variation of the main parameters and different models on the pricing of the European maximal call option are analyzed by numerical simulation.

The study of group actions and the corresponding entropy theory in topological spaces is very important in mathematical physics and topology. Factor map is a key concept in the study of the structural theory of measure preserving dynamical systems. Conditional entropy, on the other hand, quantitatively characterises the complexity of factor map. In this paper, we calculate the conditional entropy of a class of extended maps in the dynamical system for amenable group actions, and its value is zero.

Bell nonlocality sharing of quantum systems is a basic feature of quantum mechanics. The nonlocality sharing ability of network quantum states is stronger than bell nonlocality, and the structure of network quantum states is more complex. In practical applications, the error of quantum entanglement generation and the noise in the process of quantum measurement will lead to the decay of nonlocality sharing. This article explores the sufficient conditions for the persistence of nonlocality sharing with noise under unilateral and multilateral measurements in star networks, and analyzes the impact of different noise conditions on the persistence of nonlocality sharing in star networks. In particular, this paper also applies the above conclusions to the specific bilocal scenarios: with two independent sources, one central party and two edge parties.