The aim of this paper is to characterize the smallest $2$-Pisot number of degree $n \geq 2$ in the case of Laurent power series $\mathbb{F}_{q}((X^{-1}))$ with $q$ distinct from the power of $2$ over a finite field $\mathbb{F}_{q}$ by giving explicitly its minimal polynomial. Indeed, we prove that its minimal polynomial is given by $\Lambda_{n}(Y) = Y^{n} - \alpha X(X+1) Y^{n-1} - \alpha^2 X^3 Y^{n-2} + \alpha^{n}$. We show in particular that the sequence of smallest $2$-pisot numbers of degree $n$ is decreasing and converges to $(\alpha X^2, \alpha X)$ where we suppose that $\alpha$ is the least element of $\mathbb{F}_{q} \setminus \{0\}$.

In this paper, we consider a problem of recovering a space-dependent source term for the Rayleigh–Stokes equation, where the additional data is the observation at a final moment $t = T$, which is ill-posed in the sense of Hadamard. Firstly, the uniqueness, ill-posedness and the conditional stability of inverse source problem is given. Next, we develop a filter regularization method to overcome the ill-posedness of the problem. Under reasonable a priori bound assumption about the source function, a Hölder-type error estimate of the regularized solution is proved for a priori regularization parameter choice rule. Furthermore, a logarithmic-type error estimate between the exact solution and the regularized solution is established based on a posteriori regularization parameter choice rule.

This article proposes an implicit discontinuous Galerkin scheme for solving the 2D model of drug release from cardiovascular drug-eluting stents, given the binding of the drug that is saturated and reversible. The scheme uses discontinuous Galerkin discretization based on bubble modal basis functions in space and the backward Euler discretization in time. For this purpose, we have divided each domain in space into several sub-domains. In each sub-domain, we have used a bubble modal basis as the local bases. These are more efficiently matched in each sub-domain than the global bases, especially in the discontinuous points of the domain. The existence and uniqueness of the discontinuous Galerkin solution are examined. The consistency, stability, and convergence of the proposed scheme are thoroughly provided. Under the stability conditions, the domains of penalty parameters and time steps are determined such that the scheme remains stable. The appropriate range of changes for the velocity, $\mathbf{u}_w$, is obtained, regarding the convergence analysis. Numerical results show high performance due to the accuracy and efficiency of the proposed scheme.

In this paper, we establish some new endpoint regularity properties of multilinear maximal operators and their fractional variants, both in the centered and uncentered cases. The main results we obtain not only answer a question of Liu, Wu, Xue and Yabuta, but also provide some new endpoint Sobolev boundedness and continuity results for multilinear fractional maximal operators. More importantly, we present a new approach to deal with the endpoint Sobolev continuity for the above operators.

We establish a slowly moving target second main theorem for meromorphic functions on a non-Archimedean field, with counting functions truncated to level $1$. As an application, we show that two meromorphic functions on a non-Archimedean field must coincide if they share $q$ ($q \geq 5$) distinct small functions, ignoring multiplicities. Thus, our work improves the results in [2].

The $h$-function $h(r)$ represents the probability that a Brownian traveller released from a fixed point $z_{0}$ in a region $\Omega$, first exits the region $\Omega$ somewhere within distance $r$ of $z_{0}$. We compute the $h$-functions of some planar simply connected regions, and investigate the asymptotics of these $h$-functions at certain values of $r$. Also, we analyse the derivatives of $h(r)$ at a value of $r$ where two regimes meet.

A spacelike surface in Minkowski space $\mathbb{R}_1^3$ is called a $K^{\alpha}$-translator of the flow by the powers of Gauss curvature if satisfies $K^{\alpha} = \langle N, \vec{v} \rangle$, $\alpha \neq 0$, where $K$ is the Gauss curvature, $N$ is the unit normal vector field and $\vec{v}$ is a direction of $\mathbb{R}_1^3$. In this paper, we classify all rotational $K^{\alpha}$-translators. This classification will depend on the causal character of the rotation axis. Although the theory of the $K^{\alpha}$-flow holds for spacelike surfaces, the equation describing $K^{\alpha}$-translators is still valid for timelike surfaces. So we also investigate the timelike rotational surfaces that satisfy the same prescribing Gauss curvature equation.

We investigate the behavior of the approximate solution of Biot's consolidation model using a weighted least-squares (WLS) finite element method. The model describes the fluid flow in a deformable porous medium, with variables for fluid pressure, velocity, and displacement. The WLS functional is defined based on the stress-displacement formulation, with the symmetry condition of the stress and the weight that depends on the time step size for the temporal discretization of the model. An a priori error estimate for the first-order linearized least squares (LS) system is analyzed, and its validity is confirmed through numerical results. By using continuous piecewise linear finite element spaces for all variables and adjusting the weight appropriately, we obtain optimal error convergence rates for all variables. Additionally, we present two numerical examples to demonstrate the implementation of the WLS method for benchmark problems.

In this paper, we focus on the effects of Beddington–DeAngelis functional response and protection zone on the positive steady state set and dynamical behavior of a predator-prey model. First, a clear and complete characterization of the structure of the constant positive steady state set is given as well as the stabilities. Second, nonexistence, existence and stability of nonconstant positive steady states are deduced, in which the Beddington–DeAngelis functional response plays a subtle role. Finally, we introduce a protection zone to the prey and give the global bifurcation of positive steady states bifurcating from the semitrivial steady states. The result reveals that the prey species can always coexist with the predator as the protection zone is suitably large or its diffusion is slow enough.

In this paper, a new three-term conjugate gradient algorithm is proposed to solve unconstrained optimization including regression problems. We minimize the distance between the search direction matrix and the self-scaling memoryless BFGS direction matrix in the Frobenius norm to determine the search direction, which has the same advantages as the quasi-Newton method. At the same time, random parameter is used so that the search direction satisfies sufficient descent condition. For uniformly convex functions and general nonlinear functions, we all establish the global convergence of the new method. Numerical experiments show that our method has nice numerical performance for solving large-scale unconstrained optimization. In addition, the application of the new method to the regression model proves that our method is effective.