Xiang-Sheng Wang’s research while affiliated with University of Louisiana at Lafayette and other places

In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality measure of these polynomials, as well as the spectral measure for the associated doubly infinite tridiagonal Jacobi matrix. Notably, while the orthogonality measure may include discrete mass points, the spectral measure(s) of the doubly infinite Jacobi matrix are absolutely continuous. Additionally, we uncover an intrinsic connection between these new orthogonal polynomials and Chebyshev polynomials through a nonlinear transformation of the polynomial variables.

Biological invasions significantly impact native ecosystems, altering ecological processes and community behaviors through predation and competition. The introduction of non-native species can lead to either coexistence or extinction within local habitats. Our research develops a lizard population model that integrates aspects of competition, intraguild predation, and the dispersal behavior of intraguild prey. We analyze the model to determine the existence and stability of various ecological equilibria, uncovering the potential for bistability under certain conditions. By employing the dispersal rate as a bifurcation parameter, we reveal complex bifurcation dynamics associated with the positive equilibrium. Additionally, we conduct a two-parameter bifurcation analysis to investigate the combined impact of dispersal and intraguild predation on ecological structures. Our findings indicate that intraguild predation not only influences the movement patterns of brown anoles but also plays a crucial role in sustaining the coexistence of different lizard species in diverse habitats.

In this survey we summarize the current state of known orthogonality relations for the q and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the q-Askey scheme. These polynomials are the continuous dual q and $q^{-1}$-Hahn polynomials, the q and $q^{-1}$-Al-Salam--Chihara polynomials, the continuous big q and $q^{-1}$-Hermite polynomials and the continuous q and $q^{-1}$-Hermite polynomials and their dual counterparts which are connected with the big q-Jacobi polynomials, the little q-Jacobi polynomials and the q and $q^{-1}$-Bessel polynomials. The $q^{-1}$-symmetric polynomials in the q-Askey scheme satisfy an indeterminate moment problem, satisfying an infinite number of orthogonality relations for these polynomials. Among the infinite number of orthogonality relations for the $q^{-1}$-symmetric families, we attempt to summarize those currently known. These fall into several classes, including continuous orthogonality relations and infinite discrete (including bilateral) orthogonality relations. Using symmetric limits, we derive a new infinite discrete orthogonality relation for the continuous big $q^{-1}$-Hermite polynomials. Using duality relations, we explore orthogonality relations for and from the dual families associated with the q and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. In order to give a complete description of the convergence properties for these polynomials, we provide the large degree asymptotics using the Darboux method for these polynomials. In order to apply the Darboux method, we derive a generating function with two free parameters for the $q^{-1}$-Al-Salam--Chihara polynomials which has natural limits to the lower $q^{-1}$-symmetric families.

On February 13, 2020, the Health Commission of Hubei Province changed the definition of confirmed cases, resulting in a reported daily case number that is significantly larger than on other dates. Such abnormal data points pose a challenge in data fitting and parameter estimation. To address this, we derive a simple formula from the classical Kermack-McKendrick model and introduce a new quantity to capture the number of unreported cases hidden in the data. We then use this new formula to fit the inconsistent data and estimate key epidemic parameters. Based on the reported cumulative case numbers until February 21, 2020, we estimate that the unreported case number in Hubei is 60856 (95% CI: [33513, 91206]), while the unreported case number in Wuhan is estimated as 29374 (95% CI: [18205, 40665]). The peak times in Hubei and Wuhan are February 6, 2020, and February 8, 2020, respectively. The basic reproduction numbers are 2.334 (95% CI: [2.053, 2.711]) for Hubei and 2.189 (95%CI: [1.992, 2.448]) for Wuhan.

We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells, while the diffusion rate of CTL is a decreasing function of the density of infected cells. We first establish the global existence and ultimate boundedness of the solution via a priori energy estimates. We then define the basic reproduction number of viral infection $R_0$ and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state $E_0$ is globally asymptotically stable if $R_0<1$. When $R_0>1$, then $E_0$ becomes unstable, and another basic reproduction number of CTL response $R_1$ becomes the dynamic threshold in the sense that if $R_1<1$, then the CTL-inactivated steady state $E_1$ is globally asymptotically stable; and if $R_1>1$, then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state $E_2$ is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.

With the rapid advancements in synthetic aperture radar (SAR) satellites and associated processing algorithms over recent decades, interferometric SAR (InSAR) has emerged as a routine method for monitoring large-scale ground deformation and interpreting geophysical processes. Statistical inference serves as a major component in InSAR technique developments and applications. This article provides an overview of InSAR deformation measurement and InSAR-constrained geophysical inversion, using a statistical inference point of view. Its objectives are to facilitate understanding of the method by addressing its underlying mathematical challenges. We begin by introducing the concept of statistical inference and the structure of our content organization framework. Next, we investigate the distinct concerns associated with statistical inference in InSAR deformation measurement and InSAR-constrained geophysical inversion. Finally, we propose several significant directions for future research. Table 1 includes abbreviations used throughout this article. Additionally, we highlight relevant resources, such as mathematical background, open source codes, and data repositories, in an appendix, which is available as supplementary material at https://doi.org/10.1109/MGRS.2023.3344159 .

In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches -1$-1$ from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity.