In this paper, we shall survey recent developments in variable-internal-stores models with multiple resources or spatial/temporal inhomogeneity, which may enhance coexistence of species and diversity in competitor communities. On the other hand, it was known that basic limiting resources for growth usually include nutrients (e.g., nitrogen and phosphorus), light, and inorganic carbon. Thus, the proposed models will involve the competition between the species for nutrients (e.g., nitrogen and phosphorus), or light, or both of nutrients and light, or inorganic carbon.

In analyzing time-to-event data, proportional hazards (PH) regression is an ubiquitous model used in many fields. PH regression, however, requires a strong assumption that is not always appropriate. Threshold regression (TR) is one of the alternative models. A first-hitting-time (FHT) survival model postulates a health status process for a patient that gradually declines until the patient dies when the health level first reaches a critical threshold. In this article, we review the development of threshold regression models and their applications.

We study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Dirichlet-Neumann boundary value problem \[ \begin{cases} u''(x) + \lambda f(u) = 0, \quad 0 \lt x \lt 1, \\ u(0) = 0, \quad u'(1) = -c \lt 0, \end{cases} \] where $\lambda \gt 0$ is a bifurcation parameter and $c \gt 0$ is an evolution parameter. We mainly prove that, under some suitable assumptions on $f$, there exists $c_{1} \gt 0$, such that, on the $(\lambda,\|u\|_{\infty})$-plane, (i) when $0 \lt c \lt c_{1}$, the bifurcation curve is $S$-shaped; (ii) when $c \geq c_{1}$, the bifurcation curve is $\subset$-shaped. Our results can be applied to the one-dimensional perturbed Gelfand equation with $f(u) = \exp \big( \frac{au}{a+u} \big)$ for $a \geq 4.37$.

We are interested in various types of localized waves in FitzHugh-Nagumo equations. Variational methods have been successfully worked out to establish the existence of traveling and standing waves. Starting with a simple planar traveling front, an ordered method is employed to demonstrate different front propagation between two stable equilibria. If these two stable equilibria are in the same energy level, a saddle-focus condition ensures that there are infinite number of standing waves with multiple fronts.

In this paper, we analyze a predator-prey chemostat system with internal storage, in which the predator not only competes for a single inorganic nutrient with the prey species but also consumes the prey for growth. The outcome for the corresponding model without intraguild predation is that the competitive exclusion holds, that is, the superior species will win the competition, and coexistence will not happen. When the mechanism of intraguild predation is added into the system, our analysis indicates that coexistence can be possible.

The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. We develop the theory of the weighted multi-parameter Hardy space $H^p_{\mathfrak{z},w}$ and prove the boundedness for these operators on $H^p_{\mathfrak{z},w}$ for certain $p \leq 1$, which provide endpoint estimates for those singular integral operators studied by Ricci-Stein [31] and Fefferman-Pipher [15]. We also establish the Calderón-Zygmund decomposition and interpolation theorem in this setting.

In this paper, we study the $L^p$ solutions of the Fredholm integral equations with Chandrasekhar kernels. The Hilbert type inequality is resorted to establish an existence and uniqueness result for the Fredholm integral equation associated with Chandrasekhar kernel. A couple of examples well support the condition and extend the classical results in the literature with one generalizing the classical Chandrasekhar kernel. In order to approximate the original solution, a truncated operator is introduced to overcome the non-compactness of the integral operator. An error estimate of the convergence is made in terms of the truncated parameter, the upper bounds of the symbolic function constituting the integral kernel and initial data to the equation.

In this article, the authors characterize the variable Besov-type spaces $B_{p(\cdot),q(\cdot)}^{s(\cdot),\phi}(\mathbb{R}^n)$, with $1/p(\cdot)$ and $1/q(\cdot)$ satisfying the globally log-Hölder continuous conditions, via Peetre maximal functions and averages on balls. The latter characterization, via averages on balls, gives one way to introduce these spaces on metric measure spaces.

In this paper, we first establish an analogue of Wiener-Itô theorem on finite-dimensional Gaussian spaces through the inverse $S$-transform, that is, the Gauss transform on Segal-Bargmann spaces. Based on this point of view, on infinite-dimensional abstract Wiener space $(H,B)$, we apply the analyticity of the $S$-transform, which is an isometry from the $L^2$-space onto the Bargmann-Segal-Dwyer space, to study the regularity. Then, by defining the Gauss transform on Bargmann-Segal-Dwyer space and showing the relationship with the $S$-transform, an analytic version of Wiener-Itô decomposition will be obtained.

Three-dimensional finite-difference frequency-domain analyses of partially periodic photonic structures result in large-scale ill-conditioned linear systems. Due to the lack of efficient preconditioner and reordering scheme, existed general-purpose iterative and direct solvers are inadequate to solve these linear systems in time or memory. We propose an efficient direct solver to tackle this problem. By exploring the physical properties, the coefficient matrix structure, and hardware computing efficiency, we extend the concepts of grid geometry manipulation and multi-level Schur method to propose the Compressed Hierarchical Schur algorithm (CHiS). The proposed CHiS algorithm can use less memory and remove redundant computational workloads due to the homogeneity and periodicity of photonic structures. Moreover, CHiS relies on dense BLAS3 operations of sub-matrices that can be computed efficiently with strong scalability by the latest multicore processors or accelerators. The implementation and benchmarks of CHiS demonstrate promising memory usage, timing, and scalability results. The feasibility of future hardware acceleration for CHiS is also addressed using computational data. This high-performance analysis tool can improve the design and modeling capability for various photonic structures.

In this paper, we improve estimations in a previous work [3] (Arch. Rat. Mech. Anal. 2003) about estimations for the action functional of the $N$-body problem. Our method is based on repeated applications of binary decompositions for the $N$-body system, and is applicable to more general particle systems.