Atherosclerosis is the main killer in the west and therefore a major health challenge today. Total serum cholesterol and lipoprotein concentrations, used as clinical markers, fail to predict the majority of cases, especially between the risk scale extremes, due to the high complexity in lipoprotein structure and composition. In particular, low-density lipoprotein (LDL) plays a key role in atherosclerosis development, with LDL size being a parameter considered for determining the risk for cardiovascular diseases. Determining LDL size and structural parameters is challenging to address experimentally under physiological-like conditions. This article describes the biochemistry and ultrastructure of normolipidemic and hypertriglyceridemic LDL fractions and subfractions using small-angle X-ray scattering. Our results conclude that LDL particles of hypertriglyceridemic compared to healthy individuals 1) have lower LDL core melting temperature, 2) have lower cholesteryl ester ordering in their core, 3) are smaller, rounder and more spherical below melting temperature, and 4) their protein-containing shell is thinner above melting temperature.

The ability of mixotrophs to combine phototrophy and phagotrophy is now well recognized and found to have important implications for ecosystem dynamics. In this paper, we examine the dynamical consequences of the invasion of mixotrophs in a system that is a limiting case of the chemostat. The model is a hybrid of a competition model describing the competition between autotroph and mixotroph populations for a limiting resource, and a predator--prey-type model describing the interaction between autotroph and herbivore populations. Our results show that mixotrophs are able to invade in both autotrophic environments and environments described by interactions between autotrophs and herbivores. The interaction between autotrophs and herbivores might be in equilibrium or cycle. We find that invading mixotrophs have the ability to both stabilize and destabilize autotroph-herbivore dynamics depending on the competitive ability of mixotrophs. The invasion of mixotrophs can also result in multiple attractors.

We have in a previous study introduced a novel elliptic operator Delta((p,q))u = vertical bar del u vertical bar(q) Delta(1)u + (p - 1) vertical bar del u vertical bar(p-2)Delta(infinity) u, p >= 1, q >= 0, as a generalization of the p-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation u(t) =vertical bar del u vertical bar(1-q) Delta((1+q,q)), where q = q(vertical bar del u vertical bar) is continuous and has range in [0, 1], in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.

The Rosenzweig-MacArthur (1963) criterion is a graphical criterion that has been widely used for elucidating the local stability properties of the Gause (1934) type predator-prey systems. It has not been stated whether a similar criterion holds for models with explicit resource dynamics (Kooi et al. (1998)), like the chemostat model. In this paper we use the implicit function theorem and implicit derivatives for proving that a similar graphical criterion holds under chemostat conditions, too.

In this work we introduce a novel operator ∆_(p;q) as an extended family of operators that generalize the p-Laplace operator. The operator is derived with an emphasis on image processing applications, and particularly, with a focus on image denoising applications. We propose a non-linear transition function, coupling p and q, which yields a non-linear ltering scheme analogous to adaptive spatially dependent total variation and linear ltering. Well-posedness of the nal parabolic PDE is established via pertubation theory and connection to classical results in functional analysis. Numerical results demonstrates the applicability of the novel operator∆_ (p;q).

In this paper, a nonlinear mathematical model is presented for the transmission dynamics of HIV/AIDS in Cuba. Due to Cuba's highly successful national prevention program, we assume that the only mode of transmission is through contact with those yet to be diagnosed with HIV. We find the equilibria of the governing nonlinear system, perform a linear stability analysis, and then provide results on global stability.

On infinitesimally short time intervals various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann, The dynamics of physiologically structured populations, 1986, p. 3). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to non-removable dependency terms potentially affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open. Our main result is a rather complete analysis of an important limiting case. Once complete knowledge of the qualitative properties of simple models is obtained, it greatly facilitates further studies of more complex models. A consequence of our analysis is that standard methods can be applied. However, the application of those methods is far from straightforward and require non-trivial estimates in order to make them valid for all values of the parameters. We focus on making these proofs as elementary as possible.

In this paper we present a model for describing the position distribution of the endocardium in the two-chamber apical long-axis view of the heart in clinical B-mode ultrasound cycles. We propose a novel Bayesian formulation, including priors for spatial and temporal smoothness, and preferred shapes and position. The shape model takes into account both endocardium, atrial region and apex. The likelihood is built using a statistical signal model, which attempts to closely model a censored signal. In addition, the use of a censored Gamma mixture model with unknown censoring point, to handle artefacts resulting from left-censoring of the in US clinical B-mode, is to our knowledge novel. The posterior density is sampled by the Gibbs method to estimate the expected latent variable representation of the endocardium, which we call the Bayesian Probability Map; the map describes the probability of pixels being classified as being within the endocardium. The regularization parameters of the model are estimated by cross-validation, and the results are compared against the two-chamber apical model of Chen et al.

An SIQ epidemic model with isolation and nonlinear incidence rate is studied. We have obtained a threshold value R and shown that there is only a disease free equilibrium point when R < 1, and there is also an endemic equilibrium point if R > 1. With the help of Liapunov function, we have shown that disease free- and endemic equilibrium point is globally stable.

In this paper, we conduct a careful global stability analysis for a generalized cholera epidemiological model originally proposed in [J. Wang and S. Liao, A generalized cholera model and epidemic/endemic analysis, J. Biol. Dyn. 6 (2012), pp. 568–589]. Cholera is a water- and food-borne infectious disease whose dynamics are complicated by the multiple interactions between the human host, the pathogen, and the environment. Using the geometric approach, we rigorously prove the endemic global stability for the cholera model in three-dimensional (when the pathogen component is a scalar) and four-dimensional (when the pathogen component is a vector) systems. This work unifies the study of global dynamics for several existing deterministic cholera models. The analytical predictions are verified by numerical simulation results.

The global stability of SEIRS models with nonlinear incidence rates was conjectured in [W. M. Liu, H. W. Hethcote and S. A. Levin, J. Math. Biol. 25 (1987), 359-380.] and has been stated as an outstanding open question for classical bilinear models in [M. Y. Li, J. S. Muldowney and P. van den Driessche, Canad. Appl. Math. Q. 7 (1999), 409-425.]. By applying the Poincar e-Bendixson property of dynamic systems in space, the authors in [M. Y. Li and J. S. Muldowney, SIAM J. Math. Anal. 27 (1996), 1070-1083.] have proven the conjecture for the bilinear model with a su fficiently long average immunity period, and in [M. Y. Li, J. S. Muldowney and P. van den Driessche, Canad. Appl. Math. Q. 7 (1999), 409-425.] the authors have shown the case with a suffi ciently long average infection period. In this paper, we solve the open problem for the bilinear case completely, and furthermore have relaxed the constraint on the general nonlinear transmission function for global stability.

In this paper we study the generalized Fucik type eigenvalue for the boundary value problem of one dimensional $p-$Laplace type differential equations -(φ(u'))' = ψ(u), u(-T) = u(T) =0, where φ(s) = α s^p, ψ(s)= λs^p, s >0; φ(s) = -β|s|^p, ψ(s)= -μ|s|^p, s <0. We say (α, β, λ, μ) belong to the Fucik spectrum, if the above boundary value problem has a non-trival solution. We have obtained a complete and explicit characterization of Fucik spectrum for such problem.

Multiplicity of solutions to the following boundary value problem -(φ(u'))' = λψ(u), T < x < T; u(-T) = u(T) = 0, is considered. There are many works devoted to odd function φ, ψ, but are very few on non-odd cases. For the prototype function φ(u) = au^p, u >0, = -b |u|^q,u < 0 several exact multiplicity results have been established.

In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent -∆u = λu - αu^p+ u^{2^*-1}, u ≥ 0, in Ω u=0, on Ω. where Ω is a bounded C^2-domain in R^n, n ≥ 3, λ > λ_1, 1 < p < 2^* -1= (n+2)/(n-2) and α > 0 is a bifurcation parameter. Brezis and Nirenberg showed that a lower order (non-negative) perturbation can contribute to regain the compactness and whence yields existence of solutions. We study the equation with an indefinite perturbation and prove a bifurcation result of two solutions for this equation.

In this paper, we present some observations of biurcation in nonlinear elliptic boundary value problem - ∆u = f(λ, u) in domain Ω, and u = 0 on the boundary ∂Ω. We are interested in how solution(s) depends on the parameter λ. Through examples, we show that multiple solutions appear, when λ cross certain critical value, i.e. bifurcation occurs.

We, in this paper, consider the semilinear elliptic boundary value problem - ∆u = f(x, u) in Ω and u = g on ∂Ω and the corresponding Bolza problem x'' + ∂V(t, x) =0, x(0)= x0, x(T)= x1, where Ω is a bounded open subset in R^n with C² boundary and g is a given continuous function on the boundary of Ω; and T is the given traveling time, x0,x1 are two fixed points in the state space Rn. Under certain conditions on f and V, we show that the above problems have infinitely many solutions

In this paper, we study under what conditions on m(x) and f(u) the problem Δu=m(x)f(u) has a solution in a bounded domain D, which tends to infinity, as x tends to the boundary. We show that if m(x) is singular at the boundary of D, except that the Keller-Osserman condition must hold, the growth of f at the infinity has to be slow for a solution to exist. Some existence results have been established.

In this paper we consider the one-dimensional elliptic boundary blow-up problem ∆p(u) = f(u), a < x < b, u(a) = u(b) = + ∞. We show that the structure of the solutions can be very rich even for a simple function f, which indicates that a similar results might hold also in higher dimensional spaces.