Abstract:
A coupled system of an ordinary differential equation (ODE) and a heat partial differential equation (PDE) with spatially varying coefficients is discussed. By using the PDE backstepping method, the state-feedback stabilizing controller is explicitly constructed with the assumptions and , respectively. The closed-loop system is proved to be exponentially stable by this controller. A simulation example is presented to illustrate the effectiveness of the proposed method. 1. Introduction Predictor-feedback control design [1, 2] has been an active area of research for PDE or PDE-ODE coupled control systems [3–5] with actuator and sensor delays that have rich physical backgrounds such as coupled electromagnetic, coupled mechanical, and coupled chemical reactions. The input delays to the ODE system can be modeled with the first-order hyperbolic PDE (transport PDE) and the boundary condition . Thus, the original ODE system with input delay can be represented as the following ODE-PDE coupled system (1) that is driven by the input from the boundary of the PDE: Control design of coupled PDE-ODE systems was considered in [6–10]. The controller design based on the backstepping method for the coupled system (1) was designed in [9, 10]. More recently in [11], a heat diffusion PDE-ODE coupled system was considered, and a wave PDE-ODE coupled system was considered in [12]. The control system with interaction for this system coupled between the ODE and the PDE was considered in [13]: In this system, the ODE acts back on the PDE by the state of the ODE; meanwhile, the PDE acts on the ODE, which models the solid-gas interaction of heat diffusion and chemical reaction. In this paper, we replace the spatially constant coefficient of the PDE subsystem in (2) by the spatially varying coefficient ; that is, , which implies that the effects from the ODE subsystem to the PDE subsystem are varying with the location . In fact, control of the coupled systems is an important subject in control theory since this type of system arises frequently in control engineering. The objective of this paper is to convert a PDE-ODE coupled system into a closed-loop target system that is exponentially stable in the sense of the norm , with a designed stable state-feedback controller by using the backstepping-based predictor design method. Under the assumptions and , respectively, we further obtain the explicit expressions of the kernel function of the backstepping transformation. This paper is organized as follows. In Section 2, we propose the interaction of PDE-ODE coupled control system. In

Abstract:
An in-host viral model with cure of infected cells and humoral immunity is studied. We prove that the stability is completely determined by the basic reproductive number and show that the infection-free equilibrium is globally asymptotically stable if and only if . Moreover, if , the infection equilibrium is locally asymptotically stable when the time delay is small and it loses stability as the length of the time delay increases past a critical value . Finally, we confirm our analysis by providing several numerical examples. 1. Introduction The humoral immunity is a kind of immunologic mechanism which uses B lymphocytes to produce antigen to prevent virus and there are evidences to prove that the humoral immunity is more effective than the cell-mediated immune in some infections such as malaria infection [1–3]. Many authors present and develop mathematical systems for the humoral immunity [4–7]. And the cure of virus is also important especially in HBV models [8, 9]. In the present paper, we analyze an in-host viral model with humoral immunity and intracellular delay, and we incorporate a “cure” of infected cells into it. We propose the following system: where , , , and represent the uninfected cells, the infected cells, the virus, and the B cells, respectively. and are assumed as the birth rate and death rate of uninfected cells. is the infection rate and represents the number of free virus which is produced during the average infected cell life span. is the death rate of infected cells and represents the death rate of virus. and represent the birth rate and death rate of B cells. The B cells neutralization rate is represented by . The following form is taken as the initial conditions: where , the space of continuous functions mapping the interval into , and The organization of this paper is as follows. In the next section, we will find threshold parameters of system (1) and it determines the existence of the equilibriums. In Section 3, by structuring suitable Lyapunov functionals and using LaSalle’s invariance principle we attain the global stability of the uninfected equilibrium if . In Section 4, we consider the stability of the infected equilibrium and the occurrence of local Hopf bifurcation. In Section 5, we present the numerical simulations to illustrate our results. Finally, we offer concluding remarks in the last section. 2. Existence of Equilibrium We can easily find that system (1) always has an uninfected equilibrium . Denote We call the basic reproductive number. It is easy to prove that if , model (1) exist an infected equilibrium ,

Abstract:
Usually, an optimal preview servo system uses the same quadratic performance index as the original optimal servo system. The design idea is to further minimize performance index. If the known system was not designed with optimal control method, people would use a performance function dependent only on the feedforward terms to design the so-called optimal FF compensating system. Usually, these two kinds of systems are handled separately. In this paper, we present a method to deal with them at the same time such that the known optimal preview servo system and optimal FF compensating system are all special cases. Computer simulation is given at the end of this paper to compare this method with optimal servo system and optimal preview servo system.

Abstract:
We describe two new ornithurine birds from the Early Cretaceous Jiufotang Formation of western Liaoning, northeast China:Yanornis martini gen. et sp. nov. andYixianornis grabaui gen. et sp. nov. They represent the best fossil record of ornithurine birds known from the Early Cretaceous. They are more advanced than the most primitive ornithurineLiaoningornis, and are more similar to the other two Chinese Early Cretaceous ornithurinesChaoyangia andSonglingornis. Compared withConfuciusornis, Liaoxiornis andEoenantiornis from the same age, the two new birds show remarkable advanced characteristics and suggest the presence of powerful flight capability like modern birds. Compared withYixianornis andChaoyangia, Yanornis is larger, with a more elongated skull and relatively long wings. The new discoveries indicate that by the Early Cretaceous both enantiornithine and ornithurine birds had already radiated significantly. The flight structures ofYanornis andYixianornis are hardly distinguishable from those of modern birds; however, both retain a few primitive traits such as teeth on the jaws, wing claws and pubic symphysis, which exclude them from being the most recent ancestor of all extant birds.

Abstract:
We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for \'etale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-\'etale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-\'etale site. Moreover, we need to prove the Poincar\'e lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

Abstract:
We prove the Breuil-Mezard conjecture for split non-scalar residual representations of Gal(Qp/Qp) by local methods. Combined with the cases previously proved in [18] and [24], this completes the proof of the conjecture (when p>3). As a consequence, the local restriction in the proof of the Fontaine-Mazur conjecture in [18] is removed.

Abstract:
We construct one parameter families of overconvergent Siegel-Hilbert modular forms. In particular, for any classical Siegel-Hilbert modular eigenform one can find a rigid analytic disc centered at this point, on which an infinite family of classical points with varying weights accumulates at the center.

Abstract:
There are two methods for GIS similarity measurement problem, one is cross-coefficient for GIS attribute similarity measurement, and the other is spatial autocorrelation that is based on spatial location. These methods can not calculate subzone similarity problem based on universal background. The rough measurement based on membership function solved this problem well. In this paper, we used rough sets to measure the similarity of GIS subzone discrete data, and used neighborhood rough sets to calculate continuous data’s upper and lower approximation. We used neighborhood particle to calculate membership function of continuous attribute, then to solve continuous attribute’s subzone similarity measurement problem.

Abstract:
This paper discussed the pathogenesis and the principle of treatment of the dry cough with yellow greasy coating. Combined the clinic observation with the TCM theory, we found that the main reasons about this dry cough are dryness in lung and stagnated phlegm. And heavy dosages herbs which can nourish Yin and clear away the heat should be adopted.