threshold dynamics of a delayed nonlocal reaction diffusion hiv infection model with both cell free and cell to cell transmissions abstract in this paper we derive and analyze an hiv infection model containing a fixed latent period and both cell free and cell to cell transmissions the model is described by a spatially non local reaction diffusion system with the zero flux condition on the boundary we first show that the model admits global solutions and possesses a global attractor following the definition of the next infection operator we obtain the expression of the basic reproduction number 0 which is shown as a threshold parameter for the model dynamics our results indicate that the clearance persistence of hiv is governed by the sign of 0 1 in particular an explicit formula of 0 is given in the homogeneous case

connecting fuzzy preorders fuzzy consequence operators and fuzzy closure and co closure systems abstract the relations between fuzzy preorders fuzzy consequence operators fuzzy closure systems and co closure systems lattice structures are studied given a continuous t norm we will prove that the complete lattice of all fuzzy preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators moreover we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices a solution to the following open problem when two families of fuzzy possible valuations produce the same fuzzy implication relation is given concept of interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through interior operator is shown

finite time semistability and consensus for nonlinear dynamical networks this paper focuses on semistability and finite time stability analysis and synthesis of systems having a continuum of equilibria semistability is the property whereby the solutions of a dynamical system converge to lyapunov stable equilibrium points determined by the system initial conditions in this paper we merge the theories of semistability and finite time stability to develop a rigorous framework for finite time semistability in particular finite time semistability for a continuum of equilibria of continuous autonomous systems is established continuity of the settling time function as well as lyapunov and converse lyapunov theorems for semistability are also developed in addition necessary and sufficient conditions for finite time semistability of homogeneous systems are addressed by exploiting the fact that a homogeneous system is finite time semistable if and only if it is semistable and has a negative degree of homogeneity unlike previous work on homogeneous systems our results involve homogeneity with respect to semistable dynamics and require us to adopt a geometric description of homogeneity finally we use these results to develop a general framework for designing semistable protocols in dynamical networks for achieving coordination tasks in finite time

analysis on a diffusive sis epidemic model with saturated incidence rate and linear source in a heterogeneous environment abstract in this paper we are concerned with a reaction diffusion sis epidemic model with saturated incidence rate and linear source in spatially heterogeneous environments we derive the uniform bounds of parabolic system and study the extinction and persistence of the infectious disease in terms of the basic reproduction number the existence of the endemic equilibrium ee is established when the basic reproduction number is greater than one we further investigate the effects of diffusion and saturation on asymptotic profiles of the endemic equilibrium our study shows that the linear source can enhance persistence of infectious disease and large saturation can help speed up the elimination of disease

long time dynamics of an sirs reaction diffusion epidemic model abstract in this paper we first deal with a diffusive sirs epidemic system in spatially heterogeneous and time periodic environment with no flux boundary condition we establish the uniform persistence if the basic reproduction number r 0 1 when the diffusion rates are equal and the system parameters are constant we obtain the global stability of the unique endemic equilibrium if r 0 1 and that of the diseases free equilibrium if r 0 1 we also study the sirs epidemic system with spontaneous infection in addition to disease transmission our results show that spontaneous infection enhances the persistence of disease the uniform bounds of solutions which are derived by a feedback argument play a fundamental role in our analysis

distributed parameter estimation in sensor networks nonlinear observation models and imperfect communication the paper studies distributed static parameter vector estimation in sensor networks with nonlinear observation models and noisy intersensor communication it introduces separably estimable observation models that generalize the observability condition in linear centralized estimation to nonlinear distributed estimation it studies two distributed estimation algorithms in separably estimable models the nu with its linear counterpart lu and the nlu their update rule combines a consensus step where each sensor updates the state by weight averaging it with its neighbors states and an innovation step where each sensor processes its local current observation this makes the three algorithms of the consensus innovations type very different from traditional consensus this paper proves consistency all sensors reach consensus almost surely and converge to the true parameter value efficiency and asymptotic unbiasedness for lu and nu it proves asymptotic normality and provides convergence rate guarantees the three algorithms are characterized by appropriately chosen decaying weight sequences algorithms lu and nu are analyzed in the framework of stochastic approximation theory algorithm nlu exhibits mixed time scale behavior and biased perturbations and its analysis requires a different approach that is developed in this paper

convergence in competition models with small diffusion coefficients abstract it is well known that for reaction diffusion 2 species lotka volterra competition models with spatially independent reaction terms global stability of an equilibrium for the reaction system implies global stability for the reaction diffusion system this is not in general true for spatially inhomogeneous models we show here that for an important range of such models for small enough diffusion coefficients global convergence to an equilibrium holds for the reaction diffusion system if for each point in space the reaction system has a globally attracting hyperbolic equilibrium this work is planned as an initial step towards understanding the connection between the asymptotics of reaction diffusion systems with small diffusion coefficients and that of the corresponding reaction systems

asymptotic profile of the positive steady state for an sis epidemic reaction diffusion model effects of epidemic risk and population movement abstract identifying the epidemic risk for infectious disease is crucial in order to effectively perform control measures in a series of our work from an analytical aspect we study the effects of epidemic risk and population movement on the spatiotemporal transmission of infectious disease via an sis epidemic reaction diffusion model proposed by allen et al 2008 in 36 in allen et al 2008 36 peng 2009 37 it was assumed that the habitat of the populations consists of only the low and high risk areas the present paper concerns a more complicated heterogeneous environment where the moderate risk area occurs and deals with two cases i only the moderate and high risk areas exist ii the low moderate and high risk areas coexist in each case we rigorously determine the asymptotic profile of the positive steady state i e the endemic equilibrium as the migration rate of either the susceptible or infected population tends to zero our results show how epidemic risk and population movement affect the spatial distribution of infectious disease and thereby suggest important implications for predicting the patterns of disease occurrence and designing optimal control strategies numerical simulations are carried out to support the theoretical results

computation of the greatest right and left invariant fuzzy quasi orders and fuzzy equivalences abstract right invariant fuzzy quasi orders for fuzzy automata are broadly studied in the recent literature as they arise as solutions to particular systems of fuzzy relation equations and inequalities some of their applications include determinization and state reduction procedures as well as simulations and bisimulations for fuzzy automata in this paper we provide a procedure for computing the greatest right invariant fuzzy quasi order for a given fuzzy automaton over a complete residuated lattice the proposed procedure terminates in a finite number of steps whenever the underlying structure of the fuzzy automaton is locally finite when the previous condition is not satisfied we show that the greatest right invariant fuzzy quasi order can be obtained by taking the limit value of the convergent array of fuzzy quasi orders for fuzzy automata over bl algebras on the real unit interval 0 1 analogous procedures for computing the greatest left invariant fuzzy quasi order as well as the greatest right and left invariant fuzzy equivalences for a fuzzy automaton are also presented in addition the faster algorithm for computing the greatest right invariant equivalence on a nondeterministic automaton is also presented

dynamics and profiles of a diffusive host pathogen system with distinct dispersal rates abstract in this paper we investigate a diffusive host pathogen model with heterogeneous parameters and distinct dispersal rates for the susceptible and infected hosts we first prove that the solution of the model exists globally and the model system possesses a global attractor we then identify the basic reproduction number r 0 for the model and prove its threshold role if r 0 1 the disease free equilibrium is globally asymptotically stable if r 0 1 the solution of the model is uniformly persistent and there exists a positive pathogen persistent steady state finally we study the asymptotic profiles of the positive steady state as the dispersal rate of the susceptible or infected hosts approaches zero our result suggests that the infected hosts concentrate at certain points which can be characterized as the pathogen s most favoured sites when the mobility of the infected host is limited

comparing fuzzy and crisp deduction systems abstract the connections between the fuzzy deduction systems and the crisp deduction systems are examined we show that there is a natural way to associate to every fuzzy deduction system a crisp deduction system and to every crisp deduction system a fuzzy deduction system

robust finite time containment control for high order multi agent systems with matched uncertainties under directed communication graphs abstractin this paper we study the robust finite time containment control problem for a class of high order uncertain nonlinear multi agent systems modelled as high order integrator systems with bounded matched uncertainties when relative state information between neighbouring agents is available an observer based distributed controller is proposed for each follower using the sliding mode control technique which solves the finite time containment control problem under general directed communication graphs when only relative output information is available robust exact differentiators and high order sliding mode controllers are employed together with the distributed finite time observers it is shown that robust finite time containment control can still be achieved in this situation an application in the coordination of multiple non holonomic mobile robots is used as an example to illustrate the effectiveness of the proposed control strategies

fuzzy logics as families of bivaluated logics abstract in this paper we will prove the equivalence between l valued logics and l indexed families of bivaluated logics it is also proved that this equivalence can be used as a tool for checking compactness and decidability properties of a l valued logic

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

dynamics of a nonlocal dispersal sis epidemic model with neumann boundary conditions abstract in this paper we study a nonlocal dispersal susceptible infected susceptible sis epidemic model with neumann boundary condition where the spatial movement of individuals is described by a nonlocal convolution diffusion operator the transmission rate and recovery rate are spatially heterogeneous and the total population number is constant we first define the basic reproduction number r 0 and discuss the existence uniqueness and stability of steady states of the nonlocal dispersal sis epidemic model in terms of r 0 then we consider the impacts of the large diffusion rates of the susceptible and infectious populations on the persistence and extinction of the disease the obtained results indicate that the nonlocal movement of the susceptible or infectious individuals will enhance the persistence of the infectious disease in particular our analytical results suggest that the spatial heterogeneity tends to boost the spread of the infectious disease

distributed finite time containment control for second order nonlinear multi agent systems this paper considers the finite time containment control problem for a class of second order nonlinear multi agent systems with external disturbances under the directed topology graph the considered nonlinear systems are composed of multiple dynamic leaders and they have only a subset of the followers with their accessing to the leaders firstly by using the one hop neighbors information a second order finite time distributed observer estimator protocol is constructed to achieve accurate estimation of the weighted average of the velocities and positions of the leaders furthermore it is proved that the proposed estimator protocol can guarantee accurate estimation of the weighted average of the velocities and positions of the leaders in the finite time then we design a new continuous nonlinear control protocol by means of the nonsingular terminal sliding mode it is shown that by using proposed control protocol the states of the followers can converge to the convex hull spanned by those of the leaders in the finite time in addition the finite convergence time can be also explicitly estimated finally the effectiveness of the obtained theoretical results is illustrated by the numerical simulation

asymptotic profiles of the positive steady state for an sis epidemic reaction diffusion model part i abstract to capture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease allen et al in l j s allen b m bolker y lou a l nevai asymptotic profiles of the steady states for an sis epidemic reaction diffusion model discrete contin dyn syst ser a 21 1 2008 1 20 proposed a spatial sis susceptible infected susceptible reaction diffusion model and studied the existence uniqueness and particularly the asymptotic behavior of the endemic equilibrium as the diffusion rate of the susceptible individuals goes to zero in the case where a so called low risk subhabitat is created in this work we shall provide further understanding of the impacts of large and small diffusion rates of the susceptible and infected population on the persistence and extinction of the disease which leads us to determine the asymptotic behaviors of the endemic equilibrium when the diffusion rate of either the susceptible or infected population approaches to infinity or zero in the remaining cases consequently our results reveal that in order to eliminate the infected population at least in low risk area it is necessary that one will have to create a low risk subhabitat and reduce at least one of the diffusion rates to zero in this case our results also show that different strategies of controlling the diffusion rates of individuals may lead to very different spatial distributions of the population moreover once the spatial environment is modified to include a low risk subhabitat the optimal strategy of eradicating the epidemic disease is to restrict the diffusion rate of the susceptible individuals rather than that of the infected ones

dynamics and asymptotic profiles of steady states of an epidemic model in advective environments abstract we study the dynamics of a sis epidemic model of reaction diffusion advection type the persistence of infected and susceptible populations and the global stability of the disease free equilibrium are established when the basic reproduction number is greater than or less than or equal to one respectively we further consider the effects of diffusion and advection on asymptotic profiles of endemic equilibrium when the advection rate is relatively large comparing to the diffusion rates of both populations then two populations persist and concentrate at the downstream end as the diffusion rate of the susceptible population tends to zero the density of the infected population decays exponentially for positive advection rate but linearly when there is no advection our results suggest that advection can help speed up the elimination of disease

permutable fuzzy consequence and interior operators and their connection with fuzzy relations fuzzy operators are an essential tool in many fields and the operation of composition is often needed in general composition is not a commutative operation however it is very useful to have operators for which the order of composition does not affect the result in this paper we analyze when permutability appears that is when the order of application of the operators does not change the outcome we characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators we prove that for these cases permutability is completely connected to the preservation of the operator type we also study the particular case of fuzzy operators induced by fuzzy relations through zadeh s compositional rule and the inf composition for this cases we connect permutability of the fuzzy relations using the sup composition with permutability of the induced operators special attention is paid to the cases of operators induced by fuzzy preorders and similarities finally we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces

finite time convergent gradient flows with applications to network consensus this paper introduces the normalized and signed gradient dynamical systems associated with a differentiable function extending recent results on nonsmooth stability analysis we characterize their asymptotic convergence properties and identify conditions that guarantee finite time convergence we discuss the application of the results to consensus problems in multi agent systems and show how the proposed nonsmooth gradient flows achieve consensus in finite time

a spatial seirs reaction diffusion model in heterogeneous environment abstract we propose a susceptible exposed infected recovered susceptible seirs reaction diffusion model where the disease transmission and recovery rates can be spatially heterogeneous the basic reproduction number r 0 is connected with the principal eigenvalue of a linear cooperative elliptic system threshold type results on the global dynamics in terms of r 0 are established the monotonicity of r 0 with respect to the diffusion rates of the exposed and infected individuals which does not hold in general is established in several cases finally the asymptotic profile of the endemic equilibrium is investigated when the diffusion rate of the susceptible individuals is small our results reveal the importance of the movement of the exposed and recovered individuals in disease dynamics as opposed to most of previous works which solely focused on the movement of the susceptible and infected individuals

global stability of the steady states of an sis epidemic reaction diffusion model abstract in this work we investigate the sis epidemic reaction diffusion model under heterogeneous environment studied by allen et al in l j s allen b m bolker y lou a l nevai asymptotic profiles of the steady states for an sis epidemic reaction diffusion model discrete contin dyn syst a 21 1 2008 1 20 in the two cases 1 the diffusion rate d s of the susceptible individuals is equal to the diffusion rate d i of the infected individuals 2 x r x for any fixed constant r 0 where x and x respectively represent the rates of disease transmission and disease recovery we completely determine the global stability of the disease free equilibrium and the unique endemic equilibrium if it exists our results partially answer the conjecture proposed by allen et al

containment control in mobile networks in this paper the problem of driving a collection of mobile robots to a given target destination is studied in particular we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader agents while the remaining agents only employ local interaction rules to this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop go rules for the leader agents non zenoness liveness and convergence of the resulting system are also analyzed

a non commutative and non idempotent theory of quantale sets in fuzzy set theory non idempotency arises when the conjunction is interpreted by arbitrary t norms there are many instances in mathematics where set theory ought to be non commutative and or non idempotent the purpose of this paper is to combine both ideas and to present a theory of non commutative and non idempotent quantale sets among other things standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases more specifically the category q set is investigated where q is an arbitrary involutive quantale objects quantale sets are pairs consisting of a set and a q valued equality with a suitable symmetry axiom three important properties of q set are shown it is complete cocomplete and has the epi extremal mono factorization property its subcategory s q set of separated quantale sets is reflective in q set and shares the same fundamental properties with q set in particular s q set is also a complete and cocomplete epi extremal mono category the objects of the category q set are interesting categories in their own right two categorical frameworks for objects of q set are exhibited first it is shown that q valued equalities arise from q valued preorders with self adjoint extents by symmetrization which leaves q valued equalities invariant here sets with quantale preorders are shown to be b categories with base b being a specific quantaloid the second approach is based on involutive quantaloids a combination of two well known things quantaloids and ordered categories with involution in this context quantale sets are precisely symmetric b categories w r t appropriately chosen quantaloid b with involution further the cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames our non commutative generalization of frames and all left continuous t norms there exist at least two monads on q set the singleton monad and quasi singleton monad which are of special interest for fuzzy set theory the kleisli category associated with the singleton monad is the non commutative and non idempotent analogue of higgs topos while the eilenberg moore category of the quasi singleton monad permits the internalization of lukasiewicz negation as truth arrow finally an application of quantale sets to c algebras is given and the change of base is treated

distributed finite time leaderless consensus control for double integrator multi agent systems with external disturbances we study the finite time consensus control of double integrator multi agent systems with external bounded disturbances under the fixed leaderless network communication topology based on the relative position and the relative velocity information a novel finite time consensus protocol is constructed by using the lyapnuov finite time theory it is shown that the proposed protocol can coordinate the states of agents to guarantee the finite time consensus under an undirected connected topology and then we extend the results onto the strongly connected directed communication topology importantly the finite convergence time of the leaderless systems is explicitly presented finally the effectiveness of the results is illustrated by numerical simulation

distributed consensus tracking of multi agent systems with nonlinear dynamics under a reference leader this paper investigates the distributed consensus tracking problem for multi agent systems with lipschitz type dynamics under a reference leader it is assumed that the leader state information is only available to a subset of followers while the bounded reference input of the leader s is unavailable to any follower to achieve consensus tracking a class of discontinuous protocols based on the relative information between the neighbouring agents are proposed furthermore as extensions of the former result the robust and adaptive consensus tracking problems are studied for the case where there exist parameter uncertainties and external disturbances in the network finally the effectiveness of the theoretical result is demonstrated through a network of single link manipulators

asymptotic profiles of steady states for a diffusive sis epidemic model with mass action infection mechanism abstract mass action and standard incidence are two major infection mechanisms in modelling spread of infectious diseases spatial heterogeneity plays an important role in spread of infectious diseases and hence motivates and advocates diffusive models for disease dynamics by analyzing a diffusive sis model with the standard incidence infection mechanism some recent works 2 12 have investigated the asymptotical profiles of the endemic steady state for large and small diffusion rates and the results show that controlling the diffusion rate of the susceptible individuals can help eradicate the infection while controlling the diffusion rate of the infectious individuals cannot this paper aims to reveal the difference between the two infection mechanisms in a spatially heterogeneous environment to this end we consider a diffusive sis model of the same structure but with the mass action infection adopted and explore the asymptotic profiles of the endemic steady state for small and large diffusion rates it turns out that the new model poses some new challenges due to the nonlocal term in the equilibrium problem and the unboundedness of the nonlinear term our results on this new model reveal some fundamental differences between the two transmission mechanisms in such spatial models which may provide some implications on disease modelling and controls