dynamics of the deterministic and stochastic siqs epidemic model with non linear incidence the deterministic and stochastic siqs models with non linear incidence are introduced for deterministic model the basic reproductive rate r 0 is derived moreover if r 0 1 there exists a unique endemic equilibrium which is globally asymptotically stable for stochastic model sufficient condition for extinction of the disease that is regardless of the value of r 0 is presented in addition if the intensities of the white noises are sufficiently small and r 0 1 then there exists a unique stationary distribution to stochastic model numerical simulations are also carried out to confirm the analytical results

a proximal point algorithm for sequential feature extraction applications we propose a proximal point algorithm to solve the laros problem that is the problem of finding a large approximately rank one submatrix this laros problem is used to sequentially extract features in data we also develop new stopping criteria for the proximal point algorithm which is based on the duality conditions of epsilon optimal solutions of the laros problem with a theoretical guarantee we test our algorithm with two image databases and show that we can use the laros problem to extract appropriate common features from these images

analysis of a hybrid switching svir epidemic model with vaccination and levy noise abstract this paper is concerned with the dynamical behavior of a hybrid switching svir epidemic model with vaccination and levy noise besides a standard geometric brownian motion another two driving processes are considered a stationary poisson point process and a continuous time finite state markov chain to begin with we obtain sufficient conditions for persistence in the mean of the disease then we establish sufficient conditions for extinction of the disease furthermore in the case of persistence we also obtain sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic lyapunov function with regime switching

stationary distribution and extinction of a stochastic tuberculosis model abstract in this paper we focus on a stochastic tuberculosis model we first obtain the existence of a stationary distribution to the positive solutions by stochastic lyapunov function method then we establish sufficient conditions for extinction of the disease finally some examples and numerical simulations are provided to illustrate our theoretical results

automata reduced words and garside shadows in coxeter groups abstract in this article we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated coxeter system w s the definition of these automata is straightforward as it only requires the notion of weak order on w s and the related notion of garside shadows in w s an analog of the notion of a garside family then we discuss the relations between this class of automata and the canonical automaton built from brink and howlett s small roots we end this article by providing partial positive answers to two conjectures 1 the automata associated to the smallest garside shadow is minimal 2 the canonical automaton is minimal if and only if the support of all small roots is spherical i e the corresponding root system is finite

subdifferentials faces and dual matrices abstract a characterization of the dual matrices for the unitarily invariant norms is given moreover the connection between the dual matrices the subdifferentials of matrix norms and the faces of the unit ball of matrices is presented

complete global stability for an sirs epidemic model with generalized non linear incidence and vaccination abstract this paper deals with global dynamics of an sirs epidemic model for infections with non permanent acquired immunity the sirs model studied here incorporates a preventive vaccination and generalized non linear incidence rate as well as the disease related death lyapunov functions are used to show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less than or equal to one and that there is an endemic equilibrium state which is globally asymptotically stable when it is greater than one

reflection subgroups of coxeter systems let w r be a coxetcr system and t ij w wrw denote the corresponding set of reflections for u e w define n w tv tlf tw l w h w ere i denotes the length function of w r the principal result 3 3 of this paper is that if w is a reflection subgroup of w i e w w n t then e ti n z n w 1 is a set of coxeter generators for w in particular w is a coxeter group in 6 v deodhar gives a geometric proof of this last fact using properties of the root system of w it is easily seen that the coxeter gcncrators for w defined by deodhar coincide with those described above the idea of the proof here is that n may be regarded as a cocycle with n r r rer and coxeter systems are characterized among groups generated by involutions r by the existence of such a cocycle one then checks that w h n w n w w e w gives a suitable cocycle for w the same argument applies to a wider class of reflection systems g x characterized by the cxistcncc of a certain cocycle and so we give the proof for these the coxeter systems are precisely the reflection systems g x in which x consists of involutions the arrangement of this paper is as follows in section 2 we describe the main facts about reflection systems closely paralleling 1 chap iv sect 11 in section 3 we prove that reflection subgroups of a reflection system are themselves reflection systems in a canonical way 3 3 and obtain some additional facts about their canonical generators 3 4 3 11 in section 4 we specialize to the case of a coxeter system and give a geometric criterion 4 4 for a set of reflections to be the set of canonical generators of a reflection subgroup in terms of the inner products of the

dynamical analysis in a bioeconomic phytoplankton zooplankton system with double time delays and environmental stochasticity abstract this paper presents a double delayed bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered environmental stochasticity is incorporated into the proposed system in form of gaussian white noises some sufficient conditions are derived to show that the proposed system has a unique global positive solution in absence of double time delays stochastic stability and existence of stochastic hopf bifurcation are studied based on invariant measure theory and singular boundary theory of diffusion process for the proposed system in presence of double time delays asymptotic behaviors of the interior equilibrium are discussed by constructing some appropriate lyapunov functions

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

on approximation problems with zero trace matrices abstract we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms we deal with special cases of approximation of a matrix by zero trace matrices moreover some characterizations of zero trace matrices are given by means of matrix approximation problems

dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations abstract in this paper we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered stochastic fluctuations are incorporated into the proposed system in form of gaussian white noises to depict stochastic environmental factors in plankton system for deterministic system without double time delays existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting and state feedback controllers are designed to eliminate singularity induced bifurcation for deterministic system with double time delays positivity and uniform persistence of solutions are studied and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated for stochastic system without double time delays stochastic stability and existence of stochastic hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory for stochastic system with double time delays existence and uniqueness of global positive solution are investigated and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate lyapunov functions numerical simulations are carried out to validate theoretical analysis

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

dynamics in a stochastic heroin model with seasonal variation abstract in this paper a stochastic non autonomous heroin model is formulated we obtain sufficient conditions for extinction and permanence in mean with probability one of the positive solutions in addition we prove that there exists at least one positive periodic solution under some conditions finally numerical simulations are performed to illustrate our theoretical results

stochastic seir model with jumps abstract the paper establishes stochastic seir models with jumps obtains system 2 and system 3 by using two different disturbance manners respectively which are used to describe the wide spread of the infectious diseases due to the medical negligence etc by adding the first random perturbations we obtain lyapunov function and examine that the solutions of model 2 are stochastically stable with the average time summarize the conditions for persistence and extinction of the infectious diseases by adding another random disturbance we get the stochastic system 3 we prove that the positive solution of the system is stochastic asymptotically stable by using lyapunov method the conditions that the infectious diseases spread continually are obtained these studies are good for the prevention and control of the infectious diseases which are with the incubation period especially with long ones

orthogonality to matrix subspaces and a distance formula abstract we obtain a necessary and sufficient condition for a matrix a to be birkhoff james orthogonal to any subspace w of m n c using this we obtain an expression for the distance of a from any unital c subalgebra of m n c

epidemic outbreaks and its control using a fractional order model with seasonality and stochastic infection abstract a fractional order sir epidemic model is proposed under the influence of both parametric seasonality and the external noise the integer order sir epidemic model originally is stable by introducing seasonality and noise force to the model behaviors of the system is changed it is shown that the system has rich dynamical behaviors with different system parameters fractional derivative order and the degree of seasonality and noise complexity of the stochastic model is investigated by using multi scale fuzzy entropy finally hard limiter controlled system is designed and simulation results show the ratio of infected individuals can converge to a small enough target which means the epidemic outbreak can be under control by the implementation of some effective medical and health measures

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

on the characterization of the extremal points of the unit sphere of matrices abstract we give a characterization of the extremal points of the unit sphere of matrices for the unitarily invariant norms we also investigate the properties of the dual matrices moreover we show that a result given by lau and riha is not correct

a semismooth newton cg based dual ppa for matrix spectral norm approximation problems we consider a class of matrix spectral norm approximation problems for finding an affine combination of given matrices having the minimal spectral norm subject to some prescribed linear equality and inequality constraints these problems arise often in numerical algebra engineering and other areas such as finding chebyshev polynomials of matrices and fastest mixing markov chain models based on classical analysis of proximal point algorithms ppas and recent developments on semismooth analysis of nonseparable spectral operators we propose a semismooth newton cg based dual ppa for solving the matrix norm approximation problems furthermore when the primal constraint nondegeneracy condition holds for the subproblems our semismooth newton cg method is proven to have at least a superlinear convergence rate we also design efficient implementations for our proposed algorithm to solve a variety of instances and compare its performance with the nowadays popular first order alternating direction method of multipliers admm the results show that our algorithm which is warm started with an initial point obtained from the admm substantially outperforms the pure admm especially for the constrained cases and it is able to solve the problems robustly and efficiently to a relatively high accuracy

finding the largest low rank clusters with ky fan 2 k norm and ell_1 norm we propose a convex optimization formulation with the ky fan 2 k norm and ell 1 norm to find k largest approximately rank one submatrix blocks of a given nonnegative matrix that has low rank block diagonal structure with noise we analyze low rank and sparsity structures of the optimal solutions using properties of these two matrix norms we show that under certain hypotheses with high probability the approach can recover rank one submatrix blocks even when they are corrupted with random noise and inserted into a much larger matrix with other random noise blocks

dynamics of a stochastic siqr epidemic model with standard incidence abstract in this paper a stochastic siqr epidemic model with quarantine and standard incidence is investigated we first consider the existence and uniqueness of the global positive solution to the stochastic system further we obtain conclusions that the large noise can make the disease die out exponentially and the stationary distribution will exist under certain conditions

optimal control and the value of information for a stochastic epidemiological sis model abstract we analyze a continuous time stochastic sis epidemiological model describing the transmission dynamics of infectious diseases where infected individuals can become susceptible again after recovery the infectivity period of infected individuals can be influenced dynamically by a decision maker with the aim to minimize the expected aggregated economic costs due to illness and drug treatment this stochastic control problem is investigated under two alternative assumptions about the information available if a complete and exact measurement of the size of the infected population is available at any time then the optimal control is sought in a state feedback form and the related hamilton jacobi bellman hjb equation can be used if no state measurement is available then the optimal control is sought in an open loop form and the problem can be reformulated as an optimal control problem for the kolmogorov forward equation in both cases deriving optimality conditions requires non standard arguments due to the degeneracy of the involved hjb and kolmogorov equations based on the obtained theoretical results the role of the information pattern is numerically investigated

characterization of best approximations in normed linear spaces of matrices by elements of finite dimensional linear subspaces abstract a general characterization theorem of best approximations in normed linear spaces is specialized to the linear space of real n n matrices endowed with the spectral norm

stability analysis of delayed sir epidemic models with a class of nonlinear incidence rates we analyze stability of equilibria for a delayed sir epidemic model in which population growth is subject to logistic growth in absence of disease with a nonlinear incidence rate satisfying suitable monotonicity conditions the model admits a unique endemic equilibrium if and only if the basic reproduction number r0 exceeds one while the trivial equilibrium and the disease free equilibrium always exist first we show that the disease free equilibrium is globally asymptotically stable if and only if r0 1 second we show that the model is permanent if and only if r0 1 moreover using a threshold parameter r0 characterized by the nonlinear incidence function we establish that the endemic equilibrium is locally asymptotically stable for 1 r0 r0 and it loses stability as the length of the delay increases past a critical value for 1 r0 r0 our result is an extension of the stability results in j j wang j z zhang z jin analysis of an sir model with bilinear incidence rate nonl anal rwa 11 2009 2390 2402

necessary and sufficient condition for extinction and persistence of sirs system with random perturbation abstract this paper characterizes the qualitative dynamics of a stochastic sirs epidemic model the study shows that the dynamics of the model are determined by a certain threshold quantity r s expressed in terms of the model parameters and the intensity of the noise if r s 1 the disease will be eliminated from the community whereas an epidemic occurs if r s 1 our results recover the known results in the earlier literature as special cases the presented results are illustrated by numerical simulations

a stochastic threshold for an epidemic model with beddington deangelis incidence delayed loss of immunity and levy noise perturbation abstract in this paper we study a stochastic epidemic model with delayed loss of immunity and levy noise perturbation we establish the uniqueness of the global positive solution we also investigate the extinction and the persistence in mean of the disease

stationary distribution of a stochastic sis epidemic model with vaccination abstract in this paper we consider a stochastic sis epidemic model with vaccination we prove that the densities of the distributions of the solution can converge in l 1 to an invariant density under appropriate conditions also we find the support of the invariant density

threshold behavior in a stochastic siqr epidemic model with standard incidence and regime switching abstract this paper is concerned with the dynamical behavior of a stochastic siqr epidemic model with standard incidence which is disturbed by both white and telegraph noises firstly we obtain sufficient conditions for persistence in the mean of the disease then we establish sufficient conditions for extinction of the disease in addition in the case of persistence we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic lyapunov function with regime switching meanwhile the threshold between persistence in the mean and extinction of the stochastic system is also obtained finally some numerical simulations are introduced to demonstrate the analytical results

the threshold of a periodic stochastic sivs epidemic model with nonlinear incidence abstract in this paper a stochastic periodic sivs epidemic model with nonlinear incidence and vaccination is investigated the threshold conditions on the existence of stochastic positive periodic solutions and the extinction of disease with probability one are established by constructing the new stochastic lyapunov functions and using the new technique to deal with the nonlinear incidence and vaccination for the stochastic epidemic model the numerical simulations are given to illustrate the main theoretical results and present some new interesting conjectures

analysis of the deterministic and stochastic sirs epidemic models with nonlinear incidence abstract in this paper the deterministic and stochastic sirs epidemic models with nonlinear incidence are introduced and investigated for deterministic system the basic reproductive number r 0 is obtained furthermore if r 0 1 then the disease free equilibrium is globally asymptotically stable and if r 0 1 then there is a unique endemic equilibrium which is globally asymptotically stable for stochastic system to begin with we verify that there is a unique global positive solution starting from the positive initial value then when r 0 1 we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent when r 0 1 a result on fluctuation of the solution around the disease free equilibrium of deterministic model is obtained under appropriate conditions at last if the intensity of the white noise is sufficiently small and r 0 1 then there is a unique stationary distribution to stochastic system

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

stochastic analysis of a novel nonautonomous periodic siri epidemic system with random disturbances abstract in this paper a new stochastic nonautonomous siri epidemic model is formulated given that the incidence rates of diseases may change with the environment we propose a novel type of transmission function the main aim of this paper is to obtain the thresholds of the stochastic siri epidemic model to this end we investigate the dynamics of the stochastic system and establish the conditions for extinction and persistence in mean of the disease by constructing some suitable lyapunov functions and using stochastic analysis technique furthermore we show that the stochastic system has at least one nontrivial positive periodic solution finally numerical simulations are introduced to illustrate our results

stochastic asymptotic analysis of a multi host model with vector transmission abstract we present a stochastic epidemic model with vectorial transmission and multi host structure to include environmental noise we stochastically perturb biting rates with general state functional intensities so we derive a stochastic differential equation sde which describes a vector disease with two types of hosts humans and animals and give conditions to assure disease extinction and persistence finally via numerical experiments we extend and illustrate our results using literature parameters for chagas

the dominance hierarchy in root systems of coxeter groups abstract if x and y are roots in the root system with respect to the standard tits geometric realization of a coxeter group w we say that x dominates y if for all w w wy is a negative root whenever wx is a negative root we call a positive root elementary if it does not dominate any positive root other than itself the set of all elementary roots is denoted by e it has been proved by b brink and r b howlett b brink r b howlett a finiteness property and an automatic structure of coxeter groups math ann 296 1993 179 190 that e is finite if and only if w is a finite rank coxeter group amongst other things this finiteness property enabled brink and howlett to establish the automaticity of all finite rank coxeter groups later brink has also given a complete description of the set e for arbitrary finite rank coxeter groups b brink the set of dominance minimal roots j algebra 206 1998 371 412 however the set of non elementary positive roots has received little attention in the literature in this paper we answer a collection of questions concerning the dominance behavior between such non elementary positive roots in particular we show that for any finite rank coxeter group and for any non negative integer n the set of roots each dominating precisely n other positive roots is finite we give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation

small roots low elements and the weak order in coxeter groups abstract in this article we provide a new finite class of elements in any coxeter system w s called low elements they are defined from brink and howlett s small roots which are strongly linked to the automatic structure of w s our first main result is to show that they form a garside shadow in w s i e they contain s and are closed under join for the right weak order and by taking suffixes these low elements are the key to prove that all finitely generated artin tits groups have a finite garside family this result was announced in a note with p dehornoy in comptes rendus mathematiques 9 in which the present article was referred to under the following working title monotonicity of dominance depth on root systems and applications the proof is based on a fundamental property enjoyed by small roots and which is our second main result the set of small root is bipodal for a natural number n we define similarly n low elements from n small roots and conjecture that the set of n small roots is bipodal implying the set of n low elements is a garside shadow we prove this conjecture for affine coxeter groups and coxeter groups whose graph is labelled by 3 and to prove the latter we extend the root poset on positive roots to a weak order on the root system and define a bruhat order on the root system and study the paths in those orders in order to establish a criterion to prove bipodality involving only finite dihedral reflection subgroups

survival and stationary distribution of a sir epidemic model with stochastic perturbations abstract in this paper the dynamics of a sir epidemic model is investigated first we show that the system admits a unique positive global solution starting from the positive initial value then when r 0 1 we show that the stochastic model has a stationary distribution under certain parametric restrictions in particular we show that random effects may lead the disease to extinction in scenarios where the deterministic model predicts persistence when r 0 1 a result on fluctuation of the solution around the disease free equilibrium of deterministic system is established under suitable conditions finally numerical simulations are carried out to illustrate the theoretical results

some aspects of the theory of norms abstract let v be a finite dimensional vector space motivated by theory or applications one might want to consider different kinds of norms on v in this paper we discuss some results and problems involving different classes of norms on a vector space studied by this author in the past few years the paper consists of five sections section 1 concerns the conditions on two vectors x y v satisfying x y for all in a certain class of norms section 2 concerns the isometry groups of g invariant norms i e norms that satisfy g x x for all x v and for all g g where g is a group of unitary orthogonal operators on v section 3 concerns g invariant norms that satisfy some special properties section 4 concerns the best approximation s x 0 t of y where y t v with respect to different kinds of norms additional open problems topics and references are mentioned in section 5

analysis of a saturation incidence sveirs epidemic model with pulse and two time delays in this paper we study a new sveirs infectious disease model with pulse and two time delays the pulse vaccination strategy is used as an effective strategy for the elimination of infectious disease the model consists of a set of integro differential equations the existence and global attractivity of infection free periodic solution permanence of an endemic model are investigated

finding approximately rank one submatrices with the nuclear norm and ell_1 norm we propose a convex optimization formulation with the nuclear norm and ell 1 norm to find a large approximately rank one submatrix of a given nonnegative matrix we develop optimality conditions for the formulation and characterize the properties of the optimal solutions we establish conditions under which the optimal solution of the convex formulation has a specific sparse structure finally we show that under certain hypotheses with high probability the approach can recover the rank one submatrix even when it is corrupted with random noise and inserted as a submatrix into a much larger random noise matrix

extinction and stationary distribution of an epidemic model with partial vaccination and nonlinear incidence rate abstract a stochastic susceptible infected vaccinated model with a general incidence rate and varying population size is considered in this paper by means of constructing a suitable lyapunov function the sufficient conditions for the extinction are derived further some conditions that guarantee the existence of a unique stationary distribution is obtained the main results are illustrated by computer simulations

the threshold of a stochastic sis epidemic model with vaccination this paper is concerned with the long time behavior of a stochastic sis epidemic model with vaccination sufficient conditions for extinction and persistence in mean are obtained we find a threshold of the stochastic model which determines the outcome of the disease in case the white noises are small at last some numerical simulations are carried out to support our results

asymptotic behavior of a stochastic delayed chemostat model with nonmonotone uptake function abstract in this paper a stochastic delay differential equations chemostat model with nonmonotone uptake function is considered and the nutrient conversion process involves time delay first we verify that there is a unique global positive solution of the stochastic system second we find that the solutions of stochastic system will oscillate around the equilibria of the corresponding deterministic model moreover results show that time delay has critical effects on the extinction and persistence of the microorganism that is to say under small noise when the time delay is small microorganism is persistent when the time delay is large microorganism will be extinct in addition we can find by the computer simulation that large noise may lead to microorganism become extinct although microorganism is persistent in the deterministic systems when the time delay is small finally computer simulations are carried out to illustrate the obtained results and the existence of bistability is observed

stability of an seir epidemic model with independent stochastic perturbations abstract for an epidemic model of the type mentioned we prove a theorem on almost sure exponential stability of the disease free equilibrium for small values of the diffusion parameter we describe the stability of the disease free equilibrium point in terms of an appropriate analogue r of the basic reproduction number r 0 of the deterministic special case whenever 0 then r r 0 for small values of the stability theorem guarantees almost sure exponential stability whenever r 1 we also discuss the effect of increasing

global stability of a delayed sirs epidemic model with a non monotonic incidence rate in this paper we investigate a disease transmission model of sirs type with latent period 0 and the specific nonmonotone incidence rate k exp d s t i t 1 i 2 t for the basic reproduction number r0 1 applying monotone iterative techniques we establish sufficient conditions for the global asymptotic stability of endemic equilibrium of system which become partial answers to the open problem in hai feng huo and zhan ping ma dynamics of a delayed epidemic model with non monotonic incidence rate commun nonlinear sci numer simulat 15 2010 459 468 moreover combining both monotone iterative techniques and the lyapunov functional techniques to an sir model by perturbation we derive another type of sufficient conditions for the global asymptotic stability of the endemic equilibrium

imaginary cones and limit roots of infinite coxeter groups let w s be an infinite coxeter system to each geometric representation of w is associated a root system while a root system lives in the positive side of the isotropic cone of its associated bilinear form an imaginary cone lives in the negative side of the isotropic cone precisely on the isotropic cone between root systems and imaginary cones lives the set e of limit points of the directions of roots in this article we study the close relations of the imaginary cone with the set e which leads to new fundamental results about the structure of geometric representations of infinite coxeter groups in particular we show that the w action on e is minimal and faithful and that e and the imaginary cone can be approximated arbitrarily well by sets of limit roots and imaginary cones of universal root subsystems of w i e root systems for coxeter groups without braid relations the free object for coxeter groups finally we discuss open questions as well as the possible relevance of our framework in other areas such as geometric group theory

mean field dispersal induced synchrony and stability in an epidemic model under patchy environment abstract in our highly connected modern world the dispersion of population is very important at the time when disease spread to evaluate its nature i e whether the disease will die out or grow therefore in this paper sir susceptible infected recovered model is framed to describe the dispersal of population for different patches communities which are interconnected using mean field diffusive coupling the effect of coupling on the dynamics of the epidemiology classes among different patches has been investigated the dynamics of the epidemiology classes is explored under the environment of symmetric coupling and asymmetric coupling it is shown that the synchronization and stability of disease free equilibrium or endemic equilibrium can be attained simultaneously through the mechanism of suppression of oscillations namely amplitude death ad and oscillation death od irrespective of coupling environment i e symmetric and asymmetric we have observed the tri stable state and its transition from different states through hopf bifurcation and transcritical bifurcation the basic reproduction number is also established for the stability of disease free equilibrium and endemic equilibrium through bifurcation analysis it has been found that the disease can become stable or it can die out through dispersal of population among patches

dynamics of stochastic seis epidemic model with varying population size abstract we introduce the stochasticity into a deterministic model which has state variables susceptible exposed infected with varying population size in this paper the infected individuals could return into susceptible compartment after recovering we show that the stochastic model possesses a unique global solution under building up a suitable lyapunov function and using generalized ito s formula the densities of the exposed and infected tend to extinction when some conditions are being valid moreover the conditions of persistence to a global solution are derived when the parameters are subject to some simple criteria the stochastic model admits a stationary distribution around the endemic equilibrium which means that the disease will prevail to check the validity of the main results numerical simulations are demonstrated as end of this contribution

a threshold of a delayed stochastic epidemic model with crowly martin functional response and vaccination abstract in this paper we study a delayed stochastic sir epidemic model with crowly martin functional response and vaccination first we prove the existence and the uniqueness of the positive solution therefore we establish a stochastic threshold r s as a sufficient condition for the extinction and persistence in mean of the stochastic epidemic system finally numerical simulations are presented to support our theoretical results

persistence and distribution of a stochastic susceptible infected removed epidemic model with varying population size abstract in this paper the dynamics of a stochastic susceptible infected removed model in a population with varying size is investigated we firstly show that the stochastic epidemic model has a unique global positive solution with any positive initial value then we verify that random perturbations lead to extinction when some conditions are being valid moreover we prove that the solution of the stochastic epidemic model is persistent in the mean by building up a suitable lyapunov function and using generalized ito s formula further the stochastic epidemic model admits a stationary distribution around the endemic equilibrium when parameters satisfy some sufficient conditions to end this contribution and to check the validity of the main results numerical simulations are separately carried out to illustrate these results

the threshold of a stochastic delayed sir epidemic model with vaccination abstract in this paper we study the threshold dynamics of a stochastic delayed sir epidemic model with vaccination we obtain sufficient conditions for extinction and persistence in the mean of the epidemic the threshold between persistence in the mean and extinction of the stochastic system is also obtained compared with the corresponding deterministic model the threshold affected by the white noise is smaller than the basic reproduction number r 0 of the deterministic system results show that time delay has important effects on the persistence and extinction of the epidemic

exponential extinction of a stochastic predator prey model with allee effect abstract in this paper we study a stochastic predator prey model with beddington deangelis functional response and allee effect whose coefficients are dependent on expectations we prove that there is a unique global positive solution to the system with the positive initial value moreover sufficient conditions for exponential extinction are established

stability and bifurcation in a delayed predator prey system with beddington deangelis functional response we consider a delayed predator prey system with beddington deangelis functional response the stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation by choosing the delay as a bifurcation parameter we show that hopf bifurcation can occur as the delay crosses some critical values the direction and stability of the hopf bifurcation are investigated by following the procedure of deriving normal form given by faria and magalhaes an example is given and numerical simulations are performed to illustrate the obtained results

the threshold of a stochastic delayed sir epidemic model with temporary immunity abstract this paper is concerned with the asymptotic properties of a stochastic delayed sir epidemic model with temporary immunity sufficient conditions for extinction and persistence in the mean of the epidemic are established the threshold between persistence in the mean and extinction of the epidemic is obtained compared with the corresponding deterministic model the threshold affected by the white noise is smaller than the basic reproduction number r 0 of the deterministic system

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

dynamics of a stochastic sir epidemic model with saturated incidence in this paper the dynamics of a stochastic sir epidemic model with saturated incidence is investigated firstly we prove that the system has a unique global positive solution with any positive initial value then we verify that random effect may lead the disease to extinction under a simple condition thirdly we establish a sufficient condition for persistence in the mean of the disease moreover we show that there is a stationary distribution to the stochastic system under certain parametric restrictions finally some numerical simulations are carried out to confirm the analytical results

asymptotic behavior of a stochastic delayed seir epidemic model with nonlinear incidence abstract this paper is concerned with a stochastic delayed seir epidemic model with nonlinear incidence firstly we verify that there exists a unique global positive solution of the system then by constructing some suitable lyapunov functions we study the asymptotic behaviors of the disease free equilibrium and the endemic equilibrium respectively

stability analysis of a novel epidemics model with vaccination and nonlinear infectious rate in this paper by considering pathogen evolution and human interventions behaviors with vaccines or drugs we build up a novel seirw model with the vaccination to the newborn children the stability of the seirw model with time varying perturbation to predict the evolution tendency of the disease is analyzed furthermore we introduce a time varying delay into the susceptible and infective stages in the model and give some global exponential stability criteria for the time varying delay system finally numerical simulations are presented to verify the results

the set of dominance minimal roots abstract if and are positive roots in the root system of a coxeter groupw we say that dominates ifw is negative wheneverw is negative forw w we say that is elementary or dominance minimal if it does not dominate any it is shown by the author and r b howlett math ann 296 1993 179 190 that the set e of dominance minimal roots is finite if and only ifwhas finite rank this is used to show thatwis automatic to limit the size of the relevant automata and possibly facilitate other coxeter group algorithms we give an explicit description of the set of elementary roots

global stability of a network based sirs epidemic model with nonmonotone incidence rate abstract this paper studies the dynamics of a network based sirs epidemic model with vaccination and a nonmonotone incidence rate this type of nonlinear incidence can be used to describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals on heterogeneous networks is getting larger using the analytical method epidemic threshold r 0 is obtained when r 0 is less than one we prove the disease free equilibrium is globally asymptotically stable and the disease dies out while r 0 is greater than one there exists a unique endemic equilibrium by constructing a suitable lyapunov function we also prove the endemic equilibrium is globally asymptotically stable if the inhibitory factor is sufficiently large numerical experiments are also given to support the theoretical results it is shown both theoretically and numerically a larger can accelerate the extinction of the disease and reduce the level of disease

modeling and dynamical analysis of a triple delayed prey predator scavenger system with levy jumps abstract in this paper a hybrid triple delayed prey predator scavenger system with commercial harvesting and levy jumps is established where both maturation delay for prey population and gestation delays for predator and scavenger population are considered for deterministic system positivity and uniform permanence of solution are discussed local stability analysis of deterministic system around interior equilibrium is investigated due to variations of triple time delays for stochastic system without time delay sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed for stochastic system with triple time delays existence and uniqueness of global positive solution are studied by constructing appropriate lyapunov functions asymptotic behavior of the hybrid stochastic system with triple time delays and levy jumps is discussed numerical simulations are supported to illustrate theoretical results

asymptotic properties of hybrid diffusion systems in response to the increasing needs for control and optimization of hybrid systems this work is concerned with such asymptotic properties as recurrence also known as weak stochastic stability in the literature and ergodicity of regime switching diffusions using liapunov functions necessary and sufficient conditions for positive recurrence are developed then ergodicity of positive recurrent regime switching diffusions is obtained by constructing cycles using the associated discrete time markov chains

global stability of multi group epidemic models with distributed delays abstract we investigate a class of multi group epidemic models with distributed delays we establish that the global dynamics are completely determined by the basic reproduction number r 0 more specifically we prove that if r 0 1 then the disease free equilibrium is globally asymptotically stable if r 0 1 then there exists a unique endemic equilibrium and it is globally asymptotically stable our proof of global stability of the endemic equilibrium utilizes a graph theoretical approach to the method of lyapunov functionals

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

dynamic analysis of a stochastic rumor propagation model abstract the rapid development of the internet especially the emergence of the social networks leads rumor propagation into a new media era in this paper we are concerned with a stochastic rumor propagation model sufficient conditions for extinction and persistence in the mean of the rumor are established the threshold between persistence in the mean and extinction of the rumor is obtained compared with the corresponding deterministic model the threshold affected by the white noise is smaller than the basic reproduction number r 0 of the deterministic system

the ergodicity and extinction of stochastically perturbed sir and seir epidemic models with saturated incidence abstract in this paper we include stochastic perturbations into sir and seir epidemic models with saturated incidence and investigate their dynamics according to the basic reproduction number r 0 the long time behavior of the two stochastic systems is studied mainly we utilize stochastic lyapunov functions to show under some conditions the solution has the ergodic property as r 0 1 while exponential stability as r 0 1 at last we make simulations to conform our analytical results

stability of sirs system with random perturbations epidemiological models with bilinear incidence rate si usually have an asymptotically stable trivial equilibrium corresponding to the disease free state or an asymptotically stable non trivial equilibrium i e interior equilibrium corresponding to the endemic state in this paper we consider an epidemiological model which is an sirs model with or without distributed time delay influenced by random perturbations we present the stability conditions of the disease free equilibrium of the associated stochastic sirs system

behavior of a stochastic sir epidemic model with saturated incidence and vaccination rules abstract in this paper the threshold behavior of a susceptible infected recovered sir epidemic model with stochastic perturbation is investigated firstly it is obtained that the system has a unique global positive solution with any positive initial value random effect may lead to disease extinction under a simple condition subsequently sufficient condition for persistence has been established in the mean of the disease finally some numerical simulations are carried out to confirm the analytical results

global analysis and numerical simulations of a novel stochastic eco epidemiological model with time delay abstract in this paper a new stochastic eco epidemiological model is proposed and investigated and the high dimensional model includes time delay and a general incidence rate which make the study more complex first the existence and uniqueness of the global positive solution with any positive initial value are verified second using stochastic analysis methods and some differential inequality techniques we explore the long time asymptotic properties of the stochastic delayed system furthermore we prove that the solution of the stochastic delayed system has a unique stationary distribution and it has ergodic property third the sufficient conditions of permanence in mean or extinction for the ecological populations are obtained finally we introduce a series of computer simulations to show the performance of the theoretical results

strong convergence of the partially truncated euler maruyama scheme for a stochastic age structured sir epidemic model abstract we study in this paper strong convergence of the partially truncated euler maruyama scheme for an age structured susceptible infected removed sir epidemic model with environmental noise using the semigroup theory the existence and uniqueness of global positive solution for the model is first proved we then define a truncated function and develop a partially truncated em numerical solutions to the stochastic age structured sir epidemic model we present the pth moment boundedness of the partially truncated em numerical approximate solutions under appropriate conditions furthermore the strong lq convergence is established for the condition of 2 q