stability of a general hiv 1 reaction diffusion model with multiple delays and immune response abstract the aim of this paper is to study a general hiv 1 reaction diffusion model with space dependent diffusion coefficients and antibody immune response the model contains four discrete time delays and three categories of infected cells latently infected cells short lived productively infected cells and long lived productively infected cells the effect of highly active antiretroviral therapies haart used to treat hiv 1 is incorporated into the model the existence non negativity and boundedness of solutions are studied in order to verify the well posedness of the model all possible equilibria of the model are evaluated and the threshold parameters needed for their existence and stability are investigated the global asymptotic stability of the corresponding equilibria are proved by building suitable lyapunov functionals the numerical simulations are conducted in order to confirm the theoretical results and show the behavior of solutions in space and time

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

global stability of an hiv 1 infection model with saturation infection and intracellular delay in this paper an hiv 1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated by analyzing the characteristic equations the local stability of an infection free equilibrium and a chronic infection equilibrium of the model is established by using suitable lyapunov functionals and the lasalle invariant principle it is proved that if the basic reproduction ratio is less than unity the infection free equilibrium is globally asymptotically stable if the basic reproduction ratio is greater than unity the chronic infection equilibrium is globally asymptotically stable

global dynamics of a reaction diffusion virus infection model with humoral immunity and nonlinear incidence abstract in this paper we propose and investigate a reaction diffusion virus infection model with humoral immunity and nonlinear incidence in spatially heterogeneous case the basic reproduction number of virus infection r 0 is calculated when r 0 1 the global asymptotical stability of the infection free steady state is established and when r 0 1 the uniform persistence of infected cells and viruses as well as the existence of antibody free infection steady state are also obtained in spatially homogeneous case the antibody response basic reproduction number r 1 is calculated by using the lyapunov functions method and the persistence theory of dynamical systems we obtain that when r 0 1 and r 1 1 the antibody free infection equilibrium is globally asymptotically stable and when r 0 1 and r 1 1 the model is uniformly persistent and the infection equilibrium exists and is also globally asymptotically stable finally the numerical examples are presented in order to verify the validity of our theoretical results

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

asymptotic behaviors of a cell to cell hiv 1 infection model perturbed by white noise abstract in this paper we analyze a mathematical model of cell to cell hiv 1 infection to c d 4 t cells perturbed by stochastic perturbations first of all we investigate that there exists a unique global positive solution of the system for any positive initial value then by using lyapunov analysis methods we study the asymptotic property of this solution moreover we discuss whether there is a stationary distribution for this system and if it owns the ergodic property numerical simulations are presented to illustrate the theoretical results

the smallest eigenvalue of the signless laplacian recently the signless laplacian matrix of graphs has been intensively investigated while there are many results about the largest eigenvalue of the signless laplacian the properties of its smallest eigenvalue are less well studied the present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless laplacian

a numerical method for delayed partial differential equations describing infectious diseases abstract in this paper we propose a numerical method for delayed partial differential equations that describe the dynamics of viral infections such as the human immunodeficiency virus hiv and the hepatitis b virus hbv we first prove that the proposed numerical method preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem by constructing appropriate discrete lyapunov functionals we show that the proposed method also preserves the global stability of equilibria of the corresponding continuous system with no restriction on the space and time step sizes moreover the discrete model and main results presented in qin et al 2014 are extended and generalized

centre manifolds for partial differential equations with delays a centre manifold theory for reaction diffusion equations with temporal delays is developed besides an existence proof we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order as an illustration this reduction procedure is applied to the hutchinson equation with diffusion

global stability for an sei model of infectious disease with immigration we study an sei model of disease transmission with immigration into all three classes for incidence we allow for a nonlinear response to the number of infectives including mass action and saturating incidence as special cases there is no disease free equilibrium and therefore no basic reproduction number for all parameter values the only equilibrium is an endemic equilibrium using a lyapunov function we show that this equilibrium is globally asymptotically stable

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

2 1 kinematical expansions from galilei to de sitter algebras expansion of a lie algebra is the opposite process to contraction starting from a lie algebra the expansion process goes to another algebra which is non isomorphic and less abelian we propose an expansion method based on the casimir invariants of the initial and expanded algebras and where the free parameters involved in the expansion are the curvatures of their associated homogeneous spaces this method is applied for expansions within the family of lie algebras of three dimensional spaces and 2 1 d kinematical algebras we show that these expansions are classed into two types the first type makes the curvature of space or spacetime different from zero i e it introduces a space or universe radius while the other has a similar interpretation for the curvature of the space of worldlines which is non positive and equal to 1 c2 in the kinematical algebras we obtain expansions which go from galilei to either newton hooke or poincarealgebras and from these to de sitter algebras as well as some other examples

a note on the positive semidefiniteness of aα g abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g let 0 g be the smallest for which a g is positive semidefinite it is known that 0 g 1 2 the main results of this paper are 1 if g is d regular then 0 min a g d min a g where min a g is the smallest eigenvalue of a g 2 g contains a bipartite component if and only if 0 g 1 2 3 if g is r colorable then 0 g 1 r

the three dimensional real lie algebras and their contractions in this paper the general contractions of the three dimensional real lie algebras are determined a short summary will be given of the history definitions properties examples and applications of contractions of lie algebras so far mostly simple or generalized inonu wigner contractions have been used the three dimensional real lie algebras are classified in such a way that their general contractions can be easily read off all these contractions can be realized by generalized inonu wigner contractions which supports the argument that these are the interesting objects to study

the aα spectral radius characterization of some digraphs abstract let d be the a spectral radius of digraph d and let g n r be the set of digraphs with order n and dichromatic number r in this paper we characterize the digraph which has the maximal a spectral radius in g n r moreover we also determine the unique digraph which attains the maximum resp minimum a spectral radius among all strongly connected bicyclic digraphs

on the aα spectral radius of a graph abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 3 defined the matrix a g as a g d g 1 a g the largest eigenvalue of a g is called the a spectral radius of g in this paper we give three edge graft transformations on a spectral radius as applications we determine the unique graph with maximum a spectral radius among all connected graphs with diameter d and determine the unique graph with minimum a spectral radius among all connected graphs with given clique number in addition some bounds on the a spectral radius are obtained

periodic trajectories of distributed parameter biochemical systems with time delay abstract this paper deals with a model of a biochemical reactor system with distributed parameters and with a time delay in the growth response time delay has been introduced in microbial growth systems to explain the time lapse between the consumption of liquid substrate and its conversion to solid biomass we study here the properties of the resulting system of partial functional differential equations we first prove the existence positivity and a compactness property of the system trajectories we then prove the existence of periodic solutions of the system for large values of the delay numerical simulations illustrate the existence of such solutions

periodic solutions in an epidemic model with diffusion and delay a spatial diffusion si model with delay and neumann boundary conditions are investigated we derive the conditions of the existence of hopf bifurcation in one dimension space moreover we analyze the properties of bifurcating period solutions by using the normal form theory and the center manifold theorem of partial functional differential pfds equations by numerical simulations we found that spatiotemporal periodic solutions can occur in the epidemic model with spatial diffusion which verifies our theoretical results the obtained results show that interaction of delay and diffusion may induce outbreak of infectious diseases

existence and asymptotic behavior of a functional differential equation in banach space abstract existence and asymptotic behavior of solutions are given for the equation u t a t u t f t u t t 0 and u 0 c r 0 x c the space x is a banach space the family a t 0 t t of unbounded linear operators defined on d a x x generates a linear evolution system and f c x is continuous with respect to a fractional power of a t 0 for some t 0 0 t

complex dynamic behavior in a viral model with delayed immune response the rich dynamics of a viral infection model is studied under the assumption that the immune response is retarded it is shown that if the basic reproductive ratio of the virus is less than one the infection free equilibrium is globally asymptotically stable analytical and numerical results show that if the basic reproductive ratio of the virus is greater than one the combined effect of the strength of the lytic component the time delay of the immune response and the birth rate of susceptible host cells is to create a rich dynamics which includes the occurrence of stable periodic solutions and chaotic dynamical behavior the route from periodic oscillations to chaos is investigated these results can be used to explain irregular real time series data on the immune state of patients

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

the general solution of the real graded contractions of the general solution of the graded contraction equations for a grading of the real compact simple lie algebra so n 1 is presented in an explicit way it turns out to depend on independent real parameters the structure of the general graded contractions is displayed for the low dimensional cases and kinematical algebras are shown to appear straightforwardly the geometrical or physical meaning of the contraction parameters as curvatures is also analysed in particular for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous spacetimes

on the aα spectra of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 8 defined the matrix a g as a g d g 1 a g in this paper we give some results on the eigenvalues of a g for 1 2 in particular we characterize the graphs with k a g n 1 for 2 k n moreover we show that n a g 2 1 if g contains no isolated vertices

central extensions of the quasi orthogonal lie algebras we determine the central extensions of a whole family of lie algebras obtained by the method of graded contractions from n arbitrary all the inhomogeneous orthogonal and pseudo orthogonal algebras are members of this family as well as a large number of other non semisimple algebras all of which have at least a semidirect structure in some cases two or more the dimensions of their second cohomology groups and the explicit expression of their central extensions are given

discretization and dynamic consistency of a delayed and diffusive viral infection model abstract a diffusive and delayed viral infection model with nonlinear incidence has been studied and the global dynamical behaviors of the original model is investigated by constructing lyapunov functionals furthermore the analysis is carried out for the discrete model which is obtained by applying the nonstandard finite difference nsfd scheme to the original continuous model the global stability for the corresponding equilibria is investigated by constructing discrete lyapunov functionals as well as the positivity and boundedness of solutions of the corresponding continuous model the results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for the original continuous model numerical experiments are carried out to support the theoretical results

a note on the aα spectral radius of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 2017 7 defined the matrix a g as a g d g 1 a g let u and v be two vertices of a connected graph g suppose that u and v are connected by a path w 0 v w 1 w s 1 w s u where d w i 2 for 1 i s 1 let g p s q u v be the graph obtained by attaching the paths p p to u and p q to v let s 0 1 nikiforov and rojo 2018 9 conjectured that g p s q u v g p 1 s q 1 u v if p q 2 in this paper we confirm the conjecture as applications firstly the extremal graph with maximal a spectral radius with fixed order and cut vertices is characterized secondly we characterize the extremal tree which attains the maximal a spectral radius with fixed order and matching number these results generalize some known results

nonstandard finite difference scheme for a diffusive within host virus dynamics model with both virus to cell and cell to cell transmissions abstract in this paper we first propose a diffusive within host virus dynamics model with both virus to cell and cell to cell transmissions then we consider the discretization of the model by using nonstandard finite difference scheme it is then followed by the investigation of the global stability of the equilibria of the discrete model our study shows that if the basic reproduction number r 0 1 then the infection free equilibrium is globally asymptotically stable however if r 0 1 then the infection equilibrium is globally asymptotically stable numerical simulations are presented to illustrate our theoretical results

global stability and periodic solution of the viral dynamics it is well known that the mathematical models provide very important information for the research of human immunodeficiency virus type 1 and hepatitis c virus hcv however the infection rate of almost all mathematical models is linear the linearity shows the simple interaction between the t cells and the viral particles in this paper we consider the classical mathematical model with saturation response of the infection rate by stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection free steady state we also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution numerical simulations are presented to illustrate the results

the aα spread of a graph abstract let a g and d g be the adjacency matrix and the degree diagonal matrix of a graph g respectively for any real number 0 1 nikiforov defined the a matrix of a graph g as a g d g 1 a g the a spread of a graph is the difference between the largest eigenvalue and the smallest eigenvalue of the a matrix of the graph in this paper we obtain some new lower bounds on a spread and characterize the extremal graphs for certain cases which extend the results of a spread and q spread moreover some graph operations on a spread are presented

a delay dependent model with hiv drug resistance during therapy abstract the use of combination antiretroviral therapy has proven remarkably effective in controlling hiv disease progression and prolonging survival however the emergence of drug resistance can occur it is necessary that we gain a greater understanding of the evolution of drug resistance here we consider an hiv viral dynamical model with general form of target cell density drug resistance and intracellular delay incorporating antiretroviral therapy the model includes two strains wild type and drug resistant the basic reproductive ratio for each strain is obtained for the existence of steady states qualitative analysis of the model such as the well posedness of the solutions and the equilibrium stability is provided global asymptotic stability of the disease free and drug resistant steady states is shown by constructing lyapunov functions furthermore sufficient conditions related to the properties of the target cell density are obtained for the local asymptotic stability of the positive steady state numerical simulations are conducted to study the impact of target cell density and intracellular delay focusing on the stability of the positive steady state the occurrence of hopf bifurcation of periodic solutions is shown to depend on the target cell density

on the aα spectra of trees abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 define the matrix a g as a g d g 1 a g this paper gives several results about the a matrices of trees in particular it is shown that if t is a tree of maximal degree then the spectral radius of a t satisfies the tight inequality a t 2 1 1 which implies previous bounds of godsil lovasz and stevanovic the proof is deduced from some new results about the a matrices of bethe trees and generalized bethe trees in addition several bounds on the spectral radius of a of general graphs are proved implying tight bounds for paths and bethe trees

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

bounds for the largest and the smallest aα eigenvalues of a graph in terms of vertex degrees abstract let g be a graph with adjacency matrix a g and with d g the diagonal matrix of its vertex degrees nikiforov defined the matrix a g with 0 1 as a g d g 1 a g the largest and the smallest eigenvalues of a g are respectively denoted by g and n g in this paper we present a tight upper bound for g in terms of the vertex degrees of g for 1 for any graph g g max u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected for 0 1 the equality holds if and only if g is regular or bipartite semi regular as an application we solve the problem raised by nikiforov in 15 for the smallest eigenvalue n g of a bipartite graph g of order n with no isolated vertices for 0 1 then n g min u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected and 1 2 the equality holds if and only if g is regular or semi regular

global attractivity of the diffusive nicholson blowflies equation with neumann boundary condition a non monotone case abstract in this paper we establish the global attractivity of the positive steady state of the diffusive nicholson s equation with homogeneous neumann boundary value under a condition that makes the equation a non monotone dynamical system to achieve this we develop a novel method combining a dynamical systems argument with maximum principle and some subtle inequalities

discretizations of nonlinear differential equations using explicit nonstandard methods abstract in a recent paper chen and solis investigated the appearance of spurious solutions when first order odes are discretized using runge kutta schemes they concluded that the reliability of the numerical solutions to a particular ode could be verified only by constructing several discrete models and comparing their numerical results with the known properties of the exact solutions we demonstrate that by using nonstandard schemes all the difficulties found by chen and solis can be eliminated and that qualitatively correct numerical solutions are obtained for all values of the step size we illustrate these issues by applying nonstandard finite difference techniques to the logistic sine cubic and monod equations

global dynamics for a class of age infection hiv models with nonlinear infection rate abstract in this paper we study the global stability of a class of hiv viral infection models with continuous age structure using the direct lyapunov method in each of the cases where the incidence rates are given by nonlinear infection rate f t g v holling type ii functional response and crowley martin functional response we define the basic reproduction number and prove that it is a sharp threshold determining whether the infection dies out or not we give a rigorous mathematical analysis on technical materials and necessary arguments including relative compactness of the orbit and uniform persistence of system by reformulating the system as a system of volterra integral equations we further investigate global behaviors of hiv viral infection models with holling type ii functional response and crowley martin functional response through numerical simulations

pattern formation of an epidemic model with time delay abstract one of the central issues in epidemiology is the study of the distribution of disease and time delay widely exists in the process of disease spread thus in this paper we presented an epidemic model with spatial diffusion and time delay by mathematical analysis we find two different types of instability one is the diffusion induced instability and the other one is delay induced instability moreover we derive the corresponding patterns by performing a series of numerical simulations the obtained results show that the interaction of diffusion and time delay may give rise to rich dynamics in epidemic systems

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

on the α index of graphs with pendent paths abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g this paper presents some extremal results about the spectral radius g of a g that generalize previous results about 0 g and 1 2 g in particular write b p q r be the graph obtained from a complete graph k p by deleting an edge and attaching paths p q and p r to its ends it is shown that if 0 1 and g is a graph of order n and diameter at least k then g b n k 2 k 2 k 2 with equality holding if and only if g b n k 2 k 2 k 2 this result generalizes results of hansen and stevanovic 5 and liu and lu 7

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

a non standard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate abstract a non standard finite difference scheme is proposed to solve a delayed and diffusive viral infection model with general nonlinear incidence rate the results show that the discrete model preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem moreover this method preserves all equilibria of the original continuous model by constructing lyapunov functionals we show that the global stability of equilibria is completely determined by the basic reproduction number 0 which implies that the proposed discrete model can efficiently blue preserve the global stability of equilibria of the corresponding continuous model numerical experiments are carried out to support the theoretical results

anti de sitter poincare symmetries and representations from poincare galilei through a classical deformation approach a classical deformation procedure based on universal enveloping algebras casimirs and curvatures of symmetrical homogeneous spaces is applied to several cases of physical relevance starting from the 3 1 d galilei algebra we describe at the level of representations the process leading to its two physically meaningful deformed neighbours the poincare algebra is obtained by introducing a negative curvature in the flat galilean phase space or space of worldlines while keeping a flat spacetime to be precise starting from a representation of the galilei algebra with both casimirs different from zero we obtain a representation of the poincare algebra with both casimirs necessarily equal to zero the poincare angular momentum pauli lubanski components position and velocity operators etc are expressed in terms of galilean operators through some expressions deforming the proper galilean ones similarly the newton hooke algebras appear by endowing spacetime with a nonzero curvature while keeping a flat phase space the same approach starting from the 3 1 d poincare algebra provides representations of the anti de sitter as poincare deformations

on the multiplicity of α as an eigenvalue of aα g of graphs with pendant vertices abstract let g be a simple undirected graph let 0 1 let a g d g 1 a g where d g and a g are the diagonal matrix of the vertex degrees of g and the adjacency matrix of g respectively let p g 0 and q g be the number of pendant vertices and quasi pendant vertices of g respectively let m g be the multiplicity of as an eigenvalue of a g it is proved that m g p g q g with equality if each internal vertex is a quasi pendant vertex if there is at least one internal vertex which is not a quasi pendant vertex the equality m g p g q g m n is determined in which m n is the multiplicity of as an eigenvalue of the matrix n this matrix is obtained from a g taking the entries corresponding to the internal vertices which are non quasi pendant vertices these results are applied to search for the multiplicity of as an eigenvalue of a g when g is a path a caterpillar a circular caterpillar a generalized bethe tree or a bethe tree for the bethe tree case a simple formula for the nullity is given

dynamics of a periodically pulsed bio reactor model abstract global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio reactor model in terms of principal eigenvalues of the periodic parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems abstract persistence asymptotically periodic semiflows and perturbation of global attractors

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

casimir invariants for the complete family of quasisimple orthogonal algebras a complete choice of generators of the centre of the enveloping algebras of real quasisimple lie algebras of orthogonal type for arbitrary dimension is obtained in a unified setting the results simultaneously include the well known polynomial invariants of the pseudo orthogonal algebras as well as the casimirs for many non simple algebras such as the inhomogeneous the newton hooke and galilei type etc which are obtained by contraction s starting from the simple algebras the dimension of the centre of the enveloping algebra of a quasisimple orthogonal algebra turns out to be the same as for the simple algebras from which they come by contraction the structure of the higher order invariants is given in a convenient pyramidal manner in terms of certain sets of pauli lubanski elements in the enveloping algebras as an example showing this approach at work the scheme is applied to recovering the casimirs for the 3 1 kinematical algebras some prospects on the relevance of these results for the study of expansions are also given

a reaction diffusion lyme disease model with seasonality this paper is devoted to the study of a reaction diffusion lyme disease model with seasonality in the case of a bounded habitat we obtain a threshold result on the global stability of either disease free or endemic periodic solution in the case of an unbounded habitat we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time periodic traveling wave solutions we also estimate parameter values based on some published data and use them to study the lyme disease transmission in port dove ontario our numerical simulations are consistent with the obtained analytic results

global properties of delayed hiv dynamics models with differential drug efficacy in cocirculating target cells the objective of this work is to investigate the qualitative behaviors of three hiv dynamical models with two types of cocirculating target cells and intracellular discrete time delay in the two types of target cells the drug efficacy is assumed to be different the models take into account both short lived infected cells and long lived chronically infected cells the incidence rate of infection is given by bilinear and saturation functional response in the first and second model respectively while it is given by a general function in the third model lyapunov functionals are constructed and lasalle s invariance principle is applied to prove the global asymptotic stability of all equilibria of the models in the first two models we have derived the basic reproduction number r0 it has been proven that the disease free equilibrium is globally asymptotically stable gas when r0 1 and the endemic equilibrium is gas when r0 1 for the third model we have derived r0 and established a set of sufficient conditions for global stability of both disease free and endemic equilibria of the model at the end theoretical results have been checked by numerical simulations