on criticality and the stein rosenberg theorem a generalization of the stein rosenberg theorem is proved in the context of a real banach space x such that x has the bounded decomposition property with respect to a normal cone when x is finite dimensional this generalization improves several versions of the stein rosenberg theorem by i marek f robert and j vandergraft and gives an answer to the question concerning strict inequalities when comparing rates of convergence these improvements are made possible by a simple device relying on any notion of irreducibility has been avoided instead a thorough although simple study of criticality has been made this work follows the lines developed in a preceding work in the context of a banach space with a normal generating cone the present context is more restricted but we deal with quasi completely continuous linear operators instead of completely continuous ones and some proofs are less simple we finally point out that our conclusions are based on a convenient version of the perron frobenius

backward bifurcation analysis of epidemiological model with partial immunity this paper presents a two stage sis epidemiological model in animal population with bovine tuberculosis btb in african buffalo as a guiding example the proposed model is rigorously analyzed the analysis reveals that the model exhibits the phenomenon of backward bifurcation where a stable disease free equilibrium dfe coexists with a stable endemic equilibrium ee when the associated reproduction number r v is less than unity it is shown under two special cases of the presented model that this phenomenon of backward bifurcation does not arise depending on vaccination coverage and efficacy of vaccine numerical simulations of the model show that the use of an imperfect vaccine can lead to effective control of the disease if the vaccination coverage and the efficacy of vaccine are high enough

block gauss elimination followed by a classical iterative method for the solution of linear systems in the last two decades many papers have appeared in which the application of an iterative method for the solution of a linear system is preceded by a step of the gauss elimination process in the hope that this will increase the rates of convergence of the iterative method this combination of methods has been proven successful especially when the matrix a of the system is an m matrix the purpose of this paper is to extend the idea of one to more gauss elimination steps consider other classes of matrices a e g p cyclic consistently ordered and generalize and improve the asymptotic convergence rates of some of the methods known so far

improving jacobi and gauss seidel iterations abstract when convergent jacobi or gauss seidel iterations can be applied to solve systems of linear equations a natural question is how convergence rates are affected if the original system is modified by performing some gaussian elimination we prove that if the initial iteration matrix is nonnegative then such elimination improves convergence our results extend those contained in 4

more on modifications and improvements of classical iterative schemes for m matrices in the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an m matrix in order to improve on the convergence rates of the classical iterative schemes jacobi gauss seidel etc the present work is a contribution towards the generalization of the most common preconditioners used so far

erratum to a note on the preconditioned gauss seidel method for m matrices j comput appl math 219 1 2008 59 71 in this note some errors in a recent article by niki et al h niki t kohno m morimoto the preconditioned gauss seidel method faster than the sor method j comput appl math 219 2008 59 71 are pointed out and a new proof for the corresponding result is presented

equivalence classes of boolean functions for first order correlation this paper presents a complete characterization of the first order correlation immune boolean functions that includes the functions that are 1 resilient the approach consists in defining an equivalence relation on the full set of boolean functions with a fixed number of variables an equivalence class in this relation called a first order correlation class provides a measure of the distance between the boolean functions it contains and the correlation immune boolean functions the key idea consists on manipulating only the equivalence classes instead of the set of boolean functions to achieve this goal a class operator is introduced to construct a class with n variables from two classes of n 1 variables in particular the class of 1 resilient functions on n variables is considered an original and efficient method to enumerate all the boolean functions in this class is proposed by performing a recursive decomposition of classes with less variables a bottom up algorithm provides the exact number of 1 resilient boolean functions with seven variables which is 23478015754788854439497622689296 a tight estimation of the number of 1 resilient functions with eight variables is obtained by performing a partial enumeration it is conjectured that the exact complete enumeration for general n is intractable

the global dynamics for a stochastic sis epidemic model with isolation abstract in this paper we investigate the dynamical behavior for a stochastic sis epidemic model with isolation which is as an important strategy for the elimination of infectious diseases it is assumed that the stochastic effects manifest themselves mainly as fluctuation in the transmission coefficient the death rate and the proportional coefficient of the isolation of infective it is shown that the extinction and persistence in the mean of the model are determined by a threshold value r 0 s that is if r 0 s 1 then disease dies out with probability one and if r 0 s 1 then the disease is stochastic persistent in the means with probability one furthermore the existence of a unique stationary distribution is discussed and the sufficient conditions are established by using the lyapunov function method finally some numerical examples are carried out to confirm the analytical results

on modified jacobi linear operators abstract by means of successive partial substitutions new fixed point linear equations can be obtained from old ones the jacobi method applied to a system in the sequence thus obtained constitutes a partial gauss seidel method applied to the original one and we analyze the behavior of the sequence of spectral radii of the successive iteration matrices the modified jacobi operators we do this under the assumption that the starting operator is nonnegative with respect to a proper cone and has spectral radius less or greater than 1 our main result is that if the jacobi operator obtained after k substitutions is irreducible then the following one either is the same or has strictly smaller or greater spectral radius this result implies that the whole sequence of spectral radii is monotone

a convergence analysis of the iterative aggregation method with one parameter a method for accelerating linear iterations in a banach space is studied as a linear iterative method in an augmented space and sufficient conditions for convergence are derived in the general case and in ordered banach spaces an acceleration of convergence takes place if an auxiliary functional is chosen sufficiently close to a dual eigenvector associated with a dominant simple eigenvalue of the iteration operator in this case the influence of this eigenvalue on the asymptotic rate of convergence is eliminated quantitative estimates and bounds on convergence are given

optimal successive overrelaxation iterative methods for p cyclic matrices we consider linear systems whose associated block jacobi matricesj p are weakly cyclic of indexp in a recent paper pierce hadjidimos and plemmons 13 proved that the block two cyclic successive overrelaxation sor iterative method is numerically more effective than the blockq cyclic sor method 2 2 finally as a counterpart to the result of pierce hadjidimos and plemmons we show that under this more general assumption the two cyclic sor method is not always superior to theq cyclic sor method 2 q p

elimination and fixed point iterations abstract we show that under convenient hypotheses partial elimination in nonlinear systems can be used to improve convergence in the jacobi gauss seidel and newton iterations

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

comparison results for solving preconditioned linear systems in this paper we present some comparison theorems between two different modified gauss seidel mgs methods the second preconditioning based on the first preconditioning is also discussed in this paper

dynamics of an seqihrs epidemic model with media coverage quarantine and isolation in a community with pre existing immunity abstract an autonomous deterministic non linear epidemic model seqihrs is proposed for the transmission dynamics of an infectious disease with quarantine and isolation control strategies in a community with pre existing immunity the model exhibits two equilibria namely the disease free and a unique endemic equilibrium the existence and local stability of the disease free and endemic equilibria are explored in terms of the effective reproduction number r c it is observed that media coverage does not affect the effective reproduction number but it helps to mitigate disease burden by lowering the number of infectious individuals at the endemic steady state and also lowering the infection peak a new approach is proposed to estimate the coefficient of media coverage using the results of central manifold theory it is established that as r c passes through unity transcritical bifurcation occurs in the system and the unique endemic equilibrium is asymptotically stable it is observed that the population level impact of quarantine and isolation depend on the level of transmission by the isolated individuals moreover the higher level of pre existing immunity in the population decreases the infection peak and causes its early arrival theoretical findings are supported by numerical simulation sensitivity analysis is performed for r c and state variables at endemic steady state with respect to model parameters

an extended gs method for dense linear systems davey and rosindale k davey i rosindale an iterative solution scheme for systems of boundary element equations internat j numer methods engrg 37 1994 1399 1411 derived the gsor method which uses an upper triangular matrix w in order to solve dense linear systems by applying functional analysis the authors presented an expression for the optimum w moreover davey and bounds k davey s bounds a generalized sor method for dense linear systems of boundary element equations siam j comput 19 1998 953 967 also introduced further interesting results in this note we employ a matrix analysis approach to investigate these schemes and derive theorems that compare these schemes with existing preconditioners for dense linear systems we show that the convergence rate of the gauss seidel method with preconditioner p g is superior to that of the gsor method moreover we define some splittings associated with the iterative schemes some numerical examples are reported to confirm the theoretical analysis we show that the egs method with preconditioner p g c o p t produces an extremely small spectral radius in comparison with the other schemes considered

an iteration method for solving the linear system ax b abstract in this paper based on a convergence splitting of the matrix a we present an inner outer iteration method for solving the linear system a x b we analyze the overall convergence of this method without any other restriction on its parameters moreover we give the accelerated inner outer iteration method and discuss how to apply the inner outer iterations as a preconditioner for the krylov subspace methods the inner outer iteration method can also be used for the solution of a x b c finally several numerical examples are given to validate the performance of our proposed algorithms

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

lu decomposition of m matrices by elimination without pivoting abstract it is shown that if a or a is a singular m matrix satisfying the generalized diagonal dominance condition y t a 0 for some vector y 0 then a can be factored into a lu by a certain elimination algorithm where l is a lower triangular m matrix with unit diagonal and u is an upper triangular m matrix the existence of lu decomposition of symmetric permutations of a and for irreducible m matrices and symmetric m matrices follow as colollaries this work is motivated by applications to the solution of homogeneous systems of linear equations ax 0 where a or a is an m matrix these applications arise e g in the analysis of markov chains input output economic models and compartmental systems a converse of the theorem metioned above can be established by considering the reduced normal form of a

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

preconditioned aor iterative methods for m matrices linear systems with m matrices often appear in a wide variety of areas in this paper we give general preconditioners for solving the systems with nonsingular m matrix we show that our preconditioners increase the convergence rate of aor iterative methods numerical results for corresponding preconditioned gmres methods are also given

comparison theorems for the convergence factor of iterative methods for singular matrices abstract iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix t i e by the quantity t max t 1 where t is the spectrum of t theorems are presented comparing the convergence factor of two iterative methods the comparison is based on the relationship between the matrices of the splittings a cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison although this cone is based on the unknown projection onto the null space of a matrix the characterization provided in the paper allows in specific instances the cone to be readily computable

enumerative encoding of correlation immune boolean functions le bars and viola have recently proposed an innovative recursive decomposition of the first order correlation immune boolean functions based on their work this paper presents the design of an enumerative encoding of these boolean functions this is the first enumerative encoding of a class of boolean functions defined by a cryptographic property in this paper we study three major milestones to do this encoding the conceptual computational tree the use of normal classes and signed permutations and a dynamic selection of the decomposition our enumerative encoding algorithm is practicable up to 8 variables which is the best result we may expect due to the combinatorial explosion of the numbers of classes

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

mathematical analysis of a disease transmission model with quarantine isolation and an imperfect vaccine a new mathematical model for the transmission dynamics of a disease subject to the quarantine of latent cases and isolation of symptomatic cases and an imperfect vaccine is designed and analyzed the model undergoes a backward bifurcation where a stable disease free equilibrium co exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity it is shown that the backward bifurcation phenomenon can be removed if the vaccine is perfect or if mass action incidence instead of standard incidence is used in the model formulation further the model has a unique endemic equilibrium when the threshold quantity exceeds unity a nonlinear lyapunov function of the goh volterra type is used to show that the endemic equilibrium is globally asymptotically stable for a special case numerical simulations of the model show that the singular use of a quarantine isolation strategy may lead to the effective disease control or elimination if its effectiveness level is at least moderately high enough the combined use of the quarantine isolation strategy with a vaccination strategy will eliminate the disease even for the low efficacy level of the universal strategy considered in this study it is further shown that the imperfect vaccine could induce a positive or negative population level impact depending on the size or sign of a certain associated epidemiological threshold

on optimal improvements of classical iterative schemes for z matrices many researchers have considered preconditioners applied to linear systems whose matrix coefficient is a z or an m matrix that make the associated jacobi and gauss seidel methods converge asymptotically faster than the unpreconditioned ones such preconditioners are chosen so that they eliminate the off diagonal elements of the same column or the elements of the first upper diagonal milaszewicz laa 93 1987 161 170 gunawardena et al laa 154 156 1991 123 143 in this work we generalize the previous preconditioners to obtain optimal methods good jacobi and gauss seidel algorithms are given and preconditioners that eliminate more than one entry per row are also proposed and analyzed moreover the behavior of the above preconditioners to the krylov subspace methods is studied

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

a calculus for the random generation of labelled combinatorial structures a systematic approach to the random generation of labelled combinatorial objects is presented it applies to structures that are decomposable i e formally specifiable by grammars involving set sequence and cycle constructions a general strategy is developed for solving the random generation problem with two closely related types of methods for structures of size n the boustrophedonic algorithms exhibit a worst case behaviour of the form o n log n the sequential algorithms have worst case o n2 while offering good potential for optimizations in the average case the complexity model is in terms of arithmetic operations and both methods appeal to precomputed numerical table of linear size that can be computed in time o n2 a companion calculus permits systematically to compute the average case cost of the sequential generation algorithm associated to a given specification using optimizations dictated by the cost calculus several random generation algorithms of the sequential type are developed most of them have expected complexity 1 2n log n and are thus only slightly superlinear the approach is exemplified by the random generation of a number of classical combinatorial structures including cayley trees hierarchies the cycle decomposition of permutations binary trees functional graphs surjections and set partitions

modified iterative methods for nonnegative matrices and m matrices linear systems the purpose of this paper is to present new preconditioning techniques for solving nonnegative matrices linear system and m matrices linear system ax b based on the i s a type preconditioning matrices provided by hadjidimos et al 1 and evans et al 2 convergence analysis of the proposed methods are given numerical results are presented which show the improvements on the convergence rate of the jacobi type and gauss seidel type preconditioned iterative methods

a general preconditioner for linear complementarity problem with an m matrix abstract in this paper we first present a general preconditioner p for solving linear complementarity problem lcp associated with an m matrix a and a vector f and prove that the lcp a f is equivalent to the lcp p a p f then based on this general preconditioner p two preconditioned ssor methods for solving the linear complementarity problems are proposed we show that this general preconditioner p accelerates the convergence of two ssor methods under the assumption that p a is a z matrix in addition we also give a practically concrete choice for the preconditioner p satisfying aforementioned assumption numerical examples are used to illustrate the theoretical results obtained

the jacobi and gauss seidel type iteration methods for the matrix equation a x b c in this paper the jacobi and gauss seidel type iteration methods are proposed for solving the matrix equation a x b c which are based on the splitting schemes of the matrices a and b the convergence and computational cost of these iteration methods are discussed furthermore we give the preconditioned jacobi and gauss seidel type iteration methods numerical examples are given to demonstrate the efficiency of these methods proposed in this paper

optimal p cyclic sor our purpose in this paper is to extend p cyclic sor theory to consistent singular systems and to apply the results to the solution of large scale systems arising em e g in queueing network problems in markov analysis markov chains and queueing models lead to structured singular linear systems and are playing an increasing role in the understanding of complex phenomena arising in computer communication and transportation systems for certain important classes of singular problems we develop a convergence theory for p cyclic sor and show how to repartition for optimal convergence results by kontovasilis plemmons and stewart on the concept of convergence of sor in an em extended sense are rigorously analyzed and applied to the solution of periodic markov chains with period p 2

convergence analysis of the preconditioned gauss seidel method for h matrices in 1997 kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 proved that the convergence rate of the preconditioned gauss seidel method for irreducibly diagonally dominant z matrices with a preconditioner i s a is superior to that of the basic iterative method in this paper we present a new preconditioner i k b which is different from the preconditioner given by kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 and prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an h matrix meanwhile two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

modified sor type iterative method for z matrices abstract in this paper we present the convergence analysis for some modified sor type iterative method for solving linear systems and provide a comparison of spectral radius among the sor iterative method and these modified methods comparison results and the numerical example show that the rate of convergence of the gauss seidel method is faster than the rate of convergence of the sor iterative method

modified gauss seidel type methods and jacobi type methods for z matrices abstract in this paper we present the convergence analysis for some modified gauss seidel and jacobi type iterative methods and provide a comparison of spectral radius among the gauss seidel iterative method and these modified methods some recent results are improved

gauss type preconditioning techniques for linear systems abstract many researchers have considered preconditioners chosen to eliminate the off diagonal elements of the coefficient matrix of a linear system in this work we generalize the left gauss type preconditioners y zhang t z huang x p liu modified iterative methods for nonnegative matrices and m matrices linear systems comput math appl 50 2005 1587 1602 which eliminate the strictly lower triangular elements right gauss type preconditioners that eliminate strictly upper triangular elements are proposed in this paper these gauss type preconditioners are partly derived from the lu factorization method theoretic analysis on spectral radii of the two kinds of gauss type preconditioners is given numerical experiments are used to show the performance of the improved inbuilt left and right gauss type preconditioning algorithms associated with jacobi type and gauss seidel type iterative methods

convergence analysis of the two preconditioned iterative methods for m matrix linear systems in this paper we present the two preconditioners i s and i s r for solving m matrix linear systems and discuss the convergence of the two preconditioned iterative methods meanwhile we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of m matrix linear systems by preconditioned krylov subspace methods numerical experiments are given to validate the performance of the preconditioners

optimality relationships forp cyclic sor the optimality question for blockp cyclic sor iterations discussed in young and varga is answered under natural conditions on the spectrum of the block jacobi matrix in particular it is shown that repartitioning a blockp cyclic matrix into a blockq cyclic form q p results in asymptotically faster sor convergence for the same amount of work per iteration as a consequence block 2 cyclic sor is optimal under these conditions