a canonical form for pencils of matrices with applications to asymptotic linear programs abstract this paper studies asymptotic pencils of linear programs in which the constraint matrix right hand sides and objective function coefficients depend linearly on a parameter and we seek a basis that is optimal for all large or small enough values of the parameter the asymptotic linear programming formulation of discounted markov decision chains has this form as do many other problems for this problem we introduce a new canonical form for bases that facilitates computation of laurent expansions of the solutions of the resulting asymptotic linear programs and show how to implement the simplex method efficiently in this setting we show that the computational work to solve the asymptotic linear program is the product of the number of rows of the problem and the work required to solve an ordinary linear program of the same size this improves computational efficiency of the asymptotic simplex method of jeroslow 1973 and a related algorithm considered by lamond 1989 in essence this work generalizes that of miller and veinott 1969 and rothblum 1974 1975 for markov ordinary and markov branching decision chains to arbitrary linear programs

reach a nonlinear consensus for mas via doubly stochastic quadratic operators abstractthis technical note addresses the new nonlinear protocol class of doubly stochastic quadratic operators dsqos for coordination of consensus problem in multi agent systems mas we derive the conditions for ensuring that every agent reaches consensus on a desired rate of the group s decision where the group decision value in its agent s initial statuses varies besides that we investigate a nonlinear protocol sub class of extreme dsqo edsqo to reach a consensus for mas to a common value with nonlinear low complexity rules and fast time convergence if the interactions for each agent are not selfish in addition to extend the results to reach a consensus and to avoid the selfish case we specify a general class of dsqo for reaching a consensus under any given case of initial states the case that mas reach a consensus by dsqo is if each member of the agent group has positive interactions of dsqo pdsqo with the others the convergence of both edsqo and pdsqo classes is found to be directed tow

multiplicative markov decision chains previous treatments of multiplicative markov decision chains eg bellman bellman r 1957 dynamic programming princeton university press princeton new jersey mandl mandl p 1967 an iterative method for maximizing the characteristic root of positive matrices rev roumaine math pures appl xii 1317 1322 and howard and matheson howard r a matheson j e 1972 risk sensitive markov decision processes management sci 8 356 369 restricted attention to stationary policies and assumed that all transition matrices are irreducible and aperiodic they also used a first term optimality criterion namely maximizing the spectral radius of the associated transition matrix we give a constructive proof of the existence of optimal policies among all policies under new cumulative average optimality criteria which are more sensitive than the maximization of the spectral radius the algorithm for finding an optimal policy first searches for a stationary policy with a nonnilpotent transition matrix provided such a rule exists otherwise the method still finds an optimal policy though in this case the set of optimal policies usually does not contain a stationary policy if a stationary policy with a nonnilpotent transition matrix exists then we develop a policy improvement algorithm which finds a stationary optimal policy

asymptotic properties of powers of nonnegative matrices with applications abstract let a be a finite square reducible nonnegative matrix the main theorem of the paper gives first order approximations of f n a for certain sequences f n of analytic functions in particular the theorem holds when f n a a n the theorem is applied to study the local behavior of large powers of a nonnegative matrix to study the limiting output vectors of a nonnegative multiplicative process and to characterize the nonnegative eigenvectors of a nonnegative matrix finally an application to absorbing markov chains is given

product of two positive contractions abstract several characterizations are given for a square matrix that can be written as the product of two positive semidefinite contractions based on one of these characterizations and the theory of alternating projections a matlab program is written to check the condition and construct the two positive contractions whose product equal to the given matrix if they exist

a soft measure of consensus in the setting of partial fuzzy preferences abstract consensus as traditionally meant to be a full and unanimous agreement is often not reachable in practice a degree of consensus for indicating how far a particular group of individuals is from consensus may be therefore very useful we propose a new measure degree of consensus which is more human consistent in the sense that it better reflects a real human perception of the essence of consensus in practice basically our consensus measure expresses the degree to which say most of the important individuals agree as to their testimonies concerning almost all of the relevant options the point of departure is the set of individual testimonies which are here the individual fuzzy preference relations as a formal tool we use a fuzzy logic based calculus of linguistically quantified propositions

nonzero sum risk sensitive stochastic games on a countable state space the infinite horizon risk sensitive discounted cost and ergodic cost nonzero sum stochastic games for controlled markov chains with countably many states are analyzed for the discounted cost game we prove the existence of nash equilibrium strategies in the class of markov strategies under fairly general conditions under an additional weak geometric ergodicity condition and a small cost criterion the existence of nash equilibrium strategies in the class of stationary markov strategies is proved for the ergodic cost game the key nontrivial contributions in the ergodic part are to prove the existence of a particular form of a relative value function solution to a player s bellman equation and the continuity of this solution with respect to the opponent s strategies

an algorithm to identify and compute average optimal policies in multichain markov decision processes this paper concerns discrete time finite state multichain mdps with compact action sets the optimality criterion is long run average cost simple examples illustrate that optimal stationary markov policies do not always exist we establish the existence of e optimal policies that are stationary markovian and develop an algorithm that computes these approximate optimal policies we establish a necessary and sufficient condition for the existence of an optimal policy that is stationary markovian and in case that such an optimal policy exists the algorithm computes it

resolvent expansions of matrices and applications abstract various explicit expansions of the resolvent of a square complex matrix in a neighborhood of the origin including the well known laurent expansion are obtained simple proofs using algebraic arguments rather than the theory of complex functions are given cesaro average expansions of the resolvent on the spectral circle are also developed several applications of these expansions are given in particular these include some new inverse nonnegativity properties of the drazin inverses of m matrices

elastic plastic unilateral contact problems for slackened systems abstract the paper deals with the mathematical modelling of a class of unilateral elastic plastic contact problems the problem is posed in the framework of the so called slackened systems the considerations are confined to small strains and piecewise linear approximations of the yield surface and the clearance surface an incremental formulation is proposed in the form of a linear complementarity problem lcp the formulation allows one to analyse a structural system which can be subjected to nonproportional loading paths the included numerical results illustrate the application of this formulation to an elastic plastic beam that comes into contact with a unilateral elastic plastic foundation

finite element approximation of a coupled contact stefan like problem arising from the time discretization in deformation theory of thermo plasticity in the paper we draw on the mathematical formulation of the coupled contact stefan like problem in deformation theory of plasticity which arises from the discretization in time the problem leads to solving the system of variational inequalities which is approximated by the fem numerical analysis of the problem is made

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

expansions of sums of matrix powers it is shown that the n fold partial sums of powers of a square complex matrix having spectral radius one or less can be approximated by matrix polynomials where the approximation is taken in the sense of cesaro averages the degree of the approximating polynomials and the order of cesaro averaging is characterized when the matrix is nonnegative the results are strengthened and it is shown that cesaro limits can be replaced by regular limits of periodic subsequences the periods of these expansions are characterized we obtain the explicit computable form of all approximating polynomials generalizing known results for stochastic and irreducible nonnegative matrices several applications of the results are discussed

products of positive semi definite matrices abstract it is known that every complex square matrix with nonnegative determinant is the product of positive semi definite matrices there are characterizations of matrices that require two or five positive semi definite matrices in the product however the characterizations of matrices that require three or four positive semi definite matrices in the product are lacking in this paper we give a complete characterization of these two types of matrices with these results we give an algorithm to determine whether a square matrix can be expressed as the product of k positive semi definite matrices but not fewer for k 1 2 3 4 5

risk sensitive optimal control for markov decision processes with monotone cost the existence of an optimal feedback law is established for the risk sensitive optimal control problem with denumerable state space the main assumptions imposed are irreducibility and anear monotonicity condition on the one step cost function a solution can be found constructively using either value iteration or policy iteration under suitable conditions on initial feedback law

a numerical algorithm for the solution of signorini problems in this paper we propose a new iterative algorithm for the solution of a certain class of signorini problems such problems arise in the modelling of a variety of physical phenomena and usually involve the determination of an unknown free boundary here we describe a way of locating the free boundary directly and provide a proof that the algorithm converges when used with analytic methods the advantage of this algorithm is that it can be used in conjunction with any numerical method with minimal development of extra code we demonstrate its application with the boundary element method to some physical problems in both two and three dimensions

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

spectral triples and wavelets for higher rank graphs abstract in this paper we present a new way to associate a finitely summable spectral triple to a higher rank graph via the infinite path space of moreover we prove that this spectral triple has a close connection to the wavelet decomposition of which was introduced by farsi gillaspy kang and packer in 2015 we first introduce the concept of stationary k bratteli diagrams in order to associate a family of ultrametric cantor sets and their associated pearson bellissard spectral triples to a finite strongly connected higher rank graph we then study the zeta function abscissa of convergence and dixmier trace associated to the pearson bellissard spectral triples of these cantor sets and show these spectral triples are regular in the sense of pearson and bellissard we obtain an integral formula for the dixmier trace given by integration against a measure and show that is a rescaled version of the measure m on which was introduced by an huef laca raeburn and sims finally we investigate the eigenspaces of a family of laplace beltrami operators associated to the dirichlet forms of the spectral triples we show that these eigenspaces refine the wavelet decomposition of l 2 m which was constructed by farsi et al

the simplex and policy iteration methods are strongly polynomial for the markov decision problem with a fixed discount rate we prove that the classic policy iteration method howard r a 1960 dynamic programming and markov processes mit cambridge and the original simplex method with the most negative reduced cost pivoting rule of dantzig are strongly polynomial time algorithms for solving the markov decision problem mdp with a fixed discount rate furthermore the computational complexity of the policy iteration and simplex methods is superior to that of the only known strongly polynomial time interior point algorithm ye y 2005 a new complexity result on solving the markov decision problem math oper res 30 3 733 749 for solving this problem the result is surprising because the simplex method with the same pivoting rule was shown to be exponential for solving a general linear programming problem klee v g j minty 1972 how good is the simplex method technical report o shisha ed inequalities iii academic press new york the simplex method with the smallest index pivoting rule was shown to be exponential for solving an mdp regardless of discount rates melekopoglou m a condon 1994 on the complexity of the policy improvement algorithm for markov decision processes informs j comput 6 2 188 192 and the policy iteration method was recently shown to be exponential for solving undiscounted mdps under the average cost criterion we also extend the result to solving mdps with transient substochastic transition matrices whose spectral radii are uniformly below one

an algorithm for a dynamic programming model of fractional flows in an earlier paper the author introduced an n dimensional dynamic programming model of fractional flows characterized its solutions under general conditions and developed methods for their computation the latter which resemble algorithms for n state markovian decision processes break down when the costs are undiscounted and the flow matrices are nontransient here an algorithm is presented that computes many of the quantities of interest more efficiently under milder assumptions in addition some of our earlier results for the undiscounted case are extended to models with nontransient flow matrices

the greedy strategy for optimizing the perron eigenvalue we address the problems of minimizing and of maximizing the spectral radius over a compact family of non negative matrices those problems being hard in general can be efficiently solved for some special families we consider the so called product families where each matrix is composed of rows chosen independently from given sets a recently introduced greedy method works very fast however it is applicable mostly for strictly positive matrices for sparse matrices it often diverges and gives a wrong answer we present the selective greedy method that works equally well for all non negative product families including sparse ones for this method we prove a quadratic rate of convergence and demonstrate its efficiency in numerical examples the numerical examples are realised for two cases finite uncertainty sets and polyhedral uncertainty sets given by systems of linear inequalities in dimensions up to 2000 the matrices with minimal maximal spectral radii in product families are found within a few iterations applications to dynamical systems and to the graph theory are considered

linear maps preserving products of positive or hermitian matrices abstract in this note we consider linear maps on the complex n n matrix algebra m n preserving products of either two positive matrices a positive matrix and a hermitian matrix or two hermitian matrices we give characterizations of all those injections

affine monotonic and risk sensitive models in dynamic programming in this paper we consider a broad class of infinite horizon discrete time optimal control models that involve a nonnegative cost function and an affine mapping in their dynamic programming equation they include as special cases several classical models such as stochastic undiscounted nonnegative cost problems stochastic multiplicative cost problems and risk sensitive problems with exponential cost we focus on the case where the state space is finite and the control space has some compactness properties and we emphasize shortest path type models we assume that the affine mapping has a semicontractive character whereby for some policies it is a contraction whereas for others it is not in one line of analysis we impose assumptions guaranteeing that the noncontractive policies cannot be optimal under these assumptions we prove strong results that resemble those for discounted markovian decision problems such as the uniqueness of solution of bellman s equation and the validity of forms of value and policy iteration in the absence of these assumptions the results are weaker and unusual in character the optimal cost function need not be a solution of bellman s equation and may not be found by value or policy iteration instead the optimal cost function over just the contractive policies is the largest solution of bellman s equation and can be computed by a variety of algorithms

a necessary and sufficient condition for reaching a consensus using degroot s method abstract degroot 1974 proposed a model in which a group of k individuals might reach a consensus on a common subjective probability distribution for an unknown parameter this paper presents a necessary and sufficient condition under which a consensus will be reached by using degroot s method this work corrects an incorrect statement in the original paper about the conditions needed for a consensus to be reached the condition for a consensus to be reached is straightforward to check and yields the value of the consensus if one is reached

totally expanding multiplicative systems abstract a single matrix multiplicative system consists of an n n nonnegative matrix q and an n 1 semi positive vector x 0 this system is said to be totally expanding if each entry of the sequence q n x 0 n 0 1 is unbounded a multiple matrix multiplicative system replaces q by a set q d of n n nonnegative matrices where d is in product form and is said to be totally expanding if for every in d each entry of the sequence q n x 0 n 0 1 is unbounded each of these systems is shown to be totally expanding if and only if it has no degenerate coordinates and a particular set of linear inequalities has a solution these sets of linear inequalities can also be used to approximate the smallest coordinate dependent growth rate of the output of the respective system

reaching consensus on rumors abstract an important contribution in sociophysics is the galam s model of rumors spreading this model provides an explanation of rumors spreading in a population and explains some interesting social phenomena such as the diffusion of hoaxes in this paper the model has been reformulated as a markov process highlighting the stochastic nature of the phenomena this formalization allows us to derive conditions for consensus to be reached and for the existence of some interesting phenomena such as the emergence of impasses the proposed formulation allows a deeper and more comprehensive analysis of the diffusion of rumors

mathematical simulation of osteotomy numerical analysis and results abstract in this paper a model problem of one vital surgical technique in orthopaedics the so called osteotomy will be analyzed numerically the model problem will be formulated mathematically the problem leads to solving the variational equality and inequality the finite element method will be used in numerical solution and the algorithm will be given

forest matrices around the laplacian matrix abstract we study the matrices q k of in forests of a weighted digraph and their connections with the laplacian matrix l of the i j entry of q k is the total weight of spanning converging forests in forests with k arcs such that i belongs to a tree rooted at j the forest matrices q k can be calculated recursively and expressed by polynomials in the laplacian matrix they provide representations for the generalized inverses the powers and some eigenvectors of l the normalized in forest matrices are row stochastic the normalized matrix of maximum in forests is the eigenprojection of the laplacian matrix which provides an immediate proof of the markov chain tree theorem a source of these results is the fact that matrices q k are the matrix coefficients in the polynomial expansion of adj i l thereby they are precisely faddeev s matrices for l

asymptotic properties of free monoid morphisms abstract motivated by applications in the theory of numeration systems and recognizable sets of integers this paper deals with morphic words when erasing morphisms are taken into account cobham showed that if an infinite word w g f a is the image of a fixed point of a morphism f under another morphism g then there exist a non erasing morphism and a coding such that w b based on the perron theorem about asymptotic properties of powers of non negative matrices our main contribution is an in depth study of the growth type of iterated morphisms when one replaces erasing morphisms with non erasing ones we also explicitly provide an algorithm computing and from f and g

successive approximation methods for solving nested functional equations in markov decision problems this paper presents a successive approximation method for solving systems of nested functional equations which arise e g when considering markov renewal programs in which policies that are maximal gain or optimal under more selective discount and average overtaking optimality criteria are to be found in particular a successive approximation method is given to find the optimal bias vector and bias optimal policies applications with respect to a number of additional stochastic control models are pointed out our method is based on systems of simultaneously generated single equation value iteration schemes

optimality of stationary halting policies and finite termination of successive approximations the existence of an optimal stopping policy that is stationary and halting in a discrete time parameter finite state finite action branching markov decision chain is here characterized by the finite termination of successive approximations we call a policy stopping if the expected population size at time n converges to zero as n approaches infinity and halting if the expected population at time n is zero for some n we show that when the rewards are real valued the nth iterate of successive approximations is a fixed point of the optimal return operator for some n when initiated with the value of a stationary halting policy if and only if there exists a halting stationary optimal stopping policy similar results are shown for a gauss seidel improvement of successive approximations

nonnegative multiplicative processes reaching stationarity in finite time abstract a nonnegative multiplicative process is a process in which an input y a nonnegative vector is transformed in a single time period to a nonnegative output vector yp where p is a square nonnegative matrix we study such processes for which a stationary vector is reached in a finite number of steps thereby extending results from a recent paper by rothblum and tan and an earlier paper by the present author

bounds on distances between eigenvalues explicit computable lower and upper bounds on the distances between a given real eigenvalue of a real square matrix and the remaining not necessarily real eigenvalues of the matrix are developed

on the fem solution of a coupled contact two phase stefan problem in thermo elasticity coercive case abstract in the paper a semi implicit discretization in time of the weak formulation of the coupled signorini type contact two phase stefan problem is numerically analyzed the problem leads to coupled elliptic variational inequalities which are approximated by the fem

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

survival in a quasi death process we consider a markov chain in continuous time with an absorbing coffin state and a finite set s of transient states when s is irreducible the limiting distribution of the chain as t to infty conditional on survival up to time t is known to equal the unique quasi stationary distribution of the chain we address the problem of generalizing this result to a setting in which s may be reducible and obtain a complete solution if the eigenvalue with maximal real part of the generator of the sub markov chain on s has multiplicity one the result is applied to pure death processes and more generally to quasi death processes

growth optimality for branching markov decision chains this paper considers a multiplicative process called branching markov decision chains in which the output at the end of the nth period equals the product of n nonnegative matrices chosen at the beginning of periods 1 n respectively times a positive fixed terminal reward vector it is assumed that the above transition matrices are drawn out of a finite set of matrices given in product form i e the rows of the matrices can be selected independently out of finite sets of nonnegative row vectors for each coordinate s we define the geometric and algebraic growth rates respectively of the sth coordinate of the stream of output these growth rates are defined so that the magnitude of the corresponding sequence is of the order nnk where is the geometric growth rate and k is the algebraic growth rate the main result of this paper is the constructive establishment of the existence of a transition matrix whose repeated use will guarantee for each coordinate the achievement of the best geometri

a characterization of jordan canonical forms which are similar to eventually nonnegative matrices with the properties of nonnegative matrices abstract in this paper we give necessary and sufficient conditions for a matrix in jordan canonical form to be similar to an eventually nonnegative matrix whose irreducible diagonal blocks satisfy the conditions identified by zaslavsky and tam and whose subdiagonal blocks with respect to its frobenius normal form are nonnegative these matrices are referred to as seminonnegative matrices and we show that they exhibit many of the same combinatorial spectral properties as nonnegative matrices this paper extends the work on jordan forms of irreducible eventually nonnegative matrices to the reducible case

multiplicative processes reaching stationarity in finite time abstract consider a multiplicative process in which a vectorial input x is transformed in a single time period into a vectorial output xp where p is a square nonnegative matrix a stationary vector of such a process is a vector y for which the vector yp is proportional to y we study multiplicative processes in which a stationary vector is reached in finite time

on comparison of the perron frobenius eigenvalues of two ml matrices abstract an ml matrix is a matrix where all off diagonal elements are nonnegative a simple inequality relating the perron frobenius eigenvalues of two ml matrices can be obtained by performing a limiting operation on an inequality by bapat amer math monthly 96 1989 137 we derive the new inequality as a special case of a more general result which compares the values of expressions x ay and b for ml matrices a and b and nonnegative vectors x y

products of symplectic normal matrices abstract a matrix a m 2 n c is symplectic if a t 0 i n i n 0 a 0 i n i n 0 we show that every symplectic matrix is a product of a symplectic unitary and a symplectic skew hermitian matrix we show that every symplectic matrix is a product of four symplectic skew hermitian matrices or a product of four symplectic hermitian matrices we give the possible jordan canonical forms of symplectic matrices which can be written as a product of a symplectic hermitian and a matrix which is either symplectic hermitian or symplectic skew hermitian

diffusion strategies outperform consensus strategies for distributed estimation over adaptive networks adaptive networks consist of a collection of nodes with adaptation and learning abilities the nodes interact with each other on a local level and diffuse information across the network to solve estimation and inference tasks in a distributed manner in this work we compare the mean square performance of two main strategies for distributed estimation over networks consensus strategies and diffusion strategies the analysis in the paper confirms that under constant step sizes diffusion strategies allow information to diffuse more thoroughly through the network and this property has a favorable effect on the evolution of the network diffusion networks are shown to converge faster and reach lower mean square deviation than consensus networks and their mean square stability is insensitive to the choice of the combination weights in contrast and surprisingly it is shown that consensus networks can become unstable even if all the individual nodes are stable and able to solve the estimation task on their own when this occurs cooperation over the network leads to a catastrophic failure of the estimation task this phenomenon does not occur for diffusion networks we show that stability of the individual nodes always ensures stability of the diffusion network irrespective of the combination topology simulation results support the theoretical findings

characterizations of products of symmetric matrices abstract characterizations are obtained for matrices c of the form c a where a are n n matrices over the real field such that a is symmetric and c is nonnegative definite among others a proof of recent generalization of cochran s theorem is given

a turnpike theorem for a risk sensitive markov decision process with stopping this paper concerns a markov decision problem with an infinite planning horizon stationary data an exponential utility function and no discounting but also with the possibility of voluntary or involuntary termination stopping this paper establishes conditions under which the expected utility of the income received by repeated use of a well chosen stationary control equals the limit as n becomes large of the optimal expected utility for the n period problem tests for these turnpike conditions are provided as is a linear program whose optimal solution determines such a control

brief on terminating markov decision processes with a risk averse objective function we consider a class of terminating markov decision processes with an exponential risk averse objective function and compact constraint sets we assume the existence of an absorbing cost free terminal state w positive transition costs and continuity of the transition probability and cost functions without discounting future costs in the argument of the exponential utility function we establish i the existence of a real valued optimal cost function which can be achieved by a stationary policy and ii the convergence of value iteration and policy iteration to the unique solution of bellman s equation we illustrate the results with two computational examples

products of positive semidefinite matrices abstract we characterize the complex square matrices which are expressible as the product of finitely many positive semidefinite matrices a matrix t can be expressed as such if and only if det t 0 moreover the number of factors can always be limited to five we also determine those matrices which can be expressed as the product of two or four positive semidefinite matrices these results are analogous to the ones obtained before by c s ballantine for products of positive definite matrices

numerical solution of the contact problem application to a simple model of the human hip joint abstract we draw on the mathematical formulation of the contact problem which leads to the variational inequality fem is used for discretization the variational inequality enables additional contact elements to be avoided where a suitable stiffness parameter is needed moreover we obtain the asymptotic estimate model of a human hip joint is analysed in the application this analysis may be useful for modelling the human hip joint replacement after surgical reconstruction of the dysplastic acetabulum

the jordan cononical form of a product of a hermitian and a positive semidefinite matrix abstract a given square complex matrix c is the product of a positive semidefinite matrix a and a hermitian matrix b if and only if c 2 is diagonalizable and has nonnegative eigenvalues this condition is equivalent to requiring that c have real eigenvalues and a jordan canonical form that is diagonal except for r copies of a 2 by 2 nilpotent jordan block we show that r is bounded from above by the rank of a the nullity of a and both the positive and negative inertia of b it follows that a product of two positive semidefinite matrices is diagonalizable and has nonnegative eigenvalues a result that leads to a characterization of the possible concanonical forms of a positive semidefinite matrix

null controllability of discrete time positive control systems abstract the discrete time linear positive and quasilinear monotone systems are considered the set of initial points is restricted to be the cone of non negative vectors the input belongs to the cone of non positive vectors the necessary and sufficient conditions of null controllability for linear systems are obtained the conditions are sufficient for quasilinear systems

inequalities of rayleigh quotients and bounds on the spectral radius of nonnegative symmetric matrices abstract given a square nonnegative symmetric matrix a the rayleigh quotient of a nonnegative vector u under a is given by q a u u t au u t u we show that qa u au is not less than qa u where denotes coordinatewise square roots and is the hadamard product but that qa au may be smaller than qa u further we examine issues of convergence

finite element analysis of contact problems in thermoelasticity the semi coercive case abstract numerical analysis of the signorini problem with friction in two dimensional quasi coupled linear thermoelasticity is investigated piecewise linear finite elements on the triangulation of the given domain r 2 with polygonal boundary are used in this contribution we establish the rate of convergence of the finite element approximate solution u h provided the exact solution is smooth enough in general the problem represents the model problem of a great number of branches such as the model problem of a high level radioactive waste disposal system as well as the model problem of geodynamcis and biomechanics etc