modeling and analysis of timed petri nets using heaps of pieces the authors show that safe timed petri nets can be represented by special automata over the max semiring which compute the height of heaps of pieces this extends to the timed case the classical representation a la mazurkiewicz of the behavior of safe petri nets by trace monoids and trace languages for a subclass including all safe free choice petri nets we obtain reduced heap realizations using structural properties of the net covering by safe state machine components the authors illustrate the heap based modeling by the typical case of safe jobshops for a periodic schedule the authors obtain a heap based throughput formula which is simpler to compute than its traditional timed event graph version particularly if one is interested in the successive evaluation of a large number of possible schedules

triangularizability of trace class operators with increasing spectrum abstract for any measurable set e of a measure space x let p e be the orthogonal projection on the hilbert space l 2 x with the range ran p e f l 2 x f 0 a e on e c that is called a standard subspace of l 2 x let t be an operator on l 2 x having increasing spectrum relative to standard compressions that is for any measurable sets e and f with e f the spectrum of the operator p e t ran p e is contained in the spectrum of the operator p f t ran p f in 2009 marcoux mastnak and radjavi asked whether the operator t has a non trivial invariant standard subspace they answered this question affirmatively when either the measure space x is discrete or the operator t has finite rank we study this problem in the case of trace class kernel operators we also slightly strengthen the above mentioned result for finite rank operators

on a class of compactly epi lipschitzian sets the paper is devoted to the study of the so called compactly epi lipschitzian sets these sets are needed for many aspects of generalized differentiation particulary for necessary optimality conditions stability of mathematical programming problems and calculus rules for subdifferentials and normal cones we present general conditions under which sets defined by general constraints are compactly epi lipschitzian this allows us to show how the compact epi lipschitzness properties behave under set intersections

a maximum principle for nonsmooth optimal control problems with state constraints abstract optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints the conditions which apply in the absence of differentiability assumptions on the data are stated in terms of clarke s generalized jacobians although not the most general available the conditions are derived by a novel method this involves removal of the state constraints by introduction of a penalty term and application of ekeland s variational principle

pmax1 and smax properties and asymptotic stability in the max algebra abstract in this work we introduce the class of p max 1 matrices for the max algebra and derive some properties of these that echo similar results for p matrices in the conventional algebra in analogy with 1 we define the row p max 1 property and the smax property of a finite set of nonnegative matrices moreover we relate these concepts to stability questions for sets of matrices and difference inclusions in the max algebra

standard triangularization of semigroups of non negative operators we show that under mild conditions a semigroup of non negative operators on lp x for 1 p of the form scalar plus compact is triangularizable via standard subspaces if and only if each operator in the semigroup is individually triangularizable via standard subspaces also in the case of operators of the form identity plus trace class we show that triangularizability via standard subspaces is equivalent to the submultiplicativity of a certain function on the semigroup

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

on the stochastic nonlinear neutron transport equation the probability that a neutron leads to a divergent chain reaction in a nuclear reactor is governed by a nonlinear integro partial differential equation 1 a model case of this equation was completely analysed by pazy and rabinowitz 2 3 the purpose of this paper is to extend their results to the general case and to tackle some related topics

subdifferential representation formula and subdifferential criteria for the behavior of nonsmooth functions abstract several kinds of behaviors of extended real valued lower semicontinuous functions are known to be equivalent to certain appropriate conditions in terms of the clarke subdifferential the paper provides a systematic study showing that any such condition with the clarke subdifferential is valid if and only if it holds with any operator representing the clarke subdifferential like in the subdifferential proximal formula

on the leading eigenvalue of neutron transport models abstract we give variational characterizations of the leading eigenvalue of neutron transport like operators the proofs rely on sub and super eigenvalues various bounds of the leading eigenvalue are derived

barabanov norms lipschitz continuity and monotonicity for the max algebraic joint spectral radius abstract we present several results describing the interplay between the max algebraic joint spectral radius jsr for compact sets of matrices and suitably defined matrix norms in particular we extend a classical result for the conventional algebra showing that the max algebraic jsr can be described in terms of induced norms of the matrices in the set we also show that for a set generating an irreducible semigroup in a cone theoretic sense a monotone barabanov norm always exists this fact is then used to show that the max algebraic jsr is locally lipschitz continuous on the space of compact irreducible sets of matrices with respect to the hausdorff distance we then prove that the max algebraic jsr is locally hoelder continuous on the space of compact sets of nonnegative matrices finally we prove a strict monotonicity property for the max algebraic jsr that echoes a fact for the classical jsr the single matrix characterisation of the max algebraic jsr plays a vital role in our proofs

various lipschitz like properties for functions and sets i directional derivative and tangential characterizations in this work we introduce for extended real valued functions defined on a banach space x the concept of k directionally lipschitzian behavior where k is a bounded subset of x for different types of sets k e g zero singleton or compact the k directionally lipschitzian behavior recovers well known concepts in variational analysis locally lipschitzian directionally lipschitzian or compactly epi lipschitzian properties respectively characterizations of this notion are provided in terms of the lower dini subderivatives we also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the bouligand tangent cones the special case of convex functions and sets is also studied

a min plus derivation of the fundamental car traffic law we give deterministic and stochastic models of the traffic on a circular road without overtaking from this model the mean speed is derived as an eigenvalue of the min plus matrix describing the dynamics of the system in the deterministic case and as the lyapunov exponent of a min plus stochastic matrix in the stochastic case the eigenvalue and the lyapunov exponent are computed explicitly from these formulas we derive the fundamental law that links the flow to the density of vehicles on the road numerical experiments using the maxplus toolbox of scilab confirm the theoretical results obtained

positivity for equations involving polyharmonic operators with dirichlet boundary conditions cranston fabes and zhao 26 5 established the uniform bound sup x y 2 x 6 y r g1 n x z g1 n z y dz g1 n x y m 1 1 where g1 n x y is the green function for the laplacian with dirichlet boundary conditions on a lipschitz domain rn with n 3 see 27 for n 2 this estimate was used in 23 and 18 to obtain positivity uniformly with respect to f 0 for noncooperative elliptic systems as

integration of subdifferentials of lower semicontinuous functions on banach spaces abstract this paper studies the lower semicontinuous functions on banach spaces whose subdifferentials are not disassociated from several conditions are given to ensure that two lower semicontinuous functions with the same subdifferential are equal up to an additive constant

a mathematical study in the theory of dynamic population abstract this article deals with a mathematical model of an age structured proliferating cell population originally proposed by lebowitz and rubinow j math biol 1 1974 17 36 individual cells are distinguished by age and by cell cycle length the cell cycle length is considered as an inherited property determined at birth here general boundary conditions are considered by means of a linear and bounded operator k after establishing the theorem of traces we show that the model is well posed in the sense of the theory of semigroup without restriction on the boundary operator k we study the positivity and the irreducibility of the generated semigroup and we calculate its essential type the asymptotic behavior is obtained in the uniform topology

the analytic hierarchy process max algebra and multi objective optimisation the analytic hierarchy process ahp is widely used for decision making involving multiple criteria elsner and van den driessche 2004 2010 10 11 introduced a max algebraic approach to the single criterion ahp we extend this to the multi criteria ahp by considering multi objective generalisations of the single objective optimisation problem solved in these earlier papers we relate the existence of globally optimal solutions to the commutativity properties of the associated matrices we relate min max optimal solutions to the generalised spectral radius and we prove that pareto optimal solutions are guaranteed to exist

on the symbiotic lotka volterra model with diffusion and transport effects abstract in this work we analyze the existence stability and multiplicity of coexistence states for a symbiotic lotka volterra model with general diffusivities and transport effects global bifurcation theory blowing up arguments for a priori bounds singular perturbation results singularity theory and fixed point index in cones are among the techniques used to get our results and to explain the drastic change of behavior exhibited by the dynamics of the model between the cases of weak and strong mutualism between the species our methodology works out to treat much more general classes of symbiotic models

irreducible subalgebras of quasi normed operator ideals it is known that the only irreducible c p closed subalgebra of c p is c p itself a generalization of this theorem is established and some extensions of recent reducibility results are obtained based on this generalization we then apply these extensions to derive some decomposability results concerning semigroup of compact quasinilpotent positive operators on banach lattices

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

uncertain dynamical systems defined by pseudomeasures this paper deals with uncertain dynamical systems in which predictions about the future state of a system are assessed by so called pseudomeasures two special cases are stochastic dynamical systems where the pseudomeasure is the conventional probability measure and fuzzy dynamical systems in which the pseudomeasure is a so called possibility measure new results about possibilistic systems and their relation to deterministic and to stochastic systems are derived by using idempotent pseudolinear algebra by expressing large deviation estimates for stochastic perturbations in terms of possibility measures we obtain a new interpretation of the freidlin wentzell quasipotentials for stochastic perturbations of dynamical systems as invariant possibility densities

a mathematical study for a rotenberg model abstract this paper deals with a mathematical model in cell dynamics population originally proposed by m rotenberg 1983 j theoret biol 103 181 199 individual cells are distinguished by their degree of maturity and maturation velocity here all biological rules are considered we bring new techniques to the discussion of this model we show that the model is well posed in the sense of the theory of semigroups we study the positivity and irreducibility of the generated semigroups and we calculate its essential type the asymptotic behavior is obtained in uniform topology

approximate mean value theorem for upper subderivatives the purpose of this paper is to give a mean value theorem for a lower semicontinuous function f on a banach space using the upper subderivatives defined by rockafellar l as will be shown by example section 4 it may happen that the subdifferential af x is empty for all x in the closed line segment a 61 where a and b belong to x therefore we cannot formulate the mean value theorem in such a form as that obtained in 2 and 3 for other generalized derivatives we prove among other factors that there exists a sequence xk converging to some point x e a 61 x b such that the sets df xj are nonempty and

convex subcones of the contingent cone in nonsmooth calculus and optimization the tangential approximants most useful in nonsmooth analysis and optimization are those which lie between the clarke tangent cone and the bouligand contigent cone a study of this class of tangent cones is undertaken here it is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone there are such convex subcones which are more accurate approximants than the clarke tangent cone and possess an associated subdifferential calculus that is equally strong in addition a large class of convex subcones of the contingent cone can replace the clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program however the clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven overall the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones

epi lipschitz like sets in banach space theorems and examples on considere un espace de banach x et on etudie quelles proprietes un ensemble c dans x doit avoir pour que le cone tangent de clarke t c x et le cone normal n c x mesurent adequatement le comportement a la frontiere de x dans c

on the continuity of the generalized spectral radius in max algebra abstract given a bounded set of n n non negative matrices let and denote the generalized spectral radius of and its max version respectively we show that sup t 0 n 1 t 1 t where t denotes the hadamard power of we apply this result to give a new short proof of a known fact that is continuous on the hausdorff metric space h of all nonempty compact collections of n n non negative matrices

finite valued distance automata abstract distance automata are a model of finite state machines which charge to an input word the expense of the cheapest successful computation consuming that word called its distance the distance of such a machine is the maximal distance of an input word recognized by it or is infinite depending on whether or not a maximum exists the valuedness of a distance automaton is introduced by considering this machine to be a transducer with unary output alphabet the distance of a finite valued distance automaton with n states is either infinite or at most 6 n 2 n n 2 1 in the former case its growth is linear in the input length the problem of deciding whether a given finite valued 2 valued respectively distance automaton has infinite distance is pspace complete it is decidable in deterministic double exponential time whether two given finite valued distance automata are equivalent i e every input word has the same distance in both machines there is an inherently infinite valued distance automaton and for each k an inherently k valued distance automaton such that the growth of the distance in all these machines is linear in the input length it is decidable in deterministic polynomial time whether a 2 valued distance automaton given as the disjoint union of two single valued distance automata is inherently 2 valued

semiclassical analysis of general second order elliptic operators on bounded domains in this work we ascertain the semiclassical behavior of the fundamental energy and the ground state of an arbitrary second order elliptic operator not necessarily selfadjoint on a bounded domain our analysis provides us with substantial improvements of many previous results found in the context of quantum mechanics for c perturbations of the laplacian

a nonlinear age dependent model with spatial diffusion abstract the main goal of this paper is the study of the existence and uniqueness of a positive solution for a nonlinear age dependent equation with spatial diffusion for that we mainly use the properties of an eigenvalue problem related to the equation and the sub supersolution method we justify that this method works for this equation in which there is a potential blow up and a nonlocal initial condition

asymptotic theory of the linear transport equation in anisotropic media we consider linear transport in an anisotropic medium with velocity dependent cross sections r v t and scattering kernel p r v v t we introduce a scaling in terms of a small parameter where the leading order term describes an equilibrium in velocity space between collisions with a cross section that is an even function of v and scattering modes even even and odd odd in v and v we show that the asymptotic solution of the transport equation leads to a diffusion equation with a drift term with an error in 2 and derive consistent initial and boundary conditions from the analysis of the initial and boundary layers the analysis of the drift terms shows that they result from anisotropic interactions with the medium and also from streaming between neighboring but different equilibria the restriction of our results to isotropic media yields back the larsen keller diffusion equation while the one speed form reduces to the result obtained by pomraning and prinja ann nucl energy 22 159 1995 for

periodic parabolic eigenvalue problems with a large parameter and degeneration we consider a periodic parabolic eigenvalue problem with a non negative potential m vanishing on a non cylindrical domain dm satisfying conditions similar to those for the parabolic maximum principle we show that the limit as leads to a periodic parabolic problem on dm having a periodic parabolic principal eigenvalue and eigenfunction which are unique in some sense we substantially improve a result from du and peng trans amer math soc 364 2012 p 6039 6070 at the same time we offer a different approach based on a periodic parabolic initial boundary value problem the results are motivated by an analysis of the asymptotic behaviour of positive solutions to semilinear logistic periodic parabolic problems with temporal and spacial degeneracies

approximating the spectral radius of sets of matrices in the max algebra is np hard the lower and average spectral radii measure respectively the minimal and average growth rates of long products of matrices taken from a finite set the logarithm of the average spectral radius is traditionally called the lyapunov exponent when one performs these products in the max algebra we obtain quantities that measure the performance of discrete event systems we show that approximating the lower and average max algebraic spectral radii is np hard

on nonnegative solvability of linear integral equations abstract let denote a finite measure space and l p 1 p p th summable real valued functions suppose moreover k is an integral operator whose nonnegative kernel k is measurable on and which maps l p into itself while possessing a compact iterate we present necessary and sufficient conditions for the integral operator equation k g to possess a nonnegative solution l p whenever g is a given nontrival and nonnegative element of l p and is any given positive parameter this analysis extends that by victory siam j algebraic discrete methods 6 406 412 1985 for the matrix case

epi lipschitzian reachable sets of differential inclusions abstract the reachable sets of a differential inclusion have nonsmooth topological boundaries in general the main result of this paper is that under the well known assumptions of filippov s existence theorem about solutions of differential inclusions every epi lipschitzian initial compact set k r n preserves this regularity for a short time i e f t k is also epi lipschitzian for all small t 0 the proof is based on rockafellar s geometric characterization of epi lipschitzian sets and uses a new result about the inner semicontinuity of clarke tangent cone t y t f t k c y r n with respect to both arguments

concentration estimates in a multi host epidemiological model structured by phenotypic traits in this work we consider an epidemic system modelling the evolution of a spore producing pathogen within a multi host population of plants here we focus our analysis on the study of the stationary states we first discuss the existence of such nontrivial states by using the theory of global attractors then we introduce a small parameter epsilon that characterises the width of the mutation kernel and we describe the asymptotic shape of steady states with respect to epsilon in particular we show that the distribution of spores converges to the singular measure concentrated on the maxima of fitness of the pathogen in each plant population this asymptotic description allows us to show the local stability of each of the positive steady states in the regime of narrow mutations from which we deduce a uniqueness result for the nontrivial stationary states by means of a topological degree argument these analyses rely on a careful investigation of the spectral properties of some non local operators

triangularizability of operators with increasing spectrum we establish finite and infinite dimensional versions of the following assertion if m is a matrix with the property that whenever p and q are diagonal projections with p q the spectrum of pmp considered as an operator on the range of p is contained in that of qmq considered as an operator on the range of q then there is a permutation matrix u such that u 1mu is triangular

links between subderivatives and subdifferentials abstract we provide formulas linking the radial subderivative to other subderivatives and subdifferentials for arbitrary extended real valued lower semicontinuous functions

asymptotic behavior in a heap model with two pieces in a heap model solid blocks or pieces pile up according to the tetris game mechanism an optimal schedule is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap in a heap model with two pieces we prove that there always exists an optimal schedule which is balanced either periodic or sturmian we also consider the model where the successive pieces are chosen at random independently and with some given probabilities we study the expected growth rate of the heap for a model with two pieces the rate is either computed explicitly or given as an infinite series we show an application for a system of two processes sharing a resource and we prove that a greedy schedule is not always optimal

supervisory control of timed discrete event systems the ramadge wonham framework for control of discrete event systems is augmented with timing features by use of ostroff s semantics for timed transition models it is shown that the rw concept of controllability and the existence of maximally permissive supervisory controls can be suitably generalized the enhanced setting admits subsystem composition and the concept of forcible event as an event that preempts the tick of a global clock an example of a simple manufacturing cell illustrates how the new framework can be used to solve synthesis problems which may include logic based temporal and quantitative optimality specifications

irreducible compact operators the purpose of the present paper is to show that a positive compact ideal irreducible operator t in a banach lattice of dimension greater than one has spectral radius r t 0 i e t is not quasi nilpotent it was shown in 6 that there exist positive irreducible operators which are quasi nilpotent and the question under what conditions a positive irreducible operator t satisfies r t 0 has been studied extensively see e g 6 and 7 sect v 6 in particular we mention the ando krieger theorem see e g 9 theorem 136 9 which states that a positive irreducible kernel operator t in a banach function space has r t 0 recently in 8 some special situations are discussed in which a compact positive irreducible operator has a strictly positive spectral radius let l be a real or complex banach lattice by 2 we denote the banach space of all bounded linear operators in l as usual we write s t in l whenever su tu for all o uel given 0 t l the closed ideal j in l is called t invariant if t j j and t is called ideal irreducible if the only closed t invariant ideals are 0 and l see 7 iii 8 for the general theory of banach lattices and positive operators we refer to the books 7 and 9 in the proof of the main result of the paper we will use some properties of the center of a banach lattice we recall some elementary facts let l be a real banach lattice the center z l of l is the subspace of 5 l consisting of all operators z for which there exists 0 2 ir such that f l 2 f for all f l the center has the structure of a vector lattice with nl v n2 u n l u v n2u and ozla zz u lu a nzu for all o u l and all l n2ez l moreover z l is a commutative algebra with respect to composition as multiplication and the identity operator i in l is a strong unit i e for any rc z l there exists 0 2eir such that zi 2i see 9 chap 20 note that if l c k for some compact hausdorff space k then z l can be identified with c k acting on itself by multiplication if l is a dedekind complete banach lattice and 0 u v in l then it is an easy consequence of the freudenthal spectral theorem 4 theorem 40 2 that there exists n z l such that zcv u and o n i however in an arbitrary banach lattice such a 7z need not exist as is shown by the banach lattice l c 0 1 the next lemma shows that under rather mild conditions we can

on principal eigenvalues for periodic parabolic problems with optimal condition on the weight function abstract let be a bounded domain in r n and let m be a t periodic function such that its restriction to 0 t belongs to ls 0 t lv for some v n 2 and s 2 v 2 v n with v 1 and s 2 we give necessary and sufficient conditions on m for the existence uniqueness and simplicity of the principal eigenvalue for the dirichlet periodic parabolic eigenvalue problem with weight m

improved limitedness theorems on finite automata with distance functions abstract let a be a finite automaton with a distance function and id a be the set of distances associated with words accepted by a this paper presents an improved upper bound of id a when the upper limit of id a is finite it also presents one necessary and sufficient condition concerning word expressions for the upper limit of id a to be infinite

sequential normal compactness versus topological normal compactness in variational analysis abstract we study relationships between two normal compactness properties of sets in banach spaces that play an essential role in many aspects of variational analysis and its applications particularly in calculus rules for generalized differentiation necessary optimality and suboptimality conditions for optimization problems etc both properties automatically hold in finite dimensional spaces and reveal principal features of the infinite dimensional variational theory similar formulations of these properties involve the weak convergence of sequences and nets respectively containing generalized normal cones in duals to banach spaces we prove that these properties agree for a large class of banach spaces that include weakly compactly generated spaces we also show that they are always different in banach spaces whose unit dual ball is not weak sequentially compact moreover the sequential and topological normal compactness properties may not coincide even in non separable asplund spaces that admit an equivalent c smooth norm

ergodic theory of stochastic petri networks stochastic petri nets are a general formalism for describing the dynamics of discrete event systems the present paper focuses on a subclass of stochastic petri nets called stochastic event graphs under the assumption that the variables used for the timing of an event graph form stationary and ergodic sequences of random variables we make use of an associated stochastic recursive equation in order to construct its stationary and ergodic regime in particular we determine the conditions under which the existence of this regime is guaranteed

a smooth variational principle with applications to hamilton jacobi equations in infinite dimensions abstract we prove that if x is a banach space which admits a smooth lipschitzian bump function then for every lower semicontinuous bounded below function there exists a lipschitzian smooth function g on x such that f g attains its strong minimum on x thus extending a result of borwein and preiss we then show how the above result can be used to obtain existence and uniqueness results of viscosity solutions of hamilton jacobi equations in infinite dimensional banach spaces a without assuming the radon nikodym property

the hypertangent cone in borwein and strojwas 5 we have observed that in baire metrizable spaces the hypertangent cone plays an important role in the theory of tangent cones and generalized subgradients because its properties relate to lipschitz behaviour of sets and functions here more accurate results for banach spaces are presented they follow from a formula which may be viewed as the discrete version of the treiman inclusion 18 we prove this in section 1 our discrete formula has many applications it implies generalizations of the bishop phelps theorem on the density of support points 11 and the dense clarke subdifferentiability theorem of mclinden 14 in section 2 we present a new characterization of the clarke derivative and new conditions characterizing directionally lipschitzian functions on a banach space in section 3 we use our discrete formula to obtain mean value theorems for lower semicontinuous functions on banach spaces in the vein of penot 16 also we prove a strengthening of lebourg s 13 mean value theorem for reflexive spaces in section 4 we prove a general inversion theorem in particular we use it to furnish conditions for the clarke tangent cone of the intersection of sets to contain the intersection of the tangent cones for these sets this generalizes the finite dimensional version of ioffe ll and an epi lipschitzian version of rockafellar 16 in a banach space the basic facts about tangent cones and generalized subgradients that we use in this paper are discussed in clarke l0

some spectral properties for the asymptotic behavior of semigroups connected to population dynamics this paper gives a short review of the mathematics involved in the study of asymptotic behavior of semigroups arising from structured models of population dynamics first are considerations on the

performance evaluation of max automata automata with multiplicities over the max semiring can be used to represent the behavior of timed discrete event systems this formalism which extends both conventional automata and max linear representations covers a class of systems with synchronization phenomena and variable schedules performance evaluation is considered in the worst mean and optimal cases a simple algebraic reduction is provided for the worst case the last two cases are solved for the subclass of deterministic series recognized by deterministic automata deterministic series frequently arise due to the finiteness properties of max linear projective semigroups the mean performance is given by the kolmogorov equation of a markov chain the optimal performance is given by a hamilton jacobi bellman equation

joint spectral radius operator semigroups and a problem of w wojtynski abstract we investigate the connections between the invariant subspace problem for operator semigroups and the joint spectral radius as a consequence it is proved that any quasinilpotent lie algebra of compact operators on a banach space is triangularizable we extend the berger wang formula to precompact sets of essentially scalar operators and prove the continuity of the joint spectral radius on them

links between directional derivatives through multidirectional mean value inequalities we prove in the general setting of lower semicontinuous functions on banach spaces the relation between the rockafellar directional derivative and the mixed lower limit of the lower dini derivatives as a byproduct we derive the famous inclusions of tangent cones of closed sets in banach spaces the results are established using as principal tool multidirectional mean value inequalities aussel et al siam j optim 9 3 690 706 1999

bisimulation relations for weighted automata bisimulation is a well known equivalence relation for discrete event systems and has been extended to probabilistic and stochastic systems this paper introduces a general definition of bisimulation which can be applied to finite automata where weights and labels are assigned to transitions it is shown that this general view contains several known bisimulations as special cases and defines naturally equivalences for different classes of models apart from the well known forward bisimulation also backward bisimulation is introduced and it is shown that both types of bisimulation preserve different types of results furthermore it is shown that forward and backward bisimulation are congruences according to commonly known composition operations for automata

nonsmooth constrained optimization and multidirectional mean value inequalities we establish a general fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a banach space our basic tool is a constrained variational principle derived from the smooth variational principle of borwein and preiss specializing the fermat rule to the case when the convex set is a drop we obtain a multidirectional rolle type inequality from which in turn we deduce a multidirectional mean value inequality in the line of clarke and ledyaev we follow the abstract approach of our previous paper trans amer math soc 347 1995 pp 4147 4161 thus covering all standard situations met in applications while stressing the links between the results and the few key properties that are needed

existence of spectral values for irreducible c0 semigroups abstract a c 0 semigroup t t of positive linear operators on a banach lattice e is called irreducible if it leaves no closed lattice ideals 0 e invariant it is shown that all known and some recently discovered conditions on e and or an irreducible cyclic semigroup t n implying that t 0 have precise analogs for irreducible semigroups to the effect that the generator a has nonvoid spectrum and if t t is eventually compact or a has compact resolvent that the spectral bound s a is an eigenvalue of a with positive eigenvector

on strong tangential transversality abstract this is the second of two closely related papers on transversality here we introduce the notion of strong tangential transversality of two closed subsets of a banach space which is a natural sufficient condition for tangential transversality some properties of uniform tangent sets are obtained a new sum rule for the clarke subdifferential is proven

nonsmooth calculus in finite dimensions the notion of subgradient originally defined for convex functions has in recent years been extended via the upper subderivative to cover functions that are not necessarily convex or even continuous a number of calculus rules have been proven for these generalized subgradients this paper develops the finite dimensional generalized subdifferential calculus for strictly lower semicontinuous functions under considerably weaker hypotheses than those previously used the most general finite dimensional convex subdifferential calculus results are recovered as corollaries other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program various chain rule formulations are considered equality in the subdifferential calculus formulae is proven under hypotheses weaker than the usual subdifferential regularity assumptions

discrete event systems with stochastic processing times discrete event dynamic systems are studied in which the underlying algebra is the max algebra and the coefficients in the system referring to processing times in practice are stochastic the processing times and or the transportation times within a network show stochastic fluctuations the restrictions are that the stochastic processing times of the nodes in the network are independent and identically distributed the asymptotic behavior of the system is investigated and the average duration of one cycle of the process is calculated a specific example of the theory is considered the state space is two dimensional and the probability distributions are exponential it is shown that the process approaches a stationary limit as time proceeds the case when the probability distributions are discrete is also treated several examples are given two dimensional systems and more generally finite dimensional systems are considered

on the spectral radius of positive operators professor j j grobler has kindly pointed out to us that theorems 4 4 and 5 3 in our paper 1 are false as stated and need correction the error in the proofs is the tacit claim that t is a nonzero positive operator to avoid this situation we need to add the minor extra assumption t 2 4 0 which is automatically satisfied in all places where theorems 4 4 or 5 3 are invoked therefore all results in 1 on the positivity of the spectral radius are correct as stated we follow the terminology and notation of 1 and start by stating and proving the correct version of theorem 4 4

asymptotic height optimization for topical ifs tetris heaps and the finiteness conjecture given an iterated function system ifs of topical maps verifying some conditions we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional the average height on some compact space of measures we give general results to determine these extrema and then apply them to two concrete problems first we give a new proof of the theorem that the densest heaps of two tetris pieces are sturmian second we construct an explicit counterexample to the lagarias wang finiteness conjecture

generalized spectral radius and its max algebra version abstract let c n n and r n n be bounded subsets and let and denote the generalized spectral radius of and the max algebra version of the generalized spectral radius of respectively we apply a single matrix description of to give a new elementary and straightforward proof of the berger wang formula in max algebra and consequently a new short proof of the original berger wang formula in the case of bounded subsets of n n non negative matrices we also obtain a new description of in terms of the schur hadamard product and prove new trace and max trace descriptions of and in particular we show that limsup m sup a m tr a 1 m limsup m sup a m tr a 1 m and limsup m sup b m tr b 1 m limsup m sup b m tr b 1 m where tr a max i 1 n a ii and b b ij