tropical linear algebra with the łukasiewicz t norm the max ukasiewicz semiring is defined as the unit interval 0 1 equipped with the arithmetics a b max a b and a b max 0 a b 1 linear algebra over this semiring can be developed in the usual way we observe that any problem of the max ukasiewicz linear algebra can be equivalently formulated as a problem of the tropical max plus linear algebra based on this equivalence we develop a theory of the matrix powers and the eigenproblem over the max ukasiewicz semiring

the structure of max plus hyperplanes a max plus hyperplane briefly a hyperplane is the set of all points x x1 xn rmaxn satisfying an equation of the form a1x1 anxn an 1 b1x1 bnxn bn 1 that is max a1 x1 an xn an 1 max b1 x1 bn xn bn 1 with ai bi rmax i 1 n 1 where each side contains at least one term and where ai bi for at least one index i we show that the complements of max plus semispaces at finite points z rn are building blocks for the hyperplanes in rmaxn recall that a semispace at z is a maximal with respect to inclusion max plus convex subset of rmaxn z namely observing that up to a permutation of indices we may write the equation of any hyperplane h in one of the following two forms a1x1 apxp ap 1xp 1 aqxq a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m n 1 are finite or a1x1 apxp ap 1xp 1 aqxq an 1 a1x1 apxp aq 1xq 1 amxm an 1 where 0 p q m n and all ai i 1 m are finite and an 1 is either finite or we give a formula that expresses a nondegenerate strictly affine hyperplane i e with m n and an 1 as a union of complements of semispaces at a point z rn called the center of h with the boundary of a union of complements of other semispaces at z using this formula we obtain characterizations of nondegenerate strictly affine hyperplanes with empty interior we give a description of the boundary of a nondegenerate strictly affine hyperplane with the aid of complements of semispaces at its center and we characterize the cases in which the boundary bd h of a nondegenerate strictly affine hyperplane h is also a hyperplane next we give the relations between nondegenerate strictly affine hyperplanes h their centers z and their coefficients ai in the converse direction we show that any union of complements of semispaces at a point z rn with the boundary of any union of complements of some other semispaces at that point z is a nondegenerate strictly affine hyperplane we obtain a formula for the total number of strictly affine hyperplanes we give complete lists of all strictly affine hyperplanes for the cases n 1 and n 2 we show that each linear hyperplane h in rmaxn i e with an 1 can be decomposed as the union of four parts where each part is easy to describe in terms of complements of semispaces some of them in a lower dimensional space

finite size scaling properties of the damage distance and dynamical critical exponent for the ising model abstract with the damage spreading method scaling properties of the damage distance on the ising model with heat bath dynamics are studied numerically with the parallel flipping scheme the scaling curves fall on two curves which depend on the odd or even lattice sizes the both scaling curves give the consistent dynamical exponent as z 2 16 0 04 for d 2 and z 2 09 0 05 for d 3 respectively by shifting one of them two curves overlap each other perfectly meanwhile all the scaling curves obtained by single spin flipping processes with different odd or even lattice sizes fall on a single curve from which the consistent dynamical critical exponent with the parallel scheme is obtained z 2 18 0 02 for d 2 and z 2 08 0 04 for d 3

invariance concepts in infinite dimensions the concepts of a b and c a invariance are examined for infinite dimensional linear systems examples such as reachability a reachability inobservability and a inobservability subspaces are given these concepts are then used to solve various disturbance decoupling problems

bases in max algebra for n tuples over the algebraic system r r max concepts such as linear dependence space and basis may be defined by analogy with classical linear algebra whenever a space is finitely generated it possesses a basis and all its bases are trivially related and therefore have the same cardinality however for any given n 2 spaces with bases of arbitrary cardinality may be constructed as well as spaces with no basis

steady states in the scheduling of discrete time systems abstract steady solutions to a two sided max min linear system of equations with real coefficients are considered the problem is motivated by the demand to schedule cyclically repeated activities with deterministic processing times the maximum steady solutions are characterized as min eigenvectors of a special matrix q this condition is a necessary but not a sufficient one it has also been proven that the general problem of the recognition of solvability of this two sided system is np complete

just in time control of timed event graphs update of reference input presence of uncontrollable input a linear system theory has been developed for the class of discrete event systems subject to synchronization this paper presents the just in time control of such systems when reference input is updated and or in the presence of uncontrollable input s the proposed controls are the solutions to an optimization problem under equality constraint

damage spreading and dynamic stability of kinetic ising models we investigate how the time evolution of different kinetic ising models depends on the initial conditions of the dynamics to this end we consider the simultaneous evolution of two identical systems subjected to the same thermal noise we derive a master equation for the time evolution of a joint probability distribution of the two systems this equation is then solved within an effective field approach by analysing the fixed points of the master equation and their stability we identify regular and chaotic phases

max algebraic attraction cones of nonnegative irreducible matrices abstract it is known that the max algebraic powers ar of a nonnegative irreducible matrix are ultimately periodic this leads to the concept of attraction cone attr a t by which we mean the solution set of a two sided system t a ar x ar t x where r is any integer after the periodicity transient t a and a is the maximum cycle geometric mean of a a question which this paper answers is how to describe attr a t by a concise system of equations without knowing t a this study requires knowledge of certain structures and symmetries of periodic max algebraic powers which are also described we also consider extremals of attraction cones in a special case and address the complexity of computing the coefficients of the system which describes attraction cone

estimate of solutions for differential and difference functional equations with applications to difference methods abstract the theorems on the estimate of solutions for nonlinear second order partial differential functional equations mainly of parabolic type with dirichlet s condition and for the suitable explicit finite difference functional schemes are proved the proofs are based on the comparison technique the convergent difference method given is considered without an assumption of the global generalized perron condition on the functional variable but with local one in some sense only it is a consequence of our estimate theorems the functional dependence is of the volterra type

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

disjunctive optimization max separable problems and extremal algebras the paper was motivated by solution methods suggested in the literature for solving linear optimization problems over max or max min algebras and certain class of so called max separable optimization problems general features of these optimization problems which play a crucial role in the optimization methods were used to formulate a general class of optimization problems with disjunctive constraints and a max separable objective function and suggest a solution procedure for solving such problems linear problems over max algebras and the max separable problems are contained in this general class of optimization problems as special cases

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

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

dynamical critical exponent for the majority vote model we investigate the dynamical behavior of the isotropic majority vote model on a square lattice using a combination of damage spreading and finite size scaling methods for initial damage d 0 1 2 the dynamical phase diagram exhibits a chaotic frozen phase transition at a critical noise parameter qc 0 0818 0 0002 while for d 0 1 2 the damage does not propagate for any value of the model s parameter 0 q 1 2 from simulations at qc we find that the dynamical critical exponent is z 0 65 0 05

the two dimensional site diluted ising ferromagnet a damage spreading analysis abstract the quenched site diluted ising ferromagnet on a square lattice for which each site is occupied or empty with probabilities p and 1 p respectively is studied numerically through damage spreading procedures by making use of the glauber dynamics the percolation threshold pc is estimated within the heat bath dynamics the damage spreading temperatures td p for several values of p pc are computed indicating a strong correlation with the corresponding critical temperatures tc p a procedure for estimating the fractal dimensions of clusters of damaged sites at low temperatures is presented as p pc our estimate is very close to 91 48 which is the fractal dimension of the infinite cluster at p pc in two dimensional site percolation whenever possible to compare our results are in good agreement with the best estimates available from other techniques in spite of a modest computational effort

on tropical fractional linear programming very recently tropical counterparts of fractional linear programs have been studied some algorithms were proposed for solving them with techniques ranging from bisection methods to homeomorphisms to formal power series in this paper some algorithms are also proposed they mainly rely in the ability of finding the greatest and smallest solutions of tropical equations a subject that was discussed in a previous work of the authors 13

damage spreading in a two dimensional potts ferromagnet in an external field we study using the damage spreading method and within the heat bath dynamics the dynamical behaviour of the square lattice three state potts ferromagnet subjected to an external field in the zero field case we verify that this very simple model presents besides the transition consistent with the equilibrium one a new dynamical chaotic phase with unusual features although the unexpected transition occurs within the error bars at the static ising critical temperature it is in the directed percolation universality class we observe that the application of a uniform magnetic field does not destroy any of the two transitions while the unexpected one is annihilated by a new kind of field which plays the role of a conjugate field to the hamming distance

a linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing a discrete event system is a system whose behavior can be described by means of a set of time consuming activities performed according to a prescribed ordering events correspond to starting or ending some activity an analogy between linear systems and a class of discrete event systems is developed following this analogy such discrete event systems can be viewed as linear in the sense of an appropriate algebra the periodical behavior of closed discrete event systems i e involving a set of repeatedly performed activities can be totally characterized by solving an eigenvalue and eigenvector equation in this algebra this problem is numerically solved by an efficient algorithm which basically consists of finding the shortest paths from one node to all other nodes in a graph the potentiality of this approach for the performance evaluation of flexible manufacturing systems is emphasized the case of a flowshop like production process is analyzed in detail

tropical polar cones hypergraph transversals and mean payoff games we discuss the tropical analogues of several basic questions of convex duality in particular the polar of a tropical polyhedral cone represents the set of linear inequalities that its elements satisfy we characterize the extreme rays of the polar in terms of certain minimal set covers which may be thought of as weighted generalizations of minimal transversals in hypergraphs we also give a tropical analogue of farkas lemma which allows one to check whether a linear inequality is implied by a finite family of linear inequalities here the certificate is a strategy of a mean payoff game we discuss examples showing that the number of extreme rays of the polar of the tropical cyclic polyhedral cone is polynomially bounded and that there is no unique minimal system of inequalities defining a given tropical polyhedral cone

optimal control synthesis of timed event graphs with interval model specifications the purpose of this technical note is the optimal control synthesis of a timed event graph when the state and control trajectories should follow the specifications defined by an interval model the problem is reformulated in the fixed point form and the spectral theory gives the conditions of existence of a solution

two cores of a nonnegative matrix we prove that the sequence of eigencones i e cones of nonnegative eigenvectors of positive powers a k of a nonnegative square matrix a is periodic both in max algebra and in nonnegative linear algebra using an argument of pullman we also show that the minkowski sum of the eigencones of powers of a is equal to the core of a defined as the intersection of nonnegative column spans of matrix powers also in max algebra based on this we describe the set of extremal rays of the core the spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible in order to unify and compare both versions of the same theory

extremals of the supereigenvector cone in max algebra a combinatorial description abstract we give a combinatorial description of extremal generators of the supereigenvector cone x a x x in max algebra

on feedback invariance properties for systems over a principal ideal domain simple necessary and sufficient conditions for the solvability of many control problems for linear systems over a field are based on the equivalence between the a b invariance property and the a bf invariance or feedback invariance property for systems over a ring this equivalence is no longer true and many results of the geometric control theory cannot be extended we present new algorithmically checkable characterizations of the a bf invariance property for systems defined over a principal ideal domain and a new solvability condition for the disturbance rejection problem

densities of idempotent measures and large deviations considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by maslov then idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure in this paper we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied these conditions depend on the lattice structure of the semiring and on the boolean algebra in which the measure is defined as an application we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle as defined by varadhan

tropical planar networks abstract we show that every tropical totally positive matrix can be uniquely represented as the transfer matrix of a canonical totally connected weighted planar network we deduce a uniqueness theorem for the factorization of a tropical totally positive in terms of elementary jacobi matrices

tropicalization of facets of polytopes it is known that any tropical polytope is the image under the valuation map of ordinary polytopes over the puiseux series field the latter polytopes are called lifts of the tropical polytope we prove that any pure tropical polytope is the intersection of the tropical half spaces given by the images under the valuation map of the facet defining half spaces of a certain lift we construct this lift explicitly taking into account geometric properties of the given polytope moreover when the generators of the tropical polytope are in general position we prove that the above property is satisfied for any lift this solves a conjecture of develin and yu

optimistic optimization for model predictive control of max plus linear systems model predictive control for max plus linear discrete event systems usually leads to a nonsmooth nonconvex optimization problem with real valued variables which may be hard to solve efficiently an alternative approach is to transform the given problem into a mixed integer linear programming problem however the computational complexity of current mixed integer linear programming algorithms increases in the worst case exponentially as a function of the prediction horizon the focus of this paper is on making optimistic optimization suited to solve the given problem optimistic optimization is a class of algorithms that can find an approximation of the global optimum for general nonlinear optimization a key advantage of optimistic optimization is that one can specify the computational budget in advance and guarantee bounds on the suboptimality with respect to the global optimum we prove that optimistic optimization can be applied for the given problem by developing a dedicated semi metric and by proving it satisfies the necessary requirements for optimistic optimization moreover we show that the complexity of optimistic optimization is exponential in the control horizon instead of the prediction horizon hence using optimistic optimization is more efficient when the control horizon is small and the prediction horizon is large

computational techniques for reachability analysis of max plus linear systems this work discusses a computational approach to reachability analysis of max plus linear mpl systems a class of discrete event systems widely used in synchronization and scheduling applications given a set of initial states we characterize and compute its reach tube namely the collection of set of reachable states regarded step wise as reach sets by an alternative characterization of the mpl dynamics we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference bound matrices and further derive worst case bounds on the complexity of these operations the approach is also extended to backward reachability analysis the concepts and results are elucidated by a running example and we further illustrate the performance of the approach by a numerical benchmark the technique comfortably handles twenty dimensional mpl systems i e with twenty continuous state variables and as such it outperforms the state of the art alternative approaches in the literature

the max plus finite element method for solving deterministic optimal control problems basic properties and convergence analysis we introduce a max plus analogue of the petrov galerkin finite element method to solve finite horizon deterministic optimal control problems the method relies on a max plus variational formulation we show that the error in the sup norm can be bounded from the difference between the value function and its projections on max plus and min plus semimodules when the max plus analogue of the stiffness matrix is exactly known in general the stiffness matrix must be approximated this requires approximating the operation of the lax oleinik semigroup on finite elements we consider two approximations relying on the hamiltonian we derive a convergence result in arbitrary dimension showing that for a class of problems the error estimate is of order delta delta x delta 1 or sqrt delta delta x delta 1 depending on the choice of the approximation where delta and delta x are respectively the time and space discretization steps we compare our method with another max plus based discretization method previously introduced by fleming and mceneaney we give numerical examples in dimensions 1 and 2

basic solutions of systems with two max linear inequalities abstract we give an explicit description of the basic solutions of max linear systems a x b x with two inequalities

damage spreading in random field systems we investigate how a quenched random field influences the damage spreading transition in kinetic ising models to this end we generalize a recent master equation approach and derive an effective field theory for damage spreading in random field systems this theory is applied to the glauber ising model with a bimodal random field distribution we find that the random field influences the spreading transition by two different mechanisms with opposite effects first the random field favours the same particular direction of the spin variable at each site in both systems which reduces the damage second the random field suppresses the magnetization which in turn tends to increase the damage the competition between these two effects leads to a rich behaviour

duality between invariant spaces for max plus linear discrete event systems we extend the notions of conditioned and controlled invariant spaces to linear dynamical systems over the max plus or tropical semiring we establish a duality theorem relating both notions which we use to construct dynamic observers these are useful in situations in which some of the system coefficients may vary within certain intervals the results are illustrated by an application to a manufacturing system

time evolution of damage in the ising model the time evolution of the damage in the ising model with the heat bath and glauber dynamics has been investigated with the damage spreading method the damage density d t and the average distance r t of damaged sites from the origin have been measured numerically from the simulational data the scaling forms have been established as d t tx r t t1z and d t r t df and the corresponding exponents x z and df have been estimated

non equilibrium critical relaxation of the three dimensional ising model abstract the non equilibrium relaxation process at the critical point is studied using monte carlo simulation for the ferromagnetic ising model on a cubic lattice it was observed that the time dependent magnetization decayed following a power law t from the all up initial condition three kinds of single spin flip dynamics were studied for several lattices up to 1536 3 the system size dependences are also studied the value of was estimated to be 0 250 2 this value of leads to the estimation of the dynamical exponent z to be 2 06 2 if we assume z where and denote the critical exponents for spontaneous magnetization and correlation length in equilibrium state respectively and 0 515 the result of the non equilibrium relaxation in the ordered phase is also given

max plus a b invariant spaces and control of timed discrete event systems the concept of a b invariant subspace or controlled invariant of a linear dynamical system is extended to linear systems over the max plus semiring although this extension presents several difficulties which are similar to those encountered in the same kind of extension to linear dynamical systems over rings it appears capable of providing solutions to many control problems like in the cases of linear systems over fields or rings sufficient conditions are given for computing the maximal a b invariant subspace contained in a given space and the existence of linear state feedbacks is discussed an application to the study of transportation networks which evolve according to a timetable is considered

a solution to the problem of a b invariance for series the aim of this article is to study a standard problem of control theory the a b invariance problem which amounts to computing a maximal element x subject to conditions of the form ax 6 x b and x 6 k we give a solution to the problem in the framework of formal series over particular complete idempotent semirings over nite idempotent semirings we show that under the assumption that b and k are recognizable series the maximal solution exists and is also recognizable we obtain a similar result for the in nite tropical semiring with additional hypothesis that the series a is a language but the notion of recognizable series has to be extended to the weaker notion of pseudo recognizable series c 2002 elsevier science b v all rights reserved

the whitney embedding theorem for tropical torsion modules classification of tropical modules abstract we prove here a tropical version of the well known whitney embedding theorem 32 stating that a smooth connected m dimensional compact differential manifold can be embedded into r 2 m 1 the tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module r p where p equals to 2 for m 2 and 3 p m m 1 otherwise is the number of rows supporting the torsion when the generators are given by the independent columns of a matrix of size n m as a corollary we get that tropical m dimensional torsion modules are classified by a m 1 m m 1 1 parameter family

interval analysis and dioid application to robust controller design for timed event graphs this paper deals with feedback controller synthesis for timed event graphs where the number of initial tokens and time delays are only known to belong to intervals we discuss here the existence and the computation of a robust controller set for uncertain systems that can be described by parametric models the unknown parameters of which are assumed to vary between known bounds each controller is computed in order to guarantee that the closed loop system behavior is greater than the lower bound of a reference model set and is lower than the upper bound of this set the synthesis presented here is mainly based on dioid interval analysis and residuation theory

a max plus based algorithm for a hamilton jacobi bellman equation of nonlinear filtering the hamilton jacobi bellman hjb equation associated with the robust hinfty filter as well as the mortensen filter is considered these filters employ a model where the disturbances have finite power the hjb equation for the filter information state is a first order equation with a term that is quadratic in the gradient yet the solution operator is linear in the max plus algebra this property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off line the precomputed solutions are stored as vectors of coefficients of the basis functions these coefficients are then used directly in the real time computations

non equilibrium relaxation and interface energy of the ising model abstract from the non equilibrium critical relaxation study of the two dimensional ising model the dynamical critical exponent z is estimated to be 2 165 0 010 for this model the relaxation in the ordered phase of this model is consistent with exp t behavior the interface energy of the three dimensional ising model is studied and the critical exponent of the correlation length and the critical amplitude of the surface tension 0 are estimated to be 0 6250 0 0025 and 1 42 0 04 respectively a dynamic monte carlo renormalization group method is successfully applied to the equilibrium properties of the three dimensional ising model

computer simulation studies of critical phenomena abstract the recent strides which have been made in the use of computer simulations to study phase transitions and critical phenomena are reviewed we describe how the use of a combination of recently developed histogram techniques and sophisticated monte carlo simulation algorithms have allowed the determination of the static critical behavior of the d 3 ising and heisenberg models with unprecedented precision we also discuss the study of dynamic critical behavior in simple spin models by both monte carlo and spin dynamics methods recent estimates for dynamic critical exponents are reviewed including the first high precision estimate for true dynamics

duality and separation theorems in idempotent semimodules abstract we consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition we introduce a nonlinear projection on subsemimodules the projection of a point is the maximal approximation from below of the point in the subsemimodule we use this projection to separate a point from a convex set we also show that the projection minimizes the analogue of hilbert s projective metric we develop more generally a theory of dual pairs for idempotent semimodules we obtain as a corollary duality results between the row and column spaces of matrices with entries in idempotent semirings we illustrate the results by showing polyhedra and half spaces over the max plus semiring

the equation a x b y over max for the two sided homogeneous linear equation system a x b y over max with no infinite rows or columns in a or b an algorithm is presented which converges to a finite solution from any finite starting point whenever a finite solution exists if the finite elements of a b are all integers convergence is in a finite number of steps for which a precise bound can be calculated if moreover one of a b has only finite elements the algorithm is thus pseudopolynomial in complexity

duality and interval analysis over idempotent semirings abstract in this paper semirings with an idempotent addition are considered these algebraic structures are endowed with a partial order this allows to consider residuated maps to solve systems of inequalities a x b see 3 the purpose of this paper is to consider a dual product denoted and the dual residuation of matrices in order to solve the following inequality a x x b x sufficient conditions ensuring the existence of a non linear projector in the solution set are proposed the results are extended to semirings of intervals such as they were introduced in 25

observer based controllers for max plus linear systems max plus algebra is a suitable algebraic setting to model discrete event systems involving synchronization and delay phenomena which are often found in transportation networks communications systems and manufacturing systems one way of controlling this kind of systems consists in choosing the dates of input events in order to achieve the desired performances e g to obtain output events in order to respect given dates this kind of control is optimal according to a just in time criterion if the input event dates are delayed as much as possible while ensuring the output events to occur before a desired reference date this paper presents the observer based controller for max plus linear systems where only estimations of system states are available for the controller as in the classical sense this is a state feedback control problem which is solved in two steps first an observer computes an estimation of the state by using the input and the output measurements then this estimated state is used to compute the state feedback control action as a main result it is shown that the optimal solution of this observer based control problem leads to a greater control input than the one obtained with the output feedback strategy a high throughput screening example in drug discovery illustrates this main result by showing that the scheduling obtained from the observer based controller is better than the scheduling obtained from the output feedback controller

evolution by damage spreading in kauffman model in kauffman s boolean automata model on the square lattice the darwinian fitness of survival can be defined as the fraction of elements which do not change from one iteration to the next biological mutations are simulated by filpping one bit in the rule of one site selection of the fitter mutant then optimizes the whole lattice completely this optimization is particularly effective near the critical point of the transition to chaos but is in itself not a critical phenomenon also a two dimensional spin glass can be optimized in this way

k formal concept analysis as linear algebra over idempotent semifields abstract we report on progress in characterizing k valued fca in algebraic terms where k is an idempotent semifield in this data mining inspired approach incidences are matrices and sets of objects and attributes are vectors the algebraization allows us to write matrix calculus formulae describing the polars and the fixpoint equations for extents and intents adopting also the point of view of the theory of linear operators between vector spaces we explore the similarities and differences of the idempotent semimodules of extents and intents with the subspaces related to a linear operator in standard algebra this allows us to shed some light into formal concept analysis from the point of view of the theory of linear operators over idempotent semimodules in the opposite direction we state the importance of fca related concepts for dual order homomorphisms of linear spaces over idempotent semifields specially congruences the lattices of extents intents and formal concepts

on the steady state control of timed event graphs with firing date constraints two algorithms for solving a specific class of steady state control problems for timed event graphs are presented in the first asymptotic convergence to the desired set is guaranteed the second algorithm which builds on from the recent developments in the spectral theory of min max functions guarantees lyapunov stability since the distance between the actual state and the desired set never increases simulation results show the efficiency of the proposed approach in a problem of moderate complexity

characterization of tropical hemispaces by p r decompositions abstract we consider tropical hemispaces defined as tropically convex sets whose complements are also tropically convex and tropical semispaces defined as maximal tropically convex sets not containing a given point we introduce the concept of p r decomposition this yields to our knowledge a new kind of representation of tropically convex sets extending the classical idea of representing convex sets by means of extreme points and rays we characterize tropical hemispaces as tropically convex sets that admit a p r decomposition of certain kind in this characterization with each tropical hemispace we associate a matrix with coefficients in the completed tropical semifield satisfying an extended rank one condition our proof techniques are based on homogenization lifting a convex set to a cone and the relation between tropical hemispaces and semispaces

the disturbance decoupling problem for systems over a ring up to now the use of geometric methods in the study of disturbance decoupling problems ddps for systems over a ring has provided only necessary conditions for the existence of solutions in this paper we study such problems considering separately the case in which only static feedback solutions are allowed and the one in which dynamic feedback solutions are admitted in the first case we give a complete geometric characterization of the solvability conditions of such problems for injective systems with coefficients in a commutative ring practical procedures for testing the solvability conditions and for constructing solutions if any exist are given in the case of systems with coefficients in a principal ideal domain pid in the second case we give a complete geometric characterization of the solvability conditions for systems with coefficients in a pid

damage spreading and critical exponents for model a ising dynamics using damage spreading and heat bath dynamics we study the ising model in 2 and 3 dimensions with non conservative dynamics our algorithm differs in some important points from previous ones which makes it rather efficient we give estimates for the exponent z which seem to be the most precise published so far 2 172 0 006 for d 2 2 032 0 004 for d 3 we also give precise estimates of the exponent introduced by janssen et al z phys b 73 1989 539 and of analogous but in principle independent exponents we find surprisingly that some of the latter agree with and give an explanation for this

on visualization scaling subeigenvectors and kleene stars in max algebra abstract the purpose of this paper is to investigate the interplay arising between max algebra convexity and scaling problems the latter which have been studied in nonnegative matrix theory are strongly related to max algebra one problem is that of strict visualization scaling defined as for a given nonnegative matrix a a diagonal matrix x such that all elements of x 1 ax are less than or equal to the maximum cycle geometric mean of a with strict inequality for the entries which do not lie on critical cycles in this paper such scalings are described by means of the max algebraic subeigenvectors and kleene stars of nonnegative matrices as well as by some concepts of convex geometry

the tropical analogue of polar cones we study the max plus or tropical analogue of the notion of polar the polar of a cone represents the set of linear inequalities satisfied by its elements we establish an analogue of the bipolar theorem which characterizes all the inequalities satisfied by the elements of a tropical convex cone we derive this characterization from a new separation theorem we also establish variants of these results concerning systems of linear equalities

in an ising model with spin exchange dynamics damage always spreads we investigate the spreading of damage in ising models with kawasaki spin exchange dynamics which conserves the magnetization we first modify a recent master equation approach to account for dynamic rules involving more than a single site we then derive an effective field theory for damage spreading in ising models with kawasaki spin exchange dynamics and solve it for a two dimensional model on a honeycomb lattice in contrast to the cases of glauber or heat bath dynamics we find that the damage always spreads and never heals in the long time limit the average hamming distance approaches that of two uncorrelated systems these results are verified by monte carlo simulations

the minkowski theorem for max plus convex sets abstract we establish the following max plus analogue of minkowski s theorem any point of a compact max plus convex subset of r n can be written as the max plus convex combination of at most n 1 of the extreme points of this subset we establish related results for closed max plus cones and closed unbounded max plus convex sets in particular we show that a closed max plus convex set can be decomposed as a max plus sum of its recession cone and of the max plus convex hull of its extreme points

idempotent version of the frechet contingency array problem in this paper we study the idempotent version of the so called frechet correlation array problem the problem is studied using an algebraic approach the major result is that there exists a unique upper bound and several lower bounds the formula for the upper bound is given an algorithm is proposed to compute one lower bound another algorithm is provided to compute all lower bounds but the number of lower bounds may be a very large number note that all these results are only based on the distributive lattice property of the idempotent algebraic structure

the dynamic critical exponent of the three dimensional ising model we measure the dynamic exponent of the three dimensional ising model using a damage spreading monte carlo approach as described by macisaac and jan we simulate systems froml 5 tol 60 at the critical temperature tc 4 5115 we report a dynamic exponent z 2 35 0 05 a value much larger than the consensus value of 2 02 whereas if we assume logarithmic corrections we find thatz 2 05 0 05

difference methods for parabolic equations with robin condition abstract classical solutions of nonlinear second order partial differential functional equations of parabolic type with the robin condition are approximated in the paper by solutions of associated boundedness preserving implicit difference functional equations it is proved that the discrete solutions uniquely exist they are uniformly bounded with respect to meshes and the numerical method is convergent and stable we also find the error estimate and its asymptotic behavior the properties of some auxiliary nonlinear discrete recurrent equations are showed the proofs are based on the comparison technique and the banach fixed point theorem

damage spreading and critical exponents for model a ising dynamics abstract recent simulations by grassberger gave very precise estimates of the dynamical critical exponent z via damage spreading techniques we confirm these results using a slightly altered method and we examine larger systems up to 18161 2 and 695 2 giving z 2 18 0 02 and 2 04 0 01 respectively

max plus control design for temporal constraints meeting in timed event graphs the aim of the presented work is the control of timed event graph to meet tight temporal constraints the problem of temporal constraints is formulated in terms of control of linear max plus models first the synthesis of a control law that ensures the satisfaction of a single constraint for a single input system is presented then the single input multi constraints problem is tackled and finally the method is extended to the multi inputs multi constraints problem the proposed method is illustrated on the example of a simple production process

on max plus linear dynamical system theory a class of timed discrete event systems can be modeled by using timed event graphs a class of timed petri nets that can have its firing dynamic described by using an algebra called max plus algebra for this kind of systems it may be desirable to enforce some timing constraints in steady state in this paper this problem is called a max plus regulation problem in this context we show a necessary condition for solving these regulation problems and in addition that this condition is sufficient for a large class of problems the obtained controller is a simple linear static state feedback and can be computed using efficient pseudo polynomial algorithms simulation results will illustrate the applicability of the proposed methodology

an integrated control strategy to solve the disturbance decoupling problem for max plus linear systems with applications to a high throughput screening system this paper presents the new investigations on the disturbance decoupling problem ddp for the geometric control of max plus linear systems the classical ddp concept in the geometric control theory means that the controlled outputs will not be changed by any disturbances in practical manufacturing systems solving for the ddp would require further delays on the output parts than the existing delays caused by the system breakdown the new proposed modified disturbance decoupling problem mddp in this paper ensures that the controlled output signals will not be delayed more than the existing delays caused by the disturbances in order to achieve the just in time optimal control furthermore this paper presents the integration of output feedback and open loop control strategies to solve for the mddp as well as for the ddp if these controls can only solve for the mddp but not for the ddp an evaluation principle is established to compare the distance between two output signals generated by controls solving for the mddp and ddp respectively this distance can be interpreted as the number of tokens or firings that are needed in order for the controls to solve for the ddp moreover another alternative approach is finding a new disturbance mapping in order to guarantee the solvability of the ddp by the same optimal control for the mddp the main results of this paper are illustrated by using a timed event graph model of a high throughput screening system in drug discovery

time dependent thermodynamic properties of the ising model from damage spreading the relationship between damage spreading and static thermodynamic properties in the ising model developed by coniglioet al is here extended to include time dependent thermodynamic quantities we exploit this new result to measure the time dependent spin correlation function from damage spreading in the ising model with heat bath and glauber dynamics until now only static thermodynamic quantities have been correctly determined from damage spreading and even then only with heat bath dynamics we also show that there are significant differences between the kinetics of damage spreading as found in heat bath and glauber dynamics

paper a new approach to the analysis of discrete event dynamic systems we present a new time domain approach to the study of discrete event dynamical systems deds typified by queueing networks and production systems a general state space representation is developed and perturbation analysis is carried out observation of a single sample realization of such a system can be used to predict behavior over other sample realizations when some parameter is perturbed without having to make additional observations conditions under which this is always possible are investigated and explicit results for some special cases are included

survivor statistics and damage spreading on social network with power law degree distributions abstract damage spreading ds of the random graph networks with power law degree distributions is investigated using glauber dynamics various subgraphs defined by the probability of acquaintance show distinct features in ds as measured by hamming distance a heuristic understanding of the long time value of damage is achieved through an analysis of the survivor statistics all survivors are dynamical flipping in unison for the controlled sample and the damaged sample verification of these dynamic survivors is achieved through the introduction of a new measure of self damage