l and r localized solvabilities of max separable interval linear equations and its applications in this paper we propose new types of solutions an l localized solution and an r localized solution to interval systems of max separable linear equations a x b where a a a and b b b we provide some small application examples to emphasize the importance of our proposed solutions we give necessary and sufficient conditions for checking l localized and r localized solvabilities and apply them to achieve the closed forms of the sets of all l localized solutions and r localized solutions respectively

strong solvability of interval max plus systems and applications to optimal control abstract this paper considers the strong solvability of interval max plus systems the solvable entry and solvable interval are introduced to establish the necessary and sufficient condition for strong solvability of interval max plus systems it is proved that such a criterion is equivalent to the criterion proposed by cechlarova and cuninghame green the interval strong solution of interval max plus systems is described and the necessary and sufficient condition for uniquely strong solvability is given the method of testing strong solvability of interval max plus systems is constructive and leads to an effective algorithm to illustrate our results a workshop optimal control problem is addressed

interval systems over idempotent semiring this paper deals with solution of inequality a x b where a x and b are interval matrices with entries dened over idempotent semiring it deals also with the computation of a pair of intervals x y which satises the equation

parameter expansion to accelerate em the px em algorithm summary the em algorithm and its extensions are popular tools for modal estimation but ar e often criticised for their slow convergence we propose a new method that can often make em much faster the intuitive idea is to use a covariance adjustment to correct the analysis of the m step capitalising on extra information captured in the imputed complete data the way we accomplish this is by parameter expansion we expand the complete data model while preserving the observed data model and use the expanded complete data model to generate em this parameter expanded ei m px em algorithm shares the simplicity and stability of ordinary em but has a faster rate of convergence since its m step performs a more efficient analysis the px em algorithm is illustrated for the multivariate t distribution a random effects model factor analysis probit regression and a poisson imaging model

lattice valued approach to closed sets under fuzzy relations theory and applications motivated by fuzzy control problems and by some investigations of eigen fuzzy sets we deal with a closedness of fuzzy sets under fuzzy relations in two ways in one sense by directly analyzing fuzzy concepts and in the other by investigating the corresponding crisp problems in the cutworthy framework our main task is to investigate particular fuzzy functional equations and inequations appearing in this context which turn out to be essentially connected with fuzzy control problems we analyze procedures and find solutions of these equations and inequations pointing to important applications

weak dual residuations applied to tropical linear equations an extension to an algorithm of r a cuninghame green and k zimmermann for solving equations with residuated functions is presented this extension relies on the concept of weak residuation and in the so called strong property it is shown that a contextualization of this method to tropical linear equations which will be denoted as primal method in contrast with the dual method another algorithm described in literature generates a non decreasing sequence which converges to the smallest solution in a special semimodule it is also shown the connections of this method with previously published works

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

estimating a population of parameter values using bayes and empirical bayes methods abstract in standard bayes and empirical bayes component decision problems estimating inidividual parameters is the primary goal in multiple comparison problems and in comparisons of histograms of estimates however the primary goal is to produce parameter estimates that can be considered as an ensemble for example the histogram of estimates should be a good estimate of the histogram of parameters standard methods of estimating by the posterior expectation do minimize symmetric componentwise losses such as squared error but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters in this article we propose new bayes and empirical bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters these estimators are weighted averages of the prior mean and the data with weight on the data being approximately the square root of that for the posterior expectation we give theoreti

estimating variance parameters from multivariate normal variables subject to limits of detection mle reml or bayesian approaches likelihood based approaches which naturally incorporate left censoring due to limit of detection are commonly utilized to analyze censored multivariate normal data however the maximum likelihood estimator mle typically underestimates variance parameters the restricted maximum likelihood estimator reml which corrects the underestimation of variance parameters cannot be easily extended to analyze censored multivariate normal data in the light of the connection between the reml and a bayesian approach discovered in 1974 by dr harville this paper describes a bayesian approach to censored multivariate normal data this bayesian approach is justified through its link to the reml via laplace s approximation and its performance is evaluated through a simulation study we consider the bayesian approach as a valuable alternative because it yields less biased variance parameter estimates than the mle and because a solid reml is technically difficult when data are left censored copyright 2009 john wiley sons ltd

analysis of weighted networks the connections in many networks are not merely binary entities either present or not but have associated weights that record their strengths relative to one another recent studies of networks have by and large steered clear of such weighted networks which are often perceived as being harder to analyze than their unweighted counterparts here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph allowing us to apply standard techniques for unweighted graphs to weighted ones as well we give a number of examples of the method including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum flow minimum cut theorem

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

interval systems of max separable linear equations for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations we define and characterize several types of solutions and solvability of interval systems of max separable linear equations

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

extended valued topical and anti topical functions on semimodules in the papers 16 and 17 we have studied functions defined on a b complete idempotent semimodule x over a b complete idempotent semifield k k with values in k where k may or may not contain a greatest element supk and the residuation x y is not defined for x x and y infx in the present paper we assume that k has no greatest element then adjoin to k an outside greatest element supk and extend the operations and from k to k k so as to obtain a meaning also for x infx for any x x and study functions with values in k in fact we consider two different extensions of the product from k to k denoted by and respectively and use them to give characterizations of topical i e increasing homogeneous defined with the aid of and anti topical i e decreasing anti homogeneous defined with the aid of functions in terms of some inequalities next we introduce and study for functions f x k their conjugates and biconjugates of fenchel moreau type with respect to the coupling functions x y x y x y x and x y d inf x y d x y x d k and use them to obtain characterizations of topical and anti topical functions in the subsequent sections we consider for the coupling functions and some concepts that have been studied in rubinov and singer 2001 11 and singer 2004 15 for the so called additive min type coupling functions rmaxn rmaxn rmax and an an a respectively where a is a conditionally complete lattice ordered group and x y inf1 i n xi yi x y rmaxn or an thus we study the polars of a set g x for the coupling functions and and we consider for a function f x k the notion of support set of f with respect to the set t of all elementary topical functions t y x x y x x y x infx and two concepts of support set of f at a point x0 x the main differences between the properties of the conjugations with respect to the coupling functions and and between the properties of the polars of a set g with respect to the coupling functions and are caused by the fact that while is symmetric with values only in rmax resp a and are not symmetric and take values also outside rmax resp a

tropical implementation of the analytical hierarchy process decision method abstract we apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the analytic hierarchy process in multi criteria problems of rating alternatives from pairwise comparison data in this framework we first consider the minimax chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale recasting this approximation problem as a problem of tropical pseudo quadratic programming we then write out a closed form solution to it this solution might be either a unique score vector up to a positive factor or a set of different score vectors to handle the problem when the solution is not unique we develop tropical optimization techniques of maximizing and minimizing the hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores and thus are well representative of the entire solution set

unique solvability of max min fuzzy equations and strong regularity of matrices over fuzzy algebra a matrix is said to have strongly linearly independent columns or in the case of square matrices to be strongly regular if for some vector b the system a x b has a unique solution we formulate a necessary and sufficient condition for a linear system of equations over a fuzzy algebra to have a unique solution and prove the equivalence of strong regularity and trapezoidal property moreover an algorithm for testing these properties is reviewed

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

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

computing behavior of finite fuzzy machines algorithm and its application to reduction and minimization we define finite fuzzy machines and investigate their behavior algorithm and software are proposed for computing behavior for establishing equivalence and redundancy of states and for solving reduction and minimization problems computational complexity of the algorithm is discussed testing examples are supplied the results are valid for finite max min min max and max product fuzzy machines

measuring explained variation in linear mixed effects models we generalize the well known r2 measure for linear regression to linear mixed effects models our work was motivated by a cluster randomized study conducted by the eastern cooperative oncology group to compare two different versions of informed consent document we quantify the variation in the response that is explained by the covariates under the linear mixed model and study three types of measures to estimate such quantities the first type of measures make direct use of the estimated variances the second type of measures use residual sums of squares in analogy to the linear regression the third type of measures are based on the kullback leibler information gain all the measures can be easily obtained from software programs that fit linear mixed models we study the performance of the measures through monte carlo simulations and illustrate the usefulness of the measures on data sets copyright 2003 john wiley sons ltd

control solvability of interval systems of max separable linear equations abstract for a given matrix interval a a a and a given vector interval b b b the notation a x b if max represents an interval system of linear max separable equations several types of solvability of interval systems are known we define control solvability weak control solvability and weak universal solvability and give necessary and sufficient condition for an interval system to be solvable according to each definition

analysis of repeatability and reproducibility studies with ordinal measurements a bayesian inferential approach with a noninformative prior is introduced to analyze ordinal repeatability and reproducibility r r data using the de mast van wieringen model this approac

scheduling with and or precedence constraints in many scheduling applications it is required that the processing of some job be postponed until some other job which can be chosen from a pregiven set of alternatives has been completed the traditional concept of precedence constraints fails to model such restrictions therefore the concept has been generalized to so called and or precedence constraints which can cope with this kind of requirement in the context of traditional precedence constraints feasibility transitivity and the computation of earliest start times for jobs are fundamental well studied problems the purpose of this paper is to provide efficient algorithms for these tasks for the more general model of and or precedence constraints we show that feasibility as well as many questions related to transitivity can be solved by applying essentially the same linear time algorithm in order to compute earliest start times we propose two polynomial time algorithms to cope with different classes of time distances between jobs

minimization algorithm of fuzzy finite automata in this paper a class of fuzzy finite automata corresponding to the mealy type of ordinary automata is formulated and also two types of statewise equivalence relations are introduced from the equivalence relations a minimal form is defined and a minimization algorithm of the mealy type of fuzzy finite automata is obtained

an algorithm for solving two sided interval system of max plus linear equations we study six types of solutions weak strong tolerance control left localized and right localized solutions to a two sided interval system of max plus linear equations with the same vector of variables on both sides of the equations and obtain their corresponding solvability conditions these conditions are in the form of two sided systems of max plus linear inequalities which can be derived as a union of a number of interval inclusion linear systems therefore we could work on the interval inclusion linear systems instead of directly solving these six types of solutions themselves an optimization problem with the two sided interval system of max plus linear constraints may be solved using the convexity of each solution set of the interval inclusion linear systems

bisimulations for weighted automata over an additively idempotent semiring weshowthatthemethodologyfordecidingtheexistenceandcomputationofthegreatestsimulationsandbisimulations developed in the framework of fuzzy automata m ciric et al fuzzy sets and systems 186 2012 100 139 208 2012 22 42 can be applied in a similar form to weighted automata over an additively idempotent semiring we define two types of simulations and four types of bisimulations for weighted automata over an additively idempotent semiring as solutions to particular systems of matrix inequalities and we provide polynomial time algorithms for deciding whether there is a simulation or bisimulation of a given type between two weighted automata and for computing the greatest one if it exists the algorithms are based on the concept of relative boolean residuation for matrices over an additively idempotent semiring that we introduce here we also prove that two weighted automata a and b are forward bisimulation equivalent i e there is a row and column complete forward bisimulation between them if and only if the factor weighted automata with respect to the greatest forward bisimulation equivalence matrices on a and b are isomorphic in addition we show that the factor weighted automaton with respect to the greatest forward bisimulationequivalence matrix on aweighted automaton a is the unique upto an isomorphism minimal automaton in the class of all weighted automata which are forward bisimulation equivalent to a

asymptotic behavior of nonoscillatory solutions to n th order nonlinear neutral differential equations abstract for a class of n th order nonlinear neutral differential equations sufficient conditions for all nonoscillatory solutions to satisfy lim t t 1 n x t a are established for another class of equations necessary and sufficient conditions for nonoscillatory solutions to satisfy the above condition are obtained our theorems complement and extend recent results reported in the literature

determinization of fuzzy automata with membership values in complete residuated lattices in this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices in comparison with the previous methods developed by belohlavek r belohlavek determinism and fuzzy automata information sciences 143 2002 205 209 and li and pedrycz y m li w pedrycz fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids fuzzy sets and systems 156 2005 68 92 our method always gives a smaller automaton and in some cases when the previous methods result in infinite automata our method can result in a finite one we also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them and in particular to the concept of the nerode s fuzzy right congruence of a fuzzy automaton which we introduce and study here

interval max plus matrix equations abstract this paper deals with the solvability of interval matrix equations in max plus algebra max plus algebra is the algebraic structure in which classical addition and multiplication are replaced by and where a b max a b and a b a b the notation a x c b represents an interval max plus matrix equation where a b and c are given interval matrices we define four types of solvability of interval max plus matrix equations i e the tolerance weak tolerance left weak tolerance and right weak tolerance solvability we derive the necessary and sufficient conditions for checking each of them whereby all can be verified in polynomial time

finite l fuzzy machines abstract for finite l fuzzy machines we develop behaviour reduction and minimization theory here l stands for l 0 1 where l is a totally ordered set with universal bounds 0 and 1 and the operations are join and meet the most essential results include investigating behaviour equivalence reduction and minimization problems and their algorithmical decidability

finitary solvability conditions for systems of fuzzy relation equations in this paper we consider the problem of solving systems of fuzzy relation equations two types of these systems with different compositions are considered and processed simultaneously a new sufficient condition and new solvability criteria are proposed for systems of both types all these conditions are finitary and thus can be easily verified the sufficient condition and criteria of solvability characterize a relationship between a skeleton nxn matrix n is a number of equations and a vector of the right hand side

interval strong solutions of interval systems of max plus linear equations abstract this paper considers the existence and uniqueness of interval strong solutions of interval systems of max plus linear equations a necessary and sufficient condition for the existence of interval strong solutions is presented the proof of the existence of interval strong solutions is constructive and results in a formula for computing such solutions a necessary and sufficient condition for the uniqueness of interval strong solutions is established by testing the unique solvability of a finite number of subsystems rather than all subsystems on this basis a polynomial algorithm is developed to verify the uniqueness of interval strong solutions

partitioning degrees of freedom in hierarchical and other richly parameterized models hodges and sargent 2001 have developed a measure of a hierarchical model s complexity degrees of freedom df that is consistent with definitions for scatterplot smoothers is interpretable in terms of simple models and enables control of a fit s complexity by means of a prior distribution on complexity but although df describes the complexity of the whole fitted model in general it remains unclear how to allocate df to individual effects here we present a new definition of df for arbitrary normal error linear hierarchical models consistent with that of hodges and sargent that naturally partitions the n observations into df for individual effects and for error the new conception of an effect s df is the ratio of the effect s modeled variance matrix to the total variance matrix this provides a way to describe the sizes of different parts of a model e g spatial clustering vs heterogeneity to place df based priors on smoothing parameters and to describe how a smoothed effect competes with

ultra discrete permanent and the consistency of max plus linear equations abstract in this paper we investigate the consistency conditions for three classes of the max plus linear equations including the case corresponding to the convex polyhedra which was the subject of the previous paper the necessary and sufficient conditions for the existence of finite solutions are expressed in the form of single equations by using the ultra discrete permanent

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

closed form of l localized solution set of max plus interval linear system and its application on optimization problem a closed form of the l localized solution set of an interval linear system with max plus algebra is a finite union of interval vectors this paper analyzes and proves some properties to reduce number of the union sets to be a smaller number of the union of interval vectors that are not subsets of each other this result helps simplify the feasible set of an optimization problem with max plus linear equation constraints

existence of positive solutions for a class of first order neutral functional differential equations where s s q1 or y1 it is assumed throughout this paper that w a t t a r is continuous and strictly increasing t t t for 0 t g t and lim t t s 0 t a w b h t t a r is continuous 0 w c g t a r is continuous and lim g t s 0 t a w w d f t 0 a 0 is continuous and f t u is nonde0 w creasing in u g 0 for any fixed t g t 0

fuzzy correspondence inequations and equations the paper deals with fuzzy correspondences i e with mappings from a direct product of sets into a complete lattice fuzzy control problems connected with fuzzy correspondence inequations and equations are considered a fuzzy correspondence is fixed while the solutions are investigated for the input and output fuzzy sets which are unknown first we prove that these solutions can be analyzed and formulated in crisp framework solving the corresponding cut problems further the space of solutions of an inequation is proved to be a complete lattice the same holds for the space of solutions of the corresponding equation in case the membership values belong to an infinitely distributive lattice frame the solutions can be found in specified intervals moreover we were able to locate in such intervals minimal and maximal solutions of these problems

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

interval systems of max separable linear equations an interval system of equations is weakly solvable if at least one of its subsystems is solvable and it is strongly solvable if all its subsystems are solvable we give necessary and sufficient conditions enabling efficient testing of weak and strong solvability over max plus and max min algebras

an algorithm to describe the solution set of any tropical linear system a x b x abstract an algorithm to give an explicit description of all the solutions to any tropical linear system a x b x is presented the given system is converted into a finite rather small number p of pairs s t of classical linear systems a system s of equations and a system t of inequalities the notion introduced here that makes p small is called compatibility the particular feature of both s and t is that each item equation or inequality is bivariate i e it involves exactly two variables one variable with coefficient 1 and the other one with 1 s is solved by gaussian elimination we explain how to solve t by a method similar to gaussian elimination to achieve this we introduce the notion of sub special matrix the procedure applied to t is therefore called sub specialization

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

some algebra and geometry for hierarchical models applied to diagnostics recent advances in computing make it practical to use complex hierarchical models however the complexity makes it difficult to see how features of the data determine the fitted model this paper describes an approach to diagnostics for hierarchical models specifically linear hierarchical models with additive normal or t errors the key is to express hierarchical models in the form of ordinary linear models by adding artificial cases to the data set corresponding to the higher levels of the hierarchy the error term of this linear model is not homoscedastic but its covariance structure is much simpler than that usually used in variance component or random effects models the re expression has several advantages first it is extremely general covering dynamic linear models random effect and mixed effect models and pairwise difference models among others second it makes more explicit the geometry of hierarchical models by analogy with the geometry of linear models third the analogy with linear models provides a rich source of ideas for diagnostics for all the parts of hierarchical models this paper gives diagnostics to examine candidate added variables transformations collinearity case influence and residuals

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

best approximation in max plus semimodules abstract we establish new results concerning projectors on max plus spaces as well as separating half spaces and derive an explicit formula for the distance in hilbert s projective metric between a point and a half space over the max plus semiring as well as explicit descriptions of the set of minimizers as a consequence we obtain a cyclic projection type algorithm to solve systems of max plus linear inequalities

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

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

fuzzy relation equations and inequalities with two unknowns and their applications abstract in this paper we study several types of systems of fuzzy relation equations and inequalities composed of a given family of fuzzy relations between two sets and two unknown fuzzy relations on these sets solutions to these systems are pairs of fuzzy relations on the underlying sets and can be ordered coordinatewise we show that solutions to each of these systems form a complete lattice and provide procedures for computing the greatest solution which is less than or equal to a given pair of fuzzy relations we also demonstrate the application of solutions to these systems in data reduction

weakly linear systems of fuzzy relation inequalities the heterogeneous case new types of systems of fuzzy relation inequalities and equations called weakly linear have been recently introduced in j ignjatovic et al on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations fuzzy sets syst 161 2010 3081 3113 the mentioned paper dealt with homogeneous weakly linear systems composed of fuzzy relations on a single set and a method for computing their greatest solutions has been provided this method is based on the computing of the greatest post fixed point contained in a given fuzzy relation of an isotone function on the lattice of fuzzy relations here we adapt this method for computing the greatest solutions of heterogeneous weakly linear systems where the unknown fuzzy relation relates two possibly different sets we also introduce and study quotient fuzzy relational systems and establish relationships between solutions to heterogeneous and homogeneous weakly linear systems besides we point out to applications of the obtained results in the state reduction of fuzzy automata and computing the greatest simulations and bisimulations between fuzzy automata as well as in the positional analysis of fuzzy social networks

counting degrees of freedom in hierarchical and other richly parameterised models summary drawing on linear model theory we rigorously extend the notion of degrees of freedom to richly parameterised models including linear hierarchical and random effect models some smoothers and spatial models and combinations of these the number of degrees of freedom is often much smaller than the number of parameters our notion of degrees of freedom is compatible with similar ideas long associated with smoothers but is applicable to new classes of models and can be interpreted using the projection theory of linear models we use an example to illustrate the two applications of setting prior distributions for variances and fixing model complexity by fixing degrees of freedom

interval max plus systems of linear equations abstract in this paper we shall deal with solvability of interval systems of linear equations in max plus algebra max plus algebra is an algebraic structure in which classical addition and multiplication are replaced by and where a b max a b a b a b the notation a x b represents an interval system of linear equations where a a a b b b are given interval matrix and interval vector respectively and a solution is from a given interval vector x x x we can define several types of solvability of interval systems in this paper we define six types of solvability of interval max plus systems of linear equations and give necessary and sufficient conditions for them

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

reduction of fuzzy automata by means of fuzzy quasi orders abstract in our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations in that paper we have also studied reductions by means of those solutions which are fuzzy equivalences in this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi orders generally fuzzy quasi orders and fuzzy equivalences are equally good in the state reduction but we show that right and left invariant fuzzy quasi orders give better reductions than right and left invariant fuzzy equivalences we also show that alternate reductions by means of fuzzy quasi orders give better results than alternate reductions by means of fuzzy equivalences furthermore we study a more general type of fuzzy quasi orders weakly right and left invariant ones and we show that they are closely related to determinization of fuzzy automatons we also demonstrate some applications of weakly left invariant fuzzy quasi orders in conflict analysis of fuzzy discrete event systems

bayesian measures of explained variance and pooling in multilevel hierarchical models explained variance r2 is a familiar summary of the fit of a linear regression and has been generalized in various ways to multilevel hierarchical models the multilevel models that we consider in this article are characterized by hierarchical data structures in which individuals are grouped into units which themselves might be further grouped into larger units and variables are measured on individuals and each grouping unit the models are based on regression relationships at different levels with the first level corresponding to the individual data and subsequent levels corresponding to between group regressions of individual predictor effects on grouping unit variables we present an approach to defining r2 at each level of the multilevel model rather than attempting to create a single summary measure of fit our method is based on comparing variances in a single fitted model rather than with a null model in simple regression our measure generalizes the classical adjusted r2 we also discuss

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

on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations in this paper we study systems of fuzzy relation inequalities and equations of the form u v i v i u i i where u is an unknown and v i i i are given fuzzy relations the dual systems v i u u v i i i their conjunctions the systems of the form u v i v i u i i and certain special types of these systems we call them weakly linear systems for each weakly linear system with a complete residuated lattice as the underlying structure of truth values we prove the existence of the greatest solution and we provide an algorithm for computing the greatest solution which works whenever the underlying complete residuated lattice is locally finite otherwise we determine some sufficient conditions under which the algorithm works the algorithm is iterative and each its single step can be viewed as solving of a particular linear system weakly linear systems emerged from the fuzzy automata theory but we show that they also have important applications in other fields e g in the concurrency theory and social network analysis

dioids and semirings links to fuzzy sets and other applications besides the classical algebraic structures of groups rings and fields which have long been the almost exclusive reference concepts used in mathematical modelling other algebraic structures such as dioids and semirings have emerged in the last two or three decades in connection with modelling and solving a rich variety of non classical problems e g in decision analysis fuzzy set theory operations research automatic control and mathematical physics the present paper aims at providing an overview of applications of dioid and semiring structures stressing links with fuzzy sets and emphasizing linear algebraic problems solving linear systems computing eigenvalues and eigenvectors non classical path finding problems using algebras of endomorphisms and connections between dioid structure and nonlinear analysis in view of solving problems in mathematical physics

on quotient machines of a fuzzy automaton and the minimal machine in this paper the concept of substitution property sp partition for a finite fuzzy automaton is formulated and the quotient automaton with respect to an sp partition is defined the state minimization problem for such automata by using the method of quotient machines is solved

on the dimension of max min convex sets we introduce a notion of dimension of max min convex sets following the approach of tropical convexity we introduce a max min analogue of the tropical rank of a matrix and show that it is equal to the dimension of the associated polytope we describe the relation between this rank and the notion of strong regularity in max min algebra which is traditionally defined in terms of unique solvability of linear systems and the trapezoidal property

on the computational complexity of coalitional resource games we study coalitional resource games crgs a variation of qualitative coalitional games qcgs in which each agent is endowed with a set of resources and the ability of a coalition to bring about a set of goals depends on whether they are collectively endowed with the necessary resources we investigate and classify the computational complexity of a number of natural decision problems for crgs over and above those previously investigated for qcgs in general for example we show that the complexity of determining whether conflict is inevitable between two coalitions with respect to some stated resource bound i e a limit value for every resource is co np complete we then investigate the relationship between crgs and qcgs and in particular the extent to which it is possible to translate between the two models we first characterise the complexity of determining equivalence between crgs and qcgs we then show that it is always possible to translate any given crg into a succinct equivalent qcg and that it is not always possible to translate a qcg into an equivalent crg we establish some necessary and some sufficient conditions for a translation from qcgs to crgs to be possible and show that even where an equivalent crg exists it may have size exponential in the number of goals and agents of its source qcg

fuzzy function as an approximate solution to a system of fuzzy relation equations this paper is mostly focused on the problem of approximate solvability of a system of fuzzy relation equations however we put a new light on this problem connecting it with the interpolation of a fuzzy function we introduce a notion of fuzzy function and its representation by fuzzy relation we demonstrate how problems of interpolation and approximation of fuzzy functions are connected with solvability of systems of fuzzy relation equations first we explain the general framework and then we prove some results related to the problem of the best approximation in particular we have shown that fuzzy relations introduced by sanchez and mamdani are the best approximations in certain approximation spaces

system of fuzzy relation equations as a continuous model of if then rules we claim that a fuzzy relation which models a system of fuzzy if then rules has the property of continuity only if the underlying system of fuzzy relation equations is solvable in view of this connection it is important to consider the problem of solvability of a system of fuzzy relation equations with respect to an unknown fuzzy relation this problem is analyzed in the present paper we propose several new necessary and sufficient criteria for the so called mamdani relation to be a solution to the system we prove that in the general case these same criteria are sufficient but not always necessary for the solvability of the system a new easy to check criterion is proposed for the solvability of the system with special fuzzy parameters

on max min linear inequalities and coalitional resource games with sharable resources abstract in a coalitional resource game crg for brief agents form coalitions to pool their resources in order to achieve certain goals requiring the expenditure of these resources a particular coalition is said to be successful if the common resources of its members enable to achieve a set of goals that satisfies all members of the coalition it is known that when resources are consumable it is np complete to decide whether a given coalition is successful in this paper we show a connection of crgs with sharable resources and max min linear systems of inequalities this correspondence leads to polynomial algorithms for checking whether a given crg admits a successful coalition and for several other problems whose counterparts for crgs with consumable resources are hard on the other hand we prove that some problems concerning the structure of successful coalitions are hard also in the case of sharable resources

local and relativized local finiteness in t norm based structures this paper discusses the property of local finiteness for t norm monoids and extensions of them the focus is i on t norm bimonoids which are of interest in the context of weighted automata ii on those extensions of t norm monoids which are reducts of t algebras i e of the basic semantic entities for the fuzzy logics of left continuous and of continuous t norms and iii on extensions of t norm monoids with their residuation based standard negations the paper introduces also a kind of relativized local finiteness and offers a short discussion of the finite model property for t norm based residuated logics

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 r2 statistic for fixed effects in the linear mixed model statisticians most often use the linear mixed model to analyze gaussian longitudinal data the value and familiarity of the r2 statistic in the linear univariate model naturally creates great interest in extending it to the linear mixed model we define and describe how to compute a model r2 statistic for the linear mixed model by using only a single model the proposed r2 statistic measures multivariate association between the repeated outcomes and the fixed effects in the linear mixed model the r2 statistic arises as a 1 1 function of an appropriate f statistic for testing all fixed effects except typically the intercept in a full model the statistic compares the full model with a null model with all fixed effects deleted except typically the intercept while retaining exactly the same covariance structure furthermore the r2 statistic leads immediately to a natural definition of a partial r2 statistic a mixed model in which ethnicity gives a very small p value as a longitudinal predictor of blood pressure bp compellingly illustrates the value of the statistic in sharp contrast to the extreme p value a very small r2 a measure of statistical and scientific importance indicates that ethnicity has an almost negligible association with the repeated bp outcomes for the study copyright 2008 john wiley sons ltd

fuzzy relational inequalities and equations fuzzy quasi orders closures and openings of fuzzy sets the paper deals with systems of fuzzy relation equations known in the literature as eigen fuzzy set equations as well as with systems of related fuzzy relation inequalities our main results are algorithms for computing the greatest and the smallest solutions to the considered systems of fuzzy relation inequalities and algorithms for computing the greatest solutions to the considered systems of fuzzy relation equations the systems are studied in the framework of complete residuated lattices as the structure of membership values by means of fuzzy quasi orders and closures and openings on a collection of fuzzy sets