brief paper the asymptotic behavior of stochastically perturbed di sir epidemic models with saturated incidences in this paper we consider a class of di sir epidemic models with saturated incidences and parameter perturbation we investigate the asymptotic behavior according to the perturbation and the reproductive number r 0 when the perturbation is large the infective in every group decays exponentially to zero while the susceptible converges weakly to stationary distribution regardless of the magnitude of r 0 when the perturbation is small we get the same exponential stability and weak convergence if r 0 1 and we use a new class of stochastic lyapunov functions to obtain the ergodicity and positive recurrence if r 0 1

partial ordering in l underdeterminate sets abstract the purpose of this paper is to improve results on fuzzy partial orderings obtained by zadeh in 9 we overcome the difficulties connected with the axioms of antisymmetry and linearity moreover if the underlying lattice l is a complete residuated lattice we establish a szpilrajn theorem i e any l fuzzy partial ordering has a linear extension in opposition to zadeh s our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless therefore we first introduce the category lus cf 2 which can be considered as a mathematical model of fuzzy equality and subsequently we specify the axioms of l fuzzy partial orderings with respect to the frame given by lus the axioms we use clearly display the usefulness of having a zadeh like complementation and as a consequence the usefulness of a positivistic and nonintuitionistic frame of study an example concerning l rn which we give clearly shows that the lus version of the szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem but provides us with additional information

exclusion and persistence in deterministic and stochastic chemostat models abstract we first introduce and analyze a variant of the deterministic single substrate chemostat model in this model microbe removal and growth rates depend on biomass concentration with removal terms increasing faster than growth terms using a comparison principle we show that persistence of all species is possible in this scenario then we turn to modelling the influence of random fluctuations by setting up and analyzing a stochastic differential equation in particular we show that random effects may lead to extinction in scenarios where the deterministic model predicts persistence on the other hand we also establish some stochastic persistence results

a stochastic model for internal hiv dynamics in this paper we analyse a stochastic model representing hiv internal virus dynamics the stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling we show that the model established in this paper possesses non negative solutions as this is essential in any population dynamics model we also carry out analysis on the asymptotic behaviour of the model we approximate one of the variables by a mean reverting process and find out the mean and variance of this process numerical simulations conclude the paper

global stability of a delayed sir epidemic model with density dependent birth and death rates an sir epidemic model with density dependent birth and death rates is formulated in our model it is assumed that the total number of the population is governed by logistic equation the transmission of infection is assumed to be of the standard form namely proportional to i t h n t h where n t is the total variable population size i t is the size of the infective population and a time delay h is a fixed time during which the infectious agents develop in the vector we consider transmission dynamics for the model stability of an endemic equilibrium is investigated the stability result is stated in terms of a threshold parameter that is a basic reproduction number r 0

a nonlinear aids epidemic model with screening and time delay abstract a nonlinear mathematical model to study the effect of time delay in the recruitment of infected persons on the transmission dynamics of hiv aids is proposed and analyzed in modeling the dynamics the population is divided into four subclasses the susceptibles the hiv positives or infectives that do not know they are infected the hiv positives that know they are infected and the aids patients susceptibles are assumed to become infected via sexual contacts with both types of infectives the model is analyzed using stability theory of delay differential equations both the disease free and the endemic equilibria are found and their stability is investigated it is shown that the introduction of time delay in the model has a destabilizing effect on the system and periodic solutions can arise by hopf bifurcation numerical simulations are also carried out to investigate the influence of key parameters on the spread of the disease to support the analytical conclusion and to illustrate possible behavioral scenario of the model

iterative learning control synthesis based on 2 d system theory an algorithm is presented for iterative learning of the control input for a linear discrete time multivariable system necessary and sufficient conditions are stated for convergence of the proposed algorithm the algorithm synthesis and analysis are based on two dimensional 2 d system theory a numerical example is given

the maximal distance between imprecise point objects in this paper we propose a new mathematical model of imprecise point objects and a new algorithm for determining the maximal distance between imprecise point objects results obtained with our model can be used in various applications such as gis imprecise spatial object modelling robotics environment models and image analysis imprecise feature extraction imprecise point objects are modelled as fuzzy points in linear fuzzy space linear fuzzy space is defined over r n where the membership function is convex upper semi continuous symmetric w r t the core and linearly decreasing w r t the core distance where the core is a single point a fuzzy convex hull is defined in linear fuzzy space the distance between objects points is modelled as an imprecise point object in linear fuzzy space defined over r 1 fuzzy ordering relations are defined in distance space the maximal distance is a fuzzy set over a set of distances the maximal distance between a set of imprecise point objects is then determined as the maximal distance of fuzzy convex hull edge points an algorithm for determining the maximal distance between imprecise points belonging to one and two dimensional linear fuzzy space is given for an exact calculation of this maximal distance computational complexity of the algorithm is of polynomial growth of the fourth degree however if the maximal distance is determined approximately then the algorithms computational complexity is of power growth of the quadratic degree with respect to the cardinality of the convex edge

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

a stochastic model of aids and condom use in this paper we introduce stochasticity into a model of aids and condom use via the technique of parameter perturbation which is standard in stochastic population modelling we show that the model established in this paper possesses non negative solutions as desired in any population dynamics we also carry out a detailed analysis on asymptotic stability both in probability one and in pth moment our results reveal that a certain type of stochastic perturbation may help to stabilise the underlying system

on similarity in fuzzy description logics this paper is a contribution to the study of similarity relations between objects represented as attribute value pairs in fuzzy description logics for this purpose we use concrete domains in the fuzzy description logic i a l c e f d associated either with a left continuous or with a finite t norm we propose to expand this fuzzy description logic by adding a similarity box sbox including axioms expressing properties of fuzzy equalities we also define a global similarity between objects from similarities between the values of each object attribute local similarities and we prove that the global similarity defined using a t norm inherits the usual properties of the local similarities reflexivity symmetry or transitivity we also prove a result relative to global similarities expressing that in the context of the logic mtl similar objects have similar properties being these properties expressed by predicate formulas evaluated in these objects

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

asymptotic properties of an hiv aids model with a time delay a mathematical model for hiv aids with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed the disease free and endemic equilibria are found and their local stability investigated we use the lyapunov functional approach to show the global stability of the endemic equilibrium qualitative analysis of the model including positivity and boundedness of solutions and persistence are also presented the hiv aids model is numerically analysed to asses the effects of incubation period on the dynamics of hiv aids and the demographic impact of the epidemic using the demographic and epidemiological parameters for zimbabwe

graded properties of unary and binary fuzzy connectives the paper studies basic graded properties of unary and binary fuzzy connectives i e unary and binary operations on the set of truth degrees of a background fuzzy logic extending the logic mtl of left continuous t norms the properties studied in this paper are graded generalizations of monotony lipschitz continuity null and unit elements idempotence commutativity and associativity the paper elaborates the initial study presented in previous papers and focuses mainly on parameterization of graded properties by conjunction multiplicities of subformulae in the defining formulae preservation of graded properties under compositions and slight variations of fuzzy connectives the values of graded properties for basic connectives of the ground logic and the dependence of the values on the ground logic the results are proved in the formal framework of higher order fuzzy logic mtl also known as fuzzy class theory fct general theorems provable in fct are illustrated on several semantic examples

traces of fuzzy binary relations abstract in this paper the right and left trace of a fuzzy binary relation is defined and investigated basic properties of a fuzzy relation such as reflexivity different kinds of transitivity linearity etc are characterized by means of its traces

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

solution of fuzzy equations with extended operations abstract assuming that is any operation defined on a product set x y and taking values on a set z it can be extended to fuzzy sets by means of zadeh s extension principle given a fuzzy subset c of z it is here shown how to solve the equation a b c or a b c when a fuzzy subset a of x or a fuzzy subset b of y is given the methodology we provide includes as a special case the resolution of fuzzy arithmetical operations i e when stands for or extended to fuzzy numbers fuzzy subsets of the real line the paper is illustrated with several examples in fuzzy arithmetic

metrics and t equalities abstract the relationship between metrics and t equalities is investigated the latter are a special case of t equivalences a natural generalization of the classical concept of an equivalence relation it is shown that in the construction of metrics from t equalities triangular norms with an additive generator play a key role conversely in the construction of t equalities from metrics this role is played by triangular norms with a continuous additive generator or equivalently by continuous archimedean triangular norms these results are then applied to the biresidual operator e t of a triangular norm t it is shown that e t is a t equality on 0 1 if and only if t is left continuous furthermore it is shown that to any left continuous triangular norm t there correspond two particular t equalities on f x the class of fuzzy sets in a given universe x one of these t equalities is obtained from the biresidual operator e t t by means of a natural extension procedure these t equalities then give rise to interesting metrics on f x

the reproductive number for an hiv model with differential infectivity and staged progression we formulate an hiv epidemic model with differential infectivity and staged disease progression to account for variations in viral loads and in the rate of disease progression in infected individuals the stability of the infection free equilibrium determines the threshold conditions under which the modeled disease either dies out or persists in the population this stability expressed in terms of the epidemic reproductive number can be determined by the spectral radius of the next generation operator or from the eigenvalues of the jacobian matrix for the model system linearized about the infection free equilibrium we derive an explicit formula for the reproductive number employing both of these techniques by investigating the spectral radius of the next generation operator and by directly applying m matrix theory with recursive forward and backward inductions to characterize the eigenvalues of the jacobian matrix in terms of the reproductive number

a mathematical approach to hiv infection dynamics in order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as hiv infection process and aids disease progression it is efficient to introduce a general class of time dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition an attempt is then made to combine the implicit approach usually adapted for convective diffusion operators and explicit approach more suited to treat continuous type operators representing various physiological interactions resulting in a semi implicit product formula decomposing the operators in this way and considering their individual properties it is seen that approximation solvability of the original model is verified under suitable conditions once appropriate terms are formulated to describe treatment by antiretroviral therapy the time dependence of the reaction terms appears and such product formula is useful for generating approximate numerical solutions to the governing equations with this knowledge a continuous model for hiv disease progression is formulated and physiological interpretations are provided the abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the hiv virus in the human body and its reaction to treatment by antiretroviral therapy the product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with hiv1 and is applied to perform numerical simulations based on the model of the hiv infection process and disease progression finally the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects

l ordered and l lattice ordered groups this paper pursues an investigation on groups equipped with an l ordered relation where l is a fixed complete heyting algebra first by the concept of join and meet on an l ordered set the notion of an l lattice a weak l lattice is introduced and some related results are obtained then we applied them to define an l lattice ordered group we also introduce convex l subgroups to construct a quotient l ordered group at last a relation between the positive cone of an l ordered group and special type of elements of l g is found where g is a group

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

a theory of vague lattices based on many valued equivalence relations ii complete lattices as a continuation of part i two sorts of vague lattices called complete vague lattices and ordinarily complete vague lattices both of which correspond to the generalization of complete lattices based on many valued equivalence relations are introduced and some of their elementary properties are presented in this paper furthermore the representations and the constructions of ordinarily complete vague lattices are studied in details

fuzzy real numbers as dedekind cuts with respect to a multiple valued logic abstract viewing fuzzy set theory as a model of set with respect to a given multiple valued logic we establish the dedekind macneille completion of a partially ordered l valued set contrary to other constructions in fuzzy set theory the completion is unique up to an order isomorphism specializing this situation to the set of all crisp rational numbers we obtain in a unique way the fuzzy dedekind real numbers

from fuzzy logic to fuzzy mathematics a methodological manifesto the paper states the problem of fragmentation of contemporary fuzzy mathematics and the need of a unified methodology and formalism we formulate several guidelines based on hajek s methodology in fuzzy logic which enable us to follow closely the constructions and methods of classical mathematics recast in a fuzzy setting as a particular solution we propose a three layer architecture of fuzzy mathematics with the layers of formal fuzzy logic a foundational theory and individual mathematical disciplines developed within its framework the ground level of logic being sufficiently advanced we focus on the foundational level the theory we propose for the foundations of fuzzy mathematics can be characterized as henkin style higher order fuzzy logic finally we give some hints on the further development of individual mathematical disciplines in the proposed framework and proclaim it a research programme in formal fuzzy mathematics

on the convergence of iterative learning control we derive frequency domain criteria for the convergence of linear iterative learning control ilc on finite time intervals that are less restrictive than existing ones in the literature in particular the former can be used to establish the convergence of ilc in certain cases where the latter are violated the results cover ilc with non causal filters and provide insights into the transient behaviors of the algorithm before convergence we also stipulate some practical rules under which ilc can be applied to a wider range of applications

on un suitable fuzzy relations to model approximate equality in this paper we state that fuzzy equivalence relations in general are not suitable to model approximate equality since then the notion of transitivity is counter intuitive to substantiate this we investigate some of the undesirable results caused by transitivity among other things in the case of approximate reasoning we then introduce a new framework to model approximate equality i e the concept of a pseudometric based resemblance relation we go into the properties of this new kind of fuzzy relation and illustrate it by means of some examples

asymptotic behavior and stability of a stochastic model for aids transmission a stochastic model for aids transmission is presented in this paper the existence and uniqueness of its solution is proved and the solution belongs to an interval satisfying the actual condition furthermore the disease equilibrium can be demonstrated to be existent and both its almost surely exponential stability and its pth moment exponential stability can be concluded based on parameter analysis the first quadrant can be integrated into several parts according to the comparison between almost surely exponential stability and pth moment exponential stability and the potential meaning of the stability is further demonstrated finally based on the above theoretical analysis some conditions for aids to decrease and then to die out are obtained by computer simulation

some results on the degree of symmetry of fuzzy relations abstract in this paper we investigate the degree of the symmetry of fuzzy relations on a set x based on the fuzzy e equality derived from uninorms with unit element e the degree of the symmetry of fuzzy relations are defined by two different approaches the type i degree of symmetry and the type ii degree of symmetry the main work of this paper include first we discuss some basic properties of those two degrees especially for the relationship between them in particular for continuous t norms we give a necessary and sufficient condition such that the type i degree of symmetry and the type ii degree of symmetry are equal and then we obtain that the type ii degree of symmetry is more appropriate to character the degree of the symmetry of fuzzy relations than the type i degree of symmetry with respect to whether they preserve the fuzzy e equalities or not in the meantime for a special case with continuous t norms we provide a necessary and sufficient condition such that the type i degree of symmetry preserves the fuzzy equalities finally for conjunctive left continuous idempotent uninorms with neutral element e 0 1 we find out a symmetric fuzzy relation s which is close enough to the given fuzzy relation r with respect to fuzzy e equality such that the degree of fuzzy e equality of s and r is the type ii degree of symmetry of r in particular we obtain analogous results for continuous t norms and one class of left continuous t norms

triangular norm based predicate fuzzy logics the paper surveys the present state of knowledge on t norm based predicate fuzzy logics with their double semantics standard the set of truth values being the real interval 0 1 and general with abstract algebras of truth functions

the logic of tied implications part 2 syntax an implication operator a is said to be tied if there is a binary operation t that ties a that is the identity a a a b z a t a b z holds for all a b z we aim at the construction of a complete predicate logic for prelinear tied adjointness algebras we realize this in three steps in the first step we establish a propositional calculus adjtpc complete for the class of all tied adjointness algebras on partially ordered sets without prelinearity and ignoring the lattice operations for that we supply a hilbert system based on seven axioms and one deduction rule modus ponens in the second and third steps we extend adjtpc to propositional and predicate calculi complete for prelinear tied adjointness algebras we apply a duality principle due to morsi in all three calculi through which we manage to cut down the number of proofs

graded dominance and related graded properties of fuzzy connectives graded properties of binary and unary fuzzy connectives valued in mtl algebras are studied including graded monotony a generalized lipschitz property commutativity associativity unit and null elements and the dominance relation between fuzzy connectives the apparatus of fuzzy class theory or higher order fuzzy logic is employed as a tool for easy derivation of graded theorems on the connectives

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

transitivity of fuzzy relations under discretization fuzzy transitivity is a key property for many fuzzy relational structures such as fuzzy preorders and equivalences theoretical models for practical problems which are based on fuzzy relations make use of continuous scales mostly the unit interval 0 1 practical implementation of these models though involves their discretization into finite scales which generally results in some loss of transitivity in this paper we study if there are any transitivity preserving discretization strategies also we evaluate the loss of transitivity in some commonly used discretization approaches

fuzzy sets and formal logics the paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics our focus is on the historical side of these developments

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

the long time behavior of di sir epidemic model with stochastic perturbation in this paper we present a di sir epidemic model with two categories stochastic perturbations the long time behavior of the two stochastic systems is studied mainly we show how the solution goes around the infection free equilibrium and the endemic equilibrium of deterministic system under different conditions

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

acyclic rationality indicators of fuzzy choice functions we introduce the degree of acyclicity of a fuzzy preference relation and two acyclic rationality indicators of a fuzzy choice function c these two indicators measure the degree to which the choice function c is acyclic g rational respectively acyclic m rational we study the fuzzy condorcet property of a fuzzy choice function and the corresponding indicators we express the indicator of g rationality in terms of the condorcet property and the indicator of a consistency

fuzzy modifiers based on fuzzy relations in this paper we introduce a new type of fuzzy modifiers i e mappings that transform a fuzzy set into a modified fuzzy set based on fuzzy relations we show how they can be applied for the representation of weakening adverbs more or less roughly and intensifying adverbs very extremely in the inclusive and the non inclusive interpretation we illustrate their use in an approximate reasoning scheme

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

global stability of multigroup epidemic model with group mixing and nonlinear incidence rates in this paper we introduce a basic reproduction number for a multigroup seir model with nonlinear incidence of infection and nonlinear removal functions between compartments then we establish that global dynamics are completely determined by the basic reproduction number r0 it shows that the basic reproduction number r0 is a global threshold parameter in the sense that if it is less than or equal to one the disease free equilibrium is globally stable and the disease dies out whereas if it is larger than one there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population finally two numerical examples are also included to illustrate the effectiveness of the proposed result

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

completely lattice l ordered sets with and without l equality a relationship between l order based on an l equality and l order based on crisp equality is explored in detail this enables to clarify some properties of completely lattice l ordered sets and generalize some related assertions namely belohlavek s main theorem of fuzzy concept lattices is generalized as well as his theorem dealing with dedekind macneille completion analogously completion of an l ordered set via completely lattice l ordered set of all down l sets is described

fuzzy closure operators closure operators and closure systems play a significant role in both pure and applied mathematics in the framework of fuzzy set theory several particular examples of closure operators and systems have been considered e g so called fuzzy subalgebras fuzzy congruences fuzzy topology etc recently fuzzy closure operators and fuzzy closure systems themselves i e operators which map fuzzy sets to fuzzy sets and the corresponding systems of closed fuzzy sets have been studied by gerla et al see e g 3 4 6 7 as a matter of fact a fuzzy set a is usually defined as a mapping from a universe set x into the real interval 0 1 in the above mentioned works therefore the structure of truth values of the logic behind is fixed to 0 1 equipped usually with minimum as the operation corresponding to logical conjunction as it appeared recently in the investigations of fuzzy logic in narrow sense 9 10 i e fuzzy logic as a many valued logical calculus there are several logical calculi formalizing the intuitive idea of fuzzy reasoning which are complete with respect to the semantics over special structures of truth values among these structures the most general one is that of a residuated lattice it is worth noticing that residuated lattices introduced originally in 12 as an abstraction in the study of ideal systems of rings have been proposed as a suitable structure of truth values by goguen in 8 from this point of view the need for a general notion of a fuzzy closure concept becomes apparent the aim of this paper is to outline a general theory of fuzzy closure operators and fuzzy closure systems in the next section we introduce the necessary concepts in section 3 fuzzy closure operators and systems are

global stability of two group sir model with random perturbation abstract in this paper we discuss the two group sir model introduced by guo li and shuai h b guo m y li z shuai global stability of the endemic equilibrium of multigroup sir epidemic models can appl math q 14 2006 259 284 allowing random fluctuation around the endemic equilibrium we prove the endemic equilibrium of the model with random perturbation is stochastic asymptotically stable in the large in addition the stability condition is obtained by the construction of lyapunov function finally numerical simulations are presented to illustrate our mathematical findings

stability analysis of delay differential equation models of hiv 1 therapy for fighting a virus with another virus considering two kinds of delays accounting respectively for i a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and ii a virus production period for new virions to be produced within and released from the infected cells we develop and analyze a mathematical model for hiv 1 therapy by fighting a virus with another virus for the different values of the basic reproduction number and the second basic reproduction number we investigate the stability of the infection free equilibrium the single infection equilibrium and the double infection equilibrium we conclude that increasing delays will decrease the values of the basic reproduction number and the second basic reproduction number our results have potential applications in hiv 1 therapy the approach we use here is a combination of analysis of characteristic equations fluctuation lemma and lyapunov function

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

dynamical behavior of a stochastic sei epidemic model with saturation incidence and logistic growth abstract in this paper we investigate the dynamical behavior of a stochastic sei epidemic model with saturation incidence and logistic growth firstly by constructing a suitable stochastic lyapunov function we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the solutions to the model then we obtain sufficient conditions for extinction of the disease in two cases that is the first case is that the susceptible population survival and infected and exposed populations extinction the second case is that all the populations extinction

reasoning about mathematical fuzzy logic and its future this paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author our goal is to think of the state of the art in mathematical fuzzy logic mfl and to outline some of the tasks on which in the author s opinion mfl should focus in the future in our discussion we will mention not only the basic theory but also its extension called fuzzy logic in broader sense flb the paradigm of the latter is to be the logic of natural human reasoning whose most essential characteristic is the use of natural language besides brief description of flb we will also mention some of its applications on the basis of that we will ponder on other possible directions for research namely the possibility of using fln as a metatheory of fuzzy mathematics as a proper tool for modeling of the main manifestations of the phenomenon of vagueness and as a reasonable tool for developing models of linguistic semantics

a theory of vague lattices based on many valued equivalence relations i general representation results the present work introduces a new and general theory of ordering relations and lattices based on many valued equivalence relations under the name vague ordering relations and vague lattices respectively representations and constructions of vague ordering relations and vague lattices are the main subjects of this paper and various desirable results in this direction are established furthermore algebraic characterizations of vague lattices are studied

stability analysis of a time delayed sir epidemic model with nonlinear incidence rate stability of sir models has been studied extensively within the framework of disease epidemiology we formulate a nonlinear mathematical model to study the role of nonlinear incidence rates and the effect of time delay in a nonlinear logistically growing time delayed sir model with variable population size the existence and stability of the possible equilibria are examined in terms of a certain threshold condition 0 the basic reproduction number for mathematical tractability the stability of the endemic equilibrium is shown using lyapunov functional approach with linear incidence rate numerical simulations are carried out to investigate the influence of the key parameters on the spread of the disease to support the analytical conclusion and illustrate possible behavioral scenarios of the model