using dual approximation algorithms for scheduling problems theoretical and practical results the problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well studied problem in the theory of approximation algorithms for np hard optimization problems in this paper the strongest possible type of result for this problem a polynomial approximation scheme is presented more precisely for each e an algorithm that runs in time o n e 1 e 2 and has relative error at most e is given in addition more practical algorithms for e 1 5 2 k and e 1 6 2 k which have running times o n k log n and o n km 4 log n are presented the techniques of analysis used in proving these results are extremely simple especially in comparison with the baroque weighting techniques used previously the scheme is based on a new approach to constructing approximation algorithms which is called dual approximation algorithms where the aim is to find superoptimal but infeasible solutions and the performance is measured by the degree of infeasibility allowed this notion should find wide applicability in its own right and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

global dynamics of a reaction diffusion virus infection model with humoral immunity and nonlinear incidence abstract in this paper we propose and investigate a reaction diffusion virus infection model with humoral immunity and nonlinear incidence in spatially heterogeneous case the basic reproduction number of virus infection r 0 is calculated when r 0 1 the global asymptotical stability of the infection free steady state is established and when r 0 1 the uniform persistence of infected cells and viruses as well as the existence of antibody free infection steady state are also obtained in spatially homogeneous case the antibody response basic reproduction number r 1 is calculated by using the lyapunov functions method and the persistence theory of dynamical systems we obtain that when r 0 1 and r 1 1 the antibody free infection equilibrium is globally asymptotically stable and when r 0 1 and r 1 1 the model is uniformly persistent and the infection equilibrium exists and is also globally asymptotically stable finally the numerical examples are presented in order to verify the validity of our theoretical results

on a categorical analysis of zadeh generalized subsets of sets ii how can fuzzy sets be viewed as generalized subsets of actual sets an answer to this question is given via a categorical analysis of the synchronic approach of fuzzy set theory starting from an abstract definition of a category with fuzzy subsets we establish necessary and sufficient conditions on a posst to be that of truth values for such a category these conditions justify zadeh s original choice of the unit segment as a set of truth values some defects concerning universal constructions in a non trivial category with fuzzy subsets are mentioned and a natural best toposophical approximation of such a category is proposed

fast approximation algorithms for uniform machine scheduling with processing set restrictions abstract we consider the problem of nonpreemptively scheduling a set of independent jobs on a set of uniform machines where each job has a set of machines to which it can be assigned this kind of restriction is called the processing set restriction in the literature there are many kinds of processing set restrictions that have been studied in this paper we consider two kinds the inclusive processing set and the tree hierarchical processing set epstein and levin 2011 have given polynomial time approximation schemes ptas to solve both classes however the running times of their ptas are rather high in this paper we give fast approximation algorithms for both cases and show that they both have a worst case performance bound of 4 3 moreover we show that the bounds are achievable

the minimal mathematical structure for a synchronic approach to fuzzy set theory the paper gives a concrete presentation of the minimal structure on the monoid of an external subset classifier of a universe of usual sets with zadeh goguen l fuzzy subsets the result was first investigated by the first author with the aim of providing a new insight into the synchronic approach to fuzzy set theory the extensive use of conceptual mathematics i e category theory in the original paper restricts the audience of readers of this analysis to persons who are already acquainted with conceptual mathematics the aim of this paper is to avoid as much as possible the use of functorial terminology in the presentation of the minimal mathematical structure on the set of truth values for a synchronic approach to fuzzy sets by doing so we hope to widen the audience of scientists interested in the foundation of zadeh goguen l fuzzy sets and to provide a new way of looking at the synchronic approach to fuzziness

fuzzy sets do not form a topos abstract the relationship between fuzzy sets and h valued sets is briefly examined and a claim of eytan s that the former comprise the objects of a topos is corrected

tight bounds for online class constrained packing we consider class constrained packing problems in which we are given a set of bins each having a capacity v and c compartments and n items of m different classes and the same unit size we need to fill the bins with items subject to capacity constraints such that items of different classes are placed in separate compartments thus each bin can contain items of at most c distinct classes we consider two optimization goals in the class constrained bin packing problem ccbp our goal is to pack all the items in a minimal number of bins in the class constrained multiple knapsack problem ccmk we wish to maximize the total number of items packed in m bins for m 1 the ccbp and ccmk problems model fundamental resource allocation problems in computer and manufacturing systems both are known to be strongly np hard in this paper we derive tight bounds for the online variants of these problems we first present a lower bound of 1 on the competitive ratio of any deterministic algorithm for the online ccbp where 0 1 depends on v c m and n we show that this ratio is achieved by the algorithm first fit we then consider the temporary ccbp in which items may be packed for a bounded time interval that is unknown in advance we obtain a lower bound of v c on the competitive ratio of any deterministic or randomized algorithm we show that this ratio is achieved by all any fit algorithms finally tight bounds are derived for the online ccmk and the temporary ccmk problems

optimal semi online preemptive algorithms for machine covering on two uniform machines in this paper we consider the semi online preemptive scheduling problem with decreasing job sizes on two uniform machines our goal is to maximize the continuous period of time starting from time zero when both machines are busy which is equivalent to maximizing the minimum machine completion time if idle time is not introduced before all the jobs are completed we design optimal deterministic semi online algorithms for every machine speed ratio s 1 and show that idle time is required during the assignment procedure of algorithms for any s 6 2 the competitive ratios of the algorithms match the randomized lower bound for every 1 s 3 the problem of whether randomization still does not help for the discussed preemptive scheduling problem remains open

two semi online scheduling problems on two uniform machines this paper considers two semi online scheduling problems one with known optimal value and the other with known total sum on two uniform machines with a machine speed ratio of s 1 for the first problem we provide an optimal algorithm for s 1 3 2 1 21 4 and improved algorithms or and lower bounds for s 1 21 4 3 over which the optimal algorithm is unknown as a result the largest gap between the competitive ratio and the lower bound decreases to 0 02192 for the second problem we also present algorithms and lower bounds for s 1 the largest gap between the competitive ratio and the lower bound is 0 01762 and the length of the interval over which the optimal algorithm is unknown is 0 47382 our algorithms and lower bounds for these two problems provide insights into their differences which are unusual from the viewpoint of the known results on these two semi online scheduling problems in the literature

approximation algorithms for knapsack problems with cardinality constraints abstract we address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected this problem arises in the solution of real life cutting stock problems by column generation and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs we focus our attention on approximation algorithms for the problem describing a linear storage polynomial time approximation scheme ptas and a dynamic programming based fully polynomial time approximation scheme fptas the main ideas contained in our ptas are used to derive ptas s for the knapsack problem and its multi dimensional generalization which improve on the previously proposed ptas s we finally illustrate better ptas s and fptas s for the subset sum case of the problem in which profits and weights coincide

scheduling parallel machines with inclusive processing set restrictions and job release times we consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule the machines have the same processing speed but each job is compatible with only a subset of those machines the machines can be linearly ordered such that a higher indexed machine can process all those jobs that a lower indexed machine can process we present an efficient algorithm for this problem with a worst case performance ratio of 2 we also develop a polynomial time approximation scheme ptas for the problem as well as a fully polynomial time approximation scheme fptas for the case in which the number of machines is fixed

periodic trajectories of distributed parameter biochemical systems with time delay abstract this paper deals with a model of a biochemical reactor system with distributed parameters and with a time delay in the growth response time delay has been introduced in microbial growth systems to explain the time lapse between the consumption of liquid substrate and its conversion to solid biomass we study here the properties of the resulting system of partial functional differential equations we first prove the existence positivity and a compactness property of the system trajectories we then prove the existence of periodic solutions of the system for large values of the delay numerical simulations illustrate the existence of such solutions

fast approximation algorithms for job scheduling with processing set restrictions we consider the problem of scheduling n independent jobs on m parallel machines where the machines differ in their functionality but not in their processing speeds each job has a restricted set of machines to which it can be assigned called its processing set preemption is not allowed our goal is to minimize the makespan of the schedule we study two variants of this problem 1 the case of tree hierarchical processing set and 2 the case of nested processing set we first give a fast algorithm for the case of tree hierarchical processing set with a worst case bound of 4 3 which is better than the best known algorithm whose worst case bound is 2 we then give a more complicated algorithm for the case of nested processing set with a worst case bound of 5 3 which is better than the best known algorithm whose worst case bound is 7 4 in both cases we will give examples achieving the worst case bounds

existence and asymptotic behavior of a functional differential equation in banach space abstract existence and asymptotic behavior of solutions are given for the equation u t a t u t f t u t t 0 and u 0 c r 0 x c the space x is a banach space the family a t 0 t t of unbounded linear operators defined on d a x x generates a linear evolution system and f c x is continuous with respect to a fractional power of a t 0 for some t 0 0 t

analysis of a diffusive virus infection model with humoral immunity cell to cell transmission and nonlinear incidence abstract in this paper we investigate a reaction diffusion virus infection model with humoral immunity and nonlinear incidence in heterogeneous and homogenous environments the model includes simultaneously the virus to cell and cell to cell transmissions of the infection the well posedness of solutions including the existence of global solutions and the ultimate boundedness is established the virus infection reproduction number r 0 is calculated when r 0 1 it is proved that the infection free steady state is globally asymptotically stable otherwise when r 0 1 then the infection with antibody free response is uniformly persistent furthermore the antibody free infection equilibrium is also globally asymptotically stable in spatially homogeneous case in order to investigate the effect of humoral immunity the virus infection with antibody response model in the spatially homogeneous environment is considered the antibody response reproduction number r 1 is calculated the threshold criteria on the uniform persistence of total model and the global asymptotic stability of the antibody response infection equilibrium are established finally the numerical examples are presented to illustrate the theoretical results and verify the conjectures

online bin stretching with bunch techniques we are given a sequence of items that can be packed into m unit size bins and the goal is to assign these items online to m bins while minimizing the stretching factor bins have infinite capacities and the stretching factor is the size of the largest bin we present an algorithm with stretching factor 26 17 1 5294 improving the best known algorithm by kellerer and kotov 2013 1 with a stretching factor 11 7 1 5714 our algorithm has 2 stages and uses bunch techniques we aggregate bins into batches sharing a common purpose

on line load balancing the setup for our problem consists of n servers that must complete a set of tasks each task can be handled only by a subset of the servers requires a different level of service and once assigned cannot be reassigned we make the natural assumption that the level of service is known at arrival time but that the duration of service is not the on line load balancing problem is to assign each task to an appropriate server in such a way that the maximum load on the servers is minimized in this paper we derive matching upper and lower bounds for the competitive ratio of the on line greedy algorithm for this problem namely 3n 23 2 1 o 1 and derive a lower bound n12 for any other deterministic or randomized on line algorithm

on line bin stretching we are given a sequence of items that can be packed into m unit size bins in the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins in contrast in the bin stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible we present two on line algorithms for the bin stretching problem that guarantee a stretching factor of 5 3 for any number m of bins we then combine the two algorithms and design an algorithm whose stretching factor is 1 625 for any m the analysis for the performance of this algorithm is tight the best lower bound for any algorithm is 4 3 for any m 2 we note that the bin stretching problem is also equivalent to the classical scheduling load balancing problem in which the value of the makespan maximum load is known in advance copyright 2001 elsevier science b v

the accommodating function a generalization of the competitive ratio a new measure the accommodating function for the quality of on line algorithms is presented the accommodating function which is a generalization of both the competitive ratio and the competitive ratio on accommodating sequences measures the quality of an on line algorithm as a function of the resources that would be sufficient for an optimal off line algorithm to fully grant all requests more precisely if we have some amount of resources n the function value at alpha is the usual ratio still on some fixed amount of resources n except that input sequences are restricted to those where the optimal off line algorithm will not obtain a better result by having more than the amount alpha n of resources the accommodating functions for three specific on line problems are investigated a variant of bin packing in which the goal is to maximize the number of items put in n bins the seat reservation problem and the problem of optimizing total flow time when preemption is allowed we also show that when trying to distinguish between two algorithms the decision as to which one performs better cannot necessarily be made from the competitive ratio or the competitive ratio on accommodating sequences alone for the variant of bin packing considered we show that worst fit has a strictly better competitive ratio than first fit while first fit has a strictly better competitive ratio on accommodating sequences than worst fit

bounds for online bin packing with cardinality constraints abstract we study a bin packing problem in which a bin can contain at most k items of total size at most 1 where k 2 is a given parameter items are presented one by one in an online fashion we analyze the best absolute competitive ratio of the problem and prove tight bounds of 2 for any k 4 additionally we present bounds for relatively small values of k with respect to the asymptotic competitive ratio and the absolute competitive ratio in particular we provide tight bounds on the absolute competitive ratio of first fit for k 2 3 4 and improve the known lower bounds on asymptotic competitive ratios for multiple values of k our method for obtaining a lower bound on the asymptotic competitive ratio using a certain type of an input is general and we also use it to obtain an alternative proof of the known lower bound on the asymptotic competitive ratio of standard online bin packing

online hierarchical scheduling an approach using mathematical programming this paper considers an online hierarchical scheduling problem on parallel identical machines we are given a set of m machines and a sequence of jobs each machine has a different hierarchy and each job also has a hierarchy associated with it a job can be assigned to a machine only if its hierarchy is no less than that of the machine the objective is to minimize the makespan i e the maximum load of all machines two models are studied in the paper for the fractional model we present an improved algorithm and lower bounds both the algorithm and the lower bound are based on solutions of mathematical programming for any given m our algorithm is optimal by numerical calculation for the integral model we present both a general algorithm for any m and an improved algorithm with better competitive ratios of 2 333 and 2 610 for m 4 and 5 respectively

the constrained compartmentalized knapsack problem mathematical models and solution methods the constrained compartmentalized knapsack problem can be seen as an extension of the constrained knapsack problem however the items are grouped into different classes so that the overall knapsack has to be divided into compartments and each compartment is loaded with items from the same class moreover building a compartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack and the compartments are lower and upper bounded the objective is to maximize the total value of the items loaded in the overall knapsack minus the cost of the compartments this problem has been formulated as an integer non linear program and in this paper we reformulate the non linear model as an integer linear master problem with a large number of variables some heuristics based on the solution of the restricted master problem are investigated a new and more compact integer linear model is also presented which can be solved by a branch and bound commercial solver that found most of the optimal solutions for the constrained compartmentalized knapsack problem on the other hand heuristics provide good solutions with low computational effort

a characterization of quasi toposes a locally presentable category 6 is the category of models of some theory f which can be described using a limits for a regular cardinal a our aim is to find conditions which ensure good properties of the category d of models when is 6 cartesian closed when is d a quasi topos when is d a topos and so on in this paper we focus our attention on some semantic conditions a locally a presentable category 8 can thus be presented as the category of cc left exact presheaves on a small category 9 with a colimits clearly each subpresheaf s of an e left exact presheaf f has an a left exact closure sc f this closure operation plays a key role in the theory of a left exact functors and we study first conditions on dz lex which ensure the universality of that closure operation we prove this universality to be equivalent to the universality of strongly epimorphic families in but also to the fact that a lex is an epireflective subcategory of a grothendieck topos this grothendieck topos will play an important role in the rest of the paper its construction depends heavily on the fact that under our assumptions a left exact subpresheaves are stable under arbitrary intersections from the point of view of the internal logic of the topos 4 of presheaves among the epireflective subcategories of a grothendieck topos we find the categories of separated objects for a given topology in this topos these

class constrained bin packing revisited we study the following variant of the bin packing problem we are given a set of items where each item has a non negative size and a color we are also given an integer parameter k and the goal is to partition the items into a minimum number of subsets such that for each subset s in the solution the total size of the items in s is at most 1 as in the classical bin packing problem and the total number of colors of the items in s is at most k which distinguishes our problem from the classical version we follow earlier work on this problem and study the problem in both offline and online scenarios

on line load balancing in a hierarchical server topology in a hierarchical server environment jobs are to be assigned in an on line fashion to a collection of servers which form a hierarchy of capability each job requests a specific server meeting its needs but the system is free to assign it either to that server or to any other server higher in the hierarchy each job carries a certain load which it imparts to the server it is assigned to the goal is to find a competitive assignment in which the maximum total load on a server is minimized we consider the linear hierarchy in which the servers are totally ordered in terms of their capabilities we investigate several variants of the problem in the unweighted as opposed to weighted problem all jobs have unit weight in the fractional as opposed to integral model a job may be assigned to several servers each receiving some fraction of its weight finally temporary as opposed to permanent jobs may depart after being active for some finite duration of time we show an optimal e competitive algorithm for the unweighted integral permanent model the same algorithm is e 1 competitive in the weighted case its fractional version is e competitive even if temporary jobs are allowed for the integral model with temporary jobs we show an algorithm which is 4 competitive in the unweighted case and 5 competitive in the weighted case we show a lower bound of e for the unweighted case both integral and fractional this bound is valid even with respect to randomized algorithms we also show a lower bound of 3 for the unweighted integral model when temporary jobs are allowed we generalize the problem and consider hierarchies in which the servers form a tree in the tree hierarchy any job assignable to a node is also assignable to the node s ancestors we show a deterministic algorithm which is 4 competitive in the unweighted case and 5 competitive in the weighted case where only permanent jobs are allowed randomizing this algorithm improves its competitiveness to e and e 1 respectively we also show an omega sqrt n lower bound when temporary jobs are allowed

global attractivity of the diffusive nicholson blowflies equation with neumann boundary condition a non monotone case abstract in this paper we establish the global attractivity of the positive steady state of the diffusive nicholson s equation with homogeneous neumann boundary value under a condition that makes the equation a non monotone dynamical system to achieve this we develop a novel method combining a dynamical systems argument with maximum principle and some subtle inequalities

a polynomial approximation scheme for scheduling on uniform processors using the dual approximation approach in this paper we present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors more specifically the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible we give a family of polynomial time algorithms a such that a delivers a solution that is within a relative error of e of the optimum the technique employed is the dual approximation approach where infeasible but superoptimal solutions for a related dual problem are converted to the desired feasible but possibly suboptimal solution

representability of a fuzzy predicate from a category with fuzzy subsets abstract the failure of existence of products in a non trivial category with fuzzy subsets raises the problem of how to talk about predicates and their representability the concept of predicate is set off in an arbitrary category in order to examine the situation in a category with fuzzy subsets the representability of the general concept of predicate is problematic two restrictions of this are natural extensions of the ordinary concept the first which we term fuzzy predicates shows the multivariant nature of fuzziness the second a restriction is exactly the concept of predicate introduced by zadeh furthermore ordinary products are representative of zadeh predicates

the class constrained bin packing problem with applications to video on demand in this paper we present approximation results for the class constrained bin packing problem that has applications to video on demand systems in this problem we are given bins of size b with c compartments and n items of q different classes each item i 1 n with class c i and size s i the problem is to pack the items into bins where each bin contains at most c different classes and has total items size at most b we present several approximation algorithms for offline and online versions of the problem

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

dynamics of a periodically pulsed bio reactor model abstract global attractivity and uniform persistence are established for both single species growth and two species competition in a periodically pulsed bio reactor model in terms of principal eigenvalues of the periodic parabolic eigenvalue problem by appealing to the theories of monotone discrete dynamical systems abstract persistence asymptotically periodic semiflows and perturbation of global attractors

topoi and categories of fuzzy sets abstract let h be a completely distributive lattice and hence a heyting algebra goguen s category of fuzzy sets set h eytan s logos fuz h and the topos of sheaves on h sh h are interconnected by pairs of adjoint functors between them each of the categories has a predicate calculus these predicate calculi are related through the functors between the categories change of base lattice gives rise to several functors which preserve or reflect specific kinds of statements in the predicate calculi this paper gives details of the categories the predicate calculi attached to them the functors between the categories and the preservation properties of those functors