a very hard log space counting class abstract we consider the logarithmic space counting and optimization classes l span l and opt l which are defined analogously to their polynomial time counterparts we obtain complete functions for these three classes in terms of graphs and finite automata we show that l and opt l are both included in nc 2 but that surprisingly span l seems to be a much harder class than l and opt l we demonstrate that span l functions can be computed in polynomial time if and only if p p and all the classes of the polynomial time hierarchy are included in p this result follows from the fact that span l and p are very similar span l p and any function in p can be represented as the difference of a function in fl and a function in span l nevertheless the inclusion p span l would imply nl p np we furthermore investigate restrictions of the classes opt l and span l

on the existence and nonexistence of positive solutions for nonlinear sturm liouville boundary value problems abstract in this paper the existence and nonexistence results of positive solutions are obtained for sturm liouville boundary value problem p x u q x u f x u x 0 1 a u 0 b p 0 u 0 0 c u 1 d p 1 u 1 0 where p c 1 0 1 q c 0 1 p x 0 q x 0 for x 0 1 f c 0 1 r a b c d 0 are constants and satisfy a b c d 0 the discussion is based on the positivity estimation for the green s function of associated linear boundary value problem and the fixed point index theory in cones

the balanced linear programming problem abstract the balanced linear programming problem blpp arises in situations which require equitable distribution of a scarce resource the blpp can be transformed to the standard form of the linear programming problem by introducing 2 n 2 additional variables and 2 n additional constraints this transformation is not desirable from the computational point of view for larger values of n as it increases the problem size substantially it is also undesirable from a theoretical perspective as it might affect the special structure of the constraint matrix in this paper we develop an algorithm for the blpp which does not require problem enlargement the algorithm is based on the relationship between the blpp and the minimax linear programming problem and solving the latter problem parametrically our algorithm in essence performs steps that are similar to those performed in the parametric simplex method with parametric right hand side we then adapt our algorithm for the network flow problem and this specialized algorithm can be applied on the network directly without maintaining the simplex tableau

positive periodic solutions for nonlinear difference equations with periodic coefficients abstract we prove the existence of positive solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions obtained results yield positive periodic solutions of the equation on the whole discrete axis provided that the coefficients are periodic

the complexity of bayesian networks specified by propositional and relational languages abstract we examine the complexity of inference in bayesian networks specified by logical languages we consider representations that range from fragments of propositional logic to function free first order logic with equality in doing so we cover a variety of plate models and of probabilistic relational models we study the complexity of inferences when network query and domain are the input the inferential and the combined complexity when the network is fixed and query and domain are the input the query data complexity and when the network and query are fixed and the domain is the input the domain complexity we draw connections with probabilistic databases and liftability results and obtain complexity classes that range from polynomial to exponential levels we identify new languages with tractable inference and we relate our results to languages based on plates and probabilistic relational models

on equitable resource allocation problems a lexicographic minimax approach in this expository paper we review a variety of resource allocation problems in which it is desirable to allocate limited resources equitably among competing activities applications for such problems are found in diverse areas including distribution planning production planning and scheduling and emergency services location each activity is associated with a performance function representing for example the weighted shortfall of the selected activity level from a specified target a resource allocation solution is called equitable if no performance function value can be improved without either violating a constraint or degrading an already equal or worse off i e larger performance function value that is associated with a different activity a lexicographic minimax solution determines this equitable solution that is it determines the lexicographically smallest vector whose elements the performance function values are sorted in nonincreasing order the problems reviewed include large scale allocation problems with multiple knapsack resource constraints multiperiod allocation problems for storable resources and problems with substitutable resources the solution of large scale problems necessitates the design of efficient algorithms that take advantage of special mathematical structures indeed efficient algorithms for many models will be described we expect that this paper will help practitioners to formulate and solve diverse resource allocation problems and motivate researchers to explore new models and algorithmic approaches

solving knapsack sharing problems with general tradeoff functions a knapsack sharing problem is a maximin or minimax mathematical programming problem with one or more knapsack constraints an inequality constraint with all non negative coefficients all knapsack sharing algorithms to date have assumed that the objective function is composed of single variable functions called tradeoff functions which are strictly increasing continuous functions this paper develops optimality conditions and algorithms for knapsack sharing problems with any type of continuous tradeoff function including multiple valued and staircase functions which can be increasing decreasing unimodal bimodal or even multi modal to do this optimality conditions are developed for a special type of multiple valued nondecreasing tradeoff function called an ascending function the optimal solution to any knapsack sharing problem can then be found by solving an equivalent problem where all the tradeoff functions have been transformed to ascending functions polynomial algorithms are developed for piecewise linear tradeoff functions with a fixed number of breakpoints the techniques needed to construct efficient algorithms for any type of tradeoff function are discussed

multiplicity of solutions to two point boundary value problems for second order impulsive differential equations abstract in this paper we obtain some sufficient condition for the existence of at least three solutions to two point boundary value problems for second order impulsive differential equations by using leray schauder degree theory we assume the existence of two pairs of strict lower and upper solutions and the nonlinear term f satisfies a nagumo s growth condition with respect to the first order derivative

functions computable in polynomial space consider nondeterministic polynomial time turing machine that on input x outputs a 3 3 matrix with entries from 1 0 1 on each of its paths define the function f where f x is the upper left entry in the product of all these matrices in an order of the paths to be made precise below we show that the class of functions f computable as just described is exactly the class fpspace of integer valued functions computable by polynomial space turing machines along the way we obtain characterizations of fpspace in terms of arithmetic circuits and straight line programs

multiple positive solutions of some boundary value problems abstract we study the existence of multiple positive solutions of the equations u t u subject to linear boundary conditions we show that there are at least two positive solutions if t u is superlinear at one end point zero or infinity and sublinear at the other applications of these results are provided to yield multiple positive solutions of some elliptic boundary value problems on an annulus

existence results for differential delay equations i abstract existence principles for systems of differential difference equations of delay type are established also applications are given to first order delay equations throughout our topological arguments are based on a nonlinear alternative

polynomial space counting problems the classes of functions textit pspace and natural textit pspace that are analogous to the class p are defined functions in textit pspace count the number of accepting computations of a nondeterministic polynomial space bounded turing machine and functions in natural textit pspace count the number of accepting computations of nondeterministic polynomial space bounded turing machines that on each computation path make at most a polynomial number of nondeterministic choices in contrast to what is known about p exact characterizations of both textit pspace and natural textit pspace are found in particular pspace fspace the class of functions computable in polynomial space and natural pspace fspace poly the class of functions computable in polynomial space with output length bounded by a polynomial both textit pspace and natural textit pspace can be characterized by counting problems related to alternating turing machines both te

positive solutions of second order two delay differential systems with twin parameter abstract the paper is concerned with the existence of positive solutions for a second order two delay differential systems with twin parameter firstly we present some sufficient conditions with the twin parameter belonging to an open rectangular region eigenvalue region to ensure the existence of positive solutions for the differential systems with singularity secondly we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems the proofs are based on a fixed point theorem in cones

proportional equity flow problem for terminal arcs the proportional equity flow problem extends a class of problems referred to as equity flow problems whose objective is to equitably distribute flow among the arcs in a flow circulation network the proportionally bounded flow circulation problem places lower and upper bounds on each arc flow that are nondecreasing continuous functions of the flow through one special arc and the objective is to maximize the flow through the special arc the proportional equity flow problem for terminal arcs problem ta is then defined as a special case where all the proportional arcs enter a sink vertex applications of both the general problem and problem ta are given two optimality conditions for problem ta are developed as well as an algorithm that is polynomially bounded for many types of nondecreasing continuous proportional bounding functions specifically the algorithm is shown to be polynomially bounded if the largest root of an equation can be found in polynomial time

existence of positive solutions for functional differential equations under suitable conditions on f t yt the boundary value problem of second order functional differential equation fde with the form fde y t f t yt 0 for t 0 1 a bc y t y t t for t 0 bvp y t y e t for t 1 1 a has at least one positive solution moreover we also apply this main result to establish several existence theorems which guarantee bvp has the multiple positive solutions

positive solutions of nonlinear singular two point boundary value problems for second order impulsive differential equations in this paper we study the positive solutions of nonlinear singular two point boundary value problems for second order impulsive differential equations the existence of one or two positive solutions are established by using the fixed point index theorem in cone

existence of multiple positive solutions for functional differential equations abstract in this paper the existence of at least three positive solutions for the boundary value problem bvp of second order functional differential equation with the form y t f t y t 0 for t 0 1 y t y t t for t 0 y t y t t for t 1 1 a is studied moreover we investigate the existence of at least three partially symmetric positive solutions for the above bvp with

existence and iteration of positive solutions for some p laplacian boundary value problems abstract this paper deals with the existence and iteration of positive solutions for the following p laplacian differential equation p u t q t f t u t u t 0 t 0 1 subject to one of the following boundary conditions p u 0 p u 0 0 p u 1 p u 1 0 or u 0 g 1 u 0 0 u 1 g 2 u 1 0 the main tool is the monotone iterative technique we obtain not only the existence of positive solutions for the problems but also establish iterative schemes for approximating the solutions

complexity of finite horizon markov decision process problems controlled stochastic systems occur in science engineering manufacturing social sciences and many other cntexts if the systems is modeled as a markov decision process mdp and will run ad infinitum the optimal control policy can be computed in polynomial time using linear programming the problems considered here assume that the time that the process will run is finite and based on the size of the input there are mny factors that compound the complexity of computing the optimal policy for instance there are many factors that compound the complexity of this computation for instance if the controller does not have complete information about the state of the system or if the system is represented in some very succint manner the optimal policy is provably not computable in time polynomial in the size of the input we analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a mdp based on a number of confounding factors including the observability of the system state the succinctness of the representation the type of policy even the number of actions relative to the number of states in almost every case we show that the decision problem is complete for some known complexity class some of these results are familiar from work by papadimitriou and tsitsiklis and others but some such as our pl completeness proofs are surprising we include proofs of completeness for natural problems in the as yet little studied classes np pp

ancestors descendants and gardens of eden in reaction systems this paper analyses several problems related to finding and counting ancestor and descendant states as well as gardens of eden i e states without predecessors in reaction systems the focus is on the complexity of finding and counting preimages and ancestors that are minimal with respect to cardinality it turns out that the problems concerning gardens of eden seem to require the presence of an np oracle to be solved all the problems studied are intractable with a complexity that ranges form fp np log n to fpspace poly

the bipartite rationing problem in the bipartite rationing problem a set of agents share a single resource available in different types each agent has a claim over only a subset of the resource types and these claims overlap in arbitrary fashion the goal is to divide fairly the various types of resources between the claimants when resources are in short supply with a single type of resource this is the standard rationing problem o neill b 1982 a problem of rights arbitration from the talmud math soc sci 2 4 345 371 of which the three benchmark solutions are the proportional uniform gains and uniform losses methods we extend these methods to the bipartite context imposing the familiar consistency requirement the division is unchanged if we remove an agent respectively a resource and take away at the same time his share of the various resources respectively reduce the claims of the relevant agents the uniform gains and uniform losses methods have infinitely many consistent extensions but the proportional me

the complexity of computing the size of an interval given a p order a over a universe of strings i e a transitive reflexive antisymmetric relation such that if x y in a then x is polynomially bounded by y an interval size function of a returns for each string x in the universe the number of strings in the interval between strings b x and t x with respect to a where b x and t x are functions that are polynomial time computable in the length of x by choosing sets of interval size functions based on feasibility requirements for their underlying p orders we obtain new characterizations of complexity classes we prove that the set of all interval size functions whose underlying p orders are polynomial time decidable is exactly mp we show that the interval size functions for orders with polynomial time adjacency checks are closely related to the class fpspace poly indeed fpspace poly is exactly the class of all nonnegative functions that are an interval size function minus a polynomial time computable function we study two important functions in relation to interval size functions the function mdiv maps each natural number n to the number of nontrivial divisors of n we show that mdiv is an interval size function of a polynomial time decidable partial p order with polynomial time adjacency checks the function mmonsat maps each monotone boolean formula f to the number of satisfying assignments of f we show that mmonsat is an interval size function of a polynomial time decidable total p order with polynomial time adjacency checks finally we explore the related notion of cluster computation

multiple positive solutions of nonlinear two point boundary value problems we improve the results obtained by erbe hu and wang in a recent paper we show that there exist at least two positive solutions of two point boundary value problems under conditions weaker than those used by erbe hu and wang

numerical method for a singularly perturbed convection diffusion problem with delay this paper deals with the singularly perturbed boundary value problem for a linear second order delay differential equation for the numerical solution of this problem we use an exponentially fitted difference scheme on a uniform mesh which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form it is shown that one gets first order convergence in the discrete maximum norm independently of the perturbation parameter numerical results are presented which illustrate the theoretical results

the complexity of optimal small policies we investigate the complexity of problems concerned with partially observable markov decision processes which are run for a finite number of steps under small policies the calculation of the expected sum of rewards of a process under a small policy is shown to be complete for the complexity class pp a class which lies intermediate between np and pspace optimal small policy computation is shown to be complete for nppp the latter contrasts results of papadimitriou and tsitsiklis papadimitriou and tsitsiklis 1987 who showed that this problem is pspace complete if no assumptions about the representability of the policy are made and that it is p complete for fully observable processes

the flow circulation sharing problem the flow circulation sharing problem is defined as a network flow circulation problem with a maximin objective function the arcs in the network are partitioned into regular arcs and tradeoff arcs where each tradeoff arc has a non decreasing tradeoff function associated with it all arcs have lower and upper bounds on their flow while the value of the smallest tradeoff function is maximized the model is useful in equitable resource allocation problems over time which is illustrated in a coal strike example and a submarine assignment example some properties including optimality conditions are developed each cut in the network defines a knapsack sharing problem which leads to an optimality condition similar to the max flow min cut theorem an efficient algorithm for both the continuous and integer versions of the flow circulation sharing problem is developed and computational experience given in addition efficient algorithms are developed for problems where some of the arcs have infinite flow upper bounds

the linear sharing problem the linear sharing problem is defined as a linear programming problem with a maximin objective function comprised of nondecreasing continuous tradeoff functions this paper develops some properties including optimality conditions for this problem it also develops an efficient algorithm that alternately calculates a new global upper bound on the objective function and then determines if a feasible solution exists that meets the new objective function global upper bound the process continues until the algorithm finds a feasible and thus an optimal solution the algorithm also determines if the problem contains no feasible solution or if the objective function is unbounded a technique to handle problems with unbounded tradeoff variables is developed and computational experience is given

eigenvalue criteria for existence of multiple positive solutions to boundary value problems of second order delay differential equations abstract criteria are established for existence of multiple positive solutions to the boundary value problems expressed by second order delay differential equations of the form p t y t q t y t f t y t y t a t 0 1 y t y t t t 0 y t y t t t 1 1 a where 0 a 1 4 are suitably small p c 1 0 1 p 0 q c 0 1 and q 0