dirichlet problem for demi coercive functionals on considere des fonctionnelles integrales du type f u f du ou f r n 0 est une fonction convexe ayant la propriete suivante il existe a 0 b 0 r n tel que f z a z b z r n

compatibility conditions for nonlinear neumann problems abstract results we recall an abstract definition introduced in the general framework of the minimization of non coercive functionals see bbgt definition 2 1 let v a be a topological vector space and let f v co co be any mapping we call a sequential recession functional the mapping f u inf lim inf i f tnun t al u p u n m t i where t and u are sequences notice that when f is convex and a lower semicontinuous f coincides with the usual recession functional see for instance r f tu u f v lim t r r to2 where u is any point in dom f in the following we set ker f u e i f u o remark that in general ker f is not a subspace but is a possibly nonconvex set of half lines starting from the origin the following necessary condition for existence was proved in bbgt proposition 2 2 assume that the problem min f u ue v 2 1

a discretization theory for a class of semi coercive unilateral problems in this paper we present a convergence analysis applicable to the approximation of a large class of semi coercive variational inequalities the approach we propose is based on a recession analysis of some regularized galerkin schema finite element approximations of semi coercive unilateral problems in mechanics are discussed in particular a signorini fichera unilateral contact model and some obstacle problem with frictions are studied the theoretical conditions proved are in good agreement with the numerical ones

nonoblivious hashing nonoblivious hashing where information gathered from unsuccessful probes is used to modify subsequent probe strategy is introduced and used to obtain the following results for static lookup on full tables 1 an o 1 time worst case scheme that uses only logarithmic additional memory and no memory when the domain size is linear in the table size which improves upon previously linear space requirements 2 an almost sure o 1 time probabilistic worst case scheme which uses no additional memory and which improves upon previously logarithmic time requirements 3 enhancements to hashing 1 and 2 are solved for multikey recors where search can be performed under any key in time o 1 these schemes also permit properties such as nearest neighbor and rank to be determined in logarithmic time

probabilistic foundations for measurement modelling with fuzzy random variables we propose fuzzy random variables as a tool for modelling measurements when both aleatory and fuzzy uncertainty have to be taken into account uncertainty propagation follows the ordinary scheme for the random part and uses a t normed extension principle for the fuzzy part we concentrate on the probabilistic theoretical underpinnings of the model particularly limit theorems and discuss their implications to the model

applications of ramsey s theorem to decision tree complexity combinatorial techniques for extending lower bound results for decision trees to general types of queries are presented problems that are defined by simple inequalities between inputs called order invariant problems are considered a decision tree is called k bounded if each query depends on at most k variables no further assumptions on the type of queries are made it is proved that one can replace the queries of any k bounded decision tree that solves an order invariant problem over a large enough input domain with k bounded queries whose outcome depends only on the relative order of the inputs as a consequence all existing lower bounds for comparison based algorithms are valid for general k bounded decision trees where k is a constant an o n log n lower bound for the element uniqueness problem and several other problems for any k bounded decision tree such that k o n c and c n 1 2 bounded decision trees of complexity o n that solve the element uniqueness problem all the lower bounds mentioned above are shown to hold for nondeterministic and probabilistic decision trees as well

new amortized cell probe lower bounds for dynamic problems abstract we build upon the recent papers by weinstein and yu 11 larsen 7 and clifford et al 3 to present a general framework that gives amortized lower bounds on the update and query times of dynamic data structures using our framework we present two concrete results 1 for the dynamic polynomial evaluation problem where the polynomial is defined over a finite field of size n 1 1 and has degree n any dynamic data structure must either have an amortized update time of lg n lg lg n 2 or an amortized query time of lg n lg lg n 2 2 for the dynamic online matrix vector multiplication problem where we get an n n matrix whose entires are drawn from a finite field of size n 1 any dynamic data structure must either have an amortized update time of lg n lg lg n 2 or an amortized query time of n lg n lg lg n 2 for these two problems the previous works by larsen 7 and clifford et al 3 gave the same lower bounds but only for worst case update and query times our bounds match the highest unconditional lower bounds known till date for any dynamic problem in the cell probe model

noncoercive variational inequalities some applications ever since the fundamental paper of lions and stampacchia io the subject of variational inequalities has become a very active area of study and become an important tool in the study of nonlinear partial differential equations which are subject to various types of constraints see e g l 5 8 111 in lo the authors consider first the case of coercive forms and then using elliptic regularization problems involving noncoercive but nonnegative forms we shall use a similar approach to study variational inequalities involving noncoercive bilinear forms and nonlinear functionals which are nice perturbations of lower semi continuous convex functionals furthermore our bilinear forms need not necessarily have a finite dimensional kernel our investigation has been motivated by some work of baiocchi gastaldi and tomarelli see 3 61 where variational inequalities involving so called compact coercive bilinear forms are studied we shall introduce here a somewhat more general concept that of a p coercive bilinear form and study variational inequalities involving such forms and nonlinear functionals which are compact and bounded perturbations of lower semi continuous convex functionals in the next section we shall introduce the necessary terminology prove some preliminary results and give the statement of the abstract problem we then shall state and prove our main abstract result and conclude the paper with applications including one to the study of a flat plate on an elastic foundation with unilateral boundary conditions see e g l for the terminology and one to the study of unilateral problems in semipermeable media see 5

error estimates for the approximation of semicoercive variational inequalities an abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution this condition turns out to be necessary as is demonstrated analytically and numerically the results are applied to the finite element approximation of poisson s equation with signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions for second order variational inequalities the condition is always satisfied whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set applying the error estimate yields optimal order of convergence in terms of the mesh size h the numerical convergence rates observed are in good agreement with the predicted ones

membership in constant time and almost minimum space this paper deals with the problem of storing a subset of elements from the bounded universe mathcal m 0 ldots m 1 so that membership queries can be performed efficiently in particular we introduce a data structure to represent a subset of n elements of mathcal m in a number of bits close to the information theoretic minimum b left lceil lg m choose n right rceil and use the structure to answer membership queries in constant time

space time tradeoffs for emptiness queries we develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries our lower bounds apply to a general class of geometric range query data structures called partition graphs informally a partition graph is a directed acyclic graph that describes a recursive decomposition of space we show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the fredman yao semigroup arithmetic model with the same asymptotic space and time bounds thus results of bronnimann chazelle and pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality st d omega n log n d d 1 d 1 using different techniques we improve previous lower bounds for hopcroft s problem given a set of points and hyperplanes does any hyperplane contain a point in dimensions four and higher using this offline result we show that for online hyperplane emptiness queries omega n d mbox polylog n space is required to achieve polylogarithmic query time and omega n d 1 d mbox polylog n query time is required if only o n polylog n space is available these two lower bounds are optimal up to polylogarithmic factors for two dimensional queries we obtain an optimal continuous tradeoff st 2 omega n 2 between these two extremes finally using a lifting argument we show that the same lower bounds hold for both offline and online halfspace emptiness queries in mathbb r d d 3 2

fuzzy valued probability in this paper the main result is introduction of fuzzy valued probability restricted with respect to probability selectors in order to treat a highly imprecise probability system the fuzzy valued probability is defined over the measurable space as a fuzzy valued set function which is not additive in classical sense but it is additive with respect to restricted arithmetics some properties and the connection with fuzzy valued measure are discussed also an application in markov decision process with imprecise information is presented

fuzzy random renewal reward process and its applications this paper studies a renewal reward process with fuzzy random interarrival times and rewards under the independence associated with any continuous archimedean t norm the interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are independent fuzzy variables under these conditions some limit theorems in mean chance measure are derived for fuzzy random renewal rewards in the sequel a fuzzy random renewal reward theorem is proved for the long run expected reward per unit time of the renewal reward process the renewal reward theorem obtained in this paper can degenerate to that of stochastic renewal theory finally some application examples are provided to illustrate the utility of the result

fuzzy random renewal process with queueing applications using extension principle associated with a class of continuous archimedean triangular norms this paper studies a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables some limit theorems in chance measure and in expected value for the sum of fuzzy random variables are proved on the basis of the continuous archimedean triangular norm based arithmetics furthermore we discuss the fuzzy random renewal process based on the obtained limit theorems and derive a fuzzy random elementary renewal theorem for the long run expected renewal rate the renewal theorem obtained in this paper can degenerate to the corresponding classical result in stochastic renewal process finally two case studies of queueing systems are provided to illustrate the application of the fuzzy random elementary renewal theorem

renewal reward processes with fuzzy rewards and their applications to t age replacement policies the application of fuzzy set theory to renewal reward processes is proposed in this paper the reward is modeled as a fuzzy random variable a theorem which presents the long run average fuzzy reward per unit time is stated a procedure to obtain the best t age replacement policy with fuzzy cost structure is developed the original problem is transformed into a nonlinear program in order to evaluate the membership of the long run average fuzzy cost per unit time

are bitvectors optimal we study the it static membership problem given a set s of at most n keys drawn from a universe u of size m store it so that queries of the form is u in s can be answered by making few accesses to the memory we study schemes for this problem that use space close to the information theoretic lower bound of omega n log frac m n bits and yet answer queries by reading a small number of bits of the memory we show that for epsilon 0 there is a scheme that stores o frac n epsilon 2 log m bits and answers membership queries using a randomized algorithm that reads just one bit of memory and errs with probability at most epsilon we consider schemes that make no error for queries in s but are allowed to err with probability at most epsilon for queries not in s we show that there exist such schemes that store o frac n epsilon 2 log m bits and answer queries using just one bitprobe if multiple probes are allowed then the number of bits stored can be reduced to o n 1 delta log m for any delta 0 the schemes mentioned above are based on probabilistic constructions of set systems with small intersections we show lower bounds that come close to our upper bounds for a large range of n and epsilon schemes that answer queries with just one bitprobe and error probability epsilon must use omega frac n epsilon log 1 epsilon log m bits of storage if the error is restricted to queries not in s then the scheme must use omega frac n 2 epsilon 2 log n epsilon log m bits of storage we also consider deterministic schemes for the static membership problem and show tradeoffs between space and the number of probes

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

tight bounds on computing error correcting codes by bounded depth circuits with arbitrary gates we bound the minimum number w of wires needed to compute any asymptotically good error correcting code c 0 1 n 0 1 n with minimum distance n using unbounded fan in circuits of depth d with arbitrary gates our main results are 1 if d 2 then w n lgn lglgn 2 2 if d 3 then w nlglgn 3 if d 2k or d 2k 1 for some integer k 2 then w n k n where 1 n lgn i 1 n i n and the operation gives how many times one has to iterate the function i to reach a value at most 1 from the argument n and 4 if d lg n then w o n for depth d 2 our n lgn lglgn 2 lower bound gives the largest known lower bound for computing any linear map the upper bounds imply that a necessarily dense generator matrix for our code can be written as the product of two sparse matrices using known techniques we also obtain similar but not tight bounds for computing pairwise independent hash functions our lower bounds are based on a superconcentrator like condition that the graphs of circuits computing good codes must satisfy this condition is provably intermediate between superconcentrators and their weakenings considered before

compressed suffix arrays and suffix trees with applications to text indexing and string matching the proliferation of online text such as found on the world wide web and in online databases motivates the need for space efficient text indexing methods that support fast string searching we model this scenario as follows consider a text t consisting of n symbols drawn from a fixed alphabet sigma the text t can be represented in n lg sigma bits by encoding each symbol with lg sigma bits the goal is to support fast online queries for searching any string pattern p of m symbols with t being fully scanned only once namely when the index is created at preprocessing time the text indexing schemes published in the literature are greedy in terms of space usage they require omega n lg n additional bits of space in the worst case for example in the standard unit cost ram suffix trees and suffix arrays need omega n memory words each of omega lg n bits these indexes are larger than the text itself by a multiplicative factor of omega smash lg sigma n which is significant when sigma is of constant size such as in textsc ascii or textsc unicode on the other hand these indexes support fast searching either in o m lg sigma time or in o m lg n time plus an output sensitive cost o mathit occ for listing the mathit occ pattern occurrences we present a new text index that is based upon compressed representations of suffix arrays and suffix trees it achieves a fast smash o m lg sigma n lg sigma epsilon n search time in the worst case for any constant 0 epsilon leq 1 using at most smash bigl epsilon 1 o 1 bigr n lg sigma bits of storage our result thus presents for the first time an efficient index whose size is provably linear in the size of the text in the worst case and for many scenarios the space is actually sublinear in practice as a concrete example the compressed suffix array for a typical 100 mb textsc ascii file can require 30 40 mb or less while the raw suffix array requires 500 mb our theoretical bounds improve emph both time and space of previous indexing schemes listing the pattern occurrences introduces a sublogarithmic slowdown factor in the output sensitive cost giving o mathit occ smash lg sigma epsilon n time as a result when the patterns are sufficiently long we can use auxiliary data structures in o n lg sigma bits to obtain a total search bound of o m lg sigma n mathit occ time which is optimal

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

electing a leader in a synchronous ring the problem of electing a leader in a synchronous ring of n processors is considered both positive and negative results are obtained on the one hand if processor ids are chosen from some countable set then there is an algorithm that uses only o n messages in the worst case on the other hand any algorithm that is restricted to use only comparisons of ids requires o n log n messages in the worst case alternatively if the number of rounds is required to be bounded by some t in the worst case and ids are chosen from any set having at least n t elements for a certain very fast growing function then any algorithm requires o n log n messages in the worst case

unilateral contact of two solids subject to large deformations formulation and existence results the approach that is currently the most widely used for formulating the unilateral frictionless contact between two deformable solids involves two key elements the orthogonal projection of one potential contact material surface on another and the action reaction principle of contact forces in the present paper we investigate this approach in the general case of large deformations and study unilateral frictionless boundary value problems of elasticity formulated by means of it the conditions for the orthogonal projection to be unique or further a c 1 diffeomorphism are established an extended action reaction principle is explicitly given for the forces acting on a pair of potential contact surfaces assuming the solids under consideration can be described by polyconvex hyperelastic laws we formulate the associated unilateral frictionless boundary value problem as a minimum principle and show the existence of a solution to it

some properties of fuzzy random renewal processes fuzzy random variable is a measure function from a probability space to a collection of fuzzy variables based on the fuzzy random theory this paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed iid fuzzy random interarrival times the relationship between the expected value of the fuzzy random renewal variable and the distribution functions of the alpha pessimistic values and alpha optimistic values of the interarrival times is discussed furthermore the fuzzy random style of renewal equation is provided finally fuzzy random blackwell s renewal theorem and smith s key renewal theorem are also given

existence theorems for generalized noncoercive equilibrium problems the quasi convex case we provide a characterization of the nonemptiness of the solution set for generalized noncoercive equilibrium problems an extension of generalized quasi variational inequalities defined in reflexive banach spaces in the quasi convex case in addition several necessary and sufficient conditions for the set of solutions to these problems to be nonempty and bounded are also given our approach is based on recession notions which proved to be very useful in the study of noncoercive minimization problems in fact we find some particular cones as estimates for the recession cone of the solution set these cones for the ones containing the latter set are proved to be sharp enough to encompass several special situations found in the literature

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

restricted isometry property for general p norms the restricted isometry property rip is a fundamental property of a matrix which enables sparse recovery informally an m times n matrix satisfies rip of order k for the ell p norm if ax p approx x p for every vector x with at most k non zero coordinates for every 1 leq p we obtain almost tight bounds on the minimum number of rows m necessary for the rip property to hold prior to this paper only the cases p 1 1 1 log n and 2 were studied interestingly our results show that the case p 2 is a singularity point the optimal number of rows m is widetilde theta k p for all p in 1 infty setminus 2 as opposed to widetilde theta k for p 2 we also obtain almost tight bounds for the column sparsity of rip matrices and discuss the implications of our results for the stable sparse recovery problem as defined by candes et al

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

unbalanced expanders and randomness extractors from parvaresh vardy codes we give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree which is polylogarithmic in the number of vertices both the degree and the number of right hand vertices are polynomially close to optimal whereas the previous constructions of ta shma et al 2007 required at least one of these to be quasipolynomial in the optimal our expanders have a short and self contained description and analysis based on the ideas underlying the recent list decodable error correcting codes of parvaresh and vardy 2005 our expanders can be interpreted as near optimal randomness condensers that reduce the task of extracting randomness from sources of arbitrary min entropy rate to extracting randomness from sources of min entropy rate arbitrarily close to 1 which is a much easier task using this connection we obtain a new self contained construction of randomness extractors that is optimal up to constant factors while being much simpler than the previous construction of lu et al 2003 and improving upon it when the error parameter is small e g 1 poly n

existence of optimal solutions and duality results under weak conditions in this paper we consider an ordinary convex program with no qualification conditions such as slater s condition and for which the optimal set is neither required to be compact nor to be equal to the sum of a compact set and a linear space it is supposed only that the infimum is finite a very wide class of convex functions is exhibited for which the optimum is always attained and is equal to the supremum of the ordinary dual program additional results concerning the existence of optimal solutions in the non convex case are also given

contact between nonlinearly elastic bodies we study the contact between nonlinearly elastic bodies by variational methods after the formulation of the mechanical problem we provide existence results based on polyconvexity and on quasiconvexity then we derive the euler lagrange equation as a necessary condition for minimizers here clarke s generalized gradients are the essential tool to treat the nonsmooth obstacle condition

succinct representation of balanced parentheses and static trees we consider the implementation of abstract data types for the static objects binary tree rooted ordered tree and a balanced sequence of parentheses our representations use an amount of space within a lower order term of the information theoretic minimum and support in constant time a richer set of navigational operations than has previously been considered in similar work in the case of binary trees for instance we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned the approach is applied to produce a succinct representation of planar graphs in which one can test adjacency in constant time

succinct encoding of arbitrary graphs we consider the problem of encoding graphs with n vertices and m edges compactly supporting adjacency neighborhood and degree queries in constant time in the q logn bit word ram model the adjacency query asks whether there is an edge between two vertices the neighborhood query reports the neighbors of a given vertex in constant time per neighbor and the degree query reports the number of incident edges to a given vertex we study the problem in the context of succinctness where the goal is to achieve the optimal space requirement as a function of n and m to within lower order terms we prove a lower bound in the cell probe model indicating it is impossible to achieve the information theory lower bound up to lower order terms unless the graph is either too sparse namely m o n d for any constant d 0 or too dense namely m w n 2 d for any constant d 0 furthermore we present a succinct encoding of graphs supporting aforementioned queries in constant time the space requirement of the encoding is within a multiplicative 1 e factor of the information theory lower bound for any arbitrarily small constant e 0 this is the best achievable space bound according to our lower bound where it applies the space requirement of the representation achieves the information theory lower bound tightly within lower order terms where the graph is very sparse m o n d for any constant d 0 or very dense m n 2 lg 1 dn for an arbitrarily small constant d 0

new lower bound techniques for dynamic partial sums and related problems we study the complexity of the dynamic partial sum problem in the cell probe model we give the model access to nondeterministic queries and prove that the problem remains hard we give the model access to the right answer pm 1 as an oracle and prove that the problem remains hard this suggests which kind of information is hard to maintain from these results we derive a number of lower bounds for dynamic algorithms and data structures we prove lower bounds for dynamic algorithms for existential range queries reachability in directed graphs planarity testing planar point location incremental parsing and fundamental data structure problems like maintaining the majority of the prefixes of a string of bits we prove a lower bound for reachability in grid graphs in terms of the graph s width we characterize the complexity of maintaining the value of any symmetric function on the prefixes of a bit string

relaxed formulation and existence result of the degenerated elliptic small disturbance model abstract we consider an elliptic parabolic pde governing the flow of a steady state two dimensional irrotational compressible fluid in a channel or around a thin profile the model is formulated in term of a noncoercive variational problem with an integral functional in a truncated domain we propose a relaxed formulation in the bounded variation functions space and we prove the existence and the uniqueness of the relaxed optimal solution we consider also the asymptotic behavior of the solution when the domain grows to infinity

a theorem of renewal process for fuzzy random variables and its application the fuzzy set was introduced by zadeh 1965 and the concept of fuzzy random variables was provided by kwakernaak 1981 sequences of independent and identical distributed fuzzy random variables were considered by kruse 1982 he also showed the strong law of large numbers for fuzzy random variables in this paper we consider the stochastic process for fuzzy random variables and prove a theorem for fuzzy rate of a fuzzy renewal process

renewal process with t related fuzzy inter arrival times and fuzzy rewards in this paper we consider a renewal process in which the inter arrival times and rewards are characterized as fuzzy variables under t norm based fuzzy operations a t related fuzzy renewal theorem and a fuzzy renewal reward theorem are proved using a law of large numbers for fuzzy variables

on the cell probe complexity of polynomial evaluation abstract we consider the cell probe complexity of the polynomial evaluation problem with preprocessing of coefficients for polynomials of degree at most n over a finite field k we show that the trivial cell probe algorithm for the problem is optimal if k is sufficiently large compared to n as an application we give a new proof of the fact that p incr time o log n log log n

strong law of large numbers for t normed arithmetics we prove a strong law of large numbers for sequences of pairwise i i d fuzzy random variables addition is given by the extension principle associated with a general continuous triangular norm our result includes the slln for levelwise sums obtained when the chosen triangular norm is the minimum

a fully dynamic algorithm for recognizing and representing proper interval graphs in this paper we study the problem of recognizing and representing dynamically changing proper interval graphs the input to the problem consists of a series of modifications to be performed on a graph where a modification can be a deletion or an addition of a vertex or an edge the objective is to maintain a representation of the graph as long as it remains a proper interval graph and to detect when it ceases to be so the representation should enable one to efficiently construct a realization of the graph by an inclusion free family of intervals this problem has important applications in physical mapping of dna we give a near optimal fully dynamic algorithm for this problem it operates in o log n worst case time per edge insertion or deletion we prove a close lower bound of omega log n log log n log b amortized time per operation in the cell probe model with word size b we also construct optimal incremental and decremental algorithms for the problem which handle each edge operation in o 1 time as a byproduct of our algorithm we solve in o log n worst case time the problem of maintaining connectivity in a dynamically changing proper interval graph

dynamic perfect hashing upper and lower bounds the dynamic dictionary problem is considered provide an algorithm for storing a dynamic set allowing the operations insert delete and lookup a dynamic perfect hashing strategy is given a randomized algorithm for the dynamic dictionary problem that takes o 1 worst case time for lookups and o 1 amortized expected time for insertions and deletions it uses space proportional to the size of the set stored furthermore lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved this class encompasses realistic hashing based schemes that use linear space such algorithms have amortized worst case time complexity omega log n for a sequence of n insertions and lookups if the worst case lookup time is restricted to k then the lower bound becomes omega k cdot n 1 k

new existence results for equilibrium problems we consider equilibrium problems in the framework of the formulation proposed by blum and oettli which includes variational inequalities nash equilibria in noncooperative games and vector optimization problems for instance as particular cases we establish new sufficient and or necessary conditions for existence of solutions of such problems our results are based upon the relation between equilibrium problems and certain auxiliary convex feasibility problems together with extensions to equilibrium problems of gap functions for variational inequalities then we apply our results to some particular instances of equilibrium problems obtaining results which include among others a new lemma of the alternative for convex optimization problems

the cell probe complexity of succinct data structures we consider time space tradeoffs for static data structure problems in the cell probe model with word size 1 the bit probe model in this model the goal is to represent n bit data with s n r bits such that queries of a certain type about the data can be answered by reading at most t bits of the representation ideally we would like to keep both s and t small but there are tradeoffs between the values of s and t that limit the possibilities of keeping both parameters small in this paper we consider the case of succinct representations where s n r for some email protected n for a boolean version of the problem of polynomial evaluation with preprocessing of coefficients we show a lower bound on the redundancy query time tradeoff of the form r 1 t email protected n logn in particular for very small redundancies r we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure up to a logarithmic factor we show similar lower bounds for problems satisfying a certain combinatorial properties of a coding theoretic flavor and obtain r 1 t email protected n for certain problems previously no w m lower bounds were known on t in the general model for explicit boolean problems even for very small redundancies by restricting our attention to systematic or index structures f satisfying f x email protected email protected x for some map f where denotes concatenation we show similar lower bounds on the redundancy query time tradeoff for the natural data structuring problems of prefix sum and substring search

implicit o 1 probe search given a set of n elements from the domain 1 cdots m this paper investigates how to arrange them in a table of size n so that searching for an element in the table can be done in constant time yao j assoc comput mach 28 1981 pp 615 628 has shown that this cannot be done when the domain is sufficiently large as a function of n this paper gives a constructive solution when the domain m is polynomial in n the number of elements as well as a nonconstructive proof for m no larger than exponential in operatorname poly n the authors improve upon a result of yao and give better bounds on the maximum m for which implicit o 1 probe search can be done the results are achieved by showing the tight relationship between hashing and certain encoding problems called rainbows

convex analysis can be helpful for the asymptotic analysis of monotone operators asymptotic analysis of monotone operators we use representations of maximal monotone operators for studying recession or asymptotic operators associated to maximal monotone operators such a concept is useful for dealing with unboundedness

hash functions for priority queues the complexity of priority queue operations is analyzed with respect to the cell probe computational model of a yao j assoc comput mach 28 no 3 1981 615 628 a method utilizing families of hash functions is developed which permits priority queue operations to be implemented in constant worst case time provided that a size constraint is satisfied the minimum necessary size of a family of hash functions for computing the rank function is estimated and contrasted with the minimum size required for perfect hashing

existence theorems for contact with friction of two linear elastic bodies abstract in this article i give an iteration method for solving the signorini problem by the idea of concentration cancellation the nonlinear quantity in the mathematical formulation of the problem caused by contact friction effect on the boundary is relaxed to a convex functional therefore the relaxed problem can be solved by using convex minimization techniques which are numerically stable finally good convergence property of the proposed method is proved

fuzziness and randomness in an optimization framework abstract a semi infinite approach for linear programming in the presence of fuzzy random variable coefficients is described as a byproduct a way for dealing with optimization problems including both fuzzy and random data is obtained numerical examples are provided for the sake of illustration

fuzzy numbers from raw discrete data using linear regression this paper focuses on modelling fuzzy numbers with meaningful membership functions more precisely it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data in many applications input information is numerical and therefore particular fuzzy sets such as fuzzy numbers hold great interest and relevance in managing data imprecision and vagueness the proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression the technique is simple fast and effective preliminary tests are performed using different types of input data a gaussian function a sigmoidal function three datasets of synthetic discrete data and an histogram obtained from a colour satellite image

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

interactive fuzzy random two level linear programming based on level sets and fractile criterion optimization in this paper assuming cooperative behavior of the decision makers two level linear programming problems under fuzzy random environments are considered by introducing a level sets of fuzzy random variables and defining an a stochastic two level linear programming problem for guaranteeing the degree of realization of the problem through a new decision making model proposed based on the fractile criterion optimization model considering fuzzy goals the a stochastic two level programming problem can be equivalently transformed into a deterministic problem interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented an illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method

orderly spanning trees with applications we introduce and study orderly spanning trees of plane graphs this algorithmic tool generalizes canonical orderings which exist only for triconnected plane graphs although not every plane graph admits an orderly spanning tree we provide an algorithm to compute an orderly pair for any connected planar graph g consisting of an embedded planar graph h isomorphic to g and an orderly spanning tree of h we also present several applications of orderly spanning trees 1 a new constructive proof for schnyder s realizer theorem 2 the first algorithm for computing an area optimal 2 visibility drawing of a planar graph and 3 the most compact known encoding of a planar graph with o 1 time query support all algorithms in this paper run in linear time

should tables be sorted we examine optimality questions in the following information retrieval problem given a set s of n keys store them so that queries of the form is x in s can be answered quickly it is shown that in a rather general model including al1 the commonly used schemes lceil lg n l rceil probes to the table are needed in the worst case provided the key space is sufficiently large the effects of smaller key space and arbitrary encoding are also explored

error correcting data structures we study data structures in the presence of adversarial noise we want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object even if the data structure has been corrupted by a constant fraction of errors we measure the efficiency of a data structure in terms of its length the number of bits in its representation and query answering time measured by the number of bit probes to the possibly corrupted representation the main issue is the trade off between these two this new model is the common generalization of static data structures and locally decodable error correcting codes ldcs we prove a number of upper and lower bounds on various natural error correcting data structure problems in particular we show that the optimal length of t probe error correcting data structures for the membership problem where we want to store subsets of size s from a universe of size n such that membership queries can be answered effic

an efficient centroid type reduction strategy for general type 2 fuzzy logic system in this paper an efficient centroid type reduction strategy for general type 2 fuzzy set is introduced this strategy makes use of the result of a plane representation and performs the centroid type reduction on each a plane simulations show that it usually needs only several resolution of a value such that the defuzzified value converges to a real value consequently comparing with the exhaustive computation approach this approach can tremendously decrease the computation complexity from exponential into linear

determining membership with 2 simultaneous queries abstract alice and bob are playing a cooperative game in which alice must devise a scheme to store n elements in an array from a universe u of size m her goal is to store in such a way that for every x u bob can observe the values of two positions dependent on x in the array and determine whether x is in the array or not alice may share her storage scheme with bob and they win if such an arrangement is made the question is how large can the universe u be in terms of n so that alice and bob can win in this paper we give upper and lower bounds on this question for general n and the special case when n 3 we also pose conjectures and further questions for research

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

succinct data structures for searchable partial sums with optimal worst case performance abstract the notion of succinct indexes can be dated back from the debut of jacobson s thesis 1988 14 and has triggered many results in the last decade in traditional indexing some given data are preprocessed so as to support online queries and updates on the data as efficiently as possible when succinctness is involved we are restricted to index the data using only an information theoretically minimum number of bits this paper concerns the succinct indexing schemes for a well studied problem called searchable partial sums sps in sps an array a of n non negative k bit integers is preprocessed so as to support online sum and search queries and possibly update operation of individual entry a succinct indexing scheme would allow only k n o k n bits to represent the array a the only known result is that when k 1 in this case it is known as the dynamic bit array problem we can support both queries in o log b n time and update in o b amortized time for any b with lg n lg lg n b n this paper shows that even for k o lg lg n we can index a succinctly such that both query and update operations can be supported using the same time complexities moreover the time for update becomes the worst case time furthermore the tradeoff between the query times and the update time is optimal as implied by pa trascu and demaine s lower bound result 2006 24 in general when k o lg u we show a lower bound of lg n lg lg n time for the search query irrespective of the update time this gives a tighter lower bound as compared to that of pa trascu and demaine s which is a consequence of the requirement of succinctness on the other hand we give a succinct index that can support sum in o log b n time search in o log b n time and update in o b time where min lg lg n lg lg u lg lg lg u lg n lg lg n the query times are optimal when b n this paper also extends the searchable partial sums with insert and delete operations and provides a succinct data structure for some cases

uncertain random alternating renewal process with application to interval availability as a mixture of a random variable and an uncertain variable an uncertain random variable is a tool to deal with the indeterminacy quantities involving randomness and human uncertainty this paper aims at proposing a concept of an uncertain random alternating renewal process to model a repairable system with random on times and uncertain off times an alternating renewal theorem is proved which gives the limit chance distribution of the interval availability of the uncertain random alternating renewal system

on a model of indexability and its bounds for range queries we develop a theoretical framework to characterize the hardness of indexing data sets on block access memory devices like hard disks we define an indexing workload by a data set and a set of potential queries for a workload we can construct an indexing scheme which is a collection of fixed sized subsets of the data we identify two measures of efficiency for an indexing scheme on a workload storage redundancy r how many times each item in the data set is stored and access overhead a how many times more blocks than necessary does a query retrieve for many interesting families of workloads there exists a trade off between storage redundancy and access overhead given a desired access overhead a there is a minimum redundancy that any indexing scheme must exhibit we prove a lower bound theorem for deriving the minimum redundancy by applying this theorem we show interesting upper and lower bounds and trade offs between a and r in the case of multidimensional range queries and set queries

upper and lower bounds on the power of advice proving superpolylogarithmic lower bounds for dynamic data structures has remained an open problem despite years of research p trascu proposed an exciting approach for breaking this barrier via a two player communication model in which one player gets private advice at the beginning of the protocol he gave reductions from the problem of solving an asymmetric version of set disjointness in his model to a diverse collection of natural dynamic data structure problems in the cell probe model he also conjectured that for any hard problem in the standard two party communication model the asymmetric version of the problem is hard in his model provided not too much advice is given in this paper we prove several surprising results about his model we show that there exist boolean functions requiring linear randomized communication complexity in the two party model for which the asymmetric versions in his model have deterministic protocols with exponentially smaller complexity for set disjointness which

a quick tour on suffix arrays and compressed suffix arrays suffix arrays are a key data structure for solving a run of problems on texts and sequences from data compression and information retrieval to biological sequence analysis and pattern discovery in their simplest version they can just be seen as a permutation of the elements in 1 2 n encoding the sorted sequence of suffixes from a given text of length n under the lexicographic order yet they are on a par with ubiquitous and sophisticated suffix trees over the years many interesting combinatorial properties have been devised for this special class of permutations for instance they can implicitly encode extra information and they are a well characterized subset of the n permutations this paper gives a short tutorial on suffix arrays and their compressed version to explore and review some of their algorithmic features discussing the space issues related to their usage in text indexing combinatorial pattern matching and data compression

compactness properties for families of quasistationary solutions of some evolution equations the following typical problem occurs in passing to the limit in some phase field models for two sequences of space time dependent functions n x n representing e g suitable approximations of the temperature and the phase variable we know that the sum n x n converges in some l p space as n and that the time integrals of a suitable space functional evaluated on n x n are uniformly bounded with respect to n can we deduce that n and x n converge separately luckhaus 1990 gave a positive answer to this question in the framework of the two phase stefan problem with gibbs thompson law for the melting temperature plotnikov 1993 proposed an abstract result employing the original idea of luckhaus and arguments of compactness and reflexivity type we present a general setting for this and other related problems providing necessary and sufficient conditions for their solvability these conditions rely on general topological and coercivity properties of the functionals and the norms involved and do not require reflexivity

noncoercive optimization problems this paper modifies a recent sufficient condition for the existence of optimal solutions proposed by buttazzo and tomarelli to obtain a condition that is necessary and sufficient applications to mathematical programming are given and stability results established

the law of large numbers for fuzzy numbers with unbounded supports we study laws of large numbers for mutually t related fuzzy numbers with unbounded supports where t is an archimedean t norm and generalize earlier result of badard

an optimal dynamic data structure for stabbing semigroup queries let s be a set of n intervals in mathbb r and let mathbf s be any commutative semigroup we assign a weight omega s in mathbf s to each interval in s for a point x in mathbb r let s x subseteq s be the set of intervals that contain x given a point q in mathbb r the stabbing semigroup query asks for computing sum s in s q omega s we propose a linear size dynamic data structure under the pointer machine model that answers queries in worst case o log n time and supports both insertions and deletions of intervals in amortized o log n time it is the first data structure that attains the optimal o log n bound for all three operations furthermore our structure can easily be adapted to external memory where we obtain a linear size structure that answers queries and supports updates in o log b n i os where b is the disk block size for the restricted case of a nested family of intervals either every pair of intervals is disjoint or one contains the other we present a simpler solution based on dynamic trees

how to deal with the unbounded in optimization theory and algorithms the aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions but also for constructing new methods

unilateral problems in nonlinear three dimensional elasticity in 1977 john ball obtained a series of very important existence results in hyperelasticity the objective of the present paper is to extend his results in two directions first by considering unilateral boundary conditions corresponding to contact without friction secondly by allowing locking constraints in the set of admissible deformations