the solution by iteration of nonlinear equations involving psi strongly accretive operators let e be an arbitrary real banach space and let k c e be a proximal subset of e suppose t k i e is a lipschitz strongly accretive operator such that for any given g e the equation tx f has a solution x e d t it is proved that modified iteration processes of the mann and ishikawa types converge strongly to x related results deal with the solution of the equation x tx f 0 for a strongly accretive or strongly dissipative t and the iterative approximation of fixed points to a strongly pseudocontractive t we also discuss the case where the operator t is only uniformly continuous miramare trieste december 1996 permanent address department of mathematics and computer science nnamdi azikiwe university p m b 5025 awka anambra state nigeria

convergence of ishikawa s iteration method for pseudocontractive mappings let c be a nonempty closed and convex subset of a real hilbert space h let ti c c i 1 2 n be a finite family of lipschitz pseudocontractive mappings it is our purpose in this paper to prove strong convergence of ishikawa s method to a common fixed point of a finite family of lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty no compactness assumption is imposed either on t or on c moreover computation of the closed convex set cn for each n 1 is not required the results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings

convergence theorems for strongly pseudo contractive and strongly accretive maps supposeeis an arbitrary real banach space andkis a nonempty closed convex and bounded subset ofe supposet k kis a uniformly continuous strong pseudo contraction with constantk 0 1 suppose an bn cn a n b n and c n are sequences in 0 1 satisfying the following conditions i an bn cn 1 a n b n c n integersn 0 ii lim bn lim b n lim c n 0 iii bn iv cn for arbitraryx0 u0 v0 k define the sequence xn n 0iteratively byxn 1 anxn bntyn cnun yn a nxn b ntxn c nvn n 0 where un vn are arbitrary sequences ink then xn converges strongly to the unique fixed point oft related results deal with the iterative solutions of nonlinear equations involving set valued strongly accretive operators

on the linearity defect of the residue field abstract given a commutative noetherian local ring r the linearity defect of a finitely generated r module m denoted ld r m is an invariant that measures how far m and its syzygies are from having a linear resolution motivated by a positive known answer in the graded case we study the question of whether ld r k implies ld r k 0 we give answers in special cases and we discuss several interpretations and refinements of the question

a double sequence iteration process for fixed points of continuous pseudocontractions abstract let c be a bounded closed convex nonempty subset of a real hilbert space h the idea of a double sequence iteration is introduced and it is proved that a mann type double sequence iteration process converges strongly to a fixed point of a continuous pseudocontractive map t which maps c into c related results deal with the strong convergence of the iteration process to fixed points of nonexpansive maps

iterative process with errors of nonlinear equations involvingm accretive operators abstract let x be a uniformly smooth and uniformly convex banach space and t d t x x be an m accretive operator with the domain d t and the range r t for any given f x we prove that the mann and ishikawa type iterative sequences with errors converge strongly to the unique solution of the equation x tx f where t may not be lipschitz some related results deal with the convergence of mann and ishikawa type iterative sequences with errors for approximating a solution of a continuous dissipative operator equation

homological criteria for minimal multiplicity abstract lower bounds on hilbert samuel multiplicity are known for several types of commutative noetherian local rings and rings with multiplicities which achieve these lower bounds are said to have minimal multiplicity the first part of this paper gives characterizations of rings of minimal multiplicity in terms of the ext algebra of the ring in particular we show that minimal multiplicity can be detected via an ext algebra which is gorenstein or koszul as regular the second part of this paper characterizes rings of minimal multiplicity via a numerical homological invariant introduced by j herzog and s b iyengar called linearity defect our characterizations allow us to answer in two special cases a question raised by herzog and iyengar

picard iterations for solution of nonlinear equations in certain banach spaces abstract let e be a real uniformly smooth banach space and let a d a e e be locally lipschitzian and strongly quasi accretive it is proved that a picard recursion process converges strongly to the unique solution of the equation ax f f r a with the convergence being at least as fast as a geometric progression related results deal with the convergence of picard iterations to the fixed point of locally lipschitzian strong hemicontractions t and to the solutions of nonlinear equations of the forms x ax f and x ax f where a is an accretive type map

betti splitting via componentwise linear ideals abstract a monomial ideal i admits a betti splitting i j k if the betti numbers of i can be determined in terms of the betti numbers of the ideals j k and j k given a monomial ideal i we prove that i j k is a betti splitting of i provided j and k are componentwise linear generalizing a result of francisco ha and van tuyl if i has a linear resolution the converse also holds we apply this result recursively to the alexander dual of vertex decomposable shellable and constructible simplicial complexes moreover we determine the graded betti numbers of the defining ideal of three general fat points in the projective space

about randomised distributed graph colouring and graph partition algorithms we present and analyse a very simple randomised distributed vertex colouring algorithm for arbitrary graphs of size n that halts in time o logn with probability 1 o n 1 each message containing 1 bit its bit complexity per channel is o logn from this algorithm we deduce and analyse a randomised distributed vertex colouring algorithm for arbitrary graphs of maximum degree d and size n that uses at most d 1 colours and halts in time o logn with probability 1 o n 1 we also obtain a partition algorithm for arbitrary graphs of size n that builds a spanning forest in time o logn with probability 1 o n 1 we study some parameters such as the number the size and the radius of trees of the spanning forest

two absolute bounds for distributed bit complexity the concept of distributed communication bit complexity was introduced by dinitz rajsbaum and moran they studied the bit complexity of consensus and leader election arriving at more or less exact bounds this paper answers two questions on leader election which remained there open the first is to close the gap between the known upper and lower bounds for electing a leader by two linked processors the second is whether the suggested algorithm sending 1 5n bits while electing a leader in a chain of even length n is optimal in the case when n is known to the processors for both problems absolutely exact bounds are found moreover the presented lower bound proofs show that there is no optimal algorithm other than the suggested ones

bit complexity of breaking and achieving symmetry in chains and rings we consider a failure free asynchronous message passing network with n links where the processors are arranged on a ring or a chain the processors are identically programmed but have distinct identities taken from 0 1 m 1 we investigate the communication costs of three well studied tasks consensus leader and maxf finding the maximum identity we show that in chain and ring topologies the message complexities of all three tasks are the same hence we study a finer measure of complexity the number of transmitted bits required to solve a task t denoted bitc t we prove several new lower bounds and some simple upper bounds that imply the following results for the two processors case bitc consensus e 2 and bitc leader e bitc maxf e 2log2 m o 1 where the gap between the lower and upper bounds is almost always 1 for a chain bitc consensus e n bitc leader e n p log m and bitc maxf e n log m for the ring topology we prove the lower bound of n log m for leader and hence maxf we consider also a chain where the intermediate processors have no identities we prove that bitc leader e n log m which is equal to n times the bit complexity of the problem for two processors for the specific case when the chain length is even we prove that bitc leader e n for both above settings in addition we show that for any algorithm solving maxf there exists an input for which every execution has the bit complexity n log m this is not the case for leader in our proofs we use both methods of distributed computing and of communication complexity theory establishing new links between the two areas

the optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors in a previous paper we showed that the distributive construction of a spanning tree in a complete network of processors can be done in o n log n messages we show in this work that if the spanning tree is required to satisfy certain properties then the complexity of its construction increases first we show that the construction of a minimum weight spanning tree requires in the worst case at least omega n 2 messages and then we show that the construction of a spanning tree where the maximum degree is at most k may require at least omega n 2 k messages in the worst case actually in both cases the lower bounds are shown for the number of edges used in the worst case moreover the results are valid for both asynchronous and synchronous networks and are independent of the lengths of the messages on the other hand there are algorithms for the above tasks which achieve these lower bounds up to a constant factor and use messages of o log n length

edge pancyclicity and path embeddability of bijective connection graphs an n dimensional bijective connection graph in brief bc graph is a regular graph with 2 n nodes and n2 n 1 edges the n dimensional hypercube crossed cube mobius cube etc are some examples of the n dimensional bc graphs in this paper we propose a general method to study the edge pancyclicity and path embeddability of the bc graphs first we prove that a path of length l with dist x n x y 2 y in x n where x n n 4 is a n dimensional bc graph satisfying the three specific conditions and v x n is the node set of x n furthermore by this result we can claim that x n is edge pancyclic lastly we show that these results can be applied to not only crossed cubes and mobius cubes but also other bc graphs except crossed cubes and mobius cubes so far the research on edge pancyclicity and path embeddability has been limited in some specific interconnection architectures such as crossed cubes mobius cubes

a simple randomized scheme for constructing low weight k connected spanning subgraphs with applications to distributed algorithms the main focus of this paper is the analysis of a simple randomized scheme for constructing low weight k connected spanning subgraphs in this paper we focus on the metric graph we use the term metric graph for a complete graph with metric weights we first show that our scheme gives a simple approximation algorithm to construct a minimum weight k connected spanning subgraph in a metric graph an np hard problem we show that our algorithm gives an approximation ratio of o klogn for a metric graph o k for a random graph with nodes uniformly randomly distributed in 0 1 2 and for a complete graph with random edge weights u 0 1 we show that our scheme is optimal with respect to the amount of local information needed to compute any connected spanning subgraph we then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate k connected spanning subgraph for any k 1 in a point to point distributed model where the processors form a complete network our algorithm takes time and an expected number of messages our result in conjunction with a result of korach et al e korach s moran s zaks the optimality of distributive constructions of minimum weight and degree restricted spanning trees in a complete network of processors siam journal on computing 16 2 1987 231 236 implies that the expected message complexity of our algorithm is significantly better than the best distributed algorithm that finds an optimal k connected subgraph we also show that for geometric instances our randomized scheme constructs low degree k connected spanning subgraphs which have o klogn maximum degree with high probability

a generalized steepest descent approximation for the zeros of m accretive operators a necessary and sufficient condition is proved for a generalized steepest descent approximation to converge to the zeros of m accretive operators related results deal with the convergence of the scheme to fixed points of pseudocontractive maps

iterative solution of nonlinear equations involving set valued uniformly accretive operators abstract let e be a real normed linear space and let a e 2 e be a uniformly continuous and uniformly quasi accretive multivalued map with nonempty closed values such that the range of i a is bounded and the inclusion f ax has a solution x e it is proved that ishikawa and mann type iteration processes converge strongly to x further if t e 2 e is a uniformly continuous and uniformly hemicontractive set valued map with bounded range and a fixed point x e it is proved that both the mann and ishikawa type iteration processes converge strongly to x the strong convergence of these iteration processes with errors is also proved

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

on the equivalence of the convergence criteria between modified mann ishikawa and multi step iterations with errors for successively strongly pseudo contractive operators abstract in this paper the equivalence of the convergence between the modified mann ishikawa and multi step noor iterations with errors is proven for the successively strongly pseudo contractive operators without lipschitzian assumption our results generalize the recent results of the paper b e rhoades s m soltuz the equivalence between mann ishikawa iterations and multi step iteration nonlinear anal 58 2004 219 228 b e rhoades s m soltuz the equivalence between the convergences of ishikawa and mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudo contractive maps j math anal appl 289 2004 266 278 by extending to the more generalized multi step iterations with errors and hence improve the corresponding results of all the references in bibliography by providing the equivalences of convergence between all of these up to date iteration schemes

distributed verification and hardness of distributed approximation we study the verification problem in distributed networks stated as follows let h be a subgraph of a network g where each vertex of g knows which edges incident on it are in h we would l

optimal bit complexity randomised distributed mis and maximal matching algorithms for anonymous rings we present and analyse las vegas distributed algorithms which compute a mis or a maximal matching for anonymous rings their bit complexity and time complexity are o logn with high probability these algorithms are optimal modulo a multiplicative constant beyond the complexity results the interest of this work stands in the description and the analysis of these algorithms which may be easily generalised furthermore these results show a separation between the complexity of the mis problem and of the maximal matching problem on the one hand and the colouring problem on the other colouring can be computed only in w logn rounds on rings with high probability while mis is shown to have a faster algorithm this is in contrast to other models in which mis is at least as hard as colouring

on the complexity of universal leader election electing a leader is a fundamental task in distributed computing in its implicit version only the leader must know who is the elected leader this article focuses on studying the message and time complexity of randomized implicit leader election in synchronous distributed networks surprisingly the most obvious complexity bounds have not been proven for randomized algorithms in particular the seemingly obvious lower bounds of m messages where m is the number of edges in the network and d time where d is the network diameter are nontrivial to show for randomized monte carlo algorithms recent results showing that even n where n is the number of nodes in the network is not a lower bound on the messages in complete networks make the above bounds somewhat less obvious to the best of our knowledge these basic lower bounds have not been established even for deterministic algorithms except for the restricted case of comparison algorithms where it was also required that nodes may not wake up spontaneously and that d and n were not known we establish these fundamental lower bounds in this article for the general case even for randomized monte carlo algorithms our lower bounds are universal in the sense that they hold for all universal algorithms namely algorithms that work for all graphs apply to every d m and n and hold even if d m and n are known all the nodes wake up simultaneously and the algorithms can make any use of node s identities to show that these bounds are tight we present an o m messages algorithm an o d time leader election algorithm is known a slight adaptation of our lower bound technique gives rise to an m message lower bound for randomized broadcast algorithms an interesting fundamental problem is whether both upper bounds messages and time can be reached simultaneously in the randomized setting for all graphs the answer is known to be negative in the deterministic setting we answer this problem partially by presenting a randomized algorithm that matches both complexities in some cases this already separates for some cases randomized algorithms from deterministic ones as first steps towards the general case we present several universal leader election algorithms with bounds that tradeoff messages versus time we view our results as a step towards understanding the complexity of universal leader election in distributed networks

monotone circuits for matching require linear depth it is proven that monotone circuits computing the perfect matching function on n vertex graphs require o n depth this implies an exponential gap between the depth of monotone and nonmonotone circuits

optimal lower bounds for some distributed algorithms for a complete network of processors abstract lower bounds for distributed algorithms for complete networks of processors i e networks where each pair of processors is connected by a communication line are discussed we first show an n log n lower bound for the number of messages required by any algorithm in a given class of distributed algorithms for such networks this class includes algorithms for problems like finding a leader or constructing a spanning tree we then show an n 2 lower bound for other problems like constructing a maximal matching or a hamiltonian circuit in proving the lower bounds we are counting the edges which carry messages during the executions of the algorithms ignoring the actual number of messages carried by each edge interestingly this number is shown to be of the same order of magnitude as the total number of messages needed by these algorithms the proofs of the lower bounds apply for synchronous networks and for arbitrarily long messages

communication steps for parallel query processing we study the problem of computing conjunctive queries over large databases on parallel architectures without shared storage using the structure of such a query q and the skew in the data we study tradeoffs between the number of processors the number of rounds of communication and the per processor load the number of bits each processor can send or can receive in a single round that are required to compute q since each processor must store its received bits the load is at most the number of bits of storage per processor when the data are free of skew we obtain essentially tight upper and lower bounds for one round algorithms and we show how the bounds degrade when there is skew in the data in the case of skewed data we show how to improve the algorithms when approximate degrees of the necessarily small number of heavy hitter elements are available obtaining essentially optimal algorithms for queries such as skewed simple joins and skewed triangle join queries for queries that we identify as treelike we also prove nearly matching upper and lower bounds for multi round algorithms for a natural class of skew free databases one consequence of these latter lower bounds is that for any v 0 using p processors to compute the connected components of a graph or to output the path if any between a specified pair of vertices of a graph with m edges and per processor load that is o m p1 v requires logp rounds of communication our upper bounds are given by simple structured algorithms using mapreduce our one round lower bounds are proved in a very general model which we call the massively parallel communication mpc model that allows processors to communicate arbitrary bits our multi round lower bounds apply in a restricted version of the mpc model in which processors in subsequent rounds after the first communication round are only allowed to send tuples

the distributed bit complexity of the ring in moran and warmuth 1993 siam j comput 22 no 2 379 399 it was shown that computing any non constant function on a ring of n of processors requires n log bits and this bound is tight this model assumed that all the processors in the ring are identical anonymous i e all processors run the same program and the only parameter of the program is the input to the processor in a relaxed model the anonymity is broken by providing each processor with a distinct identity which becomes a second parameter of the program that is executed by all processors if the set of possible identities grows doubly exponentially in n then by a reduction to the anonymous case one can show that the lower bound holds as well in this paper we show that the lower bound of n log n bits for computing any non constant function holds even if the set of possible identities is very small that is of sizer n1 for any positive

the solution by iteration of nonlinear equations in uniformly smooth banach spaces abstract letebe a uniformly smooth banach space and lett d t e ebe a strong pseudocontraction with an open domaind t ineand a fixed pointx d t we establish the strong convergence of the mann and ishikawa iterative processes with errors to the fixed point oft related results deal with the iterative solution of operator equations of the formsf txandf x tx 0 whentis a set valued strongly accretative operator our theorems include the cases in which the operatortis defined only locally explicit error estimates are also given

finite regularity and koszul algebras we determine the positively graded commutative algebras over which the residue field modulo the homogeneous maximal ideal has finite castelnuovo mumford regularity they are the polynomial rings in finitely many indeterminates over koszul algebras this proves a conjecture of avramov and eisenbud we also show that if the residue field of a finitely generated graded algebras has finite regularity then so do all finitely generated graded modules

iteration processes for nonlinear lipschitzian strongly accretive mappings in lp spaces abstract suppose x l p or l p 1 p 2 let t x x be a lipschitzian and strongly accretive map with constant k 0 1 and lipschitz constant l satisfying p p 1 l 2 p 2 p 1 1 k define s x x by sx tx x for arbitrary x 0 x the sequence x n n 1 is defined by x n 1 1 n x n n sy n y n 1 n x n n sx n n 0 where n n 1 n n 1 are two real sequences satisfying i 0 n 2 1 p 2 p 1 1 k k n l 2 n p 2 p l 2 1 w n w l 2 1 2 n p 2 p 1 1 k k n l 2 n p 1 2 l 2 1 w n w l 2 1 2 n 1 1 for each n ii 0 n min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 for each n iii n n where w p 2 p 1 1 k p 2 p l 2 then x n 1 converges strongly to the unique solution of tx moreover if n 2 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 and n for each n and some 0 min w w l 2 1 1 w 4 p 1 l 2 k p 1 2 l 2 w 1 then x n 1 q p n 2 x 1 q where q denotes the solution of tx and 1 4 1 p 2 p 1 1 k k l 2 p 2 p l 2 1 w w l 2 1 2 2 p 2 p 1 1 k k l 2 p 1 2 l 2 1 w w l 2 1 2 1 1 0 1 a related result details with the iterative approximation of lipschitz strongly pseudocontractive maps in x

all pairs almost shortest paths let g v e be an unweighted undirected graph on n vertices a simple argument shows that computing all distances in g with an additive one sided error of at most 1 is as hard as boolean matrix multiplication building on recent work of aingworth et al siam j comput 28 1999 pp 1167 1181 we describe an ot min n 3 2 m 1 2 n 7 3 time algorithm apasp2 for computing all distances in g with an additive one sided error of at most 2 algorithm apasp2 is simple easy to implement and faster than the fastest known matrix multiplication algorithm furthermore for every even k 2 we describe an tilde o min n 2 2 k 2 m 2 k 2 n 2 2 3k 2 time algorithm apaspk for computing all distances in g with an additive one sided error of at most k we also give an tilde o n 2 time algorithm bf apasp infty for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices no constant stretch factor was previously achieved in tilde o n 2 time we say that a weighted graph f v e k emulates an unweighted graph g v e if for every u v in v we have delta g u v le delta f u v le delta g u v k we show that every unweighted graph on n vertices has a 2 emulator with tilde o n 3 2 edges and a 4 emulator with tilde o n 4 3 edges these results are asymptotically tight finally we show that any weighted undirected graph on n vertices has a 3 spanner with tilde o n 3 2 edges and that such a 3 spanner can be built in tilde o mn 1 2 time we also describe an tilde o n m 2 3 n time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3

all pairs shortest paths using bridging sets and rectangular matrix multiplication we present two new algorithms for solving the all pairs shortest paths apsp problem for weighted directed graphs both algorithms use fast matrix multiplication algorithms the first algorithm solves the apsp problem for weighted directed graphs in which the edge weights are integers of small absolute value in o n2 time where satisfies the equation 1 1 1 2 and 1 1 is the exponent of the multiplication of an n n matrix by an n n matrix currently the best available bounds on 1 1 obtained by coppersmith imply that 0 is an error parameter and w is the largest edge weight in the graph after the edge weights are scaled so that the smallest non zero edge weight in the graph is 1 it returns estimates of all the distances in the graph with a stretch of at most 1 e corresponding paths can also be found efficiently

iterative solution of nonlinear equations with strongly accretive operators abstract let e be a real banach space with a uniformly convex dual suppose t e e is a strongly accretive map with bounded range such that for each e the equation tx has a solution in e it is proved that each of the two well known fixed point iteration methods the mann and ishikawa iteration methods under suitable conditions converges strongly to a solution of the equation tx furthermore our method shows that such a solution is necessarily unique explicit error estimates are given our results resolve in the affirmative an open problem j math anal appl 151 no 2 1990 460 and generalize important known results

a distributed enumeration algorithm and applications to all pairs shortest paths diameter we consider the standard message passing model we assume the system is fully synchronous all processes start at the same time and time proceeds in synchronised rounds in each round each vertex can transmit a different message of size o 1 to each of its neighbours this paper proposes and analyses a distributed enumeration algorithm of vertices of a graph having a distinguished vertex which satisfies that two vertices with consecutive numbers are at distance at most 3 we prove that its time complexity is o n where n is the number of vertices of the graph furthermore the size of each message is o 1 thus its bit complexity is also o n we provide some links between this enumeration and hamiltonian graphs from which we deduce that this enumeration is optimal in the sense that there does not exist an enumeration which satisfies that two vertices with consecutive numbers are at distance at most 2 we deduce from this enumeration algorithms which compute all pairs shortest paths and the diameter with a time complexity and a bit complexity equal to o n

on rules with existential variables walking the decidability line we consider positive rules in which the conclusion may contain existentially quantified variables which makes reasoning tasks such as conjunctive query answering or entailment undecidable these rules called email protected rules have the same logical form as tuple generating dependencies in databases and as conceptual graph rules the aim of this paper is to provide a clearer picture of the frontier between decidability and non decidability of reasoning with these rules previous known decidable classes were based on forward chaining on the one hand we extend these classes on the other hand we introduce decidable classes based on backward chaining a side result is the definition of a backward mechanism that takes the complex structure of email protected rule conclusions into account we classify all known decidable classes by inclusion then we study the question of whether the union of two decidable classes remains decidable and show that the answer is negative except for one class and a still open case this highlights the interest of studying interactions between rules we give a constructive definition of dependencies between rules and widen the landscape of decidable classes with conditions on rule dependencies and a mixed forward backward chaining mechanism finally we integrate rules with equality and negative constraints to our framework

ishikawa iteration process for nonlinear lipschitz strongly accretive mappings abstract let e lp p 2 and let t e e be a lipschitzian and strongly accretive mapping let s e e be defined by sx tx x it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process x0 e xn 1 1 n xn ns 1 n xn nsxn n 0 converges strongly to the unique solution of tx a related result deals with the iterative approximation of fixed points for lipschitz strongly pseudocontractive mappings in e a consequence of our result gives an affirmative answer to a problem posed by c e chidume j math anal appl 151 no 2 1990 453 461

average and randomized complexity of distributed problems yao proved that in the decision tree model the average complexity of the best deterministic algorithm is a lower bound on the complexity of randomized algorithms that solve the same problem here it is shown that a similar result does not always hold in the common model of distributed computation the model in which all the processors run the same program which may depend on the processors input we therefore construct a new technique that together with yao s method enables us to show that in many cases a similar relationship does hold in the distributed model this relationship enables us to carry over known lower bounds on the complexity of deterministic computations to the realm of randomized computations thus obtaining new results the new technique can also be used for obtaining results concerning algorithms with bounded error

steepest descent method for equilibrium points of nonlinear systems with accretive operators let e be a real normed linear space and let a e 2e be a bounded uniformly continuous strongly accretive multi valued map with nonempty closed values such that the inclusion 0 ax has a solution x e it is proved that both the ishikawa and the mann iteration processes are bounded and converge strongly to x related results deal with the convergence of the iteration processes to fixed points of strongly pseudocontractive maps and the iterative solution of the equation 0 x fx for an accretive map f furthermore the ishikawa and mann iteration methods with errors are discussed and proofs are sketched on how our theorems extend to these methods our method of proof is of independent interest

new lower bound techniques for distributed leader finding and other problems on rings of processors several new lower bounds are derived for deterministic and randomized extrema finding and some other problems on asynchronous non anonymous rings of processors where the ring size n is known in advance to the processors with a new technique using results from extremal graph theory an n log n lower bounds is obtained for the average number of messages for distributed leader finding on rings where the processors know the ring size n and processors take identities from a set i with size as small as cn for any constant c 1 formerly this bound was only known for special values of n and exponential size of i also improvements are made on the constant factor of the n log n bound an elementary but powerful result shows that the same bounds hold for randomized algorithms it is shown the n log n lower bounds can be derived for the expected message complexity for computing and on an input 1n or on an input 0n or xor over all inputs even when processors have unique identities this confirms a conjecture of abrahamson et al 2

on the time and the bit complexity of distributed randomised anonymous ring colouring we present and analyse a very simple randomised distributed vertex colouring algorithm for ring graphs its time complexity is log 2n o logn on average and 2log 2n o logn with probability 1 o n 1 since each message contains one bit we deduce the same values for its bit complexity then we combine this algorithm with another and we obtain a 3 colouring algorithm for ring graphs thanks to an overlapping we obtain once more the same values for the time complexities on average and with probability 1 o n 1 the same results hold for the bit complexity these results are obtained using the mellin transform we establish lower bounds on average and with probability 1 o n 1 for the distributed randomised anonymous ring colouring problem we prove that our algorithms match these lower bounds modulo a negligible additive function negligible with respect to log 2n we assume that the ring is anonymous unique identities are not available to distinguish the processes we only assume that each vertex distinguishes between its neighbours furthermore we do not assume that the size or an upper bound on the size of the ring is known

lower bounds on communication complexity in distributed computer networks the main result of this paper is a general technique for determining lower bounds on the communication complexity of problems on various distributed computer networks this general technique is derived by simulating the general network by a linear array and then using a lower bound on the communication complexity of the problem on the linear array applications of this technique yield optimal bounds on the communication complexity of merging ranking uniqueness and triangle detection problems on a ring of processors nontrivial near optimal lower bounds on the communication complexity of distinctness merging and ranking on meshes and complete binary trees are also derived

characteristic inequalities of uniformly convex and uniformly smooth banach spaces abstract let x be a real banach space with dual x and moduli of convexity and smoothness x e and x respectively for 1 p denotes the duality mapping from x into 2 x with gauge function t p 1 and j p denotes an arbitrary selection for j p let a r 0 0 t is strictly increasing and there exists c 0 such that t c x t 2 and f r 0 0 t is convex nondecreasing and there exists k 0 such that k x it is proved that x is uniformly convex if and only if there is a a such that x y p x p p j p x y x y x y x and x is uniformly smooth if and only if there is a f such that x y p x p p j p x y x y x y x where for given function f f x y is defined by f x y p 0 1 x ty x p t f t y x ty x dt these inequalities which have various applications can be regarded as general banach space versions of the well known polarization identity occurring in hilbert spaces

viscosity approximation methods for pseudocontractive mappings in banach spaces abstract let k be a closed convex subset of a banach space e and let t k e be a continuous weakly inward pseudocontractive mapping then for t 0 1 there exists a sequence yt k satisfying yt 1 t f yt tt yt where f k f k k a contraction with a suitable contractive constant suppose further that f t and e is reflexive and strictly convex which has uniformly g teaux differentiable norm then it is proved that yt converges strongly to a fixed point of t which is also a solution of certain variational inequality moreover an explicit iteration process which converges strongly to a fixed point of t and hence to a solution of certain variational inequality is constructed provided that t is lipschitzian

homological properties of powers of the maximal ideal of a local ring abstract it is known that the powers m n of the maximal ideal of a local noetherian ring share certain homological properties for all sufficiently large integers n for example the natural homomorphisms r r m n are golod respectively small for all large n we give effective bounds on the smallest integers n for which such properties begin to hold

the equivalence between the convergence of ishikawa and mann iterations with errors for strongly successively pseudocontractive mappings without lipschitzian assumption abstract in this paper we prove some new equivalences between convergence of the ishikawa and mann iteration sequences with errors in two schemes by xu y g xu ishikawa and mann iteration process with errors for nonlinear strongly accretive operator equations j math anal appl 224 1998 91 101 and liu l s liu ishikawa and mann iterative process with errors for nonlinear strongly accretive mappings in banach spaces j math anal appl 194 1995 114 125 respectively for strongly successively pseudocontractive mappings our main results improve and extend the corresponding results of the all references listed in this article

sublinear bounds for randomized leader election this paper concerns randomized leader election in synchronous distributed networks a distributed leader election algorithm is presented for complete n node networks that runs in o 1 rounds and with high probability uses only o n log 3 2 n messages to elect a unique leader with high probability when considering the explicit variant of leader election where eventually every node knows the identity of the leader our algorithm yields the asymptotically optimal bounds of o 1 rounds and o n messages this algorithm is then extended to one solving leader election on any connected non bipartite n node graph g in o g time and o g n log 3 2 n messages where g is the mixing time of a random walk on g the above result implies highly efficient sublinear running time and messages leader election algorithms for networks with small mixing times such as expanders and hypercubes in contrast previous leader election algorithms had at least linear message complexity even in complete graphs moreover super linear message lower bounds are known for time efficient deterministic leader election algorithms finally we present an almost matching lower bound for randomized leader election showing that n messages are needed for any leader election algorithm that succeeds with probability at least 1 e e for any small constant e 0 we view our results as a step towards understanding the randomized complexity of leader election in distributed networks

steepest descent approximations for accretive operator equations a necessary and sufficient condition is established for the strong convergence of the steepest descent approximation to a solution of equations involving quasi accretive operators defined on a uniformly smooth banach space miramare trieste march 1993

homomorphism preservation theorems the homomorphism preservation theorem h p t a result in classical model theory states that a first order formula is preserved under homomorphisms on all structures finite and infinite if and only if it is equivalent to an existential positive formula answering a long standing question in finite model theory we prove that the h p t remains valid when restricted to finite structures unlike many other classical preservation theorems including the o tarski theorem and lyndon s positivity theorem applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential positive formulas and unions of conjunctive queries a further result of this article strengthens the classical h p t we show that a first order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential positive formula of equal quantifier rank

the price of query rewriting in ontology based data access we give a solution to the succinctness problem for the size of first order rewritings of conjunctive queries in ontology based data access with ontology languages such as owl 2 ql linear datalog and sticky datalog we show that positive existential and nonrecursive datalog rewritings which do not use extra non logical symbols except for intensional predicates in the case of datalog rewritings suffer an exponential blowup in the worst case while first order rewritings can grow superpolynomially unless np p poly we also prove that nonrecursive datalog rewritings are in general exponentially more succinct than positive existential rewritings while first order rewritings can be superpolynomially more succinct than positive existential rewritings on the other hand we construct polynomial size positive existential and nonrecursive datalog rewritings under the assumption that any data instance contains two fixed constants

iterative solution of nonlinear equations of the φ strongly accretive type abstract letq 1 and letebe a realq uniformly smooth banach space lett e ebe a lipschitz strongly accretive operator and suppose the equationtx f f e has a solution it is proved that under suitable conditions on the real sequences n n 0 and n n 0 the iteration process formula converges strongly to the unique solution of the equationtx f a related result deals with the approximation of fixed points of hemicontractive operators a class of operators which is much more general than the important class of strongly pseudocontractive operators

ishikawa and mann iterative processes with errors for nonlinear strongly accretive operator equations abstract the purpose of this paper is to revise the definitions of ishikawa and mann iterative processes with errors by using a new inequality to study the unique solution of the nonlinear strongly accretive operator equationtx fand the convergence problem of ishikawa and mann iterative sequences for strongly pseudo contractive mappings without the lipschitz condition the results presented in this paper improve and extend the corresponding results in 4 5 7 10 12 15 16 in the more general setting in particular the open problem mentioned by chidume in 5 has been given an affirmative answer

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

strong convergence of averaged approximants for lipschitz pseudocontractive maps abstract let t be a lipschitzian pseudocontractive self mapping of a closed convex and bounded subset k of a banach space e which is both uniformly convex and q uniformly smooth such that the set f t of fixed points of t is nonempty then f t is a sunny nonexpansive retract of k if u is the sunny nonexpansive retraction of k onto f t is any point of k and an n 0 a real sequence in 0 1 then the sequence xn n 0 in k defined by x n a n 1 a n 1 n 1 j 0 n 1 a j i a j t x n for n 0 1 2 converges strongly to u no compactness assumption is made on k

equivalency of convergence between one step iteration algorithm and two step iteration algorithm of variational inclusions for h monotone mappings the aim of this paper is to establish the equivalence of convergence for h monotone mappings of variational inclusions between the one step iteration algorithm of the paper y p fang n j huang h monotone opertor and resolvent operator technique for variational inclusions appl math comput 145 2003 795 803 and the two step iteration algorithm of the paper l c zeng s m guu j c yao characterization of h monotone operators with applications to variational inclusions comput math appl 50 2005 329 337 it is clear from this paper that the one step iteration algorithm with easy implement and small computational workload has more advantages than the two step iteration algorithm

iterative process with errors for fixed points of multivalued φ hemicontractive operators in uniformly smooth banach spaces abstract let e be a uniformly smooth banach space and t e 2e be a not necessarily continuous multivalued o hemicontractive mapping this paper proves that under suitable conditions the multivalued ishikawa iterative sequence with errors strongly converges to the unique fixed point of t the related result deals with the strong convergence of the ishikawa iterative sequence with errors to the unique solution of the equation f tx when t e 2e is multivalued o strongly accretive these results generalize the results of the author 1 ding 2 3 osilike 4 5 zhou 6 7 and chidume 8 9 into more general multivalued o hemicontractive operators without the continuous assumption on operators t