average case intractability vs worst case intractability we show that not all sets in np or other levels of the polynomial time hierarchy have efficient average case algorithms unless the arthur merlin classes ma and am can be derandomized to np and various subclasses of p poly collapse to p furthermore other complexity classes like p pp and pspace are shown to be intractable on average unless they are easy in the worst case

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

strong convergence of a general iterative method for variational inequality problems and fixed point problems in hilbert spaces abstract in this paper we introduce a new iterative scheme to investigate the problem of finding a common element of the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods our results improve and extend the recent ones announced by chen et al j m chen l j zhang t g fan viscosity approximation methods for nonexpansive mappings and monotone mappings doi 10 1016 j jmaa 2006 12 088 iiduka and tahakshi h iiduka w takahashi strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings nonlinear anal 61 2005 341 350 yao and yao y h yao j c yao on modified iterative method for nonexpansive mappings and monotone mappings appl math comput doi 10 1016 j amc 2006 08 062 and many others

strong convergence of modified mann iterations abstract the mann iterations for nonexpansive mappings have only weak convergence even in a hilbert space we propose two modifications of the mann iterations in a uniformly smooth banach space one for nonexpansive mappings and the other for the resolvent of accretive operators the two modified mann iterations are proved to have strong convergence

locating p poly optimally in the extended low hierarchy the low hierarchy in np and the extended low hierarchy have turned out to be very useful in classifying many interesting language classes and almost all of them could be located optimally therein however until now the exact location of p poly remained open

on hard instances instance complexity was introduced by orponen ko schoning and watanabe 1994 as a measure of the complexity of individual instances of a decision problem comparing instance complexity to kolmogorov complexity they introduced the notion of p hard instances and conjectured that every set not in p has p hard instances whereas this conjecture is still unsettled fortnow and kummer 6 proved that np hard sets have p hard instances unless p np the unbounded version of the conjecture was proven wrong by kummer 1995 we introduce a slightly weaker notion of hard instances in the unbounded version we characterize the classes of recursive enumerable resp recursive sets by hard instances in bounded versions we characterize the class p hard instances are shown to be stronger than complexity cores introduced by lynch 1975 nevertheless np hard sets must have super polynomially dense hard instances unless p np

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

on relativized exponential and probabilistic complexity classes an oracle x is constructed such that the exponential complexity class ep x 2 equals the probabilistic class r r x this shows that it will be difficult to prove that ep 2 is different from r r although it seems very unlikely that these two classes are equal the result subsumes several known results about relativized computations i the existence of relativized polynomial hierarchies extending two levels long t 1978 dissertation purdue univ lafayette ind heller h 1984 a siam j comput 13 717 725 heller h 1984 b math systems theory 17 71 84 ii the existence of an oracle x such that bpp x p x 2 stockmeyer l 1983 proc 15th stoc pp 118 126 iii the existence of an oracle x such that np x is polynomially turing reducible to a sparse set wilson c 1983 proc 24th focs pp 329 334 immerman n and mahaney s 1983 conference on computational complexity theory santa barbara march 21 25 the result shows possible inclusion relations for nonrelativized complexity classes and points out that certain results about probabilistic complexity classes and about polynomial size circuits cannot be improved unless methods are applied which do not relativize

strong convergence and certain control conditions for modified mann iteration abstract in this paper we propose a new modified mann iteration for computing fixed points of nonexpansive mappings in a banach space setting this new iterative scheme combines the modified mann iteration introduced by kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 and the viscosity approximation method introduced by moudafi a moudafi viscosity approximation methods for fixed points problems j math anal appl 241 2000 46 55 we give certain different control conditions for the modified mann iteration strong convergence in a uniformly smooth banach space is established

on sparse hard sets for counting classes abstract in this paper we study one word decreasing self reducible sets which are introduced by lozano and toran 1991 these are the usual self reducible sets with the peculiarity that the self reducibility machine makes at most one query and this is lexicographically smaller than the input we show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one word decreasing self reducible using this fact we can prove that for any class k chosen from pp np c p mod2p mod3 p it holds that 1 if there is a sparse pbtt hard set for k then k p and 2 if there is a sparse snbtt hard set for k then k np co np this generalizes the result of ogiwara and watanabe 1991 to the mentioned complexity classes

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

random self reducibility of complete sets this paper generalizes the previous formal definitions of random self reducibility it is shown that even under a very general definition sets that are complete for any level of the polynomial hierarchy are not nonadaptively random self reducible unless the hierarchy collapses in particular np complete sets are not nonadaptively random self reducible unless the hierarchy collapses at the third level by contrast we show that sets complete for the classes pp and text mod m text p are random self reducible

sources sinks and wavenumber selection in coupled cgl equations and experimental implications for counter propagating wave systems we study the coupled complex ginzburg landau cgl equations for traveling wave systems and show that sources and sinks are the important coherent structures that organize much of the dynamical properties of traveling wave systems we focus on the regime in which sources and sinks separate patches of left and right traveling waves i e the case that these modes suppress each other we present in detail the framework to analyze these coherent structures and show that the theory predicts a number of general properties which can be tested directly in experiments our counting arguments for the multiplicities of these structures show that independently of the precise values of the coefficients in the equations there generally exists a symmetric stationary source solution which sends out waves with a unique frequency and wave number sinks on the other hand occur in two parameter families and play an essentially passive role being sandwiched between the sources these simple but general results imply that sources are important in organizing the dynamics of the coupled cgl equations simulations show that the consequences of the wavenumber selection by the sources is reminiscent of a similar selection by spirals in the 2d complex ginzburg landau equations sources can send out stable waves convectively unstable waves or absolutely unstable waves we show that there exists an additional dynamical regime where both single and bimodal states are unstable the ensuing chaotic states have no counterpart in single amplitude equations a third dynamical mechanism is associated with the fact that the width of the sources does not show simple scaling with the growth rate e this is related to the fact that the standard coupled cgl equations are not uniform in e in particular when the group velocity term dominates over the linear growth term no stationary source can exist however sources displaying nontrivial dynamics can often survive here our results for the existence multiplicity wavelength selection dynamics and scaling of sources and sinks and the patterns they generate are easily accessible by experiments we therefore advocate a study of the sources and sinks as a means to probe traveling wave systems and compare theory and experiment in addition they bring up a large number of new research issues and open problems which are listed explicitly in the concluding section

nonuniform proof systems a new framework to describe nonuniform and probabilistic complexity classes the concept of non uniform proof systems is introduced this notion allows a uniform description of non uniform complexity classes 10 probabilistic classes e g bpp 8 15 27 32 am 2 3 and language classes defined by simultaneous non uniform and nondeterministic time bounds non uniform proof systems provide a better understanding of many results concerning these classes particularly their connections to uniform complexity measures we give an uniform approach to lowness results 19 20 for various complexity classes for instance we show that co np poly np is contained in the third level of the low hierarchy and that np c np co np poly implies that the polynomial time hierarchy collapses to its second level see also 1 finally some evidence is given that the low hierarchy cannot be extended beyond its third level by the current techniques

sparse sets lowness and highness we develop the notions of generalized lowness for sets in ph the union of the polynomial time hierarchy and of generalized highness for arbitrary sets also we develop the notions of extended lowness and extended highness for arbitrary sets these notions extend the decomposition of np into low sets and high sets developed by schoning 15 and studied by ko and schoning 9 we show that either every sparse set in ph is generalized high or no sparse set in ph is generalized high further either every sparse set is extended high or no sparse set is extended high in both situations the former case corresponds to the polynomial time hierarchy having only finitely many levels while the latter case corresponds to the polynomial time hierarchy extending infinitely many levels

relations among mod classes abstract mod classes are those subclasses of p p 1 where the deterministic polynomial time computation consists only of checking whether the oracle answer for the input lies in a certain residue class modulo a fixed natural number k for the so defined classes modk p we prove that modk p modm p if k and m have the same prime divisors and as a consequence of this modk p is closed under union for all k

quantitative relativizations of complexity classes consider the following open problems i text p text np ii text np text co np iii text p text pspace iv text np text pspace in this paper we study these four problems from a particular point of view to illustrate our approach consider the first problem it is known that there exist recursive sets a and b such that text p a text np a and text p b ne text np b we study restrictions r on both the deterministic and also the nondeterministic polynomial time bounded oracle machines such that the following holds text p text np if and only if for every set a text p r a text np r a the restrictions are quantitative in the sense that the size of the set of strings queried by the oracle in computations of a machine on an input is bounded by a polynomial in the length of the input we study several different ways of specifying such quantitative restrictions each of which has the desire

strong convergence of krasnoselskii and mann s type sequences for one parameter nonexpansive semigroups without bochner integrals abstract in this paper we prove krasnoselskii and mann s type convergence theorems for nonexpansive semigroups without using bochner integral and without assuming the strict convexity of banach spaces one of our main results is the following let c be a compact convex subset of a banach space e and let t t t 0 be a one parameter strongly continuous semigroup of nonexpansive mappings on c let t n be a sequence in 0 satisfying lim inf n t n lim sup n t n and lim n t n 1 t n 0 let 0 1 define a sequence x n in c by x 1 c and x n 1 t t n x n 1 x n for n n then x n converges strongly to a common fixed point of t t t 0

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

expected computation time for hamiltonian path problem one way to cope with an np hard problem is to find an algorithm that is fact on average with respect to a natural probability distribution on inputs we consider from that point of view the hamilto

new collapse consequences of np having small circuits we show that if a self reducible set has polynomial size circuits then it is low for the probabilistic class zpp np as a consequence we get a deeper collapse of the polynomial time hierarchy ph to zpp np under the assumption that np has polynomial size circuits this improves on the well known result in karp and lipton proceedings of the 12th acm symposium on theory of computing acm press new york 1980 pp 302 309 stating a collapse of ph to its second level sigmap 2 under the same assumption furthermore we derive new collapse consequences under the assumption that complexity classes like up fewp and c p have polynomial size circuits finally we investigate the circuit size complexity of several language classes in particular we show that for every fixed polynomial s there is a set in zpp np which does not have o s n size circuits

computation times of np sets of different densities abstract in this paper we study the computational complexity of sets of different densities in np we show that the deterministic computation time for sets in np can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bounded complexity classes we show also that for np sets of different densities there exist complete sets of the corresponding density under polynomial time turing reductions finally we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems this interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about p np conp and pspace discussed in this paper

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

traveling waves in diffusive predator prey equations periodic orbits and point to periodic heteroclinic orbits we demonstrate the existence of periodic traveling wave train and traveling front solutions for a diffusive predator prey system the analysis is in the three dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator prey system in the traveling wave variable the analysis shows the existence of periodic orbits heteroclinic orbits and a heteroclinic connection of a point and a periodic orbit the proof uses shooting techniques invariant manifold theory and the qualitative theory of ordinary differential equations

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

the complexity of malign measures this paper analyzes the concept of malignness which is the property of probability ensembles making the average case running time equal to the worst case running time for a class of algorithms the author derives lower and upper bounds on the complexity of malign ensembles which are tight for exponential time algorithms and which show that no polynomial time computable malign ensemble exists for the class of polynomial time algorithms furthermore it is shown that for no class of superlinear algorithms a polynomial time samplable malign ensemble exists unless every language in p has an expected polynomial time constructor

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

strong convergence of an iterative algorithm for an infinite family of strict pseudo contractions in banach spaces in this paper we introduce a new w mapping and present an iterative algorithm for an infinite family of strict pseudo contractions strong convergence theorems are proved in banach spaces our results improve and extend the corresponding result announced by cai and hu g cai c hu strong convergence theorems of modified ishikawa iterative process with errors for an infinite family of strict pseudo contractions nonlinear anal 71 12 2009 6044 6053

strong convergence theorems of modified ishikawa iterative process with errors for an infinite family of strict pseudo contractions abstract in this paper we study modified ishikawa iterative process with errors to have strong convergence for an infinite family of strict pseudo contractions in the framework of q uniformly smooth banach spaces our results improve and extend the recent ones announced by many others

cook versus karp levin separating completeness notions if np is not small under the hypothesis that np does not have p measure 0 roughly that np contains more than a negligible subset of exponential time it is shown that there is a language that is t p complete cook complete but not m p complete karp levin complete for np this conclusion widely believed to be true is not known to follow from p np or other traditional complexity theoretic hypotheses

numerical continuation methods for studying periodic travelling wave wavetrain solutions of partial differential equations abstract periodic travelling waves wavetrains are an important solution type for many partial differential equations in this paper i review the use of numerical continuation for studying these solutions i discuss the calculation of the form and stability of a given periodic travelling wave and the calculation of boundaries in a two dimensional parameter plane for wave existence and stability i also describe the automated implementation of these numerical continuation procedures via the software package wavetrain http www ma hw ac uk wavetrain i conclude by discussing ongoing work on numerical continuation methods for determining the absolute stability of periodic travelling waves

convergence of a general iterative method for nonexpansive mappings in hilbert spaces in this paper we introduce a modified ishikawa iterative process for approximating a fixed point of nonexpansive mappings in hilbert spaces we establish some strong convergence theorems of the general iteration scheme under some mild conditions the results improve and extend the corresponding results of many others

robustness of probabilistic computational complexity classes under definitional perturbations various types of probabilistic algorithms play an increasingly important role in computer science especially in computational complexity theory probabilistic polynomial time complexity classes are defined and compared to each other emphasizing some structural relationships to the known complexity classes p np pspace the classes r and zpp corresponding to the so called las vegas polynomial time bounded algorithms are given special attention it is shown that many definitions that arise naturally from different types of algorithms are equivalent in defining the same class r or zpp resp these robustness results justify finally the tractability of the above probabilistic classes

relative to a random oracle a bf p a ne bf np a ne text co bf np a with probability 1 let a be a language chosen randomly by tossing a fair coin for each string x to determine whether x belongs to a with probability 1 each of the relativized classes textbf logspace a bf p a bf np a bf pp a and textbf pspace a is properly contained in the next also bf np a ne text co bf np a with probability 1 by contrast with probability 1 the class bf p a coincides with the class bf bpp a of languages recognized by probabilistic oracle machines with error probability uniformly bounded below tfrac 1 2 bf np a is shown with probability 1 to contain a bf p a immune set i e a set having no infinite subset in bf p a the relationship of bf p a immunity to p sparseness and bf np a completeness is briefly discussed bf p a immune sets in bf np a can be sparse or moderately dense but not co sparse relativization with respect to a random length preserving permutation pi instead of a rand

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

on polynomial time bounded truth table reducibility of np sets to sparse sets it is proved that if text p ne text np then there exists a set in text np that is not polynomial time bounded truth table reducible in short leqq text btt text p reducible to any sparse set in other words it is proved that no sparse leqq text btt text p hard set exists for text np unless text p text np by using the technique proving this result the intractability of several number theoretic decision problems i e decision problems defined naturally from number theoretic problems is investigated it is shown that for these number theoretic decision problems if it is not in text p then it is not leqq text btt text p reducible to any sparse set

wave train selection behind invasion fronts in reaction diffusion predator prey models abstract wave trains or periodic travelling waves can evolve behind invasion fronts in oscillatory reaction diffusion models for predator prey systems although there is a one parameter family of possible wave train solutions in a particular predator invasion a single member of this family is selected sherratt 1998 13 has predicted this wave train selection using a system that is a valid approximation near a supercritical hopf bifurcation in the corresponding kinetics and when the predator and prey diffusion coefficients are nearly equal away from a hopf bifurcation or if the diffusion coefficients differ somewhat these predictions lose accuracy we develop a more general wave train selection prediction for a two component reaction diffusion predator prey system that depends on linearizations at the unstable homogeneous steady states involved in the invasion front this prediction retains accuracy farther away from a hopf bifurcation and can also be applied when the predator and prey diffusion coefficients are unequal we illustrate the selection prediction with its application to three models of predator invasions

random generation of combinatorial structures from a uniform distribution abstract the class of problems involving the random generation of combinatorial structures from a uniform distribution is considered uniform generation problems are in computational difficulty intermediate between classical existence and counting problems it is shown that exactly uniform generation of efficiently verifiable combinatorial structures is reducible to approximate counting and hence is within the third level of the polynomial hierarchy natural combinatorial problems are presented which exhibit complexity gaps between their existence and generation and between their generation and counting versions it is further shown that for self reducible problems almost uniform generation and randomized approximate counting are inter reducible and hence of similar complexity

approximation of common fixed points of an infinite family of nonexpansive mappings in banach spaces let c be a closed convex subset of a real uniformly smooth and strictly convex banach space e consider the following iterative algorithm given by x 0 x carbitrarily chosen y n b nx n 1 b n w nx n x n 1 a nf x n 1 a n y n n 0 where f is a contraction on c and w n is a mapping generated by an infinite family of nonexpansive mappings t i i 1 assume that the set of common fixed points of this infinite family of nonexpansive mappings is not empty in this paper we prove that the sequence x n generated by the above iterative algorithm converges strongly to a common fixed point of t i i 1 which solves some variational inequality our results improve and extend the results announced by many others

invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave abstract periodic wave trains are the generic solution from for oscillatory reaction diffusion equations in one space dimension it has been shown previously that invasive wavefronts generate behind them a wave train with a different speed from that of the invasion sherratt physica d 70 1994 370 382 in this paper the mechanism of wave train generation is studied in detail for systems of two reaction diffusion equations close to a supercritical hopf bifurcation in the kinetics with equal diffusion coefficients a combination of analytical and numerical evidence is presented suggesting that the invasive front and wave train are separated by a modulated travelling wave of phase gradient in which phase singularities occur periodically this calculation leads to a prediction of the amplitude and speed of the wave train generated by invasion

average case complete problems many interesting combinatorial problems were found to be np complete since there is little hope to solve them fast in the worst case researchers look for algorithms which are fast just on average this matter is sensitive to the choice of a particular np complete problem and a probability distribution of its instances some of these tasks are easy and some not but one needs a way to distinguish the difficult on average problems such negative results could not only save positive efforts but may also be used in areas like cryptography where hardness of some problems is a frequent assumption it is shown in 1 that the tiling problem with uniform distribution of instances has no polynominal on average algorithm unless every np problem with every simple probability distribution has it

viscosity approximation methods for fixed points problems the aim of this work is to propose viscosity approximation methods which amounts to selecting a particular fixed point of a given nonexpansive self mapping

circuit size lower bounds and non reducibility to sparse sets as remarked in cook towards a complexity theory of synchronous parallel computation univ of toronto 1980 a nonlinear lower bound on the circuit size of a language in p or even in np is not known the best known published lower bound seems to be due to paul proceedings 7th acm symposium on theory of computing 1975 in this paper it is shown first that for each nonnegative integer k there is a language l k in 2 2 of the meyer and stockmeyer proceedings 13th ieee symposium on switching and automata theory 1972 hierarchy which does not have o n k size circuits using the same techniques one is able to prove several similar results for example it is shown that for each nonnegative integer k there is a language l k in np that does not have o n k size uniform circuits this follows as a corollary of a stronger result shown in the paper this result like the others to follow is not provable by direct diagonalization it thus points to the most interesting feature of the techniques used hereby using the polynomial time hierarchy they are able to prove results about np that cannot seem to proved by direct diagonalization finally it is noted that existence of small circuits is in suitable contexts equivalent to being reducible to sparse sets using this one is able to prove for example that for any time constructible superpolynomial function f n ntime f n contains a language which is not many to one p time reducible to any sparse set

the shrinking property for np and conp we study the shrinking and separation properties two notions well known in descriptive set theory for np and conp and show that under reasonable complexity theoretic assumptions both properties do not hold for np and the shrinking property does not hold for conp in particular we obtain the following results 1 np and conp do not have the shrinking property unless ph is finite in general s n p and p n p do not have the shrinking property unless ph is finite this solves an open question posed by selivanov 1994 33 2 the separation property does not hold for np unless email protected conp 3 the shrinking property does not hold for np unless there exist np hard disjoint np pairs existence of such pairs would contradict a conjecture of even et al 1984 6 4 the shrinking property does not hold for np unless there exist complete disjoint np pairs moreover we prove that the assumption np conp is too weak to refute the shrinking property for np in a relativizable way for this we construct an oracle relative to which email protected conp np conp and np has the shrinking property this solves an open question posed by blass and gurevich 1984 3 who explicitly ask for such an oracle

computing absolute and essential spectra using continuation a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented the advantage of this approach compared with direct eigenvalue computations for the discretized operator are the ecient and accurate computation of selected parts of the spectrum typically those near the imaginary axis and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously in order to locate accurately the onset to instability we also discuss the implementation and usage of this approach with the software package auto and provide example computations for the fitzhugh nagumo and the complex ginzburg landau equation

convergence theorems for strict pseudo contractions in q uniformly smooth banach spaces abstract in this paper we discuss some properties of iterates generated by a strict pseudo contraction or a finite family of strict pseudo contractions in a real q uniformly smooth banach space we prove the weak convergence and strong convergence of pseudo contractive mappings the results presented in this paper are interesting extensions and improvements upon those known ones of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 349 acedo and xu g l acedo h k xu iterative methods for strict pseudo contractions in hilbert spaces nonlinear anal 67 2007 2258 2271 and zhou h zhou convergence theorems for strict pseudo contractions in 2 uniformly smooth banach spaces nonlinear anal 69 2008 3160 3173

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

strong nondeterministic polynomial time reducibilities abstract building on the work of adleman and manders 1 we define a new class of reducibilities the strong nondeterministic polynomial time reducibilities we show that several reducibilities in this class differ over the recursive sets and assuming np conp that they each determine different classes of np hard sets comparisons in strength are also made between the strong nondeterministic polynomial time reducibilities and the classes of deterministic and non deterministic polynomial time reducibilities in ladner lynch and selman 6

convergence theorems for λ strict pseudo contractions in 2 uniformly smooth banach spaces abstract in this paper we discuss some properties of iterates generated by a strict pseudo contraction or a finite family of strict pseudo contractions in a real 2 uniformly smooth banach space the results presented in this paper are interesting extensions and improvements upon those known ones of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 324 2007 336 349 in order to get a strong convergence theorem we modify the normal mann s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in c this result extends a recent result of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 both from nonexpansive mappings to strict pseudo contractions and from hilbert spaces to 2 uniformly smooth banach spaces

the effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction diffusion systems abstract many oscillatory biological systems show periodic travelling waves these are often modelled using coupled reaction diffusion equations however the effects of different movement rates diffusion coefficients of the interacting components on the predictions of these equations are largely unknown here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed time period wavelength amplitude and stability of periodic wave solutions we focus on two sets of kinetics that are commonly used in ecological applications lambda omega equations which are the normal form of an oscillatory coupled reaction diffusion system close to a supercritical hopf bifurcation and a standard predator prey model our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one parameter family of periodic travelling wave solutions the position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients in all cases stability changes through an eckhaus sideband instability these effects are always symmetrical in the two diffusion coefficients for the lambda omega equations but are asymmetric in the predator prey equations especially when the limit cycle of the kinetics is of large amplitude in particular there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator prey scenario our results also show the existence of a one parameter family of travelling waves but not necessarily a hopf bifurcation for all values of the diffusion coefficients simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms and our analysis of the travelling wave families assists in the understanding of these effects

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

towards efficient universal planning a randomized approach one of the most widespread approaches to reactive planning is schoppers universal plans we propose a stricter definition of universal plans which guarantees a weak notion of soundness not present in the original definition and isolate three different types of completeness that capture different behaviors exhibited by universal plans we show that universal plans which run in polynomial time and are of polynomial size cannot satisfy even the weakest type of completeness unless the polynomial hierarchy collapses by relaxing either the polynomial time or the polynomial space requirement the construction of universal plans satisfying the strongest type of completeness becomes trivial as an alternative approach we study randomized universal planning by considering a randomized version of completeness and a restricted but nontrivial class of problems we show that there exists randomized universal plans running in polynomial time and using polynomial space which are sound and complete for the restricted class of problems we also report experimental results on this approach to planning showing that the performance of a randomized planner is not easily compared to that of a deterministic planner

the approximation of fixed points of compositions of nonexpansive mappings in hilbert space determining fixed points of nonexpansive mappings is a frequent problem in mathematics and physical sciences an algorithm for finding common fixed points of nonexpansive mappings in hilbert space essentially due to halpern is analyzed the main theorem extends wittmann s recent work and partially generalizes a result by lions algorithms of this kind have been applied to the convex feasibility problem

complexity classes of equivalence problems revisited to determine if two lists of numbers are the same set we sort both lists and see if we get the same result the sorted list is a canonical form for the equivalence relation of set equality other canonical forms arise in graph isomorphism algorithms to determine if two graphs are cospectral have the same eigenvalues we compute their characteristic polynomials and see if they are equal the characteristic polynomial is a complete invariant for cospectrality finally an equivalence relation may be decidable in p without either a complete invariant or canonical form blass and gurevich 1984 asked whether these conditions on equivalence relations having an fp canonical form having an fp complete invariant and being in p are distinct they showed that this question requires non relativizing techniques to resolve we extend their results and give new connections to probabilistic and quantum computation

on pseudorandomness and resource bounded measure abstract in this paper we extend a key result of nisan and wigderson j comput system sci 49 1994 149 167 to the nondeterministic setting for all 0 we show that if there is a language in e dtime 2 o n that is hard to approximate by nondeterministic circuits of size 2 n then there is a pseudorandom generator that can be used to derandomize bp np in symbols bp np np by applying this extension we are able to answer some open questions in lutz theory comput systems 30 1997 429 442 regarding the derandomization of the classes bp p k and bp p k under plausible measure theoretic assumptions as a consequence if p 2 does not have p measure 0 then am coam is low for p 2 thus in this case the graph isomorphism problem is low for p 2 by using the nisan wigderson design of a pseudorandom generator we unconditionally show the inclusion ma zpp np and that ma coma is low for zpp np