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

one way communication complexity and the neciporuk lower bound on formula size in this paper the neciporuk method for proving lower bounds on the size of boolean formulas is reformulated in terms of one way communication complexity we investigate the settings of probabilistic formulas nondeterministic formulas and quantum formulas in all cases we can use results about one way communication complexity to prove lower bounds on formula size the main results regarding formula size are as follows we show a polynomial size gap between probabilistic quantum and deterministic formulas a near quadratic gap between the sizes of nondeterministic formulas with limited access to nondeterministic bits and nondeterministic formulas with access to slightly more such bits and a near quadratic lower bound on quantum formula size furthermore we give a polynomial separation between the sizes of quantum formulas with and without multiple read random inputs the lower bound methods for quantum and probabilistic formulas employ a variant of the neciporuk bound in terms of the vapnik chervonenkis dimension to establish our lower bounds we show optimal separations between one way and two way protocols for limited nondeterministic and quantum communication complexity and we show that zero error quantum one way communication complexity asymptotically equals deterministic one way communication complexity for total functions

interaction in quantum communication in some scenarios there are ways of conveying information with many fewer even exponentially fewer qubits than possible classically moreover some of these methods have a very simple structure they involve only few message exchanges between the communicating parties it is therefore natural to ask whether every classical protocol may be transformed to a simpler quantum protocol one that has similar efficiency but uses fewer message exchanges we show that for any constant k there is a problem such that its k 1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity this in particular proves a round hierarchy theorem for quantum communication complexity and implies via a simple reduction an omega n1k lower bound for k message quantum protocols for set disjointness for constant k enroute we prove information theoretic lemmas and define a related measure of correlation the informational distance that we believe may be of significance in other contexts as well

robust characterizations of polynomials withapplications to program testing the study of self testing and self correcting programs leads to the search for robust characterizations of functions here the authors make this notion precise and show such a characterization for polynomials from this characterization the authors get the following applications simple and efficient self testers for polynomial functions are constructed the characterizations provide results in the area of coding theory by giving extremely fast and efficient error detecting schemes for some well known codes this error detection scheme plays a crucial role in subsequent results on the hardness of approximating some np optimization problems

new limits to classical and quantum instance compression given an instance of a hard decision problem a limited goal is to compress that instance into a smaller equivalent instance of a second problem as one example consider the problem where given

sbil nikov hopf bifurcation abstract we derive a 2 parameter poincare map modeling a 3 dimensional vector flow near the codimension 2 point defined by a sil nikov homoclinic orbit to a fixed point undergoing a hopf bifurcation in one parameter regime the map reproduces glendinning and sparrow s results for the sil nikov bifurcation while in another regime it provides new results for a small hopf cycle with a global reinjection mechanism the map is analyzed finding primary homoclinic tangencies to the small hopf cycle parameter dependence of periodic orbits associated with these tangencies is described global behavior of the tube like unstable manifold of the hopf cycle is studied in depth analytically and numerically locating its simplest subsidiary tangencies

the space complexity of recognizing well parenthesized expressions in the streaming model the index function revisited we show an p n t lower bound for the space required by any unidirectional constanterror randomized t pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well parenthesized this proves a conjecture due to magniez mathieu and nayak 2009 and rigorously establishes the peculiar power of bi directional streams over unidirectional ones observed in the algorithms they present the lower bound is obtained by analysing the information that is necessarily revealed by the players about their respective inputs in a two party communication protocol for a variant of the index function

ip pspace using error correcting codes the mathbf ip theorem which asserts that mathbf ip mathbf pspace lund et al j acm 39 1992 pp 859 868 shamir j acm 39 1992 pp 878 880 is one of the major achievements of complexity theory the known proofs of the theorem are based on the arithmetization technique which transforms a quantified boolean formula into a related polynomial the intuition that underlies the use of polynomials is commonly explained by the fact that polynomials constitute good error correcting codes however the known proofs seem tailored to the use of polynomials and do not generalize to arbitrary error correcting codes in this work we show that the mathbf ip theorem can be proved by using general error correcting codes and their tensor products we believe that this establishes a rigorous basis for the aforementioned intuition and sheds further light on the mathbf ip theorem

undirected connectivity in log space we present a deterministic log space algorithm that solves st connectivity in undirected graphs the previous bound on the space complexity of undirected st connectivity was log4 3 obtained by armoni ta shma wigderson and zhou jacm 2000 as undirected st connectivity is complete for the class of problems solvable by symmetric nondeterministic log space computations the class sl this algorithm implies that sl e l where l is the class of problems solvable by deterministic log space computations independent of our work and using different techniques trifonov stoc 2005 has presented an o log n log log n space deterministic algorithm for undirected st connectivity our algorithm also implies a way to construct in log space a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph specifically we give log space constructible universal traversal sequences for graphs with restricted labeling and log space constructible universal exploration sequences for general graphs

property testing and its connection to learning and approximation in this paper we consider the question of determining whether a function f has property p or is e far from any function with property p a property testing algorithm is given a sample of the value of f on instances drawn according to some distribution in some cases it is also allowed to query f on instances of its choice we study this question for different properties and establish some connections to problems in learning theory and approximation in particular we focus our attention on testing graph properties given access to a graph g in the form of being able to query whether an edge exists or not between a pair of vertices we devise algorithms to test whether the underlying graph has properties such as being bipartite k colorable or having a p clique clique of density p with respect to the vertex set our graph property testing algorithms are probabilistic and make assertions that are correct with high probability while making a number of queries that is independent of the size of the graph moreover the property testing algorithms can be used to efficiently i e in time linear in the number of vertices construct partitions of the graph that correspond to the property being tested if it holds for the input graph

attractors in a sbil nikov hopf scenario and a related one dimensional map abstract we study a situation close to the sil nikov hopf bifurcation placing special emphasis on the attractors a model is proposed for the return map to a suitable surface which explains some numerical results concerning a particular physical system the model is quite general and can give reasonable explanations to many other examples later on we introduce a circle map as an additional simplification if the return map is strongly dissipative near the attractor this circle map has very interesting properties it plays a role similar to the logistic map in the case of maps on the interval however it doesn t have the disadvantage of escapes to infinity produced in the logistic map when the parameter is large enough

on the observability of time discrete conservative linear systems abstract we consider various time discretization schemes of abstract conservative evolution equations of the form z a z where a is a skew adjoint operator we analyze the problem of observability through an operator b more precisely we assume that the pair a b is exactly observable for the continuous model and we derive uniform observability inequalities for suitable time discretization schemes within the class of conveniently filtered initial data the method we use is mainly based on the resolvent estimate given by burq and zworski in n burq m zworski geometric control in the presence of a black box j amer math soc 17 2 2004 443 471 electronic we present some applications of our results to time discrete schemes for wave schrodinger and kdv equations and fully discrete approximation schemes for wave equations

designing fpt algorithms for cut problems using randomized contractions we introduce a new technique for designing fixed parameter algorithms for cut problems called randomized contractions we apply our framework to obtain the first fixed parameter algorithms fpt algorithms with exponential speed up for the steiner cut and node multiway cut uncut problems we prove that the parameterized version of the unique label cover problem which is the base of the unique games conjecture can be solved in 2 o k 2 log sigma n 4 log n deterministic time even in the stronger vertex deletion variant where k is the number of unsatisfied edges and sigma is the size of the alphabet as a consequence we show that one can in polynomial time solve instances of unique games where the number of edges allowed not to be satisfied is upper bounded by o sqrt log n to optimality which improves over the trivial o 1 upper bound we prove that the steiner cut problem can be solved in 2 o k 2 log k n 4 log n deterministic time and tilde o 2 o k 2 log k n 2 random

tight lower bounds for certain parameterized np hard problems based on the framework of parameterized complexity theory we derive tight lower bounds on the computational complexity for a number of well known np hard problems we start by proving a general result namely that the parameterized weighted satisfiability problem on depth t circuits cannot be solved in time n o k m o 1 where n is the circuit input length m is the circuit size and k is the parameter unless the t 1 st level w t 1 of the w hierarchy collapses to fpt by refining this technique we prove that a group of parameterized np hard problems including weighted sat hitting set set cover and feature set cannot be solved in time n o k m o 1 where n is the size of the universal set from which the k elements are to be selected and m is the instance size unless the first level w 1 of the w hierarchy collapses to fpt we also prove that another group of parameterized problems which includes weightedq sat for any fixed q 2 clique independent set and dominating set cannot be solved in time n o k unless all search problems in the syntactic class snp introduced by papadimitriou and yannakakis are solvable in subexponential time note that all these parameterized problems have trivial algorithms of running time either n km o 1 or o n k

the learnability of quantum states traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n but using ideas from computational learning theory we show that one can do exponentially better in a statistical setting in particular to predict the outcomes of most measurements drawn from an arbitrary probability distribution one needs only a number of sample measurements that grows linearly with n this theorem has the conceptual implication that quantum states despite being exponentially long vectors are nevertheless reasonable in a learning theory sense the theorem also has two applications to quantum computing first a new simulation of quantum one way communication protocols and second the use of trusted classical advice to verify untrusted quantum advice

uniform controllability properties for space time discretized parabolic equations this article is concerned with the analysis of semi discrete in space and fully discrete approximations of the null controllability and controllability to the trajectories for parabolic equations we propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters for time discretization we use schemes with theta in frac 1 2 1 for the proofs of controllability we rely on the strategy introduced by lebeau and robbiano comm partial differ equ 20 335 356 1995 for the null controllability of the heat equation which is based on a spectral inequality we obtain relaxed uniform observability estimates in both the semi discrete and fully discrete frameworks and associated uniform controllability properties for the practical computation of the control functions we follow j l lions hilbert uniqueness method strategy exploiting the relaxed uniform observability estimate algorithms for the computation of the controls are proposed and analysed in the semi discrete and fully discrete cases additionally we prove an error bound between the fully discrete and the semi discrete control functions this bound is however not uniform with respect to the space discretization the theoretical results are illustrated through numerical experimentations

discrete carleman estimates for elliptic operators in arbitrary dimension and applications in arbitrary dimension we consider the semidiscrete elliptic operator partial t 2 mathcal a scriptscriptstyle mathfrak m where mathcal a scriptscriptstyle mathfrak m is a finite difference approximation of the operator nabla x gamma x nabla x for this operator we derive a global carleman estimate in which the usual large parameter is connected to the discretization step size we address discretizations on some families of smoothly varying meshes we present consequences of this estimate such as a partial spectral inequality of the form of that proven by g lebeau and l robbiano for mathcal a scriptscriptstyle mathfrak m and a null controllability result for the parabolic operator partial t mathcal a scriptscriptstyle mathfrak m for the lower part of the spectrum of mathcal a scriptscriptstyle mathfrak m with the control function that we construct whose norm is uniformly bounded we prove that the l 2 norm of the final state converges to zero exponentially as the step size of the discretization goes to zero a relaxed observability estimate is then deduced

satisfiability allows no nontrivial sparsification unless the polynomial time hierarchy collapses consider the following two player communication process to decide a language l the first player holds the entire input x but is polynomially bounded the second player is computationally unbounded but does not know any part of x their goal is to decide cooperatively whether x belongs to l at small cost where the cost measure is the number of bits of communication from the first player to the second player for any integer d 3 and positive real e we show that if satisfiability for n variable d cnf formulas has a protocol of cost o nd e then conp is in np poly which implies that the polynomial time hierarchy collapses to its third level the result even holds when the first player is conondeterministic and is tight as there exists a trivial protocol for e 0 under the hypothesis that conp is not in np poly our result implies tight lower bounds for parameters of interest in several areas namely sparsification kernelization in parameterized complexity lossy compression and probabilistically checkable proofs by reduction similar results hold for other np complete problems for the vertex cover problem on n vertex d uniform hypergraphs this statement holds for any integer d 2 the case d 2 implies that no np hard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of o k2 e edges unless conp is in np poly where k denotes the size of the deletion set kernels consisting of o k2 edges are known for several problems in the class including vertex cover feedback vertex set and bounded degree deletion

tranversality of the invariant manifolds associated to the lyapunov family of periodic orbits near l2 in the restricted three body problem abstract the restricted three body problem is considered for values of the jacobi constant c near the value c 2 associated to the euler critical point l 2 a lyapunov family of periodic orbits near l 2 the so called family c is born for c c 2 and exists for values of c less than c 2 these periodic orbits are hyperbolic the corresponding invariant manifolds meet transversally along homoclinic orbits in this paper the variation of the transversality is analyzed as a function of the jacobi constant c and of the mass parameter asymptotical expressions of the invariant manifolds for c c 2 and 0 are found several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions symbolic dynamic techniques are used to show the existence of a large class of motions in particular the existence of orbits passing in a random way in a given sense from the region near one primary to the region near the other is proved

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

homoclinic orbits and mixed mode oscillations in far from equilibrium systems nonlinear autonomous dynamical systems with ahomoclinic tangency to a periodic orbit are investigated we study the bifurcation sequences of the mixed mode oscillations generated by the homoclinicity which are shown to belong to two different types depending on the nature of the liapunov numbers of the basic periodic orbit a detailed numerical analysis is carried out to show how the existence of a tangent homoclinic orbit allows us to understand in a quantitative way a particular and regular sequence of cool flame ignition oscillations observed in a thermokinetic model of hydrocarbon oxidation chaotic cool flame oscillations are also observed in the same model when the control parameter crosses a critical value this chaotic set of trajectories becomes globally unstable and forms a cantor like hyperbolic repellor and the ignition mechanism generates ahomoclinic tangency to the cantor set of trajectories the complex bifurcation diagram may be globally reconstructed from a one dimensional dynamical system thanks to the strong contractivity of thermokinetics it is found that a symbolic dynamics with three symbols is necessary to classify the periodic windows of the complex bifurcation sequence observed numerically in this system

exact controllability for multidimensional semilinear hyperbolic equations in this paper we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients for this purpose we establish an observability estimate for the linear hyperbolic equation with an unbounded potential in which the crucial observability constant is estimated explicitly by a function of the norm of the potential such an estimate is obtained by a combination of a pointwise estimate and a global carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem

quantum conditional mutual information and approximate markov chains a state on a tripartite quantum system a otimes b otimes c forms a markov chain if it can be reconstructed from its marginal on a otimes b by a quantum operation from b to b otimes c we show that the quantum conditional mutual information i a c b of an arbitrary state is an upper bound on its distance to the closest reconstructed state it thus quantifies how well the markov chain property is approximated

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

kernel bounds for disjoint cycles and disjoint paths in this paper we show that the problems disjoint cycles and disjoint paths do not have polynomial kernels unless np conp poly thus these problems do not allow polynomial time preprocessing that results in instances whose size is bounded by a polynomial in the parameter at hand we build upon recent results by bodlaender et al 6 and fortnow and santhanam 20 that show that np complete problems that are or compositional do not have polynomial kernels unless np conp poly to this machinery we add a notion of transformation and obtain that disjoint cycles and disjoint paths do not have polynomial kernels unless np conp poly for the proof we introduce a problem on strings called disjoint factors and first show that this problem has no polynomial kernel unless np conp poly we also show that the related disjoint cycles packing problem has a kernel of size o klogk

parameterized complexity of critical node cuts we consider the following graph cut problem called critical node cut cnc given a graph g on n vertices and two positive integers k and x determine whether g has a set of k vertices whose removal leaves g with at most x connected pairs of vertices we analyze this problem in the framework of parameterized complexity that is we are interested in whether or not this problem is solvable in f n o 1 time i e whether or not it is fixed parameter tractable for various natural parameters we consider four such parameters the size k of the required cut the upper bound x on the number of remaining connected pairs the lower bound y on the number of connected pairs to be removed the treewidth w of g we determine whether or not cnc is fixed parameter tractable for each of these parameters we determine this also for all possible aggregations of these four parameters apart from w k moreover we also determine whether or not cnc admits a polynomial kernel for all these parameterizations that is whether or not there is an algorithm that reduces each instance of cnc in polynomial time to an equivalent instance of size o 1 where is the given parameter

list matrix partitions of chordal graphs it is well known that a clique with k 1 vertices is the only minimal obstruction to k colourability of chordal graphs a similar result is known for the existence of a cover by l cliques both of these problems are in fact partition problems restricted to chordal graphs the first seeks partitions into k independent sets and the second is equivalent to finding partitions into l cliques in an earlier paper we proved that a chordal graph can be partitioned into k independent sets and l cliques if and only if it does not contain an induced disjoint union of l 1 cliques of size k 1 a linear time algorithm for finding such partitions can be derived from the proof in this paper we expand our focus and consider more general partitions of chordal graphs for each symmetric matrix m over 0 1 the m partition problem seeks a partition of the input graph into independent sets cliques or arbitrary sets with certain pairs of sets being required to have no edges or to have all edges joining them as encoded in the matrix m moreover the vertices of the input chordal graph can be equipped with lists restricting the parts to which a vertex can be placed such list partitions generalize list colourings and list homomorphisms and arise frequently in the theory of graph perfection we show that many m partition problems that are np complete in general become solvable in polynomial time for chordal graphs even in the presence of lists on the other hand we show that there are m partition problems without lists that remain np complete for chordal graphs it is not known whether or not each list m partition problem is np complete or polynomial but it has been shown that each is np complete or quasi polynomial no log n for chordal graphs even this quasi dichotomy is not known but we do identify large families of matrices m for which dichotomy or at least quasi dichotomy holds we also discuss forbidden subgraph characterizations of graphs admitting an m partition such characterizations have recently been investigated for partitions of perfect graphs and we focus on highlighting the improvements one can obtain for the class of chordal rather than just perfect graphs

multimodal regimes in a compartmental model of the dopamine neuron we study chains of strongly electrically coupled relaxation oscillators modeling dopamine neurons when individual oscillators are in the regime close to an andronov hopf bifurcation ahb the coupled system exhibits a variety of oscillatory behavior we show that the proximity of individual oscillators to the ahb has a significant impact on the system dynamics in a wide range of parameters it manifests itself through a family of stable multimodal periodic solutions that are composed out of large amplitude relaxation oscillations and small amplitude oscillations this family of solutions has a rich bifurcation structure the waveform and the period vary greatly across the family the structure and bifurcations of the stable periodic solutions of the coupled system are investigated using numerical and analytic techniques

divide and conquer approximation algorithms via spreading metrics we present a novel divide and conquer paradigm for approximating np hard graph optimization problems the paradigm models graph optimization problems that satisfy two properties first a divide and conquer approach is applicable second a fractional spreading metric is computable in polynomial time the spreading metric assigns lengths to either edges or vertices of the input graph such that all subgraphs for which the optimization problem is nontrivial have large diameters in addition the spreading metric provides a lower bound t on the cost of solving the optimization problem we present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is o min log t log log t log k log log k where k denotes the number of interesting vertices in the problem instance and is at most the number of vertices we present seven problems that can be formulated to fit the paradigm for all these problems our algorithm improves previous results the problems are 1 linear arrangement 2 embedding a graph in a d dimensional mesh 3 interval graph completion 4 minimizing storage time product 5 subset feedback sets in directed graphs and multicuts in circular networks 6 symmetric multicuts in directed networks 7 balanced partitions and p separators for small values of p in directed graphs

nondeterminism within p classes of machines using very limited amounts of nondeterminism are studied the p np question is related to questions about classes lying within p complete sets for these classes are given

on the parameterized complexity of multiple interval graph problems multiple interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it many applications of interval graphs also generalize to multiple interval graphs often allowing for more robustness in the modeling of the specific application with this motivation in mind a recent systematic study of optimization problems in multiple interval graphs was initiated in this sequel we study multiple interval graph problems from the perspective of parameterized complexity the problems under consideration are k independent set k dominating set and k clique which are all known to be w 1 hard for general graphs and np complete for multiple interval graphs we prove that k clique is in fpt while k independent set and k dominating set are both w 1 hard we also prove that k independent dominating set a hybrid of the two above problems is also w 1 hard our hardness results hold even when each vertex is associated with at most two intervals and all intervals have unit length furthermore as an interesting byproduct of our hardness results we develop a useful technique for showing w 1 hardness via a reduction from the k multicolored clique problem a variant of k clique we believe this technique has interest in its own right as it should help in simplifying w 1 hardness results which are notoriously hard to construct and technically tedious

parameterized graph separation problems we consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible the following five problems are studied delete k vertices such that a each of the given l terminals is separated from the others b each of the given l pairs of terminals is separated c exactly l vertices are cut away from the graph d exactly l connected vertices are cut away from the graph e the graph is separated into at least l components we show that if both k and l are parameters then a b and d are fixed parameter tractable while c and e are w 1 hard

polynomial kernels for hitting forbidden minors under structural parameterizations abstract we investigate polynomial time preprocessing for the problem of hitting forbidden minors in a graph using the framework of kernelization for a fixed finite set of connected graphs f the f deletion problem is the following given a graph g and integer k is it possible to delete k vertices from g to ensure the resulting graph does not contain any graph from f as a minor earlier work by fomin lokshtanov misra and saurabh focs 12 showed that when f contains a planar graph an instance g k can be reduced in polynomial time to an equivalent one of size k o 1 in this work we focus on structural measures of the complexity of an instance with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large motivated by several impossibility results we parameterize the f deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth we prove that for each set f of connected graphs and constant the f deletion problem parameterized by the size of a treedepth modulator has a polynomial kernel our kernelization is fully explicit and does not depend on protrusion reduction or well quasi ordering which are sources of algorithmic non constructivity in earlier works on f deletion our main technical contribution is to analyze how models of a forbidden minor in a graph g with modulator x interact with the various connected components of g x using the language of labeled minors we analyze the fragments of potential forbidden minor models that can remain after removing an optimal f deletion solution from a single connected component of g x by bounding the number of different types of behavior that can occur by a polynomial in x we obtain a polynomial kernel using a recursive preprocessing strategy our results extend earlier work for specific instances of f deletion such as vertex cover and feedback vertex set it also generalizes earlier preprocessing results for f deletion parameterized by a vertex cover which is a treedepth one modulator

stabilizing unstable periodic orbits in the lorenz equations using time delayed feedback control for many years it was believed that an unstable periodic orbit with an odd number of real floquet multipliers greater than unity cannot be stabilized by the time delayed feedback control mechanism of pyragas a recent paper by fiedler et al phys rev lett 98 114101 2007 uses the normal form of a subcritical hopf bifurcation to give a counterexample to this theorem using the lorenz equations as an example we demonstrate that the stabilization mechanism identified by fiedler et al for the hopf normal form can also apply to unstable periodic orbits created by subcritical hopf bifurcations in higher dimensional dynamical systems our analysis focuses on a particular codimension two bifurcation that captures the stabilization mechanism in the hopf normal form example and we show that the same codimension two bifurcation is present in the lorenz equations with appropriately chosen pyragas type time delayed feedback this example suggests a possible strategy for choosing the feedback gain matrix in pyragas control of unstable periodic orbits that arise from a subcritical hopf bifurcation of a stable equilibrium in particular our choice of feedback gain matrix is informed by the fiedler et al example and it works over a broad range of parameters despite the fact that a center manifold reduction of the higher dimensional problem does not lead to their model problem

fixed parameter tractability of multicut parameterized by the size of the cutset given an undirected graph g a collection s 1 t 1 dots s k t k of pairs of vertices and an integer p the edge multicut problem asks if there is a set s of at most p

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

meta kernelization in a parameterized problem every instance i comes with a positive integer k the problem is said to admit a polynomial kernel if in polynomial time one can reduce the size of the instance i to a polynomial in k while preserving the answer in this work we give two meta theorems on kernelization the first theorem says that all problems expressible in counting monadic second order logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus these theorems unify and extend all previously known kernelization results for planar graph problems

multimodal oscillations in systems with strong contraction one and two parameter families of flows in r 3 near an andronov hopf bifurcation ahb are investigated in this work we identify conditions on the global vector field which yield a rich family of multimodal orbits passing close to a weakly unstable saddle focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle focus our analysis covers both cases of sub and supercritical ahb for the supercritical case we find that the periodic orbits born from the ahb are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium if the ahb is subcritical it is accompanied by the appearance of multimodal orbits which consist of long series of nearly harmonic oscillations separated by large amplitude spikes we analyze the dependence of the interspike intervals which can be extremely long on the control parameters in particular we show that the interspike intervals grow logarithmically as the boundary between regions of sub and supercritical ahb is approached in the parameter space we also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub and supercritical ahb and explain the mechanism generating these oscillations this work is motivated by the numerical results for a finite dimensional approximation of a free boundary problem modeling solid fuel combustion c 2007 elsevier b v all rights reserved

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

streaming universal distortion free entanglement concentration this paper presents a streaming sequential protocol for universal entanglement concentration at the shannon bound alice and bob begin with n identical but unknown two qubit pure states each containing e ebits of entanglement they each run a reversible algorithm on their qubits and end up with y perfect epr pairs where y ne o n our protocol is streaming so the n input systems are fed in one at a time and perfect epr pairs start popping out almost immediately it matches the optimal block protocol exactly at each stage so the average yield after n inputs is y ne o logn so somewhat surprisingly there is no tradeoff between yield and lag our protocol optimizes both in contrast the optimal n qubit block protocol achieves the same yield but since no epr pairs are produced until the entire input block is read its lag is o n finally our algorithm runs in o log n space so a lot of entanglement can be efficiently concentrated using a very small e g current or near future technology quantum processor along the way we find an optimal streaming protocol for extracting randomness from classical i i d sources and a more space efficient implementation of the schur transform

constant round interactive proofs for delegating computation the celebrated sf ip sf pspace theorem lund fortnow karloff and nisan j acm 39 1992 pp 859 868 shamir j acm 39 1992 pp 869 877 allows an all powerful but untrusted pro

dense quantum coding and quantum finite automata we consider the possibility of encoding m classical bits into many fewer n quantum bits qubits so that an arbitrary bit from the original m bits can be recovered with good probability we show that nontrivial quantum codes exist that have no classical counterparts on the other hand we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding the proof is based on an entropy coalescence principle that is obtained by viewing holevo s theorem from a new perspective in the existing implementations of quantum computing qubits are a very expensive resource moreover it is difficult to reinitialize existing bits during the computation in particular reinitialization is impossible in nmr quantum computing which is perhaps the most advanced implementation of quantum computing at the moment this motivates the study of quantum computation with restricted memory and no reinitialization that is of quantum finite automata it was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata here we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata

monadic second order evaluations on tree decomposable graphs abstract every graph generated by a hyperedge replacement graph grammar can be represented by a tree namely the derivation tree of the derivation sequence that produced it certain functions on graphs can be computed recursively on the derivation trees of these graphs by using monadic second order logic and semiring homomorphisms we describe in a single formalism a large class of such functions polynomial and even linear algorithms can be constructed for some of these functions we unify similar results obtained by takamizawa 1982 bern 1987 arnborg 1991 and habel 1989

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies a population of oscillatory hodgkin huxley hh model neurons is shown numerically to exhibit a behavior in which the introduction of excitatory synaptic coupling synchronizes and dramatically slows firing this effect contrasts with the standard theory that recurrent synaptic excitation promotes states of rapid sustained activity independent of intrinsic neuronal dynamics the observed behavior is not due to simple depolarization block nor to standard elliptic bursting although it is related to these phenomena we analyze this effect using a reduced model for a single self coupled hh oscillator the mechanism explained here involves an extreme form of delayed bifurcation in which the development of a vortex structure through interaction of fast and slow subsystems pins trajectories near a surface that consists of unstable equilibria of a certain reduced system in a canard like manner using this vortex structure a new passage time calculation is used to approximate the interspike time interval we

representative sets and irrelevant vertices new tools for kernelization we continue the development of matroid based techniques for kernelization initiated by the present authors 47 we significantly extend the usefulness of matroid theory in kernelization by showing applications of a result on representative sets due to lovasz 51 and marx 53 as a first result we show how representative sets can be used to derive a polynomial kernel for the elusive almost 2 sat problem where the task is to remove at most k clauses to make a 2 cnf formula satisfiable solving a major open problem in kernelization this result also yields a new o log opt approximation for the problem improving on the o log n approximation of agarwal et al 3 and an implicit o log opt approximation due to even et al 24 we further apply the representative sets tool to the problem of finding irrelevant vertices in graph cut problems that is vertices that can be made undeletable without affecting the answer to the problem this gives the first significant progress towards a polynomial kernel for the multiway cut problem in particular we get a kernel of o ks 1 vertices for multiway cut instances with at most s terminals both these kernelization results have significant spin off effects producing the first polynomial kernels for a range of related problems more generally the irrelevant vertex results have implications for covering min cuts in graphs for a directed graph g v e and sets s t v let r be the size of a minimum s t vertex cut which may intersect s and t we can find a set z v of size o s t r that contains a minimum a b vertex cut for every a s b t similarly for an undirected graph g v e a set of terminals x v and a constant s we can find a set z v of size o x s 1 that contains a minimum multiway cut for every partition of x into at most s pairwise disjoint subsets both results are polynomial time we expect this to have further applications in particular we get direct reduction rule based kernelizations for all problems above in contrast to the indirect compression based kernel previously given for odd cycle transversal 47 all our results are randomized with failure probabilities that can be made exponentially small in n due to needing a representation of a matroid to apply the representative sets tool

a property of quantum relative entropy with an application to privacy in quantum communication we prove the following information theoretic property about quantum states substate theorem let and be quantum states in the same hilbert space with relative entropy s c tr log log c then for all 0 there is a state such that the trace distance tr tr 2 and 2o c 2 it states that if the relative entropy of and is small then there is a state close to i e with small trace distance tr that when scaled down by a factor 2o c sits inside or becomes a substate of this result has several applications in quantum communication complexity and cryptography using the substate theorem we derive a privacy trade off for the set membership problem in the two party quantum communication model here alice is given a subset a n bob an input i i n and they need to determine if i i a privacy trade off for set membership in any two party quantum communication protocol for the set membership problem if bob reveals only k bits of information about his input then alice must reveal at least n 2o k bits of information about her input we also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem

convergence of an inverse problem for discrete wave equations it is by now well known that one can recover a potential in the wave equation from the knowledge of the initial waves the boundary data and the flux on a part of the boundary satisfying the gamma conditions of j l lions we are interested in proving that trying to fit the discrete fluxes given by discrete approximations of the wave equation with the continuous one one recovers at the limit the potential of the continuous model in order to do that we shall develop a lax type argument usually used for convergence results of numerical schemes which states that consistency and uniform stability imply convergence in our case the most difficult part of the analysis is the one corresponding to the uniform stability that we shall prove using new uniform discrete carleman estimates where uniform means with respect to the discretization parameter we shall then deduce a convergence result for the discrete inverse problems our analysis will be restricted to the 1 d case for space semi discrete wave equations discretized on a uniform mesh using a finite differences approach

locally testable codes and pcps of almost linear length we initiate a systematic study of locally testable codes that is error correcting codes that admit very efficient membership tests specifically these are codes accompanied with tests that make a constant number of random queries into any given word and reject non codewords with probability proportional to their distance from the code locally testable codes are believed to be the combinatorial core of pcps however the relation is less immediate than commonly believed nevertheless we show that certain pcp systems can be modified to yield locally testable codes on the other hand we adapt techniques that we develop for the construction of the latter to yield new pcps our main results are locally testable codes and pcps of almost linear length specifically we prove the existence of the following constructs locally testable binary linear codes in which k information bits are encoded by a codeword of length k exp o log k this improves over previous results that either yield codewords of exponential length or obtained almost quadratic length codewords for sufficiently large nonbinary alphabet pcp systems of almost linear length for sat the length of the proof is n exp o log n and verification in performed by a constant number i e 19 of queries as opposed to previous results that used proof length n 1 p o 1 q for verification by q queries the novel techniques in use include a random projection of certain codewords and pcp oracles that preserves local testability an adaptation of pcp constructions to obtain linear pcp oracles for proving conjunctions of linear conditions and design of pcps with some new soundness properties a direct construction of locally testable linear codes of subexponential length

singular hopf bifurcations and mixed mode oscillations in a two cell inhibitory neural network abstract recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators r curtu a shpiro n rubin j rinzel mechanisms for frequency control in neuronal competition models siam j appl dyn syst 7 2 2008 609 649 showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics several types of behavior were identified such as fusion equivalent to a steady state of identical activity levels for both neural units oscillations due to either an escape or a release mechanism and a winner take all state of bistability the model consists of two neural populations interacting through reciprocal inhibition each endowed with a slow negative feedback process in the form of spike frequency adaptation in this paper we report the occurrence of another complex oscillatory pattern the mixed mode oscillations mmos they exist in the model at the transition between the relaxation oscillator dynamical regime and the winner take all regime the system distinguishes itself from other neuronal models where mmos were found by the following interesting feature there is no autocatalysis involved as in the examples of voltage gated persistent inward currents and or intrapopulation recurrent excitation and therefore the two cells in the network are not intrinsic oscillators the oscillations are instead a combined result of the mutual inhibition and the adaptation we prove that the mmos are due to a singular hopf bifurcation point situated in close distance to the transition point to the winner take all case we also show that in the vicinity of the singular hopf other types of bifurcations exist and we construct numerically the corresponding diagrams

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1

asymptotic analysis of subcritical hopf homoclinic bifurcation abstract this paper discusses the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical hopf bifurcation with a homoclinic orbit of the hopf equilibrium our work is motivated by our previous analysis of a hodgkin huxley neuron model which possesses a subcritical hopf bifurcation j guckenheimer r harris warrick j peck a willms j comput neurosci 4 1997 257 277 in this model the hopf bifurcation has the additional feature that trajectories beginning near the unstable manifold of the equilibrium point return to pass through a small neighborhood of the equilibrium that is the hopf bifurcation appears to be close to a homoclinic bifurcation as well this model of the lateral pyloric lp cell of the lobster stomatogastric ganglion was analyzed for its ability to explain the phenomenon of spike frequency adaptation in which the time intervals between successive spikes grow longer until the cell eventually becomes quiescent the presence of a subcritical hopf bifurcation in this model was the one identified mechanism for oscillatory trajectories to increase their period and finally collapse to a non oscillatory solution the analysis presented here explains the apparent proximity of homoclinic and hopf bifurcations we also develop an asymptotic theory for the scaling properties of the interspike intervals in a singularly perturbed system undergoing subcritical hopf bifurcation that may be close to a codimension two subcritical hopf homoclinic bifurcation

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

robust pcps of proximity shorter pcps and applications to coding we continue the study of the trade off between the length of probabilistically checkable proofs pcps and their query complexity establishing the following main results which refer to proofs of satisfiability of circuits of size n 1 we present pcps of length exp o log log n 2 cdot n that can be verified by making o log log n boolean queries 2 for every epsilon 0 we present pcps of length exp log epsilon n cdot n that can be verified by making a constant number of boolean queries in both cases false assertions are rejected with constant probability which may be set to be arbitrarily close to 1 the multiplicative overhead on the length of the proof introduced by transforming a proof into a probabilistically checkable one is just quasi polylogarithmic in the first case of query complexity o log log n and is 2 log n epsilon for any epsilon 0 in the second case of constant query complexity our techniques include the introduction of a new variant of pcps that we call robust pcps of proximity these new pcps facilitate proof composition which is a central ingredient in the construction of pcp systems a related notion and its composition properties were discovered independently by dinur and reingold our main technical contribution is a construction of a length efficient robust pcp of proximity while the new construction uses many of the standard techniques used in pcp constructions it does differ from previous constructions in fundamental ways and in particular does not use the parallelization step of arora et al j acm 45 1998 pp 501 555 the alternative approach may be of independent interest we also obtain analogous quantitative results for locally testable codes in addition we introduce a relaxed notion of locally decodable codes and present such codes mapping k information bits to codewords of length k 1 epsilon for any epsilon 0

half integrality lp branching and fpt algorithms a recent trend in parameterized algorithms is the application of polytope tools to fixed parameter tractable fpt algorithms e g cygan et al focs 2011 52nd annual symposium on foundations of computer science ieee 2011 pp 150 159 narayanaswamy et al stacs 2012 symposium on theoretical aspects of computer science 2012 pp 338 349 although this approach has yielded significant speedups for a range of important problems it requires the underlying polytope to have very restrictive properties including half integrality and nemhauser trotter style persistence properties to date these properties are essentially known to hold only for two classes of polytopes covering the cases of vertex cover nemhauser and trotter math program 8 1975 pp 232 248 and node multiway cut garg et al j alg 50 2004 pp 49 61 taking a slightly different approach we view half integrality as a discrete relaxation of a problem e g a relaxation of the search space from 0 1 v to 0

on the parameterized complexity of recognizing well covered r l graph an r ell partition of a graph g is a partition of its vertex set into r independent sets and ell cliques a graph is r ell if it admits an r ell partition a graph is well covered if every maximal independent set is also maximum a graph is r ell well covered if it is both r ell and well covered in this paper we consider two different decision problems in the r ell well covered graph problem r ell wcg for short we are given a graph g and the question is whether g is an r ell well covered graph in the well covered r ell graph problem wc r ell g for short we are given an r ell graph g together with an r ell partition of v g into r independent sets and ell cliques and the question is whether g is well covered we classify most of these problems into p conp complete np complete np hard or conp hard only the cases wc r 0 g for r ge 3 remain open in addition we consider the parameterized complexity of these problems for several choices of parameters such as the size alpha of a maximum independent set of the input graph its neighborhood diversity or the number ell of cliques in an r ell partition in particular we show that the parameterized problem of deciding whether a general graph is well covered parameterized by alpha can be reduced to the wc 0 ell g problem parameterized by ell and we prove that this latter problem is in xp but does not admit polynomial kernels unless mathsf conp subseteq mathsf np mathsf poly

explicit observability inequalities for the wave equation with lower order terms by means of carleman inequalities in this paper by means of carleman estimates and the usual energy estimate we obtain directly two observability inequalities for the linear wave equation with time variant nonsmooth lower order terms we do not need any unique continuation property of the linear equation a priori since this is actually one of the by products of our analysis furthermore the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation also we apply our observability estimates to exact controllability for wave equations

does ignorance of the whole imply ignorance of the parts large violations of noncontextuality in quantum theory a central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts one aspect of this question is the following to what extent does ignorance about a whole preclude knowledge of at least one of its parts relying purely on classical intuition one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts indeed we show that this reasoning holds in any noncontextual nc hidden variable model hv curiously however such a conjecture is false in quantum theory we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts more specifically we provide a simple information theoretic inequality satisfied in any nc hv but which can be arbitrarily violated by quantum mechanics

on the compressibility of mathcal np instances and cryptographic applications we study compression that preserves the solution to an instance of a problem rather than preserving the instance itself our focus is on the compressibility of mathcal np decision problems we consider mathcal np problems that have long instances but relatively short witnesses the question is whether one can efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not we want the length of the compressed instance to be polynomial in the length of the witness and polylog in the length of original input such compression enables succinctly storing instances until a future setting will allow solving them either via a technological or algorithmic breakthrough or simply until enough time has elapsed in this paper we first develop the basic complexity theory of compression including reducibility completeness and a stratification of mathcal np with respect to compression we then show that compressibility say of sat would have vast implications for cryptography including constructions of one way functions and collision resistant hash functions from any hard on average problem in mathcal np and cryptanalysis of key agreement protocols in the bounded storage model when mixed with time complexity based cryptography

boundary controllability of a linear semi discrete 1 d wave equation derived from a mixed finite element method in this article one discusses the controllability of a semi discrete system obtained by discretizing in space the linear 1 d wave equation with a boundary control at one extremity it is known that the semi discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero see 8 here we introduce a new semi discrete model based on a mixed finite element method with two different basis functions for the position and velocity we show that the controls obtained with these semi discrete systems can be chosen uniformly bounded in l 2 0 t and in such a way that they converge to the hum control of the continuous wave equation i e the minimal l 2 norm control we illustrate the mathematical results with several numerical experiments