matching cutsets in graphs of diameter 2 we say that a graph has a matching cutset if its vertices can be coloured in red and blue in such a way that there exists at least one vertex coloured in red and at least one vertex coloured in blue and every vertex has at most one neighbour coloured in the opposite colour in this paper we study the algorithmic complexity of a problem of recognizing graphs which possess a matching cutset in particular we present a polynomial time algorithm which solves this problem for graphs of diameter two

regular random attractors for non autonomous stochastic evolution equations with time varying delays on thin domains this paper deals with the limiting dynamical behavior of non autonomous stochastic reaction diffusion equations with time varying delays on thin domains first we prove the existence and uniqueness of the regular random attractor then we prove the upper semicontinuity of the regular random attractors for the equations on a family of n 1 dimensional thin domains that collapses onto an n dimensional domain

a complexity dichotomy for matching cut in bipartite graphs of fixed diameter abstract in a graph a matching cut is an edge cut that is a matching matching cut is the problem of deciding whether or not a given graph has a matching cut which is known to be np complete even when restricted to bipartite graphs it has been proved that matching cut is polynomially solvable for graphs of diameter two in this paper we show that for any fixed integer d 3 matching cut is np complete in the class of graphs of diameter d this resolves an open problem posed by borowiecki and jesse jozefczyk 2008 6 we then show that for any fixed integer d 4 matching cut is np complete even when restricted to the class of bipartite graphs of diameter d complementing the hardness results we show that matching cut is polynomial time solvable in the class of bipartite graphs of diameter at most three and point out a new and simple polynomial time algorithm solving matching cut in graphs of diameter 2

computational complexity of solving polynomial differential equations over unbounded domains in this paper we investigate the computational complexity of solving ordinary differential equations odes y p y over unbounded time domains where p is a vector of polynomials contrarily to the bounded compact time case this problem has not been well studied apparently due to the intuition that it can always be reduced to the bounded case by using rescaling techniques however as we show in this paper rescaling techniques do not seem to provide meaningful insights on the complexity of this problem since the use of such techniques introduces a dependence on parameters which are hard to compute we present algorithms which numerically solve these odes over unbounded time domains these algorithms have guaranteed accuracy i e given some arbitrarily large time t and error bound e as input they will output a value y which satisfies y t y e we analyze the complexity of these algorithms and show that they compute y in time polynomial in several quantities including the time t the accuracy of the output e and the length of the curve y from 0 to t assuming it exists until time t we consider both algebraic complexity and bit complexity

dynamics of stochastic 2d navier stokes equations abstract in this paper we study the dynamics of a two dimensional stochastic navier stokes equation on a smooth domain driven by linear multiplicative white noise we show that solutions of the 2d navier stokes equation generate a perfect and locally compacting c 1 1 cocycle using multiplicative ergodic theory techniques we establish the existence of a discrete non random lyapunov spectrum for the cocycle the lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium stationary solution we give sufficient conditions on the parameters of the navier stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium uniqueness of the stationary solution viz ergodicity local almost sure asymptotic stability of the cocycle and the existence of global invariant foliations of the energy space

computability aspects for 1st order partial differential equations via characteristics in this paper it is shown that the method of characteristics can be used to compute local solutions of the boundary value problems for the first order partial differential equations at feasible instances under the natural definition of computability from the view point of application but the maximal region of existence of a computable local solution may not be computable it is also shown that the problem whether a boundary value problem has a global solution is not algorithmically decidable the negative results retain even within the class of quasilinear equations defined by analytic computable functions over particularly simple domains quasilinear equations are among the simplest first order nonlinear partial differential equations this fact shows that the algorithmic unsolvability is intrinsic

an impact of noise on invariant manifolds in nonlinear dynamical systems invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems however these nonlinear systems are often subject to random fluctuations or noises it is thus desirable to quantify the impact of noises on the invariant manifolds when the noise intensity is small this impact is estimated via asymptotic analysis in the context of liapunov perron formulation namely the random invariant manifold is represented as a perturbation of the deterministic invariant manifold with a well defined bound for the deviation

existence stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations abstract in this paper we study the existence stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise we first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions we then examine the existence and stability of random complete quasi solutions and establish the relations of these solutions and the structures of tempered attractors when the stochastic equations are incorporated with periodic forcing we obtain the existence and stability of random periodic solutions for the stochastic chafee infante equation we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions

computability noncomputability and undecidability of maximal intervals of ivps let c r denote the maximal interval of existence of solutions for the initial value problem dx dt f t x x t 0 x 0 where e is an open subset of r m 1 f is continuous in e and t 0 x 0 e we show that under the natural definition of computability from the point of view of applications there exist initial value problems with computable f and t 0 x 0 whose maximal interval of existence is noncomputable the fact that f may be taken to be analytic shows that this is not a lack of the regularity phenomenon moreover we get upper bounds for the degree of noncomputability by showing that is r e recursively enumerable open under very mild hypotheses we also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable

a random fixed point theorem and the random graph transformation in the first part of this article we formulate a random fixed point theorem for random dynamical systems where a contraction condition is formulated as an expectation of a particular expression in the following we are going to formulate a random graph transformation this transformation defines a random dynamical system under certain conditions this random dynamical system has a random fixed point which can be found by the mentioned random fixed point theorem in one application we show that random dynamical systems given by particular differential equations have a random unstable invariant manifold

the wave equation with computable initial data such that its unique solution is not computable we consider the three dimensional wave equation it is well known that the solution u x y z t is uniquely determined by two initial conditions the values of u and au at at time t 0 our question is can computable initial data give rise to noncomputable solutions the answer is yes and two quite different types of noncomputability can occur theorem 1 below gives an example in which the solution u x y z t takes a noncomputable real value at a computable point in space time by contrast theorem 2 provides an example in which the solution maps each computable sequence of points in space time into a computable sequence nevertheless u x y z t is not a computable function we note in passing that computability as used in this paper is a technical term familiar to mathematical logicians the precise definitions are spelled out in the next section the proof of theorem 1 is quite short that for theorem 2 is considerably more intricate it should be mentioned that these noncomputable solutions of the wave equation are of the type commonly referred to as weak solutions i e although continuous they are not twice differentiable at all points weak solutions describe creases cusps and other non differentiable patterns which frequently appear in models of wave phenomena the use of weak solutions is inevitable we will prove that no cz solutions of the desired kind exist for the convenience of readers unfamiliar with weak solutions we have put them into a coherent framework in an addendum basing our presentation on the energy integral associated with the wave just as the initial conditions are computable the energy integral in our examples is also a computable real finally the proof that all noncomputable solutions must be of weak type is sketched at the end of this addendum we note however that the main part of the paper may be read without reference to the addendum the results in this paper are related to comments of kreisel in 4 kreisel asks whether existing physical theories e g classical mechanics or quantum mechanics can predict theoretically the existence of a physical constant which is not a recursive real previous work of the authors in this area 9 was concerned with ordinary differential equations it was proved that there exists a computable and hence continuous function f such that 215 oool 8708 81 030215 25905 00 o

a characterization of computable analysis on unbounded domains using differential equations the functions of computable analysis are defined by enhancing normal turing machines to deal with real number inputs we consider characterizations of these functions using function algebras known as real recursive functions since there are numerous incompatible models of computation over the reals it is interesting to find that the two very different models we consider can be set up to yield exactly the same functions bournez and hainry used a function algebra to characterize computable analysis restricted to the twice continuously differentiable functions with compact domains in our earlier paper we found a different and apparently more natural function algebra that also yields computable analysis with the same restriction in this paper we improve earlier work finding three function algebras characterizing computable analysis removing the restriction to twice continuously differentiable functions and allowing unbounded domains one of these function algebras is built upon the widely studied real primitive recursive functions furthermore the proof of this paper uses our previously developed method of approximation whose applicability is further evidenced by this paper

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

computability noncomputability and hyperbolic systems abstract in this paper we study the computability of the stable and unstable manifolds of a hyperbolic equilibrium point these manifolds are the essential feature which characterizes a hyperbolic system having many applications in physical sciences and other fields we show that i locally these manifolds can be computed but ii globally they cannot since their degree of computational unsolvability lies on the second level of the borel hierarchy we also show that smale s horseshoe the first example of a hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit is computable

on stable cutsets in line graphs we answer a question of brandstadt et al by showing that deciding whether a line graph with maximum degree 5 has a stable cutset is np complete conversely the existence of a stable cutset in a line graph with maximum degree at most 4 can be decided efficiently the proof of our np completeness result is based on a refinement on a result due to chvatal that recognizing decomposable graphs with maximum degree 4 is an np complete problem here a graph is decomposable if its vertices can be colored red and blue in such a way that each color appears on at least one vertex but each vertex v has at most one neighbor having a different color from v we also discuss some open problems on stable cutsets

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

pullback attractors for reaction diffusion delay equations on unbounded domains with non autonomous deterministic and stochastic forcing terms the existence of a pullback attractor in a space of higher regularity is established for the non autonomous non compact dynamical system associated with the reaction diffusion delay equation with both non autonomous deterministic and stochastic forcing terms defined on all n dimensional space the asymptotical compactness of the solutions is demonstrated by using the uniform estimates for far field values of solutions and a new method

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

autonomous and non autonomous attractors for differential equations with delays abstract the asymptotic behaviour of some types of retarded differential equations with both variable and distributed delays is analysed in fact the existence of global attractors is established for different situations with and without uniqueness and for both autonomous and non autonomous cases using the classical notion of attractor and the recently new concept of pullback one respectively

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

flattening squeezing and the existence of random attractors the study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems in the deterministic case the properties of flattening and squeezing in infinite dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor the flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories it is shown here that the flattening property is implied by the squeezing property and is in fact weaker since the attractor in a system with the flattening property can be infinite dimensional whereas it is always finite dimensional in a system with the squeezing property the flattening property is then generalized to random dynamical systems for which it is called the pullback flattening property it is shown to be weaker than the random squeezing property but equivalent to pullback asymptotic compactness and pullback limit set compactness and thus implies the existence of a random attractor the results are also valid for deterministic non autonomous dynamical systems formulated as skew product flows

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

upper semicontinuity of attractors of stochastic delay reaction diffusion equations in the delay a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero

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

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

the connection between computability of a nonlinear problem and its linearization the hartman grobman theorem revisited as one of the seven open problems in the addendum to their 1989 book computability in analysis and physics pour el and richards 1989 17 pour el and richards asked what is the connection between the computability of the original nonlinear operator and the linear operator which results from it yet at present systematic studies of the issues raised by this question seem to be missing from the literature in this paper we study one problem in this direction the hartman grobman linearization theorem for ordinary differential equations odes we prove roughly speaking that near a hyperbolic equilibrium point x 0 of a nonlinear ode email protected f x there is a computable homeomorphism h such that email protected email protected email protected h where f is the solution to the ode and l is the solution to its linearization email protected df x 0 x

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

algorithms solving the matching cut problem in a graph a matching cut is an edge cut that is a matching matching cut is the problem of deciding whether or not a given graph has a matching cut which is known to be np complete this paper provides a first branching algorithm solving matching cut in time o 2 n 2 o 1 4143 n for an n vertex input graph and shows that matching cut parameterized by the vertex cover number g can be solved by a single exponential algorithm in time 2 g o n 2 moreover the paper also gives a polynomially solvable case for matching cut which covers previous known results on graphs of maximum degree three line graphs and claw free graphs

non computable mandelbrot like sets for a one parameter complex family abstract we show the existence of computable complex numbers for which the bifurcation locus of the one parameter complex family f b z z b z 2 z 3 is not turing computable

random attractor for stochastic reaction diffusion equation with multiplicative noise on unbounded domains abstract in this paper we study the asymptotic dynamics for stochastic reaction diffusion equation with multiplicative noise defined on unbounded domains we investigate the existence of a random attractor for the random dynamical system associated with the equation the asymptotic compactness of the random dynamical system is established by using uniform a priori estimates for far field values of solutions and a cut off technique

non computable julia sets polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system apart from the beautiful mathematics one of the reasons for their popularity is the beauty of the computer generated images of such sets the algorithms used to draw these pictures vary the most naive work by iterating the center of a pixel to determine if it lies in the julia set milnor s distance estimator algorithm mil uses classical complex analysis to give a one pixel estimate of the julia set this algorithm and its modifications work quite well for many examples but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution moreover there are examples even in the family of quadratic polynomials when no satisfactory pictures of the julia set exist in this paper we study computability properties of julia sets of quadratic polynomials under the definition we use a set is computable if roughly speaking its image can be generated by a computer with an arbitrary precision under this notion of computability we show

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

random attractors for stochastic reaction diffusion equations on unbounded domains the existence of a pullback attractor is established for a stochastic reaction diffusion equation on all n dimensional space the nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise the reaction diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far field values of solutions

asymptotic behaviour of nonautonomous difference inclusions abstract nonautonomous difference inclusions are formulated as difference inclusion cocycles which generalize set valued semigroups corresponding to autonomous difference inclusions and the existence of a difference inclusion cocycle attractor is established under the assumption of upper semi continuity of the cocycle mappings if the difference inclusion cocycle has an absorbing set several illustrative examples are provided

computational bounds on polynomial differential equations in this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations we consider elementary in the sense of analysis discrete time dynamical systems satisfying certain criteria of robustness we show that those systems can be simulated with elementary and robust continuous time dynamical systems which can be expanded into fully polynomial ordinary differential equations in q p this sets a computational lower bound on polynomial odes since the former class is large enough to include the dynamics of arbitrary turing machines we also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial value problem defined with polynomial odes is bounded or not is in general undecidable even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56 combined with earlier results on the computability of solutions of polynomial odes one can conclude that there is from a computational point of view a close connection between these systems and turing machines