lyapunov instabilities of lennard jones fluids recent work on many particle systems reveals the existence of regular collective perturbations corresponding to the smallest positive lyapunov exponents les called hydrodynamic lyapunov modes until now however these modes have been found only for hard core systems here we report results on lyapunov spectra and lyapunov vectors lvs for lennard jones fluids by considering the fourier transform of the coordinate fluctuation density u ensuremath alpha x t it is found that the lvs with ensuremath lambda ensuremath approx 0 are highly dominated by a few components with low wave numbers these numerical results provide strong evidence that hydrodynamic lyapunov modes do exist in soft potential systems although the collective lyapunov modes are more vague than in hard core systems in studying the density and temperature dependence of these modes it is found that when the value of the lyapunov exponent ensuremath lambda ensuremath alpha is plotted as function of the dominant wave number k mathit max of the corresponding lv all data from simulations with different densities and temperatures collapse onto a single curve this shows that the dispersion relation ensuremath lambda ensuremath alpha vs k mathit max for hydrodynamical lyapunov modes appears to be universal for the low density cases studied here despite the wavelike character of the lvs no steplike structure exists in the lyapunov spectrum of the systems studied here in contrast to the hard core case further numerical simulations show that the finite time les fluctuate strongly we have also investigated localization features of lvs and propose a length scale to characterize the hamiltonian spatiotemporal chaotic states

dynamic localization of lyapunov vectors in hamiltonian lattices the convergence of the lyapunov vector toward its asymptotic shape is investigated in two different one dimensional hamiltonian lattices the so called fermi pasta ulam and phi 4 chains in both cases we find an anomalous behavior i e a clear difference from the previously conjectured analogy with the kardar parisi zhang equation the origin of the discrepancy is eventually traced back to the existence of nontrivial long range correlations both in space and time as a consequence of this anomaly we find that in a hamiltonian lattice the largest lyapunov exponent is affected by stronger finite size corrections than standard space time chaos

hydrodynamic lyapunov modes and strong stochasticity threshold in fermi pasta ulam models the existence of a strong stochasticity threshold sst has been detected in many hamiltonian lattice systems including the fermi pasta ulam fpu model which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density correspondingly the relaxation time to energy equipartition and the largest lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value in this paper we attempt to go one step further in this direction to explore further changes in the energy density dependence of other lyapunov exponents and of hydrodynamic lyapunov modes hlms in particular we find that for the fpu ensuremath beta and fpu ensuremath alpha ensuremath beta models the scalings of the energy density dependence of all lyapunov exponents experience a similar change at the sst as that of the largest lyapunov exponent in addition the threshold values of the crossover of all lyapunov exponents are nearly identical these facts lend support to the point of view that the crossover in the system dynamics at the sst manifests a global change in the geometric structure of phase space they also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi isotropic works quite well in the analytical calculation of the largest lyapunov exponent furthermore the fpu ensuremath beta model is used as an example to show that hlms exist in hamiltonian lattice models with continuous symmetries some measures are defined to indicate the significance of hlms numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in lyapunov exponents at the sst in particular our numerical results indicate that strong chaos is essential for the appearance of hlms and those modes become more significant with increasing degree of chaoticity

hydrodynamic lyapunov modes and strong stochasticity threshold in the dynamic xy model an alternative scenario crossover from weak to strong chaos in high dimensional hamiltonian systems at the strong stochasticity threshold sst was anticipated to indicate a global transition in the geometric structure of phase space our recent study of fermi pasta ulam models showed that corresponding to this transition the energy density dependence of all lyapunov exponents is identical apart from a scaling factor the current investigation of the dynamic xy model discovers an alternative scenario for the energy dependence of the system dynamics at ssts though similar in tendency the lyapunov exponents now show individually different energy dependencies except in the near harmonic regime such a finding restricts the use of indices such as the largest lyapunov exponent and the ricci curvatures to characterize the global transition in the dynamics of high dimensional hamiltonian systems these observations are consistent with our conjecture that the quasi isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities moreover numerical simulations demonstrate the existence of hydrodynamical lyapunov modes hlms in the dynamic xy model and show that corresponding to the crossover in the lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of hlms in particular our numerical results confirm that strong chaos is essential for the appearance of hlms

boundary effects in the stepwise structure of the lyapunov spectra for quasi one dimensional systems boundary effects in the stepwise structure of the lyapunov spectra and corresponding wavelike structure of the lyapunov vectors are discussed numerically in quasi one dimensional systems of many hard disks four different types of boundary conditions are constructed by combinations of periodic boundary conditions and hard wall boundary conditions and each leads to different stepwise structures of the lyapunov spectra we show that for some lyapunov exponents in the step region the spatial y component of the corresponding lyapunov vector deltaq yj divided by the y component of momentum p yj exhibits a wavelike structure as a function of position q xj and time t for the other lyapunov exponents in the step region the y component of the corresponding lyapunov vector deltaq yj exhibits a time independent wavelike structure as a function of q xj these two types of wavelike structure are used to categorize the type and sequence of steps in the lyapunov spectra for each different type of boundary condition

liapunov spectra for infinite chains of nonlinear oscillators we argue that the spectrum of liapunov exponents for long chains of nonlinear oscillators at large energy per mode may be well approximated by the liapunov exponents of products of independent random matrices if in addition statistical mechanics applies to the system the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone under a certain isotropy hypothesis which is not always satisfied we argue that the liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix this construction uses an integral equation first derived by newman we then derive and discuss a method to compute the spectrum of a typical random matrix putting the pieces together we see that the liapunov spectrum can be computed from the potential between the oscillators

on syntactic versus computational views of approximability we attempt to reconcile the two distinct views of approximation classes syntactic and computational syntactic classes such as max snp permit structural results and have natural complete problems while computational classes such as apx allow us to work with classes of problems whose approximability is well understood our results provide a syntactic characterization of computational classes and give a computational framework for syntactic classes we compare the syntactically defined class max snp with the computationally defined class apx and show that every problem in apx can be placed i e has approximation preserving reduction to a problem in max snp our methods introduce a simple yet general technique for creating approximation preserving reductions which shows that any well approximable problem can be reduced in an approximation preserving manner to a problem which is hard to approximate to corresponding factors the reduction then follows easily from the recent nonapproximability results for max snp hard problems we demonstrate the generality of this technique by applying it to other classes such as max snp rmax 2 and min f pi 2 1 which have the clique problem and the set cover problem respectively as complete problems the syntactic nature of max snp was used by papadimitriou and yannakakis j comput system sci 43 1991 pp 425 440 to provide approximation algorithms for every problem in the class we provide an alternate approach to demonstrating this result using the syntactic nature of max snp we develop a general paradigm nonoblivious local search useful for developing simple yet efficient approximation algorithms we show that such algorithms can find good approximations for all max snp problems yielding approximation ratios comparable to the best known for a variety of specific max snp hard problems nonoblivious local search provably outperforms standard local search in both the degree of approximation achieved and the efficiency of the resulting algorithms

approximation algorithms for optimal decision trees and adaptive tsp problems we consider the problem of constructing optimal decision trees given a collection of tests that can disambiguate between a set of m possible diseases each test having a cost and the a priori likelihood of any particular disease what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease this problem has been studied in several works with o log m approximations known in the special cases when either costs or probabilities are uniform in this paper we settle the approximability of the general problem by giving a tight o log m approximation algorithm we also consider a substantial generalization the adaptive traveling salesman problem given an underlying metric space a random subset s of vertices is drawn from a known distribution but s is initially unknown we get information about whether any vertex is in s only when it is visited what is a good adaptive strategy to visit all vertices in the random subset s while minimizing the expected dista

liapunov exponents in high dimensional symplectic dynamics the existence of a thermodynamic limit of the distribution of liapunov exponents is numerically verified in a large class of symplectic models ranging from hamiltonian flows to maps and products of random matrices in the highly chaotic regime this distribution is approximately model independent near an integrable limit only a few exponents give a relevant contribution to the kolmogorov sinai entropy

stepwise structure of lyapunov spectra for many particle systems using a random matrix dynamics the structure of the lyapunov spectra for the many particle systems with a random interaction between the particles is discussed the dynamics of the tangent space is expressed as a master equation which leads to a formula that connects the positive lyapunov exponents and the time correlations of the particle interaction matrix applying this formula to one and two dimensional models we investigate the stepwise structure of the lyapunov spectra that appear in the region of small positive lyapunov exponents long range interactions lead to a clear separation of the lyapunov spectra into a part exhibiting stepwise structure and a part changing smoothly the part of the lyapunov spectrum containing the stepwise structure is clearly distinguished by a wave like structure in the eigenstates of the particle interaction matrix the two dimensional model has the same step widths as found numerically in a deterministic chaotic system of many hard disks

dynamical behavior of hydrodynamic lyapunov modes in coupled map lattices in our previous study of hydrodynamic lyapunov modes hlms in coupled map lattices we found that there are two classes of systems with different ensuremath lambda text ensuremath k dispersion relations for coupled circle maps we found the quadratic dispersion relations ensuremath lambda ensuremath sim k 2 and ensuremath lambda ensuremath sim k for coupled standard maps here we carry out further numerical experiments to investigate the dynamic lyapunov vector lv structure factor which can provide additional information on the lyapunov vector dynamics the dynamic lv structure factor of coupled circle maps is found to have a single peak at ensuremath omega 0 and can be well approximated by a single lorentzian curve this implies that the hydrodynamic lyapunov modes in coupled circle maps are nonpropagating and show only diffusive motion in contrast the dynamic lv structure factor of coupled standard maps possesses two visible sharp peaks located symmetrically at ifmmode pm else textpm fi ensuremath omega u the spectrum can be well approximated by the superposition of three lorentzian curves centered at ensuremath omega 0 and ifmmode pm else textpm fi ensuremath omega u respectively in addition the ensuremath omega text ensuremath k dispersion relation takes the form ensuremath omega u c u k for k ensuremath rightarrow 2 ensuremath pi l these facts suggest that the hydrodynamic lyapunov modes in coupled standard maps are propagating the hlms in the two classes of systems are shown to have different dynamical behavior besides their difference in spatial structure moreover our simulations demonstrate that adding damping to coupled standard maps turns the propagating modes into diffusive ones alongside a change of the ensuremath lambda text ensuremath k dispersion relation from ensuremath lambda ensuremath sim k to ensuremath lambda ensuremath sim k 2 in cases of weak damping there is a crossover in the dynamic lv structure factors i e the spectra with smaller k are akin to those of coupled circle maps while the spectra with larger k are similar to those of coupled standard maps

time dependent mode structure for lyapunov vectors as a collective movement in quasi one dimensional systems the time dependent mode structure of the lyapunov vectors associated with the stepwise structure of the lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi one dimensional many hard disk systems we obtain the complete mode structures lyapunov modes for all components of the lyapunov vectors including the longitudinal and transverse components of both the spatial and momentum parts and their phase relations these mode structures are suggested by the form of the lyapunov vectors for the zero lyapunov exponents the spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models our main result is that the largest time oscillating period of the lyapunov modes is twice as long as the time oscillating period of the longitudinal momentum autocorrelation function this relation is satisfied irrespective of the number of particles and the boundary conditions a simple explanation for this relation is given based on the form of the time dependent lyapunov mode

completeness in approximation classes we introduce a formal framework for studying approximation properties of np optimization npo problems the classes we consider are those appearing in the literature namely the class of approximable problems within a constant e apx the class of problems having a polynomial time approximation scheme pas and the class of problems having a fully polynomial time approximation scheme fpas we define natural approximation preserving reductions and obtain completeness results for these classes a complete problem in a class can not have stronger approximation properties unless p np we also show that the degree structure of npo allows intermediate degrees that is if p np there are problems which are neither complete nor belong to a lower class

time oscillating lyapunov modes and the momentum autocorrelation function received 29 april 2004 published 20 april 2005 the lyapunov vectors corresponding to the steps of lyapunov spectra for many particle systems showtime oscillating behavior in two types of lyapunov modes one associated with time translationalinvariance and the other with spatial translational invariance our result is that for each coordinatedirection the longest period of the lyapunov modes is twice as long as the period of the momentumautocorrelation function a simple explanation for this relation is proposed and we argue that this result isgenerally true for many particle systems this gives the rst quantitative connection between thelyapunov modes and an experimentally accessible quantity

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

hydrodynamic lyapunov modes in coupled map lattices in this paper numerical and analytical results are presented which indicate that hydrodynamic lyapunov modes hlms also exist for coupled map lattices cmls the dispersion relations for the hlms of cmls are found to fall into two different universality classes it is characterized by lambda approximately k for coupled standard maps and lambda approximately k2 for coupled circle maps the conditions under which hlms can be observed are discussed the role of the hamiltonian structure conservation laws translational invariance and damping is elaborated our results are as follows 1 the hamiltonian structure is not a necessary condition for the existence of hlms 2 conservation laws or the translational invariance alone cannot guarantee the existence of hlms 3 including a damping term in the system of coupled hamiltonian maps does not destroy the hlms the lambda k dispersion relation of hlms however changes to the universality class with lambda k2 under damping in contrast no hlms survives in the system of coupled circle maps under damping 4 an on site potential destroys the hlms 5 the study of zero value lyapunov exponents les and associated lyapunov vectors lvs shows that translational invariance and conservation laws play different roles in the tangent space dynamics 6 the dynamics of the coordinate and momentum parts of lvs in hamiltonian systems are related but different furthermore numerical results for a two dimensional system show that the appearance of hlms in cmls is not restricted to the one dimensional case

localized behavior in the lyapunov vectors for quasi one dimensional many hard disk systems we introduce a definition of a localization width whose logarithm is given by the entropy of the distribution of particle component amplitudes in the lyapunov vector different types of localization widths are observed for example a minimum localization width where the components of only two particles are dominant we can distinguish a delocalization associated with a random distribution of particle contributions a delocalization associated with a uniform distribution and a delocalization associated with a wavelike structure in the lyapunov vector using the localization width we show that in quasi one dimensional systems of many hard disks there are two kinds of dependence of the localization width on the lyapunov exponent index for the larger exponents one is exponential and the other is linear differences due to these kinds of localizations also appear in the shapes of the localized peaks of the lyapunov vectors the lyapunov spectra and the angle between the spatial and momentum parts of the lyapunov vectors we show that the krylov relation for the largest lyapunov exponent ensuremath lambda ensuremath sim ensuremath ensuremath rho mathrm ln ensuremath rho as a function of the density ensuremath rho is satisfied apart from a factor in the same density region as the linear dependence of the localization widths is observed it is also shown that there are asymmetries in the spatial and momentum parts of the lyapunov vectors as well as in their x and y components

approximating the stochastic knapsack problem the benefit of adaptivity we consider a stochastic variant of the np hard 0 1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known arbitrary distributions items are placed in the knapsack sequentially and the act of placing an item in the knapsack instantiates its size our goal is to compute a solution policy that maximizes the expected value of items successfully placed in the knapsack where the final overflowing item contributes no value we consider both nonadaptive policies that designate a priori a fixed sequence of items to insert and adaptive policies that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far an important facet of our work lies in characterizing the benefit of adaptivity for this purpose we advocate the use of a measure called the adaptivity gap the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy we bound the adaptivity

goldstone modes in lyapunov spectra of hard sphere systems in the study of chaotic behavior lyapunov exponents play an important part in this paper we demonstrate how the lyapunov exponents close to zero of a system of many hard spheres can be described as goldstone modes by using a boltzmann type of approach at low densities the correct form is found for the wave number dependence of the exponents as well as for the corresponding eigenvectors in tangent space the predicted values for the lyapunov exponents belonging to the transverse mode are within a few percent of the values found in recent simulations the propagation velocity for the longitudinal mode is within 1 but the value for the lyapunov exponent belonging to the longitudinal mode deviates from the simulations by 30 for higher densities the predicted values deviate more from the values calculated in the simulations these deviations may be due to contributions from ring collisions and similar terms which even at low densities can contribute to the leading order

a linear recognition algorithm for cographs cographs are the graphs formed from a single vertex under the closure of the operations of union and complement another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms furthermore it is known that cographs have a unique tree representation called a cotree using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general such problems include chromatic number clique determination clustering minimum weight domination isomorphism minimum fill in and hamiltonicity in this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation

some apx completeness results for cubic graphs four fundamental graph problems minimum vertex cover maximum independent set minimum dominating set and maximum cut are shown to be apx complete even for cubic graphs therefore unless p np these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three

perturbed phase space dynamics of hard disk fluids abstract the lyapunov spectrum describes the exponential growth or decay of infinitesimal phase space perturbations the perturbation associated with the maximum lyapunov exponent is strongly localized in space and only a small fraction of all particles contributes to the perturbation growth at any instant of time this fraction converges to zero in the thermodynamic large particle number limit for hard disk and hard sphere systems the perturbations belonging to the smallest of the non vanishing exponents are coherently spread out and form orthogonal periodic structures in space the lyapunov modes there are two types of mode polarizations transverse and longitudinal the transverse modes do not propagate but the longitudinal modes do with a speed about one third of the sound speed we characterize the symmetry and the degeneracy of the modes in the thermodynamic limit the lyapunov spectrum has a diverging slope near the intersection with the abscissa no positive lower bound exists for the positive exponents the mode amplitude scales with the inverse square root of the particle number as expected from the normalization of the perturbation vectors

lyapunov modes in hard disk systems we consider simulations of a two dimensional gas of hard disks in a rectangular container and study the lyapunov spectrum near the vanishing lyapunov exponents to this spectrum are associated eigen directions called lyapunov modes we carefully analyze these modes and show how they are naturally associated with vector fields over the container we also show that the lyapunov exponents and the coupled dynamics of the modes where it exists follow linear laws whose coefficients only depend on the density of the gas but not on aspect ratio and very little on the boundary conditions

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

origin of the hydrodynamic lyapunov modes recent studies of the lyapunov spectrum of the hard sphere fluid reveal that there are hydrodynamic lyapunov exponents corresponding to collective perturbations in phase space we show that these collective perturbations are due to the conservation of certain quantities during collisions these new conservation laws generate new hydrodynamic fields just as the conservation of mass momentum and energy generate the density velocity and temperature fields we then construct a detailed theory of the new hydrodynamic fields using a kinetic theory approach this theory predicts several properties of the modes but not all of them this suggests that the underlying idea is correct but a detailed theory must be elaborated in another way the hydrodynamic exponents are not related in a simple way to the transport coefficients

on off intermittency in random map lattices in this paper we study numerically an ensemble of random driving logistic maps there are two accessible cases of on off intermittency in the system for different values of parameters the first one corresponds to the loss of stability of the fixed point while the second one is due to the instability of the synchronous motion of the ensemble it shows that the two cases of intermittency belong to the same class of universality

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

when can one observe good hydrodynamic lyapunov modes inspired by recent results on differences in fluctuations of finite time lyapunov exponents between hard core and soft potential systems we surmise that partial domination of the oseledec splitting dos with respect to subspaces associated with near zero lyapunov exponents is essential for observing good hydrodynamic lyapunov modes hlms numerical results for coupled map lattices are presented to show the importance of dos for observing good hlms this is achieved by relating splitting parameters to the maximum value of the lyapunov mode structure factor

lyapunov hydrodynamics in the dilute limit we study the hydrodynamic lyapunov exponents of the hard disk fluid in the dilute limit these exponents appear in discrete groups and their values depend on the system size in a previous paper phys rev e 64 2001 051103 we presented a theory for these exponents explaining them as the growth rate of collective organized perturbations in phase space that theory successfully described the basic features of the hydrodynamic exponents but it had some difficulties many of these difficulties can be removed by considering the dilute limit

a priori solution of a travelling salesman problem in which a random subset of the customers are visited consider a problem of routing through a set of n points on any given instance of the problem only a subset consisting of k out of n points 0 k n has to be visited with the number k random with known probability distribution we wish to find a priori a tour through all n points on any given instance the k points present will then be visited in the same order as they appear in the a priori tour the problem of finding such a tour of minimum length in the expected value sense is defined as a probabilistic traveling salesman problem ptsp what distinguishes one ptsp from another is the probability distribution or more generally the probability law that specifies the number k and the identity of the points that need to be visited on any given instance of the problem after motivating the problem by applications we first derive closed form expressions for computing efficiently the expected length of any given tour under very general probabilistic assumptions we then provide in a unified way

a review of the hydrodynamic lyapunov modes of hard disk systems a review of the results obtained for hard disk fluids confined to a quasi one dimensional qod system is presented one of the main achievements in recent years has been determining the hydrodynamic lyapunov modes hlms as covariant stable and unstable manifolds defined by k vector analogues of the zero exponent subspace of conserved quantities the tangent space dynamics allows an interpretation of the hlms as reduced hydrodynamic fields over the perturbations and is expanded upon in this paper the time evolution of these modes is governed by the dynamics of conjugate pairs of stable and unstable directions each pair is seen to interact in a subspace almost completely separated from other vectors the modes undergo a rotation until they are collinear with the stable or unstable manifold the angle distributions between these covariant stable and unstable hlms are also determined and tangencies are not observed for a generic chaotic trajectory this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

relationship between directed percolation and the synchronization transition in spatially extended systems we study the nature of the synchronization transition in spatially extended systems by discussing a simple stochastic model an analytic argument is put forward showing that in the limit of discontinuous processes the transition belongs to the directed percolation dp universality class the analysis is complemented by a detailed investigation of the dependence of the first passage time for the amplitude of the difference field on the adopted threshold we find the existence of a critical threshold separating the regime controlled by linear mechanisms from that controlled by collective phenomena as a result of this analysis we conclude that the synchronization transition belongs to the dp class also in continuous models the conclusions are supported by numerical checks on coupled map lattices too

on the concept of stationary lyapunov basis we propose the concept of stationary lyapunov basis the basis of tangent vectors e i x defined at every point x of the attractor of the dynamical system and show that one can reformulate some algorithms for calculation of lyapunov exponents i so that eachi can be treated as the average of a function si x this enables one to use measure averaging in theoretical arguments thus proposing the rigorous basis for a number of ideas for calculation of lyapunov exponents from time series we also study how the lyapunov vectors in benettin s algorithm converge to the stationary basis and show that this convergence rate determines continuity of the field of stationary lyapunov vectors 1998 elsevier science b v

finding a potential community in networks abstract an independent 2 clique of a graph is a subset of vertices that is an independent set and such that any two vertices inside have a common neighbor outside in this paper we study the complexity of finding an independent 2 clique of maximum size in several graph classes and we compare it with the complexity of maximum independent set we prove that this problem is np hard on apex graphs apx hard on line graphs not n 1 2 approximable on bipartite graphs and not n 1 approximable on split graphs while it is polynomial time solvable on graphs of bounded degree and their complements graphs of bounded treewidth planar graphs c 3 c 6 free graphs threshold graphs interval graphs and cographs

the complexity and approximability of finding maximum feasible subsystems of linear relations we study the combinatorial problem which consists given a system of linear relations of finding a maximum feasible subsystem that is a solution satisfying as many relations as possible the computational complexity of this general problem named max fls is investigated for the four types of relations and various constrained versions of max fls where a subset of relations must be satisfied or where the variables take bounded discrete values are also considered we establish the complexity of solving these problems optimally and whenever they are intractable we determine their degree of approximability max fls with or relations is np hard even when restricted to homogeneous systems with bipolar coefficients whereas it can be solved in polynomial time for relations with real coefficients the various np hard versions of max fls belong to different approximability classes depending on the type of relations and the additional constraints we show that the range of approximability stretches from apx complete problems which can be approximated within a constant but not within every constant unless p np to npo pb complete ones that are as hard to approximate as all np optimization problems with polynomially bounded objective functions while max fls with equations and integer coefficients cannot be approximated within pe for some e 0 where p is the number of relations the same problem over gf q for a prime q can be approximated within q but not within qe for some e 0 max fls with strict or nonstrict inequalities can be approximated within 2 but not within every constant factor our results also provide strong bounds on the approximability of two variants of max fls with and relations that arise when training perceptrons which are the building blocks of artificial neural networks and when designing linear classifiers

the geometry of generalized binary search this paper investigates the problem of determining a binary valued function through a sequence of strategically selected queries the focus is an algorithm called generalized binary search gbs gbs is a well known greedy algorithm for determining a binary valued function through a sequence of strategically selected queries at each step a query is selected that most evenly splits the hypotheses under consideration into two disjoint subsets a natural generalization of the idea underlying classic binary search this paper develops novel incoherence and geometric conditions under which gbs achieves the information theoretically optimal query complexity i e given a collection of n hypotheses gbs terminates with the correct function after no more than a constant times logn queries furthermore a noise tolerant version of gbs is developed that also achieves the optimal query complexity these results are applied to learning halfspaces a problem arising routinely in image processing and machine learning

lyapunov analysis and information flow in coupled map lattices abstract lyapunov analysis for the coupled map lattices is presented the co moving lyapunov exponent is calculated which is related with the propagation of the disturbance in space the propagation speed agrees with the zero crossing point of the co moving lyapunov exponent the co moving mutual information flow is introduced which shows the selective transmission of the information at some speed lyapunov spectra and vectors are calculated spatial structures of the vectors are investigated possible analogy with the anderson localization problem is discussed

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

intrinsic stochasticity with many degrees of freedom we study a simple model equation describing a system with an infinity of degrees of freedom which displays an intrinsically chaotic behavior some concepts of fully developed turbulence are discussed in relation to this model we also develop an approach based on lyapunov exponent measurements numerical results on the distribution of lyapunov numbers and the power spectrum of the associated lyapunov vectors are presented and briefly discussed