edge decompositions of graphs with high minimum degree abstract a fundamental theorem of wilson states that for every graph f every sufficiently large f divisible clique has an f decomposition here a graph g is f divisible if e f divides e g and the greatest common divisor of the degrees of f divides the greatest common divisor of the degrees of g and g has an f decomposition if the edges of g can be covered by edge disjoint copies of f we extend this result to graphs g which are allowed to be far from complete in particular together with a result of dross our results imply that every sufficiently large k 3 divisible graph of minimum degree at least 9 n 10 o n has a k 3 decomposition this significantly improves previous results towards the long standing conjecture of nash williams that every sufficiently large k 3 divisible graph with minimum degree at least 3 n 4 has a k 3 decomposition we also obtain the asymptotically correct minimum degree thresholds of 2 n 3 o n for the existence of a c 4 decomposition and of n 2 o n for the existence of a c 2 l decomposition where l 3 our main contribution is a general iterative absorption method which turns an approximate or fractional decomposition into an exact one in particular our results imply that in order to prove an asymptotic version of nash williams conjecture it suffices to show that every k 3 divisible graph with minimum degree at least 3 n 4 o n has an approximate k 3 decomposition

an extension of pointwise circumscription abstract in this paper we generalize lifschitz s pointwise circumscription ander the first order framework the generalized version has the ability for simultaneously minimizing several predicates at finitely many pinpoints in a pointwise manner we show that if an anderlying first order theory is almost existential then the extended pointwise circumscription is complete with respect to minimal model semantics almost existential formulas are in the dual form of almost universal formulas which was proposed by lifschitz to investigate the satisfiability of circumscription this completeness result is a generalization of the result by kolaitis and papadimitriou who regarded the case of existential formulas we also give a partial answer to the question for exponential growth of the size of first order formulas equivalent to circumscription moreover we clarify that lifschitz s pointwise circumscription is complete in a slightly wider class of positive formulas

almost reducibility and non perturbative reducibility of quasi periodic linear systems in this paper we prove that a quasi periodic linear differential equation in sl 2 with two frequencies 1 is almost reducible provided that the coefficients are analytic and close to a constant in the case that is diophantine we get the non perturbative reducibility we also obtain the reducibility and the rotations reducibility for an arbitrary irrational under some assumption on the rotation number and give some applications for schrodinger operators our proof is a generalized kam type iteration adapted to all irrational

degenerate lower dimensional tori in reversible systems abstract in this paper we prove the persistence of lower dimensional invariant tori with prescribed frequencies and singular normal matrices in reversible systems the normal variable is two dimensional and the unperturbed nonlinear terms in the differential equation for this variable have a special structure

construction of response solutions for two classes of quasi periodically forced four dimensional nonlinear systems with degenerate equilibrium point under small perturbations abstract by developing two kam theorems in this paper we show that two classes of quasi periodically forced four dimensional nonlinear systems with degenerate equilibrium point a reversible system and a non conservative system admit a response solution under small perturbations for the degenerate reversible system applying special structure of unperturbed nonlinear term and herman method we successfully control the shift of equilibrium point which is difficult in view of the degenerate linear term for the degenerate non conservative system kam method is brought into force even in completely degenerate case because of the restrictions on the smallness and average of perturbation moreover arithmetic condition on the frequency is assumed to satisfy the brjuno russmann s non resonant condition by the poschel russmann kam method we prove that these two kinds of perturbed systems can be reduced to a suitable normal form with zero as equilibrium point by a quasi periodic transformation

existence of quasi periodic solutions of fast excited van der pol mathieu duffing equation the van der pol mathieu duffing equation x 02 h1cos 1t h2cos 2t x x2 x h3x3 h4 32cosxcos 3t is considered in this paper where h1 h2 h3 h4 1 2 are small parameters 0 the frequency 3 is large compared to 1 and 2 the above parameters are real for 0 we use kam kolmogorov arnold moser theory to prove that the van der pol mathieu duffing equation possesses quasi periodic solutions for most of the parameters 0 1 2 3 it verifies some phenomenon of fahsi and belhaq commun nonlinear sci 14 244 253 2009 and can be regarded as a extension of abouhazim et al nonlinear dyn 39 395 409 2005

a generalization of collapsible cases of circumscription abstract circumscription is a second order formula expressing the minimality of the extension of a particular predicate or set of predicates it was fashioned to capture the concept of nonmonotonic logic of human beings in some instances circumscription is equivalent to a first order formula we show that the known cases are special instances of a more general case

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

elliptic type degenerate invariant tori for quasi periodically forced four dimensional non conservative systems abstract in this paper we will focus on the persistence of elliptic type degenerate invariant tori with the same frequency vector r d as the forcing in a class of quasi periodically forced four dimensional non conservative systems it is shown that under suitable hypothesis of smoothness with respect to and the brjuno russmann s non resonant condition instead of ordinary diophantine condition with respect to the frequency vector r d the invariant tori persist under small perturbations for most of the sufficiently small parameters in the sense of the lebesgue measure hence the system has a quasi periodic solution with the same frequency vector as the forcing the proof is based on the poschel russmann kam method which is a kam type method in which one uses the polynomial structure of function to truncate introduce a parameter q and make the steps of kam iteration infinitely small in the speed of function q n 0 q 1 rather than super exponential function like traditional kam method

circumscription completeness reviewed abstract in this paper we demonstrate that some results on the completeness of p defining theories published earlier are incorrect we point out that by restricting the original propositions to well founded theories results somewhat weaker than the original ones can be retained we also present a theorem that provides some insight into the relation between completeness and reducibility and helps to identify the theories whose minimal models can be adequately handled with circumscription

on small perturbation of two dimensional quasi periodic systems with hyperbolic type degenerate equilibrium point abstract in this paper we consider small quasi periodic perturbation of two dimensional nonlinear quasi periodic system with hyperbolic type degenerate equilibrium point by kam method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi periodic transformation hence the perturbed system has a quasi periodic solution near the equilibrium point

the suprem architecture a new intelligent paradigm abstract we investigate the model theory of the notion of circumscription and find completeness theorems that provide a partial converse to a result of mccarthy we show that the circumscriptive theorems are precisely the truths of the minimal models in the case of various classes of theories and for various versions of circumscription we also present an example of commonsense reasoning in which first order circumscription does not achieve the intuitive and desired minimization

an asymptotic existence result on compressed sensing matrices abstract for any rational number h and all sufficiently large n we give a deterministic construction for an n h n compressed sensing matrix with l 1 t recoverability where t o n our method uses pairwise balanced designs and complex hadamard matrices in the construction of equiangular frames which we introduce as a generalisation of equiangular tight frames the method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design the l 1 t recoverability performance is specified as a simple function of the parameters of the design to obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

a parallel algorithm for the computation of invariant tori in large scale dissipative systems a parallelizable algorithm to compute invariant tori of high dimensional dissipative systems obtained upon discretization of pdes is presented the size of the set of equations to be solved is only a small multiple of the dimension of the original system the sequential and parallel implementations are compared with a previous method sanchez et al 2010 11 showing that important savings in wall clock time can be achieved in order to test it a thermal convection problem of a binary mixture of fluids has been used the new method can also be applied to problems with very low rotation numbers for which the previous is not suitable this is tested in two examples of two dimensional maps

persistence of lower dimensional tori of general types in hamiltonian systems this work is a generalization to a result of j you 1999 we study the persistence of lower dimensional tori of general type in hamiltonian systems of general normal forms by introducing a modified linear kam iterative scheme to deal with small divisors we shall prove a persistence result under a melnikov type of non resonance condition which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori

on the reducibility of linear differential equations with quasiperiodic coefficients the system x a eq t x in rd is considered where a is a constant matrix and q a quasiperiodic analytic matrix with r basic frequencies the eigenvalues of a are arbitrary including the purely imaginary case suppose that the set formed by the eigenvalues of a and the basic frequencies of q satisfies a nonresonant condition then there is a positive measure cantorian set e such that for e e the system is reducible to constant coefficients by means of a quasiperiodic change of variables provided a nondegeneracy condition holds this condition prevents locking at resonance

on the satisfiability of circumscription abstract etherington mercer and reiter showed on the basis of ideas of bossu and siegel that circumscription cannot lead to inconsistency for universal formulas we extend this result in three directions to formulas of a more general syntactic form to circumscription with some predicate symbols allowed to vary and to prioritized circumscription

some computational aspects of circumscription the effects of circumscribing first order formulas are explored from a computational standpoint first extending work of v lifschitz it is shown that the circumscription of any existential first order formula is equivalent to a first order formula after this it is established that a set of universal horn clauses has a first order circumscription if and only if it is bounded when considered as a logic program thus it is undecidable to tell whether such formulas have first order circumscription finally it is shown that there arefirst order formulas whode circumscription has a conp complete model checking problem

a kam theorem for hyperbolic type degenerate lower dimensional tori in hamiltonian systems a kam theorem for degenerate lower dimensional tori in nearly integrable hamiltonian systems is given in this paper for the non degenerate cases both hyperbolic and elliptic the kam theorem has been well established by many authors 8 9 11 13 14 17

reducibility for wave equations of finitely smooth potential with periodic boundary conditions abstract in the present paper the reducibility is derived for the wave equations with finitely smooth and time quasi periodic potential subject to periodic boundary conditions more exactly the linear wave equation u t t u x x m u e v 0 t u x x v t x u 0 x r 2 z can be reduced to a linear hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set where v is finitely smooth in t x quasi periodic in time t with diophantine frequency r n and v 0 is finitely smooth and quasi periodic in time t with diophantine frequency r n moreover it is proved that the corresponding wave operator possesses the property of pure point spectra and zero lyapunov exponent

resummation of perturbation series and reducibility for bryuno skew product flows we consider skew product systems on d l sl 2 for bryuno base flows close to constant coefficients depending on a parameter in any dimension d and we prove reducibility for a large measure set of values of the parameter the proof is based on a resummation procedure of the formal power series for the conjugation and uses techniques of renormalisation group in quantum field theory

an efficient method for eliminating varying predicates from a circumscription abstract circumscription appears to be the most powerful and well studied technique used in formalizing common sense reasoning the general form of predicate circumscription allows for fixed and varying floating predicates we show that the inference problem under this form of circumscription is efficiently reducible to inferencing under circumscription without varying predicates in fact we transform this problem even into circumscription without fixed and varying predicates that is where all predicates are minimized thus any theorem prover or algorithm for inferencing under circumscription without fixed and varying predicates is able to handle inferencing under the general form of predicate circumscription as a consequence algorithms that compute circumscription for an inference task can be simplified

smoothness of spectral subbundles and reducibility of quasi periodic linear differential systems abstract in this paper we study linear differential systems 1 x a t x where a is an n n matrix valued function defined on the k torus t k and t t is a given irrational twist flow on t k first we show that if a c n t k where n 0 1 2 then the spectral subbundles are of class c n on t k next we assume that a is sufficiently smooth on t k and satisfies a suitable small divisors inequality we show that if 1 satisfies the full spectrum assumption then there is a quasi periodic linear change of variables x p t y that transforms 1 to a constant coefficient system y by finally we study the case where the matrix a t in 1 is the jacobian matrix of a nonlinear vector field x evaluated along a quasi periodic solution x t of 2 x x we give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system z defined in the vicinity of h the hull of so that the linearized system 1 can be represented in the form z dz where d is a constant matrix our results represent substantial improvements over known methods because we do not require that a be close to a constant coefficient system

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps rigorous results abstract in this paper we prove rigorous results on persistence of invariant tori and their whiskers the proofs are based on the parameterization method of x cabre e fontich r de la llave the parameterization method for invariant manifolds i manifolds associated to non resonant subspaces indiana univ math j 52 2 2003 283 328 x cabre e fontich r de la llave the parameterization method for invariant manifolds ii regularity with respect to parameters indiana univ math j 52 2 2003 329 360 the invariant manifolds results proved here include as particular cases of the usual strong stable and strong unstable manifolds but also include other non resonant manifolds the method lends itself to numerical implementations whose analysis and implementation is studied in a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical algorithms preprint 2005 a haro r de la llave a parameterization method for the computation of invariant tori and their whiskers in quasi periodic maps numerical implementation and examples preprint 2005 the results are stated as a posteriori results namely that if one has an approximate solution which is not degenerate then one has a true solution not too far from the approximate one this can be used to validate the results of numerical computations

on the reducibility of a class of nonlinear quasi periodic system with small perturbation parameter near zero equilibrium point abstract this work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system x a x f t x x r 2 where a is a real 2 2 constant matrix and f t 0 o and x f t 0 o as 0 with some non resonant conditions of the frequencies with the eigenvalues of a and without any nondegeneracy condition with respect to by an affine analytic quasiperiodic transformation we change the system to a suitable norm form at the zero equilibrium for most of the sufficiently small perturbation parameter

an algorithm to compute circumscription abstract mccarthy s theory of circumscription appears to be the most powerful among various formalizations of nonmonotonic reasoning designed to handle incomplete and negative information in knowledge representation systems in this paper we describe a query answering algorithm for circumscriptive theories the closed world assumption cwa and its generalizations the generalized closed world assumption gcwa and the extended closed world assumption ecwa can be considered as special forms of circumscription consequently our algorithm also applies to answering queries in theories using the closed world assumption and its generalizations similarly since prioritized circumscription is equivalent to a conjunction of parallel circumscriptions our algorithm can be used to answer queries in theories circumscribed by prioritized circumscription

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients