polynomial bound on the number of scattering poles on montre que dans la diffusion par un potentiel infiniment differentiable a support compact les poles de l operateur de diffusion sont a distribution polynomiale

transient and cyclic behavior of cellular automata with null boundary conditions one dimensional cellular automata ca over finite fields are studied in which each interior cell is updated to contain the sum of the previous values of its two nearest neighbors boundary cells are updated according to null boundary conditions for a given initial configuration the ca evolves through transient configurations to an attracting cycle the dependence of the maximal transient length and maximal cycle length on the number of cells is investigated both can be determined from the minimal polynomial of the update matrix which in this case satisfies a useful recurrence relation with cell values from a field of characteristic two the explicit dependence of the maximal transient length on the number of cells is determined extensions and directions for future work are presented

invertible cellular automata a review abstract in the light of recent developments in the theory of invertible cellular automata we attempt to give a unified presentation of the subject and discuss its relevance to computer science and mathematical physics

cylindrical cellular automata a one dimensional cellular automaton with periodic boundary conditions may be viewed as a lattice of sites on a cylinder evolving according to a local interaction rule a technique is described for finding analytically the set of attractors for such an automaton given any one dimensional automaton rule a matrixa is defined such that the number of fixed points on an arbitrary cylinder size is given by the trace ofa n where the powern depends linearly on the cylinder size more generally the number of strings of arbitrary length that appear in limit cycles of any fixed period is found as the solution of a linear recurrence relation derived from the characteristic equation of an associated matrix the technique thus makes it possible for any rule to compute the number of limit cycles of any period on any cylinder size to illustrate the technique closed form expressions are provided for the complete attractor structure of all two neighbor rules the analysis of attractors also identifies shifts as a basic mechanism underlying periodic behavior every limit cycle can be equivalently defined as a set of strings on which the action of the rule is a shift of sizes h i e each string cyclically shifts bys sites inh iterations of the rule the study of shifts provides detailed information on the structure and number of limit cycles for one dimensional automata

graph theoretical characterization of invertible cellular automata abstract a cellular automaton is defined to be invertible if its induced global rule is surjective on the set of all finite configurations it is shown that all one dimensional invertible cellular automata are obtainable from invertible nearest pair ones i e cellular automata given by a rule matrix for these an algorithm is proposed which runs in a time which is a polynomial function of the number of letters of the alphabet the result is expressed in a graph theoretical way the cellular automaton is invertible if and only if two graphs constructed by the algorithm have no edges in common it follows that if the number of letters in the alphabet is a prime the only invertible cellular automata are those for which every row or column of the rule matrix is a permutation of the alphabet for composite alphabet cardinalities several methods analytical and computational ones to obtain more complicated invertible cellular automata are proposed and used to construct examples all four letter invertible automata are constructed

spectral sets and tiles on vector space over local fields abstract in this article we prove that a set which is of positive and finite haar measure in k d is a spectral set with quasi lattice as a spectrum if and only if it tiles k d by the dual quasi lattice as a translation set moreover we obtain some results connected with the dual spectral set conjecture on vector space over local fields

resonances for steplike potentials forward and inverse results we consider resonances associated to the one dimensional schrodinger operator d 2 d x 2 v x where v x v if x x m and v x v if x x m with v v we obtain asymptotics of the resonance counting function for several regions moreover we show that in several situations the resonances v and v determine v uniquely up to translation

asymptotics of resonances for a schrodinger operator with vector values abstract we show that the resonance counting function for a schrodinger operator in dimension one has an asymptotic expansion and calculate an explicit expression for the leading term in some situations

resonances for euler bernoulli operator on the half line abstract we consider resonances for fourth order differential operators on the half line with compactly supported coefficients we determine asymptotics of a counting function of resonances in complex discs at large radius describe the forbidden domain for resonances and obtain trace formulas in terms of resonances we apply these results to the euler bernoulli operator on the half line the coefficients of this operator are positive and constants outside a finite interval we show that this operator does not have any eigenvalues and resonances iff its coefficients are constants on the whole half line

sharp polynomial bounds on the number of scattering poles of radial potentials abstract it is shown that for scattering by a radially symmetric potential in r n n odd the number of poles in a disk of radius r satisfies an estimate n r c n r 1 n this bound is sharp as shown by the special case of potentials nonvanishing at the boundary where n r k n a n r n 1 o 1 a being the diameter of the support

on characterizations of spectra and tilings abstract in a recent paper lagarias reeds and wang established a characterization of spectra and tilings that can be used to prove a conjecture of jorgensen and pedersen by keller s criterion different techniques to prove these facts have also been developed by kolountzakis iosevich and pedersen the primary aim of this paper is to present an elementary method of describing certain characterizations of spectra and tilings to illustrate this method we first give a simple proof of this characterization we then use the method to derive some characteristic results connected with the dual fuglede s spectral set conjecture the results here extend several known conclusions in a simple manner

exact results for deterministic cellular automata with additive rules deterministic cellular automata ca with additive rules are studied by exploiting the properties of circulant matrices on finite fields complete state transition diagrams for higher order and multidimensional ca on finite lattices are analyzed conditions on the rules which make them reversible are obtained it is shown that all state transition diagrams of the ca have identical trees rooted on cycles general formulae for cycle lengths and multiplicities are given

fuglede s conjecture holds in mathbb q _ p q p we prove fuglede s conjecture in mathbb q p which states that a borel set of positive and finite haar measure in mathbb q p is a spectral set if and only if it tiles mathbb q p by translations

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

spectral theory for borel sets in rn of finite measure abstract let be a pair of subsets in r n such that has finite positive lebesgue measure the pair is called a spectral pair if the functions e i x for form an orthogonal total family in l 2 i e when restricted to our present work continues earlier papers on simultaneous diagonalization of vector fields for the special case when is also assumed open however the setting here is more general and the results are stated in the measure theoretic category in considering fundamental domains for lattices for example we allow distinct lattice point translates of the domain to overlap on sets of measure zero rather than the traditional condition of empty overlap two classes of examples of spectral pairs are known fundamental domains and simple factors the latter may be identified in terms of a certain finite group action we show that every spectral pair can be factored in the sense of cartesian products as a product of two pairs of lower dimension where the first factor is a fundamental domain and the second a simple factor the dimensions add a simple factor yields in a natural way a finite covering of an associated compact torus but not every such covering corresponds to a spectral pair we give a complete spectral characterization of the finite coverings and identify those which correspond to simple factors the latter identification is phrased in terms of generators and relations for certain c algebras which are analogous to but different from algebras studied earlier by cuntz and arveson

precise estimates of tunneling and eigenvalues near a potential barrier abstract we consider a semiclassical schrodinger operator in one dimension with an analytic potential presenting a barrier between two potential wells we give precise estimates for eigenvalues eigenvalue splittings and avoided crossings of eigenvalues near the barrier level these estimates include bounds of the coefficients in the asymptotic expansions

linear cellular automata and recurring sequences in finite fields a one dimensional linear cellular automaton with periodic boundary conditions consists of a lattice of sites on a cylinder evolving according to a linear local interaction rule limit cycles for such a system are studied as sets of strings on which the rule acts as a shift of sizes h i e each string in the limit cycle cyclically shifts bys sites inh iterations of the rule for any given rule the size of the shift varies with the cylinder sizen the analysis of shifts establishes an equivalence between the strings of values appearing in limit cycles for these automata and linear recurring sequences in finite fields specifically it is shown that a string appears in a limit cycle for a linear automaton rule on a cylinder sizen iff its values satisfy a linear recurrence relation defined by the shift value for thatn the rich body of results on recurring sequences and finite fields can then be used to obtain detailed information on periodic behavior for these systems topics considered here include the inverse problem of identifying the set of linear automata rules for which a given string appears in a limit cycle and the structure under operations such as addition and complementation of sets of limit cycle strings

geometry of p adic siegel discs abstract we survey recent advances in the study of regular motions over p adic fields show its varied connections with dynamics and number theory and illustrate its significance to an important class of discrete dynamical systems we also show that mappings supporting quasi periodic motions can be naturally interpreted as flows with p adic time

tiling the line with translates of one tile a region t is a closed subset of the real line of positive finite lebesgue measure which has a boundary of measure zero call a region t a tile if bbb r can be tiled by measure disjoint translates of t for a bounded tile all tilings of bbb r with its translates are periodic and there are finitely many translation equivalence classes of such tilings the main result of the paper is that for any tiling of bbb r by a bounded tile any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling from it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups

existence of resonances in three dimensions ifp is an elliptic self adjoint perturbation of the laplacian on 3 and the coefficients ofp decay super exponentially then we show thatp has infinitely many resonances the resonances are defined here as the poles of the meromorphic continuation of p 2 1

period lengths of cellular automata on square lattices with rule 90 this paper studies two dimensional cellular automata ca 90 m n having states 0 and 1 and working on a square lattice of size m 1 n 1 all their dynamics driven by the local transition rule 90 can be simply formulated by representing their configurations with laurent polynomials over a finite field f2 0 1 the initial configuration takes the next configuration to a particular configuration whose cells all have the state 1 this paper answers the question of whether the initial configuration lies on a limit cycle or not and if that is the case some properties on period lengths of such limit cycles are studied

gabor orthonormal bases generated by the unit cubes abstract we consider the problem in determining the countable sets in the time frequency plane such that the gabor system generated by the time frequency shifts of the window 0 1 d associated with forms a gabor orthonormal basis for l 2 r d we show that if this is the case the translates by elements of the unit cube in r 2 d must tile the time frequency space r 2 d by studying the possible structure of such tiling sets we completely classify all such admissible sets of time frequency shifts when d 1 2 moreover an inductive procedure for constructing such sets in dimension d 3 is also given an interesting and surprising consequence of our results is the existence for d 2 of discrete sets with g 0 1 d forming a gabor orthonormal basis but with the associated time translates of the window 0 1 d having significant overlaps

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

computation theoretic aspects of cellular automata abstract cellular automata may be viewed as computational models of complex systems they also can be seen as continuous functions on a particular family of compact metric spaces this paper surveys results largely motivated by an attempt to answer questions posed by the latter approach by means of techniques from the former

additive cellular automata and global injectivity abstract we employ algebraic methods to investigate the global injectivity of additive cellular automata when the state alphabet is a finite commutative ring the set of local rules of such additive cellular automata in d 1 dimensions itself forms a ring the invertible elements in this ring correspond to the globally injective additive cellular automata we indicate how one determines the invertible elements and thus the globally injectibe cellular automata in particular we show that in contrast to the general situation the global injectivity of additive cellular automata in d 1 dimensions is decidable construction of local rules for the inverse cellular automata is illustrated through examples a strict upper bound on the radius of the inverse cellular automata is obtained in the special case of linear cellular automata with state alphabet the ring of integers modulo m we count the globally injective cellular automata of up to a given radius we show that in one dimension a non globally injective additive cellular automaton is either not injective on spatially periodic configurations of any period n n or the set of n 0 such that the cellular automation is injective on spatially periodic configurations of period n has positive density in n

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

cellular automata and finite fields abstract we rewrite some concepts in the theory of one dimensional periodic cellular automata in the language of finite fields the state space of an automaton with n cell and q pf possible values for each cell p prime is identified with the finite field of qn elements represented by means of a normal basis the dynamics is given by a polynomial mapping with coefficients in the field of q elements we illustrate the dynamical significance of several constructs of the theory of finite fields and we suggest that cellular automata may provide a novel framework for the study of the dynamics of polynomials over finite fields

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

asymptotics for a resonance counting function for potential scattering on cylinders abstract we study certain resonance counting functions for potential scattering on infinite cylinders or half cylinders under certain conditions on the potential we obtain asymptotics of the counting functions with an explicit formula for the constant appearing in the leading term

compact open spectral sets in qp abstract in this article we prove that a compact open set in the field q p of p adic numbers is a spectral set if and only if it tiles q p by translation and also if and only if it is p homogeneous which is easy to check we also characterize spectral sets in z p n z p 2 prime n 1 integer by tiling property and also by homogeneity moreover we construct a class of singular spectral measures in q p some of which are self similar measures

spectrum is periodic for n intervals abstract in this paper we study spectral sets which are unions of finitely many intervals in r we show that any spectrum associated with such a spectral set is periodic with the period an integral multiple of the measure of as a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case