beta integers as natural counting systems for quasicrystals recently discrete sets of numbers the integers have been proposed as numbering tools in quasicrystalline studies indeed there exists a unique numeration system based on the irrational in which the integers are all real numbers with no fractional part these integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely is equal to golden mean to or to i e when is one of the self similarity ratios observed in quasicrystalline structures as a matter of fact integers are natural candidates for coordinating quasicrystalline nodes and also the bragg peaks beyond a given intensity in corresponding diffraction patterns they could play the same role as ordinary integers do in crystallography in this paper we prove interesting algebraic properties of the sets when is a quadratic unit pv number a class of algebraic integers which includes the quasicrystallographic cases we completely characterize their respective meyer additive and multiplicative properties where f and g are finite sets and also their respective galois conjugate sets these properties allow one to develop a notion of a quasiring we hope that in this way we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be module on a quasiring in we give also some two dimensional examples with

systems of numeration and fractal functions relating to substitutions french abstract let a be a finite alphabet a substitution over a un n n a fixed point for and for each a a a a real number we establish under some assumptions an asymptotic formula concerning the sum s n i n u i n n this result generalizes some previous results from coquet or brillhart erdos and morton moreover relations with self affine functions in a sense which generalizes a definition from kamae are proved the calculi leave over systems of representation of integers and real numbers

singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

symmetrized β integers this article deals with b numeration systems which are numeration systems with a non integral base b 1 in this framework there exist elements which naturally play the role of integers which are called b integers the set of non negative b integers denoted by z b has various equivalent definitions which arise from different points of view nevertheless these definitions may be generalized on negative real numbers in a non unique way depending on the chosen framework we focus in this article on confluent parry units they are the positive roots of the polynomials x d i 1 d 1kx i 1 where the integers d and k satisfy d 2 and k 1 for any of these numbers we prove that there exists a discrete subset of r that we denote by z b s which is locally isomorphic to z b and such that z b s z b s moreover z b s is a model set and satisfies the inflation property there exists l 1 such that lz b s z b s l is then called the inflation factor for z b s finally we compute inflation factors for z b s of the form b i i being a positive integer

a framework for iterative analysis of non classically damped dynamical systems abstract in this paper we propose a general iterative framework to solve the dynamic problem for linear systems with non classical viscous damping a systematic approach is used to derive families of stationary iterative schemes that as an instance of particular interest decouple the equations of motion for numerical study of the system response for such schemes we present a detailed convergence analysis and propose several solution strategies suitable for a broad class of systems these techniques are based on spectral analysis of particular iteration matrices arising in the derivation and aim at optimizing the convergence performance of the method we demonstrate that the proposed systematic framework based on a novel application of the homotopy analysis method generalizes iterative schemes previously reported in the literature and importantly provides a unified perspective for the study of iterative solutions of dynamic problems further we establish a connection between our results and the theory of iterative schemes for algebraic linear systems thus providing insights on convergence results and applicability of the method numerical examples illustrate the effectiveness of the approach and indicate future research directions

sturmian words β shifts and transcendence consider the minimal shift containing the shift space generated by a given sturmian word in this paper we characterize such and investigate their combinatorial dynamical and measure theoretical properties and prove that such are transcendental numbers

almost asymptotically optimal flag encoding of the integers a simple prefix free encoding scheme for the positive integers is proposed in which a flag of f zeros is used to mark the end of a codeword and bit stuffing is used to prevent premature appearance of the flag in the conventional binary coding of the integers it is shown that this coding scheme is universal in the sense defined by p elias 1975 and that for large f its asymptotic efficiency is virtually 1

on the application of the wald statistic to order estimation of arma models consistent criteria for order estimation of autoregressive moving average arma processes based on the wald statistic are presented the new criteria require only the estimation of the model parameters at the largest order unlike alternative methods in the literature that require the estimation of the model parameters at all possible orders

equidistribution from fractal measures we give a fractal geometric condition for a measure on 0 1 to be supported on points x that are normal in base n i e such that n kx k in mathbb n equidistributes modulo 1 this condition is robust under c 1 coordinate changes and it applies also when n is a pisot number rather than an integer as applications we obtain new results and strengthen old ones about the prevalence of normal numbers in fractal sets and new results on measure rigidity specifically completing host s theorem to multiplicatively independent integers and proving a rudolph johnson type theorem for certain pairs of beta transformations

application of homotopy perturbation methods for solving systems of linear equations abstract in this paper homotopy perturbation methods hpms are applied to obtain the solution of linear systems and conditions are deduced to check the convergence of the homotopy series moreover we have adapted the richardson method the jacobi method and the gauss seidel method to choose the splitting matrix the numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius

the multifractal spectra for the recurrence rates of beta transformations abstract in this paper we show a handy approximate approach to provide a lower bound of the hausdorff dimension of a given subset in 0 1 related to transformation dynamical system here approximation means from special class with shift satisfying the specification property or being subshift of finite type to general 1 as an application we obtain the multifractal spectra for the recurrence rate of the first return time of transformation including the cases returning to the ball and cylinder

diophantine analysis in beta dynamical systems and hausdorff dimensions abstract let x n n 1 0 1 be a sequence of real numbers and let n 0 1 be a positive function using the mass transference principle established by beresnevich and velani 1 we prove that for any x 0 1 the hausdorff dimension of the set 1 t n x x n n for infinitely many n n satisfies a so called 0 1 law according to lim sup n log n n or not where t is the transformation

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

probability one homotopies in computational science probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that under mild assumptions are globally convergent for a wide range of problems in science and engineering convergence theory robust numerical algorithms and production quality mathematical software exist for general nonlinear systems of equations and special cases such as brouwer fixed point problems polynomial systems and nonlinear constrained optimization using a sample of challenging scientific problems as motivation some pertinent homotopy theory and algorithms are presented the problems considered are analog circuit simulation for nonlinear systems reconfigurable space trusses for polynomial systems and fuel optimal orbital rendezvous for nonlinear constrained optimization the mathematical software packages hompack90 and polsys plp are also briefly described

a result on the approximation properties of the orbit of 1 under the β transformation abstract for any real number 1 we denote by t the transformation on the unit interval 0 1 given by t x x x where denotes the integer part of we consider the size of the set of 1 where the orbit of 1 under t ultimately has a positive distance from a given point in 0 1 for any x 0 0 1 and any 0 1 1 we obtain the set of 0 1 such that x 0 is not an accumulation point of the orbit of 1 under t with full hausdorff dimension specifically dim h 0 1 lim inf n t n 1 x 0 0 1 this is a generalization of the result of schmeling for the case where x 0 0 and 0 1 1

wald statistic for model order selection in superposition models a consistent model selection algorithm is presented for superimposed signal models the proposed method is motivated by the wald statistic and reduces the computational complexity of procedures based on the minimum description length mdl principle the procedure is suggested when a noncyclostationary signal model or short data length prevents use of covariance rank test for maximum model order k the procedure provides o k computational savings over an mdl test additionally a proof establishes the consistency of a least squares estimator using overparametrized models finite sample performance of the proposed model selection method is studied via monte carlo simulations for estimating the multipath delays and amplitudes of a chirp signal

on dichotomy law for beta dynamical system in parameter space let varphi mathbb n rightarrow 0 1 be a positive function and t beta be the beta transformation for any beta 1 we prove that the set begin aligned e 0 varphi beta 1 t n beta 1 0 varphi n text for infinitely many n in mathbb n end aligned is of zero or full lebesgue measure in 1 infty according to sum varphi n infty or not as an application we determine the exact lebesgue measure of the set begin aligned mathfrak e 0 l n beta 1 t n beta 1 0 beta l n text for infinitely many n in mathbb n end aligned where l n n ge 1 is a sequence of non negative real numbers

kms states entropy and the variational principle in full c dynamical systems to any periodic and full c dynamical system an invertible operator s acting on the banach space of trace functionals of the fixed point algebra is canonically associated kms states correspond to positive eigenvectors of s a perron frobenius type theorem asserts the existence of kms states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s extremal kms states examples arising from subshifts in symbolic dynamics self similar sets in fractal geometry and noncommutative metric spaces are discussed

a construction on finite automata that has remained hidden abstract we show how a construction on matrix representations of two tape automata proposed by schutzenberger to prove that rational functions are unambiguous can be given a central role in the theory of relations and functions realized by finite automata in such a way that the other basic results such as the cross section theorem its dual the theorem of rational uniformisation or the decomposition theorem of rational functions into sequential functions appear as direct and formal consequences of it

on alpha adic expansions in pisot bases we study a adic expansions of numbers that is to say left infinite representations of numbers in the positional numeration system with the base a where a is an algebraic conjugate of a pisot number b based on a result of bertrand and schmidt we prove that a number belongs to q a if and only if it has an eventually periodic a adic expansion then we consider a adic expansions of elements of the ring z a 1 when b satisfies the so called finiteness property f we give two algorithms for computing these expansions one for positive and one for negative numbers in the particular case that b is a quadratic pisot unit satisfying f we inspect the unicity and or multiplicity of a adic expansions of elements of z a 1 we also provide algorithms to generate a adic expansions of rational numbers in that case

on the computation of the cramer rao bound for arma parameter estimation the cramer rao lower bound crlb provides a useful tool for evaluating the performance of parameter estimation techniques several techniques for the computation of the asymptotic form of the crlb for arma models are presented it is shown that the asymptotic crlb can be expressed as an explicit function of the model parameters

b expansions and symbolic dynamics given a real number p i it is easy to expand x e o i in a canonical way into a series of negative powers of p with integer coefficients less than or equal to p the form of the set of such p expansions which is a matter of symbolic dynamics or language theory varies according to arithmetical properties of p but there are many common features known facts which are few and open questions much more numerous are reviewed in this paper

chaotic and topological properties of β transformations article history in this paper we prove that the transformations are chaotic in the sense of both li yorke and devaney the topological and metric properties of the sets of points with dense or non dense orbits are investigated we also prove the result that the set of points with non dense orbits under the transformation is of full hausdorff dimension for any 1

dimensions of recurrent sets in β symbolic dynamics abstract given 1 let 0 1 t be the dynamical system moreover for any n n and x 0 1 let n x inf j 1 t j x i n x where i n x is the n th cylinder containing x then given any a b 0 with a b the hausdorff dimension of the following recurrent set e a b x 0 1 lim inf n log n x log n a lim sup n log n x log n b admits a dichotomy it is either one or zero according as a 1 or not

fibonacci representations and finite automata finite state automata are used as a simple model of computation since only a finite memory is needed the problem of passing from any representation to the normal representation of an integer within the fibonacci numeration system which is called the process of normalization is addressed it is shown that the normalization can be realized by means of infinite automata more precisely this function can be obtained by the composition of two subsequential transducers that are simply obtained from the linear recurrence definition of the basis of the fibonacci system one processing words from left to right and the other from right to left the normalization although not a sequential process can be obtained in two sequential passes it is proved that it is possible to add to integers written in the fibonacci numeration system of order m by means of a finite state automaton the conversion from a fibonacci representation to the standard binary representation cannot be realized by a finite state automaton

diophantine approximation of the orbit of 1 in the dynamical system of beta expansions we consider the distribution of the orbits of the number 1 under the transformations as varies mainly the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its hausdorff dimension the dimension of the following set is determined where is a given point in and is a sequence of integers tending to infinity as for the proof of this result the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders the set of with a common prefix in the expansion of 1 in the parameter space

deterministic synchronization of automations with bounded delay resume nous montrons qu un automate a delai borne sous sequentiel i e deterministe sur les entrees est equivalent a un automate en ligne c est a dire lettre a lettre apres une periode transitoire fixe ou l automate lit le debut de l entree sans rien sortir et sous sequentiel les resultats de synchronisation etablis anterieurement pour les automates a delai borne ne preservaient pas la sous sequentialite

confluent linear numeration systems abstract numeration systems where the basis is defined by a linear recurrence with integer coefficients are considered a rewriting system is associated with the recurrence in this paper we study the case when it is confluent we prove that the function of normalization which transforms any representation of an integer into the normal one obtained by the usual algorithm can be realized by a finite automation which is the composition of a left subsequential transducer and of a right subsequential transducer associated with the rewriting system addition of integers in a confluent linear numeration system is also computable by a finite automaton these results extend to the representation of real numbers in basis where is the dominant root of the characteristic polynomial of the recurrence

ergodic universality of some topological dynamical systems the krieger generator theorem says that every invertible ergodic measure preserving system with finite measure theoretic entropy can be embedded into a full shift with strictly greater topological entropy we extend krieger s theorem to include toral automorphisms and more generally any topological dynamical system on a compact metric space that satisfies almost weak specification asymptotic entropy expansiveness and the small boundary property as a corollary one obtains a complete solution to a natural generalization of an open problem in halmos s 1956 book regarding an isomorphism invariant that he proposed

maximum likelihood and lower bounds in system identification with non gaussian inputs we consider the problem of estimating the parameters of an unknown discrete linear system driven by a sequence of independent identically distributed i i d random variables whose probability density function pdf may be non gaussian we assume a general system structure that may contain causal and noncausal poles and zeros the parameters characterizing the input pdf may also be unknown we derive an asymptotic expression for the cramer rao lower bound and show that it is the highest worst in the gaussian case indicating that the estimation accuracy can only be improved when the input pdf is non gaussian it is further shown that the asymptotic error variance in estimating the system parameters is unaffected by lack of knowledge of the pdf parameters and vice verse computationally efficient gradient based algorithms for finding the maximum likelihood estimate of the unknown system and pdf parameters which incorporate backward filtering for the identification of non causal parameters are presented the dual problem of blind deconvolution equalization is considered and asymptotically attainable lower bounds on the equalization performance are derived these bounds imply that it is preferable to work with compact equalizer structures characterized by a small number of parameters as the attainable performance depend only on the total number of equalizer parameters

recursive estimation of mixed autoregressive moving average order summary the order p q of an autoregressive moving average sequence y t may be estimated by minimizing a criterion log 2 p q log t t with respect to p and q where 2 is the maximum likelihood estimate of the variance of the innovations t it is suggested that instead a2 be estimated from a series of regressions of y t on y t 1 y t 2 y t p 5 t 1 t q where the t are obtained by fitting a long autoregression to the data it is shown how the sequence of regressions may for p q be economically recursively calculated by embedding them in a sequence of bivariate autoregressions asymptotic properties of the procedure are established under very general conditions

language complexity of rotations and sturmian sequences given a rotation of the circle we study the complexity of formal languages that are generated by the itineraries of interval covers these languages are regular iff the rotation is rational in the case of irrational rotations our study reduces to that of the language complexity of the corresponding sturmian sequences we show that for a large class of irrationals including e all quadratic numbers and more generally all hurwitz numbers the corresponding languages can be recognized by a nondeterministic turing machine in linear time in other words belongs to nlin

robust transmission of unbounded strings using fibonacci representations families of fibonacci codes and fibonacci representations are defined their main attributes are robustness manifesting itself by the local containment of errors and simple encoding and decoding the main application explored is the transmission of binary strings in which the length is in an unknown range using robust fibonacci representations instead of the conventional error sensitive logarithmic ramp representation though the former is asymptotically longer than the latter the former is actually shorter for very large initial segments of integers

regular universal codeword sets corresp proof is given that no regular or synchronizable universal codeword sets can achieve optimality some asymptotic properties of a remarkable class of regular codeword sets related to the fibonacci representation of integers are also obtained

application of homotopy perturbation and variational iteration methods to nonlinear heat transfer and porous media equations perturbation methods depend on a small parameter which is difficult to be found for real life nonlinear problems to overcome this shortcoming two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article one is he s variational iteration method vim and the other is the homotopy perturbation method hpm the vim is to construct correction functionals using general lagrange multipliers identified optimally via the variational theory and the initial approximations can be freely chosen with unknown constants the hpm deforms a difficult problem into a simple problem which can be easily solved nonlinear convective radiative cooling equation nonlinear heat equation porous media equation and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures comparison of the applied methods with exact solutions reveals that both methods are tremendously effective

approximation orders of real numbers by beta expansions we prove that almost all real numbers with respect to lebesgue measure are approximated by the convergents of their beta expansions with the exponential order beta n moreover the hausdorff dimensions of sets of the real numbers which are approximated by all other orders are determined these results are also applied to investigate the orbits of real numbers under beta transformation the shrinking target type problem the diophantine approximation and the run length function of beta expansions