on the complexity of algebraic numbers i expansions in integer bases let b 2 be an integer we prove that the b ary expansion of every irrational algebraic number cannot have low complexity furthermore we establish that irrational morphic numbers are transcendental for a wide class of morphisms in particular irrational automatic numbers are transcendental our main tool is a new combinatorial transcendence criterion

monte carlo simulations of wave scattering from lossy dielectric random rough surfaces using the physics based two grid method and the canonical grid method in using the method of moments to solve scattering by lossy dielectric surfaces usually a single dense grid sdg with 30 points per wavelength is required for accurate results a single coarse grid scg of ten points per wavelength does not give accurate results however the central processing unit cpu and memory requirements of sdg are much larger than that of scg in a physics based two grid method pbtg two grids are used a dense grid and a coarse grid the method is based on the two observations 1 green s function of the lossy dielectric is attenuative and 2 the free space green s function is slowly varying on the dense grid in this paper the pbtg method is combined with the banded matrix iterative approach canonical grid method to solve rough surface scattering problem for both te and tm cases and also for near grazing incidence we studied cases of dielectric permittivities as high as 25 i spl epsiv sub 0 and incidence angle up to 85 spl deg salient features of the numerical results are 1 an scg has poorer accuracy for tm case than te case 2 pbtg banded matrix iterative approach canonical grid bmia cag method speeds up cpu and preserves the accuracy it has an accuracy comparable to single dense grid and yet has cpu comparable to single coarse grid 3 pbtg bmia cag gives accurate results for emissivity calculations and also for low grazing backscattering problems lgba and 4 the computational complexity and the memory requirements of the present algorithm are o n log n and o n respectively where n is the number of surface unknowns on the coarse grid

reversals and palindromes in continued fractions several results on continued fractions expansions are on indirect consequences of the mirror formula we survey occurrences of this formula for sturmian real numbers for simultaneous diophantine approximation and for formal power series

mesures de transcendance et aspects quantitatifs de la methode de thue siegel roth schmidt a proof of the transcendence of a real number based on the thue siegel roth schmidt method involves generally a sequence n n 1 of algebraic numbers of bounded degree or a sequence xn n 1 of integer r tuples in the present paper we show how such a proof can produce a transcendence measure for if one is able to quantify the growth of the heights of the algebraic numbers n or of the points xn our method rests on the quantitative schmidt subspace theorem we further give several applications including to certain normal numbers and to the extremal numbers introduced by roy

scattering by anisotropic rough layered 2d interfaces we propose a statistical study of the scattering of an incident plane wave by a stack of two two dimensional rough interfaces the interfaces are characterized by gaussian height distributions with zero mean values and gaussian correlation functions the electromagnetic fields are represented by rayleigh expansions and a perturbation method is used for solving the boundary value problem and determining the first order scattering amplitudes for slightly rough interfaces with a finite extension we show that the modulus of the co and cross polarized scattering amplitudes follows a hoyt law and that the phase is not uniformly distributed for interfaces with an infinite extension the modulus follows a rayleigh law and the phase is uniformly distributed we show that these results are true for correlated or uncorrelated interfaces and for isotropic or anisotropic interfaces

periodicity repetitions and orbits of an automatic sequence we revisit a technique of s lehr on automata and use it to prove old and new results in a simple way we give a very simple proof of the 1986 theorem of honkala that it is decidable whether a given k automatic sequence is ultimately periodic we prove that it is decidable whether a given k automatic sequence is overlap free or squarefree or cubefree etc we prove that the lexicographically least sequence in the orbit closure of a k automatic sequence is k automatic and use this last result to show that several related quantities such as the critical exponent irrationality measure and recurrence quotient for sturmian words with slope a have automatic continued fraction expansions if a does

on the maillet baker continued fractions we use the schmidt subspace theorem to establish the transcendence of a class of quasi periodic continued fractions this improves earlier works of maillet and of a baker we also improve an old result of davenport and roth on the rate of increase of the denominators of the convergents to any real algebraic number

electromagnetic scattering from a rough layer propagation inside layer expansion method combined to an updated bmia cag approach an efficient method is developed to calculate the bistatic scattering coefficient bsc from a stack of two one dimensional rough interfaces separating homogeneous media the propagation inside layer expansion pile method recently published by dechamps was efficient with a complexity o n2 n being the number of samples per interface to reduce this complexity a fast method valid for a single rough surface the banded matrix iterative approach canonical grid bmia cag is combined to the pile method furthermore the calculation of the coupling interactions between both interfaces are also accelerated using a new method similar to the bmia cag the pile method reaches then a complexity of only o nlogn a study of the convergence of the pile is done and compared to another efficient method the forward backward

infinite words with linear subword complexity abstract in section 1 we study the relations among some combinatorial properties of infinite words especially in the case of infinite words with linear subword complexity the main result of section 1 concerns a permutation property of infinite words with linear complexity in section 2 we investigate the special case of the sturmian infinite words the main result is that a sturmian infinite word associated to a real number contains no k th powers iff has a continued fraction expansion with bounded partial quotients in section 3 it is shown that some of the results of sections 1 and 2 are optimal

relation between powers of factors and the recurrence function characterizing sturmian words in this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of sturmian words as a by product we give a new proof of the theorem describing the index of a sturmian word in terms of the continued fraction expansion of its slope this theorem was independently proved in a carpi a de luca special factors periodicity and an application to sturmian words acta inform 36 2000 983 1006 and d damanik d lenz the index of sturmian sequences european j combin 23 2002 23 29

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

a fast hybrid method for scattering from a large object with dihedral effects above a large rough surface a new hybrid numerical method is described for scattering from an electrically large perfectly conducting object with dihedral effects above a very long one dimensional rough surface two dimensional problem such a problem involves a large number of unknowns and cannot be solved easily with a conventional method of moments by using a direct lu inversion thus to solve this issue the extended pile method is combined with the forward backward spectral acceleration fbsa for the local interactions on the rough surface and with the second order physical optics po2 approximation for the local interactions on the object classically objects under test do not present dihedral effects and in the high frequency domain the first order po po1 provides the main contribution in opposite in this paper since a cross is considered the second order inner reflections contribute significantly and the po2 must be included by assuming a gaussian process with a gaussian height spectrum this new hybrid method e pile fbsa po2 is tested against the rigorous e pile fbsa method direct lu inversion on the object as functions of the object inclination the polarization and the incidence angle

on a problem in simultaneous diophantine approximation littlewood s conjecture a consequence of hurwitz s theorem is that the right hand side of the above inequality cannot be improved by an arbitrary positive constant s more precisely for s l x 5 there exist real numbers a e i for which the inequality ilqall cq 1 has at most a finite number of solutions these a are the badly approximable numbers and we will denote by b a d the set of all such numbers that is

on the least number of palindromes contained in an infinite word we investigate the least number of palindromic factors in an infinite word we first consider general alphabets and give answers to this problem for periodic and non periodic words closed or not under reversal of factors we then investigate the same problem when the alphabet has size two

an iterative solution of one dimensional rough surface scattering problems based on a factorization of the helmholtz operator an iterative series solution for two dimensional scattering problems derived from a minimally coupled factorization of the helmholtz equation is presented the helmholtz operator is represented as a product of one way wave operators and a corresponding coupling operator coupling between oppositely propagating fields is shown to be predominantly localized and this permits an accurate and numerically efficient approximation of the coupling operator the resulting iterative series is shown to be rapidly convergent for the problem of scattering from impenetrable rough surfaces

limit values of the recurrence quotient of sturmian sequences abstract the recurrence quotient is a real number associated to any symbolic sequence in the case of sturmian sequences morse and hedlund proved that the smallest possible value is 5 5 2 we study here the structure of the set of values taken by the recurrence quotient for a sturmian sequence

on the complexity of algebraic numbers ii continued fractions the continued fraction expansion of an irrational number alpha is eventually periodic if and only if alpha is a quadratic irrationality however very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three because of some numerical evidence and a belief that these numbers behave like most numbers in this respect it is often conjectured that their partial quotients form an unbounded sequence more modestly we may expect that if the sequence of partial quotients of an irrational number alpha is in some sense simple then alpha is either quadratic or transcendental the term simple can of course lead to many interpretations it may denote real numbers whose continued fraction expansion has some regularity or can be produced by a simple algorithm by a simple turing machine for example or arises from a simple dynamical system the aim of this paper is to present in a unified way several new results on these different approaches of the notion of simplicity complexity for the continued fraction expansion of algebraic real numbers of degree at least three

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

approximation to real numbers by cubic algebraic integers i in 1969 h davenport and w m schmidt studied the problem of approximation to a real number xi by algebraic integers of degree at most 3 they did so using geometry of numbers by resorting to the dual problem of finding simultaneous approximations to xi and xi 2 by rational numbers with the same denominator in this paper we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers xi we give several properties of these numbers including measures of approximation by rational numbers by quadratic real numbers and by algebraic integers of degree at most 3

on the co polarized phase difference of rough layered surfaces formulae derived from the small perturbation method we determine the statistical distribution of the co polarized phase difference of fields scattered from a stack of two two dimensional rough interfaces in the incidence plane the electromagnetic fields are represented by rayleigh expansions and a perturbation method is used to solve the boundary value problem and to determine the first order scattering amplitudes for slightly rough interfaces with infinite length and gaussian height distributions we show that the probability density function is only a function of two parameters for a sand layer on a granite surface in backscattering configurations we study the influence of the incidence angle the layer thickness the cross spectral density and the wave frequency upon both parameters of the probability law

monte carlo simulations of large scale problems of random rough surface scattering and applications to grazing incidence with the bmia canonical grid method scattering of a te incident wave from a perfectly conducting one dimensional random rough surface is studied with the banded matrix iterative approach canonical grid bmia cag method the bmia cag is an improvement over the previous bmia the key idea of bmia cag is that outside the near field interaction the rest of the interactions can be translated to a canonical grid by taylor series expansion the use of a flat surface as a canonical grid for a rough surface facilitates the use of the fast fourier transform for nonnear field interaction the method can be used for monte carlo simulations of random rough surface problems with a large surface length including all the coherent wave interactions within the entire surface we illustrate results up to a surface length of 2500 wavelengths with 25000 surface unknowns the method is also applied to study scattering from random rough surfaces at near grazing incidence the numerical examples illustrate the importance of using a large surface length for some backscattering problems

sturmian words lyndon words and trees abstract we prove some new combinatorial properties of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 4 3 we show that aperb a b st lynd where st is the set of the finite factors of all infinite sturmian words and lynd is the set of the lyndon words on the alphabet a b it is also shown that aperb a b cp where cp is the set of christoffel primitive words such words can be defined in terms of the slope of the words and of their prefixes 1 from this result one can derive in a different way by using a theorem of borel and laubie that the elements of the set aperb are lyndon words we prove the following correspondence between the ratio p q of the periods p q p q of w per a a b and the slope w b 1 w a 1 of the corresponding christoffel primitive word awb if p q has the development in continued fractions 0 h 1 h n 1 h n 1 then has the development in continued fractions 0 h n h 2 h 1 1 this and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the stern brocot and the raney numbers of a node in a complete binary tree however one needs some further results more precisely we label the binary tree with standard pairs standard tree christoffel pairs christoffel tree and the elements of per farey tree the main theorem is the following if the node w is labeled by the standard pair u v by the christoffel pair x y and by w per then uv wab xy awb the stern brocot number sb w is equal to the slope of the standard word uv and of the christoffel word xy while the raney number ra w is equal to the ratio of the minimal period of wa and the minimal period of wb some further auxiliary results are also derived

on the canonical grid method for two dimensional scattering problems the banded matrix iterative approach with a canonical grid expansion bmia cag has been shown by tsang et al see ibid vol 43 p 851 9 1995 to be an efficient method for the calculation of scattering for near planar two dimensional 2 d geometries such as one dimensional rough surfaces however in the article of tsang et al only the first three terms in the canonical grid series for te polarization above a perfectly conducting surface were discussed and methods for implementing only a portion of these terms were presented in this paper a general form for all terms in the canonical grid series is provided for both te and tm polarizations above an impedance surface and an efficient algorithm for calculating their contributions is described the relationship between the canonical grid and operator expansion methods is also discussed a sample surface scattering problem is shown to illustrate the utility of higher order terms in the canonical grid method

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

approximation to real numbers by cubic algebraic integers ii it has been conjectured for some time that for any integer n 2 any real number e 0 and any transcendental real number there would exist infinitely many algebraic integers a of degree at most n with the property that h n e where h denotes the height of a although this is true for n 2 we show here that for n 3 the optimal exponent of approximation is not 3 but 3 5 2 2 618

a new numerical method for rough surface scattering calculations a new approach to solving the magnetic field integral equation mfie for the current induced on a infinite perfectly conducting rough surface is presented by splitting the propagator matrix into contributions from the left and from the right of the point of observation a second kind integral equation can be formed with a new born term and a new kernel following discretization of this new integral equation the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional lu decomposition where the time saving factor is roughly n 3 where n is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated while the new born term is usually adequate for scattered field calculations the new discretized integral equation can be iterated to amy desired accuracy with no apparent convergence problems results are presented for one dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40 which illustrate the robustness of the new born term

forward backward method with spectral acceleration for scattering from layered rough surfaces a fast method of moments is presented to calculate electromagnetic wave scattering from layered one dimensional rough surfaces the formulation is provided for m stratified homogeneous regions separated by m 1 rough surfaces and solved using point matching and pulse basis functions compared to the single surface case the solution of scattering from m 1 surfaces requires significantly more memory and computational time to facilitate the solution the forward backward method with spectral acceleration is applied as an example a dielectric layer on a perfect electric conductor surface is studied the numerical results are compared with the analytical solution for layered flat surfaces to partly validate the formulation the accuracy efficiency and convergence of the method are then studied for various rough surfaces and layer permittivities

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

sturmian words and words with a critical exponent let s be a standard sturmian word that is a fixed point of a non trivial homomorphism associated to the infinite word s is a unique irrational number with 0 0 it contains a fractional power with exponent greater than e here is a constant that depends on the constant is given explicitly using these results we are able to give a short proof of mignosi s theorem and give an exact evaluation of the maximal power that can occur in a standard sturmian word

some properties of the factors of sturmian sequences in this paper we introduce the singular words of sturmian sequences which play an important role in studying the properties of the factors of sturmian sequence we also completely determine the powers of the factors the overlaps of the factors and the structure of the palindromes of the factors

scattering from multilayer rough surfaces based on the extended boundary condition method and truncated singular value decomposition electromagnetic scattering from rough surfaces has been investigated extensively for the past decades there exist analytical solutions to rough surface scattering such as small perturbation method spm or kirchhoff approximation ka these solutions however are limited to either small or large roughness regimes recent efforts have been put forth to study electromagnetic scattering of multilayer rough surface in the application of deep soil moisture and ice property estimation in this paper multilayer rough surface scattering is analyzed based on the extended boundary condition method ebcm and truncated singular value decomposition the method of extended boundary condition can accurately handle scattering from a slightly rough surface for mild and large roughness regime ill conditioned matrices can arise from ebcm in this paper we present a regularization scheme to mitigate this ill conditioning effect via truncated singular value decomposition tsvd the truncation parameters are set to enforce power conservation relations across all rough interfaces bistatic scattering coefficients are then obtained by incoherently averaging the power computed from the resulting floquet modes in the scattering directions therefore the electromagnetic wave interactions in multilayer rough surfaces are analyzed in a systematic way finally the results are validated against the analytical spm solution to two rough interface rough surface scattering in addition the formulation is extended to solve scattering from three rough interface rough surfaces several results are reported

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

a numerical strategy to discretize and solve the poisson equation on dynamically adapted multiresolution grids for time dependent streamer discharge simulations we develop a numerical strategy to solve multi dimensional poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts the method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time dependent pdes such an approach guarantees a numerical solution of the poisson equation within a user defined accuracy tolerance most adaptive meshing approaches in the literature solve elliptic pdes level wise and hence at uniform resolution throughout the set of adapted grids here we introduce a numerical procedure to represent the elliptic operators on the adapted grid strongly coupling inter grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes the discrete poisson equation is solved at once over the entire computational domain as a completely separate process the accuracy and numerical performance of the method are assessed in the context of streamer discharge simulations