bounds for fourth order 0 1 difference equations this note examines bounds for fourth order linear difference equations with coefficients restricted to the unit interval it is shown that all solutions are of order strictly less than 3 2 n the bound is shown to be nearly best possible applications to zero one banded matrices are also discussed

hausdorff dimension of univoque sets and devil s staircase abstract we fix a positive integer m and we consider expansions in arbitrary real bases q 1 over the alphabet 0 1 m we denote by u q the set of real numbers having a unique expansion completing many former investigations we give a formula for the hausdorff dimension d q of u q for each q 1 furthermore we prove that the dimension function d 1 0 1 is continuous and has bounded variation moreover it has a devil s staircase behavior in q where q denotes the komornik loreti constant although d q d q for all q q we have d 0 a e in q during the proofs we improve and generalize a theorem of erd s et al on the existence of large blocks of zeros in expansions and we determine for all m the lebesgue measure and the hausdorff dimension of the set u of bases in which x 1 has a unique expansion

a mixed variational formulation based on exact intrinsic equations for dynamics of moving beams abstract a nonlinear intrinsic formulation for the dynamics of initially curved and twisted beams in a moving frame is presented the equations are written in a compact matrix form without any approximations to the geometry of the deformed beam reference line or to the orientation of the intrinsic cross section frame in accordance with previously published work when effects of warping on the dynamics and local constraints on the cross sectional deformation are ignorable the in and out of plane st venant warping displacements which are fully coupled for nonhomogeneous anisotropic beams need only be included explicitly in the determination of a suitable elastic law a two dimensional problem and need not be considered explicitly in the one dimensional equations governing global deformation in this paper it is presupposed that such an elastic law is given as a one dimensional strain energy function thus the present equations which are based on only six generalized strain variables are valid for beams with closed cross sections and for which warping is unrestrained when simplified for various special cases they agree with similar intrinsic equations in the literature although the resulting equations are newtonian in structure closely resembling euler s dynamical equations for a rigid body the formulation adheres to a variational approach throughout thus providing a link between newtonian and energy based methods in particular the present development provides substantial insight into the relationships among variational formulations in which different displacement and rotational variables are used as well as between these formulations and newtonian ones for computational purposes a compact and complete mixed variational formulation is presented that is ideally suited for development of finite element analyses finally a specialized version of the intrinsic equations is developed in which shear deformations is suppressed

on beta expansions for pisot numbers given a number 1 the beta transformation t t is defined for x 0 1 by tx x mod 1 the number is said to be a beta number if the orbit t n 1 is finite hence eventually periodic in this case is the root of a monic polynomial r x with integer coefficients called the characteristic polynomial of if p x is the minimal polynomial of then r x p x q x for some polynomial q x it is the factor q x which concerns us here in case is a pisot number it is known that all pisot numbers are beta numbers and it has often been asked whether q x must be cyclotomic in this case particularly if 1 2 we answer this question in the negative by an examination of the regular pisot numbers associated with the smallest 8 limit points of the pisot numbers by an exhaustive enumeration of the irregular pisot numbers in 1 1 9324 u 1 9333 1 96 an infinite set by a search up to degree 50 in 1 9 2 to degree 60 in 1 96 2 and to degree 20 in 2 2 2 we find the smallest counterexample the counterexample of smallest degree examples where q x is nonreciprocal and examples where q x is reciprocal but noncyclotomic we produce infinite sequences of these two types which converge to 2 from above and infinite sequences of with q x nonreciprocal which converge to 2 from below and to the 6th smallest limit point of the pisot numbers from both sides we conjecture that these are the only limit points of such numbers in 1 2 the pisot numbers for which q x is cyclotomic are related to an interesting closed set of numbers f introduced by flatto lagarias and poonen in connection with the zeta function of t our examples show that the set s of pisot numbers is not a subset of f

intrinsic equations for the nonlinear dynamics of space beams equations are developed for the fully intrinsic nonlinear dynamics of a spatially curved pretwisted beam direct deformation measures are used as variables the equations are not referred to any global frame of references fixed moving or floating large deformations can be accommodated the bernoulli euler assumption is introduced for the gross dynamics but not for the constitutive relations the latter may be also inelastic and time dependent initial and edge conditions force and kinematic types are presented specific equations are also given for the plane case and for the linear case some of the merits and drawbacks of this approach are discussed the extension to a continuum is also briefly discussed

an intrinsic beam model based on a helicoidal approximation part i formulation the focus of the present work is directed towards the one dimensional non linear analysis of space curved and twisted beams undergoing large displacements and finite rotations according to cosserat s model a beam is intended here as a continuum generated by the rigid motion of a cross section along a curve the novelty of the proposed methodology is implied in the assumption that the reference line of the beam both in the undeformed and in the deformed configurations is a helicoid in space this formulation radically departs from the classical beam models based on polynomial interpolations of the independent fields in fact within the framework of helicoidal geometry a powerful description of space curved beam kinematics with large displacements and finite rotations can be developed furthermore the beam model developed herein can be proved to enjoy several invariance properties that make it particularly attractive for non linear analysis the formulation discussed in this work adheres to a mixed variational principle the independent fields being the generalized displacements and the sectional stress resultants in a companion paper we address the linearization of the governing weak form the finite element implementation of the proposed methodology with a novel incremental treatment of finite rotations and the development of a helicoidal model for rigid body dynamics using finite elements in time

explicit bounds for multidimensional linear recurrences with restricted coefficients this note employs path counting techniques to extend recent results on bounds for odd order linear recurrences to higher dimensions the results imply optimal zero free polydisks for multivariable power series with 0 1 coefficients among the applications is a result that states that the optimal zero free polydisk has radius approximately 1 v 1 for large dimensions v

cross sectional analysis of composite beams including large initial twist and curvature effects an asymptotically exact methodology based on geometrically nonlinear three dimensional elasticity is presented for cross sectional analysis of initially curved and twisted nonhomogeneous anisotropic beams the analysis is subject only to the restrictions that the strain is small relative to unity and that the maximum dimension of the cross section is small relative to the wavelength of the deformation and to the minimum radius of curvature and or twist the final one dimensional strain energy per unit length exhibits asymptotically correct second order dependence of the initial curvature and twist parameters cross sectional constants of the one dimensional theory are obtained via finite element discretization over the cross sectional plane numerical results obtained for both isotropic and composite beams are compared with published results from special purpose analyses for initially twisted straight beams as well as initially curved untwisted beams the agreement with previously published results is excellent

on asymptotically correct timoshenko like anisotropic beam theory abstract this paper presents a finite element cross sectional beam analysis capable of capturing transverse shear effects the approach uses the variational asymptotic method and can handle beams of general cross sectional shape and arbitrary anisotropic material the work builds on previous works which deal with development of the classical beam theory which includes only extension torsion and bending a timoshenko like formulation is sought to achieve a refined theory with simple boundary conditions apart from some simple special cases it is shown that this problem is overdetermined however it is possible to obtain the best possible solution using the least squares minimization technique the geometrical meaning of the shear variable for this formulation is also derived results are found to be in good agreement with published results for the shear stiffness coefficients for both isotropic and anisotropic beams

statically and kinematically exact nonlinear theory of rods and its numerical verification abstract the entirely general nonlinear theory of rods is developed two vector equilibrium equations valid on the rod axis are derived exactly by direct integration of the mechanical balance laws of continuum mechanics over the rod cross section then the energetically conjugate exact 6 field kinematics of the rod axis is implied in an integral identity which expresses the principle of the virtual work the kinematics consists of the displacement vector of the axis describing the average translation and the proper orthogonal tensor representing the average rotation of the cross section the integral identity also suggests the required general form of the constitutive relations they are already approximate from the physical side are then the best place to make geometric approximations and abbreviations to a certain constrained theory the constitutive relations the only nonexact segment of the theory are treated here as an interchangeable element supplementing the exactly formulated principle of virtual work the solution of the nonlinear rod problem becomes nonstandard here for the configuration space is the nonlinear manifold containing the rotation group so 3 applying the finite element method the proper parameterization linearization interpolation accumulation and discretization procedures in so 3 group are discussed a library of 2 3 4 node 1 d isoparametric elements is developed and the computer program ram is worked out the presented numerical tests contain several problems of highly nonlinear plane and spatial deformations of rods with constitutive relations taken at the moment from the classical isotropic elastic beams in all the tests the developed elements in comparison with the solutions known in the literature show a good performance

monotone convex sequences and cholesky decomposition of symmetric toeplitz matrices this paper studies off diagonal decay in symmetric toeplitz matrices it is shown that if the generating sequence of the matrix is monotone positive and convex then the monotonicity and positivity are maintained through triangular decomposition the work is motivated by recent results on explicit bounds for inverses of triangular matrices

second order bounds for linear recurrences with negative coefficients this paper introduces a generalization of fibonacci and pell polynomials in order to obtain optimal second order bounds for general linear recurrences with negative coefficients an important aspect of the derived bounds is that they are applicable and easily computable the results imply bounds on all entries in inverses of triangular matrices as well as on coefficients of reciprocals of power series

nonlinear modal condensation of large finite element models application of hodges s intrinsic theory a method to obtain geometrically nonlinear reduced order descriptions of structures defined by a large finite element model is presented the full model is used to identify all the coefficients in

a bound for linear recurrence relations with unbounded order this paper considers linear recurrence relations with unbounded order where the coefficients are restricted to intervals of negative real numbers the optimal inequalities under the given constraints exhibit a second order structure

computing yapunov constants for random recurrences with smooth coefficients in recent years there has been much interest in the growth and decay rates lyapunov constants of solutions to random recurrences such as the random fibonacci sequence x n 1 x n x n 1 many of these problems involve nonsmooth dynamics nondi1erentiable invariant measures making computations hard here however we consider recurrences with smooth random coe cients and smooth invariant measures by computing discretised invariant measures and applying richardson extrapolation we can compute lyapunov constants to 10 digits of accuracy in particular solutions to the recurrence x n 1 x n cn 1x n 1 where the cn are independent standard normal variables increase exponentially almost surely at the asymptotic rate 1 0574735537 n solutions to the related recurrences x n 1 cn 1x n x n 1 and x n 1 cn 1x n dn 1x n 1 where the dn are also independent standard normal variables increase decrease at the rates 1 1149200917 n and 0 9949018837 n respectively c 2001 elsevier science b v all rights reserved

stability analysis of columns with imperfection in this paper geometrically exact fully intrinsic equations and variational asymptotic beam sections are used to study the buckling and postbuckling of a column under self weight initial curvature effects both generalized one dimensional beam equations and the cross sectional stiffness properties the effects of initial curvature have been studied on buckling and postbuckling in this paper this problem is a valuable test case for the utility of the fully intrinsic equations and variational asymptotic beam sections

on timoshenko like modeling of initially curved and twisted composite beams abstract a generalized finite element based cross sectional analysis for nonhomogenous initially curved and twisted anistropic beams is formulated from geometrically nonlinear three dimensional 3 d elasticity the 3 d strain field is formulated based on the concept of decomposition of the rotation tensor and is given in terms of one dimensional 1 d generalized strains and a 3 d warping displacement that is obtained from the formulation not assumed the warping is found in terms of the 1 d strains via the variational asymptotic method vam in this paper a timoshenko like model is presupposed for a beam with cross sectional characteristic length h wavelength of deformation given by l and the magnitude of the radius of initial curvature and or twist is taken to be of the order r first a solution for the asymptotically correct refinement of classical anisotropic beam theory for initially curved and twisted beams through o h2 r2 is obtained next the o h2 l2 correction is computed it is known that timoshenko like theory is not capable of capturing all the o h2 l2 corrections for generally anisotropic beams however if all the o h2 l2 terms are known then the corresponding timoshenko like theory is uniquely defined numerical results are presented to illustrate the trends of the various classical extension twist bending twist and extension bending and nonclassical couplings extension shear bending shear and shear torsion as the initial twist and curvatures are varied

littlewood type problems on 0 1 we consider the problem of minimizing the uniform norm on 0 1 over non zero polynomials p of the form p x sum j 0 n a jx j quad text with a j le 1 a j in bbb c where the modulus of the first non zero coefficient is at least delta 0 essentially sharp bounds are given for this problem an interesting related result states that there are absolute constants c 1 0 and c 2 0 such that exp c 1 sqrt n le inf 0 ne p in cal f n p 0 1 le exp c 2 sqrt n for every n ge 2 where cal f n denotes the set of polynomials of degree at most n with coefficients from 1 0 1 this chebyshev type problem is closely related to the question of how many zeros a polynomial from the above classes can have at 1 we also give essentially sharp bounds for this problem em inter alia we prove that there is an absolute constant c 0 such that every polynomial p of the form p x sum j 0 n a j x j quad text with a j le 1 a 0 a n 1 a j in bbb c has at most c sqrt n real zeros this improves the old bound c sqrt n log n given by schur in 1933 as well as more recent related bounds of bombieri and vaaler and up to the constant c this is the best possible result all the analysis rests critically on the key estimate stating that there are absolute constants c 1 0 and c 2 0 such that f 0 c 1 a le exp c 2 a f 1 a 1 for every f in cal s and a in 0 1 where cal s denotes the collection of all analytic functions f on the open unit disk d z in bbb c z f z le frac 1 1 z quad text for z in d

on the continuity of the hausdorff dimension of the univoque set abstract in a recent paper komornik et al 2017 19 proved a long conjectured formula for the hausdorff dimension of the set u q of numbers having a unique expansion in the non integer base q and showed that this hausdorff dimension is continuous in q unfortunately their proof contained a gap which appears difficult to fix this article gives a completely different proof of these results using a more direct combinatorial approach

validation of the variational asymptotic beam sectional analysis the computer program vabs variational asymptotic beam section analysis uses the variational asymptotic method to split a three dimensional nonlinear elasticity problem into a twodimensional linear cross sectional analysis and a one dimensional nonlinear beam problem this is accomplished by taking advantage of certain small parameters inherent to beam like structures vabs is able to calculate the one dimensional cross sectional stiffness constants with transverse shear and vlasov refinements for initially twisted and curved beams with arbitrary geometry and material properties several validation cases are presented first an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method and the solution is shown to be identical to that obtained from the theory of elasticity the shear center locations calculated by vabs for various cross sections agree well with those obtained from common engineering analyses comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model for numerical validation values of the one dimensional variables are extracted from an abaqus model and compared with results from a one dimensional beam analysis using cross sectional constants from vabs furthermore point wise three dimensional stress and strain fields are recovered using vabs and the correlation with the three dimensional results from abaqus is excellent finally classical theory is shown to be insufficient for general purpose beam modeling appropriate refined theories are recommended for some classes of problems

multiple expansions of real numbers with digits set 0 1 q for q 1 we consider expansions in base q with digits set left 0 1 q right let mathcal u q be the set of points which have a unique q expansion for k 2 3 ldots aleph 0 let mathcal b k be the set of bases q 1 for which there exists x having precisely k different q expansions and for q in mathcal b k let mathcal u q k be the set of all such x s which have exactly k different q expansions in this paper we show that begin aligned mathcal b aleph 0 2 infty quad text and quad mathcal b k q c infty quad text for any quad k ge 2 end aligned where q c approx 2 32472 is the appropriate root of x 3 3x 2 2x 1 0 moreover we show that for any integer k ge 2 and any q in mathcal b k the hausdorff dimensions of mathcal u q k and mathcal u q are the same i e begin aligned dim h mathcal u q k dim h mathcal u q quad text for any quad k ge 2 end aligned finally we conclude that the set of points having a continuum of q expansions has full hausdorff dimension

a solution of the eigenproblem for undamped gyroscopic systems with the lanczos algorithm this paper presents an efficient numerical solution of the quadratic eigenproblem arising in the analysis of gyroscopic systems such problems are known to reduce to a generalized linear eigenproblem defined by two real non singular matrices one symmetric and one skew symmetric for this class of problems the general lanczos algorithm for unsymmetric matrices is shown to simplify considerably and yields an efficient solution of the problem full advantage can be taken of the sparsity of the matrices and of the specific nature of gyroscopic systems numerical examples are presented which demonstrate the efficiency and accuracy of the solution procedure

a fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three dimensional stress effects abstract a new approach is used to develop a geometrically exact nonlinear beam model for naturally curved and twisted solid composite rotor blades undergoing large vibrations in three dimensional space a combination of the new concepts of local displacements and local engineering stresses and strains a new interpretation and manipulation of the virtual local rotations an exact coordinate transformation and the extended hamilton principle is used to derive six fully nonlinear equations of motion describing one extension two bending one torsion and two shearing vibrations of composite beams the formulation is based on an energy approach but the derivation is fully correlated with the newtonian approach and provides a straightforward explanation of all nonlinear structural terms without using complex tensor operations or asymptotic expansions the theory accounts for in plane and out of plane warpings due to bending extensional shearing and torsional loadings elastic couplings among warpings and three dimensional stress effects by using the results of a two dimensional static sectional finite element analysis also the theory fully accounts for extensionality initial curvatures and geometric nonlinearities the equations display linear elastic couplings due to structural anisotropy and initial curvatures and nonlinear geometric couplings the theory contains most of the existing beam theories as special cases and the final equations of motion are put in compact matrix form

aeroelastic stability of hingeless rotor blade in hover using large deflection theory the coupled flap lag torsion aeroelastic stability of a hingeless rotor blade in hover is investigated using finite elements based on large deflection beam theory the finite element equations of motion for beams undergoing arbitrary large displacements and rotations but small strains are obtained from hamilton s principle the stability boundary is calculated assuming blade motions to be small perturbations about the nonlinear steady equilibrium deflections which are obtained through an iterative newton raphson method the p k mudal flutter analysis based on coupled rotating natural modes is used various unsteady two dimensional strip theories are used to evaluate the aerodynamic loads

entropy topological transitivity and dimensional properties of unique q expansions let m be a positive integer and q in 1 m 1 we consider expansions of real numbers in base q over the alphabet 0 ldots m in particular we study the set mathcal u q of real numbers with a unique q expansion and the set mathbf u q of corresponding sequences it was shown by komornik kong and li that the function h which associates to each q in 1 m 1 the topological entropy of mathcal u q is a devil s staircase in this paper we explicitly determine the plateaus of h and characterize the bifurcation set mathscr e of q s where the function h is not locally constant moreover we show that mathscr e is a cantor set of full hausdorff dimension we also investigate the topological transitivity of a naturally occurring subshift mathbf v q sigma which has a close connection with open dynamical systems finally we prove that the hausdorff dimension and box dimension of mathcal u q coincide for all q in 1 m 1

error on the constitutive relation in the nonlinear formulation of elastic beams the plane case the error measure we propose in this study is based on the mechanics concept of error on the constitutive relation the error on the constitutive relation has already been used for the updating of elastic structures in dynamics under the small perturbation assumption here we extend this concept to the context of geometrically nonlinear elastic beams large rotations and finite strains it is shown that the proposed strategy which uses measured experimental data enables us to locate damaged substructures or if the quality of the model is to be checked the method provides the location of the zones of modelling errors the formulation and its derivation are developed and a complete algorithmic treatment is presented the implications of the formulation of the errors location discussed in this paper are illustrated through a set of numerical examples

on newman polynomials which divide no littlewood polynomial recall that a polynomial p x g z x with coefficients 0 1 and constant term 1 is called a newman polynomial whereas a polynomial with coefficients 1 1 is called a littlewood polynomial is there an algebraic number ol which is a root of some newman polynomial but is not a root of any littlewood polynomial in other words but not equivalently is there a newman polynomial which divides no littlewood polynomial in this paper for each newman polynomial p of degree at most 8 we find a littlewood polynomial divisible by p moreover it is shown that every trinomial l uxa vxb where a b are positive integers and u v 1 1 so in particular every newman trinomial 1 xa xb divides some littlewood polynomial nevertheless we prove that there exist newman polynomials which divide no littlewood polynomial e g x9 x6 x2 x l this example settles the problem 006 07 posed by the first named author at the 2006 west coast number theory conference it also shows that the sets of roots of newman polynomials vjv littlewood polynomials vc and 1 0 1 polynomials v are distinct in the sense that between them there are only trivial relations v v c v and vc c v moreover v jz vc u vjsf the proofs of several main results after some preparation are computational

geometrically exact intrinsic theory for dynamics of curved and twisted anisotropic beams a formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams when the applied loads at the ends of and distributed along the beam are independent of the deformation neither displacement nor rotation variables appear an intrinsic formulation like well known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two advantages of such a formulation are demonstrated with a simple example when the initial curvature and twist are constant along the beam two space time conservation laws are shown to exist one being a work energy relation and the other a generalized impulse momentum relation these laws can be used for example as benchmarks to check the accuracy of any proposed solution including time marching and finite element schemes the structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization a particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end loaded beams that by virtue of the kirchhoff analogy leads to a time marching scheme for the dynamics of a pivoted rigid body in a gravity field this time marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy whereas the analogous spatial discretization scheme for the nonlinear static behavior of end loaded beams satisfies analogous integrals of deformation along the beam span remarkably a straightforward generalization of these discretization schemes is shown to satisfy both space time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space time element

large deformation analysis of flexible beams presented here are numerical verifications of a geometrically exact curved beam model which fully accounts for large rotations large displacements initial curvatures and extensionality a multiple shooting method is used to solve the two point boundary value problem of flexible beams undergoing large elastic rotations and displacements in three dimensional space numerically exact large static deformations of eight beams subjected to different loading and or boundary conditions are obtained these solutions can be used to verify the performance of general finite element codes in analyzing large structural deformations

corotational total lagrangian formulation for three dimensional beamelement a corotational total lagrangian formulation of beam element is presented for the nonlinear analysis of three dimensional beam structures with large rotations but small strains the nonlinear coupling among the bending twisting and stretching deformations is considered all of the element deformations and equations are defined in body attached element coordinates three rotation parameters are proposed to determine the orientation of the element cross section two sets of element nodal parameters termed explicit nodal parameters and implicit nodal parameters are introduced the explicit nodal parameters are used in the assembly of the system equations from the element equations and chosen to be three components of the incremental translation vector and three components of the incremental rotation vector the implicit nodal parameters are used to determine the deformations of the beam element and chosen to be three components of the total displacement vector and nodal values of the three rotation parameters the element internal nodal forces corresponding to the implicit nodal parameters are obtained from the virtual work principle numerical examples are presented and compared with the numerical and experimental results reported in the literature to demonstrate the accuracy and efficiency of the proposed method hree dimensional beams are very important structural elements in all types of engineering systems in many applications these beam elements undergo finite rotations that require a nonlinear formulation to their structural analysis the development of new and efficient formulations for nonlinear analysis of beam structures has attracted the study of many researchers in recent years based on different kinematic assumptions different alternative formulation strategies and procedures to accommodate large rotation capability during the large deformation process have been presented 1 20 the kinematic assumptions used in refs 4 5 7 11 and 17 19 are based on timoshenko s hypothesis the effect of stretching bending torsion and transverse shear are taken into account

random fibonacci sequences and the number 1 13198824 begin abstract for the familiar fibonacci sequence defined by f 1 f 2 1 and f n f n 1 f n 2 for n 2 f n increases exponentially with n at a rate given by the golden ratio 1 sqrt 5 2 1 61803398 ldots but for a simple modification with both additions and subtractions the it random fibonacci sequences defined by t 1 t 2 1 and for n 2 t n pm t n 1 pm t n 2 where each pm sign is independent and either or with probability 1 2 it is not even obvious if abs t n should increase with n our main result is that begin equation sqrt n abs t n rightarrow 1 13198824 ldots text as n rightarrow infty end equation with probability 1 finding the number 1 13198824 ldots involves the theory of random matrix products stern brocot division of the real line a fractal like measure a computer calculation and a rounding error analysis to validate the computer calculation end abstract

a performance analysis of subspace based methods in the presence of model errors i the music algorithm application of subspace based algorithms to narrowband direction of arrival doa estimation requires that both the array response in all directions of interest and the spatial covariance of the noise must be known in practice however neither of these quantities is known precisely depending on the degree to which they deviate from their nominal values serious performance degradation can result the performance of the music algorithm is examined for situations in which the noise covariance and array response are perturbed from their assumed values theoretical expressions for the error in the music doa estimates are derived and compared with simulations performed for several representative cases and with the appropriate cramer rao bound an optimally weighted version of music is proposed for a particular class of array errors

analytical predictions of chaos in a non linear rod the resonant and chaotic conditions for non dampened non linear planar rods are developed through the chirikov criterion and the subharmonic bifurcation conditions for weakly dampened non linear planar rods are also presented through the melnikov method the analytical conditions are based on a simply supported geometrically non linear planar rod model with a specific single mode response but these conditions are applicable to geometrically non linear planar rods with different supports chaos and transient motion from chaos to periodic motion in the non linear rod are simulated through the approximate rod model and they are illustrated through the poincare mapping section

on the beta expansion for salem numbers of degree 6 for a given 1 the beta transformation t t is defined for x 0 1 by tx x mod 1 the number is said to be a beta number if the orbit t n 1 n 1 is finite hence eventually periodic it is known that all pisot numbers are beta numbers and it is conjectured that this is true for salem numbers but this is known only for salem numbers of degree 4 here we consider some computational and heuristic evidence for the conjecture in the case of salem numbers of degree 6 by considering the set of 11836 such numbers of trace at most 15 although the orbit is small for the majority of these numbers there are some examples for which the orbit size is shown to exceed 10 9 and for which the possibility remains that the orbit is infinite there are also some very large orbits which have been shown to be finite an example is given for which the preperiod length is 39420662 and the period length is 93218808 this is in contrast to salem numbers of degree 4 where the orbit size is bounded by 2 3 an heuristic probabilistic model is proposed which explains the difference between the degree 4 and degree 6 cases the model predicts that all salem numbers of degree 4 and 6 should be beta numbers but that degree 6 salem numbers can have orbits which are arbitrarily large relative to the size of furthermore the model predicts that a positive proportion of salem numbers of any fixed degree 8 will not be beta numbers this latter prediction is not tested here

pisot numbers in the neighborhood of a limit point ii let s denote the set of real algebraic integers greater than one all of whose other conjugates lie within the unit circle in an earlier paper we introduced the notion of width of a limit point a of s and showed that if the width of a is smaller than 1 28 then there is an algorithm for determining all members of s in a neighborhood of a recently we introduced the derived tree in order to deal with limit points of greater width here we apply these ideas to the study of the limit point a3 the zero of z4 2z3 1 outside the unit circle we determine the smallest neighborhood 9x a3 92 of a3 in which all elements of s other than a3 satisfy one of the equations z z4 2z3 z 1 a z 0 where a z is one of z z2 1 z3 z 1 or z4 z3 z 1 the endpoints 0 and 62 are elements of s of degrees 23 and 42 respectively

direct and inverse dynamics of a very flexible beam abstract the problem of direct and inverse dynamics of a very flexible plane beam was discussed and analyzed direct dynamics is referred to as an ordinary dynamics problem where initial conditions are known and where three conditions either forces or motion are prescribed at each end of the beam as boundary conditions partially or fully inverse dynamics are defined to be the case when some or all of the conditions are moved from one end to the other so that one end is overprescribed including both forces and motion and the constraints at the other end of the beam are partially or fully unspecified a new model for direct and inverse dynamics of a straight or curved planar beam at small strains and large deflections has been derived lagrange equations in a local frame of reference and a finite element method were utilized for the formulation the direct dynamics problems were solved by a fast and relatively simple linear iteration method and verified by results of existing programs for specific cases this formulation is particularly useful for more complicated constitutive relations on the other hand inverse dynamics solutions were stable for only short periods of time this result demonstrates the need for further research in the area of inverse dynamics practical application of the study may be considered in the field of robotics where very flexible beams may be used in order to reduce the number of arms and actuators in a robotics system

on univoque pisot numbers we study pisot numbers beta in 1 2 which are univoque i e such that there exists only one representation of 1 as 1 sum n geq 1 s n beta n with s n in 0 1 we prove in particular that there exists a smallest univoque pisot number which has degree 14 furthermore we give the smallest limit point of the set of univoque pisot numbers

music algorithm for vector sensors array using biquaternions in this paper we use a biquaternion formalism to model vector sensor signals carrying polarization information this allows a concise and elegant way of handling signals with eight dimensional 8 d vector valued samples using this model we derive a biquaternionic version of the well known array processing music algorithm and we show its superiority to classically used long vector approach new results on biquaternion valued matrix spectral analysis are presented of particular interest for the biquaternion music bq music algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces we propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition evd of a hermitian biquaternionic matrix by means of a newly defined quantity the quaternion adjoint matrix the bq music estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing lv music and psa music the proposed algorithm is shown to be superior in several aspects to the existing approaches compared with lv music the bq music algorithm is more robust to modelization errors and coherent noise while it can detect less sources in comparaison with psa music our approach exhibits more accurate estimation of direction of arrival doa for a small number of sources while keeping the polarization information accessible

dynamic response of intrinsic continua for use in biological and molecular modeling explicit finite element formulation an intrinsic beam formulation has recently appeared aiaa j 2003 41 6 1131 1137 which presents the three dimensional equations of motion governing spatial and temporal changes in a one dimensional continua s curvature strain rectilinear velocity and angular velocity the formulation would suggest several computational advantages over more traditional displacement based continua approaches low order interpolation functions can describe generally curved and twisted continua configurations inter element displacements slopes strains and curvatures can be matched and finite rotational variables and their complexities are absent here we present a completed intrinsic continua finite element development and critical analysis which follows from an earlier preliminary treatment as applied to carbon nanotubes int j solids struct 2007 44 874 894 modeling of nodal displacements and rotations are included explicit time stepping with desired high frequency damping is accomplished using an implementation of the generalized method zero energy modes inherent in the formulation are also identified and rectified finally we document very good agreement between results predicted with the intrinsic continua finite element simulator and results generated using more traditional finite element simulations copyright 2009 john wiley sons ltd

growth and decay of random fibonacci sequences for 0 they grow exponentially by formulating the problem as a markov chain involving random matrix products and computing its invariant measure a fractal the lyapunov constant limn x n 1 n is determined numerically for a wide range of values and its dependence on is observed to be non smooth the limit is defined in the almost sure sense this generalizes recent work of viswanath who proved 1 1 131 988 24 by a simple rescaling these results also apply to the more general random recurrence x n 1 x n x n 1 for fixed and these random recurrence relations have links with many fields including ergodic theory dynamical systems heavy tailed statistics spectral theory continued fractions and condensed matter physics

nonlinear response and stability analysis of beams using finite elements in time the dynamic response and stability analysis of beams undergoing large deflections and rotations is analyzed using the finite element method in time this formulation provides an efficient and consistent approach to predicting the dynamic response of nonlinear periodic systems as well as their stability boundaries based on floquet s theory this paper has two goals 1 to present the formulation of the finite element in time equations for naturally curved and twisted beams undergoing large deflections and rotations and 2 to discuss the predictions of this method when applied to several classical nonlinear beam problems for which analytical solutions exist and in some cases experimental results are available these examples are 1 the natural vibration frequencies of a rotating beam 2 the harmonic and superharmonic response of a clamped clamped beam undergoing largeamplitude vibrations 3 the dynamic instability of a cantilevered beam under a tip follower force and 4 the parametric excitation of a beam under an axial pulsating load with and without the presence of viscous damping forces in all cases close agreement is found between the analytical results and the predictions of the finite element in time approach which appears to be an efficient and reliable technique for nonlinear dynamic response and stability analysis of periodic systems

unique expansions of real numbers it was discovered some years ago that there exist non integer real numbers q 1 for which only one sequence ci of integers ci 0 q satisfies the equality i 1 ciq i 1 the set of such univoque numbers has a rich topological structure and its study revealed a number of unexpected connections with measure theory fractals ergodic theory and diophantine approximation in this paper we consider for each fixed q 1 the set uq of real numbers x having a unique representation of the form i 1 ciq i x with integers ci belonging to 0 q we carry out a detailed topological study of these sets for instance we characterize their closures and we determine those bases q for which uq is closed or even a cantor set we also study the set uq consisting of all sequences ci of integers ci 0 q such that i 1 ciq i uq we determine the numbers r 1 for which the map q uq defined on 1 is constant in a neighborhood of r and the numbers q 1 for which uq is a subshift or a subshift of finite type

vector sensor array processing for electromagnetic source localization the authors present a new approach for localizing electromagnetic sources using sensors where the output of each is a vector consisting of the complete six electric and magnetic field components two types of source transmissions are considered 1 single signal transmission sst and 2 dual signal transmission dst the model is given in terms of several parameters including the wave direction of arrival doa and state of polarization a compact expression is derived for the cramer rao bound crb on the estimation errors of these parameters for the multi source multi vector sensor model quality measures including mean square angular error msae and covariance of vector angular error cvae are introduced and their lower bounds are derived the advantage of using vector sensors is highlighted by explicit evaluation of the msae and cvae bounds for source localization with a single vector sensor a simple algorithm for estimating the source doa with this sensor is presented along with its statistical performance analysis

free vibration analysis of rotating beams by a variable order finite element method the free vibration of rotating beams is analyzed by means of a finite element method of variable order this method entails displacement functions that are a complete power series of a variable number of terms the terms are arranged so that the generalized coordinates are composed of displacements and slopes at the element extremities and additionally displacements at certain points within the element the displacement is assumed to be analytic within an element and thus can be approximated to any degree of accuracy desired by a complete power series numerical results are presented for uniform beams with zero and nonzero hub radii tapered beams and a nonuniform beam with discontinuities since the present method reduces to a conventional beam finite element method for a cubic displacement function the results are compared and found to be superior to the conventional results in terms of accuracy for a given number of degrees of freedom indeed essentially exact eigenvalues and eigenvectors are obtained with this technique which is far more rapidly convergent than other approaches in the literature

power series with restricted coefficients and a root on a given ray we consider bounds on the smallest possible root with a specified argument o of a power series f z 1 n 1 a i z i with coefficients a i in the interval g g we describe the form that the extremal power series must take and hence give an algorithm for computing the optimal root when o 2 is rational when g 2 2 3 we show that the smallest disc containing two roots has radius g 1 1 coinciding with the smallest double real root possible for such a series it is clear from our computations that the behaviour is more complicated for smaller g we give a similar procedure for computing the smallest circle with a real root and a pair of conjugate roots of a given argument we conclude by briefly discussing variants of the beta numbers where the defining integer sequence is generated by taking the nearest integer rather than the integer part we show that the conjugates a of these pseudobeta numbers either lie inside the unit circle or their reciprocals must be roots of 1 2 1 2 power series in particular we obtain the sharp inequality 3 2

geometric algebra of euclidean 3 space for electromagnetic vector sensor array processing part i modeling a new mathematical tool the geometric algebra of euclidean 3 space g3 is introduced for electromagnetic vector sensor array processing herein this paper focuses on modeling the six component outputs of a vector sensor holistically by an entry called as a multivector in g3 a compact polarized model for the array termed as a geometric algebra model g model is then presented using the g model a novel data covariance matrix model is defined by the geometric products in g3 and then analyzed the analytical results show that the six component measurement noise of a vector sensor can naturally be whitened if the noise cross correlations between the different axial electric and magnetic components are equal to one another compared with the known best quad quaternion model the new covariance matrix model results in a reduction of half memory requirements while the amount of divisions is reduced to 1 2 multiplications and additions reduced to almost 1 7

formulation and solution of inverse spaghetti problem application to beam deployment dynamics a finite element model of the dynamics of axially moving highly flexible beams is presented the model is based on a geometrically nonlinear beam formulation that allows for large overall motions to account for the varying length of a deploying beam a moving finite element reference grid is incorporated within the nonlinear beam formulation such that the number of finite element nodes remains fixed and the finite element length is allowed to vary hamilton s law is used to formulate the equations of motion and a transient integration solution procedure is derived from a space time finite element discretization of the hamiltonian variational principle computational results of the methodology are presented for a planar inverse spaghetti problem he dynamics of axially moving beamlike structures are becoming increasingly important for the analysis of spacecraft and large space structures that deploy flexible appendages such as antenna stabilizing booms solar arrays and long trusslike structures as well as for other applications such as magnetic tape drives printing machines traveling cables and band saws an extensive amount of research has focused on the modeling of axially moving continua to analyze the dynamic behavior of such structures one of the most common models of axially moving continua that has received the most attention is the traveling string problem in what he termed the spaghetti problem carrier1 analyzed the motion of a string being accelerated upward into a fixed orifice in other works since then analytical expressions representing both linear and nonlinear vibrations of axially moving strings subject to various support conditions have been derived 2 9 the dynamics of beamlike structures traveling between or over fixed supports have also been studied to address industrial applications such as high speed tape drives and band saws 10 13 to address spacecraft applications in which antenna or other long flexible structures are deployed from a host satellite the dynamics of a cantilever beam that is ejected from a single fixed guide are modeled using bernoulli euler beam theory in ref 14 in a quite different approach the extrusion of a beam from a rotating base is modeled using a series of elastically connected rigid links in which the number of rigid links are continuously increased decreased to account for deployment retrieval of the beam 15 general formulations of axially moving bernoulli euler beam models cantilevered to a rotating rigid host body have also been developed to study the effect of appendage deployment on the attitude dynamics of orbiting spacecraft 16 20 all of these given analyses are based on an assumption of small elastic deformations with the exception of the discrete spring mass model of ref 15 all of the other models that have been referenced thus far use linear combinations of space depende nt admissible functions of a time varying length weighted by time dependen t generalized coordi

a weak weak formulation for large displacements beam statics a finite volumes approximation this paper addresses the problem of the numerical solution of beam statics undergoing large displacements a kinematic analysis outlines the beam geometrical model through the definition of its lagrangian co ordinate and strain parameters a definition of the stress parameters a constitutive law and an expression for the strain energy of the beam are then provided under the hypothesis of small strain the equations governing the beam equilibrium are introduced and their weak form is derived these equations are then proved to be equivalent to the primal and mixed form of principle of virtual work the numerical approximation is introduced by applying the bidiscontinuous finite elements method on the linearized weak form the weak weak formulation is attained by using the lowest interpolation order both for test and trial functions on two staggered decompositions of the space domain some numerical examples prove the capability of present formulation in handling actual problems

a representation of the solution of the nth order linear difference equation with variable coefficients abstract the linear difference equation of the n th order with variable coefficients and a related difference equation are considered it is shown that for each problem the determinants of submatrices of a single solution matrix provides the solution sequence applications to sequences and series are shown results on toeplitz matrices involving the roots of associated polynomial equations are obtained

geometrically exact intrinsic theory for dynamics of moving composite plates a geometrically exact and fully intrinsic theory is presented for dynamics of composite plates undergoing large deformation to say that the formulation is intrinsic means that it is without displacement and rotation variables although the equations are geometrically exact the highest degree nonlinearities are quadratic there are no singularities associated with finite rotation methods for posing problems in this framework along with advantages of the formulation are discussed

refined theory of composite beams the role of short wavelength extrapolation the present paper presents an asymptotically correct beam theory with nonclassical sectional degrees of freedom the basis for the theory is the variational asymptotical method a mathematical technique by which the three dimensional analysis of composite beam deformation can be split into a linear two dimensional cross sectional analysis and a nonlinear one dimensional beam analysis the elastic constants used in the beam analysis are obtained from the cross sectional analysis which also yields approximate closed form expressions for three dimensional distributions of displacement strain and stress such theories are known to be valid when a characteristic dimension of the cross section is small relative to the wavelength of the deformation however asymptotically correct refined theories may differ according to how they are extrapolated into the short wavelength regime thus there is no unique asymptotically correct refined theory of higher order than classical euler bernoulli like theory different short wavelength extrapolations can be obtained by changing the meaning of the theory s one dimensional variables numerical results for the stiffness constants of a refined beam theory and for deformations from the corresponding one dimensional theory are presented it is shown that a theory can be asymptotically correct and still have non positive definite strain energy density which is completely inappropriate mathematically and physically a refined beam theory which appropriately possesses a positive definite strain energy density and agrees quite well with experimental results is constructed by using a certain short wavelength extrapolation

generalized variational principle on nonlinear theory of naturally curved and twisted beams the nonlinear behavior for naturally curved and twisted beams undergoing large displacements and rotations but small strains has been investigated in this paper an incomplete generalized variational functional for the beams with complete constrained boundaries at two ends is established by means of the lagrange multiplier method a set of equilibrium equations which takes into account the effects of transverse shear deformations as well as torsion related warping has been derived from the stationary conditions of the functional the various elastic couplings of bending twisting and or extension twisting are also included in this analysis the present theory can also be easily extended to other various incomplete constrained boundaries in case of small displacement the equilibrium equations obtained by the present theory reduce to the well known results in literature

lateral torsional buckling of cantilevered elastically coupled composite strip and i beams the lateral torsional buckling of composite strip and i beams is considered the geometrically exact governing equations are simplified by consistently regarding certain configuration parameters as small the assumption that these parameters are indeed small is equivalent to the assumption that the square of the maximum prebuckling cross sectional rotation due to bending is small compared to unity the analysis takes into account various refinements of previously published results including the vlasov eaect elastic coupling the oaset of the load from the centroid and of course prebuckling deflections the analysis is thereby reduced to a single fourth order diaerential equation and boundary conditions all of which are derivable from a corresponding energy expression from the form of matched asymptotic expansions of the buckling mode when small parameters are ignored altogether a single comparison function is found which gives the correct buckling load to within 1 for a wide range of the warping rigidity using this comparison function a formula for the buckling load as a function of the small parameters of the problem is found and validated with certain exceptions regarding the load oaset parameter the formula provides results which agree quite well with the numerical solution of the exact equations as long as all the small parameters remain small however the load oaset parameter always appears in the governing equations as multiplied by a ratio of stianesses which can become large especially for composite i beams for this case a special treatment is required 2001 elsevier science ltd all rights reserved

quaternion music for vector sensor array processing this paper considers the problem of direction of arrival doa and polarization parameters estimation in the case of multiple polarized sources impinging on a vector sensor array the quaternion model is used and a data covariance model is proposed using quaternion formalism a comparison between long vector orthogonality and quaternion vector orthogonality is also performed and its implications for signal subspace estimation are discussed consequently a music like algorithm is presented allowing estimation of wave s doas and polarization parameters the algorithm is tested in numerical simulations and performance analysis is conducted when compared with other music like algorithms for vector sensor array the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort for equivalent performance this paper also illustrates a compact and elegant way of dealing with multicomponent complex valued data

differentiability of a two parameter family of self affine functions abstract this paper highlights an unexpected connection between expansions of real numbers to noninteger bases so called expansions and the infinite derivatives of a class of self affine functions precisely we extend okamoto s function itself a generalization of the well known functions of perkins and katsuura to a two parameter family f n a n n 1 n 1 a 1 we first show that for each x f n a x is either 0 or undefined we then extend okamoto s theorem by proving that for each n depending on the value of a relative to a pair of thresholds the set x f n a x 0 is either empty uncountable but lebesgue null or of full lebesgue measure we compute its hausdorff dimension in the second case the second result is a characterization of the set d a x f n a x which enables us to closely relate this set to the set of points which have a unique expansion in the typically noninteger base 1 a recent advances in the theory of expansions are then used to determine the cardinality and hausdorff dimension of d a which depend qualitatively on the value of a relative to a second pair of thresholds

anti hadamard matrices abstract an anti hadamard matrix may be loosely defined as a real 0 1 matrix which is invertible but only just let a be an invertible 0 1 matrix with eigenvalues i singular values i and inverse b bij we are interested in the four closely related problems of finding n mina i i n mina i i n maxa i j bij and n maxa ijb2ij then a is an anti hadamard matrix if it attains n we show that n n are between 2n 1 n 4 n 2 and c n 2 274 n where c is a constant c 2 274 n n 2 n 4 n 2 and c 5 172 n n 4n 2 n 4 n we also consider these problems when a is restricted to be a toeplitz triangular circulant or 1 1 matrix besides the obvious application to finding the most ill conditioned 0 1 matrices there are connections with weighing designs number theory and geometry

large displacement analysis of naturally curved and twisted composite beams this paper presents an efficient procedure for analyzing naturally curved and twisted beams undergoing large displacements and rotations but small strains special care is taken to accurately model the structural behavior of the beam shearing deformations as well as torsion related warping are taken into account the various elastic couplings e g bending twisting or extention twisting that can arise in composite beams are also modeled in a rational fashion the equations are derived in curvilinear material coordinates and euler angles are used to represent the arbitrarily large rotations the strain energy in the beam is reduced to a quadratic expression by means of a quasilinearization procedure this expression of the strain energy is the basis for a finite element analysis of the structure and standard iteration techniques are used to obtain nonlinear solutions to the problem the predictions of this model are found to be in good agreement with the experimental results and the computational efficiency is greatly improved as compared with other available analyses

double roots of 1 1 power series and related matters for a given collection of distinct arguments vec theta theta 1 ldots theta t multiplicities vec k k 1 ldots k t and a real interval i u v containing zero we are interested in determining the smallest r for which there is a power series f x 1 sum n 1 infty a i x i with coefficients a i in i and roots alpha 1 re 2 pi i theta 1 ldots alpha t re 2 pi i theta t of order k 1 ldots k t respectively we denote this by r vec theta vec k i we describe the usual form of the extremal series we give a sufficient condition which is also necessary when the extremal series possesses at least left sum i 1 t delta theta i k i right 1 coefficients strictly inside i where delta theta i is 1 or 2 as alpha i is real or complex we focus particularly on r theta 2 1 1 the size of the smallest double root of a 1 1 power series lying on a given ray of interest in connection with the complex analogue of work of boris solomyak on the distribution of the random series sum pm lambda n we computed the value of r theta 2 1 1 for the rationals theta in 0 1 2 of denominator less than fifty the smallest value we encountered was r 4 29 2 1 1 0 7536065594 for the one sided intervals i 0 1 and 1 0 the corresponding smallest values were r 11 30 2 0 1 8237251991 and r 1 3 2 1 0 8656332072

chebyshev collocation method for static intrinsic equations of geometrically exact beams abstract a chebyshev collocation method is presented for the static analysis of the geometrically exact beams undergoing arbitrary large deflections using intrinsic formulation in the intrinsic formulation of the governing equations of the spatial beams neither displacement nor rotation variables appear the proposed collocation technique is based on the chebyshev points as the collocation points and the orthogonal chebyshev polynomials as the trial functions this method is successfully applied to the static solution of the strong form of the intrinsic governing equations of the nonlinear spatial beam and in order to show the applicability accuracy and computational efficiency of the proposed method the numerical results are compared to the experimental analytical and other numerical results for a number of test cases