explicit merit factor formulae for fekete and turyn polynomials we give explicit formulas for the l 4 norm or equivalently for the merit factors of various sequences of polynomials related to the fekete polynomials

the merit factor of legendre sequences corresp turyn has shown by numerical computation that the merit factor of long legendre sequences offset by a quarter of their length has in all likelihood the asymptotic value 6 the ergodicity postulate is used in this correspondence to calculate that the merit factor f of a legendre sequence offset by a fraction f of its length has an asymptotic value given by 1 f 2 3 4 f 8f 2 f leq 1 2 which gives f 6 for f 1 4

dynamic stiffness formulation and free vibration analysis of centrifugally stiffened timoshenko beams abstract the dynamic stiffness matrix of a centrifugally stiffened timoshenko beam has been developed and used to carry out a free vibration analysis the governing differential equations of motion of the beam in free vibration are derived using hamilton s principle and include the effect of an arbitrary hub radius for harmonic oscillation the derivation leads to two different but of similar form fourth order ordinary differential equations with variable coefficients that govern the amplitudes of bending displacement and bending rotation respectively an outboard force at the end of the beam is taken into account which makes possible the free vibration analysis of rotating non uniform or tapered timoshenko beams using the frobenius method of series solution and imposing boundary conditions the dynamic stiffness matrix which relates amplitudes of harmonically varying forces with the amplitudes of harmonically varying displacements at the ends of the element is formulated applying the wittrick williams algorithm to the resulting dynamic stiffness matrix the natural frequencies of a few carefully chosen illustrative examples are obtained the results are compared with those available in the literature

hardy s inequality and the l1 norm of exponential sums in this paper we generalize hardy s inequality 3 for measures of analytic type and obtain a proof of the littlewood conjecture 4 for the l norm of exponential sums as a simple consequence let t be the circle group z the additive group of integers and m t the customary convolution algebra of borel measures on t for pe e m t and n g z put m e ino dp o denote by h t the classical space of all measures e e m t such that j n 0 for all n 0 and let c denote the complex numbers we now state hardy s inequality for measures of analytic type theorem 1 if m c h t then

blind channel estimation in orthogonally coded mimo ofdm systems a semidefinite relaxation approach a new blind channel estimation approach for orthogonally coded multiple input multiple output orthogonal frequency division multiplexing mimo ofdm systems is proposed the main idea of our technique is using specific properties of the orthogonal space time block codes ostbcs to estimate the finite impulse response fir channel parameters in the time domain instead of doing this in the frequency domain independently for each subcarrier this results in a substantially improved parsimony of the channel parametric model as compared to the direct per subcarrier channel estimation methods and allows coherent processing across the subcarriers it is shown that using the semidefinite relaxation sdr technique our channel estimation problem can be approximated as a convex semidefinite programming sdp problem and hence can be solved efficiently using modern convex optimization methods our simulations validate the performance and or computational complexity advantages of the proposed method as compared to some current state of the art blind mimo ofdm channel estimation methods

on the performance of a four degree of freedom per node element for stepped beam analysis and higher frequency estimation the frequency values obtained by using two degree of freedom per node and four degree of freedom per node elements for uniform stepped and continuous beams are compared for various boundary conditions to demonstrate the superior performance of the four degree of freedom per node element the stiffness and mass matrices for the element are given in explicit form

flapwise bending vibration analysis of a rotating tapered cantilever bernoulli euler beam by differential transform method abstract this paper studies the vibration characteristics of a rotating tapered cantilever bernoulli euler beam with linearly varying rectangular cross section of area proportional to xn where n equals to 1 or 2 covers the most practical cases in this work the differential transform method dtm is used to find the nondimensional natural frequencies of the tapered beam numerical results are tabulated for different taper ratios nondimensional angular velocities and nondimensional hub radius the effects of the taper ratio nondimensional angular velocity and nondimensional hub radius are discussed the accuracy is assured from the convergence of the natural frequencies and from the comparisons made with the studies in the open literature it is shown that the natural frequencies of a rotating tapered cantilever bernoulli euler beam can be obtained with high accuracy by using dtm

rudin shapiro like polynomials in l 4 we examine sequences of polynomials with 1 1 coefficients constructed using the iterations p x p x x d 1 p x where d is the degree of p and p is the reciprocal polynomial of p if po 1 these generate the rudin shapiro polynomials we show that the l4 norm of these polynomials is explicitly computable we are particularly interested in the case where the iteration produces sequences with smallest possible asymptotic l4 norm or equivalently with largest possible asymptotic merit factor the rudin shapiro polynomials form one such sequence we determine all p 0 of degree less than 40 that generate sequences under the iteration with this property these sequences have asymptotic merit factor 3 the first really distinct example has a p 0 of degree 19

a signal perturbation free whitening rotation based semiblind approach for mimo channel estimation it was shown in our previous work that in the noise free case the whitening rotation wr based mimo channel estimation algorithm is subject to a signal perturbation error justifying that the wr based method is efficient only in the low signal to noise ratio snr case in this paper a very efficient signal perturbation free wr based approach is proposed for semiblind channel estimation of mimo systems a novel transmit scheme is developed based on the eigenvalue decomposition of the correlation matrix of the transmitted signal the new scheme is to send a small volume of data bearing the information of the correlation matrix to the receiver for the cancellation of the signal perturbation error so as to improve the performance of the wr based method in the case of high snrs then a perturbation analysis of the proposed wr based semiblind method with the new transmit scheme is conducted leading to a closed form expression for the mean square error mse of the channel estimate computer simulations show that the proposed approach significantly outperforms the original wr based method as well as some other channel estimation methods for all snr levels

rudin shapiro like sequences with maximum asymptotic merit factor borwein and mossinghoff investigated the rudin shapiro like sequences which are infinite families of binary sequences usually represented as polynomials each family of rudin shapiro like sequences is obtained from a starting sequence which we call the seed by a recursive construction that doubles the length of the sequence at each step and many sequences produced in this manner have exceptionally low aperiodic autocorrelation borwein and mossinghoff showed that the asymptotic autocorrelation merit factor for any such family is at most 3 and found the seeds of length 40 or less that produce the maximum asymptotic merit factor of 3 the definition of rudin shapiro like sequences was generalized by katz lee and trunov to include sequences with arbitrary complex coefficients among which are families of low autocorrelation polyphase sequences katz lee and trunov proved that the maximum asymptotic merit factor is also 3 for this larger class here we show that a family of such rudin shapiro like sequences achieves asymptotic merit factor 3 if and only if the seed is either of length 1 or is the interleaving of a pair of golay complementary sequences for small seed lengths where this is not possible the optimal seeds are interleavings of pairs that are as close as possible to being complementary pairs and the idea of an almost complementary pair makes sense of remarkable patterns in previously unexplained data on optimal seeds for binary rudin shapiro like sequences

algorithms to enumerate codewords for dc sup 2 constrained channels we introduce a new method for enumerating codewords that can be applied to dc sup 2 constrained channels based on this method two efficient algorithms for evaluating the number of codewords with specified characteristics are developed computer calculation results show that these algorithms are significantly more computationally efficient than other techniques developed to date

the merit factor of binary sequences related to difference sets long binary sequences related to cyclic difference sets are investigated among all known constructions of cyclic difference sets it is shown that only sequences constructed from hadamard difference sets can have an asymptotic nonzero merit factor maximal length shift register sequences legendre and twin prime sequences are all constructed from hadamard difference sets the authors prove that the asymptotic merit factor of any maximal length shift register sequence is three for twin prime sequences it is shown that the best asymptotic merit factor is six this value is obtained by shifting the twin prime sequence one quarter of its length it turns out that legendre sequences and twin prime sequences have similar behavior jacobi sequences are investigated on the basis of the jacobi symbol the best asymptotic merit factor is shown to be six through the introduction of product sequences it is argued that the maximal merit factor among all sequences of length n is at least six when n is large the authors also demonstrate that it is fairly easy to construct sequences of moderate composite length with a merit factor close to six

equal moments division of a set let n q m be the minimal positive integer n for which there exists a splitting of the set 0 n 1 into q subsets so s 1 s q 1 whose first m moments are equal similarly let m q n be the maximal positive integer m such that there exists a splitting of 0 n 1 into q subsets whose first m moments are equal for q 2 these functions were investigated by several authors and the values of n 2 m and m 2 n have been found for m 8 and n 167 respectively in this paper we deal with the problem for any prime q we demonstrate our methods by finding m 3 n for any n 90 and n 3 m for m 6

on a problem of byrnes concerning polynomials with restricted coefficients as in the earlier paper with this title we consider a question of byrnes concerning the minimal length n m of a polynomial with all coefficients in 1 1 which has a zero of a given order m at x 1 in that paper we showed that n m 2 m for all m 5 and showed that the extrernal polynomials for were those conjectured by byrnes but for m 6 that n 6 48 rather than 64 a polynomial with n 48 was exhibited for m 6 but it was not shown there that this extremal was unique here we show that the extremal is unique in the previous paper we showed that n 7 is one of the 7 values 48 56 64 72 80 88 or 96 here we prove that n 7 96 without determining all extremal polynomials we also make some progress toward determining n 8 as in the previous paper we use a combination of number theoretic ideas and combinatorial computation the main point is that if p is a primitive pth root of unity where p m 1 is a prime then the condition that all coefficients of p be in 1 1 together with the requirement that p x be divisible by x 1 m puts severe restrictions on the possible values for the cyclotomic integer p p

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

counting zeros of cosine polynomials on a problem of littlewood abstract we show that if a is a finite set of non negative integers then the number of zeros of the function f a a a cos a in 0 2 is at least log log log a 1 2 e this gives the first unconditional lower bound on a problem of littlewood and solves a conjecture of borwein erdelyi ferguson and lockhart we also prove a result that applies to more general cosine polynomials with few distinct rational coefficients one of the main ingredients in the proof is perhaps of independent interest we show that if f is an exponential polynomial with few distinct integer coefficients and f correlates with a low degree exponential polynomial p then f has a very particular structure this result allows us to prove a structure theorem for trigonometric polynomials with few zeros in 0 2 and restricted coefficients

newman polynomials with prescribed vanishing and integer sets with distinct subset sums we study the problem of determining the minimal degree d m of a polynomial that has all coefficients in 0 1 and a zero of multiplicity m at 1 we show that a greedy solution is optimal precisely when m 5 mirroring a result of boyd on polynomials with 1 coefficients we then examine polynomials of the form e e xe 1 where e is a set of m positive odd integers with distinct subset sums and we develop algorithms to determine the minimal degree of such a polynomial we determine that d m satisfies inequalities of the form 2m c1m d m 103 96 2m c2 last we consider the related problem of finding a set of m positive integers with distinct subset sums and minimal largest element and show that the conway guy sequence yields the optimal solution for m 9 extending some computations of lunnon

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

flatness of conjugate reciprocal unimodular polynomials abstract a polynomial is called unimodular if each of its coefficients is a complex number of modulus 1 a polynomial p of the form p z j 0 n a j z j is called conjugate reciprocal if a n j a j a j c for each j 0 1 n let d be the unit circle of the complex plane we prove that there is an absolute constant e 0 such that max z d f z 1 e 4 3 m 1 2 for every conjugate reciprocal unimodular polynomial of degree m we also prove that there is an absolute constant e 0 such that m q f exp e q 2 q 1 3 m 3 2 1 q 2 and m q f exp e q 2 q 1 3 m 3 2 2 q for every conjugate reciprocal unimodular polynomial of degree m where m q g 1 2 0 2 g e i t q d t 1 q q 0

free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method abstract starting from the governing differential equations of motion in free vibration the dynamic stiffness matrix of a uniform rotating bernoulli euler beam is derived using the frobenius method of solution in power series the derivation includes the presence of an axial force at the outboard end of the beam in addition to the existence of the usual centrifugal force arising from the rotational motion this makes the general assembly of dynamic stiffness matrices of several elements possible so that a non uniform or tapered rotating beam can be analyzed for its free vibration characteristics by idealizing it as an assemblage of many uniform rotating beams the application of the derived dynamic stiffness matrix is demonstrated by investigating the free vibration characteristics of uniform and non uniform tapered rotating beams with particular reference to the wittrick williams algorithm the results from the present theory are compared with published results it is shown that the proposed dynamic stiffness method offers an accurate and effective method of free vibration analysis of rotating beams

shape functions for mixed p version finite elements in the time domain abstract the advantages of mixed formulations are well known in that they simplify the form of the governing equations and can lead to potentially significant savings in computational effort for given accuracy likewise the p version of the finite element method is known to offer savings in computational effort for some problems in this paper shape functions for a mixed p version finite element formulation in the time domain are developed these shape functions produce a high percentage of zeros in certain coefficient matrices the mixed formulation in time used here is based on a weak form of hamilton s extended principle the present formulation allows for accurate and efficient solution of time domain problems the accuracy and power of this formulation are shown for initial value problems such as time marching and for directly obtaining the periodic response of linear systems the method is shown to yield a family of accurate unconditionally stable algorithms for initial value problems based on a simple linear oscillator

dynamic instability of a beam undergoing periodic motions over supports abstract this paper investigates the dynamic stability of a beam moving over two bi lateral supports using finite element analysis with essential conditions applied via the penalty function approach computational advantages of the penalty based approach compared to that of the lagrangian multipliers are highlighted in the context of such unique problems in dynamics penalty based numerical formulation for the moving beam results in a system of second order differential equations with periodic coefficients the governing equations are reduced to state space form and floquet lyapunov theory is applied to investigate dynamic stability of the moving beam the instability characteristics are studied for a range of amplitudes and frequencies for sinusoidal longitudinal motions of the beam in addition to the predictions using floquet lyapunov theory further instability regions based on first and higher approximations are identified the instability results for periodic motion compare well with previous research and new results are presented taking into consideration the effect of damping the penalty based finite element formulation is found to be effective when applied to this class of dynamics problems the avenues for further research are also highlighted

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

efficient encoding algorithm for third order spectral null codes an efficient algorithm is presented for encoding unconstrained information sequences into a third order spectral null code of length n and redundancy 9log sub 2 n o log log n the encoding can be implemented using o n integer additions and o nlog n counter increments

exhaustive search for low autocorrelation binary sequences binary sequences with low autocorrelations are important in communication engineering and in statistical mechanics as ground states of the bernasconi model computer searches are the main tool in the construction of such sequences owing to the exponential size of the configuration space exhaustive searches are limited to short sequences we discuss an exhaustive search algorithm with run time characteristic and apply it to compile a table of exact ground states of the bernasconi model up to n 48 the data suggest f 9 for the optimal merit factor in the limit

integer sets with distinct subset sums in section 1 we introduce the problem of finding minimal height sets of n natural numbers with distinct subset sums ssd and in section 2 review the well known conway guy sequence u conjectured to yield a minimal ssd set for every n we go on section 3 to prove that u certainly cannot be improved upon by any greedy sequence to verify numerically section 4 that it does yield ssd sets for n 80 and section 5 by direct search to show that these are minimal for n s 8 there is a brief interlude section 6 on the problem of decoding the subset from its sum in section 7 generalizations of u are constructed which are asymptotically smaller defining the limit ratio of a sequence w to be a 1imn wn 2l the atkinson negro santoro sequence v known to give ssd sets has a 0 6334 conwayguy conjectured to has a 0 4703 and our best generalization has a 0 4419 we also section 8 discuss when such sequences have the same a and section 9 how a may efficiently be computed to high accuracy 1 the distinct subset sum problem a well known problem in combinatorial number theory 2 pp 64 65 involves the construction for given n of a set p pi i 1 n of natural numbers possessing the property we shall call subset sum distinctness or ssd for short that is l l f pi p e pi s t c l n i s i t or distinct subsets of p have distinct sums evidently choosing pi 21 satisfies 1 1 the interest lies in how much the maximum element pn can be reduced below 2n 1 in order to explore the ssd property it is convenient to consider a more general concept we say that x has a representation by p with length k and signature 1 when for some s and t as before 1 2 x e pe po ies iet where s n t 0 si it k isi iti 1 that is x is the difference of two disjoint subset sums of which the first has 1 terms more than the second and the two together have k terms alternatively n 1 3 x eip i l received june 23 1986 revised october 20 1986 1980 mathematics subject classification 1985 revision primary 11b13 05 04 05a99 11j13 94a60 65b05

stiff string approximations in rayleigh ritz method for rotating beams the governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant the solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the rayleighs method numerical experiments show better agreement with converged finite element solutions compared to polynomials using this as an estimate for the first mode shape higher mode shape approximations are obtained using gram schmidt orthogonalization estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order rayleigh ritz approximation

free vibration of rotating tapered beams using the dynamic stiffness method abstract the free bending vibration of rotating tapered beams is investigated by using the dynamic stiffness method the range of problems considered includes beams for which the depth and or width of the cross section vary linearly along the length first the governing differential equation of motion of the rotating tapered beam in free flap bending vibration is derived for the most general case using hamilton s principle allowing for the effects of centrifugal stiffening an arbitrary outboard force and the hub radius term for harmonic oscillation the differential equation is solved for bending displacement by applying the frobenius method of series solution the expressions for bending rotation shear force and bending moment at any cross section of the beam are also obtained in explicit analytical form then the dynamic stiffness matrix is developed by relating the amplitudes of forces and moments to those of the displacements and rotations at the ends of the harmonically vibrating tapered beam next the wittrick williams algorithm is used as a solution technique to the resulting dynamic stiffness matrix to compute the natural frequencies and mode shapes of some illustrative examples a parametric study is carried out to demonstrate the effects of rotational speed taper ratio and hub radius on the results which are discussed and compared with the published ones finally some conclusions are drawn

on the performance of semi blind subspace based channel estimation this paper is devoted to the analysis of a semi blind estimation framework in which the standard least squares estimator based on a known training sequence is enhanced by using the statistical structure of the observations more specifically we consider the case of a general time division multiple access tdma frame based receiver equipped with multiple sensors and restrict our attention to second order based subspace methods that are suitable for most standard communication applications due to their moderate computational cost the semi blind channel estimator is obtained as a regularized least squares solution where a blind subspace criterion plays the role of the regularization constraint the main contribution of the paper consists of showing by asymptotic analysis how to optimally tune the balance between the blind criterion and the least squares fit depending on the design parameters of the system simulations show that the proposed solutions are robust and significantly improve the efficiency of the equalization

the phase problem of ultraflat unimodular polynomials the resolution of the conjecture of saffari where e 0 is an absolute constant independent of n yet combining some probabilistic lemmas from korner s paper ko with some constuctive methods gauss polynomials etc which were completely unrelated to the deterministic part of korner s paper kahane ka proved that there exists a sequence pn with pn kn which is en ultraflat where en o n 1 17 log n thus the erdos conjecture was disproved for the classes kn in this paper we study ultraflat sequences pn of unimodular polynomials pn kn in general not necessarily those produced by kahane in his paper ka

littlewood polynomials with high order zeros let n m be the minimal length of a polynomial with 1 coefficients divisible by x 1 m byrnes noted that n m 2 m for each m and asked whether in fact n m 2 m boyd showed that n m 2 m for all m 5 but n 6 48 he further showed that n 7 96 and that at 8 is one of the 5 numbers 96 144 160 176 or 192 here we prove that n 8 144 similarly let m n be the maximal power of x 1 dividing some polynomial of degree n 1 with 1 coefficients boyd was able to find m n for n 88 in this paper we determine m n for n 168

blind channel estimation for mimo ofdm systems via nonredundant linear precoding based on the assumption that the transmitted symbols are independent and identically distributed i i d we develop a simple subspace based blind channel estimation technique for orthogonal frequency division multiplexing ofdm systems by utilizing nonredundant linear block precoding a novel contribution is that the proposed method can be applied for scenarios where the number of receive antennas is less than the number of transmit antennas e g multiple input single output miso transmissions in which case the traditional subspace based methods could not be applied further consideration that can eliminate the multidimensional ambiguity in channel estimation under multiple transmitter scenarios is also proposed the numerical results clearly show the effectiveness of our proposed algorithm

blind channel equalization with colored sources based on second order statistics a linear prediction approach we consider the blind equalization and estimation of single user multichannel models from the second order statistics of the channel output when the channel input statistics are colored but known by exploiting certain results from linear prediction theory we generalize the algorithm of tong et al 1994 which was derived under the assumption of a white transmitted sequence in particular we show that all one needs to estimate the channel to within an unitary scaling constant and thus to find its equalizers is a that a standard channel matrix have full column rank and b a vector of the input signal and its delays have positive definite lag zero autocorrelation an algorithm is provided to determine the equalizer under these conditions we argue that because this algorithm makes explicit use of the input statistics the equalizers thus obtained should outperform those obtained by other methods that neither require nor exploit the knowledge of the input statistics simulation results are provided to verify this fact

determination of the merit factor of legendre sequences m j e golay ibid vol it 23 no 1 p 43 51 1977 has used the ergodicity postulate to calculate that the merit factor f of a legendre sequence offset by a fraction f of its length has an asymptotic value given by 1 f 2 3 4 mod f mod 8f sup 2 mod f mod

on the second order statistics of the weighted sample covariance matrix the second order statistics of the sample covariance are encountered in many covariance based processing algorithms this paper derives closed form expressions for the covariance of the weighted sample covariance matrix with an arbitrary weight for both a real system and complex system given a system model the results explicitly rely on the second order and fourth order statistics of the channel noise and inputs they are shown to coincide with the existing results when the channel inputs and noise are gaussian distributed our results can be directly applied to analyze the statistical properties of subspace based channel estimation methods for single input multiple output simo systems and code division multiple access cdma systems numerical examples are provided to further verify analyses

a semiblind channel estimation approach for mimo ofdm systems in this paper a very efficient semiblind approach that uses a training based least square criterion along with a blind constraint is proposed for multiple input multiple output orthogonal frequency division multiplexing mimo ofdm channel estimation the blind constraint is derived from the linear prediction of the received mimo ofdm signal and is used in conjunction with a weighting factor in the semiblind cost function an appealing scheme for the determination of the weighting factor is presented as a part of the proposed approach a perturbation analysis of the proposed method is conducted to justify the superiority of the semiblind solution and to obtain a closed form expression for the mean square error mse of the blind constraint further facilitating the calculation of the weighting factor the proposed method is validated through computer simulation based experimentations showing a very high estimation accuracy of the semiblind solution in terms of the mse of the channel estimate

classical jacobi polynomials closed form solutions for transverse vibrations this paper deals with transverse vibrations of nonuniform homogeneous beams and plates classes of beams and axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations respectively can be reduced to an eigenvalue singular problem singularities occur at both ends of orthogonal polynomials are reported exact natural frequencies and jacobi polynomials as exact mode shapes which result directly from eigenvalues and eigenfunctions of eigenvalue singular problems of classical orthogonal polynomials are reported for these classes the above classes of beams and plates hereafter called jacobi classes are given by geometry and boundary conditions the geometry consists of parabolic thickness variation with respect to the axial coordinate for beams and with respect to the radius for plates beams belonging to this class have either one or two sharp ends singularities along with certain boundary conditions plates have zero thickness at zero and outer radii the boundary value problems associated with plates and beams of two sharp ends are free boundary problems two other boundary value problems hinged free and sliding free are reported for beams with one sharp end also exact natural frequencies and mode shapes for uniformly rotating beams with hinged free boundary are reported

perturbation analysis for subspace decomposition with applications in subspace based algorithms subspace decomposition has been exploited in different applications due to perturbations from various sources such as finite data samples and measurement noise perturbations arise in subspaces therefore some loss is introduced to performance of subspace based algorithms although first order perturbation results have been proposed in the literature and applied to various problems up to second order perturbation analysis can provide more accurate analytical results and is studied in this paper based on the orthogonality principle perturbations of subspaces and singular values or eigenvalues are derived explicitly as functions of a perturbation in the objective matrix up to the second order respectively all in closed forms it is shown that by keeping only the first order terms the derived results reduce to those from existing approaches examples to apply the proposed results to both matrix computation and subspace based channel estimation are provided to verify our analysis

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

on codes satisfying mth order running digital sum constraints multi level sequences with a spectral null of order m at frequency f meaning that the power spectral density and its first 2m 1 derivatives vanish at f are characterized by finite state transition diagrams fstds whose edge labels satisfy bounds on the variation of the mth order running digital sum rds necessary and sufficient conditions for fstds with higher order null constraints at dc and at an arbitrary submultiple of the symbol frequency are derived analytical results are given concerning the performance of codes satisfying an mth order rds constraint on partial response channels specific code designs for quaternary channel inputs are presented the euclidean distance properties of this new class of codes aside from their spectral shaping properties are demonstrated

eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method abstract in this paper free vibrations of a rotating clamped euler bernoulli beams with uniform cross section are studied using continuation method namely asymptotic numerical method the governing equations of motion are derived using lagrange s method the kinetic and strain energy expression are derived from rayleigh ritz method using a set of hybrid variables and based on a linear deflection assumption the derived equations are transformed in two eigenvalue problems where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms while the second is standard eigenvalue problem and corresponds to the flapping motion those two eigenvalue problems are transformed into two functionals treated by continuation method the asymptotic numerical method new method proposed for the solution of the linear gyroscopic system based on an augmented system which transforms the original problem to a standard form with real symmetric matrices by using some techniques to resolve these singular problems by the continuation method evolution curves of the natural frequencies against dimensionless angular velocity are determined at high angular velocity some singular points due to the linear elastic assumption are computed numerical tests of convergence are conducted and the obtained results are compared to the exact values results obtained by continuation are compared to those computed with discrete eigenvalue problem

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

on the zeros of cosine polynomials solution to a problem of littlewood littlewood in his 1968 monograph some problems in real and complex analysis 12 problem 22 poses the following research problem which appears to be still open problem if the nj are integral and all different what is the lower bound

binary transmission codes with higher order spectral zeros at zero frequency corresp a method is presented for designing binary channel codes in such a way that both the power spectral density function and its low order derivatives vanish at zero frequency the performance of the new codes is compared with that of channel codes designed with a constraint on the unbalance of the number of transmitted positive and negative pulses some remarks are made on the error correcting capabilities of these codes

null steering employing polynomials and restricted coefficients several novel formulations of analytic null steering algorithms for linear arrays are described two of them the beta technique and the positive coefficient model allow for placing an arbitrary number of nulls in arbitrary directions while maintaining main beam and sidelobe level control

on the second order statistics of the eigenvectors of sample covariance matrices eigenvectors of sample covariance matrices are used in a variety of statistical signal processing problems the second order statistics of these eigenvectors are needed to compute the variance of estimates based on these eigenvectors formulas for the second order statistics of the eigenvectors have been derived in the statistical literature and are widely used we point out a discrepancy between the statistics observed in numerical simulations and the theoretical formulas due to the nonuniqueness of the definition of eigenvectors we present two ways to resolve this discrepancy the first involves modifying the theoretical formulas to match the computational results the second involved a simple modification of the computations to make them match existing formulas

synthesis of antenna patterns with prescribed nulls the least mean square pattern synthesis method is extended to include constraints such as pattern nulls or pattern derivative nulls at a given set of angles the problem is formulated as a constrained approximation problem which is solved exactly and a clear geometrical interpretation of the solution in a multidimensional vector space is given the relation of the present method to those of constrained gain maximization and signal to noise ratio snr maximization is discussed and conditions for their equivalence stated for a linear uniform n element array it is shown that when m single nulls are imposed on a given quiescent pattern the optimum solution for the constrained pattern is the initial pattern and a set of m weighted sin nx sin x beams each beam is centered exactly at the corresponding pattern null irrespective of its relative location for the case of higher order nulls the n th pattern derivative is similarly canceled by the n th derivative of a sin nx sin x beam in addition simple quantitative expressions are derived for the pattern change and gain cost associated with the forced pattern nulls several illustrative examples are included

minesweeper on graphs minesweeper is a popular single player game it has been shown that the minesweeper consistency problem is np complete and the minesweeper counting problem is p complete in this paper we present efficient algorithms for solving these problems for minesweeper graphs with bounded treewidth our algorithms turn out to be much better than those based directly on dynamic programming the algorithms mostly use of algebraic operations on multivariate polynomials so that one may use existing software to implement them easily

equalization and semi blind channel estimation for space time block coded signals over a frequency selective fading channel in this paper we investigate the equalization and channel identification for space time block coded signals over a frequency selective multiple input multiple output mimo channel the equalization has been considered by taking into account the cyclostationarity of space time block coded signals the minimum mean square error mmse solutions have been derived for the linear and decision feedback df equalizers the channel estimation is required for the equalization with known symbols as pilot symbols mimo channels can be estimated in addition due to the redundancy induced by space time block code it is possible to identify mimo channels blindly using the subspace method we consider both blind and semi blind channel estimation for mimo channels it is shown that the semi blind channel estimate has fewer estimation errors and it results in less bit error rate performance degradation of the mmse linear and df equalizers

multistep linear predictors based blind identification and equalization of multiple input multiple output channels channel estimation and blind equalization of multiple input multiple output mimo communications channels is considered using primarily the second order statistics of the data such models arise when single receiver data from multiple sources is fractionally sampled assuming that there is excess bandwidth or when an antenna array is used with or without fractional sampling we consider the estimation of partial channel impulse response and design of finite length minimum mean square error mmse blind equalizers we extend the multistep linear prediction approach to mimo channels where the multichannel transfer function need not be column reduced moreover we allow infinite impulse response iir channels as well as the case where the subchannel transfer functions have common zeros in the past this approach has been confined to simo finite impulse response fir channels with no common subchannel zeros a related existing approach applicable to mimo channels is restricted to fir column reduced systems with equal length subchannels in our approach the knowledge of the nature of the underlying model fir or iir or the model order is not required our approach works when the subchannel transfer functions have common zeros as long as the common zeros are minimum phase zeros the sources are recovered up to a unitary mixing matrix and are further unmixed using higher order statistics of the data illustrative computer simulation examples are provided

asymptotically exact bounds on the size of high order spectral null codes the spectral null code s n k of kth order and length n is the union of n tuples with spl plusmn 1 components having kth order spectral null at zero frequency we determine the exact asymptotic in n behavior of the size of such codes in particular we prove that for n satisfying some divisibility conditions log sub 2 s n k n k sup 2 2log sub 2 n c sub k o 1 where c sub k is a constant depending only on k and o 1 tends to zero when n grows this is an improvement on the earlier known bounds due to roth siegel and vardy see ibid vol40 p 1826 40 1994

polynomials with coefficients from a finite set in 1945 duffin and schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of c is already a rational function their proof is relatively indirect it is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem this will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of c cannot tend to zero uniformly on an arc of the unit circle another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients let n be so large that v satisfies n 1 we show that any polynomial in has at least zeros in any disk with center on the unit circle and radius n

bilinear formulation applied to the response and stability of helicopter rotor blade a time finite element method based on hamilton s law of varying action is applied to the analysis of helicopter blade dynamics the bilinear formulation of hamilton s law of varying action is used to discretize the temporal dependence of the equations of motion variable order shape and test functions based on legendre polynomials are used two approaches for the numerical implementation of the bilinear formulation are used one approach obtains the response of a general dynamic system using marching in time the other approach is based on the assumption that the solution is identical at the beginning and end of one period therefore this approach is suitable for periodic systems the bilinear formulation in imposed periodicity mode is used to compute the response of a rigid blade with flap and lag degrees of freedom in forward flight the bilinear formulation in marching mode is applied to find the floquet transition matrix necessary to test the stability of blade motion

the modulus of polynomials with zeros on the unit circle a problem of erdos must tend to infinity note that this is in fact problem 21 in erdds very first some unsolved problems paper see michigan math j 4 1957 291 300 let us first say a few words on the motivation behind this problem if the points 21 i an are just the nth roots of unity i e tj e27ij ni i 1 n then pn z n1 andsoan 2 thisshowsthatthe conjecture lim sup an must be a consequence of the impossibility of getting every segment 1 2 n close to uniform distribution there seems therefore to be an intimate connection with the classical van der corput problem of discrepancy of sequences in the unit interval by changing variables one can easily pass from the unit circle to the unit interval as follows consider the function

codes with a multiple spectral null at zero frequency codes whose spectral density and its lower order derivatives vanish at zero frequency are investigated and necessary and sufficient conditions for the spectral properties are given in particular for the most common balanced alphabets a complete characterization is furnished for the class of symbol by symbol codes with vanishing spectrum and second derivative the fundamental results are extended to block codes and suitable criteria for their design are given specific examples of binary and ternary block codes are presented their performance in terms of low power content at low frequencies compares advantageously with that of some conventional codes

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