synchronization in a semiclassical kuramoto model synchronization is a ubiquitous phenomenon occurring in social biological and technological systems when the internal rythms of their constituents are adapted to be in unison as a result of their coupling this natural tendency towards dynamical consensus has spurred a large body of theoretical and experimental research in recent decades the kuramoto model constitutes the most studied and paradigmatic framework in which to study synchronization in particular it shows how synchronization appears as a phase transition from a dynamically disordered state at some critical value for the coupling strength between the interacting units the critical properties of the synchronization transition of this model have been widely studied and many variants of its formulations have been considered to address different physical realizations however the kuramoto model has been studied only within the domain of classical dynamics thus neglecting its applications for the study of quantum synchronization phenomena based on a system bath approach and within the feynman path integral formalism we derive equations for the kuramoto model by taking into account the first quantum fluctuations we also analyze its critical properties the main result being the derivation of the value for the synchronization onset this critical coupling increases its value as quantumness increases as a consequence of the possibility of tunneling that quantum fluctuations provide

on the kz reduction the korkine zolotareff kz reduction is one of the often used reduction strategies for lattice decoding in this paper we first investigate some important properties of kz reduced matrices specifically we present a linear upper bound on the hermit constant which is around 7 8 times of the existing sharpest linear upper bound and an upper bound on the kz constant which is polynomially smaller than the existing sharpest one we also propose upper bounds on the lengths of the columns of kz reduced matrices and an upper bound on the orthogonality defect of kz reduced matrices which are even polynomially and exponentially smaller than those of boosted kz reduced matrices respectively then we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem svp when its basis matrix is lll reduced these upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a kz reduction algorithm finally we propose a new kz reduction algorithm by modifying the commonly used schnorr euchner search strategy for solving svps and the basis expansion method proposed by zhang et al simulation results show that the new kz reduction algorithm is much faster and more numerically reliable than the kz reduction algorithm proposed by zhang et al especially when the basis matrix is ill conditioned

reconstructing algebraic functions from mixed data we consider a variant of the traditional task of explicitly reconstructing algebraic functions from black box representations in the traditional setting for such problems one is given access to an unknown function f that is represented by a black box or an oracle which can be queried for the value of f at any input given a guarantee that this unknown function f is some nice algebraic function say a polynomial in its input of degree bound d the goal of the reconstruction problem is to explicitly determine the coefficients of the unknown polynomial all work on polynomial interpolation especially sparse ones are or may be presented in such a setting the work of kaltofen and trager computing with polynomials given by black boxes for their evaluations greatest common divisors factorization separation of numerators and denominators in proc 29th ann ieee symp on foundations of computer science 1988 pp 296 305 for instance highlights the utility of this setting by performing numerous manipulations on polynomials presented as black boxes the variant considered in this paper differs from the traditional setting in that our black boxes represent several algebraic functions f1 fk where at each input x the box arbitrarily chooses a subset of f1 x fk x to output and we do not know which subset it outputs we show how to reconstruct the functions f1 fk from the black box provided the black box outputs according to these functions often this allows us to group the sample points into sets such that for each set all outputs to points in the set are from the same algebraic function our methods are robust in the presence of a small fraction of arbitrary errors in the black box our model and techniques can be applied in the areas of computer vision machine learning curve fitting and polynomial approximation self correcting programs and bivariate polynomial factorization

multi stage robust chinese remainder theorem it is well known that the traditional chinese remainder theorem crt is not robust in the sense that a small error in a remainder may cause a large reconstruction error a robust crt was recently proposed for a special case when the greatest common divisor gcd of all the moduli is more than 1 and the remaining integers factorized by the gcd are co prime it basically says that the reconstruction error is upper bounded by the remainder error level tau if tau is smaller than a quarter of the gcd of all the moduli in this paper we consider the robust reconstruction problem for a general set of moduli we first present a necessary and sufficient condition on the remainder errors with a general set of moduli and also a corresponding robust reconstruction method this can be thought of as a single stage robust crt we then propose a two stage robust crt by grouping the moduli into several groups as follows first the single stage robust crt is applied to each group then with these robust reconstructions from all the groups the single stage robust crt is applied again across the groups this is easily generalized to multi stage robust crt with this two stage robust crt the robust reconstruction holds even when the remainder error level tau is above the quarter of the gcd of all the moduli and an algorithm on how to group a set of moduli for a better reconstruction robustness is proposed in some special cases

generalized quantum anharmonic oscillator using an operator ordering approach we construct a generalized expression for the normal ordering of a a m for integral values of m and use the result to study the quantum anharmonic oscillator problem in the heisenberg approach in particular we derive generalized expressions for energy eigenvalues and frequency shifts for the hamiltonian h x2 2 2 2 m xm we also derive a closed form first order multiscale perturbation theoretic operator solution of this hamiltonian with a view to generalize some recent results of bender and bettencourt 1996 phys rev lett 77 4114 1996 phys rev d 54 1710

selecting wavelengths for least squares range estimation we consider the problem of estimating the distance or range between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods more recently estimators based on the chinese remainder theorem crt and least squares have appeared recent research suggests the least squares estimator to be the most accurate in many cases the accuracy of all of these range estimators depends upon the wavelengths chosen this leads to the problem of selecting wavelengths that maximize accuracy procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described but procedures for the crt and least squares estimators are yet to be developed in this paper we develop an algorithm to automatically select wavelengths for use with the least square range estimator the algorithm minimizes an optimization criterion connected with the mean square error interesting properties of a particular class of lattices simplify the criterion allowing minimization by depth first search monte carlo simulations indicate that wavelengths that minimize the criterion can result in considerably more accurate range estimates than wavelengths selected by ad hoc means

the caldeira leggett quantum master equation in wigner phase space continued fraction solution and application to brownian motion in periodic potentials the continued fraction method of solving classical fokker planck equations has been adapted to tackle quantum master equations of the caldeira leggett type this was done taking advantage of the phase space wigner representation of the quantum density matrix the approach differs from those in which some continued fraction expression is found for a certain quantity in that the full solution of the master equation is obtained by continued fraction methods this allows us to study in detail the effects of the environment fluctuations and dissipation on several classes of nonlinear quantum systems we apply the method to the canonical problem of quantum brownian motion in periodic potentials both for cosine and ratchet potentials lacking inversion symmetry

asymptotic integration of ordinary different systems an extension and a new proof of hartman and wintner s theorem on asymptotic integration are given

quantum brownian motion and generalizations of hermite polynomials abstract we introduce new families of orthogonal polynomials h d n motivated by the non equilibrium evolution of a quantum brownian particle qbp the h d n s generalize non trivially the standard hermite polynomials employed for classical brownian motion we treat several models labelled by d for a non equilibrium qbp by means of the wigner function w in the presence of a heat bath at thermal equilibrium with and without ab initio friction for long times for a suitable class of initial conditions the non equilibrium wigner function w should approach in some sense the time independent equilibrium wigner function w e q d which describes the thermal equilibrium of the qbp with the heat bath and plays a central role w e q d is chosen to be the weight function which orthogonalizes the h d n s new results on w e q d and on the h d n s are reported we justify the key role of the h d n s as follows using the h d n s moments w e q d n and w n are introduced for w e q d and w respectively at equilibrium all moments w e q d n except the lowest one w e q d 0 vanish identically off equilibrium one expects that for long times for suitable initial conditions i all non equilibrium moments w n except the lowest moment w 0 will approach zero while ii the lowest non equilibrium moment w 0 will tend to w e q d 0 0 to complete the justification we outline how the approximate long time non equilibrium theories determined by w 0 for the different models d yield smoluchowski equations and irreversible evolutions of the qbp towards thermal equilibrium

exact solution of smoluchowski s equation for reorientational motion in maier saupe potential abstract the analytic treatment of the non inertial rotational diffusion equation i e of smoluchowski s one se in a symmetric genuinely double well maier saupe uniaxial potential of mean torque is considered such potential may find applications to reorientations of the fragments of structure in polymers and proteins we obtain the exact solution of se via the confluent heun s function the solution is uniformly valid for any barrier height we apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height in the intermediate to high barrier low temperature region the results of our approach are in full agreement with those of the approach developed by coffey kalmykov dejardin and their coauthors in the low barrier high temperature region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well the reason is that the above mentioned formula is obtained in the stationary limit we conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly when this requirement is satisfied as e g at the calculation of the longitudinal correlation time we obtain absolute identity of our results with the literature formula in the whole range of barrier heights the drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non vanishing eigenvalue

relaxation dynamics of a particle in the presence of an external potential exact solution in terms of matrix continued fractions exact expressions for the laplace transform of the after effect function arising from the solution of the langevin or underlying fokker planck equation for brownian motion in an external potential are obtained as a sum of products of infinite matrix continued fractions this is accomplished by reducing the scalar multiterm recurrence relations associated with the fokker planck equation to matrix three term recurrence relations the solution is illustrated by considering the problem of dielectric relaxation of a single axis rotator subjected to both constant and crystalline anisotropy fields

asymptotic formula for solutions of l2 perturbed linear delay difference systems abstract the aim of this paper is to obtain an asymptotic formula for each solution of a l 2 perturbed linear delay difference system the results obtained here extend some earlier results for ordinary difference equations and are parallel to some recent results for delay differential equations

quantum smoluchowski equation for driven systems we consider a driven quantum harmonic oscillator strongly coupled to a heat bath starting from the exact quantum langevin equation we use a green s function approach to determine the corresponding semiclassical equation for the wigner phase space distribution in the limit of high friction and high temperature we apply brinkman s method to derive the quantum smoluchowski equation for the probability distribution in position space we further determine the range of validity of the equation and discuss the special case of a brownian parametric oscillator

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

time scales and diffusion mechanisms in the kramers equation with periodic potentials i abstract noise assisted diffusion on a periodic substrate is studied solving the klein kramers equation kke by the matrix continued fraction method assuming a periodic cosine potential and a homogeneous friction the one particle dynamics is completely characterized both at short and long times calculating up to large momentum transfer the dynamic structure factor its full width at half maximum the velocity self correlation spectrum and the mean square displacement four different dynamical regimes are found as the potential strength or the friction vary at high potential barriers diffusion proceeds by single or multiple jumps at high and low friction respectively while two different kinds of continuous unactivated diffusion are found at lower barriers these dynamical features are discussed in terms of three characteristic time scales whose relations determine the diffusion mechanisms

error correction in polynomial remainder codes with non pairwise coprime moduli and robust chinese remainder theorem for polynomials this paper investigates polynomial remainder codes with non pairwise coprime moduli we first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small namely the robust chinese remainder theorem crt for polynomials it basically says that a polynomial can be reconstructed from its erroneous residues such that the degree of the reconstruction error is upper bounded by tau whenever the degrees of all residue errors are upper bounded by tau where a sufficient condition for tau and a reconstruction algorithm are obtained by relaxing the constraint that all residue errors have small degrees another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues we finally obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result some errors with small degrees can be also corrected in the residues with this newly obtained result improvements in uncorrected error probability and burst error correction capability in data transmission are illustrated

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

asymptotic integration of delay differential systems on considere des equations non autonomes qui sont asymptotiquement autonomes et on cherche des formules pour les solutions aux grandes valeurs de la variable independante

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

exact and robust reconstructions of integer vectors based on multidimensional chinese remainder theorem md crt the robust chinese remainder theorem crt has been recently proposed for robustly reconstructing a large nonnegative integer from erroneous remainders it has found many applications in signal processing including phase unwrapping and frequency estimation under sub nyquist sampling motivated by the applications in multidimensional md signal processing in this article we propose the md crt and robust md crt for integer vectors specifically by rephrasing the abstract crt for rings in number theoretic terms we first derive the md crt for integer vectors with respect to a general set of integer matrix moduli which provides an algorithm to uniquely reconstruct an integer vector from its remainders if it is in the fundamental parallelepiped of the lattice generated by a least common right multiple of all the moduli for some special forms of moduli we present explicit reconstruction formulae moreover we derive the robust md crt for integer vectors when the remaining integer matrices of all the moduli left divided by their greatest common left divisor gcld are pairwise commutative and coprime two different reconstruction algorithms are proposed and accordingly two different conditions on the remainder error bound for the reconstruction robustness are obtained which are related to a quarter of the minimum distance of the lattice generated by the gcld of all the moduli or the smith normal form of the gcld

semiclassical klein kramers and smoluchowski equations for the brownian motion of a particle in an external potential the quantum brownian motion of a particle in an external potential v x is treated using the master equation for the wigner distribution function w x p t in phase space x p a heuristic method of determination of diffusion coefficients in the master equation is proposed the time evolution equation so obtained contains explicit quantum correction terms up to o 4 and in the classical limit 0 reduces to the klein kramers equation for a quantum oscillator the method yields an evolution equation for w x p t coinciding with that of agarwal 1971 phys rev a 4 739 in the non inertial regime by applying the brinkman expansion of the momentum distribution in weber functions brinkman 1956 physica 22 29 the corresponding semiclassical smoluchowski equation is derived

a robust generalized chinese remainder theorem for two integers a generalized chinese remainder theorem crt for multiple integers from residue sets has been studied recently where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known a robust crt has also been proposed lately to robustly reconstruct a single integer from its erroneous remainders in this paper we consider the reconstruction problem of two integers from their residue sets where the remainders not only are out of order but also may have errors we prove that two integers can be robustly reconstructed if their remainder errors are less than m 8 where m is the greatest common divisor of all the moduli we also propose an efficient reconstruction algorithm finally we present some simulations to verify the efficiency of the proposed algorithm this paper is motivated from and has applications in the determination of multiple frequencies from multiple undersampled waveforms

on asymptotically autonomous retarded functional differential equations the aim of the present paper is to investigate the asymptotic behavior of the solutions of the asymptotically autonomous retarded functional differential equation t lxt m t xt we obtain a convergence result without assuming that m t is in lp 1 p moreover we give bounds on the rate of convergence as well

resolution doubled co prime spectral analyzers for removing spurious peaks co prime dft filter banks provide an effective tool of spectral analysis in case that the sampling rate is far lower than the nyquist sampling rate however since the practical prototype filters are not ideal spurious peaks will occur in multiple locations when the frequency offsets of the stimulus components are close to half of the frequency resolution of the co prime spectrum as a result the quality of the original co prime panorama spectrum will degrade greatly this paper proposes a modified analyzer by synthesizing the original co prime analyzer and a 0 5 frequency resolution shifted analyzer which is derived from the original one by applying two simple operations on each low rate data channel following some decision criterion between the original co prime analyzer and the shifted analyzer the synthesized analyzer not only removes all the annoying spurious peaks but also doubles the spectral resolution numerical comparisons are presented throughout to illustrate the above improvements

exact solution for the extended debye theory of dielectric relaxation of nematic liquid crystals abstract the exact solution for the transverse i e in the direction perpendicular to the director axis component of a nematic liquid crystal and the corresponding correlation time t is presented for the uniaxial potential of martin et al symp faraday soc 5 1971 119 the corresponding longitudinal i e parallel to the director axis quantities t may be determined by simply replacing magnetic quantities by the corresponding electric ones in our previous study of the magnetic relaxation of single domain ferromagnetic particles coffey et al phys rev e 49 1994 1869 the calculation of is accomplished by expanding the spatial part of the distribution function of permanent dipole moment orientations on the unit sphere in the fokker planck equation in normalised spherical harmonics this leads to a three term recurrence relation for the laplace transform of the transverse decay functions the recurrence relation is solved exactly in terms of continued fractions the zero frequency limit of the solution yields an analytic formula for the transverse correlation time t which is easily tabulated for all nematic potential barrier heights a simple analytic expression for t which consists of the well known meier saupe formula mol cryst 1 1966 515 with a substantial correction term which yields a close approximation to the exact solution for all and the correct asymptotic behaviour is also given the effective eigenvalue method is shown to yield a simple formula for t which is valid for all it appears that the low frequency relaxation process for both orientations of the applied field is accurately described in each case by a single debye type mechanism with corresponding relaxation times t t

analytical inversion of general tridiagonal matrices in this paper we give a complete analysis for general tridiagonal matrix inversion for both non block and block cases and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or toeplitz tridiagonal matrices

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

asymptotic behavior and oscillation of functional differential equations asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established in the scalar case it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation the proofs are based on a recent perron type theorem for functional differential equations

short pulse multi frequency phase based time delay estimation an approach for time delay estimation based on phase difference detection is presented a multiple frequency short continuous wave pulse is used to solve the well known phase ambiguity problem when the maximum distance exceeds a full wavelength within an unambiguous range defined with the lowest frequency difference between components the corresponding phase difference is unique and any distance within this range can be determined phase differences between higher frequency components are used to achieve a finer resolution the concept will be presented and the effectiveness of the approach will be investigated through theoretical and practical examples the method will be validated using underwater acoustic measurements simulating noisy environments demonstrating resolutions better than a 50th of a wavelength even in the presence of high levels 5db of additive gaussian noise furthermore the algorithm is simple to use and can be easily implemented being based on phase detection using the disc

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

l2 perturbation of a linear delay differential equation abstract in this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation t a 0 a t x t b 0 b t x t assuming that it is an l 2 perturbation of the autonomous equation t a 0 x t b 0 x t our theorem is an essential generalization of the corresponding result in the classical book by r bellman and k l cooke the proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation

the hartman wintner theorem for functional differential equations abstract in this paper we present a generalization of the hartman wintner theorem about the asymptotic behavior of the solutions of linear ordinary differential equations to functional differential equations

fault tolerant convolution via chinese remainder codes constructed from non coprime moduli this paper develops a framework for performing fault tolerant convolution via error correcting codes based on the chinese remainder theorem crt with non coprime moduli in contrast to convolution that is protected through crt codes with coprime moduli our scheme allows errors to be detected and located in a highly parallel manner and has the advantage of being robust to faults that occur during the error handling stage in addition for certain important classes of errors the codes developed in this paper require less redundancy than crt codes constructed from coprime moduli we demonstrate the applicability and advantages of our codes by comparing them to both crt codes with coprime moduli and repetition codes we focus on codes over polynomial rings but our constructions apply naturally to integer codes we discuss this extension and show that it generalizes the distance properties and error correction strategies associated with previously developed approaches for codes based on non coprime integer moduli

a robust chinese remainder theorem with its applications in frequency estimation from undersampled waveforms the chinese remainder theorem crt allows to reconstruct a large integer from its remainders modulo several moduli in this paper we propose a robust reconstruction algorithm called robust crt when the remainders have errors we show that using the proposed robust crt the reconstruction error is upper bounded by the maximal remainder error range named remainder error bound if the remainder error bound is less than one quarter of the greatest common divisor gcd of all the moduli we then apply the robust crt to estimate frequencies when the signal waveforms are undersampled multiple times it shows that with the robust crt the sampling frequencies can be significantly reduced

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

a closed form robust chinese remainder theorem and its performance analysis chinese remainder theorem crt reconstructs an integer from its multiple remainders that is well known not robust in the sense that a small error in a remainder may cause a large error in the reconstruction a robust crt has been recently proposed when all the moduli have a common factor and the robust crt is a searching based algorithm and no closed from is given in this paper a closed form robust crt is proposed and a necessary and sufficient condition on the remainder errors for the closed form robust crt to hold is obtained furthermore its performance analysis is given it is shown that the reason for the robustness is from the remainder differential process in both searching based and our proposed closed form robust crt algorithms which does no exist in the traditional crt we also propose an improved version of the closed form robust crt finally we compare the performances of the traditional crt the searching based robust crt and our proposed closed form robust crt and its improved version algorithms in terms of both theoretical analysis and numerical simulations the results demonstrate that the proposed closed form robust crt its improved version has the best performance has the same performance but much simpler form than the searching based robust crt

compact sequences of co primes and their applications to the security of crt based threshold schemes crt based threshold secret sharing schemes use sequences of pairwise co prime positive integers in their construction if these sequences are not compact then the crt based threshold secret sharing schemes may have a weak security in the sense of a massive loss of entropy or an arbitrarily large information rate in this paper compact sequences of co primes are introduced and their applications to the security of crt based threshold secret sharing schemes is throughout investigated it is shown that all the results regarding the security of crt based threshold secret sharing schemes that use sequences of consecutive primes also hold for threshold secret sharing schemes that use compact sequences of co primes moreover compact sequences of co primes may be significantly denser than sequences of consecutive primes of the same length and their use in the construction of crt based threshold secret sharing schemes may lead to better security properties

escape times for rigid brownian rotators in a bistable potential from the time evolution of the green function and the characteristic time of the probability evolution abstract the greatest relaxation time for an assembly of three dimensional rigid rotators in an axially symmetric bistable potential is obtained exactly in terms of continued fractions as a sum of the zero frequency decay functions averages of the legendre polynomials of the system this is accomplished by studying the entire time evolution of the green function transition probability by expanding the time dependent distribution as a fourier series and proceeding to the zero frequency limit of the laplace transform of that distribution the procedure is entirely analogous to the calculation of the characteristic time of the probability evolution the integral of the configuration space probability density function with respect to the position co ordinate for a particle undergoing translational diffusion in a potential a concept originally used by malakhov and pankratov physica a 229 1996 109 this procedure allowed them to obtain exact solutions of the kramers one dimensional translational escape rate problem for piecewise parabolic potentials the solution was accomplished by posing the problem in terms of the appropriate sturm liouville equation which could be solved in terms of the parabolic cylinder functions the method as applied to rotational problems and posed in terms of recurrence relations for the decay functions i e the brinkman approach c f blomberg physica a 86 1977 49 as opposed to the sturm liouville one demonstrates clearly that the greatest relaxation time unlike the integral relaxation time which is governed by a single decay function albeit coupled to all the others in non linear fashion via the underlying recurrence relation is governed by a sum of decay functions the method is easily generalized to multidimensional state spaces by matrix continued fraction methods allowing one to treat non axially symmetric potentials where the distribution function is governed by two state variables

a modular approach to key safeguarding a method is proposed for a key safeguarding scheme threshold scheme in which the shadows are congruence classes of a number associated with the original key a variation of this scheme provides efficient error detection and even exposes deliberate tampering certain underlying similarities of this scheme with shamir s interpolation method make it possible to incorporate these protective features in that method as well

convergence theorems and asymptotic integration for functional differential equations abstract this paper is devoted to the investigation on convergence of solutions and asymptotic integration of some functional differential equations fdes in the first part of the paper some convergence criteria are obtained and an invariance principle is established for the convergence of solutions of linear fdes under perturbations and disturbances with l 1 integrable coefficients these theorems are applicable to some nonlinear or nonhomogeneous equations in the second part the convergence theorems are applied to investigate the asymptotic integration of some linear fdes this research is made mainly to discuss a situation that was avoided by some previous literature some examples are given to illustrate the theorems

error correcting data structures we study data structures in the presence of adversarial noise we want to encode a given object in a succinct data structure that enables us to efficiently answer specific queries about the object even if the data structure has been corrupted by a constant fraction of errors we measure the efficiency of a data structure in terms of its length the number of bits in its representation and query answering time measured by the number of bit probes to the possibly corrupted representation the main issue is the trade off between these two this new model is the common generalization of static data structures and locally decodable error correcting codes ldcs we prove a number of upper and lower bounds on various natural error correcting data structure problems in particular we show that the optimal length of t probe error correcting data structures for the membership problem where we want to store subsets of size s from a universe of size n such that membership queries can be answered effic

distance estimation using wrapped phase measurements in noise measuring a distance using phase measurements is a common practice in many areas of engineering almost inevitably these measurements are accompanied by noise and are always subject to ambiguity resulting from the phase of modulo 2 pi in the presence of phase ambiguity where for instance the unknown distance is far longer than the wavelength of the signal carrying the phase measurement the distance cannot be uniquely determined one way to resolve this phase ambiguity is to measure the signal phase at multiple frequencies converting the phase ambiguity problem into one of solving a family of diophantine equations typically under some reasonable assumptions the diophantine problems can be solved using the chinese reminder theorem as documented in the literature however the existing algorithms can experience significant computational overhead for a given application because an exhaustive search is required in this paper a novel method addressing the phase ambiguity issue using lattice theoretic ideas is proposed and a closed form algorithm is presented for the estimation of the number of wavelengths in the unknown distance using the phase measurements taken at multiple wavelengths the algorithm is extremely efficient as the diophantine equations are solved without searching the unknown distance can then be estimated via a maximum likelihood method using the unwrapped phase measurement a statistical bound of the measurement noise which ensures that the number of whole wavelengths in the unknown distance can be found with a probability close to unity is derived the robustness efficiency and estimation accuracy of the proposed method are demonstrated by the simulated results presented

optimal estimates of two common remainders for a robust generalized chinese remainder theorem estimation of multiple common remainders from a sequence of erroneous residue sets is an important step for the robust generalized chinese remainder theorem crt this paper considers the problem of how to estimate the two common remainders from their residue sets modulo a set of moduli to measure the errors properly under the modular operation we introduce two type circular distances based on these circular distances two estimation methods are proposed by properly grouping the erroneous remainders into two ordered clusters both of the two methods perform better with lower computational complexities than the existing methods in this paper theoretical analysis as well as analytical results for the two proposed methods are obtained for the first method the two optimal estimates are proved to be in a finite set with no more than l candidates where l is the number of the given moduli the second method have closed forms simulation results show that the two proposed methods have nearly the same performance these optimal estimates can improve the performance of the robust generalized crt significantly

asymptotic integration of nonautonomous delay differential systems we prove existence theorems and asymptotic formulas for the solutions of a class of delay differential equations with time state dependent lag

the security of all rsa and discrete log bits we study the security of individual bits in an rsa encrypted message en x we show that given en x predicting any single bit in x with only a nonnegligible advantage over the trivial guessing strategy is through a polynomial time reduction as hard as breaking rsa moreover we prove that blocks of o log log n bits of x are computationally indistinguishable from random bits the results carry over to the rabin encryption scheme considering the discrete exponentiation function gx modulo p with probability 1 o 1 over random choices of the prime p the analog results are demonstrated the results do not rely on group representation and therefore applies to general cyclic groups as well finally we prove that the bits of ax b modulo p give hard core predicates for any one way function f all our results follow from a general result on the chosen multiplier hidden number problem given an integer n and access to an algorithm px that on input a random a i zn returns a guess of the ith bit of ax mod n recover x we show that for any i if px has at least a nonnegligible advantage in predicting the ith bit we either recover x or obtain a nontrivial factor of n in polynomial time the result also extends to prove the results about simultaneous security of blocks of o log log n bits

brownian motion quantum corrections and a generalization of the hermite polynomials the nonequilibrium evolution of a brownian particle in the presence of a heat bath at thermal equilibrium without imposing any friction mechanism from the outset is considered using a suitable family of orthogonal polynomials moments of the nonequilibrium probability distribution for the brownian particle are introduced which fulfill a recurrence relation we review the case of classical brownian motion in which the orthogonal polynomials are the hermite ones and the recurrence relation is a three term one after having performed a long time approximation in the recurrence relation the approximate nonequilibrium theory yields irreversible evolution of the brownian particle towards thermal equilibrium with the heat bath for quantum brownian motion which is the main subject of the present work we restrict ourselves to include the first quantum correction this leads us to introduce a new family of orthogonal polynomials which generalize the hermite ones some general properties of the new family are established the recurrence relation for the new moments of the nonequilibrium distribution including the first quantum correction turns out to be also a three term one which justifies the new family of polynomials a long time approximation on the new three term recurrence relation describes irreversible evolution towards equilibrium for the new moment of lowest order the standard smoluchowski equations for the lowest order moments are recovered consistently both classically and quantum mechanically

chinese remaindering with errors the chinese remainder theorem states that a positive integer m is uniquely specified by its remainder module k relatively prime integers p sub 1 spl middot spl middot spl middot p sub k provided m spl pi sub i 1 sup k p sub i thus the residues of m module relatively prime integers p sub 1 p sub 2 spl middot spl middot spl middot p sub n form a redundant representation of m if m spl pi sub i 1 sup k p sub i and k n this gives a number theoretic construction of an error correcting code that has been considered often in the past in this code a message integer m spl pi sub i 1 sup k p sub i is encoded by the list of its residues module p sub 1 spl middot spl middot spl middot p sub n by the chinese remainder theorem if a codeword is corrupted in e n k 2 coordinates then there exists a unique integer m whose corresponding codesword differs from the corrupted word in at most e places furthermore mandelbaum 1976 1978 shows how m can be recovered efficiently given the corrupted word provided that the p sub i s are very close to one another to deal with arbitrary p sub i s we present a variant of his algorithm that runs in almost linear time and recovers from e log p sub 1 log p sub 1 log p sub n spl middot n k errors our main contribution is an efficient decoding algorithm for the case in which the error e may be larger than n k 2 specifically given n residues r sub 1 spl middot spl middot spl middot r sub n and an agreement parameter t we find a list of all integers m spl pi sub i 1 sup k p sub i such that m mod p sub i r sub i for at least t values of i spl isin 1 spl middot spl middot spl middot n provided t spl omega spl radic kn log p sub n log p sub 1 for n spl gt k and p sub n spl les p sub 1 sup o 1 the fraction of error corrected by the algorithm is almost twice that corrected by the previous work more significantly the algorithm recovers the message even when the amount of agreement between the received word and the codeword is much smaller than the number of errors

how to calculate the wigner function from phase space we present a method for the direct computation of the wigner function by solving a coupled system of linear partial differential equations in phase space our modified spectral method relies on chebyshev polynomials since this approach allows us to include arbitrary high orders of partial derivatives our procedure is applicable to arbitrary binding potentials we apply our scheme to wigner functions of the harmonic oscillator the morse oscillator and an asymmetric double well potential

multiple error detection and correction based on redundant residue number systems this paper presents some results on multiple error detection and correction based on the redundant residue number system rrns rrns is often used in parallel processing environments because of its ability to increase the robustness of information passing between the processors the proposed multiple error correction scheme utilizes the chinese remainder theorem crt together with a novel algorithm that significantly simplifies the error correcting process for integers an extension of the scheme further reduces the computational complexity without compromising its error correcting capability proofs and examples are provided for the coding technique

bounded distance 1 soft decision reed solomon decoding we present a new reed solomon decoding algorithm which embodies several refinements of an earlier algorithm some portions of this new decoding algorithm operate on symbols of length lgq bits other portions operate on somewhat longer symbols in the worst case the total number of calculations required by the new decoding algorithm is proportional to nr where n is the code s block length and r is its redundancy this worst case workload is very similar to prior algorithms but in many applications average case workload and error correcting performance are both much better the input to the new algorithm consists of n received symbols from gf q and n nonnegative real numbers each of which is the reliability of the corresponding received symbol any conceivable errata pattern has a score equal to the sum of the reliabilities of its locations with nonzero errata values a max likelihood decoder would find the minimum score over all possible errata patterns our new decoding algorithm finds the minimum score only over a subset of these possible errata patterns the errata within any candidate errata pattern may be partitioned into errors and erasures depending on whether the corresponding reliabilities are above or below an erasure threshold different candidate errata patterns may have different thresholds each chosen to minimize its corresponding errata count which is defined as 2 spl middot number of errors number of erasures the new algorithm finds an errata pattern with minimum score among all errata patterns for which errata count spl les r 1 where r is the redundancy of the rs code conventional algorithms also require that the erasure threshold be set a priori the new algorithm obtains the best answer over all possible settings of the erasure threshold conventional cyclic rs codes have length n q 1 and their locations correspond to the nonzero elements of gf q the new algorithm also applies very naturally to rs codes which have been doubly extended by the inclusion of 0 and spl infin as additional locations

continued fraction as a discrete nonlinear transform the connection between a taylor series and a continued fraction involves a nonlinear relation between the taylor coefficients an and the continued fraction coefficients bn in many instances it turns out that this nonlinear relation transforms a complicated sequence an into a very simple one bn this simplification is illustrated in the context of graph combinatorics

maximum likelihood estimation based robust chinese remainder theorem for real numbers and its fast algorithm robust chinese remainder theorem crt has been recently investigated for both integers and real numbers where the folding integers are accurately recovered from erroneous remainders in this paper we consider the crt problem for real numbers with noisy remainders that follow wrapped gaussian distributions we propose the maximum likelihood estimation mle based crt when the remainder noises may not necessarily have the same variances furthermore we present a fast algorithm for the mle based crt algorithm that only needs to search for the solution among l elements where l is the number of remainders then a necessary and sufficient condition on the remainder errors for the mle crt to be robust is obtained which is weaker than the existing result finally we compare the performances of the newly proposed algorithm and the existing algorithm in terms of both theoretical analysis and numerical simulations the results demonstrate that the proposed algorithm not only has a better performance especially when the remainders have different error levels variances but also has a much lower computational complexity

distributive weighted threshold secret sharing schemes the concept of distributive weighted threshold access structure is introduced which is an weighted threshold access structure where the participants are distributed on levels the participants on the same level are assigned the same weight and the threshold of the access structure is 1 the weight of the participants on the ith level is of the form 1 ki and therefore the ith level induces a standard threshold access structure with threshold ki we propose a crt based realization of distributive weighted threshold access structures which is asymptotically perfect and perfect zero knowledge we also show that distributive weighted threshold access structures do not generally have ideal realizations in case of just one level our scheme can be viewed as an asymptotically perfect and perfect zero knowledge variation of the asmuth bloom secret sharing scheme

lattice based threshold changeability for standard shamir secret sharing schemes we consider the problem of increasing the threshold parameter of a secret sharing scheme after the setup share distribution phase without further communication between the dealer and the shareholders previous solutions to this problem require one to start off with a nonstandard scheme designed specifically for this purpose or to have communication between shareholders in contrast we show how to increase the threshold parameter of the standard shamir secret sharing scheme without communication between the shareholders our technique can thus be applied to existing shamir schemes even if they were set up without consideration to future threshold increases our method is a new positive cryptographic application for lattice reduction algorithms inspired by recent work on lattice based list decoding of reed solomon codes with noise bounded in the lee norm we use fundamental results from the theory of lattices geometry of numbers to prove quantitative statements about the information theoretic security of our construction these lattice based security proof techniques may be of independent interest

probability distribution function for reorientations in maier saupe potential abstract exact analytic solution for the probability distribution function of the non inertial rotational diffusion equation i e of the smoluchowski one in a symmetric maier saupe uniaxial potential of mean torque is obtained via the confluent heun s function both the ordinary maier saupe potential and the double well one with variable barrier width are considered thus the present article substantially extends the scope of the potentials amenable to the treatment by reducing smoluchowski equation to the confluent heun s one the solution is uniformly valid for any barrier height we use it for the calculation of the mean first passage time also the higher eigenvalues for the relaxation decay modes in the case of ordinary maier saupe potential are calculated the results obtained are in full agreement with those of the approach developed by coffey kalmykov dejardin and their coauthors in the whole range of barrier heights

towards robustness in residue number systems the problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli namely the robust remaindering problem may occur in many applications including phase unwrapping frequency detection from several undersampled waveforms wireless sensor networks etc assuming that the dynamic range of the large number is the maximal possible one i e the least common multiple lcm of all the moduli a method called robust chinese remainder theorem for solving the robust remaindering problem has been recently proposed in this paper by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli a tradeoff between the dynamic range and the robustness bound for two modular systems is studied it basically says that a decrease in the dynamic range may lead to an increase of the robustness bound we first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed form formula for the robustness to hold we then propose simple closed form reconstruction algorithms furthermore the newly obtained two modular results are applied to the robust reconstruction for multimodular systems and generalized to real numbers finally some simulations are carried out to verify our proposed theoretical results