random polynomials having few or no real zeros consider a polynomial of large degree n whose coefficients are independent identically distributed nondegenerate random variables having zero mean and finite moments of all orders we show that such a polynomial has exactly k real zeros with probability n b o 1 as n through integers of the same parity as the fixed integer k 0 in particular the probability that a random polynomial of large even degree n has no real zeros is n b o 1 the finite positive constant b is characterized via the centered stationary gaussian process of correlation function sech t 2 the value of b depends neither on k nor upon the specific law of the coefficients under an extra smoothness assumption about the law of the coefficients with probability n b o 1 one may specify also the approximate locations of the k zeros on the real line the constant b is replaced by b 2 in case the i i d coefficients have a nonzero mean

the gauss bonnet chern theorem a probabilistic perspective we prove that the euler form of a metric connection on a real oriented vector bundle e over a compact oriented manifold m can be identified as a current with the expectation of the random current defined by the zero locus of a certain random section of the bundle we also explain how to reconstruct probabilistically the metric and the connection on e from the statistics of random sections of e contents

complex roots of a class of random algebraic polynomials abstract the average density of the distribution of complex roots of equation 0 g 0 1 g 1 z 2 g 2 z 2 n 1 g n 1 z n 1 k with random coefficients j and for z c g j 1 j 0 1 n 1 and k a complex constant with equal real and imaginary parts is known the present paper provides a formula for such an average density for any constant real g j and j a j b j where the a j s and b j s j 0 1 n 1 are real standard normal independent random variables and k k 1 ik 2 where k 1 and k 2 are constants independent of z the limiting behaviour of this density function as n tends to infinity for several special selected g j s is obtained

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

lyapunov exponents and rotation number of two dimensional systems with telegraphic noise exact expressions for the exponential growth rates lyapunov exponents and rotation numbers of two dimensional systems perturbed by telegraphic noise are derived asymptotic expansions for small large slow and fast noise and the white noise limit are constructed then results are applied to the random oscillator stochastic stability in particular stabilization by noise and stochastic bifurcation

exact results for the lyapunov exponents of certain linear ito systems the lyapunov exponents of certain linear ito systems are considered the approach exploits the relations between the lyapunov exponents and the p moment lyapunov exponents as has been observed and studied by arnold siam j appl math 44 1984 pp 793 802 using the method that was developed in a leizarowitz explicit expressions for the lyapunov exponents of certain markov processes preprint the differential equations that are satisfied by the function t x 0 to e x t x 0 p and its laplace transform are derived in the two dimensional case this study yields explicit expressions for the lyapunov exponents

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

on the stability properties of a van der pol duffing oscillator that is driven by a real noise abstract in this paper we consider a van der pol duffing oscillator that is excited parametrically by a small intensity real noise which is assumed to be an integrable function of an n dimensional ornstein uhlenbeck vector process that is an output of a linear filter system the stability properties include the moment lyapunov exponent g p x 0 and the maximal lyapunov exponent and the stability in probability are examined to study a model of enhanced generality we remove both the detailed balance condition and the strong mixing condition in the case of an arbitrary finite real number p we employ the perturbation method and a spectrum representation of the fokker planck operator of the linear filter system to construct asymptotic expansions of the pth moment lyapunov exponent and the top lyapunov exponent the same methods are also used for a nonlinear stochastic system to obtain the fpk fokker planck kolmogonov equation for the amplitude process which is identical to the one that is derived from the stochastic averaging method in the case of a broadband noise excitation on the basis of this fpk equation we also examine the almost sure stability condition of the ito stochastic differential equation for the amplitude process which matches the result that is derived from the maximal lyapunov exponent finally the method proposed by lin and cai probabilistic structural dynamics advanced theory and application mcgraw hill new york 1995 is adopted to examine the stability in probability of the amplitude process for the nonlinear ito differential equation

optimal reconstruction of images from localized phase the importance of localized phase in signal representation is investigated the convergence rate of the pocs algorithm projection onto convex sets used for image reconstruction from spectral phase is defined and analyzed and the characteristics of images optimally reconstructed from phase only information are presented it is concluded that images of geometric form are most efficiently reconstructed from their spectral phase whereas images of symmetric form have the poorest convergence characteristics the transition between the two extremes is shown to be continuous the results provide a new approach and analysis of the previously reported advantages of the localized phase representation over the global approach and suggest possible compression schemes

almost sure instability of the random harmonic oscillator the problem of almost sure stability of the random harmonic oscillator with a jump process coefficient is studied it is shown that the sample paths are almost surely unstable and an asymptotic series expansion in the powers of the noise strength for the asymptotic exponential growth rate is obtained

lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle abstract let t x n 1 j 0 eijx where stands for a random choice of sign with equal probability the first author recently showed that for any 0 and most choices of sign minx 0 2 t x 0 and large n most choices of sign satisfy minx 0 2 t x n 1 2 furthermore we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials

adaptive coded modulation for fading channels we apply coset codes to adaptive modulation in fading channels adaptive modulation is a powerful technique to improve the energy efficiency and increase the data rate over a fading channel coset codes are a natural choice to use with adaptive modulation since the channel coding and modulation designs are separable therefore trellis and lattice codes designed for additive white gaussian noise awgn channels can be superimposed on adaptive modulation for fading channels with the same approximate coding gains we first describe the methodology for combining coset codes with a general class of adaptive modulation techniques we then apply this methodology to a spectrally efficient adaptive m ary quadrature amplitude modulation mqam to obtain trellis coded adaptive mqam we present analytical and simulation results for this design which show an effective coding gain of 3 db relative to uncoded adaptive mqam for a simple four state trellis code and an effective 3 6 db coding gain for an eight state trellis code more complex trellis codes are shown to achieve higher gains we also compare the performance of trellis coded adaptive mqam to that of coded modulation with built in time diversity and fixed rate modulation the adaptive method exhibits a power savings of up to 20 db

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

on the capacity of time varying channels with periodic feedback the capacity of time varying channels with periodic feedback at the transmitter is evaluated it is assumed that the channel state information csi is perfectly known at the receiver and is fed back to the transmitter at the regular time intervals the system capacity is investigated in two cases 1 finite state markov channel and 2 additive white gaussian noise channel with time correlated fading in the first case it is shown that the capacity is achievable by multiplexing multiple codebooks across the channel in the second case the channel capacity and the optimal adaptive coding is obtained it is shown that the optimal adaptation can be achieved by a single gaussian codebook while adaptively allocating the total power based on the side information at the transmitter

the complex zeros of random polynomials mark kac gave an explicit formula for the expectation of the number vn a of zeros of a random polynomial n i pn z e tj j o in any measurablc subset q of the reals here in i are independent standard normal random variables in fact for each n 1 he obtained an explicit intensity function gn for which e vn l jgn x dx here we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset q of the complex plane c namely we show that e vn ki j hn x y dxdy j gn x dx where hn is an explicit intensity function we also study the asymptotics of hn showing that for large n its mass lies close to and is uniformly distributed around the unit circle

the zeros of random polynomials further results and applications we consider the density of the zeros of random polynomials with nonzero mean correlated gaussian coefficients we show that for most communication problems the original result by hammersley 1956 can be expressed in a simpler form in order to illustrate the usefulness of the presented theory two applications are considered first the zeros of frequency selective ricean fading channels are investigated second the zeros of the transfer function corresponding to channel impulse response estimates obtained by a least squares approach are calculated for both applications implications for equalizer design are discussed

density of complex critical points of a real random so m 1 polynomial we study the density of complex critical points of a real random so m 1 polynomial in m variables in a previous paper macdonald in j stat phys 136 5 807 2009 the author used the poincare lelong formula to show that the density of complex zeros of a system of these real random polynomials rapidly approaches the density of complex zeros of a system of the corresponding complex random polynomials the su m 1 polynomials in this paper we use the kac rice formula to prove an analogous result the density of complex critical points of one of these real random polynomials rapidly approaches the density of complex critical points of the corresponding complex random polynomial in one variable we give an exact formula and a scaling limit formula for the density of critical points of the real random so 2 polynomial as well as for the density of critical points of the corresponding complex random su 2 polynomial

distribution of zeros of random and quantum chaotic sections of positive line bundles we study the limit distribution of zeros of certain sequences of holomorphic sections of high powers m n of a positive holomorphic hermitian line bundle l over a compact complex manifold m our first result concerns random sequences of sections using the natural probability measure on the space of sequences of orthonormal bases s n j of h 0 m l n we show that for almost every sequence s n j the associated sequence of zero currents 1 n z s n j tends to the curvature form of l thus the zeros of a sequence of sections s n h 0 m l n chosen independently and at random become uniformly distributed our second result concerns the zeros of quantum ergodic eigenfunctions where the relevant orthonormal bases s n j of h 0 m l n consist of eigensections of a quantum ergodic map we show that also in this case the zeros become uniformly distributed

stochastic dynamics of parametrically excited two d o f systems with symmetry in this paper we investigate the non linear dynamics of a two degree of freedom system with symmetries subject to random parametric excitation the study of this non linear near hamiltonian system is simplified by using the symmetry and separation of scales present in the problem to this end we study the equations as a random perturbation of a four dimensional weakly dissipative hamiltonian system we achieve the model reduction through stochastic averaging and the reduced process is simply a markov process on a line examination of the reduced markov process on the line yields many important results namely probability density functions and stochastic bifurcations the steady state dynamics is computed explicitly phenomenological and dynamical bifurcations are investigated the approach adopted in this paper can in principle be applied to any four dimensional integrable system

exploiting randomness in acoustic impulse responses to achieve headphone compensation through deconvolution this paper presents a method of headphone earphone equalization based upon deconvolution of the headphone impulse response from other acoustic filters in the processing chain the methods presented are thus applicable to areas such as spatial audio where input signals are processed with binaural impulse responses the extraction of low order from higher order acoustic impulse responses is justified based upon an application of the theory pertaining to the clustering of the zeros of random coefficient polynomials about the unit circle an argument is presented supporting the applicability of this theory to acoustic impulse responses such an equalization method guarantees a considerable reduction in computational effort over traditional separate compensating filters a comparison is presented of the implementation of the equalization using the homomorphic method and least squares method with and without regularization to a traditional separate least squares inverse filter the proposed approach is shown to have a performance comparable to a separate least squares designed compensator often giving superior performance despite a large reduction in complexity

correlation free forms for nonlinear stochastic systems abstract an asymptotic method for analysis of nonlinear systems with wide band and high frequency stochastic perturbation is developed correlation free ito limiting equations are derived and a generalization of the wong zakai correction for the drift term is obtained a correction for the diffusion term is also shown to exist several illustrative examples are considered

equidistribution and convergence speed for zeros of holomorphic sections of singular hermitian line bundles abstract we establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular hermitian line bundle with an estimate of the convergence speed

roots of random polynomials whose coefficients have logarithmic tails it has been shown by ibragimov and zaporozhets in prokhorov and contemporary probability theory 2013 springer that the complex roots of a random polynomial gn z nk 0 kzk with i i d coefficients 0 n concentrate a s near the unit circle as n if and only if elog 0 we study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like l log t log t as t where 0 and l is a slowly varying function under this assumption the structure of complex and real roots of gn is described in terms of the least concave majorant of the poisson point process on 0 1 0 with intensity v 1 dudv

equidistribution of zeros of random holomorphic sections for moderate measures we establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle the equidistribution is associated with a family of singular moderate measures we also give a convergence speed for the equidistribution

stability of a stochastic two dimensional non hamiltonian system we study the top lyapunov exponent of the response of a two dimensional non hamiltonian system driven by additive white noise the origin is not a fixed point for the system however there is an invariant measure for the one point motion of the system in this paper we consider the stability of the two point motion by khasminskii s method of linearization along trajectories the specific system we consider is the third order truncated normal form of the unfolding of a hopf bifurcation we show that in the small noise limit the top lyapunov exponent always approaches zero from below and is thus negative for noise sufficiently small we also show that there exist open sets of parameters for which the top lyapunov exponent is positive thus the two point motion can be either stable or unstable while the stationary density that describes the one point motion always exists

the local structure of the spectrum of the one dimensional schrodinger operator let h v frac d text 2 dt text 2 q t omega be an one dimensional random schrodinger operator in 2 v v with the classical boundary conditions the random potentialq t has a formq t f x t wherex t is a brownian motion on the compact riemannian manifoldk andf k r 1 is a smooth morse function mathop min limits k f 0 let n v delta sum limits ei v in delta 1 where 0 e i v are the eigenvalues ofh v the main result theorem 1 of this paper is the following ifv e 0 0 k z anda 0 a is a fixed constant then p left n v left e 0 frac a 2v e 0 frac a 2v right k right xrightarrow v to infty e an e 0 an e 0 k k wheren e 0 is a limit state density ofh v v this theorem mean that there is no repulsion between energy levels of the operatorh v v the second result theorem 2 describes the phenomen of the repulsion of the corresponding wave functions

long time dynamics of korteweg de vries solitons driven by random perturbations this paper deals with the transmission of a soliton in a random medium described by a randomly perturbed korteweg de vries equation different kinds of perturbations are addressed depending on their specific time or position dependences with or without damping we derive effective evolution equations for the soliton parameter by applying a perturbation theory of the inverse scattering transform and limit theorems of stochastic calculus original results are derived that are very different compared to a randomly perturbed nonlinear schrodinger equation first the emission of a soliton gas is proved to be a very general feature second some perturbations are shown to involve a speeding up of the soliton instead of the decay that is usually observed in random media

exact statistics of complex zeros for gaussian random polynomials with real coefficients k point correlations of complex zeros for gaussian ensembles of random polynomials of order n with real coefficients grprc are calculated exactly following an approach of hannay for the case of gaussian random polynomials with complex coefficients grpcc it is shown that in the thermodynamic limit of gaussian random holomorphic functions all the statistics converge to their grpcc counterparts as one moves off the real axis while close to the real axis the two cases are essentially different special emphasis is given to one and two point correlation functions in various regimes

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

on the functional central limit theorem and the law of the iterated logarithm for markov processes let xt t 0 be an ergodic stationary markov process on a state space s if is its infinitesimal generator on l2 s dm where m is the invariant probability measure then it is shown that for all f in the range of hat a n 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 int limits 0 nt f x s text ds text t underline underline 0 converges in distribution to the wiener measure with zero drift and variance parameter 2 2 f g 2 g g where g is some element in the domain of such that g f theorem 2 1 positivity of 2 is proved for nonconstant f under fairly general conditions and the range of is shown to be dense in 1 a functional law of the iterated logarithm is proved when the 2 th moment of f in the range of is finite for some 0 theorem 2 7 a under the additional condition of convergence in norm of the transition probability p t x d y to m dy as t for each x the above results hold when the process starts away from equilibrium theorems 2 6 2 7 b applications to diffusions are discussed in some detail

complex roots of a random algebraic polynomial abstract this paper for any constantk provides an exact formula for the average density of the distribution of the complex roots of equation 0 1z 2z2 n 1zn 1 kwhere j aj ibjand aj n 1j 0and bj n 1j 0are sequences of independent identically and normally distributed random variables andkis a complex number withkas its real and imaginary parts the case of real roots of the above equation with real coefficients andk z ris well known further we obtain the limiting behaviour of this distribution function asntends to infinity

maximal lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise asymptotic expansions for the exponential growth rate known as the lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise are constructed such systems arise naturally in the investigation of the stability of steady state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems almost sure stability or instability of dynamical systems depends on the sign of the maximal lyapunov exponent stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable the results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging where approximate ito equations in amplitudes and phases are obtained in the sense of weak convergence

complex zeros of trigonometric polynomials with standard normal random coefficients abstract in this paper we obtain an exact formula for the average density of the distribution of complex zeros of a random trigonometric polynomial 0 1 cos 2 cos 2 n cos n in 0 2 where the coefficients j a j b j and a j j 1 n and b j j 1 n are sequences of independent normally distributed random variables with mean 0 and variance 1 we also provide the limiting behaviour of the zeros density function as n tends to infinity the corresponding results for the case of random algebraic polynomials are known

on roots of random polynomials we study the distribution of the complex roots of random polynomials of degree n with i i d coefficients using techniques related to rice s treatment of the real roots question we derive under appropriate moment and regularity conditions an exact formula for the average density of this distribution which yields appropriate limit average densities further using a different technique we prove limit distribution results for coefficients in the domain of attraction of the stable law

comparison of some approaches for analysis of stochastic systems with short correlation time noise abstract we give a strict comparison of the diffusion bourret and van kampen approximations in the analysis of stochastic equations with short correlation time disturbances through the example of a random harmonic oscillator we show that the diffusion approximation does not always gives satisfactory results

root statistics of random polynomials with bounded mahler measure the mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coecient times the modulus of the product of its roots outside the unit circle the roots of a real degree n polynomial chosen uniformly from the set of polynomials of mahler measure at most 1 yields a pfaan point process on the complex plane when n is large with probability tending to 1 the roots tend to the unit circle and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle when this point is away from the real axis on which there is a positive probability of nding a root the scaled process degenerates to a determinantal point process with the same local statistics i e scalar kernel as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of mahler measure at most 1 three new matrix kernels appear in a neighborhood of 1 which encode information about the correlations between real roots between complex roots and between real and complex roots away from the unit circle the kernels converge to new limiting kernels which imply among other things that the expected number of roots in any open subset of c disjoint from the unit circle converges to a positive number we also give ensembles with identical statistics drawn from two dimensional electrostatics with potential theoretic weights and normal matrices chosen with regard to their topological entropy as actions on euclidean space

lyapunov exponents for two nonlinear systems driven by real noises for a van der pol duffing oscillator and a codimension two bifurcation system excited parametrically by a small real noise a model of enhanced generality is developed by assuming that the real noise is the first component of an output vector of a linear filter system which conforms with the detailed balance condition the strong mixing condition and the hypothesis i e the infinitesimal generator of the noise has only one isolated simple zero eigenvalue employed by arnold and co workers in 1986 are not considered to tackle the complexity encountered in the present work the asymptotic analysis approach and the eigenfunction expansion for the solution to the fokker planck equations are applied to calculate the asymptotic expansions of the invariant measures and the maximal lyapunov exponents for the relevant systems

on the complexity of real root isolation using continued fractions we present algorithmic complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers one motivation is to explain the method s good performance in practice we derive an expected complexity bound of email protected b d 6 d email protected 2 where d is the polynomial degree and t bounds the coefficient bit size using a standard bound on the expected bit size of the integers in the continued fraction expansion thus matching the current worst case complexity bound for real root isolation by exact methods sturm descartes and bernstein subdivision moreover using a homothetic transformation we improve the expected complexity bound to email protected b d email protected we compute the multiplicities within the same complexity and extend the algorithm to non square free polynomials finally we present an open source c implementation in the algebraic library synaps and illustrate its completeness and efficiency as compared to some other available software for this we use polynomials with coefficient bit size up to 8000 bits and degree up to 1000

real roots of random polynomials universality close to accumulation points we identify the scaling region of a width o n 1 in the vicinity of the accumulation points t 1 of the real roots of a random kac like polynomial of large degree n we argue that the density of the real roots in this region tends to a universal form shared by all polynomials with independent identically distributed coefficients ci as long as the second moment e c2i is finite in particular we reveal a gradual in contrast to the previously reported abrupt and quite nontrivial suppression of the number of real roots for coefficients with a nonzero mean value n e ci scaled as n n 1 2

almost sure asymptotic stability of a general four dimensional system driven by real noise in the first part of this paper we construct an asymptotic expansion for the maximal lyapunov exponent the exponential growth rate of solutions to a linear stochastic system and the rotation numbers for a general four dimensional dynamical system driven by a small intensity real noise process stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue this work is an extension of the work of sri namachchivaya and van roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered

sampling algebraic sets in local intrinsic coordinates numerical data structures for positive dimensional solution sets of polynomial systems are sets of generic points cut out by random planes of complementary dimension we may represent the linear spaces defined by those planes either by explicit linear equations or in parametric form these descriptions are respectively called extrinsic and intrinsic representations while intrinsic representations lower the cost of the linear algebra operations we observe worse condition numbers in this paper we describe the local adaptation of intrinsic coordinates to improve the numerical conditioning of sampling algebraic sets local intrinsic coordinates also lead to a better step size control we illustrate our results with maple experiments and computations with phcpack on some benchmark polynomial systems

channel capacity and error exponents of variable rate adaptive channel coding for rayleigh fading channels we have evaluated the information theoretical performance of variable rate adaptive channel coding for rayleigh fading channels the channel states are detected at the receiver and fed back to the transmitter by means of a noiseless feedback link based on the channel state informations the transmitter can adjust the channel coding scheme accordingly coherent channel and arbitrary channel symbols with a fixed average transmitted power constraint are assumed the channel capacity and the error exponent are evaluated and the optimal rate control rules are found for rayleigh fading channels with feedback of channel states it is shown that the variable rate scheme can only increase the channel error exponent the effects of additional practical constraints and finite feedback delays are also considered finally we compare the performance of the variable rate adaptive channel coding in high bandwidth expansion systems cdma and high bandwidth efficiency systems tdma

the asymptotic stability of a noisy non linear oscillator abstract the purpose of this work is to obtain an approximation for the top lyapunov exponent the exponential growth rate of the response of a single well kramers oscillator driven by either a multiplicative or an additive white noise process to this end we consider the equations of motion as dissipative and noisy perturbations of a two dimensional hamiltonian system a perturbation approach is used to obtain explicit expressions for the exponent in the presence of small intensity noise and small dissipation we show analytically that the top lyapunov exponent is positive and for small values of noise intensity e and dissipation e the exponent grows in proportion with e1 3

numerical simulation of focusing stochastic nonlinear schrodinger equations abstract in this paper we numerically investigate nonlinear schrodinger equations with a stochastic contribution which is of white noise type and acts either as a potential multiplicative noise or as a forcing term additive noise in the subcritical case we recover similar results as in the case of the korteweg de vries equation in the critical or supercritical case we observe that depending on its smoothness the noise may have different effects spatially smooth noises amplify blow up phenomena whereas delta correlated multiplicative noises prevent blow up formation note that in this latter case very few results are known both from a theoretical and a numerical point of view

time delay in random scattering this work treats the time delay in scattering for the schrodinger equation with a random potential the time delay is related to the energy derivative of the phase shift in the white noise limit the phase shift and time delay satisfy a system of stochastic differential equations in the high frequency limit the time delay is governed by a single stochastic differential equation in this limit the expected time delay is zero but the variance of the time delay is the square of a kinematical time multiplied by a function of the ratio of the length of the scatterer to the localization length

scattering of a wave packet by an interval of random medium this paper deals with the scattering theory for the one dimensional discrete schrodinger equation with a random potential having large support the main result is that a fluctuation deep within the scattering region has a very small effect on the scattering of wave packets the region of random potential is effectively opaque

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

joint distribution of conjugate algebraic numbers a random polynomial approach abstract we count the algebraic numbers of fixed degree by their w weighted l p norm which generalizes the naive height the length the euclidean and the bombieri norms for non negative integers k l such that k 2 l n and a borel subset b r c l denote by p w k l q b the number of ordered k l tuples in b of conjugate algebraic numbers of degree n and w weighted l p norm at most q we show that lim q p w k l q b q n 1 vol n 1 b p w n 1 2 n 1 b p w k l x z d x d z where vol n 1 b p w n 1 is the volume of the unit w weighted l p ball and p w k l will denote the correlation function of k real and l complex zeros of the random polynomial j 1 n j w j z j where j are i i d random variables with density c p e t p for 0 p and with constant density on 1 1 for p if the boundary of b is of lipschitz type we also estimate the rate of convergence we give an explicit formula for p w k l which in the case k 2 l n has a very simple form to this end we obtain a general formula for the correlations between real and complex zeros of a random polynomial with arbitrary independent absolutely continuous coefficients

lyapunov exponent and rotation number of two dimensional linear stochastic systems with small diffusion by means of a singular perturbation scheme and large deviation estimates we expand the lyapunov exponent and rotation number in terms of the small noise intensity varepsilon for all 2 times 2 systems dx t varepsilon ax t varepsilon dt sqrt varepsilon sum nolimits k 1 r b k x t varepsilon circ dw t k for which a is complex diagonalizable examples are given

lyapunov exponents for small random perturbations of hamiltonian systems consider the stochastic nonlinear oscillator equation ddot x x x 3 varepsilon 2 beta dot x varepsilon sigma x dot w t with beta 0 and sigma neq 0 if 4 beta sigma 2 0 then for small enough varepsilon 0 the system x dot x is positive recurrent in bf r 2 setminus 0 0 now let overline lambda varepsilon denote the top lyapunov exponent for the linearization of this equation along trajectories the main result asserts that overline lambda varepsilon varepsilon 2 3 overline lambda o varepsilon 4 3 qquad mbox as varepsilon to 0 with overline lambda 0 this result depends crucially on the fact that the system above is a small perturbation of a hamiltonian system the method of proof can be applied to a more general class of small perturbations of two dimensional hamiltonian systems the techniques used include i an extension of results of pinsky and wihstutz for perturbations of nilpotent linear systems and ii a stochastic averaging argument involving

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

blow up for the stochastic nonlinear schrodinger equation with multiplicative noise we study the influence of a multiplicative gaussian noise white in time and correlated in space on the blow up phenomenon in the supercritical nonlinear schrodinger equation we prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability

asymptotic transmission of solitons through random media this paper contains a study of the transmission of a soliton through a slab of nonlinear and random media a random nonlinear schrodinger equation is considered where the randomness holds in the potential and the nonlinear coefficient using the inverse scattering transform we exhibit several asymptotic behaviors corresponding to the limit when the amplitudes of the random fluctuations go to zero and the size of the slab goes to infinity the mass of the transmitted soliton may tend to zero exponentially as a function of the size of the slab or following a power law or else the soliton may keep its mass while its velocity decreases at a logarithmic rate or even more slowly numerical simulations are in good agreement with the theoretical results

on the almost sure stability condition for a co dimension two bifurcation system under the parametric excitation of a real noise abstract in this paper the almost sure stability condition for a co dimension two bifurcation system on a three dimensional central manifold which is parametrically excited by a real noise is investigated a model of enhanced generality is developed by assuming the real noise as the first component of an output of a linear filter system a zero mean stationary gaussian diffusion vectoral process which conforms to the detailed balance condition the strong mixing condition which is the essential theoretic basis for the stochastic averaging method is removed in the present study to solve the complicated problem encountered in this work the asymptotic analysis approach and the eigenfunction expansion of the solutions to the relevant fokker planck equations are employed in the construction of the asymptotic expansions of the invariant measures and the maximal lyapunov exponents for the relevant system

density of complex zeros of a system of real random polynomials we study the density of complex zeros of a system of real random so m 1 polynomials in m variables we show that the density of complex zeros of this random polynomial system with real coefficients rapidly approaches the density of complex zeros in the complex coefficients case we also show that the behavior the scaled density of complex zeros near m of the system of real random polynomials is different in the m 2 case than in the m 1 case the density approaches infinity instead of tending linearly to zero

qr factoring to compute the gcd of univariate approximate polynomials we present a stable and practical algorithm that uses qr factors of the sylvester matrix to compute the greatest common divisor gcd of univariate approximate polynomials over spl ropf x or spl copf x an approximate polynomial is a polynomial with coefficients that are not known with certainty the algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle by splitting and reversal if necessary the algorithm has been tested on thousands of examples including pairs of polynomials of up to degree 1000 and is now distributed as the program qrgcd in the snap package of maple 9

numerical modeling of nonlinear acoustic waves in a tube connected with helmholtz resonators acoustic wave propagation in a one dimensional waveguide connected with helmholtz resonators is studied numerically finite amplitude waves and viscous boundary layers are considered the model consists of two coupled evolution equations a nonlinear pde describing nonlinear acoustic waves and a linear ode describing the oscillations in the helmholtz resonators the thermal and viscous losses in the tube and in the necks of the resonators are modeled by fractional derivatives a diffusive representation is followed the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations a splitting method is then applied to the evolution equations their propagative part is solved using a standard tvd scheme for hyperbolic equations whereas their diffusive part is solved exactly various strategies are examined to compute the coefficients of the diffusive representation finally an optimization method is preferred to the usual quadrature rules the numerical model is validated by comparisons with exact solutions the properties of the full nonlinear solutions are investigated numerically in particular the existence of acoustic solitary waves is confirmed

modulational instability induced by randomly varying coefficients for the nonlinear schrodinger equation abstract we introduce the theory of modulational instability mi of electromagnetic waves in optical fibers the model at hand is the one dimensional nonlinear schrodinger equation with random group velocity dispersion and random nonlinear coefficient we compute the mi gain which reads as the lyapunov exponent of a random linear system we show that the distribution of the mi gain can be expressed in terms of the log normal statistics the heavy tail of this probability distribution function involves very different behaviors for the sample and moment mi gains in the anomalous dispersion regime random fluctuations of the nonlinear coefficient reduces the sample mi gain peak although the moment mi peak is enhanced and the unstable bandwidth is widened still in the anomalous dispersion regime random fluctuations of the group velocity dispersion reduces both the sample mi gain peak and the moment mi peak finally in the normal dispersion regime randomness extends the mi domain to the whole spectrum of modulations and increases the mi gain peak the linear stability analysis is confirmed by numerical simulations of the full stochastic nonlinear schrodinger equation

global bounds for the lyapunov exponent and the integrated density of states of random schrodinger operators in one dimension in this paper we prove an upper bound for the lyapunov exponent e and a two sided bound for the integrated density of states n e at an arbitrary energy e 0 of random schrodinger operators in one dimension these schrodinger operators are given by potentials of identical shape centred at every lattice site but with non overlapping supports and with randomly varying coupling constants both types of bounds only involve scattering data for the single site potential they show in particular that both e and n e e 1 2 decay at infinity at least like 1 e 1 2 as an example we consider the random kronig penney model

moment lyapunov exponent for two coupled oscillators driven by real noise in this paper an approximation for the moment lyapunov exponent the asymptotic growth rate of the moments of the response of two coupled oscillators driven by real noise is constructed a perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise the results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation

exact statistical properties of the zeros of complex random polynomials the zeros of complex gaussian random polynomials with coefficients such that the density in the underlying complex space is uniform are known to have the same statistical properties as the zeros of the coherent state representation of one dimensional chaotic quantum systems we extend the interpretation of these polynomials by showing that they also arise as the wavefunction for a quantum particle in a magnetic field constructed from a random superposition of states in the lowest landau level a study of the statistical properties of the zeros is undertaken using exact formulae for the one and two point distribution functions attention is focused on the moments of the two point correlation in the bulk the variance of a linear statistic and the asymptotic form of the two point correlation at the boundary a comparison is made with the same quantities for the eigenvalues of complex gaussian random matrices

explicit formulas for the distribution of complex zeros of a family of random sums abstract the present paper provides an explicit formula for the average intensity of the distribution of complex zeros of a family of random sums of the form s n z j 0 n j f j z where z is the complex variable x i y j j i j and j j 0 n and j j 0 n are sequences of standard normal independent random variables and f j j 0 n is a sequence of given analytic functions that are real valued on the real number line in addition the numerical computations of the intensity functions and the empirical distributions for the special cases of random weyl polynomials random taylor polynomials and random truncated fourier cosine series are included as examples

time reversal symmetry and random polynomials we analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed deterministic phase the density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients this means that contrary to earlier belief eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation moreover a class of time reverse invariant quantum systems is shown for which spectra display fluctuations characteristic of orthogonal ensemble while eigenvectors confer to predictions of unitary ensemble