stochastic precoding for miso interference channels with channel mean feedback this work considers linear precoding strategies for multiple input single out miso interference channels with channel mean feedback at transmitters where the interference at each receiver is treated as additive noise the challenge here is that previous precoder designs with perfect channel state information csi at transmitters do not apply and new approaches are required based on the laplace transform order an altruistic non equilibrium strategy i e the stochastic zero forcing is first proposed under practical assumptions generalizing the traditional zero forcing which requires perfect csi interestingly the precoding matrices here are all rank one beamformers as in the traditional zero forcing the competitive use of the common physical media in miso interference channels is also formulated as a strategic noncooperative game in contrast to the perfect csi case with a unique rank one nash equilibrium with channel mean feedback the nash equilibria here are not necessarily rank one in general nevertheless when achieved by the rank one beamforming the equilibrium is unique and convenient for implementation accordingly the condition for beamforming to achieve the equilibrium is derived comparisons of the above two strategies reveal no overall dominance of one over the other thereby establishing stochastic zero forcing as an alternative to the nash equilibrium designs

extracting macroscopic stochastic dynamics model problems the purpose of this work is to shed light on an algorithm designed to extract effective macroscopic models from detailed microscopic simulations the particular algorithm we study is a recently developed transfer operator approach due to schutte et al 20 the investigations involve the formulation and subsequent numerical study of a class of model problems the model problems are ordinary differential equations constructed to have the property that when projected onto a low dimensional subspace the dynamics is approximately that of a stochastic differential equation exhibiting a finite state space markov chain structure the numerical studies show that the transfer operator approach can accurately extract finite state markov chain behavior embedded within high dimensional ordinary differential equations in so doing the studies lend considerable weight to existing applications of the algorithm to the complex systems arising in applications such as molecular dynamics the algorithm is predicated on the assumption of markovian input data further numerical studies probe the role of memory effects although preliminary these studies of memory indicate interesting avenues for further development of the transfer operator methodology

higher order wong zakai approximations of stochastic reaction diffusion equations on rn abstract in this paper we consider the higher order wong zakai approximations of the non autonomous stochastic reaction diffusion equation driven by additive multiplicative white noises the solutions between the approximation equation and stochastic reaction diffusion equation are compared in higher order spaces in terms of the initial data based on these results and the known l 2 upper semi continuity we prove that the random attractor of the approximation random system converges to that of the non autonomous stochastic reaction diffusion equation with additive multiplicative white noises in l p r n h 1 r n when the size of the approximation shrinks to zero

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

numerical algorithms for undamped gyroscopic systems abstract the solutions of a gyroscopic vibrating system oscillating about an equilibrium position with no external applied forces and no damping forces are completely determined by the quadratic eigenvalue problem 2 i m i g k x i 0 for i 1 2 n where m g and k are real n n matrices and m is symmetric positive definite denoted by m 0 g is skew symmetric and either k 0 or k 0 gyroscopic system in motion about a stable equilibrium position with k 0 are well understood two lanczos type algorithms the pseudo skew symmetric lanczos algorithm and the j lanczos algorithm are studied for computing some extreme eigenpairs for solving gyroscopic systems in motion about an unstable equilibrium position with k 0 shift and invert strategies error bounds implementation issues and numerical results for both algorithms are presented in details

error bounds for the lanczos methods for approximating matrix exponentials in this paper we present new error bounds for the lanczos method and the shift and invert lanczos method for computing e tau a v for a large sparse symmetric positive semidefinite matrix a compared with the existing error analysis for these methods our bounds relate the convergence to the condition numbers of the matrix that generates the krylov subspace in particular we show that the lanczos method will converge rapidly if the matrix a is well conditioned regardless of what the norm of tau a is numerical examples are given to demonstrate the theoretical bounds

an analysis of the hr algorithm for computing the eigenvalues of a matrix abstract the hr algorithm a method of computing the eigenvalues of a matrix is presented it is based on the fact that almost every complex square matrix a can be decomposed into a product a hr of a so called pseudo hermitian matrix h and an upper triangular matrix r this algorithm is easily seen to be a generalization of the well known qr algorithm it is shown how it is related to the power method and inverse iteration and for special matrices the connection between the lr and hr algorithms is indicated

statistical properties for flows with unbounded roof function including the lorenz attractor for geometric lorenz attractors including the classical lorenz attractor we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors we also obtain the functional central limit theorem and moment estimates as well as iterated versions of these results a consequence is deterministic homogenisation convergence to a stochastic differential equation for fast slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two

a vector valued almost sure invariance principle for hyperbolic dynamical systems we prove an almost sure invariance principle approximation by d dimensional brownian motion for vector valued holder observables of large classes of nonuniformly hyperbolic dynamical systems these systems include axiom a dieomorphisms and flows as well as systems modelled by young towers with moderate tail decay rates in particular the position variable of the planar periodic lorentz gas with finite horizon approximates a 2 dimensional brownian motion

smooth random functions random odes and gaussian processes the usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of brownian motion which is the limit of a random walk as the step length goes to zero a browni

stability of mixing and rapid mixing for hyperbolic flows we obtain general results on the stability of mixing and rapid mixing superpolynomial decay of correlations for hyperbolic flows amongst c r axiom a flows r 2 we show that there is a c 2 open c r dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing this is the first general result on the stability of rapid mixing or even mixing for axiom a

ergodic capacity and beamforming optimality for multi antenna relaying with statistical csi we investigate the capacity and beamforming optimality of multi antenna relaying systems given access to statistical channel information at the relay and source multi antenna relay configurations are considered for which the source as well as either the relay or destination have multiple antennas and the relay operates with amplify and forward we first compute the optimal transmission strategies at both the source and relay and derive necessary and sufficient conditions for which beamforming achieves capacity to gain more insight we then employ tools from stochastic majorization theory to characterize the impact of the destination correlation and the relay gain on the capacity and beamforming optimality range these results demonstrate some intriguing behavior which arises due to the joint interplay between the transmit power relay gain and level of correlation at the destination it is shown among other things that for certain transmit powers and relay gains the beamforming optimality region becomes independent of the destination correlation by relaxing the beamforming optimality condition we also derive a simple explicit upper bound which gives further insights into the joint effect of the snr and the transmit correlation on the beamforming optimality range

symmetric matrix pencils abstract a significant number of matrix eigenvalue problems quadratic or linear are best reformulated as pencils a m in which both a and m are real and symmetric some examples are given and then the canonical forms are re examined to explain the role of the sign characteristic attached to real eigenvalues in addition we examine the limitations on the use of the rayleigh quotient functional x ax x mx in describing the eigenvalues this sheds new light on the class of definite pencils and the stability of their eigenvalues under perturbations the reduction of indefinite pencils to useful sparse forms is mentioned

complex shift and invert strategies for real matrices abstract when using an iterative method for solving a generalized nonsymmetric eigenvalue problem of the form fu mu where f and m are real banded matrices it is often desirable to work with the shifted and inverted operator b f m 1 m in order to enhance the eigenvalue separation and improve efficiency unfortunately the shift is generally complex and so is the matrix b the question then is whether it is possible to avoid complex arithmetic while preserving any advantages of bandedness of the pair f m for the classical problem where m i and f is banded complex arithmetic can be avoided by using double shifts i e by working with the real matrix bb whose bandwidth is double that of f this satisfactory solution extends to the case where m is diagonal as well in the generalized case the answer to the above question is negative in the sense that complex arithmetic can be avoided only at the expense of losing the advantage of bandedness one solution is to factor the shifted matrix f m in complex arithmetic but employ real arithmetic subsequently in the iterative procedure this paper examines several approaches and discusses their respective merits under various circumstances

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

large deviations for nonuniformly hyperbolic systems we obtain large deviation estimates for a large class of nonuniformly hyperbolic systems namely those modelled by young towers with summable decay of correlations in the case of exponential decay of correlations we obtain exponential large deviation estimates given by a rate function in the case of polynomial decay of correlations we obtain polynomial large deviation estimates and exhibit examples where these estimates are essentially optimal in contrast with many treatments of large deviations our methods do not rely on thermodynamic formalism hence for holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton as well as polynomial estimates in situations where there is not a unique equilibrium measure

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

slow rates of mixing for dynamical systems with hyperbolic structures we consider invertible discrete time dynamical systems having a hyperbolic product structure in some region of the phase space with infinitely many branches and variable return time we show that the decay of correlations of the srb measure associated to that hyperbolic structure is related to the tail of the recurrence times we also give sufficient conditions for the validity of the central limit theorem this extends previous results by young in ann math 147 585 650 1998 israel j math 110 153 188 1999

optimal transmitter eigen beamforming and space time block coding based on channel mean feedback optimal transmitter designs obeying the water filling principle are well documented they are widely applied when the propagation channel is deterministically known and regularly updated at the transmitter because channel state information is impossible to be known perfectly at the transmitter in practical wireless systems we design in this paper an optimal multiantenna transmitter based on the knowledge of mean values of the underlying channels our optimal transmitter design turns out to be an eigen beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the correlation matrix of the estimated channel at the transmitter and with proper power loading across beams the optimality pertains to minimizing an upper bound on the symbol error rate which leads to better performance than maximizing the expected signal to noise ratio snr at the receiver coupled with orthogonal space time block codes two directional eigen beamforming emerges as a more attractive choice than conventional one directional beamforming with uniformly improved performance without rate reduction and without essential increase in complexity with multiple receive antennas and reasonably good feedback quality the two directional eigen beamformer is also capable of achieving the best possible performance in a large range of transmit power to noise ratios without a rate penalty

wong zakai approximations and center manifolds of stochastic differential equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of the stochastic differential equation driven by a l dimensional brownian motion we prove that the solutions of wong zakai approximations converge in the mean square to the solutions of the stratonovich stochastic differential equation we also show that for a simple multiplicative noise the center manifold of the wong zakai approximations converges to the center manifold of the stratonovich stochastic differential equation

differential equations driven by rough paths with jumps abstract we develop the rough path counterpart of ito stochastic integration and differential equations driven by general semimartingales this significantly enlarges the classes of ito forward stochastic differential equations treatable with pathwise methods a number of applications are discussed

superdiffusive limits for deterministic fast slow dynamical systems we consider deterministic fast slow dynamical systems on mathbb r m times y of the form begin aligned left begin array ll x k 1 n x k n n 1 a big x k n big n 1 alpha b big x k n big v y k y k 1 f y k end array right end aligned where alpha in 1 2 under certain assumptions we prove convergence of the m dimensional process x n t x lfloor nt rfloor n to the solution of the stochastic differential equation begin aligned mathrm d x a x mathrm d t b x diamond mathrm d l alpha end aligned where l alpha is an alpha stable levy process and diamond indicates that the stochastic integral is in the marcus sense in addition we show that our assumptions are satisfied for intermittent maps f of pomeau manneville type

linear and numerical linear algebra in control theory some research problems abstract this paper presents an informal survey of theoretical and computational aspects of three linear algebra problems arising in control theory namely the sylvester observer matrix equation the matrix eigenvalue assignment problem and the stability and stabilization of large second order systems difficulties are highlighted recent developments are mentioned some research problems in these areas which should be of interest to both linear algebraists and control theorists are described

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

analysis of the finite precision bi conjugate gradient algorithm for nonsymmetric linear systems in this paper we analyze the bi conjugate gradient algorithm in finite precision arithmetic and suggest reasons for its often observed robustness by using a tridiagonal structure which is preserved by the finite precision bi conjugate gradient iteration we are able to bound its residual norm by a minimum polynomial of a perturbed matrix i e the residual norm of the exact gmres applied to a perturbed matrix multiplied by an amplification factor this shows that occurrence of near breakdowns or loss of biorthogonality does not necessarily deter convergence of the residuals provided that the amplification factor remains bounded numerical examples are given to gain insights into these bounds

iterative solution of indefinite symmetric linear systems by methods using orthogonal polynomials over two disjoint intervals in the classical chebyshev algorithm for solving symmetric positive definite linear systems ax b the approximate solution is of the form x k x 0 s k a r 0 where r 0 b ax 0 is the initial residual and s k is a polynomial of degree k 1 which minimizes 1 lambda s k lambda infty where cdot infty is the uniform norm on some interval s containing the spectrum of a but not the origin if a is not positive definite the set s can be taken to be the union of two disjoint intervals a b and c d with b 0 c and a generalization of the chebyshev algorithm based on the minimax polynomial has been considered by several authors in this paper we propose a new technique based upon the least squares polynomial in the set s i e the polynomial t k which minimizes 1 lambda t k lambda w where cdot w is an l 2 norm with respect to a weight function w lambda defined on s with a suitable weight function w lambda the po

outer bounds on the capacity of gaussian interference channels two outer bounds on the capacity region of the two user gaussian interference channel ifc are derived the idea of the first bound is to let a genie give each receiver just enough information to decode both messages this bound unifies and improves the best known outer bounds of sato and carleial furthermore the bound extends to discrete memoryless ifcs and is shown to be equivalent to another bound of carleial the second bound follows directly from existing results of costa and sato and possesses certain optimality properties for weak interference

a breakdown free variation of the nonsymmetric lanczos algorithms the nonsymmetric lanczos tridiagonalization algorithm is essentially the gram schmidt biorthogonalization method for generating biorthogonal bases of a pair of krylov subspaces it suffers from breakdown and instability when a pivot at some step is zero or nearly zero which is often the result of mismatch of the two krylov subspaces in this paper we propose to modify one of the two krylov subspaces by introducing a new start vector when a pivot is small this new start vector generates another krylov subspace which we add to the old one in an appropriate way so that the gram schmidt method for the modified subspaces yields a recurrence similar to the lanczos algorithm our method enforces the pivots to be above a certain threshold and can handle both exact breakdown and near breakdown in particular we recover lookahead lanczos algorithms and arnoldi s algorithm as two special cases we also discuss theoretical and practical issues concerning the new start procedure and present a convergence analysis as well as some numerical examples

lanczos algorithms and field of value rotations for symmetric matrix pencils abstract iterative algorithms for the eigensolution of symmetric pencils of matrices are considered it is shown that the symmetric lanczos algorithm the nonsymmetric lanczos algorithm and the arnoldi algorithm are closely related in this case the applicability of this class of algorithms to indefinite pencils is discussed a new field of values concept is used to describe the symmetric pencil problem then spectral transformation corresponds to a rotation in the complex plane and the inertia count gives the number of eigenvalues corresponding to points in one of two half planes

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

a spectral approach for quenched limit theorems for random expanding dynamical systems we prove quenched versions of i a large deviations principle ldp ii a central limit theorem clt and iii a local central limit theorem lclt for non autonomous dynamical systems a key advance is the extension of the spectral method commonly used in limit laws for deterministic maps to the general random setting we achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter while some versions of the ldp and clt have previously been proved with other techniques the local central limit theorem is to our knowledge a completely new result and one that demonstrates the strength of our method applications include non autonomous piecewise expanding maps defined by random compositions of the form t n 1 t t an important aspect of our results is that we only assume ergodicity and invertibility of the random driving in particular no expansivity or mixing properties are required

orthogonality and partial pole assignment for the symmetric definite quadratic pencil abstract the eigenvectors of a symmetric matrix can be chosen to form an orthogonal set with respect to the identity and to the matrix itself similarly the eigenvectors of a symmetric definite linear pencil can be chosen to be orthogonal with respect to the pair this paper presents the three sets of matrix weights with respect to which the eigenvectors of the symmetric definite quadratic pencil are orthogonal one of these is used to derive an explicit solution of the partial pole assignment problem by state feedback control for a control system modeled by a system of second order differential equations the solution may be of particular interest in the stabilization and control of flexible large space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum is required to remain unchanged

complete characterization of the pareto boundary for the miso interference channel in this correspondence we study the achievable rate region of the multiple input single output miso interference channel under the assumption that all receivers treat the interference as additive gaussian noise our main result is an explicit parametrization of the pareto boundary for an arbitrary number of users and antennas the parametrization describes the boundary in terms of a low dimensional manifold for the two user case we show that a single real valued parameter per user is sufficient to achieve all points on the pareto boundary and that any point on the pareto boundary corresponds to beamforming vectors that are linear combinations of the zero forcing zf and maximum ratio transmission mrt beamformers we further specialize the results to the miso broadcast channel bc a numerical example illustrates the result

error analysis of the lanczos algorithm for the nonsymmetric eigenvalue problem this paper presents an error analysis of the lanczos algorithm in finite precision arithmetic for solving the standard nonsymmetric eigenvalue problem if no breakdown occurs an analog of paige s theory on the relationship between the loss of orthogonality among the lanezos vectors and the convergence of ritz values in the symmetric lanczos algorithm is discussed the theory developed illustrates that in the nonsymmetric lanczos scheme if ritz values are well conditioned then the loss of biorthogonality among the computed lanczos vectors implies the convergence of a group of ritz triplets in terms of small residuals numerical experimental results confirm this observation

robust mimo cognitive radio via game theory cognitive radio cr systems improve the spectral efficiency by allowing the coexistence in harmony of primary users pus the legacy users with secondary users sus this coexistence is built on the premises that no su can generate interference higher than some prescribed limits against pus the system design based on perfect channel state information csi can easily end up violating the interference limits in a realistic situation where csi may be imperfect in this paper we propose a robust design of cr systems composed of multiple pus and multiple noncooperative sus in either single input single output siso frequency selective channels or more general multiple input multiple output mimo channels we formulate the design of the su network as a noncooperative game where the sus compete with each other over the resources made available by the pus by maximizing their own information rates subject to the transmit power and robust interference constraints following the philosophy of the worst case robustness we take explicitly into account the imperfectness of su to pu csi by adopting proper interference constraints that are robust with respect to the worst channel errors relying on the variational inequality theory we study the existence and uniqueness properties of the nash equilibria of the resulting robust games and devise totally asynchronous and distributed algorithms along with their convergency properties we also propose efficient numerical methods based on decomposition techniques to compute the robust transmit strategy for each su

almost sure invariance principle for nonuniformly hyperbolic systems we prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems discrete time systems and flows are covered by this result in particular the result applies to the planar periodic lorentz flow with finite horizon statistical limit laws such as the central limit theorem the law of the iterated logarithm and their functional versions are immediate consequences

convergence en loi des suites d integrales stochastiques sur l espace mathbb d 1 de skorokhod for sequences of stochastic integrals k n x n functional limit theorems are presented these results hold under simple natural conditions

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

optimizing multiple input single output miso communication systems with general gaussian channels nontrivial covariance and nonzero mean we consider a narrow band point to point communication system with many input transmitters and a single output receiver i e a multiple input single output miso system we assume the receiver has perfect knowledge of the channel but the transmitter only knows the channel distribution we focus on two canonical classes of gaussian channel models a the channel has zero mean with a fixed covariance matrix and b the channel has nonzero mean with covariance matrix proportional to the identity in both cases we are able to derive simple analytic expressions for the ergodic average and the cumulative distribution function c d f of the mutual information for arbitrary input transmission signal covariance with minimal numerical effort we then determine the ergodic and outage capacities and the corresponding capacity achieving input signal covariances interestingly we find that the optimal signal covariances for the ergodic and outage cases have very different behavior in particular under certain conditions the outage capacity optimal covariance is a discontinuous function of the parameters describing the channel such as strength of the correlations or the nonzero mean of the channel

wong zakai approximations and attractors for stochastic reaction diffusion equations on unbounded domains abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated long term behavior of the stochastic reaction diffusion equation driven by a white noise we first prove the existence and uniqueness of tempered pullback attractors for the wong zakai approximations of stochastic reaction diffusion equation then we show that the attractors of wong zakai approximations converges to the attractor of the stochastic reaction diffusion equation for both additive and multiplicative noise

spectra of a class of non self adjoint matrices we consider a new class of non self adjoint matrices that arise from an indefinite self adjoint linear pencil of matrices and obtain the spectral asymptotics of the spectra as the size of the matrices diverges to infinity we prove that the spectrum is qualitatively different when a certain parameter c equals 0 and when it is non zero and that certain features of the spectrum depend on diophantine properties of c

convergence analysis of lanczos type methods for the linear response eigenvalue problem two different lanczos type methods for the linear response eigenvalue problem are analyzed the first one is a natural extension of the classical lanczos method for the symmetric eigenvalue problem while the second one was recently proposed by tsiper specially for the linear response eigenvalue problem our analysis leads to bounds on errors for eigenvalue and eigenvector approximations by the two methods these bounds suggest that the first method can converge significantly faster than tsiper s method numerical examples are presented to support this claim

almost sure invariance principle for dynamical systems by spectral methods we prove the almost sure invariance principle for stationary d valued random processes with very precise dimension independent error terms solely under a strong assumption concerning the characteristic functions of these processes this assumption is easy to check for large classes of dynamical systems or markov chains using strong or weak spectral perturbation arguments

eigenvalues of ax λbx for real symmetric matrices a and b computed by reduction to a pseudosymmetric form and the hr process abstract the paper presents a method for solving the eigenvalue problem ax bx where a and b are real symmetric but not necessarily positive definite matrices and b is nonsingular the method reduces the general case into a form cz z where c is a pseudosymmetric matrix a further reduction of c produces a tridiagonal pseudosymmetric form to which the iterative hr process is applied the tridiagonal pseudosymmetric form is invariant under the hr transformations the amount of computation is significantly less than in treating the problem by a general method

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

the capacity of the gaussian interference channel under strong interference corresp the capacity region of a gaussian interference channel with two separate messages is obtained for the case of moderately strong interference it is shown that the region coincides with the one where both messages are required in both receiving terminals

feedback stabilization of a second order system a nonmodal approach a novel approach for feedback stabilization of a second order model is proposed specifically two nonmodal algorithms are described and it is shown mathematically that under some mild assumptions on the damping matrix the feedback matrix obtained by each algorithm indeed stabilizes a closed loop system the first algorithm requires the solution of a symmetric positive definite linear system and the inversion of a small matrix a remarkable feature of this algorithm is that it does not require knowledge of the stiffness and damping for implementation which makes it very feasible for practical application the second algorithm requires the solution of a small linear least squares problem and an estimate of the stability index which can be obtained just by computing the extremal eigenvalues of the data matrices which are symmetric for almost all practical applications these minimal computational requirements make the proposed algorithms suitable for practical implementations even for large and sparse systems using the state of the art techniques of matrix computations this is in sharp contrast with the traditional modal approach ordinarily used by practicing engineers which requires the solution of a quadratic eigenvalue problem or equivalently the solution of a 2n x 2n generalized eigenvalue problem and therefore may not be practical for very large and sparse problems besides the proposed algorithms the paper also contains some mathematical results on the bounds of the eigenvalues of a second order pencil which may be useful for investigating stability of second order systems and are of independent interest

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

rough flows and homogenization in stochastic turbulence abstract we provide in this work a tool kit for the study of homogenisation of random ordinary differential equations under the form of a friendly user black box based on the technology of rough flows we illustrate the use of this setting on the example of stochastic turbulence

rayleigh ritz and lanczos methods for symmetric matrix pencils we are concerned with eigenvalue problems for definite and indefinite symmetric matrix pencils first rayleigh ritz methods are formulated and using krylov subspaces a convergence analysis is presented for definite pencils second generalized symmetric lanczos algorithms are introduced as a special rayleigh ritz method in particular an a posteriori convergence criterion is demonstrated by using residuals local convergence to real and nonreal eigenvalues is also discussed numerical examples concerning vibrations of damped cantilever beams are included

smooth approximation of stochastic differential equations consider an ito process x satisfying the stochastic differential equation dx a x dt b x dw where a b are smooth and w is a multidimensional brownian motion suppose that wn has smooth sample paths and that wn converges weakly to w a central question in stochastic analysis is to understand the limiting behavior of solutions xn to the ordinary differential equation dxn a xn dt b xn dwn the classical wong zakai theorem gives sufficient conditions under which xn converges weakly to x provided that the stochastic integral b x dw is given the stratonovich interpretation the sufficient conditions are automatic in one dimension but in higher dimensions the correct interpretation of b x dw depends sensitively on how the smooth approximation wnis chosen in applications a natural class of smooth approximations arise by setting wn t n 1 2 nt0v sds where t is a flow generated e g by an ordinary differential equation and v is a mean zero observable under mild conditions on t we give a definitive answer to the interpretation question for the stochastic integral b x dw our theory applies to anosov or axiom a flows t as well as to a large class of nonuniformly hyperbolic flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on t the methods used in this paper are a combination of rough path theory and smooth ergodic theory

optimal linear precoding strategies for wideband noncooperative systems based on game theory part i nash equilibria in this two part paper we propose a decentralized strategy based on a game theoretic formulation to find out the optimal precoding multiplexing matrices for a multipoint to multipoint communication system composed of a set of wideband links sharing the same physical resources i e time and bandwidth we assume as optimality criterion the achievement of a nash equilibrium and consider two alternative optimization problems 1 the competitive maximization of mutual information on each link given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies and 2 the competitive maximization of the transmission rate using finite order constellations under the same constraints as above plus a constraint on the average error probability in this first part of the paper we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies then we prove that the optimal precoding multiplexing scheme for both games leads to a channel diagonalizing structure so that both matrix valued problems can be recast in a simpler unified vector power control game with no performance penalty thus we study this simpler game and derive sufficient conditions ensuring the uniqueness of the nash equilibrium interestingly although derived under stronger constraints incorporating for example spectral mask constraints our uniqueness conditions have broader validity than previously known conditions finally we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a pareto optimal centralized scheme to reach the nash equilibria of the game in part ii we propose alternative distributed algorithms along with their convergence conditions

rapid decay of correlations for nonuniformly hyperbolic flows we show that superpolynomial decay of correlations rapid mixing is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which young proved exponential decay of correlations the proof combines techniques of dolgopyat and operator renewal theory it follows from our results that planar periodic lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing

rate of convergence in the weak invariance principle for deterministic systems we obtain the first results on convergence rates in the prokhorov metric for the weak invariance principle functional central limit theorem for deterministic dynamical systems our results hold for uniformly expanding hyperbolic axiom a systems as well as nonuniformly expanding hyperbolic systems such as dispersing billiards henon like attractors viana maps and intermittent maps as an application we obtain convergence rates for deterministic homogenization in multiscale systems

numerical solutions for large sparse quadratic eigenvalue problems abstract we study the quadratic eigenvalue problem a b 2c x 0 where a b and c are symmetric real n n matrices and a c are positive definite we propose an efficient numerical algorithm to compute a few of the smallest positive eigenvalues of the problem and their associated eigenvectors the new algorithm includes two parts the first part gives iterative methods which can be used to compute the smallest positive eigenvalue we develop a globally linearly convergent basic iteration and two locally quadratically convergent iterations the second part uses the nonequivalence deflation technique this technique allows us to transform the original problem to a new problem with different a b and c the new problem has the same eigenvalues as the old problem except that the smallest positive eigenvalue of the old problem is replaced by zero therefore the second smallest positive eigenvalue of the old problem becomes the smallest positive one for the new problem then the above proposed iterative methods can be applied again to find the second smallest positive eigenvalue proceeding in this way we can find out the 3rd 4th smallest positive eigenvalues our algorithm utilizes the symmetry and positivity of the given matrices and avoids computing the undesired complex conjugate eigenvalues under some mild conditions is efficient and reliable the above process can also be used to find a few of the largest negative eigenvalues

how to implement the spectral transformation the general linear eigenvalue equations h x m z 0 where h and m are real symmetric matrices with m positive semidefinite must be transformed if the lanczos algorithm is to be used to compute eigenpairs x z when the matrices are large and sparse but not diagonal some factorization must be performed as part of the transformation step if we are interested in only a few eigenvalues x near a specified shift then the spectral transformation of ericsson and ruhe 1 proved itself much superior to traditional methods of reduction the purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it it arises in the use of inverse iteration a second goal is to introduce a systematic modification of the computed ritz vectors which improves the accuracy when m is ill conditioned or singular we confine our attention to the simple lanczos algorithm although the first two sections apply directly to the block algorithms as well 1 overview this contribution is an addendum to the paper by ericsson and ruhe 1 and also 7 the value of the spectral transformation is reiterated in a later section here we outline our implementation of this transformation the equation to be solved for an eigenvalue a and eigenvector z is 1 h xm z 0 h and m are real symmetric n x n matrices and m is positive semidefinite a practical instance of 1 occurs in dynamic analysis of structures where h and m are the stiffness and mass matrices respectively we assume that a linear combination of h and m is positive definite it then follows that all eigenvalues x are real in addition one has a real scalar a distinct from any eigenvalue and we seek a few eigenvalues a close to a together with their eigenvectors z ericsson and ruhe replace 1 by a standard eigenvalue equation 2 c h am ct v y 0 where c is the choleski factor of m m ctc and y cz if m is singular then so is c but fortunately the eigenvector z can be recovered from y via z h am icty of course there is no intention to invert h a m explicitly the received may 14 1984 revised december 20 1985 1980 mathematics subject classification primary 65f15 this research was supported in part by the afosr contract f49620 84 c 0090 the third author was also supported in part by the swedish natural science research council the paper was written while this author was visiting the center for pure and applied mathematics university of california berkeley california 94720 1987 american mathematical society 0025 5718 87 1 00 25 per page

fast frequency response analysis of large scale structures with non proportional damping modal frequency response analysis is an economical approach for large and complex structural systems since there is an enormous reduction in dimension from the finite element frequency response problem however when non proportional damping exists the modal frequency response problem is expensive to solve at many frequencies because modal damping matrices are fully populated this paper presents a new algorithm to solve the modal frequency response problem for large and complex structural systems with structural and viscous damping the newly developed algorithm fast frequency response analysis ffra algorithm solves the damped modal frequency response problem with o m2 operations at each frequency then the ffra algorithm is extended for solving a system of equations in optimization application with the modal correction approach in which the mass stiffness and damping matrices of a modified configuration differ from the original configuration numerical results show that the ffra algorithm dramatically improves the performance of the modal frequency response analysis compared to conventional methods in industry while obtaining the same accuracy copyright 2006 john wiley sons ltd

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

optimal linear precoding strategies for wideband non cooperative systems based on game theory part ii algorithms in this two part paper we address the problem of finding the optimal precoding multiplexing scheme for a set of noncooperative links sharing the same physical resources e g time and bandwidth we consider two alternative optimization problems p l the maximization of mutual information on each link given constraints on the transmit power and spectral mask and p 2 the maximization of the transmission rate on each link using finite order constellations under the same constraints as in p l plus a constraint on the maximum average error probability on each link aiming at finding decentralized strategies we adopted as optimality criterion the achievement of a nash equilibrium and thus we formulated both problems p l and p 2 as strategic noncooperative matrix valued games in part i of this two part paper after deriving the optimal structure of the linear transceivers for both games we provided a unified set of sufficient conditions that guarantee the uniqueness of the nash equilibrium in this part ii of the paper we focus on the achievement of the equilibrium and propose alternative distributed iterative algorithms that solve both games specifically the new proposed algorithms are the following 1 the sequential and simultaneous iterative waterfilling based algorithms incorporating spectral mask constraints and 2 the sequential and simultaneous gradient projection based algorithms establishing an interesting link with variational inequality problems our main contribution is to provide sufficient conditions for the global convergence of all the proposed algorithms which although derived under stronger constraints incorporating for example spectral mask constraints have a broader validity than the convergence conditions known in the current literature for the sequential iterative waterfilling algorithm

monotonic optimization framework for the two user miso interference channel resource allocation and transmit optimization for the multiple antenna gaussian interference channel are important but difficult problems the spatial degrees of freedom can be exploited to avoid align or utilize the interference in recent literature the upper boundary of the achievable rate region has been characterized however the resulting programming problems for finding the sum rate proportional fair and minimax egalitarian operating points are non linear and non convex in this paper we develop a non convex optimization framework based on monotonic optimization by outer polyblock approximation first the objective functions are represented in terms of differences of monotonic increasing functions next the problems are reformulated as maximization of increasing functions over normal constraint sets finally the idea to approximate the constraint set by outer polyblocks is explained and the corresponding algorithm is derived numerical examples illustrate the advantages of the proposed framework compared to an exhaustive grid search approach

a novel nonsymmetric k lanczos algorithm for the generalized nonsymmetric k eigenvalue problems in this article we present a novel algorithm named nonsymmetric k lanczos algorithm for computing a few extreme eigenvalues of the generalized eigenvalue problem mx al where the matrices m and l have the so called k structures we demonstrate a k tridiagonalization procedure preserves the k structures an error bound for the extreme k ritz value obtained from this new algorithm is presented when compared with the class nonsymmetric lanczos approach this method has the same order of computational complexity and can be viewed as a special 2 x 2 block nonsymmetric lanczos algorithm numerical experiments with randomly generated k matrices show that our algorithm converges faster and more accurate than the nonsymmetric lanczos algorithm 0 ekevier science inc 1997

the shift inverted j lanczos algorithm for the numerical solutions of large sparse algebraic riccati equations abstract the goal of solving an algebraic riccati equation is to find the stable invariant subspace corresponding to all the eigenvalues lying in the open left half plane the purpose of this paper is to propose a structure preserving lanczos type algorithm incorporated with shift and invert techniques named shift inverted j lanczos algorithm for computing the stable invariant subspace for large sparse hamiltonian matrices the algorithm is based on the j tridiagonalization procedure of a hamiltonian matrix using symplectic similarity transformations we give a detailed analysis on the convergence behavior of the j lanczos algorithm and present error bound analysis and paige type theorem numerical results for the proposed algorithm applied to a practical example arising from the position and velocity control for a string of high speed vehicles are reported

use of indefinite pencils for computing damped natural modes abstract a quadratic eigenvalue problem with symmetric positive definite coefficient matrices may be reduced to linear form while retaining symmetry in the new coefficients but neither of them will be positive definite formally the symmetric lanczos algorithm and subspace iteration may be used to compute some eigenpairs of the linear problem the trouble is that the basis vectors are orthogonal with respect to an indefinite inner product so there is no assurance that they will be linearly independent nevertheless this is an attractive way to solve the original problem and we discuss how to implement it and how it relates to the unsymmetric lanczos procedures we discuss complex origin shifts reorthogonalization and error bounds several methods for solving the reduced problem are mentioned but we have no fully satisfactory technique some dangers are described and examples are given comparing our lanczos program with a modified subspace iteration

the wong zakai approximations of invariant manifolds and foliations for stochastic evolution equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise we prove that the solutions of wong zakai approximations almost surely converge to the solutions of the stratonovich stochastic evolution equation we also show that the invariant manifolds and stable foliations of the wong zakai approximations converge to the invariant manifolds and stable foliations of the stratonovich stochastic evolution equation respectively

existence of periodic solutions in distribution for stochastic newtonian systems periodic phenomena such as oscillation have been studied for many years in this paper we verify the stochastic version of levinson s conjecture which confirmed the existence of stochastic periodic solutions for second order newtonian systems with dissipativeness first we provide a stochastic duffing s equation to display our result then we apply wong zakai approximation method and lyapunov s method to stochastic second order newtonian systems driven by brownian motions with the help of horn s fixed point theorem we prove that this kind of systems is stochastic dissipative and admits periodic solutions in distribution

an explicit formula for lanczos polynomials abstract the lanczos algorithm for tridiagonalizing a given matrix a generates a sequence of approximating matrices a p that can naturally be obtained as restrictions to subspaces the eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of a in this paper we produce explicit formulas for the characteristics polynomials of the a p in terms of the eigenvalues of a these formulas can be used to explain heuristically why these approximations are often quite good at present we have no concrete analytic argument that explains the quality of the approximation the main result draws on the formal relationship between the lanczos algorithm and pade approximations to the moment generating function of a this result is one of the few analytic results available for the unsymmetric lanczos algorithm