stabilization of a class of stochastic differential equations with markovian switching stability of stochastic differential equations with markovian switching has been studied quite extensively for a number of years for example by basak et al j math anal appl 202 1996 604 622 ji and chizeck ieee trans automat control 35 1990 777 788 mariton jump linear systems in automatic control marcel dekker new york 1990 mao et al stochastic process appl 79 1999 45 67 bernoulli 6 2000 73 90 and yuan and lygeros in r alur g pappas eds hybrid systems computation and control seventh international workshop hscc 2004 lecture notes in computer science vol 2993 springer berlin 2004 pp 646 659 by contrast the problem of designing controllers to stabilize systems of this type has received relatively little attention in this paper we study the problem of mean square exponential stabilization for a class of stochastic differential equations with markovian switching

on stabilization of bilinear uncertain time delay stochastic systems with markovian jumping parameters in this paper we investigate the stochastic stabilization problem for a class of bilinear continuous time delay uncertain systems with markovian jumping parameters specifically the stochastic bilinear jump system under study involves unknown state time delay parameter uncertainties and unknown nonlinear deterministic disturbances the jumping parameters considered here form a continuous time discrete state homogeneous markov process the whole system may be regarded as a stochastic bilinear hybrid system that includes both time evolving and event driven mechanisms our attention is focused on the design of a robust state feedback controller such that for all admissible uncertainties as well as nonlinear disturbances the closed loop system is stochastically exponentially stable in the mean square independent of the time delay sufficient conditions are established to guarantee the existence of desired robust controllers which are given in terms of the solutions to a set of either linear matrix inequalities lmis or coupled quadratic matrix inequalities the developed theory is illustrated by numerical simulation

gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

solution of toeplitz normal equations by sine transform based preconditioning abstract the normal equations constructed by a toeplitz matrix are studied in order to find a suitable preconditioner related to the discrete sine transform new results are given about the structure of the product of two toeplitz matrices which allow the cgn method to achieve a superlinear rate of convergence this preconditioner outperforms the circulant one for the iterative solution of toeplitz least squares problems such strategy can also be applied to nonsymmetric linear systems a block generalization is discussed

a korovkin type theory for finite toeplitz operators via matrix algebras preconditioned conjugate gradients pcg are widely and successfully used methods for solving a toeplitz linear system a n vec x vec b 59 9 20 5 34 62 6 10 28 45 44 46 49 frobenius optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the frobenius distance from a n the convergence features of these pcg have been naturally studied by means of the weierstrass jackson theorem 17 36 45 owing to the profound relationship between the spectral features of the matrices a n generated by the fourier coefficients of a continuous function f and the analytical properties of the symbol f itself in this paper we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f on the other hand by modifying the korovkin theorem to study the frobenius optimal preconditioning problem we provide a new and unifying tool for analyzing all frobenius optimal preconditioners in any generic matrix algebra related to trigonometric transforms finally the multilevel case is sketched and discussed by showing that a korovkin type theory also holds in a multivariate sense

delay dependent exponential stability of stochastic systems with time varying delay nonlinearity and markovian switching the problem of delay dependent stability in the mean square sense for stochastic systems with time varying delays markovian switching and nonlinearities is investigated both the slowly time varying delays and fast time varying delays are considered based on a linear matrix inequality approach delay dependent stability criteria are derived by introducing some relaxation matrices which can be chosen properly to lead to a less conservative result numerical examples are given to illustrate the effectiveness of the method and significant improvement of the estimate of stability limit over some existing results in the literature

spectral and computational properties of band symmetric toeplitz matrices abstract we are investigating spectral properties of band symmetric toeplitz matrices bst matrices by giving a suitable representation of a bst matrix we achieve separation results and multiplicity conditions for the eigenvalues of a 7 r 5 diagonal bst matrix and also structural properties of the eigenvectors we give eigenvalue bounds for a 2k 1 diagonal bst matrix and also necessary and sufficient conditions for positive definiteness which are easy to check the same conditions apply in the case of block bst matrices either with full blocks or with bst blocks we exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a bst matrix either for sequential or for parallel computations two algorithms based on the bisection technique and newton s method are shown to be very fast for computing the eigenvalues of a 7 or 5 diagonal bst matrix

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

singular values and eigenvalues of non hermitian block toeplitz matrices in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices ftn f g generated by a p variate lebesgue integrable matrix valued function f q p c hk q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of ftn f g by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of ftn f g second order ergodic formulas and localization of eigenvalues of preconditioned matrices ft 1 n g tn f g are presented some examples of applications to the preconditioning of these results are also discussed c 1999 elsevier science b v all rights reserved msc 15a12 15a18 65f10

brief paper sliding mode control for ito stochastic systems with markovian switching this paper deals with the problem of sliding mode control smc for a class of nonlinear uncertain stochastic systems with markovian switching by introducing some specified matrices the connections among the designed sliding surfaces corresponding to every mode are established furthermore the present sliding mode controller including the transition rates of modes can cope with the effect of markovian switching by means of linear matrix inequalities lmis with equality constraint sufficient conditions are derived such that the sliding motions on the specified sliding surfaces are stochastically stable with c disturbance attenuation level finally a numerical example is given to illustrate the applicability of the present method

preconditioning strategies for hermitian indefinite toeplitz linear systems in this paper we propose and analyze preconditioning strategies for hermitian indefinite linear systems by using indefinite preconditioners under very elementary assumptions we show that the eigenvalues are real moreover in the case of multilevel toeplitz structures we prove distributional and localization results these techniques used in connection with the cg gmres bicgstab and qmr algorithms allow us to solve in an optimal way the corresponding linear systems a wide numerical experimentation confirms the efficiency of the proposed procedures

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

technical communique an explicit formula of linear sliding surfaces for a class of uncertain dynamic systems with mismatched uncertainties in this note the problem of designing a variable structure control vsc law for uncertain systems where the uncertainty in the state model does not satisfy the matching condition is considered based on the lmi approach we develop a new design method of linear sliding surfaces which are linear to the state we give a sufficient condition for the existence of linear sliding surfaces in terms of lmis and we give an explicit formula of linear sliding surfaces guaranteeing the quadratic stability of the reduced order equivalent system dynamics restricted to the sliding surfaces we also give sufficient conditions for the existence of linear sliding surfaces insuring some pole clustering constraints thus one can easily design vsc laws for a class of uncertain systems with mismatched uncertainties which are difficult to deal with by using conventional vsc design methods finally we give a design example

simplification of a result on banded toeplitz matrices and bvm methods in a recent paper motivated by the analysis of some bvm methods aceto and trigiante consider tests for proving the positive definiteness of real banded toeplitz matrices here we furnish a new test and we show that the analysis provided by aceto and trigiante can be simplified by using known facts on the symbol

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

degrees of freedom of mimo cellular networks decomposition and linear beamforming design this paper investigates the symmetric degrees of freedom dof of multiple input multiple output mimo cellular networks with g cells and k users per cell having n antennas at each base station and m antennas at each user in particular we investigate techniques for achievability that are based on either decomposition with asymptotic interference alignment or linear beamforming schemes and show that there are distinct regimes of g k m n where one outperforms the other we first note that both one sided and two sided decomposition with asymptotic interference alignment achieve the same dof we then establish specific antenna configurations under which the dof achieved using decomposition based schemes is optimal by deriving a set of outer bounds on the symmetric dof using these results we completely characterize the optimal dof of any g cell network with single antenna users for linear beamforming schemes we first focus on small networks and propose a structured approach to linear beamforming based on a notion called packing ratios packing ratio describes the interference footprint or shadow cast by a set of transmit beamformers and enables us to identify the underlying structures for aligning interference such a structured beamforming design can be shown to achieve the optimal spatially normalized dof sdof of two cell two user cell network and the two cell three user cell network for larger networks we develop an unstructured approach to linear interference alignment where transmit beamformers are designed to satisfy conditions for interference alignment without explicitly identifying the underlying structures for interference alignment the main numerical insight of this paper is that such an approach appears to be capable of achieving the optimal sdof for mimo cellular networks in regimes where linear beamforming dominates asymptotic decomposition and a significant portion of sdof elsewhere remarkably polynomial identity test appears to play a key role in identifying the boundary of the achievable sdof region in the former case

degrees of freedom for mutually interfering broadcast channels in this paper we study spatial degree of freedom dof for two mutually interfering broadcast channels ibc as the demand for space division multiple access sdma increases the ibc where each link has a single transmit node and multiple receive nodes becomes important this paper presents the results of the lower and upper bounds on the dof of the ibc from the derived results it is shown that for most cases zero forcing zf beamforming can achieve the optimal dof of the ibc except for some special cases also we identify a condition that disabling receive cooperation in the multiple input multiple output interference channels causes no dof loss it is confirmed that we cannot expect a dof improvement by enabling in cell receive cooperation if in at least one of two bss the number of antennas is greater than or equal to that of users per cell furthermore we observe a positive result that as the number of users goes to infinity the total dof of the ibc converges to the interference free dof which is the maximum achievable dof in the absence of the inter cell interference

the extension of the concept of the generating function to a class of preconditioned toeplitz matrices abstract it is well known that the generating function f l 1 of a class of hermitian toeplitz matrices a n f describes very precisely the spectrum of each matrix of the class recently this theory has been extended in various senses in three directions the non hermitian case the block case and the case of preconditioned toeplitz matrices here by using some results of the first and the second extension we deduce new properties of distribution of the spectrum of preconditioned matrices and we define the concept of generating function for these matrices moreover we discuss the consequences of these results on the analysis of convergence of preconditioned conjugate gradient methods previously defined in the literature finally we perform some numerical experiments which confirm the theoretical idea

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

a circulant preconditioner for the systems of lmf based ode codes in this paper a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations odes is investigated most ode codes based on implicit formulas at each integration step need the solution of one or more unsymmetric linear systems that are often large and sparse here the boundary value methods a class of implicit methods for the numerical integration of odes based on linear multistep formulas are considered more in detail for initial value problems theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems moreover the p circulants a recently introduced circulant approximation for unsymmetric toeplitz matrices are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner

spectral analysis and structure preserving preconditioners for fractional diffusion equations fractional partial order diffusion equations are a generalization of classical partial differential equations used to model anomalous diffusion phenomena when using the implicit euler formula and the shifted grunwald formula it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a toeplitz like structure in this paper we focus our attention on the case of variable diffusion coefficients under appropriate conditions we show that the sequence of the coefficient matrices belongs to the generalized locally toeplitz class and we compute the symbol describing its asymptotic eigenvalue singular value distribution as the matrix size diverges we employ the spectral information for analyzing known methods of preconditioned krylov and multigrid type with both positive and negative results and with a look forward to the multidimensional setting we also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system with krylov methods such as cgnr and gmres a number of numerical examples show that our proposal is more effective than recently used circulant preconditioners

c g preconditioning for toeplitz matrices abstract we consider the problem of solving a toeplitz system of equations by conjugate gradient method when a sequence of nested toeplitz matrices is associated to a function the spectral behaviour of the matrices involved is closely related to the analytical properties of the generating function thus it is possible to devise efficient preconditioning techniques by using various functional approximation strategies this approach leads to attractive results in the case of ill conditioned matrices for which a wide class of preconditioners are proposed

essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices essential spectral equivalence via multiple step preconditioning and applications to ill conditioned toeplitz matrices

superlinear convergence for pcg using band plus algebra preconditioners for toeplitz systems the paper studies fast and efficient solution algorithms for nxn symmetric ill conditioned toeplitz systems t n f x b where the generating function f is known a priori real valued nonnegative and has isolated roots of even order the preconditioner that we propose is a product of a band toeplitz matrix and matrices that belong to a certain trigonometric algebra the basic idea behind the proposed scheme is to combine the advantages of all components of the product that are well known when every component is used as a stand alone preconditioner as a result we obtain a flexible preconditioner which can be applied to the system t n f x b infusing superlinear convergence to the pcg method the important feature of the proposed technique is that it can be extended to cover the 2d case i e ill conditioned block toeplitz matrices with toeplitz blocks we perform many numerical experiments whose results confirm the theoretical analysis and effectiveness of the proposed strategy

exponential stability of stochastic delay interval systems with markovian switching in the past few years a lot of research has been dedicated to the stability of interval systems as well as the stability of systems with markovian switching however little research has been on the stability of interval systems with markovian switching which is the topic of this paper the system discussed is the stochastic delay interval system with markovian switching it is a very advanced system and takes all the features of interval systems ito equations and markovian switching as well as time lag into account the theory developed is applicable in many different and complicated situations so the importance of the paper is clear

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

two level toeplitz preconditioning approximation results for matrices and functions large 2 level toeplitz systems arise in a variety of applications see e g r h chan and m ng siam rev 38 1996 pp 427 482 for which efficient numerical methods for their solution are required some successful numerical techniques need the explicit knowledge of the generating function f of the considered system t n f rm x b an assumption that usually is not fulfilled in real applications in this paper we analyze and complete the procedure proposed in d noutsas s serra capizzano and p vassalos numer linear algebra appl 12 2005 pp 231 239 for the 2 level case in such a way from the knowledge of the coefficients of t n f we determine optimal preconditioning strategies for the solution of our systems finally some numerical experiments are performed and discussed in connection with our theoretical analysis

preconditioners for ill conditioned toeplitz systems constructed from positive kernels in this paper we are interested in the iterative solution of ill conditioned toeplitz systems generated by continuous nonnegative real valued functions f with a finite number of zeros we construct new w circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f kn with a suitable positive reproducing kernel kn by the restriction to positive kernels we obtain positive definite preconditioners moreover if f has only zeros of even order le 2s then we can prove that the property int pi pi t 2k k n t mbox d t le c n 2k k 0 hspace 1 5pt ldots s of the kernel is necessary and sufficient to ensure the convergence of the pcg method in a number of iteration steps independent of the dimension n of the system our theoretical results were confirmed by numerical tests

spectral behavior of preconditioned non hermitian multilevel block toeplitz matrices with matrix valued symbol this note is devoted to preconditioning strategies for non hermitian multilevel block toeplitz linear systems associated with a multivariate lebesgue integrable matrix valued symbol in particular we consider special preconditioned matrices where the preconditioner has a band multilevel block toeplitz structure and we complement known results on the localization of the spectrum with global distribution results for the eigenvalues of the preconditioned matrices in this respect our main result is as follows let i k k let m s be the linear space of complex s i s matrices and let f g i k m s be functions whose components f ij g ij i k c i j 1 s belong to l consider the matrices t n 1 g t n f where n n 1 n k varies in n k and t n f t n g are the multilevel block toeplitz matrices of size n 1 n k s generated by f g then t n 1 g t n f n n k g 1 f i e the family of matrices t n 1 g t n f n n k has a global asymptotic spectral distribution described by the function g 1 f provided g possesses certain properties which ensure in particular the invertibility of t n g for all n and the following topological conditions are met the essential range of g 1 f defined as the union of the essential ranges of the eigenvalue functions j g 1 f j 1 s does not disconnect the complex plane and has empty interior this result generalizes the one obtained by donatelli neytcheva serra capizzano in a previous work concerning the non preconditioned case g 1 the last part of this note is devoted to numerical experiments which confirm the theoretical analysis and suggest the choice of optimal gmres preconditioning techniques to be used for the considered linear systems

a classification scheme for regularizing preconditioners with application to toeplitz systems preconditioning techniques for linear systems are widely used in order to speed up the convergence of iterative methods if the linear system is generated by the discretization of an ill posed problem preconditioning may lead to wrong results since components related to noise on input data are amplified using basic concepts from the theory of inverse problems we identify a class of preconditioners which acts as a regularizing tool in this paper we study relationships between this class and previously known circulant preconditioners for ill conditioned hermitian toeplitz systems in particular we deal with the low pass filtered optimal preconditioners and with a recent family of superoptimal preconditioners we go on to describe a set of preconditioners endowed with particular regularization properties whose effectiveness is supported by several numerical tests

a unifying approach to abstract matrix algebra preconditioning in earlier papers tyrtyshnikov 42 and the first author 14 considered the analysis of clustering properties of the spectra of specific toeplitz preconditioned matrices obtained by means of the best known matrix algebras here we generalize this technique to a generic banach algebra of matrices by devising general preconditioners related to convergent approximation processes 36 finally as case study we focus our attention on the tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric toeplitz systems can be constructed in this class

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

degrees of freedom of the k user m times n mimo interference channel we provide inner bound and outer bound for the total number of degrees of freedom of the k user multiple input multiple output mimo gaussian interference channel with m antennas at each transmitter and n antennas at each receiver if the channel coefficients are time varying and drawn from a continuous distribution the bounds are tight when the ratio max m n min m n r is equal to an integer for this case we show that the total number of degrees of freedom is equal to min m n k if k r and min m n r r 1 k if k r achievability is based on interference alignment we also provide examples where using interference alignment combined with zero forcing can achieve more degrees of freedom than merely zero forcing for some mimo interference channels with constant channel coefficients

generalized channel inversion methods for multiuser mimo systems block diagonalization bd is a well known precoding method in multiuser multi input multi output mimo broadcast channels this scheme can be considered as a extension of the zero forcing zf channel inversion to the case where each receiver is equipped with multiple antennas one of the limitation of the bd is that the sum rate does not grow linearly with the number of users and transmit antennas at low and medium signal to noise ratio regime since the complete suppression of multi user interference is achieved at the expense of noise enhancement also it performs poorly under imperfect channel state information in this paper we propose a generalized minimum mean squared error mmse channel inversion algorithm for users with multiple antennas to overcome the drawbacks of the bd for multiuser mimo systems we first introduce a generalized zf channel inversion algorithm as a new approach of the conventional bd applying this idea to the mmse channel inversion for identifying orthonormal basis vectors of the precoder and employing the mmse criterion for finding its combining matrix the proposed scheme increases the signal to interference plus noise ratio at each user s receiver simulation results confirm that the proposed scheme exhibits a linear growth of the sum rate as opposed to the bd scheme for block fading channels with four transmit antennas the proposed scheme provides a 3 db gain over the conventional bd scheme at 1 frame error rate also we present a modified precoding method for systems with channel estimation errors and show that the proposed algorithm is robust to channel estimation errors

stability of the notion of approximating class of sequences and applications given an approximating class of sequences b n m n m for a n n we prove that b n m n m x being the pseudo inverse of moore penrose is an approximating class of sequences for a n n where a n n is a sparsely vanishing sequence of matrices a n of size d n with d k d q for k q k email protected n as a consequence we extend distributional spectral results on the algebra generated by toeplitz sequences by including the pseudo inversion operation in the case where the sequences that are pseudo inverted are distributed as sparsely vanishing symbols applications to preconditioning and a potential use in image signal restoration problems are presented

degrees of freedom for the mimo interference channel in this correspondence we show that the exact number of spatial degrees of freedom dof for a two user nondegenerate full rank channel matrices multiple input multiple output mimo gaussian interference channel with m1 m2 antennas at transmitters 1 2 and n1 n2 antennas at the corresponding receivers and perfect channel knowledge at all transmitters and receivers is min m1 m2 n1 m2 max m1 n2 max m2 n1 a constructive achievability proof shows that zero forcing is sufficient to achieve all the available dof on the two user mimo interference channel we also show through an example of a share and transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available dof are not increased

band preconditioners for block toeplitz toeplitz block systems abstract preconditioned conjugate gradient methods are employed to solve symmetric positive definite m by m block toeplitz with n by n toeplitz block systems am nx b where am n are generated by 2 periodic nonnegative functions with zeros serra has proposed using band block toeplitz with band toeplitz block matrices bm n with their external and internal bandwidths independent of m and n as preconditioners serra showed that if the hessians of the generating function at the zeros are positive definite then the condition number of b 1m nam n is uniformly bounded by a constant independent of m and n whereas the condition number of am n tends to infinity as m and n tend to infinity in this paper we provide a method for deriving band preconditioners for block toeplitz toeplitz block matrices numerical examples are given to illustrate the performance of the method

the analysis of a chebyshev problem via spectral matrix theory abstract some chebyshev type problems 1 are introduced here and an interpretation and related solution in terms of the extremes of certain toeplitz rational forms 2 are given also shown is how matrix analysis can be successfully used in this specific approximation theory context

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

on designing of sliding mode control for stochastic jump systems in this note we consider the problems of stochastic stability and sliding mode control for a class of linear continuous time systems with stochastic jumps in which the jumping parameters are modeled as a continuous time discrete state homogeneous markov process with right continuous trajectories taking values in a finite set by using linear matrix inequalities lmis approach sufficient conditions are proposed to guarantee the stochastic stability of the underlying system then a reaching motion controller is designed such that the resulting closed loop system can be driven onto the desired sliding surface in a limited time it has been shown that the sliding mode control problem for the markovian jump systems is solvable if a set of coupled lmis have solutions a numerical example is given to show the potential of the proposed techniques

how to choose the best iterative strategy for symmetric toeplitz systems in the last decade many efficient iterative solvers for n x n symmetric also hermitian or generic toeplitz systems have been devised r h chan and g strang siam j sci stat comput 10 1989 pp 104 119 t f chan siam j sci stat comput 9 1988 pp 766 771 d bini and f di benedetto proceedings 2nd spaa conference 1990 pp 220 223 d bini and p favati siam j matrix anal appl 14 1993 pp 500 507 t huckle siam j matrix anal appl 13 1992 pp 767 777 f di benedetto g fiorentino and s serra comput math appl 26 1993 pp 35 45 s serra siam j matrix anal appl 17 1996 pp 1007 1019 g fiorentino and s serra calcolo 28 1991 pp 283 305 s serra siam j matrix anal appl 20 1998 pp 31 44 in all these methods it is assumed that the considered systems have the form an f x b where the symbol f the generating function is an l1 function and the entries of an f along the kth diagonal coincide with the kth fourier coefficient of f the proof of convergence and sometimes even the definition of these solvers are derived by using some symbolic or analytic properties of f while this function is often unknown in real applications in this paper by means of the knowledge of the coefficients of an f only we propose and discuss an algorithm to economically determine the minimal features of f allowing us to choose and define the best iterative strategy as a consequence of the analysis of the proposed procedure we find some results on the structure of quasi algebra of the set a n f f in l 1 finally we exhibit several numerical experiments which confirm the effectiveness of the proposed idea

superlinear pcg methods for symmetric toeplitz systems in this paper we deal with the solution by means of preconditioned conjugate gradient pcg methods of n x n symmetric toeplitz systems a n f x b with nonnegative generating function f here the function f is assumed to be continuous and strictly positive or is assumed to have isolated zeros of even order in the first case we use as preconditioner the natural and the optimal approximation of a n f proposed by bini and di benedetto and we prove that the related pcg method has a superlinear rate of convergence and a total arithmetic cost of o n log n ops under the second hypothesis we cannot guarantee that the natural matrix is positive definite while for the optimal we show that in the ill conditioned case this can be really a bad choice consequently we define a new matrix for preconditioning the given system then by applying the sherman morrison woodbury inversion formula to the preconditioned system we introduce a small constant number of subsidiary systems which can be solved again by means of the previous pcg method finally we perform some numerical experiments that show the effectiveness of the devised technique and the adherence with the theoretical analysis

sum capacity of gaussian vector broadcast channels this paper characterizes the sum capacity of a class of potentially nondegraded gaussian vector broadcast channels where a single transmitter with multiple transmit terminals sends independent information to multiple receivers coordination is allowed among the transmit terminals but not among the receive terminals the sum capacity is shown to be a saddle point of a gaussian mutual information game where a signal player chooses a transmit covariance matrix to maximize the mutual information and a fictitious noise player chooses a noise correlation to minimize the mutual information the sum capacity is achieved using a precoding strategy for gaussian channels with additive side information noncausally known at the transmitter the optimal precoding structure is shown to correspond to a decision feedback equalizer that decomposes the broadcast channel into a series of single user channels with interference pre subtracted at the transmitter

optimal quasi optimal and superlinear band toeplitz preconditioners for asymptotically ill conditioned positive definite toeplitz systems in this paper we are concerned with the solution of n x n hermitian toeplitz systems with nonnegative generating functions f the preconditioned conjugate gradient pcg method with the well known circulant preconditioners fails in the case where f has zeros in this paper we consider as preconditioners band toeplitz matrices generated by trigonometric polynomials g of fixed degree l we use different strategies of approximation of f to devise a polynomial g which has some analytical properties of f is easily computable and is such that the corresponding preconditioned system has a condition number bounded by a constant independent of n for each strategy we analyze the cost per iteration and the number of iterations required for the convergence within a preassigned accuracy we obtain different estimates of l for which the total cost of the proposed pcg methods is optimal and the related rates of convergence are superlinear finally for the most economical strategy we perform various numerical experiments which fully confirm the effectiveness of approximation theory tools in the solution of this kind of linear algebra problems

on the asymptotic spectrum of hermitian block toeplitz matrices with toeplitz blocks we study the asymptotic behaviour of the eigenvalues of hermitian n x n block toeplitz matrices a n m with m x m toeplitz blocks such matrices are generated by the fourier coefficients of an integrable bivariate function f and we study their eigenvalues for large n and m relating their behaviour to some properties of f as a function in particular we show that for any fixed k the first k eigenvalues of a n m tend to inf f while the last k tend to sup f so extending to the block case a well known result due to szego in the case the a n m s are positive definite we study the asymptotic spectrum of p 1 n m a n m where p n m is a block toeplitz preconditioner for the conjugate gradient method applied to solve the system a n m x b obtaining strict estimates when n and m are fixed and exact limit values when n and m tend to infinity for both the condition number and the conjugate gradient convergence factor of the previous matrices extensions to the case of a deeper nesting level of the block structure are also discussed

the effect of non optimal bases on the convergence of krylov subspace methods there are many examples where non orthogonality of a basis for krylov subspace methods arises naturally these methods usually require less storage or computational effort per iteration than methods using an orthonormal basis optimal methods but the convergence may be delayed truncated krylov subspace methods and other examples of non optimal methods have been shown to converge in many situations often with small delay but not in others we explore the question of what is the effect of having a non optimal basis we prove certain identities for the relative residual gap i e the relative difference between the residuals of the optimal and non optimal methods these identities and related bounds provide insight into when the delay is small and convergence is achieved further understanding is gained by using a general theory of superlinear convergence recently developed our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important only the need to maintain linear independence is numerical examples illustrate our theoretical results

extreme singular values and eigenvalues of non hermitian block toeplitz matrices abstract in this paper we are concerned with the analysis of the distribution and localization of the singular values of toeplitz matrices tn f generated by a p variate lebesgue integrable matrix valued function f q p c h k q we prove that the union of the essential ranges of the singular values of f is a proper weak cluster for the whole set of the singular values of tn f by showing that the number of outliers is strongly depending on the regularity features of the underlying function f in particular if f is continuous or from the krein algebra and p 1 then the cluster is proper other results concerning the extreme spectral behavior of tn f second order ergodic formulas and localization of eigenvalues of preconditioned matrices tn 1 g tn f are presented some examples of applications to the preconditioning of these results are also discussed

design and analysis of toeplitz preconditioners the solution of symmetric positive definite toeplitz systems ax b by the preconditioned conjugate gradient pcg method was recently proposed by strang 1986 and analyzed by chan 1989 and chan and strang 1989 the convergence rate of the pcg method depends on the choice of preconditioners for the given toeplitz matrices the authors present a general approach to the design of toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions all resulting preconditioners can therefore be inverted via various fast transform algorithms with o n log n operations for a wide class of problems the pcg method converges in a finite number of iterations independent of n so that the computational complexity for solving these toeplitz systems is o n log n

robust integral sliding mode control for uncertain stochastic systems with time varying delay this paper is concerned with sliding mode control for uncertain stochastic systems with time varying delay both time varying parameter uncertainties and an unknown nonlinear function may appear in the controlled system an integral sliding surface is first constructed then by means of linear matrix inequalities lmis a sufficient condition is derived to guarantee the global stochastic stability of the stochastic dynamics in the specified switching surface for all admissible uncertainties the synthesized sliding mode controller guarantees the reachability of the specified sliding surface finally a simulation example is presented to illustrate the proposed method

absolute equal distribution of the spectra of hermitian matrices abstract if x if x x if x if x let a n b n p n where b n and p n are n n hermitian matrices we show that if p n f 2 o n then for any a i 1 n f mid i a n f mid i b n o n if f c eigenvalues numbered in nondecreasing order we consider the special case where p n are real hankel matrices we also show that if rank p n o n then a holds for every and f c combining these results yields a result concerning c n b n e n r n where e n 2 f o n and rank r n o n we also consider the case where the conditions on e n are stated in terms of schatten p norms finally we show that if t n are hermitian toeplitz matrices generated by f c with minimum m f and maximum m f 2 i 1 n n in 2 i n n 1 i n and n is a permutation of 1 2 n such that f n 1 n f n 2 n f n n n then i 1 n f i t n f f n i n o n if f c m f m f

output feedback control design for strict feedback stochastic nonlinear systems under a risk sensitive cost in this note we study the problem of output feedback control design for a class of strict feedback stochastic nonlinear systems under an infinite horizon risk sensitive cost criterion the controller designed can guarantee an arbitrary small long term average cost for arbitrary risk sensitivity parameter and achieve boundedness in probability for the closed loop system using the integrator backstepping methodology furthermore the controller preserves the equilibrium at the origin of the nonlinear system

some theorems on linear positive operators and functionals and their applications abstract in this paper we prove two theorems concerning linear positive operators and functionals in a very general sense we show how these results can be used to derive new results in a wide range of applications from approximation theory to the numerical treatment of differential equations and to the numerical analysis connected with signals and images

an ergodic theorem for classes of preconditioned matrices abstract we consider abstract classes of matrices a satisfying some structural conditions and in particular satisfying a crucial assumption about the asymptotic distribution of eigenvalues we prove a similar distribution property for classes of preconditioned matrices constructed by using representants of a as a particular case this result applies to preconditioned matrices coming from several important contexts finite differences and faedo ritz galerkin linear systems associated with elliptic and semielliptic boundary value problems very general hermitian toeplitz structures generated by multivariate ll functions this result answers in the positive some structural questions raised by tyrtyshnikov e tyrtyshnikov linear algebra appl 207 1994 225 249 and by the author s serra linear algebra appl 267 1997 139 161 s serra siam j numer anal in press in the toeplitz context

carrier cooperation can reduce the transmit power in parallel mimo broadcast channels with zero forcing even though parallel multiple input multiple output mimo broadcast channels are known to be separable from an information theoretic point of view performing separate encoding and decoding on each of the parallel channels has been shown to be potentially suboptimal in broadcast channels with linear transceivers in this paper we show that suboptimality of such a carrier noncooperative transmission also occurs in broadcast channels with zero forcing and quality of service constraints if time sharing is not allowed the proof is given by constructing a minimal example and identifying a rate tuple that is achievable using carrier cooperative zero forcing with a certain sum power but requires a higher sum power with carrier noncooperative zero forcing this observation is of practical relevance since zero forcing without time sharing is a popular assumption in the design of low complexity optimization algorithms

matrix algebra preconditioners for multilevel toeplitz systems do not insure optimal convergence rate in the last decades several matrix algebra optimal and superlinear preconditioners those assuring a strong clustering at the unity have been proposed for the solution of polynomially ill conditioned toeplitz linear systems the corresponding generalizations for multilevel structures are neither optimal nor superlinear see e g contemp math 281 2001 193 concerning the notion of superlinearity it has been recently shown that the proper clustering cannot be obtained in general see linear algebra appl 343 344 2002 303 siam j matrix anal appl 22 1 1999 431 math comput 72 2003 1305 in this paper by exploiting a proof technique previously proposed by the authors see contemp math 323 2003 313 we prove that the spectral equivalence and the essential spectral equivalence up to a constant number of diverging eigenvalues are impossible too in conclusion optimal matrix algebra preconditioners in the multilevel setting simply do not exist in general and therefore the search for optimal iterative solvers should be oriented to different directions with special attention to multilevel multigrid techniques

the linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and markovian jumping in this paper the linear quadratic optimization problem for a class of linear stochastic systems subject both to multiplicative white noise and markovian jumping is investigated two classes of admissible controls are considered one of these classes contains controls with additional property that corresponding trajectories tend to zero in mean square when tends to spl infin while concerning the controls contained in the second class of admissible controls there is not any stability assumption in the optimization problem over the first class of admissible controls the cost functional could have indefinite sign of weights matrices an iterative procedure to compute the maximal solution of the systems of generalized riccati equations is provided a numerical example to illustrate the applicability of the iterative procedure is given

robust h sub spl infin control of uncertain markovian jump systems with time delay this paper is concerned with the robust stochastic stabilizability and robust h sub spl infin disturbance attenuation for a class of uncertain linear systems with time delay and randomly jumping parameters the transition of the jumping parameters is governed by a finite state markov process sufficient conditions on the existence of a robust stochastic stabilizing and spl gamma suboptimal h sub spl infin state feedback controller are presented using the lyapunov functional approach it is shown that a robust stochastically stabilizing h sub spl infin state feedback controller can be constructed through the numerical solution of a set of coupled linear matrix inequalities

on the achievable degrees of freedom of two cell multiuser mimo interference networks in this paper we study the sum degrees of freedom dof of an uplink two cell multiuser mimo interference network with asymmetric number of users in the cells the achievable dof is devised based on a two dimensional space time spreading code framework with linear precoding decoding design and finite channel extension the derivation of the achievable dof is shown related to a rank minimization problem which corresponds to the minimization of the dimension of the interference subspace the problem is solved by the proposed grouping algorithm ga based on aligning interfering signals into a low dimensional subspace as a group and attaining the minimum number of groups the achievable sum dof derived based on the proposed ga is shown to be greater than prior arts and achieves the theoretic upper bound in several cases we also give a closed form expression of the max imum achievable sum dof when there is the maximum number of admissible users in the considered finite diversity environment

robust h sub spl infin filtering for nonlinear stochastic systems this paper describes the robust h sub spl infin filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance dependent noise we assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by ito spl circ type stochastic differential equations for general nonlinear stochastic systems the h sub spl infin filter can be obtained by solving second order nonlinear hamilton jacobi inequalities when the worst case disturbance is considered in the design procedure a mixed h sub 2 h sub spl infin filtering problem is also solved by minimizing the total estimation error energy it is found that for a class of special nonlinear stochastic systems the h sub spl infin filtering design can be given via solving several linear matrix inequalities instead of hamilton jacobi inequalities a few examples show that the proposed methods are effective

komleva type expansions and asymptotics for linear operators abstract in this paper we prove asymptotic komleva type expansions valid for sequences of linear operators t n approximating the identity in supremum norm over the space of the continuous functions in particular under suitable mild conditions on the sequence t n we obtain rational expansions for t n f that are of special interest in a numerical analysis context as special cases of these results we find asymptotic expansions for exponential type and de la vallee poussin polynomial operators the case of the cesaro sums is discussed in connection with the komleva theory but the main asymptotic results are proved by using other tools coming from a context of structured linear algebra some numerical applications of the theoretical part are then discussed