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

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

application of approximation theory methods to recursive digital filter design in this paper the e herglotz tansform or e outer function is utilized to yield an analytic function hat h epsilon z on the unit disk from the given amplitude characteristic of a digital filter this is motivated by a two fold objective 1 the analytic function can be written in a power series expansion so that pade approximants can be obtained and 2 hat h epsilon z is a zero free function in the hardy space h2so that its polynomial least squares inverse exists and does not vanish on the unit disk hence we can immediately obtain recursive filters by pade approximants and all pole filters by means of least squares inverses there are two advantages in these methods 1 we only need to solve linear systems of algebraic equations to obtain the filter coefficients and 2 the phases of these filters exhibit almost linear in the passband while the amplitude characteristics of these filters closely approximate the given one if some adjustments are needed for the all pole filters we can use a standard least squares technique to change an all pole filter into a pole zero filter the detailed procedures are presented and the stability and approximation of these methods are discussed several examples are used for illustration purposes in this paper

observer based control design of semi markovian jump systems with uncertain probability intensities and mode transition dependent sojourn time distribution abstract this paper deals with the problem of h observer based control for a class of continuous time semi markovian jump systems s mjss with more detailed observational information first to explore the impact of uncertain probability intensities a mathematical analysis is accomplished from which some useful inequality conditions on the sum of transition rates trs are obtained further to come up with more accurate bounds of trs the mode transition dependent probability distribution of sojourn time is imposed on the mechanism of forming trs lastly by devising a compatible relaxation process that can embrace all the conditions found in our derivation the resultant observer based stabilization conditions are formulated in terms of linear matrix inequalities

2 correcting lee codes quasi perfect spectral conditions and some constructions let p be an odd prime recently camarero and martinez in quasi perfect lee codes of radius 2 and arbitrarily large dimension ieee trans inform theory vol 62 no 3 2016 constructed some p ary 2 quasi perfect lee codes for p equiv pm 5 pmod 12 in this paper some infinite classes of p ary 2 quasi perfect lee codes for any odd prime p with flexible length and dimension are presented more specifically we provide a new method for constructing quasi perfect lee codes our approach uses subsets derived from some quadratic curves over finite fields in odd characteristic to obtain two classes of 2 quasi perfect lee codes defined in the space pmb mathbb z p n for n frac p k 1 2 with p equiv 1 5 pmod 12 and k is any integer or p equiv 1 5 pmod 12 and k is an even integer and n frac p k 1 2 with p equiv 1 5 pmod 12 k is an odd integer and p k 12 our codes encompass the p ary p equiv pm 5 pmod 12 2 quasi perfect lee codes constructed by camarero and martinez furthermore we prove that the related cayley graphs are ramanujan or almost ramanujan using kloosterman sums this generalizes the work of bibak kapron and srinivasan in the cayley graphs associated with some quasi perfect lee codes are ramanujan graphs ieee trans inform theory vol 62 no 11 2016 from the case p equiv 3 pmod 4 and k 1 to the case of any odd prime p and positive integer k finally we derive some necessary conditions with the exponential sums of all 2 perfect codes and 2 quasi perfect codes and present a heuristic algorithm for constructing 2 perfect codes and 2 quasi perfect codes our results show that in general the cayley graphs associated with 2 perfect codes are ramanujan from the algorithm some new 2 quasi perfect lee codes different from those constructed from quadratic curves are given the lee codes presented in this paper have applications in constrained and partial response channels flash memories and decision diagrams

circulant and skew circulant splitting methods for toeplitz systems we study efficient iterative methods for toeplitz systems based on the circulant and skew circulant splitting cscs of the toeplitz matrix theoretical analysis show that if the circulant and the skew circulant splitting matrices are positive definite then the cscs method converges to the unique solution of the system of linear equations moreover we derive an upper bound of the contraction factor of the cscs iteration which is dependent solely on the spectra of the circulant and the skew circulant matrices involved numerical examples are presented to demonstrate the method

robust h_ infty fuzzy control approach for a class of markovian jump nonlinear systems this paper addresses the problem of stabilizing a class of nonlinear systems subject to markovian jump parameters using a robust stochastic fuzzy controller with hinfin performance the class of jump nonlinear systems considered is described by a fuzzy model composed of two levels a crisp level which represents the jumps and a fuzzy level which represents the system nonlinearities considering the approximation error between the fuzzy model and the jump nonlinear system as norm bounded uncertainties we develop a systematic technique based on a coupled lyapunov function to obtain a robust stochastic fuzzy controller which guarantees the l2 gain of the closed loop system in respect to external inputs to be equal to or less than a prescribed value a simulation example on an industrial power plant operating in a cogeneration scheme is presented to illustrate the effectiveness of the proposed stabilizing controller in reducing oscillations as well as maintaining a desired operation condition in the presence of fluctuations in the local load

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

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

stability and stabilization of discrete time semi markov jump linear systems via semi markov kernel approach this technical note is concerned with the problems of stability and stabilization for a class of discrete time semi markov jump linear systems s mjlss the discrete time semi markov kernel smk is introduced where the probability density function of sojourn time is dependent on both current and next system mode as a consequence different types of distributions and or different parameters in a same type of distribution of sojourn time depending on the target mode towards which the system jumps can coexist in each mode of a smk the underlying s mjlss are therefore more general than those considered in existing studies a new stability concept generalizing the traditional mean square stability is proposed such that numerically testable criteria on the basis of smk are obtained numerical examples are presented to illustrate the validity and advantage of the developed theoretical results

circulant preconditioners constructed from kernels circulant preconditioners for hermitian toeplitz systems are considered from the viewpoint of function theory it is shown that some well known circulant preconditioners can be derived from convoluting the generating function f of the toeplitz matrix with famous kernels like the dirichlet and the fejer kernels several circulant preconditioners are then constructed using this approach finally it is proven that if the convolution product converges to f uniformly then the circulant preconditioned toeplitz systems will have a clustered spectrum

h filtering for networked systems with partly known distribution transmission delays this paper studies the h filtering problem for a class of discrete time networked systems with partly known distribution transmission delays it is assumed that the random sensor filter delay satisfies markovian characteristics but the transition probabilities of the delay transferring from one value to others are partly known a novel measurement model is proposed to reflect the random delay of the transmission modes and the resulting filtering error system is modeled as a markovian jump system mjs with multiple modes and state delay sufficient conditions are derived for the filtering error system to be stochastically stable and ensure a prescribed h disturbance attenuation level bound by using mode dependent lyapunov functional approach based on the derived conditions a design procedure for the h filter is also presented a numerical example is provided to show the design process of the filter and the effectiveness of the proposed method

improved delay dependent stability criteria for time delay systems this note provides an improved asymptotic stability condition for time delay systems in terms of a strict linear matrix inequality unlike previous methods the mathematical development avoids bounding certain cross terms which often leads to conservatism when time varying norm bounded uncertainties appear in a delay system an improved robust delay dependent stability condition is also given examples are provided to demonstrate the reduced conservatism of the proposed conditions

new results on output feedback h_ infty control for linear discrete time systems this note investigates the problem of output feedback h control for linear discrete time systems three types of h controllers are considered namely static output feedback controllers dynamic output feedback controllers and observer based output feedback controllers new design conditions for the three type of output feedback controllers are introduced in terms of unified linear matrix inequality lmi representations which guarantee the prescribed h performances of the closed loop systems in contrast to the existing lmi conditions for designing the output feedback h controllers the improvement of the proposed results over the existing ones is shown by theoretical proof and numerical example

stochastic stability of ito differential equations with semi markovian jump parameters in this note the problem of stochastic stability for linear systems with jump parameters being semi markovian rather than full markovian is further investigated in particular the system under consideration is described by ito type nonlinear stochastic differential equations with phase type semi markovian jump parameters stochastic stability conditions are presented

brief paper markovian jump delayed hopfield networks with multiplicative noise a non symmetric version of hopfield networks subject to state multiplicative noise pure time delay and markov jumps is considered such networks arise in the context of visuo motor control loops and may therefore be used to mimic their complex behavior in this paper we adopt the lur e postnikov systems approach to analyze the stochastic stability and the l 2 gain of generalized hopfield networks including these effects

bttb preconditioners for bttb least squares problems abstract in this paper we consider solving the least squares problem min x b t x 2 by using preconditioned conjugate gradient pcg methods where t is a large rectangular matrix which consists of several square block toeplitz toeplitz block bttb matrices and b is a column vector we propose a bttb preconditioner to speed up the pcg method and prove that the bttb preconditioner is a good preconditioner we then discuss the construction of the bttb preconditioner numerical examples including image restoration problems are given to illustrate the efficiency of our bttb preconditioner numerical results show that our bttb preconditioner is more efficient than the well known level 1 and level 2 circulant preconditioners

toeplitz preconditioners for hermitian toeplitz systems abstract we propose a new type of preconditioners for hermitian positive definite toeplitz systems a n x b where a n are assumed to be generated by functions f that are positive and 2 periodic our approach is to precondition a n by the toeplitz matrix a n generated by 1 f we prove that the resulting preconditioned matrix a n a n will have clustered spectrum when a n cannot be formed efficiently we use quadrature rules and convolution products to construct nearby approximations to a n we show that the resulting approximations are toeplitz matrices which can be written as sums of circulant matrices as a side result we prove that any toeplitz matrix can be written as a sum of circulant matrices we then show that our toeplitz preconditioners t n are generalizations of circulant preconditioners and the way they are constructed is similar to the approach used in the additive schwarz method for elliptic problems we finally prove that the preconditioned systems t n a n will have clustered spectra around 1

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

four short stories about toeplitz matrix calculations the stories told in this paper are dealing with the solution of finite infinite and biinfinite toeplitz type systems a crucial role plays the off diagonal decay behavior of toeplitz matrices and their inverses classical results of gelfand et al on commutative banach algebras yield a general characterization of this decay behavior we then derive estimates for the approximate solution of bi infinite toeplitz systems by the finite section method showing that the approximation rate depends only on the decay of the entries of the toeplitz matrix and its condition number furthermore we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems finally some quantitative results on the construction of preconditioners via circulant embedding are derived which allow us to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems

the cayley graphs associated with some quasi perfect lee codes are ramanujan graphs let mathbb z n i be the ring of gaussian integers modulo a positive integer n very recently camarero and martinez et al showed that for every prime number p 5 such that p equiv pm 5 pmod 12 the cayley graph mathcal g p mathrm cay mathbb z p i s 2 where s 2 is the set of units of mathbb z p i induces a two quasi perfect lee code over mathbb z p m where m 2 lfloor p 4 rfloor they also conjectured that mathcal g p is a ramanujan graph for every prime p such that p equiv 3 pmod 4 in this paper we solve this conjecture our main tools are deligne s bound from 1977 for estimating a particular kind of trigonometric sum and a result of lovasz from 1975 or of babai from 1979 which gives the eigenvalues of cayley graphs of finite abelian groups our proof techniques may motivate more work in the interactions between spectral graph theory character theory and coding theory and may provide new ideas toward the famous golomb welch conjecture on the existence of perfect lee codes

construction of preconditioners for wiener hopf equations by operator splitting abstract in this paper we propose a new type of preconditioners for solving finite section wiener hopf integral equations i a x g by the preconditioned conjugate gradient algorithm we show that for an integer u 1 the operator i a can be decomposed into a sum of operators i p u v for 0 v 1 u v i p u v 1 thus the way the preconditioners are constructed is very similar to the approach used in the additive schwarz method for elliptic problems as for the convergence rate we prove that the spectra of the resulting preconditioned operators 1 u v i p u v 1 i a are clustered around 1 and thus the algorithm converges sufficiently fast finally we discretize the resulting preconditioned equations by rectangular rule numerical results show that our methods converges faster than those preconditioned by using circulant integral operators

robust finite time h control for discrete time singular markovian jump systems with time varying delay and actuator saturation in this paper the problem of robust finite time h control for discrete time singular markovian jump systems with time varying delay and actuator saturation is studied first sufficient conditions that guarantee the systems singular stochastic finite time boundedness are derived via constructing a delay dependent lyapunov krasonskii functional and linear matrix inequalities lmi approaches then the results are extended to singular stochastic finite time h boundedness furthermore with these conditions and convex optimization arithmetic the design method of h controller and conditions that ensure the h disturbance attenuation level are obtained finally some numerical examples are presented to illustrate the validity and efficiency of the proposed methods

on the hss iteration methods for positive definite toeplitz linear systems we study the hss iteration method for large sparse non hermitian positive definite toeplitz linear systems which first appears in bai golub and ng s paper published in 2003 z z bai g h golub m k ng hermitian and skew hermitian splitting methods for non hermitian positive definite linear systems siam j matrix anal appl 24 2003 603 626 and hss stands for the hermitian and skew hermitian splitting of the coefficient matrix a in this note we use the hss iteration method based on a special case of the hss splitting where the symmetric part h 12 a a t is a centrosymmetric matrix and the skew symmetric part s 12 a a t is a skew centrosymmetric matrix for a given toeplitz matrix hence fast methods are available for computing the two half steps involved in the hss and ihss iteration methods some numerical results illustrate their effectiveness

robust h sub infinity control for linear systems with norm bounded time varying uncertainty robust h sub infinity control design for linear systems with uncertainty in both the state and input matrices is treated a state feedback control design which stabilizes the plant and guarantees an h sub infinity norm bound constraint on disturbance attenuation for all admissible uncertainties is presented the robust h sub infinity control problem is solved via the notion of quadratic stabilization with an h sub infinity norm bound necessary and sufficient conditions for quadratic stabilization with an h sub infinity norm bound are derived the results can be regarded as extensions of existing results on h sub infinity control and robust stabilization of uncertain linear systems

fast band toeplitz preconditioners for hermitian toeplitz systems this paper considers the solutions of hermitian toeplitz systems where the toeplitz matrices are generated by nonnegative functions f the preconditioned conjugate gradient method with well known circulant preconditioners fails in the case when f has zeros this paper employs toeplitz matrices of fixed bandwidth as preconditioners their generating functions g are trigonometric polynomials of fixed degree and are determined by minimizing the maximum relative error f g f infty it is shown that the condition number of systems preconditioned by the band toeplitz matrices are o 1 for f with or without zeros when f is positive the preconditioned systems converge at the same rate as other well known circulant preconditioned systems an a priori bound of the number of iterations required for convergence is also given

explicit inverse matrices of tribonacci skew circulant type matrices the determinants and inverses of tribonacci skew circulant type matrices are discussed in the paper firstly tribonacci skew circulant type matrices are defined in addition we show the invertibility of the tribonacci skew circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices by utilizing the relation between circulant and left circulant the invertibility of the tribonacci skew left circulant are also discussed finally the determinants and the inverse matrices of these matrices are given respectively

inverse toeplitz preconditioners for ill posed problems abstract it has been shown recently that iterative regularization using conjugate gradient type methods for image restoration problems can be effectively preconditioned with circulant approximations here it is shown that the theoretical properties of this approach are not restricted to circulant matrices specifically a toeplitz approximate inverse preconditioning scheme for discrete ill posed problems is considered it is proved that the preconditioned system approximates the prolate matrix and that this property implies that fast convergence of conjugate gradient type methods can be expected in addition it is shown that these results can be generalized to two dimensional problems an image restoration application is used to demonstrate the properties of the preconditioner

perfect and quasi perfect codes under the l_ p metric a long standing conjecture of golomb and welch raised in 1970 states that there is no perfect r error correcting lee code of length n for n geq 3 and r 1 in this paper we study perfect codes in mathbb z n under the l p metric where 1 leq p we show some nonexistence results of linear perfect l p codes for p 1 and 2 leq p r 2 1 p 3 1 p we also give an algebraic construction of quasi perfect l p codes for p 1 r 2 and 2 leq p

analysis of the structured perturbation for the bsccb linear system abstract in this paper based on block style spectral decomposition of the block skew circulant with circulant blocks bsccb matrix the structure perturbation is discussed which includes the condition number and relative error of the bsccb linear system then the optimal backward perturbation bound of the bsccb linear system is analyzed simultaneously the algorithm for the optimal backward perturbation bound is presented at the end of the paper a numerical example is provided to verify the effectiveness of the algorithm

a decomposition of toeplitz matrices and optimal circulant preconditioning abstract a factorization of the given toeplitz matrix t obtained in this work is used to derive an optimal circulant preconditioner c in the sense of minimizing the frobenius norm of i ct by use of the fast fourier transform the computation of the new preconditioner can be performed in o n log n operations