two dimensional period estimation by ramanujan s sum period estimation in one dimension 1 d has been studied for years however 2 d period estimation is still a hard problem since it has three parameters to determine length width and direction recently a special kind of 2 d function called 2d gcd delta function is proposed it has close relationship with ramanujan s sum and the 2 d periodicity matrix in this paper we will describe how to use this function to decompose an image into subband signals each subband signal will have its own periodicity matrix so we also provide a simple algorithm to calculate the least common multiple lcm of those subband signal periodicity matrices concrete experiments are given to prove the robustness of the proposed 2 d period estimation method

exponential stability and solution bounds for systems with bounded nonlinearities estimation of lyapunov exponents of systems with bounded nonlinearities plays an essential part in their robust control known results in this field are based on the gronwall inequality yielding relatively conservative bounds for lyapunov exponents in this note we obtained more accurate upper bounds for the general lyapunov exponent for systems consisting of a known linear time varying part and an unknown nonlinear component with a bounded lipschitz constant at zero consequently a sufficient condition for exponential stability of this system is formulated the systems are indicated for which the obtained bound is precise i e cannot be improved without additional information on the nonlinear term in the presence of a persisting perturbation an upper bound for the solution norm is derived and expressed in the norm of the solution of the corresponding linear system using the obtained results a condition for exponential stability of a linear time varying control system with a nonlinear feedback is derived numerical results are obtained for a second order time varying system and for the lienard equation in the latter case they are favorably compared with stability conditions previously obtained using the lyapunov function method

cauchy problems for fractional differential equations with riemann liouville fractional derivatives abstract in this paper we are concerned with cauchy problems of fractional differential equations with riemann liouville fractional derivatives in infinite dimensional banach spaces we introduce the notion of fractional resolvent obtain some its properties and present a generation theorem for exponentially bounded fractional resolvents moreover we prove that a homogeneous order cauchy problem is well posed if and only if its coefficient operator is the generator of an order fractional resolvent and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous order cauchy problem

brief paper a way to stabilize linear systems with delayed state a simple method is proposed for stabilizing linear systems with delayed state using linear feedback the method requires checking the negativity of a matrix containing two free parameters and solving a matrix lyapunov equation with these parameters a known stabilizability criterion and a simple stability theorem for the open loop system are obtained as special cases of the main result

coprimality of certain families of integer matrices commuting coprime pairs of integer matrices have been of interest in multidimensional multirate systems and more recently in array processing in multirate systems they arise for example in the design of interchangeable cascades of decimator and expander matrices in array processing they arise in the construction of dense coarrays from sparse sensors located on a pair of lattices for the important case of two dimensional signals these matrices have size 2 2 in this paper the condition for coprimality is derived for several classes of 2 2 integer matrices namely circulant skew circulant and triangular families the first two are also commuting families for each class the special case of adjugate pairs which automatically commute is also elaborated it is also shown that the problem of testing coprimality of two 2 2 matrices is equvialent to testing coprimality of a pair of triangular matrices which can be done almost by inspection also considered is the case of 3 3 triangular matrices and their adjugates which have potential applications in three dimensional signal processing

fractional kinetic equations and universal decoupling of a memory function in mesoscale region it is proved that kinetic equations containing non integer integrals and derivatives appear in the result of reduction of a set of micromotions to some averaged collective motion in the mesoscale region in other words it means that after a proper statistical average the microscopic dynamics is converted into a collective motion in the mesoscopic regime a fractal medium containing weakly and strongly correlated relaxation units has been considered it is shown that in most cases the original of the memory function recovers the riemann liouville integral for a strongly correlated fractal medium a generalization of the riemann liouville integral is obtained for the fractal branched processes one can derive the stretched exponential law of relaxation that is widely used for description of relaxation phenomena in disordered media it is shown that the generalized stretched exponential function describes the averaged collective motion in the fractal branched complex systems it is confirmed that the generalized stretched exponential law can be definitely applicable for description of a set of spontaneous evoked postsynaptic signals in the living matter the application of these fractional kinetic equations for the description of the dielectric relaxation phenomena is also discussed these kinetic equations and their generalizations can be applicable for the description of different relaxation or diffusion processes in the intermediate mesoscale region

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

generalized aizerman myshkis problem for abstract differential delay equations abstract we consider a class of semilinear functional differential equations in a hilbert space conditions for the absolute stability are established moreover it is shown that these conditions separate equations that satisfy the generalized aizerman myshkis hypothesis the suggested approach is based on a combined usage of the properties of operators on tensor products of hilbert spaces and recent estimates for the norm of the resolvent our results are new even in the finite dimensional case we also discuss applications of the mentioned results to coupled systems of parabolic equations with delay and integro differential equations with delay

invertible deinterlacing with sampling density preservation theory and design a novel class of deinterlacing for intra frame field based motion picture coding such as motion jpeg2000 as well as an inter frame based coding without the support of interlaced scanning video such as mpeg 1 is developed this technique has two features sampling density preservation and invertibility these features mean that the amount of deinterlaced pictures is not increased and the original pictures can be perfectly reconstructed this deinterlacing technique is a kind of sampling lattice alteration and is regarded as a generalization of conventional field interleaving and field separation with the help of multidimensional m d multirate theory it is shown that the design problem of such a system that is invertible deinterlacer with sampling density preservation can be replaced to finding a 2 spl times 2 multivariable polynomial matrix with a monomial determinant this problem resembles the design of two channel m d maximally decimated perfect reconstruction finite impulse response fir filterbanks the inverse system which is referred to as a reinterlacer is given as fir when the dual deinterlacing system is fir a practical design procedure is provided by suggesting three constraints considered to be preferable normalization regularity and vertical symmetry the significance of the procedure is verified by showing some design examples of deinterlacing and reinterlacing filters simulation results show that the developed method causes fewer comb shaped artifacts than conventional field interleaving

statistical spectrum of the non markovity parameter for simple model systems the spectra of the non markovity parameter for three model systems are found an ideal gas weakly non ideal gas and a medium in hydrodynamic limits it is shown that the non markovian effects play the dominant role in all three above mentioned cases the effects of molecular memory are connected with the statistical appearance of intermolecular interactions in statistical systems

mittag leffler analysis ii application to the fractional heat equation abstract mittag leffler analysis is an infinite dimensional analysis with respect to non gaussian measures of mittag leffler type which generalizes the powerful theory of gaussian analysis and in particular white noise analysis in this paper we further develop the mittag leffler analysis by characterizing the convergent sequences in the distribution space moreover we provide an approximation of donsker s delta by square integrable functions then we apply the structures and techniques from mittag leffler analysis in order to show that a green s function to the time fractional heat equation can be constructed using generalized grey brownian motion ggbm by extending the fractional feynman kac formula from schneider moreover we analyse ggbm show its differentiability in a distributional sense and the existence of corresponding local times

theory of two dimensional multirate filter banks results are presented on 2 d fir two dimensional finite impulse response filter banks for multirate applications the theory is valid for all sampling lattices conditions for alias free and perfect signal reconstruction are derived synthesis structures for paraunitary and nonparaunitary polynomial matrices are derived which yield perfect reconstruction filter banks the degrees of freedom are given for these systems linear phase conditions are posed on the polyphase form of filter banks which is used to derive a design structure for the restricted but important case of linear phase filter banks

brief paper preservation of exponential stability for linear non autonomous functional differential systems we consider preservation of exponential stability for a system of linear equations with a distributed delay under the addition of new terms and a delay perturbation as particular cases the system includes models with concentrated delays and systems of integrodifferential equations our method is based on bohl perron type theorems

time fractional wave diffusion equation in an inhomogeneous half space we consider the fundamental solution of the time fractional wave diffusion equation in a three dimensional half space medium which contains an inhomogeneity in form of a plane parallel layer the corresponding green s function which is derived by means of the fourier and laplace transforms can be accurately and efficiently evaluated without recourse to the mittag leffler or the fox h function moreover it is shown that in the one dimensional case the fundamental solution in an inhomogeneous half space is no longer a probability density function in addition we consider the advection equation for the fractional laplacian and the caputo time fractional derivative of orders on a bounded domain simple algorithms for accurate evaluation of the m wright function and the mittag leffler function are enclosed at the end of this article

the brownian snake and solutions of δu u2 in a domain we investigate the connections between the path valued process called the brownian snake and nonnegative solutions of the partial differential equation u u2 in a domain of d in particular we prove two conjectures recently formulated by dynkin the first one gives a complete characterization of the boundary polar sets which correspond to boundary removable singularities for the equation u u2 the second one establishes a one to one correspondence between nonnegative solutions that are bounded above by a harmonic function and finite measures on the boundary that do not charge polar sets this correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the brownian snake our proofs combine probabilistic and analytic arguments an important role is played by a new version of the special markov property which is of independent interest

fractional non linear linear and sublinear death processes this paper is devoted to the study of a fractional version of non linear open image in new window t 0 linear m t t 0 and sublinear mathfrak m nu t t 0 death processes fractionality is introduced by replacing the usual integer order derivative in the difference differential equations governing the state probabilities with the fractional derivative understood in the sense of dzhrbashyan caputo we derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values a useful subordination relation is also proved allowing us to express the death processes as compositions of their classical counterparts with the random time process t2 t t 0 this random time has one dimensional distribution which is the folded solution to a cauchy problem of the fractional diffusion equation

delay independent stability conditions for time varying nonlinear uncertain systems a new stability criterion for time varying systems consisting of linear and norm bounded nonlinear terms with uncertain time varying delays is formulated an explicit delay independent sufficient stability condition is formulated in the terms of the transition matrix of the given linear part without delay and the bounds for the uncertain terms the obtained condition turns out to be also necessary if the matrix of the linear part is time invariant and symmetric it is shown that these systems satisfy the well known aizerman s conjecture the obtained criterion is contrasted by some of stability estimates available in the literature for these kinds of systems in all cases the proposed criterion provides less conservative stability bounds

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

fluctuation noise spectroscopy and a universal fitting function of amplitudes of random sequences abstract a universal fitting function has been recognized that makes use of the eigen coordinates method physica a 285 2000 547 to accurately describe the distribution of ordered amplitudes within random sequences taken from a diversity of sources it is shown that sequences with a discrete structure can be described in terms of specific distributions of relative frequencies with respect to the number of quantum levels an investigation of these quantum distributions leads to an increase in both sensitivity and selectivity when attempting the statistical detection of various predominant factors of the hidden signals the physical meaning of this new function is discussed and a proposal is made as the basis of a new fluctuation noise spectroscopy in which the recognized function can be used for the detection of small signals and or the forerunners of strong signals that are hidden within the random sequences analyzed

log periodic oscillations for biased diffusion on random lattice abstract random walks with a fixed bias direction on randomly diluted cubic lattices far above the percolation threshold exhibit log periodic oscillations in the effective exponent versus time a scaling argument accounts for the numerical results in the limit of large biases and small dilution and shows the importance of the interplay of these two ingredients in the generation of the log periodicity these results show that log periodicity is the dominant effect compared to previous predictions of and reports on anomalous diffusion

on upsampling downsampling and rational sampling rate filter banks solutions to the problem of design of rational sampling rate filter banks in one dimension has previosly been proposed the ability to interchange the operations of upsampling downsampling and filtering plays an important role in these solutions the present paper develops a complete theory for the analysis of arbitrary combinations of upsamplers downsamplers and filters in multiple dimensions although some of the simpler results are well known the more difficult results concerning swapping upsamplers and downsamplers and variations thereof are new as an application of this theory the authors obtain algebraic reductions of the general multidimensional rational sampling rate problem to a multidimensional uniform filter bank problem however issues concerning the design of the filters themselves are not addressed in multiple dimensions upsampling and downsampling operators are determined by integer matrices as opposed to scalars in one dimension and the noncommutativity of matrices makes the problem considerably more difficult cascades of upsamplers and downsamplers in one dimension are easy to analyze the new results for the analysis of multidimensional upsampling and downsampling operators are derived using the aryabhatta bezout identity over integer matrices as a fundamental tool a number of new results in the theory of integer matrices that a relevant to the filter bank problem are also developed special cases of some of the results pertaining to the commutativity of upsamplers downsamplers have been obtained in parallel by several authors

fractional dynamics and mds visualization of earthquake phenomena this paper analyses earthquake data in the perspective of dynamical systems and fractional calculus fc this new standpoint uses multidimensional scaling mds as a powerful clustering and visualization tool fc extends the concepts of integrals and derivatives to non integer and complex orders mds is a technique that produces spatial or geometric representations of complex objects such that those objects that are perceived to be similar in some sense are placed on the mds maps forming clusters in this study over three million seismic occurrences covering the period from january 1 1904 up to march 14 2012 are analysed the events are characterized by their magnitude and spatiotemporal distributions and are divided into fifty groups according to the flinn engdahl f e seismic regions of earth several correlation indices are proposed to quantify the similarities among regions mds maps are proven as an intuitive and useful visual representation of the complex relationships that are present among seismic events which may not be perceived on traditional geographic maps therefore mds constitutes a valid alternative to classic visualization tools for understanding the global behaviour of earthquakes

fractional diffusions with time varying coefficients this paper is concerned with the fractionalized diffusion equations governing the law of the fractional brownian motion bh t we obtain solutions of these equations which are probability laws extending that of bh t our analysis is based on mcbride fractional operators generalizing the hyper bessel operators l and converting their fractional power l into erdelyi kober fractional integrals we study also probabilistic properties of the random variables whose distributions satisfy space time fractional equations involving caputo and riesz fractional derivatives some results emerging from the analysis of fractional equations with time varying coefficients have the form of distributions of time changed random variables

brownian time processes the pde connection ii and the corresponding feynman kac formula we delve deeper into our study of the connection of brownian time processes btps to fourth order parabolic pdes which we introduced in a recent joint article with w zheng probabilistically btps and their cousins btps with excursions form a unifying class of interesting stochastic processes that includes the celebrated ibm of burdzy and other new intriguing processes and is also connected to the markov snake of le gall btps also offer a new connection of probability to pdes that is fundamentally different from the markovian one they solve fourth order pdes in which the initial function plays an important role in the pde itself not only as initial data we connect two such types of interesting and new pdes to btps the first is obtained by running the btp and then integrating along its path and the second type of pdes is related to what we call the feynman kac formula for btps a special case of the second type is a step towards a probabilistic solution to linearized cahn hilliard and kuramoto sivashinsky type pdes which we tackle in an upcoming paper

conditional propagation of chaos and a class of quasilinear pde s we consider conditional propagation of chaos and use it to solve a class of quasilinear equations of parabolic type in addition we construct a class of continuous stochastic processes associated with the above nonlinear equations our method imposes fewer smoothness conditions on the coefficients and allows a degenerate nonlinear weight before a divergence form operator we hope this probabilistic approach will introduce a better microscopic picture for understanding some stefan type problems

a novel general formulation of up downsampling commutativity the paper establishes the commutativity conditions of upsampling and downsampling for multidimensional signals defined on discrete abelian groups lattices the general condition includes results available in the literature as particular cases complete generality is achieved by giving an abstract definition of up downsampling and by working in the signal domain instead of the traditional approach based on z and fourier transforms examples of applications are outlined 1 for the one dimensional 1 d case verifying existing results 2 for television scanning 3 for degenerate reduced dimensionality lattices and 4 for multiplicative groups

an lmi approach to stability of systems with severe time delay this note describes the stability problems of uncertain systems with arbitrarily time varying and severe time delay using new lyapunov krasovskii functionals less conservative stability conditions are obtained for such systems the results are illustrated using the numerical examples based on simple linear matrix inequalities

vector space framework for unification of one and multidimensional filter bank theory a number of results in filter bank theory can be viewed using vector space notations this simplifies the proofs of many important results in this paper we first introduce the framework of vector space and then use this framework to derive some known and some new filter bank results as well for example the relation among the hermitian image property orthonormality and the perfect reconstruction pr property is well known for the case of one dimensional 1 d analysis synthesis filter banks we can prove the same result in a more general vector space setting this vector space framework has the advantage that even the most general filter banks namely multidimensional nonuniform filter banks with rational decimation matrices become a special case many results in 1 d filter bank theory are hence extended to the multidimensional case with some algebraic manipulations of integer matrices some examples are the equivalence of biorthonormality and the pr property the interchangeability of analysis and synthesis filters the connection between analysis synthesis filter banks and synthesis analysis transmultiplexers etc furthermore we obtain the subband convolution scheme by starting from the generalized parseval s relation in vector space several theoretical results of wavelet transform can also be derived using this framework in particular we derive the wavelet convolution theorem

absolute stability criteria for a generalized lur e problem with delay in the feedback a nonautonomous linear system controlled by a nonlinear sector restricted feedback with a time varying delay is considered delay independent sufficient conditions for absolute stability and instability expressed in the transfer function of the linear part and the sector bounds are established for a system with an exponentially stable linear part an upper bound for the lyapunov exponent is found it is shown that if the transfer function is sign constant asymptotic stability of the system with the margin linear feedback guarantees absolute stability of the considered system thus such systems satisfy the known aizerman conjecture they include in particular a closed loop system consisting of any number of time varying first order links and feedback with arbitrary delay under some additional condition which is certainly true for a time invariant linear block the obtained stability criterion is precise the approach employed in the proofs is based on a direct analysis of the corresponding volterra equation which contains only the transfer function of the linear block and therefore embraces a wide range of control systems as an example a second order system is considered it is shown that here the obtained stability bound is reached for a linear feedback with a periodic delay function

log periodic corrections to the cole cole expression in dielectric relaxation abstract a model of the self similar process of relaxation is given and a method of derivation of the kinetic equations for the total polarization based on the ideas of fractional kinetics is suggested the derived kinetic equations contain integro differential operators having non integer order they lead to the cole cole expression for the complex dielectric permittivity it is shown rigorously that the power law exponent in the cole cole expression coincides with the dimension of the mixed space temporal fractal ensemble if the discrete scale invariance for the temporal space structure of the dielectric medium considered becomes important then the expression for the complex dielectric permittivity contains log periodic corrections oscillations and hence it generalizes the conventional cole cole expression the corrections obtained in this model suggest another way of interpretation and analysis of dielectric spectra for different complex materials

brownian time processes the pde connection and the half derivative generator we introduce a class of interesting stochastic processes based on brownian time processes these are obtained by taking markov processes and replacing the time parameter with the modulus of brownian motion they generalize the iterated brownian motion ibm of burdzy and the markov snake of le gall and they introduce new interesting examples after defining brownian time processes we relate them to fourth order parabolic partial differential equations pde s we then study their exit problem as they exit nice domains in r d and connect it to elliptic pde s we show that these processes have the peculiar property that they solve fourth order parabolic pde s but their exit distribution at least in the standard brownian time process case solves the usual second order dirichlet problem we recover fourth order pde s in the elliptic setting by encoding the iterative nature of the brownian time process through its exit time in a standard brownian motion we also show that it is possible to assign a formal generator to these non markovian processes by giving such a generator in the half derivative sense

maximally decimated perfect reconstruction fir filter banks with pairwise mirror image analysis and synthesis frequency responses structures are presented for the perfect reconstruction quadrature mirror filter bank that are based on lossless building blocks these structures ensure that the frequency responses of the analysis and synthesis filters have pairwise symmetry with respect to pi 2 and require fewer parameters than recently reported structures also based on lossless building blocks the design time for the proposed structures is correspondingly much less than for the earlier methods which did not incorporate such symmetry

fundamentals of higher order stochastic equations stochastic processes that correspond to equations of arbitrary differential order are obtained we show that stochastic paths in a complex extension of the original phase space allow an implementation of such higher order derivative terms the resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments however the correspondence has unusual properties only the analytic moments are convergent while non analytic moments such as the complex moduli are non convergent these results unify previous approaches that transform higher derivative equations into probabilistic stochastic equations larger ensembles are required as time steps are reduced giving these equations unusual convergence properties this type of process is relevant to the question of how to obtain stochastic quantum simulations using the wigner representation

vibrations and fractional vibrations of rods plates and fresnel pseudo processes different initial and boundary value problems for the equation of vibrations of rods also called fresnel equation are solved by exploiting the connection with brownian motion and the heat equation the equation of vibrations of plates is considered and the case of circular vibrating disks cr is investigated by applying the methods of planar orthogonally reflecting brownian motion within cr the analysis of the fractional version of order of the fresnel equation is also performed and in detail some specific cases like 1 2 1 3 2 3 are analyzed by means of the fundamental solution of the fresnel equation a pseudo process f t t 0 with real sign varying density is constructed and some of its properties examined the composition of f with reflecting brownian motion b yields the law of biquadratic heat equation while the composition of f with the first passage time tt of b produces a genuine probability law strictly connected with the cauchy process

statistical mechanics of nonlinear hydrodynamic fluctuations abstract the paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics the generalized fokker planck equation for the distribution function of coarse grained densities of conserved quantities is derived from the liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix we have obtained the functional differential fokker planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse grained fluctuating fields an outline of the derivation of fokker planck equations containing the burnett terms is also given the explicit coordinate representation for the hydrodynamic fokker planck equation is discussed in the case of one component simple fluid the general scheme of a change of coarse grained functional variables is developed for hydrodynamic fokker planck equations the corresponding transformation rules are found for drift terms diffusion coefficients and thermodynamic forces the dynamical equations and stationary conditions for averages of functions functionals of hydrodynamic fields are discussed by using the fokker planck operators acting on such functions the explicit form of these operators are found for various sets of fluctuating fields as an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state

asymptotic properties of some space time fractional stochastic equations consider non linear time fractional stochastic heat type equations of the following type begin aligned partial beta tu t x nu delta alpha 2 u t x i 1 beta t lambda sigma u mathop f limits cdot t x end aligned in d 1 dimensions where nu 0 beta in 0 1 alpha in 0 2 the operator partial beta t is the caputo fractional derivative while delta alpha 2 is the generator of an isotropic stable process and i 1 beta t is the riesz fractional integral operator the forcing noise denoted by mathop f limits cdot t x is a gaussian noise and the multiplicative non linearity sigma mathbb r rightarrow mathbb r is assumed to be globally lipschitz continuous mijena and nane stochastic process appl 125 9 3301 3326 2015 have introduced these time fractional spdes these types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory under suitable conditions on the initial function we study the asymptotic behaviour of the solution with respect to time and the parameter lambda in particular our results are significant extensions of those in ann probab to appear foondun and khoshnevisan electron j probab 14 21 548 568 2009 mijena and nane 2015 and mijena and nane potential anal 44 295 312 2016 along the way we prove a number of interesting properties about the deterministic counterpart of the equation

brief on aizerman myshkis problem for systems with delay absolute stability conditions for nonlinear retarded systems are derived it is proved that systems with positive green s functions impulse response satisfy the generalized aizerman conjecture

fractional diffusion equations and processes with randomly varying time in this paper the solutions u u x t to fractional diffusion equations of order 0 v 2 are analyzed and interpreted as densities of the composition of various types of stochastic processes for the fractional equations of order v 1 2 n n 1 we show that the solutions u 1 2 n correspond to the distribution of the n times iterated brownian motion for these processes the distributions of the maximum and of the sojourn time are explicitly given the case of fractional equations of order v 2 3 n n 1 is also investigated and related to brownian motion and processes with densities expressed in terms of airy functions in the general case we show that u coincides with the distribution of brownian motion with random time or of different processes with a brownian time the interplay between the solutions u and stable distributions is also explored interesting cases involving the bilateral exponential distribution are obtained in the limit

on necessary and sufficient conditions for perfect reconstruction multidimensional delay chain systems we present some necessary and sufficient conditions for a perfect reconstruction pr modified multidimensional md delay chain system with these results one is able to systematically check if a d dimensional delay chain system with a d spl times d sampling matrix m and a d spl times d delay matrix l has a pr property in a simpler way than before where the matrix module operations are avoided moreover given a d spl times d sampling matrix m in many cases one can determine all possible d spl times d delay matrices l so that the delay chain systems with the sampling matrix m have a pr property several examples are provided we also present a method to generate d spl times d sampling and delay matrices m and l such that their corresponding traditional md delay chain systems are pr

robust stability criteria for dynamical systems including delayed perturbations considers the problem of robust stability of uncertain time delay dynamical systems a new robust stability criteria for linear dynamical systems subject to delayed time varying and nonlinear perturbations is derived the results obtained in this note are less conservative than the ones reported so far in the literature some analytical methods are employed to investigate the bound on the perturbations so that the systems are stable a numerical example is given to demonstrate the utilization of the authors results

scalar criteria for exponential stability of functional differential equations abstract using a novel approach we present some scalar criteria for the exponential stability of general functional differential equations both delay independent and delay dependent stability criteria are provided a discussion of the obtained results is given

robust stability of interval time delay systems with delay dependence abstract this paper proposes a sufficient robust stability condition for interval time delay systems with delay dependence the properties of the comparison theorem and matrix measure are employed to investigate the problem the stability criteria are delay dependent and less conservative than delay independent stability criteria 5 6 12 16 and delay dependent stability criteria 1 14 15 17 when delay is small however the results of this paper indeed give us one more choice for the stability examination of the interval time delay systems simulation examples are given to demonstrate the application of our result

dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part abstract systems with periodically nonstationary linear part are considered inputs and outputs of nonlinearities satisfy a certain integral quadratic constraint the conditions necessary and sufficient of dichotomy and absolute stability are obtained

finding commutative multidimensional downsamplers and upsamplers the commutativity of multidimensional downsamplers and upsamplers has been discussed very intensively for the past few years this is due to its importance in sampling structure conversion e g the conversion between conventional television signals and high definition television hdtv signals among many other results a test for such commutativity was found to be that the two matrices that define the multidimensional downsampling and upsampling should be commutative and coprime however the problem of finding multidimensional downsamplers and upsamplers that satisfy these conditions has remained open we develop a systematic procedure to solve this open problem

a new approach to the lur e problem in the theory of absolute stability the classical lur e problem consists of finding conditions for absolute stability of a linear system with a nonlinear feedback contained within a prescribed sector most of the results obtained on this problem are based on the frequency domain or lyapunov functions methods which are applied to systems with a time invariant or periodic linear block this paper develops a new approach providing a sufficient stability criterion for systems with time variable coefficients which is expressed in the transfer function of the linear block and the sector margins of the nonlinear block the systems for which this criterion is precise are found it is shown that stability of a system with a sign constant transfer function is guaranteed by stability of the system with a limit linear feedback so that for such systems the famous aizerman conjecture is true this in particular is the case for systems with a linear block consisting of an arbitrary number of first order time dependent links as an example the stability criterion is applied to a second order system for which the obtained results are contrasted with ones delivered by the popov criterion

levy mixing related to distributed order calculus subordinators and slow diffusions abstract the study of distributed order calculus usually concerns fractional derivatives of the form 0 1 u m d for some measure m eventually a probability measure in this paper an approach based on levy mixing is proposed non decreasing levy processes associated with levy triplets of the form a y b y d s y are considered and the parameter y is randomized by means of a probability measure the related subordinators are studied from different points of view some distributional properties are obtained and the interplay with inverse local times of markov processes is explored distributed order integro differential operators are introduced and adopted in order to write explicitly the governing equations of such processes an application to slow diffusions delayed brownian motion is discussed

l kuramoto sivashinsky spdes in one to three dimensions l ks kernel sharp holder regularity and swift hohenberg law equivalence generalizing the l kuramoto sivashinsky l ks kernel from our earlier work we give a novel explicit kernel formulation useful for a large class of fourth order deterministic stochastic linear and nonlinear pdes in multispatial dimensions these include pattern formation equations like the swift hohenberg and many other prominent and new pdes we first establish existence uniqueness and sharp dimension dependent spatio temporal holder regularity for the canonical zero drift l ks spde driven by white noise on r rd d 13 the spatio temporal holder exponents are exactly the same as the striking ones we proved for our recently introduced brownian time brownian motion btbm stochastic integral equation associated with time fractional pdes the challenge here is that unlike the positive btbm density the l ks kernel is the gaussian average of a modified highly oscillatory and complex schrodinger propagator we use a combination of harmonic and delicate analysis to get the necessary estimates second attaching order parameters e1 to the l ks spatial operator and e2 to the noise term we show that the dimension dependent critical ratio e2 e1d 8 controls the limiting behavior of the l ks spde as e1 e2 0 and we compare this behavior to that of the less regular second order heat spdes finally we give a change of measure equivalence between the canonical l ks spde and nonlinear l ks spdes in particular we prove uniqueness in law for the swift hohenberg and the law equivalence and hence the same holder regularity of the swift hohenberg spde and the canonical l ks spde on compacts in one to three dimensions

block sampling rate conversion systems using real valued fast cyclic convolution algorithms the real valued transform is related to the polyphase decomposition of a sequence this observation is applied for deriving sampling rate conversion systems that are implemented by the real valued fast cyclic convolution algorithms the systems include interpolation by an integer factor decimation by an integer factor and sampling rate conversion by a rational factor the proposed implementations are useful when the signals and impulse responses of filters are restricted to be real

the processing of periodically sampled multidimensional signals this paper discusses algorithms for processing multidimensional signals which are sampled on regular but nonrectangular sampiing lattices such sampling lattices are dictated by some applications and may be chosen for others because of their resulting symmetric responses or computational efficiencies we show that any operation which can be performed on a rectangular lattice can be performed on any regular periodic lattice including fir and iir filtering dft calculation and decimation and interpolation this paper also discusses how generalized decimators and interpolators can be used to convert from one sampling lattice to another

a straightforward memory function calculation with the non equilibrium statistical operator method abstract a straightforward function calculation in the non equilibrium statistical operator nso method is suggested as an example the relaxation of a two level system to a thermostat is considered

the commutativity of up downsampling in two dimensions the authors state and prove a theorem solving the problem of commutativity in two dimensions it is shown under which conditions upsampling and downsampling can be interchanged in two dimensions this is the generalization to arbitrary two dimensional lattices of the result that one dimensional upsampling and downsampling commute if and only if their sampling rates are coprime some illustrative examples are given the result holds for arbitrary sampling lattices

strang type preconditioners for solving fractional diffusion equations by boundary value methods the finite difference scheme with the shifted grunwald formula is employed to semi discrete the fractional diffusion equations this spatial discretization can reduce to the large system of ordinary differential equations odes with initial values recently the boundary value method bvm was developed as a popular algorithm for solving the large systems of odes this method requires the solutions of one or more nonsymmetric and large scale linear systems in this paper the gmres method with the block circulant preconditioner is proposed to solve relevant linear systems some conclusions about the convergence analysis and spectrum of the preconditioned matrices are also drawn if the diffusion coefficients are constant finally extensive numerical experiments are reported to show the performance of our method for solving the fractional diffusion equations

generalized diffusion equation with fractional derivatives within renyi statistics by using the zubarev nonequilibrium statistical operator method and the liouville equation with fractional derivatives a generalized diffusion equation with fractional derivatives is obtained within the renyi statistics averaging in generalized diffusion coefficient is performed with a power distribution with the renyi parameter q

fast algorithms for the multidimensional discrete fourier transform in this paper the prime factor algorithm for the evaluation of a one dimensional discrete fourier transform is generalized to the evaluation of multidimensional discrete fourier transforms defined on arbitrary periodic sampling lattices it is shown that such an algorithm is equivalent in computational complexity to the evaluation of a rectangular discrete fourier transform as a sidelight to the derivation of the algorithm a chinese remainder theorem is derived for integer lattices

multirate filter banks with block sampling multirate filter banks with block sampling were recently studied by khansari and leon garcia 1993 in this paper we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices respectively we show that multirate filter banks with block sampling studied by khansari and leon garcia are special vector filter banks where the transfer matrices are pseudocirculant we present some fundamental properties for the basic building blocks such as noble identities interchangeability of down up sampling polyphase representations of m channel vector filter banks and multirate filter banks with block sampling we then present necessary and sufficient conditions for the alias free property finite impulse response fir systems with fir inverses paraunitariness and lattice structures for paraunitary vector filter banks we also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling as an application of this theory we present all possible perfect reconstruction delay chain systems with block sampling we also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling in this paper we also present a connection between vector filter banks and vector transforms studied by li vector filter banks also play important roles in multiwavelet transforms and vector subband coding

solution estimates for abstract nonlinear time variant differential delay equations abstract semilinear time variant functional differential equations in a banach space are considered a solution estimate is derived by that estimate explicit conditions for boundedness of solutions and stability are established

sampling rate conversion systems using a new generalized form of the discrete fourier transform a recursive factorization of the polynomial 1 z sup n leads to an efficient algorithm for the computation of the discrete fourier transform dft and the cyclic convolution the paper introduces a new recursive polynomial factorization of the polynomial when n is highly composite the factorization is used to define a generalized form of the dft and to derive an efficient algorithm for the computation the generalized form of the dft is shown to be closely related to the polyphase decomposition of a sequence and is applied for the design of sampling rate conversion systems it gives not only alternative derivations for the polyphase interpolation and the polyphase decimation by an integer factor but also a new sampling rate conversion system by a rational factor which is more efficient than the known rational polyphase implementation when the filter length is large

recognition of nonextensive statistical distributions by the eigencoordinates method nonextensive statistical distributions suggested by tsallis are studied in the framework of the so called eigencoordinates ecs method which allows to recognize them with high level of authenticity at first the possibilities of the ecs method have been demonstrated on model files to exhibit some peculiarities of this new method then the amplitude distributions of the real noise tracks before and after earthquakes have been analyzed it has been shown that this type of noise is fractal and strongly correlated histograms of the noise amplitudes with high level of reliability are really described by a distribution coming from nonextensive statistics the possibilities and recommendations for the application of the ecs method to identify theoretical curves claimed for the description of experimental data are discussed

a general approach to coprime pairs of matrices based on minors in signal processing coprime pairs of integer matrices have an important role in the study of multidimensional multirate systems and in multidimensional array processing in this paper a general theorem for coprimality of pairs of matrices is first presented based on the minors of an associated composite matrix first a necessary and sufficient set of conditions for coprimality is presented based on minors this result applies to matrices of any size several useful consequences of the result are discussed and it is first applied to the 2 2 case including special cases such as toeplitz matrices commuting toeplitz matrices circulants and skew circulants the condition under which a 2 2 integer matrix and its ajdugate are coprime is also derived the minor based result is then applied extensively for the 3 3 case to derive new coprimality conditions for several matrix pairs in particular adjugates of matrices are considered in detail because adjugates always commute with the parent matrices the so called generalized 3 3 circulant matrix and its adjugate are then analyzed in detail generalized circulants also form a commuting family special cases such as 3 3 circulant pairs and skew circulant pairs are also elaborated

fractional integral associated to generalized cookie cutter set and its physical interpretation this paper is based on nigmatullin s study when the residual memory set is a generalized cookie cutter set on using various hypotheses it is proved that the fractional exponent of a fractional integral is not uniquely determined by the fractal dimension of the generalized cookie cutter set it is determined by of self similar measure or infinite self similar measure on this generalized cookie cutter set and can run over all positive real numbers

imposing structure on smith form decompositions of rational resampling matrices imposing structure on the smith form of an integer periodicity matrix n u spl lambda v leads to efficient multidimensional m d dft implementations guessoum 1984 for nonsingular integer and rational matrices the authors introduce algorithms to generate u or v matrices with minimum norm and spl lambda matrices whose diagonal elements exhibit minimum variance such structure simplifies nonuniform m d filter bank design gardos et al 1992

sharp bounds for lyapunov exponents and stability conditions for uncertain systems with delays stability of a set of systems with norm bounded nonlinear terms and arbitrary time varying as well as distributed delays is studied a novel approach to this problem based on deriving bounds for the norms of system solutions is developed a sharp estimate for the maximal lyapunov exponent of the solutions expressed in the bounds for the uncertain parameters is found the subsystems for which the obtained estimate is attained are indicated using these results a delay independent necessary and sufficient stability condition for the considered set of systems is derived for a system with prescribed parameters sufficient conditions for exponential stability and upper bound for the maximal lyapunov exponent are obtained the proposed approach is applied to illustrative examples which contrast its efficiency

quantitative measures of robustness for systems including delayed perturbations asymptotically stable linear systems subject to delayed time varying and nonlinear perturbations are considered razumikhin type theorems are used to obtain easy to compute bounds on the perturbations so that the systems remain stable results indicate that if delayed perturbations are included then the bound is reduced as compared to the one for nondelayed perturbations however in certain cases previously obtained bounds for the nondelayed perturbations guarantee stability even when delayed perturbations are in effect

damping characteristics of a fractional oscillator abstract a study of sinusoidally forced oscillations of a fractional oscillator shows that the system exhibits a rich variety of damping characteristics while some aspects of the damping mimic the characteristic features of a damped harmonic oscillator there are others which do not find any parallel in the damped harmonic oscillator system it is clearly demonstrated that the free and forced oscillations of a fractional oscillator are characterized by different damping parameters while both depend on the fractional index the free oscillation damping depends on the natural frequency 0 of the oscillator the forced oscillation damping depends in addition on the driving frequency furthermore there is a different power law tail associated with each of these cases

theory of sparse coprime sensing in multiple dimensions coprime sampling and coprime sensor arrays have been introduced recently for the one dimensional 1 d case and applications in beamforming and direction finding discussed a pair of coprime arrays can be used to sample a wide sense stationary signal sparsely and then reconstruct the autocorrelation at a significantly denser set of points all applications based on autocorrelation e g spectrum and doa estimation benefit from this property it was also shown in the past that coprimality can be exploited in the frequency domain by using a pair of coprime dft filter banks to produce the effect of a much denser tiling in the frequency domain compared to what the two filter banks can individually achieve this paper extends these ideas to multiple dimensions in the 1 d case the samples or sensors lie on a pair of uniform grids whereas in the multidimensional case they lie on a pair of multidimensional lattices not necessarily rectangular this makes the developments mathematically more intricate first several properties of coarrays of lattices are derived it is shown how one can get dense coarrays from sparse arrays on non rectangular lattices this requires that the lattice generating matrices m and n be commuting and coprime to be defined multidimensional dft filter banks for applications such as beamforming with commuting coprime lattice arrays are then described and it is shown that a very dense tiling of the frequency plane can be obtained from the two sparse lattice arrays a particular family of commuting coprime matrices called adjugate pairs are considered in some detail and shown to have attractive properties a brief review of the 1 d case is included at the beginning for convenience

multidimensional multirate filters and filter banks derived from one dimensional filters a method by which every multidimensional m d filter with an arbitrary parallelepiped shaped passband support can be designed and implemented efficiently is presented it is shown that all such filters can be designed starting from an appropriate one dimensional prototype filter and performing a simple transformation with d denoting the number of dimensions the complexity of design and implementation of the m d filter are reduced from o n sup d to o n using the polyphase technique an implementation with complexity of only 2n is obtained in the two dimensional even though the filters designed are in general nonseparable they have separable polyphase components one special application of this method is in m d multirate signal processing where filters with parallelepiped shaped passbands are used in decimation interpolation and filter banks some generalizations and other applications of this approach including m d uniform discrete fourier transform dft quadrature mirror filter banks that achieve perfect reconstruction are studied several design example are given

fractional kinetics in kac zwanzig heat bath models we study a variant of the kac zwanzig model of a particle in a heat bath the heat bath consists of n particles which interact with a distinguished particle via springs and have random initial data as n the trajectories of the distinguished particle weakly converge to the solution of a stochastic integro differential equation a generalized langevin equation gle with power law memory kernel and driven by 1 f noise the limiting process exhibits fractional sub diffusive behaviour we further consider the approximation of non markovian processes by higher dimensional markovian processes via the introduction of auxiliary variables and use this method to approximate the limiting gle in contrast we show the inadequacy of a so called fractional fokker planck equation in the present context all results are supported by direct numerical experiments

the role of integer matrices in multidimensional multirate systems various theoretical issues in multidimensional m d multirate signal processing are formulated and solved in the problems considered the decimation matrix and the expansion matrix are nondiagonal so that extensions of 1 d results are nontrivial the m d polyphase implementation technique for rational sampling rate alterations the perfect reconstruction properties for the m d delay chain systems and the periodicity matrices of decimated m d signals both deterministic and statistical are treated the discussions are based on several key properties of integer matrices including greatest common divisors and least common multiples these properties are reviewed