strong laws and limit theorems for local time of markov processes the local time of a markov process can be constructed from the length of small excursions we obtain the exact almost sure rate of this construction we also prove a functional limit theorem for the difference between the length of small excursions and the local time at zero

image representation by one bit fourier phase theory sampling and coherent image model the problem of recovering an image from its fourier transform phase quantized to one bit or equivalently finding the locations of the zero crossings of the real part of the fourier transform is addressed theoretical results are presented that set an algebraic condition under which real zero crossings uniquely specify a band limited image it is then shown that sampling in the frequency domain presents a major obstacle to obtaining good reconstruction results the one bit fourier phase reconstruction problem is then considered when the original image is coherent i e the image phase is random and highly uncorrelated examples are given which demonstrate that the information loss produced by frequency sampling is not as severe as that in the nonclassical problem motivated by digital phase only holograms a known random diffuser is used as the image phase and a well known iterative reconstruction procedure is extended to incorporate the knowledge of the image phase at each stage of the iteration this reconstruction method produces good image quality by using a few iterations unlike its noncoherent counterpart

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

direct phase retrieval a direct noniterative approach to retrieving a multidimensional complex image i e its phase can vary from pixel to pixel from the magnitude of its fourier transform is developed the uniqueness of the reconstruction is shown to be a direct consequence of the existence of zero surfaces or sheets in the multidimensional z transforms of the image the analytic properties of these zero sheets enable one to distinguish the zero sheet belonging to the image from that belonging to the complex conjugate of its reflection in the coordinate origin it is thereby possible to recover the unique most compact image form of the original image two dimensional examples are presented

matrix cyclization over complex polynomials abstract let a be an n n b an n m complex polynomial matrix the following open problem is investigated does there exist a complex polynomial m n matrix f such that a bf has a cyclic vector in the image of b an explicit solution of the problem is given for the following generic situation n 4 b ab a n 1 b is rightinvertible necessary the entries of a specific normalized form of a b do not satisfy certain polynomial equations and one specific entry has at least one simple zero outside a certain finite set the exceptional equations are given explicitely and can easily be checked for

how big are the lag increments of a gaussian process abstract in this paper the limit theorems on lag increments of a wiener process due to chen kong and lin 1 are developed to the case of a gaussian process via estimating upper bounds of large deviation probabilities on suprema of the gaussian process

cyclic accessibility of reachable states characterizes von neumann regular rings abstract the class of commutative von neumann regular rings is characterized by a generalization of the feedback cyclization property to non necessarily reachable systems for any system a b there exist a matrix k and a vector u such that a b and the single input system a bk bu have the same submodule of reachable states an explicit algorithm is presented to obtain k u for a given system a b

invariance principles for renewal processes on presente une methodologie generale pour demontrer des principes d invariance pour des processus de renouvellement on obtient des approximations presque sures et d inegalites de probabilites

relaxation procedure for phase retrieval of nonnegative signals the paper presents an iterative procedure called the relaxation of autocorrelation equations rae for solving the phase retrieval problem for nonnegative signals first the phase retrieval problem is formulated in the spatial domain as a set of polynomial equations with autocorrelations as known data and signal values as unknowns then the rae procedure solves these equations by recognizing one unknown at a time while other unknowns are held constant at previously estimated values a single unknown is varied inside the nonnegative region to globally minimize the sum of squared residuals of the equations with respect to the unknown in every iteration this procedure is repeated for each signal value since the sum of squared residuals is nonincreasing the algorithm will either converge to a solution or stagnate ways to overcome stagnation are suggested the key feature of the rae procedure is that unlike iterative transform algorithms it allows direct control over bounding values of the signal at all times several numerical examples illustrate the rae procedure

a b cyclic submodules abstract an a b cyclic submodule m is generated by the states of one single trajectory of a linear control system whose parameters come from a commutative ring m is finite when it is generated by the states of a deadbeat control process motivations and basic properties of such modules are given and among several further results it is shown that the family of finite a b cyclic submodules is an invariant which e g over polynomials can be determined by an appropriate grobner basis computation

some limit results for lag sums of independent non i i d random variables let x k be an independent but not necessarily i i d sequence of random variables suppose ex k 0 and that nk 2 e s n s n k 2 where s n x1 xn we obtain conditions under which mathop lim sup limits n to infty mathop max limits n alpha leqq k leqq n frac s n s n k sigma n k 2 log n k log log k 1 2 1 text a text s text and obtain other related results we assume either that p x k t mt r for all k and all t 0 or that e expx k b for all k and all t0 the results are like earlier results for the i i d case but the method of proof is entirely different

image representation by sign information image representation by sign data in its most general context is considered for the case when f is a two dimensional signal the authors study conditions under which f is determined by its sign or in other words by its quantization to 1 bit of information this study is carried out in two different directions first theoretical results are presented that set an algebraic condition under which real zero crossings uniquely specify a band limited image an interesting paradigm arising in theoretical computer vision is then posed are the zero crossing of f convolved with a laplacian of a gaussian at a single channel enough for unambiguously representing f second the problem of the completeness of the representation when the position of the zero crossings is known only approximately is addressed it is shown that when sign f is sampled significant ambiguities are introduced in the representation experimental results are presented which were obtained from an iterative algorithm devised to reconstruct real images from multiscale sign information

a new direct algorithm for image reconstruction from fourier transform magnitude in this paper we present a new direct solution to the problem of reconstructing a two dimensional discrete signal of finite support from knowledge of only its fourier transform magnitude and support using the autocorrelation function of the unknown signal zeros are calculated of a polynomial whose coefficients correspond to the unknown image or the image reversed from these zeros a set of linear equations is developed whose solution yields either the original image or the image rotated by 180

sojourns and future infima of planar brownian motion let w t 0 t 1 be a two dimensional wiener process starting from 0 we are interested in the almost sure asymptotic behaviour asr tends to 0 of the processesx r andy r wherex r denotes the total time spent byw in the ball centered at 0 with radiusr andy r the distance between 0 and the curve w t r t 1 while a characterization of the lower functions ofy was previously established by spitzer s we characterize via integral tests its upper functions as well as the upper and lower functions ofx

novel criteria of uniqueness for signal reconstruction from phase an approach for ascertaining whether a signal is uniquely determined by its fourier transform phase is proposed it is shown that uniqueness corresponds to the nonsingularity of a matrix which can be formed from the finite length real sequence the criterion of uniqueness for reconstructing a one dimensional finite length sequence from its phase is discussed the criterion of uniqueness for reconstructing a multidimensional finite length sequence is presented

a note on iterative fourier transform phase reconstruction from magnitude in this correspondence a well known iterative procedure for fourier transform phase reconstruction from magnitude usually referred to as the gerchberg saxton fienup procedure is revisited the convergence of this algorithm has raised some controversy in the recent literature in this correspondence we first point out a potential source of error in the numerical implementation of the algorithm then we present a conjecture which would explain why this algorithm sometimes fails

some results on increments of the wiener process with applications to lag sums of i i d random variables obtained for the wiener process but with n replacing t and sn replacing w t using the work of koml6s major and tusnady on the invariance principle we obtain the desired results for i i d sequences as immediate corollaries to our work for the wiener process 1 introduction the motivation for the work done here came from a specific statistical problem suppose xk is a sequence of random variables whose distribution functions approach in some appropriate sense a fixed distribution with mean it if the sequence is independent or is appropriately dependent then under mild regularity conditions sn n t almost surely there will usually be bias or non random error associated with the earlier xk s one might hope to reduce this bias by throwing away

some more results on increments of the wiener process let w t for 0 leq t infty be a standard weiner process and suppose that c k and b k are fixed sequences of real numbers satisfying 0 leq c k b k infty let k omega be the set of limit points as t rightarrow infty of frac w b k omega w c k omega 2 b k c k lbrack log b k b k c k log log b k rbrack 1 2 where omega is a point in the probability space on which w t is defined we give necessary conditions on b k and c k to have k omega lbrack 1 1 rbrack a s we also give some related results and discuss sharpness

on the future infima of some transient processes let x t t s be a real valued stochastic process with x 0 0 1 and mathbb p mathop lim limits t to infty x t infty 1 in this paper we are interested in the reluctance of such a process to tend to infinity this entails determining the rate of escape of the associated process mathop inf limits s underset raise0 3em hbox smash scriptscriptstyle geqslant t x s t in s the so calledfuture infima process

strong feedback cyclization for systems over rings abstract for nonnecessarily reachable systems over a commutative ring r we define a strong form of feedback cyclization fc with this natural generalization of the fc property we obtain a feedback reduced form for systems over strong fc rings i e rings for which every system verifies the fc property in the particular case of reachable single input systems this gives the usual control canonical form of bumby et al remarks on the pole shifting problem over rings j pure appl algebra 20 1981 113 127 also it is proved that a commutative ring with 1 in its stable range has the strong fc property if and only if it has the ucu property which is the natural parallel form of the result given by brewer et al pole assignability in polynomial rings power series rings and prufer domains j algebra 106 1987 265 286 for the reachable case many classes of rings which were known to be fc rings are in fact strong fc rings but there are fc rings which are not strong fc rings

strong approximations of open queueing networks we consider a queueing system with one or more stations we show that the queue length processes can be approximated with brownian motions or reflected brownian motions using the approximations we obtain rates of convergence of the distributions and expected values of queue lengths

on the increments of the wiener process let w t t 0 be a standard wiener process and 0 a t t a nondecreasing function of t the asymptotic behaviour of several increment processes obtained from w and a t is investigated in terms of their upper classes in some cases we characterize these classes by means of an integral test two such processes are w t a t w t a t 1 2 and mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbf9f4rqqrpu0dxde9lqfhe9lq pg0di9veea0 yr0ryxir jbba9q8aq0 yq he9q8qqi frfve9fve9 ff0dmeaabaqacigacagaaeqabawaaeaaeaaakeaadawfqaqaaigaco hacagg1bgaaiicaawcbaacbagaa83bamrr1ngbprwthrhayaqeguud jxwakbsthrhagq1dvbacfagae4nziyoae4ha3thcbigaa0hdaiaa b cacqgfmjihcqgfgaaiiewacaafubaabeaakiaa bcadawfqaqaaiga cohacagg1bgaaiicaawcbagaa83baiaa bcatcvaufkttlearcwql5 gd1jxegm1yo92bsj2bvtgaiyaacqwe8aapcqgfacpdcaqfzbgaa8hi aiab zmigkab bcagiaa9fgadawgaaadbagaa8hdaaqabawccawfga gaey4jiktaa8hiaiaa hcacaqfubqeduudjxwakbyu51myvxgaicaa cqgesislcqgegaaicaqf0bgaa8xkaaqabagccawfgagaa8hkaiaa9d facawfoagaa0hdaiaa9tcacaqfzbgaa8xkaiaa bcacqgesislcqge gaaicaqfxbgaa8hkaiaa9rhacawfpagaa8xkaiaa9fgadaqhaawcba gaa0hdaaqaaiab6jhitiab6fdaxiab69caviab6jdayaaaaaa 83f2 mathop sup limits o leqq t leqq t mathop sup limits o s leqq a t wedge t t w t s w t a t 1 2

on stability of reconstruction from fourier transform modulus the lower bound on the condition number of reconstruction of two classes of sequences from their fourier transform magnitude ftm is derived the lower bound for one class is shown to be 1 and for the other class to be 1 2 n sup q 2 where q is the dimensionality of the sequence and n is the number of nonzero elements in each dimension stability of reconstruction from the ftm and the space domain phase is discussed it is found that randomizing the space domain phase improves reconstruction robustness experimental results are presented to verify theoretical predictions

partitioning algorithms for 1 d and 2 d discrete phase retrieval problems with disconnected support the discrete phase retrieval problem with disconnected support is to reconstruct a discrete time signal whose support is the union of disjoint intervals from the magnitude of its discrete fourier transform we use the good thomas 1958 fft mapping to transform the one dimensional 1 d version of this problem into a two dimensional 2 d discrete phase retrieval problem we then solve the latter problem by partitioning it into a set of 1 d phase retrieval problems the discrete and modulated radon transforms are used to formulate two coupled 1 d problems the solution to which specifies solutions to the other 1 d problems this effectively partitions the original disconnected support 1 d problem into smaller 1 d problems which may be solved in parallel small amounts of noise in the data can be rejected by using more than two coupled 1 d problems the 2 d phase retrieval problem with disconnected support is also considered numerical examples are provided including comparison with the hybrid i o algorithm

one and two dimensional minimum and nonminimum phase retrieval by solving linear systems of equations the discrete phase retrieval problem is to reconstruct a discrete time signal whose support is known and compact from the magnitude of its discrete fourier transform we formulate the problem as a linear system of equations our methods do not require polynomial rooting tracking zero curves of algebraic functions or any sort of iteration like previous methods our solutions obviate the stagnation problems associated with iterative algorithms and our solutions are computationally simpler and more stable than alternative noniterative algorithms furthermore our methods can explicitly accommodate noisy fourier magnitude information through the use of total least squares type techniques we assume either of the following two types of a priori knowledge of the signal 1 a band of known values which may be zeros or 2 some known values of a subminimum phase signal whose zeros lie inside a disk of radius greater than unity we illustrate our methods with nonminimum phase one dimensional 1 d and two dimensional 2 d signals

a geometric approach to dynamic feedback cyclization over commutative rings a different geometric point of view is introduced in the study of dynamic feedback cyclization over rings this considerably simplifies the theory in the free case in the sense that the main results can now be obtained by simple proofs the non free case remains to be examined

rate of convergence of transport processes with an application to stochastic differential equations we obtain a rate of convergence of uniform transport processes to brownian motion which we apply to the wong and zakai approximation of stochastic integrals

computation of maximal reachability submodules abstract a new and conceptually simple procedure is derived for the computation of the maximal reachability submodule of a given submodule of the state space of a linear discrete time system over a noethenian ring r the procedure is effective if r is effective and if kernels and intersections can be computed the procedure is compared with a rather different procedure by assan e a published recently

multidimensional signal representation by zero crossings and algebraic study multidimensional signal representation by zero crossings arises in many problems in physics and engineering its applications span several areas such as signal modulation computer vision and data compression in this paper mathematical conditions under which signals are uniquely specified in terms of their real zero crossings are presented these results apply to multidimensional signals i e the dimension n geqq 2 modeled by polynomials this model is appropriate for the z transform of discrete signals and can also be used for certain two dimensional images the central theorems shown in this paper establish conditions under which a multivariate polynomial p is uniquely determined from its real zero crossings furthermore the regularity of a real zero of pwill be shown equivalent to the existence of zero crossings this property is interesting since it shows a deep relationship between the theory of polynomials and the representations used in practical applications finally this paper generalizes

dynamic feedback a b invariant submodules for linear systems over commutative noetherian domains abstract in this paper some basic characterizations of a b invariant submodules for linear systems over commutative noetherian domains are studied first a notion of dynamic feedback a b invariant submodules for such systems is introduced and then it is shown that this notion is equivalent to a b invariance

polynomial system of equations and its applications to the study of the effect of noise on multidimensional fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent signal from the magnitude of its fourier transform we will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued signal x from the magnitude of the output of a linear distortion hx j j 1 n an important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval conditioning of the problem and stability of the essentially unique solution will be addressed

strong invariance principles for partial sums of independent random vectors on obtient une estimation dans le theoreme central limite qui est utilisee avec le theoreme de strassen dudley pour demontrer une approximation forte pour des sommes partielles de vecteurs aleatoires i i d a d dimensions

three dimensional feedback cyclization over c y abstract this paper answers in the affirmative the following open question for 3 dimensional linear systems over the ring of complex polynomials in one variable if the system is reachable does there exist a feedback that makes it reachable from a single input channel the result given remains valid for slightly more general coefficient domains

feedback classification of linear systems over von neumann regular rings abstract it is proved that feedback classification of a linear system over a commutative von neumann regular ring r can be reduced to the classification of a finite family of systems each of which is properly split into a reachable and a non reachable part where the reachable part is in a brunovski type canonical form while the non reachable part can only be altered by similarity if a canonical form is known for similarity of matrices over r then it can be used to construct a canonical form for feedback equivalence an explicit algorithm is given to obtain the canonical form in a computable context together with an example over a finite ring

answers to some questions about increments of a wiener process on obtient des resultats qui donnent une reponse affirmative aux questions posees par hanson et russo sans hypothese additionnelle 1983

pole assignability in polynomial rings power series rings and prufer domains the topics treated in this paper have their origins in the area of algebraic systems theory however the paper itself should be classified as pure commutative algebra and we shall present it as such in the body of the text still it is appropriate to give a brief paragraph of motivation if a physical system is governed by a pair f g of matrices then the stability of the system can be determined by examining the eigenvalues of the matrix f if the system is unstable a feedback matrix k can sometimes be employed in such a way that the eigenvalues of the matrix f gk measure the stability of the modified system f gk g in this manner an unstable system can be rendered stable the pole assignability problem over commutative rings is one method of attacking the problem of finding such matrices k the paper is divided into four sections section 1 is given over almost entirely to defining the properties in which we shall be interested it concludes with a theorem about residuating and lifting the properties section 2 is concerned with the preservation of the properties under polynomial ring and power series ring formation section 3 is concerned with feedback cyclization a strong form of pole assignability section 4 is concerned with pole assignability over priifer domains a complete elaboration of our results must await the introduction of the necessary terminology for now we mention the following results in somewhat vague language if r is a zero dimensional ring then the pole assignability problem is solvable in r x and in r xl from a ringtheoretic standpoint almost any class of commutative rings contains members for which the feedback cyclization problem is solvable on the other hand if r is a ring with 1 in its stable range then the feedback cyclization problem is solvable in r if and only if a certain nice matricial property

signal reconstruction from fourier transform sign information in this paper we present new results on the reconstruction of signals from only the sign of the real part of the fourier transform specifically we develop new theoretical results which state conditions under which two dimensional signals are uniquely specified to within a scale factor with this information and show that these conditions include a broad class of signals furthermore we apply this result to the problem of reconstructing two dimensional signals from their zero crossings we also present two algorithms for reconstructing a signal from sign information in either the time or frequency domain