uniform characterizations of non uniform complexity measures non uniform complexity measures originated in automata and formal languages theory are characterized in terms of well known uniform complexity classes the initial index of languages is introduced by means of several computational models it is shown to be closely related to context free cost boolean circuits straight line programs and turing machines with sparse oracles and time or space bounds

double series representation of bounded signals series representations of the form f t approximately sigma sub n infinity sup infinity sigma sub k infinity sup infinity a sub n k nu t n sub e sup 2 pi kts for bounded signals f t are studied as are conditions on the unit function nu t such that coefficients a sub n k reveal the energy content of f t in the time interval n 1 2

functions analytic on the half plane as quantum mechanical states a transform between the state space of one dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half plane is constructed it gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple generalizations to higher dimensions are also discussed

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

on the hopcroft s minimization technique for dfa and dfca we show that the absolute worst case time complexity for hopcroft s minimization algorithm applied to unary languages is reached only for deterministic automata or cover automata following the structure of the de bruijn words a previous paper by berstel and carton gave the example of de bruijn words as a language that requires o nlogn steps in the case of deterministic automata by carefully choosing the splitting sets and processing these sets in a fifo mode for the list of the splitting sets in the algorithm we refine the previous result by showing that the berstel carton example is actually the absolute worst case time complexity in the case of unary languages for deterministic automata we show that the same result is valid also for the case of cover automata and an algorithm based on the hopcroft s method used for minimization of cover automata we also show that a lifo implementation for the splitting list will not achieve the same absolute worst time complexity for the case of unary languages both in the case of regular deterministic finite automata or in the case of the deterministic finite cover automata as defined by s yu

analytical footprints compact representation of elementary singularities in wavelet bases we introduce a family of elementary singularities that are point holder regular these singularities are self similar and are the green functions of fractional derivative operators i e by suitable fractional differentiation one retrieves a dirac function at the exact location of the singularity we propose to use fractional operator like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain we show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed form expression termed the analytical footprint practically this means that the dictionary of wavelet footprints is embodied in a single analytical form we show that the wavelet coefficients of the nonredundant decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity we propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations

linear systems in gabor time frequency space a new approach for the description and operation of linear time varying systems is presented the proposed formalism implements the superposition integral describing a general linear operation in gabor time frequency space enabling the use of its joint properties the method is based on transforming both the input and the time varying impulse response function into the gabor space the 2 and 4 d arrays corresponding to the gabor spectra of the input signal and the impulse response respectively are multiplied yielding a 2 d array corresponding to the output gabor spectrum from which the output time signal is subsequently reconstructed unlike other time varying operation methods carried out in different mixed time frequency spaces e g wigner space the suggested approach when employing the critically sampled gabor expansion generates a legal output spectrum and eliminates the need to resort to cumbersome and numerically taxing approximate synthesis techniques the proposed formalism can be implemented in the areas of time variant filtering speech scrambling and any other linear operations that vary their characterizations in time

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

the analysis and design of windowed fourier frame based multiple description source coding schemes in this paper the windowed fourier encoding decoding scheme applied to the multiple description compression problem is analyzed in the general case four window functions are needed to define the encoder and decoder although this number can be reduced to three or two by using time shift or frequency shift division schemes the encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index the distortion on each channel is analyzed using the zak transform for the optimal windows explicit representation formulas are obtained and nonlocalization results are proved asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two

a general description of linear time frequency transforms and formulation of a fast invertible transform that samples the continuous s transform spectrum nonredundantly examining the frequency content of signals is critical in many applications from neuroscience to astronomy many techniques have been proposed to accomplish this one of these the s transform provides simultaneous time and frequency information similar to the wavelet transform but uses sinusoidal basis functions to produce frequency and globally referenced phase measurements it has shown promise in many medical imaging applications but has high computational requirements this paper presents a general transform that describes fourier family transforms including the fourier short time fourier and s transforms a discrete nonredundant formulation of this transform as well as algorithms for calculating the forward and inverse transforms are also developed these utilize efficient sampling of the time frequency plane and have the same computational complexity as the fast fourier transform when configured appropriately this new algorithm samples the continuous s transform spectrum efficiently and nonredundantly allowing signals to be transformed in milliseconds rather than days as compared to the original s transform algorithm the new and efficient algorithms make practical many existing signal and image processing techniques both in biomedical and other applications

the quaternion formalism for mobius groups in four or fewer dimensions abstract the mobius group of r n defines n dimensional inversive geometry this geometry can serve as an alternative to projective geometry in providing a common foundation for spherical euclidean and hyperbolic geometry accordingly the mobius group plays an important role in geometry and topology the modern emphasis on low dimensional topology makes it timely to discuss a useful quaternion formalism for the mobius groups in four or fewer dimensions the present account is self contained it begins with the representation of quaternions by 2 x 2 matrices of complex numbers it discusses 2 x 2 matrices of quaternions and how a suitably normalized subgroup of these matrices extended by a certain involution related to sense reversal is 2 1 homomorphic to the mobius group acting on r 4 it provides details of this action and the relation of this action to various models of the classical geometries in higher dimensions n 5 the best description of the mobius group is probably by means of n 2 n 2 lorentz matrices in the lower dimensions covered by the quaternion formalism this alternative lorentz formalism is a source of interesting homomorphisms a sampling of these homomorphisms is computed explicitly both for intrinsic interest and for an illustration of the ease with which one can handle the quaternion formalism

improved impedance matrix localization method em problems moment method calculations have the well known limitations of requiring excessive storage and execution times for even modestly large electromagnetics problems the impedance matrix localization iml method was introduced as a modification to standard moment method calculations to ease these limitations it utilizes a matrix transformation which effectively changes the basis testing functions into ones resembling traveling waves an improved method that uses an orthogonal transformation to generate standing wave like basis functions is presented here remarkable improvements are achieved in the numerical stability of the method and in its compatibility with iterative solvers furthermore the correspondence of the large elements in this matrix to geometrical theory of diffraction gtd terms is strengthened as is the possibility of further increasing the speed of iterative solutions by constructing preconditioners based on the pattern of nonzero matrix elements

spiraling spectra of geodesic lines in negatively curved manifolds given a negatively curved geodesic metric space m we study the asymptotic penetration behaviour of geodesic lines of m in small neighbourhoods of closed geodesics and of other compact convex subsets of m we define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects we prove analogs of the theorems of dirichlet hall and cusick in this context as a consequence we obtain diophantine approximation results of elements of mathbb r mathbb c or the heisenberg group by quadratic irrational ones

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

frames for exact inversion of the rank order coder our goal is to revisit rank order coding by proposing an original exact decoding procedure for it rank order coding was proposed by thorpe who stated that the order in which the retina cells are activated encodes for the visual stimulus based on this idea the authors proposed in a rank order coder decoder associated to a retinal model though it appeared that the decoding procedure employed yields reconstruction errors that limit the model bit cost quality performances when used as an image codec the attempts made in the literature to overcome this issue are time consuming and alter the coding procedure or are lacking mathematical support and feasibility for standard size images here we solve this problem in an original fashion by using the frames theory where a frame of a vector space designates an extension for the notion of basis our contribution is twofold first we prove that the analyzing filter bank considered is a frame and then we define the corresponding dual frame that is necessary for the exact image reconstruction second to deal with the problem of memory overhead we design a recursive out of core blockwise algorithm for the computation of this dual frame our work provides a mathematical formalism for the retinal model under study and defines a simple and exact reverse transform for it with over than 265 db of increase in the peak signal to noise ratio quality compared to furthermore the framework presented here can be extended to several models of the visual cortical areas using redundant representations

image reconstruction from projections under wavelet constraints first the authors discuss how the wavelet transform can be used to perform spatially varying filtering of an image suppressing noise locally in smooth regions of the image and discuss detection of such regions in a noise corrupted image second they show how to compute the minimum mean square estimate of an image given 1 noisy projections of the image 2 statistics of additive noise in the projections and 3 constraints on wavelet coefficients of the image examples illustrate the resulting procedure

more on deterministic and nondeterministic finite cover automata finite languages are an important sub regular language family which were intensively studied during the last two decades in particular from a descriptional complexity perspective an important contribution to the theory of finite languages are the deterministic and the recently introduced nondeterministic finite cover automata dfcas and nfcas respectively as an alternative representation of finite languages by ordinary finite automata we compare these two types of cover automata from a descriptional complexity point of view showing that these devices have a lot in common with ordinary finite automata in particular we study how to adapt lower bound techniques for nondeterministic finite automata to nfcas such as e g the biclique edge cover technique solving an open problem from the literature moreover the trade off of conversions between dfcas and nfcas as well as between finite cover automata and ordinary finite automata are investigated finally we present some results on the average size of finite cover automata

quaternionic modular groups abstract matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n space or hyperbolic n 1 space h n 1 in particular groups operating in the hyperbolic plane or hyperbolic 3 space may be represented by 2 2 matrices whose entries are rational integers or real or imaginary quadratic integers the theory is extended here to groups operating in h 4 or h 5 and matrices over one of the three basic systems of quaternionic integers quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of h 4 and h 5 for four dimensional groups the division ring of quaternions is treated as a clifford algebra results in hyperbolic 5 space derive from the homeomorphism of inversive 4 space and the quaternionic projective line

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

minimal cover automata for finite languages abstract a cover automaton a of a finite language l is a finite deterministic automaton dfa that accepts all words in l and possibly other words that are longer than any word in l a minimal deterministic finite cover automaton dfca of a finite language l usually has a smaller size than a minimal dfa that accept l thus cover automata can be used to reduce the size of the representations of finite languages in practice in this paper we describe an efficient algorithm that for a given dfa accepting a finite language constructs a minimal deterministic finite cover automaton of the language we also give algorithms for the boolean operations on deterministic cover automata i e on the finite languages they represent

on the average state and transition complexity of finite languages we investigate the average case state and transition complexity of deterministic and nondeterministic finite automata when choosing a finite language of a certain size n uniformly at random from all finite languages of that particular size here size means that all words of the language are either of length n or of length at most n it is shown that almost all deterministic finite automata accepting finite languages over a binary input alphabet have state complexity q 2 nn while nondeterministic finite automata are shown to perform better namely the nondeterministic state complexity is in q 2 n interestingly in both cases the aforementioned bounds are asymptotically like in the worst case however the nondeterministic transition complexity is shown to be again q 2 nn the case of unary finite languages is also considered moreover we develop a framework that allows us to investigate the average case complexity of operations like e g union intersection complementation and reversal on finite languages in this setup

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

enhanced frequency analysis using wavelets an enhanced frequency analysis method is introduced using a phase shift wavelet representation to analyse individual frequency components and to reveal features of a given signal the frequency representation provides details at each time location without the loss caused by local orthogonality between the wavelet basis functions and the given signal which occurs when a normal wavelet decomposition is employed movement of the local orthogonality locations corresponding to the phase shift of the wavelet basis functions are studied and it is shown that these movements are important and can be used to reveal features of the signal in the frequency domain such as continuities smoothness and trends several examples are presented to illustrate the method and an application to a permanent magnet synchronous drive system is provided to demonstrate the approach

separating exponentially ambiguous finite automata from polynomially ambiguous finite automata we resolve an open problem raised by ravikumar and ibarra siam j comput 18 1989 pp 1263 1282 on the succinctness of representations relating to the types of ambiguity of finite automata we show that there exists a family of nondeterministic finite automata an over a two letter alphabet such that for any positive integer n an is exponentially ambiguous and has n states whereas the smallest equivalent deterministic finite automaton has 2n states and any smallest equivalent polynomially ambiguous finite automaton has 2n 1 states

incremental construction of minimal deterministic finite cover automata we present a fast incremental algorithm for constructing minimal deterministic finite cover automata dfca for a given language since it was shown that the minimal dfca for a language l has less states than the minimal deterministic finite automata dfa for the same language l this technique seems to be the best choice for incrementally building the automaton for a large language especially when the number of states in the dfca is significantly less than the number of states in the corresponding minimal dfa we have implemented the proposed algorithm and have tested it against the best known dfca minimization technique

tensor least angle regression for sparse representations of multidimensional signals sparse signal representations have gained much interest recently in both signal processing and statistical communities compared to orthogonal matching pursuit omp and basis pursuit which solve

error correction with frames the burt laplacian pyramid the study of multiresolution representations of images led to the development of the pyramid and wavelet representations the redundancy in the pyramid representation bestows on it the properties of an error correcting code the error correcting properties of the pyramids studied are a property of the representation itself and not a result of making any assumptions about the original images the article extends the error reduction properties of frames to an error correcting routine it leads to complete noise removal for sparse noise for a class of frames the results for images using laplacian pyramids show an additional 10 db improvement over the previously demonstrated noise reduction for frames

orientation preserving mobius transformations in r 4 and quaternionic determinants abstract this paper explores m r 4 the group of orientation preserving mobius transformations acting in r 4 on the one hand m r 4 is given by the group of 2 2 matrices over the quaternions h with determinant d derived from the corresponding 4 4 matrices over the complex numbers c on the other hand we know that in general m r n may be given in terms of 2 2 matrices over the clifford algebra c n with n 1 generators thus when n 4 m r 4 is given in terms of 2 2 matrices with entries drawn from c 4 and determinant defined in terms of the entries of the given matrix we note that the skew field h may be considered as a clifford algebra c 3 based on two generators i and j or more generally i 1 and i 2 where i j k or i 1 i 2 k while the set of elements 1 i j k form a basis of h regarded as a 4 dimensional real vector space thus h is embedded in c 4 in the paper we reconcile the two representations of m r 4 by comparing the generating sets of the underlying groups of matrices a relationship between a determinants d and is also exposed

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

a sufficient condition to polynomially compute a minimum separating dfa the computation of a minimal separating automaton msa for regular languages has been studied from many different points of view from synthesis of automata or grammatical inference to the minimization of incompletely specified machines or compositional verification in the general case the problem is np complete but this drawback does not prevent the problem from having a real application in the above mentioned fields in this paper we propose a sufficient condition that guarantees that the computation of the msa can be carried out with polynomial time complexity

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

on some identities and inequalities for frames in hilbert spaces we give new parseval type identities and inequalities for frames in hilbert spaces our results generalize the remarkable results obtained recently by r balan p g casazza d edidin and g kutyniok

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

minimal consistent dfa revisited we address the classic problem of polynomial computation of a minimal finite state representation compatible with two finite input sets that contain those strings that have to be accepted and rejected we extend the previously defined uniformly complete sample and prove that it is a special case of the sufficient condition we propose

conjugacy invariants of sl 2 h abstract in this paper we use the quaternionic formalism of mobius transformations on r 4 to derive conjugacy invariants on sl 2 h and hence on psl 2 h we then use these invariants to distinguish between the various conjugacy classes of psl 2 h

balian low type theorems in finite dimensions we formulate and prove finite dimensional analogs for the classical balian low theorem and for a quantitative balian low type theorem that in the case of the real line we obtained in a previous work moreover we show that these results imply their counter parts on the real line

on the compressibility of finite languages and formal proofs abstract we consider the minimal number of productions needed for a grammar to cover a finite language l as the grammatical complexity of l we study this measure for several types of word and tree grammars and show that it is closely connected to well known measures for the complexity of formal proofs in first order predicate logic we construct an incompressible sequence of finite word languages and transfer this and several other results about the complexity of word and tree languages to formal proofs

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

testing based on identifiable p systems using cover automata and x machines we represent a significant advance on the issue of testing for implementations specified by p systems with transformation and communicating rules using the x machine framework and the concept of cover automaton we devise a testing approach for such systems that under well defined conditions our approach ensures that the implementation conforms to the specification we investigate the issue of identifiability for p systems that is an essential prerequisite for testing these systems and establishes a fundamental set of properties for identifiable p systems this paper represents a significant advance on the issue of testing for implementations specified by p systems with transformation and communicating rules using the x machine framework and the concept of cover automaton it devises a testing approach for such systems that under well defined conditions it ensures that the implementation conforms to the specification it also investigates the issue of identifiability for p systems that is an essential prerequisite for testing implementations based on such specifications and establishes a fundamental set of properties for identifiable p systems

ondelettes and phase cell cluster expansions a vindication y meyer has recently developed a particularly useful o n basis forl2 r d expansions using this basis i e expansions into ondelettes or wavelets have yielded important new results in soft and hard analysis the expansion into ondelettes of a boson scalar field naturally leads to phase cell cluster expansions a formalism already developed by the authors using other related bases adoption of ondelettes expansions into the phase cell program gives improvements of some extant results and excises an early error

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

delambre gauss formulas for augmented right angled hexagons in hyperbolic 4 space abstract we study the geometry of right angled hexagons in the hyperbolic 4 space h 4 via clifford numbers or quaternions we show how to augment alternate sides of such a hexagon and arbitrarily orient each line and plane involved so that for the non augmented sides we can define quaternion half side lengths whose angular parts are obtained from half the euler angles associated to a certain orientation preserving isometry of the euclidean 3 space we also define appropriate complex half side lengths for the augmented sides of the augmented hexagon we further explain how to geometrically read off the quaternion half side lengths for a given oriented augmented right angled hexagon in h 4 our main result is a set of generalized delambre gauss formulas for oriented augmented right angled hexagons in h 4 involving the quaternion half side lengths and the complex half side lengths we also show in the appendix how the same method gives delambre gauss formulas for oriented right angled hexagons in h 3 from which the well known laws of sines and of cosines can be deduced these formulas generalize the classical delambre gauss formulas for spherical hyperbolic triangles

higher dimensional analogues of the modular and picard groups clifford algebras are used to describe arithmetic groups which are generalizations of the modular and picard groups

acceleration of the frame algorithm shows how polynomial acceleration techniques which have been developed for the solution of large linear systems can be employed to improve and accelerate the frame algorithm these methods permit a reduction in the number of necessary iterations by an order of magnitude when the frame algorithm is slow the author gives several examples from the theory of irregular sampling from wavelet theory and from gabor theory where these methods are probably mandatory for efficient reconstruction

an nlogn algorithm for hyper minimizing a minimized deterministic automaton we improve a recent result a badr hyper minimization in o n 2 internat j found comput sci 20 4 2009 735 746 for hyper minimized finite automata namely we present an o nlogn algorithm that computes for a given deterministic finite automaton dfa an almost equivalent dfa that is as small as possible such an automaton is called hyper minimal here two finite automata are almost equivalent if and only if the symmetric difference of their languages is finite in other words two almost equivalent automata disagree on acceptance on finitely many inputs in this way we solve an open problem stated in a badr v geffert i shipman hyper minimizing minimized deterministic finite state automata rairo theor inf appl 43 1 2009 69 94 and by badr moreover we show that minimization linearly reduces to hyper minimization which shows that the time bound o nlogn is optimal for hyper minimization independently similar results were obtained in p gawrychowski a jez hyper minimisation made efficient in proc 34th int symp mathematical foundations of computer science in lncs vol 5734 springer 2009 pp 356 368

frames in the bargmann space of entire functions we look at the decomposition of arbitrary f in l2 r in terms of the family of functions cp x w exp iimnab iman i x nb 2 with a b 0 we derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab 2w n n an integer 2 2 transported to the hilbert space f of entire functions introduced by v bargmann these results are expressed as inequalities of the form

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

a time complexity gap for two way probabilistic finite state automata it is shown that if a two way probabilistic finite state automaton 2pfa m recognizes a nonregular language l with error probability bounded below frac 1 2 then there is a positive constant b depending on m such that for infinitely many inputs x the expected running time of m on input x must exceed 2 n b where n is the length of x this complements a result of freivalds showing that 2pfa s can recognize certain nonregular languages in exponential expected time it also establishes a time complexity gap for 2pfa s since any regular language can be recognized by some 2pfa in linear time other results give roughly exponential upper and lower bounds on the worst case increase in the number of states when converting a polynomial time 2pfa to an equivalent two way nondeterministic finite state automaton or to an equivalent one way deterministic finite state automaton

pattern analysis with two dimensional spectral localisation applications of two dimensional s transforms abstract an image is a function f x y of the independent space variables x and y the global fourier spectrum of the image is a complex function f k x k y of the wave numbers k x and k y the global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of f x y leading to the concept of a local spectrum at every point of f x y the two dimensional s transform is introduced here as a method of computation of the local spectrum at every point of an image in addition to the variables x and y the 2 d s transform retains the variables k x and k y being a complex function of four variables visualisation of a function of four variables is difficult we skirt around this by removing one degree of freedom through examination of slices each slice of the 2 d s transform would then be a complex function of three variables with separate amplitude and phase components by ranging through judiciously chosen slice locations the entire s transform can be examined images with strictly periodic patterns are best analysed with a global fourier spectrum on the other hand the 2 d s transform would be more useful in spectral characterisation of aperiodic or random patterns

hierarchical active shape models using the wavelet transform active shape models asms are often limited by the inability of relatively few eigenvectors to capture the full range of biological shape variability this paper presents a method that overcomes this limitation by using a hierarchical formulation of active shape models using the wavelet transform the statistical properties of the wavelet transform of a deformable contour are analyzed via principal component analysis and used as priors in the contour s deformation some of these priors reflect relatively global shape characteristics of the object boundaries whereas some of them capture local and high frequency shape characteristics and thus serve as local smoothness constraints this formulation achieves two objectives first it is robust when only a limited number of training samples is available second by using local statistics as smoothness constraints it eliminates the need for adopting ad hoc physical models such as elasticity or other smoothness models which do not necessarily reflect true biological variability examples on magnetic resonance images of the corpus callosum and hand contours demonstrate that good and fully automated segmentations can be achieved even with as few as five training samples

commensurability classes of hyperbolic coxeter groups in this paper we classify all the hyperbolic coxeter n simplex reflection groups up to widecommensurability for all n 3 we also determine all the subgroup relationships among the groups 2002 elsevier science inc all rights reserved ams classification primary 20f55

generalized gabor expansions of discrete time signals in l sup 2 z via biorthogonal like sequences in this paper a biorthogonal like sequences bls theory and its application to the generalized gabor expansions equivalently the generalized short time fourier transform filterbank summation are presented a pair of bls s are defined to be two sequences satisfying a biorthogonal like condition blc which is a moment equation and equivalent to a linear difference equation we show that two collections in a hilbert space generated by a pair of bls s in the joint time frequency domain are complete either can be used as an analysis filter and the other can be used as a synthesis filter for a generalized gabor expansion of discrete time signals a sufficient and necessary condition on the existence of bls s based on the moment equation is presented which is simpler to use than frame theory given a filter generating a frame its bls s also generate frames the dual frame is one of them given a fir analysis synthesis filter there is a fir synthesis analysis filter if bls s exist the algorithm to compute fir analysis and synthesis filters based on the linear difference equation is presented in this paper which is simpler than frame operator

quaternions further contributions to a matrix oriented approach abstract by representing quaternions as four dimensional vectors and the multiplication of quaternions as matrix by vector product we employ a matrix oriented approach to the topic we investigate properties of the fundamental real matrix associated with a quaternion and examine a frequently considered quaternion equation from which the n th power of a quaternion can be determined

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

bounded sequence testing from deterministic finite state machines the w andwp methods are the basis for conformance testing from a deterministic finite state machine dfsm when the conformance relation considered is equivalence however many dfsm applications use only input sequences of limited length in such cases the test data only need to establish that the implementation under test produces the specified responses for sequences of length less than or equal to the upper bound l this paper extends the w and wp methods to the case in which only bounded sequences are allowed the methods for bounded sequences are stronger than the originals since test suites for the unbounded case can be obtained as a particular case in which the upper bound l is sufficiently large from the new formulae furthermore the generalization is not straightforward as it is not sufficient to extract the sequences of length at most l from the test suites produced in the unbounded case or even all prefixes of length at most l of the original test sequences the practicality of the methods is also improved in comparison to the unbounded case the size of the test suites may be considerably reduced while the complexity of the test generation algorithms remains basically unchanged

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed