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

correlation immunity of nonlinear combining functions for cryptographic applications corresp pseudonoise generators for cryptographic applications consisting of several linear feedback shift registers with a nonlinear combining function have been proposed as running key generators in stream ciphers these running key generators eau sometimes be broken by ciphertext only correlation attacks on individual subsequences a new class of combining functions is presented which provides better security against such attacks the security is quantified by the smallest number m 1 of subsequences that must be simultaneously considered in a correlation attack a necessary condition for such m th order correlation immunity is proved a recursive construction is given that permits the construction of an m th order immune combining function for n subsequences for any m and n with 1 leq m finally the trade off between the length of the linear equivalent of the nonlinear generator and the order m of its immunity against correlation attacks is considered

the distribution of zeros of all solutions of first order neutral differential equations this paper is concerned with the distribution of zeros of all solutions of the first order neutral differential equation x t p t x t q t x t 0 t t 0 where p c t 0 0 q c t 0 0 and r new estimations for the distance between adjacent zeros of this neutral equation are obtained via comparison with a corresponding differential inequality these results extend some known results from the non neutral to the neutral case and improve other published results as well

upper bounds for the distances between consecutive zeros of solutions of first order delay differential equations abstract by introducing a new class of sequences involving iterates of the delay we are able to derive a sharper estimate on the ratio x t x t based on this we get some new upper bounds for the distance between consecutive zeros of solutions of the first order delay differential equation x t p t x t 0 in particular our results are not covered by previously known results some examples and a table are given to illustrate our results

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

design and properties of a new pseudorandom generator based on a filtered fcsr automaton feedback with carry shift registers fcsr was introduced by goresky and klapper in 1993 it is similar to the classical linear feedback shift registers lfsr used in many pseudorandom generators the main difference is that the elementary additions are not additions modulo 2 but with propagation of carries the main problem for the use of an fcsr automaton is the fact that the generated sequences are predictable in order to remove this weakness of fcsr based generators we propose filtering the state of the fcsr with a linear function this method is efficient since the fcsr structure is not related to a linear property this paper presents an extensive study of fcsr automata a security analysis of our generator concerning linear and 2 adic cryptanalysis algebraic attack correlation attack etc and a practical example of parameters in order to design this generator an important point concerning this generator is the fact that it is simple and efficient both in hardware and software implementation

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

the distribution of zeros of solutions for a class of neutral delay differential equations abstract in this paper the distribution of zeros of solutions of neutral differential equations x t cx t p t x t 0 where c 0 0 and p c t0 r are studied and some estimates for the distance between adjacent zeros of the oscillatory solutions of the above equation are obtained

some results on discrete gabor transforms for finite periodic sequences in this correspondence using linear system theory and frame theory we address computation methods for analysis and synthesis sequences of the discrete gabor transform dgt where all sequences are periodic with the same period l and show that the minimum energy solution to the wexler and raz 1990 biorthogonal condition and the solution to the frame operator are the same via linear system theory and frame theory

the distribution of zeros of solutions of differential equations with a variable delay this paper is concerned with the distribution of zeros of solutions of the first order linear differential equations with a variable delay of the form begin aligned x t p t x left tau t right 0 quad t ge t 0 nonumber end aligned where p tau in c t 0 infty 0 infty tau t le t tau t is nondecreasing and lim limits t rightarrow infty tau t infty by introducing a class of new series we are able to derive sharper upper bounds on the distance between zeros of solutions of the above delay differential equations some examples and a table are given to support our accomplishment

regular articlethe distribution of zeros of solutions of differential equations with a variable delay in this paper the distribution of zeros of solutions of the first order differential equation with a variable delayx t ptx t 0is studied the estimate for the distance between adjacent zeros of the oscillatory solution of the above equation is obtained

on the distance between consecutive zeros of solutions of first order delay differential equations this paper is concerned with the distribution of zeros of solutions of first order linear delay differential equations with variable coefficients of the formx t p t x email protected 0 t t where t 0 p t c t 0 by introducing a class of polynomial functions we are able to derive new estimates for the lower and upper bounds of the distance between consecutive zeros of solutions of the above equations we illustrate the obtained results with several examples

on the distribution of zeros of solutions of first order delay differential equations abstract this paper contains new estimates for the distance between adjacent zeros of solutions of the first order delay differential equation x t p t x t 0 where p is a nonnegative continuous function our results are new and improve some known criteria illustrative examples are given to show the strength of some obtained results over the known ones

some results on fcsr automata with applications to the security of fcsr based pseudorandom generators this article describes new theoretical results concerning the general behavior of a feedback with carry shift register fcsr automaton they help to better understand how the initial parameters must be chosen to use this automaton as a basic block of a filtered stream cipher these results especially concern the structure of the transition graph of an fcsr automaton and the number of iterations of the fcsr transition function required to reach the main part of the graph a potential linear weakness and a easy way to prevent the corresponding attack are also given

oversampling in the gabor scheme a method for calculating the coefficients of the gabor expansion in the case of oversampling is presented the method is based on the concept of frames and utilizes the zak transform as such the zak transform highlights the meaning and importance of frames in the context of oversampling and other aspects of signal representation by means of nonorthogonal bases

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

fibonacci and galois representations of feedback with carry shift registers a feedback with carry shift register fcsr with fibonacci architecture is a shift register provided with a small amount of memory which is used in the feedback algorithm like the linear feedback shift register lfsr the fcsr provides a simple and predictable method for the fast generation of pseudorandom sequences with good statistical properties and large periods in this paper we describe and analyze an alternative architecture for the fcsr which is similar to the galois architecture for the lfsr the galois architecture is more efficient than the fibonacci architecture because the feedback computations are performed in parallel we also describe the output sequences generated by the d fcsr a slight modification of the fibonacci fcsr architecture in which the feedback bit is delayed for d clock cycles before being returned to the first cell of the shift register we explain how these devices may be configured so as to generate sequences with large periods we show that the d fcsr also admits a more efficient galois architecture

the generalized gabor transform the generalized gabor transform for image representation is discussed for a given function f t t spl isin r the generalized gabor transform finds a set of coefficients a sub mr such that f t spl sigma sub m spl infin sup spl infin spl sigma sub r spl infin sup spl infin spl alpha sub mr g t mt exp i2 spl pi rt t the original gabor transform proposed by d gabor 1946 is the special case of t t the computation of the generalized gabor transform with biorthogonal functions is discussed the optimal biorthogonal functions are discussed a relation between a window function and its optimal biorthogonal function is presented based on the zak 1967 transform when t t is rational the finite discrete generalized gabor transform is also derived methods of computation for the biorthogonal function are discussed the relation between a window function and its optimal biorthogonal function derived for the continuous variable generalized gabor transform can be extended to the finite discrete case efficient algorithms for the optimal biorthogonal function and generalized gabor transform for the finite discrete case are proposed

arithmetic crosscorrelations of feedback with carry shift register sequences an arithmetic version of the crosscorrelation of two sequences is defined generalizing mandelbaum s 1967 arithmetic autocorrelations large families of sequences are constructed with ideal vanishing arithmetic crosscorrelations these sequences are decimations of the 2 adic expansions of rational numbers p q such that 2 is a primitive root module q

the distribution of zeros of solutions of first order delay differential equations abstract we give two estimates of the distance between adjacent zeros of the solutions of the first order delay differential equation x t p t x t 0

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

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

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

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

upper bounds for the distances between adjacent zeros of solutions of delay differential equations abstract this paper is concerned with upper bounds for the distances between adjacent zeros of solutions of first order linear delay differential equations with variable coefficients of the form x t p t x t 0 t t where 0 and p t c t r by introducing a sequence of testing functions we are able to derive upper bounds that are better than those in the literature

distribution of zeros of solutions of functional differential equations abstract by improving the bracketing techniques for a functional differential inequality in a previous paper h w wu y t xu the distribution of zeros of solutions of neutral differential equations appl math comput 156 3 2004 665 677 we are able to derive sharper upper bounds on the distance between zeros of solutions of a class of neutral functional differential equations with delays and hence improve many known bounds in the literature

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

multiwindow real valued discrete gabor transform and its fast algorithms based on the biorthogonal analysis approach a multiwindow real valued discrete gabor transform m rdgt for periodic sequences is presented to efficiently analyze the dynamic time frequency content of a signal containing components with multiple and or time varying frequencies the m rdgt offers a computationally efficient implementation as well as a real valued formulation of the multiwindow complex valued discrete gabor transform m cdgt the completeness condition of the m rdgt is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows the m rdgt can utilize the fast discrete hartley transform algorithms for fast computation and has a simple relationship with the m cdgt such that its coefficients can be directly computed from the m rdgt coefficients therefore the m rdgt offers an efficient method to compute the m cdgt since the analyzed sequence analysis and synthesis windows in the existing m cdgt must have an equal period if the period of a sequence is very long solving its windows requires a huge amount of computation and memory and could lead to numerical instability to overcome this problem a modified m rdgt for long periodic or even infinite sequences is presented and its corresponding biorthogonality constraint between analysis windows and synthesis windows is modified in which the period of the analysis and synthesis windows is independent of the period of a analyzed sequence so that one can apply short windows to process any long periodic or even in finite sequence finally the multirate based parallel implementation of the m rdgt is presented which has shown to be effective and fast for time frequency analysis

the distribution of zeros of solutions of neutral differential equations in this paper the distribution of zeros of solutions of the first order neutral differential equation with two variable delays x t p t x g t q t x h t 0is studied the estimate for the distance between adjacent zeros of oscillatory solutions of the above equation is obtained

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

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

period and complementarity properties of fcsr memory sequences in this correspondence we investigate feedback with carry shift register fcsr memory sequences for an sequence generated by an fcsr with connection integer and initial memory we prove that for where and is the memory sequence generally speaking the period of a memory sequence is a factor of that of the fcsr output binary sequence we show there are a large number of connection integers with which an fcsr can generate sequences that have the same period as their memory sequences especially including all connection integers for sequences