design of two dimensional digital filters using singular value decomposition and balanced approximation method it is shown that the singular value decomposition svd of the sampled amplitude response of a two dimensional 2 d digital filter possesses a special structure every singular vector is either mirror image symmetric or antisymmetric with respect to its midpoint consequently the svd can be applied along with 1 d finite impulse response fir techniques for the design of linear phase 2 d filters with arbitrary prescribed amplitude responses which are symmetrical with respect to the origin of the omega psi omega sub 2 plane the balanced approximation method is applied to linear phase 2 d fir filters of the type that may be obtained by using the svd method the method leads to economical and computationally efficient filters usually infinite impulse response filters which have prescribed amplitude responses and whose phase responses are approximately linear

finite sums of products of functions in single variables abstract a characterization is given of smooth functions h of k variables that admit decompositions into a finite sum of products of k functions of single variables only this sufficient and necessary condition is given in terms of certain ordinary linear differential equations formed from a given h which also serve for the construction of the functions in single variables occuring in such a decomposition conditions for the existence of special decompositions are also given including the case of three variables when h x y t n i 1 i x g i y h i t

the zeros of derivatives of entire functions and the polya wiman conjecture an entire function f x which assumes only real values on the real axis is said to be a real entire function thus if f is a real entire function then its maclaurin coefficients are all real and consequently the zeros of f are symmetrically located with respect to the real axis one of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by polya p4 in his celebrated survey article entitled on the zeros of the derivatives of a function and its analytic character in this work p4 p 181 polya states the real axis seems to exert an influence on the complex zeros of f n z it seems to attract these zeros when the order is less than 2 and it seems to repel them when the order is greater than 2 indeed the first confirmation of this principle was made by alander a2 see also p6lya p2 who in 1930 showed that the polya wiman conjecture is true if the order of f x is less than 2 3 in wil and wi2 wiman established the validity of this conjecture for functions f x of order at most 1 while in 1937 polya p3 proved it for functions of order less than 4 3 a general survey of these results and related conjectures may be found in p2 and p4 for more recent surveys of this and related areas of research see boas b2 and prather pr it seems that no progress has been made on the p6lya wiman conjecture since the publication of polya s 4 3 theorem in 1937 our proof of the conjecture is self contained and does not rely on the results cited above in section 2 we will 1 introduce some definitions and notations 2 state some of the properties of functions in the laguerre p6lya class and 3 prove the polya wiman conjecture theorem 3 in the proof of theorem 3 we

the lee yang and polya schur programs i linear operators preserving stability in 1952 lee and yang proposed the program of analyzing phase transitions in terms of zeros of partition functions linear operators preserving non vanishing properties are essential in this program and various contexts in complex analysis probability theory combinatorics and matrix theory we characterize all linear operators on finite or infinite dimensional spaces of multivariate polynomials preserving the property of being non vanishing whenever the variables are in prescribed open circular domains in particular this solves the higher dimensional counterpart of a long standing classification problem originating from classical works of hermite laguerre hurwitz and polya schur on univariate polynomials with such properties

renyi divergence and kullback leibler divergence renyi divergence is related to renyi entropy much like kullback leibler divergence is related to shannon s entropy and comes up in many settings it was introduced by renyi as a measure of information that satisfies almost the same axioms as kullback leibler divergence and depends on a parameter that is called its order in particular the renyi divergence of order 1 equals the kullback leibler divergence we review and extend the most important properties of renyi divergence and kullback leibler divergence including convexity continuity limits of sigma algebras and the relation of the special order 0 to the gaussian dichotomy and contiguity we also show how to generalize the pythagorean inequality to orders different from 1 and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs for all orders and present several other minimax results

mixed radix fast filter bank approach for the design of variable digital filters with simultaneously tunable bandedge and fractional delay a popular technique to design variable fractional delay vfd filters or variable bandedge filters is the polynomial based finite impulse response fir filters where each filter coefficient is approximated as a polynomial of the variable parameter however if the filter is required to have both vfd and variable bandedge the computational complexity becomes very high because two dimensional polynomials have to be used provided that the same polynomial idea is followed in this paper a filter bank approach is proposed for the design of vfd filters the basic idea of the approach is to split the full band signals into subbands and each subband is respectively shifted by a phase which is determined by the required fractional delay the overall fractional delay is achieved by combining all subbands using this technique the variable bandedge could be incorporated into the vfd filter with trivial extra computations design examples show that the proposed technique achieves high performance with low computational complexity

renyi divergence measures for commonly used univariate continuous distributions abstract probabilistic distances also called divergences which in some sense assess how close two probability distributions are from one another have been widely employed in probability statistics information theory and related fields of particular importance due to their generality and applicability are the renyi divergence measures this paper presents closed form expressions for the renyi and kullback leibler divergences for nineteen commonly used univariate continuous distributions as well as those for multivariate gaussian and dirichlet distributions in addition a table summarizing four of the most important information measure rates for zero mean stationary gaussian processes namely renyi entropy differential shannon entropy renyi divergence and kullback leibler divergence is presented lastly a connection between the renyi divergence and the variance of the log likelihood ratio of two distributions is established thereby extending a previous result by song j stat plan infer 93 2001 on the relation between renyi entropy and the log likelihood function a table with the corresponding variance expressions for the univariate distributions considered here is also included

h infty optimal fractional delay filters fractional delay filters are digital filters to delay discrete time signals by a fraction of the sampling period since the delay is fractional the intersample behavior of the original analog signal becomes crucial in contrast to the conventional designs based on the shannon sampling theorem with the band limiting hypothesis the present paper proposes a new approach based on the modern sampled data h optimization that aims at restoring the intersample behavior beyond the nyquist frequency by using the lifting transform or continuous time blocking the design problem is equivalently reduced to a discrete time h optimization which can be effectively solved by numerical computation softwares moreover a closed form solution is obtained under an assumption on the original analog signals design examples are given to illustrate the advantage of the proposed method

weighted least squares design of variable fractional delay fir filters using coefficient symmetry our previous work has shown that the coefficient symmetry can be efficiently exploited in designing variable finite impulse response fir filters with simultaneously tunable magnitude and fractional delay responses this paper presents the optimal solutions for the weighted least squares wls design of variable fractional delay vfd fir filters with same order and different order subfilters through utilizing the coefficient symmetry along with an imposed coefficient constraint in deriving the closed form error functions since the taylor series expansions of sin omegap and cos omegap are used the numerical integrals using conventional quadrature rules can be completely removed which speeds up the wls design and guarantees the optimality of the final solution two design examples are given to illustrate that the proposed wls methods can achieve better design with significantly reduced vfd filter complexity and computational cost than the existing ones including the wls svd approach consequently the proposed wls design is the best among all the existing wls methods so far

a geometric approach of the generalized d alembert equation abstract by using a geometric approach we prove that the set of solutions of the generalized d alembert equation n log f x 1 x n 0 considered in domain of the x1 xn space r n is larger than the set of the functions that can be represented in the form as f x1 xn f1 x2 xn fn x1 xn 1

a new look at q exponential distributions via excess statistics q exponential distributions play an important role in nonextensive statistics they appear as the canonical distributions i e the maximum generalized q entropy distributions under mean constraint their relevance is also independently justified by their appearance in the theory of superstatistics introduced by beck and cohen in this paper we provide a third and independent rationale for these distributions we indicate that q exponentials are stable by a statistical normalization operation and that pickands extreme values theorem plays the role of a clt like theorem in this context this suggests that q exponentials can arise in many contexts if the system at hand or the measurement device introduces some threshold moreover we give an asymptotic connection between excess distributions and maximum q entropy we also highlight the role of generalized pareto distributions in many applications and present several methods for the practical estimation of q exponential parameters

some applications of nevanlinna theory to mathematical logic identities of exponential functions in this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition x y multiplication x v and two place exponentiation x for a restricted class of such functions we show that every true identity follows from the natural set of eleven axioms the rates of growth of such functions in the case of a single independent variable x as x are also studied and we give an algorithm for the hardy relation of eventual domination again for a restricted class of functions value distribution of analytic functions of one and of several complex variables especially the nevanlinna characteristic plays a major role in our proofs introduction to make precise the questions we consider it is necessary to introduce a formal language containing the expressions which define the elementary functions to be studied this language has variables x x2 a set s of constants representing a given set of fixed numbers and function symbols for addition multiplication and exponentiation the terms that is the function defining expres sions are obtained inductively starting with the variables and constants and continuing according to the rule if t and s are terms then so are t s t s and t we follow some standard conventions for dropping parentheses and below let us refer to these as the exponential terms over s in most cases considered here the constants in s represent positive real numbers in such a case every term t represents a function which is defined for all positive real values of the variables which occur in it we write t s to mean that t and s are terms which define exactly the same function for positive real values of the variables appearing in t or s it is necessary to distinguish between this function equality relation t s and a formal expression of equality between t and s which we write in the form t s the relation t s is a mathematical relation between certain elementary functions the expression t s is a purely formal expression which is susceptible to formal proof within various axiomatic systems a closely related topic concerns the orders of growth of functions of one variable defined by exponential terms if t and s are such terms which contain only the single

the lee yang and polya schur programs ii theory of stable polynomials and applications in the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains for such sets this completes the multivariate generalization of the classification program initiated by polya and schur for univariate real polynomials we build on these classification theorems to develop here a theory of multivariate stable polynomials applications and examples show that this theory provides a natural framework for dealing in a uniform way with lee yang type problems in statistical mechanics combinatorics and geometric function theory in one or several variables in particular we answer a question of hinkkanen on multivariate apolarity

on the renyi divergence joint range of relative entropies and a channel coding theorem this paper starts by considering the minimization of the renyi divergence subject to a constraint on the total variation distance based on the solution of this optimization problem the exact locus of the points bigl d q p 1 d q p 2 bigr is determined when p 1 p 2 and q are arbitrary probability measures which are mutually absolutely continuous and the total variation distance between p 1 and p 2 is not below a given value it is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet this characterization yields a geometric interpretation of the minimal chernoff information subject to a constraint on the variational distance this paper also derives an exponential upper bound on the performance of binary linear block codes or code ensembles under maximum likelihood decoding its derivation relies on the gallager bounding technique and it reproduces the shulman feder bound as a special case the bound is expressed in terms of the renyi divergence from the normalized distance spectrum of the code or the average distance spectrum of the ensemble to the binomially distributed distance spectrum of the capacity achieving ensemble of random block codes this exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes or code ensembles as a function of the deviation of their distance spectra from the binomial distribution an efficient use of this bound is considered

computation and estimation of generalized entropy rates for denumerable markov chains we study entropy rates of random sequences for general entropy functionals including the classical shannon and renyi entropies and the more recent tsallis and sharma mittal ones in the first part we obtain an explicit formula for the entropy rate for a large class of entropy functionals as soon as the process satisfies a regularity property known in dynamical systems theory as the quasi power property independent and identically distributed sequence of random variables naturally satisfy this property markov chains are proven to satisfy it too under simple explicit conditions on their transition probabilities all the entropy rates under study are thus shown to be either infinite or zero except at a threshold where they are equal to shannon or renyi entropy rates up to a multiplicative constant in the second part we focus on the estimation of the marginal generalized entropy and entropy rate for parametric markov chains estimators with good asymptotic properties are built through a plug in procedure using a maximum likelihood estimation of the parameter

on the number of nonreal zeros of real entire functions and the fourier polya conjecture this paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions j fourier formulated the theorem in his work analyse des equations determineesin 1831 but he did not give a proof roughly speaking the theorem states that if a real entire function f x can be expressed as a product of linear factors then we can count the nonreal zeros of f x by observing the behavior of the derivatives of f x on the real axis alone as we shall see in the sequel this theorem completely justifies his former argument by which he tried to prove that the function j0 2 x has only real zeros it seems that no complete proof of the theorem is known and no general theorem has been published that justifies the argument later in 1930 g polya published a paper entitled some problems connected with fourier s work on transcendental equations p3 in this paper polya conjectured two hypothetical theorems that are closely related to fourier s unproved theorem in fact he conjectured three but he proved that two of them are equivalent to each other the first hypothetical theorem is a modernized formulation of the theorem and it justifies fourier s argument completely the second conjecture was proved in 1990 but it is impossible to justify the argument using the conjecture alone in the present paper we prove polya s formulation of the theorem his first conjecture as well as its extensions give a very simple and direct proof of the second conjecture mentioned above and exhibit some applications of the results in particular we completely justify fourier s argument by our general theorems acknowledgments professor fefferman has encouraged and helped us to publish this paper the authors truly thank him for this the authors also thank professor csordas for his kind interest in the results and his valuable suggestions on the proof of the polya wiman conjecture 1 historical introduction in this section we briefly explain our results as well as their background a real entire function is an entire function that assumes only real values on the real axis fourier s unproven theorem asserts that we can know the number of nonreal zeros of such real entire functions by counting their critical points which are defined as follows let f x be a real analytic function defined in an open

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

monotonically equivalent entropies and solution of additivity equation generalized entropies are studied as lyapunov functions for the master equation markov chains three basic properties of these lyapunov functions are taken into consideration universality independence of the kinetic coefficients trace form the form of sum over the states and additivity for composition of independent subsystems all the entropies which have all three properties simultaneously and are defined for positive probabilities are found they form a one parametric family we consider also pairs of entropies s1 s2 which are connected by the monotonous transformation s2 f s1 equivalent entropies all classes of pairs of universal equivalent entropies one of which has a trace form and another is additive these entropies can be different one from another were found these classes consist of two one parametric families the family of entropies which are equivalent to the additive trace form entropies and the family of renyi tsallis entropies

non real zeros of real differential polynomials the main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non real poles and certain combinations of derivatives of f have few non real zeros msc 2000 30d20 30d35

fractal surfaces from simple arithmetic operations fractal surfaces patchwork quilts are shown to arise under most general circumstances involving simple bitwise operations between real numbers a theory is presented for all deterministic bitwise operations on a finite alphabet it is shown that these models give rise to a roughness exponent h that shapes the resulting spatial patterns larger values of the exponent leading to coarser surfaces

generalized kolmogorov entropy in the dynamics of multifractal generation abstract we point out that applying a maximization principle on a tsallis like generalized form of the kolmogorov entropy for iterated function systems naturally provides a canonical statistical frame for the description of the multifractal measures generated by such dynamical processes multifractal spectra can then be characterized by usual statistical parameters in particular the temperature we show that in the limit of zero temperature the multifractal measure collapses to a homogeneous distribution over a fractal support for finite temperatures multifractal spectra are studied numerically in an illustrative example

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

renyi entropy guesswork moments and large deviations for a large class of stationary probability measures on a sup n where a is a finite alphabet we compute the specific renyi entropy of order spl alpha and the specific guesswork moments of order spl beta 1 we show that the specific guesswork moment of order spl beta equals the specific renyi entropy of order spl alpha 1 1 spl beta multiplied by spl beta the method is based on energy entropy estimates suggested by statistical physics the technique also yields a simple proof of the large deviation principle for the empirical measure on the space of an irreducible sofic shift with reference probability measure spl nu which is stationary and satisfies a rate condition on the probability of allowed words

proof of a conjecture of polya on the zeros of successive derivatives of real entire functions we prove polya s conjecture of 1943 for a real entire function of order greater than 2 with finitely many non real zeros the number of non real zeros of the nth derivative tends to infinity as n to infty we use the saddle point method and potential theory combined with the theory of analytic functions with positive imaginary part in the upper half plane

selective influence through conditional independence let each of several generally interdependent random vectors taken separately be influenced by a particular set of external factors under what kind of the joint dependence of these vectors on the union of these factor sets can one say that each vector is selectively influenced by its own factor set the answer proposed and elaborated in this paper is one can say this if and only if one can find a factor independent random vector given whose value the vectors in question are conditionally independent with their conditional distributions selectively influenced by the corresponding factor sets equivalently the random vectors should be representable as deterministic functions of their factor sets and of some mutually independent and factor independent random variables some of which may be shared by several of the functions

differentiation evens out zero spacings if f is a polynomial with all of its roots on the real line then the roots of the derivative f are more evenly spaced than the roots of f the same holds for a real entire function of order 1 with all its zeros on a line in particular we show that if f is entire of order 1 and has sufficient regularity in its zero spacing then under repeated differentiation the function approaches after normalization the cosine function we also study polynomials with all their zeros on a circle and we find a close analogy between the two situations this sheds light on the spacing between zeros of the riemann zeta function and its connection to random matrix polynomials

low complexity and high modularity structure for implementing transient free pascal delay filter the maximally flat variable delay vd digital filter can be derived by using the discrete pascal transform dpt and the dpt based pascal polynomial interpolation in this paper we develop an efficient structure for the low complexity and high modularity implementation of this pascal type vd filter as compared to other existing filter structures this new structure greatly reduces the multiplication operations required in the filtering process roughly speaking the proposed new structure can reduce the multiplication operations by almost one third thus this new filter structure has the lowest filter complexity among all the existing filter structures for the implementation of the maximally flat vd filter as a result signal interpolation can be efficiently performed through utilizing this new filter structure as opposed to other existing structure with transient problem we will show that the proposed new structure does not involve any transient problem that is it is a transient free structure consequently the proposed new filter structure can achieve more accurate signal interpolation results than other structures that have the transient problem finally we show that this proposed structure also has high modularity and it is suited for modularized hardware implementation

some properties of q logarithm and q exponential functions in tsallis statistics we present some formulae which q logarithmic and q exponential functions introduced in tsallis statistics satisfy exact expression of the mellin transform of q exponential function is given moreover the q generalization of the digamma function is presented

convergence in reaction diffusion systems an information theory approach abstract we have applied an information theory approach in order to study the problem of convergence to point like or extended attractors in reaction diffusion systems a distance between two states based on the kullback leibler relative information was defined different forms of the probability distribution some of them based on the knowledge of the nonequilibrium potential when accesible give the possibility to look for a faster and or more accurate convergence this approach offers the chance to estimate the attraction basins of the different attractors as well as detecting limit circles together with an easy evaluation of their periods

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

coefficient symmetries for implementing arbitrary order lagrange type variable fractional delay digital filters in this paper we first derive an explicit formula for expressing the coefficients of an arbitrary order lagrange type variable fractional delay vfd digital filter as polynomials of the vfd parameter and then develop some useful symmetries for both even and odd order lagrange type vfd filter coefficients the coefficient symmetries facilitate the evaluations of vfd filter coefficients as well as variable frequency responses with reduced computational complexity more importantly the coefficient symmetries can be exploited for efficiently implementing both even and odd order lagrange type vfd filters as farrow structure and a more efficient one called even odd structure such that the subfilters have symmetric or anti symmetric coefficients which saves the storage for vfd filter coefficients and reduces the number of multiplications required in vfd filtering process by almost 50 therefore exploiting the coefficient symmetries not only speeds up the vfd filtering but also reduces implementation cost

convolution based trigonometric interpolation of band limited signals this paper presents a method to obtain a trigonometric polynomial that accurately interpolates a given band limited signal from a finite sequence of samples the polynomial delivers accurate approximations in the range covered by the sequence except for a short frame close to the range limits besides its accuracy increases exponentially with the frame width the method is based on using a band limited window in order to reduce the truncation error of a convolution series it is shown that the polynomial can be efficiently constructed and evaluated using algorithms designed for the discrete fourier transform dft specifically two basic procedures are presented one based on the fast fourier transform fft and another based on a recursive update algorithm for the short time fft the paper contains three applications the first is a variable fractional delay vfd filter which consists of a short time fft combined with the evaluation of a trigonometric polynomial this filter has low complexity and can be implemented using cordic rotations the second is the interpolation of nonuniform fourier summations where the proposed method eliminates the need to interpolate any kernel sample finally the third can be viewed as a generalization of the fft convolution algorithm and makes it possible to interpolate the output of an finite impulse response fir filter efficiently

differential operators of infinite order and the distribution of zeros of entire functions whenever the right hand side of 1 2 represents an analytic function in a neighborhood of the origin when x is an entire function the operator d has been studied by several authors see for example 5 11 19 chapter ix 22 and 32 the conjecture of polya and wiman proved in 8 9 and 17 states that if f x l p then df x is in the laguerre polya class for all sufficiently large positive integers m in this paper we analyze the more general situation when d is replaced by the operator d in section 2 we consider real power series with zero linear term i e 1 0 in 1 1 and 0 2 0 and show that if f x is any real polynomial then d f x l p for all sufficiently large positive integers m theorem 2 4 if the linear term in 1 1 is nonzero

the fox wright functions and laguerre multiplier sequences abstract linear operators which 1 preserve the reality of zeros of polynomials having only real zeros and 2 map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain fox wright functions and generalized mittag leffler functions the paper includes several open problems and questions

student s t and r distributions unified derivation from an entropic variational principle abstract the optimization of the recently generalized entropy of the form s q 1 dx p x q q 1 with the constraints dx p x 1 and x 2 q dx x 2 p x q 1 yields the student s t distribution for q 1 and the r distribution for q

interpretations of renyi entropies and divergences abstract in this paper a new operational definition of renyi divergence and renyi entropy is presented other operational definitions are mentioned

polya schur master theorems for circular domains and their boundaries we characterize all linear operators on finite or infinite dimensional polynomial spaces that preserve the property of having the zero set inside a prescribed region omega subset of c for arbitrary

svd based design and new structures for variable fractional delay digital filters this paper shows that the problem of designing one dimensional 1 d variable fractional delay vfd digital filter can be elegantly reduced to the easier subproblems that involve one dimensional 1 d constant filter subfilter designs and 1 d polynomial approximations by utilizing the singular value decomposition svd of the variable design specification we prove that both 1 d constant filters and 1 d polynomials possess either symmetry or anti symmetry simultaneously therefore a vfd filter can be efficiently obtained by designing 1 d constant filters with symmetrical or antisymmetrical coefficients and performing 1 d symmetrical or antisymmetrical approximations to perform the weighted least squares wls vfd filter design a new wls svd method is also developed moreover an objective criterion is proposed for selecting appropriate subfilter orders and polynomial degrees our computer simulations have shown that the svd based design and wls svd design can achieve much higher design accuracy with significantly reduced filter complexity than the existing wls design method another important part of the paper proposes two new structures for efficiently implementing the resulting vfd filter which require less computational complexity than the so called farrow structure

fractal structure with a typical scale in order to understand the characteristics common to distributions which have both fractal and non fractal scale regions in a unified framework we introduce the concept of a typical scale we employ a model of 2d gravity modified by the r2 term as a tool to understand such distributions through the typical scale this model is obtained by adding an interaction term with a typical scale to a scale invariant system a distribution derived in the model provides power law one in the large scale region but weibull like one in the small scale region as examples of distributions which have both fractal and non fractal regions we take those of personal income and citation number of scientific papers we show that these distributions are fitted fairly well by the distribution curves derived analytically in the r2 2d gravity model as a result we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way we also point out that the r2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way

multiwavelet packet entropy and its application in transmission line fault recognition and classification multiwavelets possess better properties than traditional wavelets multiwavelet packet transformation has more high frequency information spectral entropy can be applied as an analysis index to the complexity or uncertainty of a signal this paper tries to define four multiwavelet packet entropies to extract the features of different transmission line faults and uses a radial basis function rbf neural network to recognize and classify 10 fault types of power transmission lines first the preprocessing and postprocessing problems of multiwavelets are presented shannon entropy and tsallis entropy are introduced and their difference is discussed second multiwavelet packet energy entropy time entropy shannon singular entropy and tsallis singular entropy are defined as the feature extraction methods of transmission line fault signals third the plan of transmission line fault recognition using multiwavelet packet entropies and an rbf neural network is proposed finally the experimental results show that the plan with the four multiwavelet packet energy entropies defined in this paper achieves better performance in fault recognition the performance with sa4 symmetric antisymmetric multiwavelet packet tsallis singular entropy is the best among the combinations of different multiwavelet packets and the four multiwavelet packet entropies

on the de bruijn newman constant abstract if 0 denotes the infimum of the set of real numbers such that the entire function represented by t 0 e 4 log x 2 i t 2 log x x 5 4 n 1 2 n 4 2 x 3 n 2 e n 2 x d x x has only real zeros then the de bruijn newman constant is defined as 4 0 the riemann hypothesis is equivalent to the inequality 0 the fact that the non trivial zeros of the riemann zeta function lie in the strip s 0 re s 1 and a theorem of de bruijn imply that 1 2 in this paper we prove that all but a finite number of zeros of are real and simple for each 0 and consequently that 1 2

a method for computing the partial singular value decomposition a method for computing the partial singular value decomposition of a matrix is described the method is appropriate to problems where the matrix is known to be of low rank and only the principal singular vectors are of interest the technique is simple easy to implement in integer arithmetic and places modest memory requirements the convergence properties of the algorithm are investigated analytically and by simulation

on the polya wiman properties of differential operators abstract let x n x n be a formal power series with real coefficients and let d denote differentiation it is shown that for every real polynomial f there is a positive integer m 0 such that d m f has only real zeros whenever m m 0 if and only if 0 0 or 2 0 2 1 2 0 and that if does not represent a laguerre polya function then there is a laguerre polya function f of genus 0 such that for every positive integer m d m f represents a real entire function having infinitely many nonreal zeros

an efficient structure for the design of variable fractional delay filters based on the windowing method a variable fractional delay vfd filter is presented that constitutes an efficient alternative to the usual farrow structure the filter is constructed by multiplying the signal to interpolate by a window with proper truncation properties and then applying the sampling theorem the resulting interpolator involves less arithmetic operations than the existing variants of the farrow structure besides if its coefficients are numerically optimized its performance is close to that of the fir equi ripple filter the final part of the correspondence includes a description of the vfd filter implementation corresponding to this interpolator

sums of products of functions of x times functions of y we study functions u of the form u x y nk 1ak x bk y mostly by means of two functional determinants wn u and dn u a sample result is that if u has the form on an open subset of a region g on which u is real analytic then u must have this form with possibly different ak and bk on all of g we also study the representability of u x y in the form on a region g in which u is either n times differentiable or is merely continuous the question whether the limit of functions of the form is again of the form is treated moreover we study those functions of the form that are solutions of some classical partial differential equations