generalized bayesian information criterion for source enumeration in array processing we investigate the problem of enumerating source signals impinging on an array of sensors in an information theoretic framework the conventional bayesian information criterion bic does not yield satisfactory performance for this problem because it only considers the density of the observations in order to remedy the limitations of the bic we propose a generalized bayesian information criterion gbic rule by incorporating the density of the sample eigenvalues or corresponding statistics such a density contains extra information and complements the density of the observations in constructing the gbic as a result two different expressions for the gbic are suggested simulation results validate the superiority of the proposed gbic over the conventional bic in terms of correctly determining the number of sources while their computational costs are comparable

inferring the eigenvalues of covariance matrices from limited noisy data the eigenvalue spectrum of covariance matrices is of central importance to a number of data analysis techniques usually the sample covariance matrix is constructed from a limited number of noisy samples we describe a method of inferring the true eigenvalue spectrum from the sample spectrum results of silverstein 1986 which characterize the eigenvalue spectrum of the noise covariance matrix and inequalities between the eigenvalues of hermitian matrices are used to infer probability densities for the eigenvalues of the noise free covariance matrix using bayesian inference posterior densities for each eigenvalue are obtained which yield error estimates the evidence framework gives estimates of the noise variance and permits model order selection by estimating the rank of the covariance matrix the method is illustrated with numerical examples

faulhaber and bernoulli polynomials and solitons abstract a relation between the classical faulhaber and bernoulli polynomials and the theory of the korteweg de vries equation is established

detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

the discrete galerkin method for integral equations a general theory is given for discretized versions of the galerkin method for solving fredholm integral equations of the second kind the discretized galerkin method is obtained from using numerical integration to evaluate the integrals occurring in the galerkin method the theoretical framework that is given parallels that of the regular galerkin method including the error analysis of the superconvergence of the iterated galerkin and discrete galerkin solutions in some cases the iterated discrete galerkin solution is shown to coincide with the nystrom solution with the same numerical integration method the paper concludes with applications to finite element galerkin methods

discrete legendre spectral projection methods for fredholm hammerstein integral equations in this paper we discuss the discrete legendre galerkin and discrete legendre collocation methods for fredholm hammerstein integral equations with smooth kernel using sufficiently accurate numerical quadrature rule we obtain optimal convergence rates for both discrete legendre galerkin and discrete legendre collocation solutions in both infinity and l 2 norm numerical examples are given to illustrate the theoretical results

using the sphericity test for source detection with narrow band passive arrays a new sphericity test statistic is proposed for the complex case with a distribution that approaches the asymptotic chi squared distribution much faster than previously proposed statistics especially for situations in which the sources are very difficult to detect such as closely spaced or weak sources consequently the probability of a correct estimate is superior to those resulting from other sphericity tests the capability of this new sphericity test to detect closely space sources is examined and compared to the resolution capabilities of bearing estimation algorithms as well as other well known source detection algorithms

superconvergence of the iterated galerkin methods for hammerstein equations in this paper the well known iterated galerkin method and iterated galerkin kantorovich regularization method for approximating the solution of fredholm integral equations of the second kind are generalized to hammerstein equations with smooth and weakly singular kernels the order of convergence of the galerkin method and those of superconvergence of the iterated methods are analyzed numerical examples are presented to illustrate the superconvergence of the iterated galerkin approximation for hammerstein equations with weakly singular kernels

projection and iterated projection methods for nonliear integral equations consider the nonlinear operator equation x mathcal k x with mathcal k a completely continuous mapping of a domain in the banach space mathcal x into mathcal x and let x denote an isolated fixed point of mathcal k let mathcal k n n geqq 1 denote a sequence of finite dimensional approximating subspaces and let p n be a projection of mathcal x onto mathcal x n the projection method for solving x mathcal k x given by x n p n mathcal k x n and the iterated projection solution is defined as tilde x n mathcal k x n we analyze the convergence of x n and tilde x n to x giving a general analysis that includes both the galerkin and collocation methods a more detailed analysis is then given for a large class of urysohn integral operators in one variable showing the superconvergence of tilde x n to x

the appell interpolation problem a general linear interpolation problem is considered we will call it the appell interpolation problem because the solution can be expressed by a basis of appell polynomials some classical and non classical examples are also considered finally numerical calculations are given

integrable maps from galois differential algebras borel transforms and number sequences abstract a new class of integrable maps obtained as lattice versions of polynomial dynamical systems is introduced these systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation in particular symmetries and integrability see tempesta 2010 40 our approach is based on the properties of a suitable galois differential algebra that we shall call a rota algebra a formulation of the procedure in terms of category theory is proposed in order to render the lattice dynamics confined a borel regularization is also adopted as a byproduct of the theory a connection between number sequences and integrability is discussed

determination of the dimension of a signal subspace from short data records a nested sequence of constant false alarm rate cfar hypothesis tests is presented for rank determination over short data records the procedure is based on the interpretation of sum of squares of singular values as energy in a particular subspace the cfar thresholds are set up based on distributions derived from matrix perturbation ideas expressions for probability of overestimation and underestimation are presented

a note on bernoulli numbers and shintani generalized bernoulli polynomials generalized bernoulli polynomials were introduced by shintani in 1976 in order to express the special values at non positive integers of dedekind zeta functions for totally real numbers the coefficients of such polynomials are finite combinations of products of bernoulli numbers which are difficult to get hold of on the other hand zagier was able to get the explicit formula for the special values in cases of real quadratic number fields in this paper we shall improve shintani s formula by proving that the special values can be determined by a finite set of polynomials this provides a convenient way to evaluate the special values of various types of dedekind functions indeed a much broader class of zeta functions considered by the author 4 admits a similar formula for its special values as a consequence we are able to find infinitely many identities among bernoulli numbers through identities among zeta functions all these identities are difficult to prove otherwise 1 identities among bernoulli numbers the bernoulli numbers bn n 0 1 2 are defined by t 00 bntn et1 e n itl 1 since the function t t et 1 2 is an even function of t by direct verification bernoulli numbers are used to express the special values of riemann zeta function

analysis of the performance and sensitivity of eigendecomposition based detectors a new framework is presented for the analysis of the performance of detection methods such as aic and mdl which are based on the eigenvalues of the sample covariance matrix it is shown that theoretical analysis of the probabilities of overestimation and underestimation can be much more conveniently carried out via a proposed particularly simple sequence of statistics also the breakdown of these detection methods in the presence of model nonidealities is explored by theory simulations and experimentation with real array data for example theoretical arguments are given to demonstrate the high degree of sensitivity of the detectors to unknown deviations of the noise from whiteness

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

estimation of the number of sources in unbalanced arrays via information theoretic criteria estimating the number of sources impinging on an array of sensors is a well known and well investigated problem a common approach for solving this problem is to use an information theoretic criterion such as minimum description length mdl or the akaike information criterion aic the mdl estimator is known to be a consistent estimator robust against deviations from the gaussian assumption and nonrobust against deviations from the point source and or temporally or spatially white additive noise assumptions over the years several alternative estimation algorithms have been proposed and tested usually these algorithms are shown using computer simulations to have improved performance over the mdl estimator and to be robust against deviations from the assumed spatial model nevertheless these robust algorithms have high computational complexity requiring several multidimensional searches in this paper which is motivated by real life problems a systematic approach toward the problem of robust estimation of the number of sources using information theoretic criteria is taken an mdl type estimator that is robust against deviation from assumption of equal noise level across the array is studied the consistency of this estimator even when deviations from the equal noise level assumption occur is proven a novel low complexity implementation method avoiding the need for multidimensional searches is presented as well making this estimator a favorable choice for practical applications

no eigenvalues outside the support of the limiting spectral distribution of large dimensional sample covariance matrices let b n 1 n t n 1 2 x n x n t n 1 2 where x n is n times n with i i d complex standardized entries having finite fourth moment and t n 1 2 is a hermitian square root of the nonnegative definite hermitian matrix t n it is known that as n to infty if n n converges to a positive number and the empirical distribution of the eigenvalues of t n converges to a proper probability distribution then the empirical distribution of the eigenvalues of b n converges a s to a nonrandom limit in this paper we prove that under certain conditions on the eigenvalues of t n for any closed interval outside the support of the limit with probability 1 there will be no eigenvalues in this interval for all n sufficiently large

nonparametric detection of signals by information theoretic criteria performance analysis and an improved estimator determining the number of sources from observed data is a fundamental problem in many scientific fields in this paper we consider the nonparametric setting and focus on the detection performance of two popular estimators based on information theoretic criteria the akaike information criterion aic and minimum description length mdl we present three contributions on this subject first we derive a new expression for the detection performance of the mdl estimator which exhibits a much closer fit to simulations in comparison to previous formulas second we present a random matrix theory viewpoint of the performance of the aic estimator including approximate analytical formulas for its overestimation probability finally we show that a small increase in the penalty term of aic leads to an estimator with a very good detection performance and a negligible overestimation probability

on information theoretic criteria for determining the number of signals in high resolution array processing an important problem in high resolution array processing is the determination of the number of signals arriving at the array information theoretic criteria provide a means to achieve this two commonly used criteria are the akaike information criterion aic and minimum descriptive length mdl criterion while the aic tends to overestimate even at a high signal to noise ratio snr the mdl criterion tends to underestimate at low or moderate snr by excluding irrelevant parameters a new log likelihood function has been chosen utilizing this new log likelihood function gives a set of more accurate estimates of the eigenvalues and in the establishment of modified information theoretic criteria which moderate the performance of the aic and the mdl criterion computer simulations confirm that the modified criteria have superior performance

group entropies correlation laws and zeta functions the notion of group entropy is proposed it enables the unification and generaliztion of many different definitions of entropy known in the literature such as those of boltzmann gibbs tsallis abe and kaniadakis other entropic functionals are introduced related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic the associated thermostatistics are discussed the mathematical structure underlying our construction is that of formal group theory which provides the general structure of the correlations among particles and dictates the associated entropic functionals as an example of application the role of group entropies in information theory is illustrated and generalizations of the kullback leibler divergence are proposed a new connection between statistical mechanics and zeta functions is established in particular tsallis entropy is related to the classical riemann zeta function

hyperfunctions formal groups and generalized lipschitz summation formulas abstract a construction relating the theory of hyperfunctions with the theory of formal groups and generalizations of the classical lipschitz summation formula is proposed it involves new polylogarithmic rational functions constructed via the fourier expansion of certain sequences of bernoulli type polynomials related to the lazard formal group related families of one dimensional hyperfunctions are also constructed

binomial convolution and transformations of appell polynomials abstract we show that the set a of appell sequences is an abelian group under the binomial convolution this is essentially equivalent to other approaches considered in the literature in particular the determinantal approach we also define a scale transformation t a a which is an isomorphism and a transformation r y a a based on expectations with respect to a random variable y using these tools we give explicit formulas for the determinantal form of appell sequences as well as for the formula of representation of powers and the srivastava pinter addition theorem various illustrative examples mainly referring to various generalizations of bernoulli and euler polynomials are discussed in detail

non parametric detection of the number of signals hypothesis testing and random matrix theory detection of the number of signals embedded in noise is a fundamental problem in signal and array processing this paper focuses on the non parametric setting where no knowledge of the array manifold is assumed first we present a detailed statistical analysis of this problem including an analysis of the signal strength required for detection with high probability and the form of the optimal detection test under certain conditions where such a test exists second combining this analysis with recent results from random matrix theory we present a new algorithm for detection of the number of sources via a sequence of hypothesis tests we theoretically analyze the consistency and detection performance of the proposed algorithm showing its superiority compared to the standard minimum description length mdl based estimator a series of simulations confirm our theoretical analysis

a fast collocation method for solving stochastic integral equations based on sparse grid multiscale piecewise polynomial bases we develop a fast collocation method for solving fredholm integral equations of the second kind with stochastic loading terms it is proved that the proposed method preserves the optimal rate of convergence and has linear up to a logarithmic factor computational complexity the reduction in computational costs come from three aspects the adoption of the collocation principle the use of the sparse grid for the solution space and the employment of a truncation strategy for the coefficient matrix of the resulting linear system numerical experiments confirm the theoretical results on approximation accuracy and computational efficiency of the proposed method

source enumeration via mdl criterion based on linear shrinkage estimation of noise subspace covariance matrix numerous methodologies have been investigated for source enumeration in sample starving environments for those having their root in the framework of random matrix theory the involved distribution of the sample eigenvalues is required instead of relying on the eigenvalue distribution this work devises a linear shrinkage based minimum description length ls mdl criterion by utilizing the identity covariance matrix structure of noise subspace components with linear shrinkage and gaussian assumption of the observations an accurate estimator for the covariance matrix of the noise subspace components is derived the eigenvalues obtained from the estimator turn out to be a linear function of the corresponding sample eigenvalues enabling the ls mdl criterion to accurately detect the source number without incurring significantly additional computational load furthermore the strong consistency of the ls mdl criterion for m n and m n c 0 is proved where m and n are the antenna number and snapshot number respectively simulation results are included for illustrating the effectiveness of the proposed criterion

legendre spectral projection methods for urysohn integral equations in this paper we consider the legendre spectral galerkin and legendre spectral collocation methods to approximate the solution of urysohn integral equation we prove that the approximated solutions of the legendre galerkin and legendre collocation methods converge to the exact solution with the same orders o n r in l 2 norm and o n 1 2 r in infinity norm and the iterated legendre galerkin solution converges with the order o n 2 r in both l 2 norm and infinity norm whereas the iterated legendre collocation solution converges with the order o n r in both l 2 norm and infinity norm n being the highest degree of the legendre polynomial employed in the approximation and r being the smoothness of the kernel we are able to obtain similar superconvergence rates for the iterated galerkin solution for urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions

information theoretic criteria for the determination of the number of signals in spatially correlated noise the problem of determining the number of signals in high resolution array processing when the noise is spatially correlated having an unknown covariance matrix is examined by considering a model in which two sensor arrays are well separated such that their noise outputs are uncorrelated the authors develop a likelihood function whose maximum can be expressed in a very simple form involving the canonical correlation coefficients this likelihood function and a choice of penalty functions constitute a number of new information theoretic criteria suitable for the determination of the number of signals in an unknown correlated noise environment furthermore it is demonstrated that the new criteria are applicable in the case when only one sensor array is available

discrete derivatives and symmetries of difference equations we show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial leibniz rule suitable for finding the symmetries of discrete equations in this way we obtain a symmetry lie algebra defined in terms of shift operators isomorphic to that of the continuous heat equation

a theorem on the existence of trace form generalized entropies an analytic technique is proposed which allows to generate many new examples of entropic functionals generalizing the standard boltzmanngibbs entropy our approach is based on the existence of a g

detection of the number of coherent signals by the mdl principle an approach is presented to the problem of detecting the number of sources impinging on a passive sensor array that is based on j rissanen s 1983 minimum description length mdl principle the approach is applicable to any type of sources including the case of sources which are fully correlated referred to as the coherent signals case two slightly different detection criteria are derived both requiring the estimation of the locations of the sources the first is tailored to the detection problem per se whereas the second is tailored to the combined detection estimation problem consistency of the two criteria is proved and their performance is demonstrated by computer simulations

superconvergence of the iterated collocation methods for hammerstein equations in this paper we analyse the iterated collocation method for hammerstein equations with smooth and weakly singular kernels the paper expands the study which began in 16 concerning the superconvergence of the iterated galerkin method for hammerstein equations we obtain in this paper a similar superconvergence result for the iterated collocation method for hammerstein equations we also discuss the discrete collocation method for weakly singular hammerstein equations some discrete collocation methods for hammerstein equations with smooth kernels were given previously in 3 18

eigen inference for energy estimation of multiple sources in this paper a new method is introduced to blindly estimate the transmit power of multiple signal sources in multiantenna fading channels when the number of sensing devices and the number of available samples are sufficiently large compared to the number of sources recent advances in the field of large dimensional random matrix theory are used that result in a simple and computationally efficient consistent estimator of the power of each source a criterion to determine the minimum number of sensors and the minimum number of samples required to achieve source separation is then introduced simulations are performed that corroborate the theoretical claims and show that the proposed power estimator largely outperforms alternative power inference techniques

an information theoretic approach to source enumeration in array signal processing in this paper a new information theoretic algorithm is proposed for signal enumeration in array processing the approach is based on predictive description length pdl that is defined as the length of a predictive code for the set of observations we assume that several models with each model representing a certain number of sources will compete the pdl criterion is computed for the candidate models and is minimized over all models to select the best model and to determine the number of signals in the proposed method the correlation matrix is decomposed into two orthogonal components in the signal and noise subspaces the maximum likelihood ml estimates of the angles of arrival are used to find the projection of the sample correlation matrix onto the signal and noise subspaces the summation of the ml estimates of these matrices is the ml estimate of the correlation matrix this method can detect both coherent and noncoherent signals the proposed method can be used online and can be applied to time varying systems and target tracking

projection and multi projection methods for nonlinear integral equations on the half line abstract in this article galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear hammerstein type integral equation on the half line for both convolution and non convolution kernels using the space of piecewise polynomial subspace we prove that the approximate and iterated approximate solution in galerkin and collocation methods converges with the order o n r and o n 2 r respectively in the infinity norm where n 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation we improve these results further by considering multi projection methods in fact we show that iterated approximate solution in multi galerkin multi collocation converges with the order o n 4 r numerical results are presented to confirm the theoretical results

umbral calculus difference equations and the discrete schrodinger equation in this paper we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries the method is then applied to the schrodinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space time in this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case moreover systems that are integrable superintegrable or exactly solvable preserve these properties in the discrete case

detection tests for array processing in unknown correlated noise fields this paper introduces two eigenvalue based rules for estimating the number of signals impinging on an array of sensors along with a spatially correlated noise field the first rule called s is derived under the assumption that the noise spatial covariance is block diagonal or banded the assumption underlying the second detection rule named t is that the temporal correlation of the noise has a shorter length than that of the signals in both cases a matrix is built from the array output data covariances the smallest eigenvalue of which is equal to zero under the hypothesis that the source number is overestimated the sample distribution of the aforementioned smallest eigenvalue is derived and used to formulate the detection rules s and t both these rules are computationally quite simple additionally they can be used with a noncalibrated array the paper includes numerical examples that lend empirical support to the theoretical findings and illustrate the kind of performance that can be achieved by using the s and t detection rules

convergence analysis of discrete legendre spectral projection methods for hammerstein integral equations of mixed type in this paper we consider the discrete legendre spectral galerkin and discrete legendre spectral collocation methods to approximate the solution of mixed type hammerstein integral equation with smooth kernels the convergence of the discrete approximate solutions to the exact solution is discussed and the rates of convergence are obtained we have shown that when the quadrature rule is of certain degree of precision the rates of convergence for the legendre spectral galerkin and legendre spectral collocation methods are preserved we obtain superconvergence rates for the iterated discrete legendre galerkin solution by choosing the collocation nodes and quadrature points to be same we also obtain superconvergence rates for the iterated discrete legendre collocation solution

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

the lazard formal group universal congruences and special values of zeta functions a connection between the theory of formal groups and arithmetic number theory is established in particular it is shown how to construct general almkvist meurman type congruences for the universal bernoulli polynomials that are related with the lazard universal formal group 31 33 their role in the theory of l genera for multiplicative sequences is illustrated as an application sequences of integer numbers are constructed new congruences are also obtained useful to compute special values of a new class of riemann hurwitz type zeta functions

approximation methods for second kind weakly singular volterra integral equations abstract projection methods are applied to obtain the convergence rates for volterra integral equations with weakly singular kernels we consider galerkin and multi galerkin methods and their iterated versions to solve volterra integral equations with weakly singular kernels in the space of piecewise polynomials subspaces based on graded mesh we will show that the iterated multi galerkin method improves over iterated galerkin method in fact we show that iterated multi galerkin solution converges with the convergence rates o n 3 m and o n 3 m log n 2 for algebraic and logarithmic type kernels respectively we prove that iterated galerkin method for algebraic kernel converges with the convergence rate o n 2 m and for logarithmic type kernel converges with the convergence rate o n 2 m log n where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations theoretical results are justified by the numerical results

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

statistical analysis of the performance of information theoretic criteria in the detection of the number of signals in array processing the performances of the akaike 1974 information criterion and the minimum descriptive length criterion methods are examined the events which lead to erroneous decisions are considered and on the basis of these events the probabilities of error for the two criteria are derived the probabilities of the first two events are derived based on the asymptotic distribution of the sample eigenvalues of an estimated hermitian matrix it is further shown that the probabilities of missing and false alarm for these two criteria can be evaluated to a close approximation although the derivation of the probabilities of error is based on an asymptotic analysis the results are confirmed to be in very close agreement with computer simulation results

detection and localization in colored noise via generalized least squares a method for detection and localization of multiple signals in spatially colored noise by an arbitrary passive sensor array is presented the method also enables exploitation of prior knowledge that the signals are uncorrelated so as to improve the performance and to allow detection and localization even if the number of signals exceeds the number of sensors the estimation based on the generalized least squares criterion is both consistent and asymptotically efficient the detection is performed via the minimum description length mdl principle and is proved to be consistent simulation results confirming the theoretical results are included

generalized higher order bernoulli number pairs and generalized stirling number pairs abstract from a delta series f t and its compositional inverse g t hsu defined the generalized stirling number pair s n k s n k in this paper we further define from f t and g t the generalized higher order bernoulli number pair b n z b n z making use of the bell polynomials the potential polynomials as well as the lagrange inversion formula we give some explicit expressions and recurrences of the generalized higher order bernoulli numbers present the relations between the generalized higher order bernoulli numbers of both kinds and the corresponding generalized stirling numbers of both kinds and study the relations between any two generalized higher order bernoulli numbers moreover we apply the general results to some special number pairs and obtain series of combinatorial identities it can be found that the introduction of generalized bernoulli number pair and generalized stirling number pair provides a unified approach to lots of sequences in mathematics and as a consequence many known results are special cases of ours

on the kummer congruences and the stable homotopy of u we study the torsion free part of the stable homotopy groups of the space bu by considering upper and lower bounds the upper bound is furnished by the ring pk bu of coaction primitives into which nrs bu is mapped by the complex k theoretic hurewicz homomorphism xs bu pk bu we characterize pk b u in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical kummer congruences among the bernoulli numbers for a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in bu from among these we define families of classes constructed from regular neighborhoods of embeddings of iterated thom complexes in euclidean space employing techniques of duality theory we deduce that these two families correspond except possibly in the lowest dimensions under the hurewicz homomorphism which thus provides a link between the algebra and the geometry in the course of this work we greatly extend certain e invariant calculations of j f adams

source enumeration in array processing using a two step test we investigate the problem of enumerating source signals impinging on an array of sensors given noisy limited observations the assumption of a large sample size is indispensable for most of the existing approaches so that they suffer significant performance degradation when only a small number of samples is available random matrix theory can capture the asymptotic distributions of the eigenvalues accurately and provide good approximations even for a finite sample size in this paper we take into account the relationship and distributions of the noise and signal eigenvalues based on random matrix theory and derive a simple two step test numerical simulations demonstrate that it is capable of correctly determining the number of sources in the case of small sample sizes

on appell sequences of polynomials of bernoulli and euler type a construction of new sequences of generalized bernoulli polynomials of first and second kind is proposed these sequences share with the classical bernoulli polynomials many algebraic and number theoretical properties a class of euler type polynomials is also presented

numerical solution of nonlinear mixed volterra fredholm integral equations in complex plane via pqws abstract this paper presents an efficient numerical method for solving nonlinear mixed volterra integral equations in the complex plane the method is based on the fixed point iteration procedure in each iteration of this method periodic quasi wavelets are used as basis functions to approximate the solution also using the banach fixed point theorem some results concerning the error analysis are obtained finally some numerical examples show the implementation and accuracy of this method

detection of sources using bootstrap techniques source detection in array processing can be viewed as a test for equality of eigenvalues such a test is proposed based on a multiple test procedure that considers all pairwise comparisons between eigenvalues using the bootstrap to estimate the null distributions of the test statistics results in a procedure with minimal assumptions on the nature of the signal simulations show that the proposed test is superior to information theoretic criteria such as the mdl which are based on gaussian signals and large sample sizes performance in most cases exceeds the more powerful sphericity test

source enumeration in a correlated signal environment the authors present a simple method for determining the number of signals impinging on a uniform linear array that is applicable even in the extreme case of fully correlated signals this technique uses what they term modified rank sequences which is a modification of the construction implicit in the matrix decomposition method of di 1981 they prove that if a particular rank sequence stabilizes the last two terms of the sequence are equal to a value strictly less then the common row size of the defining block matrices then this value equals the number of signals provided that the number of signals has not exceeded a bresler macovski 1986 type bound using the above characterization of stability they formulate an algorithm that either determines the number of signals or indicates that the resolution capability of the algorithm has been exceeded they also provide theorems that show that under certain conditions a rank sequence can stabilize to a value strictly less than the number of signals this result allows them to find simple counterexamples to all of the existing rank sequence methods

flexible detection criterion for source enumeration in array processing two new procedures are proposed using random matrix theory to evaluate theoretically the performance of the bayesian information criterion bic for source enumeration in array processing the proposed procedures can predict precisely the estimation results of the bic also they shed light on the behavior of the bic in order to remedy the limitation of the bic that is the bic tends to underestimate the number of sources by one in most cases an extra parameter is introduced into the bic formulation which results in a new criterion for source enumeration referred to as the flexible detection criterion fdc by carefully choosing this parameter the fdc is capable of substantially reducing the probability of underestimation which is validated by theoretical analysis and simulations note that the fdc is consistent and has low computational complexity as the bic

detection of number of sources via exploitation of centro symmetry property forward backward fb averaging is a technique which can be employed in sensor array signal processing for exploiting a so called centro symmetry property of the signal subspace to improve parameter estimation accuracy although fb averaging effectively doubles the number of data samples it also introduces correlation among these data vectors this correlation complicates not only the performance analysis but also the detection of the number of sources when fb averaging is employed existing detection procedures such as sequential hypothesis sh likelihood ratio tests minimum description length mdl and an information criterion aic must be modified to correctly account for this preprocessing of the data herein proper exploitation of the centro symmetry property of the fb averaged covariance is described and a rigorous analysis is conducted it is shown that the general scheme of existing methods is still applicable to the fb averaging case but that the parameters e g degrees of freedom need to be adjusted

formal groups and z entropies we shall prove that the celebrated renyi entropy is the first example of a new family of infinitely many multi parametric entropies we shall call them the z entropies each of them under suitable hypotheses generalizes the celebrated entropies of boltzmann and renyi a crucial aspect is that every z entropy is composable tempesta 2016 ann phys 365 180 197 doi 10 1016 j aop 2015 08 013 1 this property means that the entropy of a system which is composed of two or more independent systems depends in all the associated probability space on the choice of the two systems only further properties are also required to describe the composition process in terms of a group law the composability axiom introduced as a generalization of the fourth shannon khinchin axiom postulating additivity is a highly non trivial requirement indeed in the trace form class the boltzmann entropy and tsallis entropy are the only known composable cases however in the non trace form class the z entropies arise as new entropic functions possessing the mathematical properties necessary for information theoretical applications in both classical and quantum contexts from a mathematical point of view composability is intimately related to formal group theory of algebraic topology the underlying group theoretical structure determines crucially the statistical properties of the corresponding entropies 1 http dx doi org 10 1016 j aop 2015 08 013

multiple scale analysis of dynamical systems on the lattice we propose a new approach to the multiple scale analysis of difference equations in the context of the finite operator calculus we derive the transformation formulae that map any given dynamical system continuous or discrete into a rescaled discrete system by generalizing a classical result due to jordan under suitable analytical hypotheses on the function space we consider the rescaled equations are of finite order our results are applied to the study of multiple scale reductions of dynamical systems and in particular to the case of a discrete nonlinear harmonic oscillator