a 2 adic approach to the analysis of cyclic codes this paper describes how 2 adic numbers can be used to analyze the structure of binary cyclic codes and of cyclic codes defined over z sub 2 a a spl ges 2 the ring of integers modulo 2 sup a it provides a 2 adic proof of a theorem of mceliece that characterizes the possible hamming weights that can appear in a binary cyclic code a generalization of this theorem is derived that applies to cyclic codes over z sub 2 a that are obtained from binary cyclic codes by a sequence of hensel lifts this generalization characterizes the number of times a residue modulo 2 sup a appears as a component of an arbitrary codeword in the cyclic code the limit of the sequence of hensel lifts is a universal code defined over the 2 adic integers this code was first introduced by calderbank and sloane 1995 and is the main subject of this paper binary cyclic codes and cyclic codes over z sub 2 a are obtained from these universal codes by reduction modulo some power of 2 a special case of particular interest is cyclic codes over z sub 4 that are obtained from binary cyclic codes by means of a single hensel lift the binary images of such codes under the gray isometry include the kerdock preparata and delsart goethals codes these are nonlinear binary codes that contain more codewords than any linear code presently known fundamental understanding of the composition of codewords in cyclic codes over z sub 4 is central to the search for more families of optimal codes this paper also constructs even unimodular lattices from the hensel lift of extended binary cyclic codes that are self dual with all hamming weights divisible by 4 the leech lattice arises in this way as do extremal lattices in dimensions 32 through 48

fuzzy rule selection by multi objective genetic local search algorithms and rule evaluation measures in data mining this paper shows how a small number of simple fuzzy if then rules can be selected for pattern classification problems with many continuous attributes our approach consists of two phases candidate rule generation by rule evaluation measures in data mining and rule selection by multi objective evolutionary algorithms in our approach first candidate fuzzy if then rules are generated from numerical data and prescreened using two rule evaluation measures i e confidence and support in data mining then a small number of fuzzy if then rules are selected from the prescreened candidate rules using multi objective evolutionary algorithms in rule selection we use three objectives maximization of the classification accuracy minimization of the number of selected rules and minimization of the total rule length thus the task of multi objective evolutionary algorithms is to find a number of non dominated rule sets with respect to these three objectives the main contribution of this paper is to propose an idea of utilizing the two rule evaluation measures as prescreening criteria of candidate rules for fuzzy rule selection an arbitrarily specified number of candidate rules can be generated from numerical data for high dimensional pattern classification problems through computer simulations we demonstrate that such a prescreening procedure improves the efficiency of our approach to fuzzy rule selection we also extend a multi objective genetic algorithm moga in our former studies to a multi objective genetic local search mogls algorithm where a local search procedure adjusts the selection i e inclusion or exclusion of each candidate rule furthermore a learning algorithm of rule weights i e certainty factors is combined with our mogls algorithm such extensions to our moga for fuzzy rule selection are another contribution of this paper

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

binary kloosterman sums modulo 256 and coefficients of the characteristic polynomial kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory we use stickelberger s theorem and the gross koblitz formula to determine the value of the binary kloosterman sum at a modulo 64 modulo 128 and modulo 256 in terms of coefficients of the characteristic polynomial of a

scalable tsk fuzzy modeling for very large datasets using minimal enclosing ball approximation in order to overcome the difficulty in takagi sugeno kang tsk fuzzy modeling for large datasets scalable tsk stsk fuzzy model training is investigated in this study based on the core set based minimal enclosing ball meb approximation technique the specified l2 norm penalty based insensitive criterion is first proposed for tsk model training and it is found that such tsk fuzzy model training can be equivalently expressed as a center constrained meb problem with this finding an stsk fuzzy model training algorithm which is called stsk for large or very large datasets is then proposed by using the core set based meb approximation technique the proposed algorithm has two distinctive advantages over classical tsk fuzzy model training algorithms the maximum space complexity for training is not reliant on the size of the training dataset and the maximum time complexity for training is linear with the size of the training dataset as confirmed by extensive experiments on both synthetic and real world regression datasets

generalized linear array models with applications to multidimensional smoothing summary data with an array structure are common in statistics and the design or regression matrix for analysis of such data can often be written as a kronecker product factorial designs contingency tables and smoothing of data on multidimensional grids are three such general classes of data and models in such a setting we develop an arithmetic of arrays which allows us to define the expectation of the data array as a sequence of nested matrix operations on a coefficient array we show how this arithmetic leads to low storage high speed computation in the scoring algorithm of the generalized linear model we refer to a generalized linear array model and apply the methodology to the smoothing of multidimensional arrays we illustrate our procedure with the analysis of three data sets mortality data indexed by age at death and year of death spatially varying microarray background data and disease incidence data indexed by age at death year of death and month of death

deep tsk fuzzy classifier with stacked generalization and triplely concise interpretability guarantee for large data although takagi sugeno kang tsk fuzzy classifier has been applied to a wide range of practical scenarios how to enhance its classification accuracy and interpretability simultaneously is still a challenging task in this paper based on the powerful stacked generalization principle a deep tsk fuzzy classifier d tsk fc is proposed to achieve the enhanced classification accuracy and triplely concise interpretability for fuzzy rules d tsk fc consists of base building units just like the existing popular deep learning d tsk fc can be built in a layer by layer way in terms of the stacked generalization principle the training set plus random shifts obtained from random projections of prediction results of current base building unit are presented as the input of the next base building unit the hidden layer in each base building unit of d tsk fc is represented by triplely concise interpretable fuzzy rules in the sense of randomly selected features with the fixed five fuzzy partitions random rule combinations and the same input space kept in every base building unit of d tsk fc the output layer of each base building unit can be learnt quickly by least learning machine llm besides benefiting from llm d tsk fc s deep learning can be well scaled up for large datasets our extensive experimental results witness the power of the proposed deep tsk fuzzy classifier

eigencurves for two parameter sturm liouville equations this paper concerns two parameter sturm liouville problems of the form p x y q x y lambda r x mu y quad a leqslant x leqslant b with self adjoint boundary conditions at a and b the set of lambda mu in bf r 2 for which there exists a nontrivial y satisfying the differential equation and the boundary conditions turns out to be a countable union of graphs of analytic functions our focus is on these graphs which are termed eigencurves in the literature although eigencurves have been used in a variety of ways for about a century they seem comparatively underdeveloped in their own right our plan is to give motivation for the topic elementary properties of eigencurves illustrations on a simple example first studied by richardson in 1918 and since then by several authors and some natural questions which may whet the reader s appetite some of these questions lead to new types of inverse sturm liouville problems

selecting fuzzy if then rules for classification problems using genetic algorithms this paper proposes a genetic algorithm based method for selecting a small number of significant fuzzy if then rules to construct a compact fuzzy classification system with high classification power the rule selection problem is formulated as a combinatorial optimization problem with two objectives to maximize the number of correctly classified patterns and to minimize the number of fuzzy if then rules genetic algorithms are applied to this problem a set of fuzzy if then rules is coded into a string and treated as an individual in genetic algorithms the fitness of each individual is specified by the two objectives in the combinatorial optimization problem the performance of the proposed method for training data and test data is examined by computer simulations on the iris data of fisher

distributions of singular values for some random matrices the singular value decomposition is a matrix decomposition technique widely used in the analysis of multivariate data such as complex space time images obtained in both physical and biological systems in this paper we examine the distribution of singular values of low rank matrices corrupted by additive noise past studies have been limited to uniform uncorrelated noise using diagrammatic and saddle point integration techniques we extend these results to heterogeneous and correlated noise sources we also provide perturbative estimates of error bars on the reconstructed low rank matrix obtained by truncating a singular value decomposition 1999 the american physical society

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

guarantees for the kronecker fast johnson lindenstrauss transform using a coherence and sampling argument abstract in the recent paper jin kolda ward arxiv 1909 04801 it is proved that the kronecker fast johnson lindenstrauss transform kfjlt is in fact a johnson lindenstrauss transform which had previously only been conjectured in this paper we provide an alternative proof of this for when the kfjlt is applied to kronecker vectors using a coherence and sampling argument our proof yields a different bound on the embedding dimension which can be combined with the bound in the paper by jin et al to get a better bound overall as a stepping stone to proving our result we also show that the kfjlt is a subspace embedding for matrices with columns that have kronecker product structure lastly we compare the kfjlt to four other sketch techniques in numerical experiments on both synthetic and real world data

a new approach for mutual information analysis of large dimensional multi antenna channels this paper adresses the behavior of the mutual information of correlated multiple input multiple output mimo rayleigh channels when the numbers of transmit and receive antennas converge to infin at the same rate using a new and simple approach based on poincare nash inequality and on an integration by parts formula it is rigorously established that the mutual information when properly centered and rescaled converges to a standard gaussian random variable simple expressions for the centering and scaling parameters are provided these results confirm previous evaluations based on the powerful but nonrigorous replica method it is believed that the tools that are used in this paper are simple robust and of interest for the communications engineering community

recursive fast algorithm and the role of the tensor product the use of the tensor product for modeling and designing fft algorithms is addressed the benefit of the tensor product approach lies in the strong connection between certain tensor product constructs and important computer architectures the scope of the tensor product approach is generalized to include a much larger class of fast recursive algorithms this greatly enhances the versatility of the tensor product technique and brings many different algorithms to the level of understanding and flexibility enjoyed by the fft

duadic codes over z sub 2k duadic codes constitute a well known family of binary cyclic codes they are generalized in this correspondence to the setting of abelian codes over the ring z sub 2k self duality isoduality and type ii properties are studied

an efficient deterministic test for kloosterman sum zeros we propose a simple deterministic test for deciding whether or not a non zero element a in mathbb f 2 n or mathbb f 3 n is a zero of the corresponding kloosterman sum over these fields and analyse its complexity the test seems to have been overlooked in the literature for binary fields the test has an expected operation count dominated by just two mathbb f 2 n multiplications when n is odd with a slightly higher cost for even extension degrees making its repeated invocation the most efficient method to date to find a non trivial kloosterman sum zero in these fields the analysis depends on the distribution of sylow p subgroups in two corresponding families of elliptic curves which we prove using a theorem due to howe

random matrix theory of multi antenna communications the ricean channel the use of multi antenna arrays in wireless communications through disordered media promises huge increases in the information transmission rate it is therefore important to analyse the information capacity of such systems in realistic situations of microwave transmission where the statistics of the transmission amplitudes channel may be coloured here we present an approach that provides analytic expressions for the statistics i e the moments of the distribution of the mutual information for general gaussian channel statistics the mathematical method applies tools developed originally in the context of coherent wave propagation in disordered media such as random matrix theory and replicas although it is valid formally for large antenna numbers this approach produces extremely accurate results even for arrays with as few as two antennas we also develop a method to analytically optimize over the input signal distribution which enables us to calculate analytic capacities when the transmitter has knowledge of the statistics of the channel the emphasis of this paper is on elucidating the novel mathematical methods used we do this by analysing a specific case when the channel matrix is a complex gaussian with arbitrary mean and unit covariance which is usually called the ricean channel

fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains a time dependent coordinate transformation of a constant coefficient hyperbolic system of equations which results in a variable coefficient system of equations is considered by applying the energy method well posed boundary conditions for the continuous problem are derived summation by parts sbp operators for the space and time discretization together with a weak imposition of boundary and initial conditions using simultaneously approximation terms sats lead to a provable fully discrete energy stable conservative finite difference scheme we show how to construct a time dependent sat formulation that automatically imposes boundary conditions when and where they are required we also prove that a uniform flow field is preserved i e the numerical geometric conservation law ngcl holds automatically by using sbp sat in time and space the developed technique is illustrated by considering an application using the linearized euler equations the sound generated by moving boundaries numerical calculations corroborate the stability and accuracy of the new fully discrete approximations

capacity scaling in mimo wireless systems under correlated fading previous studies have shown that single user systems employing n element antenna arrays at both the transmitter and the receiver can achieve a capacity proportional to n assuming independent rayleigh fading between antenna pairs we explore the capacity of dual antenna array systems under correlated fading via theoretical analysis and ray tracing simulations we derive and compare expressions for the asymptotic growth rate of capacity with n antennas for both independent and correlated fading cases the latter is derived under some assumptions about the scaling of the fading correlation structure in both cases the theoretic capacity growth is linear in n but the growth rate is 10 20 smaller in the presence of correlated fading we analyze our assumption of separable transmit receive correlations via simulations based on a ray tracing propagation model results show that empirical capacities converge to the limit capacity predicted from our asymptotic theory even at moderate n 16 we present results for both the cases when the transmitter does and does not know the channel realization

p adic estimates of hamming weights in abelian codes over galois rings a generalization of mceliece s theorem on the p adic valuation of hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by wilson along with a technique known as trace averaging introduced here the original theorem of mceliece concerned cyclic codes over prime fields delsarte and mceliece later extended this to abelian codes over finite fields calderbank li and poonen extended mceliece s original theorem to cover cyclic codes over the rings zopf2 d wilson strengthened their results and extended them to cyclic codes over zopf p d and katz strengthened wilson s results and extended them to abelian codes over zopfp d it is natural to ask whether there is a single analogue of mceliece s theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers in this paper this question is answered affirmatively a single theorem for abelian codes over galois rings is presented this theorem contains all previously mentioned results and more

the shannon s mutual information of a multiple antenna time and frequency dependent channel an ergodic operator approach consider a random non centered multiple antenna radio transmission channel assume that the deterministic part of the channel is itself frequency selective and that the random multipath part is represented by an ergodic stationary vector process in the hilbert space l2 one can associate to this channel a random ergodic self adjoint operator having a so called integrated density of states ids shannon s mutual information per receive antenna of this channel coincides then with the integral of a log function with respect to the ids in this paper it is shown that when the numbers of antennas at the transmitter and at the receiver tend to infinity at the same rate the mutual information per receive antenna tends to a quantity that can be identified and in fact is closely related to that obtained within the random matrix approach i telatar eur trans telecommun 10 585 1999 this result can be obtained by analyzing the behavior of the stieltjes transform of the ids in the regime of the larg

first principles model of time dependent variations in transmission through a fluctuating scattering environment fading is the time dependent variation in transmitted signal strength through a complex medium due to interference or temporally evolving multipath scattering in this paper we use random matrix theory rmt to establish a first principles model for fading including both universal and nonuniversal effects this model provides a more general understanding of the most common statistical models rayleigh fading and rice fading and provides a detailed physical basis for their parameters we also report experimental tests on two ray chaotic microwave cavities the results show that our rmt model agrees with the rayleigh and rice models in the high loss regime but there are strong deviations in low loss systems where the rmt approach describes the data well

p adic valuation of weights in abelian codes over spl zopf p sup d counting polynomial techniques introduced by wilson are used to provide analogs of a theorem of mceliece mceliece s original theorem relates the greatest power of p dividing the hamming weights of words in cyclic codes over gf p to the length of the smallest unity product sequence of nonzeroes of the code calderbank li and poonen presented analogs for cyclic codes over zopf 2d using various weight functions hamming lee and euclidean weight as well as count of occurrences of a particular symbol some of these results were strengthened by wilson who also considered the alphabet zopf pd for p an arbitrary prime these previous results new strengthened versions and generalizations are proved here in a unified and comprehensive fashion for the larger class of abelian codes over zopf pd with p any prime for abelian codes over zopf 4 combinatorial methods for use with counting polynomials are developed these show that the analogs of mceliece s theorem obtained by wilson for hamming weight lee weight and symbol counts and the analog obtained here for euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose fourier transforms have a specified support

constructing l2 svm based fuzzy classifiers in high dimensional space with automatic model selection and fuzzy rule ranking in this paper a new scheme for constructing parsimonious fuzzy classifiers is proposed based on the l2 support vector machine l2 svm technique with model selection and feature ranking performed simultaneously in an integrated manner in which fuzzy rules are optimally generated from data by l2 svm learning in order to identify the most influential fuzzy rules induced from the svm learning two novel indexes for fuzzy rule ranking are proposed and named as alpha values and omega values of fuzzy rules in this paper the alpha values are defined as the lagrangian multipliers of the l2 svm and adopted to evaluate the output contribution of fuzzy rules while the omega values are developed by considering both the rule base structure and the output contribution of fuzzy rules as a prototype based classifier the l2 svm based fuzzy classifier evades the curse of dimensionality in high dimensional space in the sense that the number of support vectors which equals the number of induced fuzzy rules is not related to the dimensionality experimental results on high dimensional benchmark problems have shown that by using the proposed scheme the most influential fuzzy rules can be effectively induced and selected and at the same time feature ranking results can also be obtained to construct parsimonious fuzzy classifiers with better generalization performance than the well known algorithms in literature

information capacity of a random signature multiple input multiple output channel a closed form expression for information capacity of the random signature multiple input multiple output channel is given a direct calculation of the capacity is provided additional to the proof involving the capacity of minimum mean square error systems provided by verdu and shamai see ieee trans inform theory vol 45 p 622 40 1999

symmetric boolean functions we present an extensive study of symmetric boolean functions especially of their cryptographic properties our main result establishes the link between the periodicity of the simplified value vector of a symmetric boolean function and its degree besides the reduction of the amount of memory required for representing a symmetric function this property has some consequences from a cryptographic point of view for instance it leads to a new general bound on the order of resiliency of symmetric functions which improves siegenthaler s bound the propagation characteristics of these functions are also addressed and the algebraic normal forms of all their derivatives are given we finally detail the characteristics of the symmetric functions of degree at most 7 for any number of variables most notably we determine all balanced symmetric functions of degree less than or equal to 7

solving nonstationary classification problems with coupled support vector machines many learning problems may vary slowly over time in particular some critical real world applications when facing this problem it is desirable that the learning method could find the correct input output function and also detect the change in the concept and adapt to it we introduce the time adaptive support vector machine ta svm which is a new method for generating adaptive classifiers capable of learning concepts that change with time the basic idea of ta svm is to use a sequence of classifiers each one appropriate for a small time window but in contrast to other proposals learning all the hyperplanes in a global way we show that the addition of a new term in the cost function of the set of svms that penalizes the diversity between consecutive classifiers produces a coupling of the sequence that allows ta svm to learn as a single adaptive classifier we evaluate different aspects of the method using appropriate drifting problems in particular we analyze the regularizing effect of changing the number of classifiers in the sequence or adapting the strength of the coupling a comparison with other methods in several problems including the well known stagger dataset and the real world electricity pricing domain shows the good performance of ta svm in all tested situations

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

a fast learning algorithm for deep belief nets we show how to use complementary priors to eliminate the explaining away effects that make inference difficult in densely connected belief nets that have many hidden layers using complementary priors we derive a fast greedy algorithm that can learn deep directed belief networks one layer at a time provided the top two layers form an undirected associative memory the fast greedy algorithm is used to initialize a slower learning procedure that fine tunes the weights using a contrastive version of the wake sleep algorithm after fine tuning a network with three hidden layers forms a very good generative model of the joint distribution of handwritten digit images and their labels this generative model gives better digit classification than the best discriminative learning algorithms the low dimensional manifolds on which the digits lie are modeled by long ravines in the free energy landscape of the top level associative memory and it is easy to explore these ravines by using the directed connections to display what the associative memory has in mind

penalized least squares estimation of volterra filters and higher order statistics volterra filters vfs and higher order statistics hos are important tools for nonlinear analysis processing and modeling despite their highly desirable properties the transfer of vfs and hos to real world signal processing problems has been hindered by the requirement of very large data records needed to obtain reliable estimates the identification of vfs and the estimation of hos both fall into the category of ill posed estimation problems we develop penalized least squares pls estimation methods for vfs and hos it is shown that pls is a very effective way to incorporate prior information of the problem at hand without directly constraining the estimation procedure hence pls produces much more reliable estimates the main contributions of this paper are the development of appropriate penalizing functionals and cross validation procedures for pls based vf identification and hos estimation

improved parallel qr method for large least squares problems involving kronecker products abstract a new algorithm is presented for the efficient solution of large least squares problems in which the coefficient matrix of the linear system is a kronecker product of two smaller dimension matrices the solution algorithm is based on qr factorizations of the smaller dimension matrices near perfect load balancing is achieved by exploiting a commutativity property of the kronecker product and communication requirements are minimized by employing a binary exchange algorithm for matrix transposition the parallel algorithm is presented and timing results are shown from test runs on an intel i 860 computer

tensor product basis approximations for volterra filters the paper studies approximations for a class of nonlinear filters known as volterra filters although the volterra filter provides a relatively simple and general representation for nonlinear filtering it is often highly overparameterized due to the large number of parameters the utility of the volterra filter is limited the overparameterization problem is addressed in the paper using a tensor product basis approximation tpba in many cases a volterra filter may be well approximated using the tpba with far fewer parameters hence the tpba offers considerable advantages over the original volterra filter in terms of both implementation and estimation complexity furthermore the tpba provides useful insight into the filter response the paper studies the crucial issue of choosing the approximation basis several methods for designing an appropriate approximation basis and error bounds on the resulting mean square output approximation error are derived certain methods are known to be nearly optimal

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

a genetic fuzzy linguistic combination method for fuzzy rule based multiclassifiers fuzzy set theory has been widely and successfully used as a mathematical tool to combine the outputs provided by the individual classifiers in a multiclassification system by means of a fuzzy aggregation operator however to the best of our knowledge no fuzzy combination method has been proposed which is composed of a fuzzy rule based system we think this can be a very promising research line as it allows us to benefit from the key advantage of fuzzy systems i e their interpretability by using a fuzzy linguistic rule based classification system as a combination method the resulting classifier ensemble would show a hierarchical structure and the operation of the latter component would be transparent to the user moreover for the specific case of fuzzy multiclassification systems the new approach could also become a smart way to allow standard fuzzy classifiers to deal with high dimensional problems avoiding the curse of dimensionality as the chance to perform classifier selection at class level is also incorporated into the method we conduct comprehensive experiments considering 20 uci datasets with different dimensionality where our approach improves or at least maintains accuracy while reducing complexity of the system and provides some interpretability insight into the multiclassification system reasoning mechanism the results obtained show that this approach is able to compete with the state of the art multiclassification system selection and fusion methods in terms of accuracy thus providing a good interpretability accuracy tradeoff

a multilayered neuro fuzzy classifier with self organizing properties a novel self organizing neuro fuzzy multilayered classifier sonefmuc is suggested in this paper which is composed of small scale interconnected fuzzy neuron classifiers fncs arranged in layers the model provides a different perspective for generating a new class of hierarchical classifiers with multilevel classifiers combination at each layer parent fncs are combined to construct a descendant fnc at the next layer with enhanced classification qualities the generic fncs exhibit an original structure including four modules 1 the fuser aggregates the decision support outputs of the parent fncs using a fusion operator 2 the data splitting module divides data into correctly classified patterns with high degree of confidence which are handled by the fuser and ambiguous ones 3 the last two modules are realized by fuzzy rule based systems and implement a neuro fuzzy classifier within each fnc used to improve the classification accuracy of ambiguous patterns the former performs feature transformations to an intermediate output space while the latter provides the decision supports of non confident patterns unlike traditional classifiers the classification mapping is accomplished in sonefmuc by performing successive decision fusions and feature transformations sonefmuc structure is determined sequentially via a self constructing learning algorithm structure learning inherently implements feature selection considering the most informative attributes as model inputs to improve classification accuracy the models obtained after structure learning are optimized using a parameter learning scheme based on genetic algorithms the effectiveness of our approach is tested on a set of benchmark classification problems experimental results show that the proposed neuro fuzzy classifier is favorably compared to other well known classification techniques of the literature

the ternary golay code the integers mod 9 and the coxeter todd lattice the 12 dimensional coxeter todd lattice can be obtained by lifting the ternary golay code to a code over the integers mod 9 and applying construction a

hadamard matrices and the spectrum of quadratic symmetric polynomials over finite fields abstract in this article we present a beautiful connection between hadamard matrices and exponential sums of quadratic symmetric polynomials over galois fields this connection appears when the recursive nature of these sequences is analyzed we calculate the spectrum for the hadamard matrices that dominate these recurrences the eigenvalues depend on the legendre symbol and the quadratic gauss sum over finite field extensions in particular these formulas allow us to calculate closed formulas for the exponential sums over galois field of quadratic symmetric polynomials finally in the particular case of finite extensions of the binary field we show that the corresponding hadamard matrix is a permutation away from a classical construction of these matrices

asymptotic mutual information statistics of separately correlated rician fading mimo channels the asymptotic probability distribution of the mutual information of a separately correlated rician fading multiple input multiple output mimo channel is addressed the mean and variance of the mutual information are derived when the number of transmit and receive antennas grows asymptotically large while their ratio approaches a finite constant this derivation is based on the replica method widely used in theoretical physics and more recently in the analysis of communication systems code division multiple access cdma and mimo though the replica method allows to analyze complex systems in a comparatively simple way some authors pointed out that its assumptions are not always rigorous it is shown that the mutual information converges asymptotically to a gaussian distribution under mild technical conditions which are tantamount to assuming that the spatial correlation structure has no asymptotically dominant eigenmodes the accuracy of the asymptotic approach is assessed by numerical results it is shown that the approximation is very accurate in a wide range of system settings even when the number of transmit and receive antennas is as small as a few units

restoring images degraded by spatially variant blur restoration of images that have been blurred by the effects of a gaussian blurring function is an ill posed but well studied problem any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation fast and effective algorithms then exist for determining the original image by preconditioned iterative methods if the blurring function is spatially variant however then the problem is more difficult in this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support this assumption leads to a discrete problem involving a banded matrix we devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm numerical examples are given related to the hubble space telescope hst wide field planetary camera the algorithms that we develop are applicable to other ill posed integral equations as well

low complexity polynomial expansion detector with deterministic equivalents of the moments of channel gram matrix for massive mimo uplink we consider a low complexity polynomial expansion pe detector in a massive multiple input multiple output mimo uplink channel in contrast to most massive mimo systems in the literature where single antenna user equipments ues are assumed multiple antenna ues are employed in this paper moreover the channel between a base station bs and a ue is a jointly correlated rician fading channel the pe detector reduces the computational complexity of the minimum mean square error mmse detector by replacing the matrix inversion with an approximate matrix polynomial the coefficients of the approximate matrix polynomial are computed from the deterministic equivalents of the moments of the channel gram matrix we use operator valued free probability which is a more general version of free probability to derive the deterministic equivalents in particular we use the operator valued moment cumulant formula the proposed low complexity pe detector is easy to compute simulation results show that the proposed detector can achieve performance close to the mmse detector

on perturbation bounds of kronecker product linear systems and their level 2 condition numbers the classical perturbation theory of linear systems ax b is extended to kronecker product linear systems email protected b x d upper bounds are derived for the normwise and componentwise condition number the nearness to singularity and the sensitivity of the condition numbers are analyzed

an improvement on the bounds of weil exponential sums over galois rings with some applications we present an upper bound for weil type exponential sums over galois rings of characteristic p sup 2 which improves on the analog of the weil carlitz uchiyama bound for galois rings obtained by kumar helleseth and calderbank 1995 a more refined bound expressed in terms of genera of function fields and an analog of mceliece s 1971 theorem on the divisibility of the homogeneous weights of codewords in trace codes over z sub p sup 2 are also derived these results lead to an improvement on the estimation of the minimum distance of certain trace codes over z sub p sup 2 and the bounds on the correlation of certain nonlinear p ary sequences

spectral efficiency of cdma with random spreading the cdma channel with randomly and independently chosen spreading sequences accurately models the situation where pseudonoise sequences span many symbol periods furthermore its analysis provides a comparison baseline for cdma channels with deterministic signature waveforms spanning one symbol period we analyze the spectral efficiency total capacity per chip as a function of the number of users spreading gain and signal to noise ratio and we quantify the loss in efficiency relative to an optimally chosen set of signature sequences and relative to multiaccess with no spreading white gaussian background noise and equal power synchronous users are assumed the following receivers are analyzed a optimal joint processing b single user matched filtering c decorrelation and d mmse linear processing

on the capacity achieving covariance matrix for frequency selective mimo channels using the asymptotic approach in this contribution an algorithm for evaluating the capacity achieving input covariance matrices for frequency selective rayleigh mimo channels is proposed in contrast with the flat fading rayleigh case no closed form expressions for the eigenvectors of the optimum input covariance matrix are available classically both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding in this paper it is proposed to optimize w r t the input covariance matrix a large system approximation of the average mutual information derived by moustakas and simon the validity of this asymptotic approximation is clarified thanks to gaussian large random matrices methods it is shown that the approximation is a strictly concave function of the input covariance matrix and that the average mutual information evaluated at the argmax of the approximation is equal to the capacity of the channel up to a o 1 t term where t is the number of transmit antennas an algorithm based on an iterative waterfilling scheme is proposed to maximize the average mutual information approximation and its convergence studied numerical simulation results show that even for a moderate number of transmit and receive antennas the new approach provides the same results as direct maximization approaches of the average mutual information

recent results on balanced symmetric boolean functions this paper focuses on the balancedness of symmetric boolean functions we prove a conjecture presented by canteaut and videau which states that the balanced symmetric boolean functions of fixed algebraic degree are trivially balanced when the number of variables is large enough denoted by sigma n d the n variable elementary symmetric boolean function of degree d as an application of this result to elementary symmetric boolean functions we show that all the trivially balanced elementary symmetric boolean functions are of the form sigma 2 t 1 l 1 2 t where t and l are any positive integers it implies that cusick et al s conjecture which claims that sigma 2 t 1 l 1 2 t is the only nonlinear balanced elementary symmetric boolean functions is equivalent to the conjecture that all the balanced elementary symmetric boolean functions are trivially balanced

diophantine equations with binomial coefficients and perturbations of symmetric boolean functions this paper presents a study of perturbations of symmetric boolean functions in particular it establishes a connection between exponential sums of these perturbations and diophantine equations of the form sum l 0 n binom n l x l 0 where x j belongs to some fixed bounded subset gamma of mathbb z the concepts of trivially balanced symmetric boolean function and sporadic balanced boolean function are extended to this type of perturbations an observation made by canteaut and videau for symmetric boolean functions of fixed degree is extended to be specific it is proved that excluding the trivial cases balanced perturbations of fixed degree do not exist when the number of variables grows some sporadic balanced perturbations are presented finally a beautiful but unexpected identity between exponential sums for perturbations of two different symmetric boolean functions is also included in this work

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

on the capacity achieving covariance matrix for rician mimo channels an asymptotic approach in this paper the capacity achieving input covariance matrices for coherent block fading correlated multiple input multiple output mimo rician channels are determined in contrast with the rayleigh and uncorrelated rician cases no closed form expressions for the eigenvectors of the optimum input covariance matrix are available classically both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding in the asymptotic regime where the number of transmit and receive antennas converge to infinity at the same rate new results related to the accuracy of the approximation of the average mutual information are provided based on the accuracy of this approximation an attractive optimization algorithm is proposed and analyzed this algorithm is shown to yield an effective way to compute the capacity achieving matrix for the average mutual information and numerical simulation results show that even for a moderate number of transmit and receive antennas the new approach provides the same results as direct maximization approaches of the average mutual information

a statistical mechanics approach to large system analysis of cdma multiuser detectors we present a theory based on statistical mechanics to evaluate analytically the performance of uncoded fully synchronous randomly spread code division multiple access cdma multiuser detectors with additive white gaussian noise awgn channel under perfect power control and in the large system limit application of the replica method a tool developed in the literature of statistical mechanics allows us to derive analytical expressions for the bit error rate as well as the multiuser efficiency of the individually optimum io and jointly optimum jo multiuser detectors over the whole range of noise levels the information theoretic capacity of the randomly spread cdma channel and the performance of decorrelating and linear minimum mean square error mmse detectors are also derived in the same replica formulation thereby demonstrating validity of the statistical mechanical approach

sharp p divisibility of weights in abelian codes over bbz p d bbz a theorem of mceliece on the p divisibility of hamming weights in cyclic codes over fp is generalized to abelian codes over zopf p dzopf this work improves upon results of helleseth kumar moreno shanbhag calderbank li poonen wilson and katz these previous attempts are not sharp in general i e do not report the full extent of the p divisibility except in special cases nor do they give accounts of the precise circumstances under which they do provide best possible results this paper provides sharp results on p divisibilities of hamming weights and counts of any particular symbol for an arbitrary abelian code over zopf p dzopf it also presents sharp results on 2 divisibilities of lee and euclidean weights for abelian codes over zopf 4zopf

the spectra of multiplicative attribute graphs abstract a multiplicative attribute graph is a random graph in which vertices are represented by random words of length t in a finite alphabet and the probability of adjacency is a symmetric function t t 0 1 these graphs are a generalization of stochastic kronecker graphs and both classes have been shown to exhibit several useful real world properties we establish asymptotic bounds on the spectra of the adjacency matrix and the normalized laplacian matrix for these two families of graphs under certain mild conditions as an application we examine various properties of the stochastic kronecker graph and the multiplicative attribute graph including the diameter clustering coefficient chromatic number and bounds on low congestion routing

cyclic codes over z sub 4 locator polynomials and newton s identities certain nonlinear binary codes contain more codewords than any comparable linear code presently known these include the kerdock 1972 and preparata 1968 codes that can be very simply constructed as binary images under the gray map of linear codes over z sub 4 that are defined by means of parity checks involving galois rings this paper describes how fourier transforms on galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes these methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of z sub 4 linear codes with length 2 sup m m odd and size 2 2 sup m 1 5m 2 the gray image of the code of length 32 is the best 64 2 sup 37 code that is presently known this paper also determines the exact minimum lee distance of the linear codes over z sub 4 that are obtained from the extended binary two and three error correcting bch codes by hensel lifting the gray image of the hensel lift of the three error correcting bch code of length 32 is the best 64 2 sup 32 code that is presently known this code also determines an extremal 32 dimensional even unimodular lattice

on multistage fuzzy neural network modeling in the past couple of years there has been increasing interest in the fusion of neural networks and fuzzy logic most of the existing fuzzy neural network fnn models have been proposed to implement different types of single stage fuzzy reasoning mechanisms and inevitably they suffer from the dimensionality problem when dealing with complex real world problems to address the problem fnn modeling based on multistage fuzzy reasoning msfr is pursued here and two hierarchical network models namely incremental type and aggregated type are introduced the new models called multistage fnn msfnn model a hierarchical fuzzy rule set that allows the consequence of a rule passed to another as a fact through the intermediate variables from the stipulated input output data pairs they can generate an appropriate fuzzy rule set through structure and parameter learning procedures proposed in this paper in addition we have particularly addressed the input selection problem of these two types of multistage network models and proposed two efficient methods for them the effectiveness of the proposed msfnn models in handling high dimensional problems is demonstrated through various numerical simulations

kronecker products unitary matrices and signal processing applications discrete unitary transforms are extensively used in many signal processing applications and in the development of fast algorithms kronecker products have proved quite useful in this semitutorial paper we briefly review properties of kronecker products and direct sums of matrices which provide a compact notation in treating patterned matrices a generalized matrix product which inherits some useful algebraic properties from the standard kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula is then introduced the notation is intimately related to sparse matrix factorizations and examples are included illustrating the utility of the new notation in signal processing applications finally some novel characteristics of hadamard transforms and polyadic permutations are derived in the framework of kronecker products

the matrix equation xa bx r and its applications abstract we study the well known sylvester equation xa bx r in the case when a and b are given and r is known up to its first n 1 rows we prove new results on the existence and uniqueness of x our results essentially state that in case a is a nonderogatory matrix there always exists a solution to this equation a solution is uniquely determined by its first row x 1 and there is an interesting relationship between x 1 and the rows of r we also give a complete characterization of the nonsingularity of x in this case as applications of our results we develop direct methods for constructing symmetrizers and commuting matrices computing the characteristic polynomial of a matrix and finding the numbers of common eigenvalues between a and b some well known important results on symmetrizers bezoutians and inertia are recovered as special cases

kronecker products and shuffle algebra relates three classical concepts viz mixed radix number system kronecker product of matrices and perfect shuffle it presents an algebra which describes the hardware organization of the computation of a product mv where m is a matrix in kronecker product form and v is a vector the algebraic formalism describes both the blockwise structure of the computation and the various possible connection patterns

impact of antenna correlation on the capacity of multiantenna channels this paper applies random matrix theory to obtain analytical characterizations of the capacity of correlated multiantenna channels the analysis is not restricted to the popular separable correlation model but rather it embraces a more general representation that subsumes most of the channel models that have been treated in the literature for arbitrary signal to noise ratios snr the characterization is conducted in the regime of large numbers of antennas for the low and high snr regions in turn we uncover compact capacity expansions that are valid for arbitrary numbers of antennas and that shed insight on how antenna correlation impacts the tradeoffs among power bandwidth and rate

tight bounds for chevalley warning ax katz type estimates with improved applications following the work of chevalley and warning ax obtained a bound on the p divisibility of exponential sums involving multi variable polynomials of fixed degree d over a finite field of characteristic p this bound was subsequently improved by katz more recently moreno and moreno and adolphson and sperber derived bounds that in many instances improved upon the ax katz result here we derive a tight bound on the p divisibility of the exponential sums while exact computation of this bound requires the solution of a system of modular equations approximations are provided which in several classes of examples improve on the results of chevalley and warning ax and katz adolphson and sperber and moreno and moreno all of the above results readily translate into bounds on the p divisibility of the number of zeros of multi variable polynomials an important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight in particular we give classes of examples for which the moreno moreno bound is tight it is important to note that we have also found applications of our results to coding theory computation of the covering radius of certain codes and to waring s problem over finite fields these will be described elsewhere

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

low rank estimation of higher order statistics low rank estimators for higher order statistics are considered in this paper the bias variance tradeoff is analyzed for low rank estimators of higher order statistics using a tensor product formulation for the moments and cumulants in general the low rank estimators have a larger bias and smaller variance than the corresponding full rank estimator and the mean squared error can be significantly smaller this makes the low rank estimators extremely useful for signal processing algorithms based on sample estimates of the higher order statistics the low rank estimators also offer considerable reductions in the computational complexity of such algorithms the design of subspaces to optimize the tradeoffs between bias variance and computation is discussed and a noisy input noisy output system identification problem is used to illustrate the results

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

time and lag recursive computation of cumulants from a state space model time and lag recursive algorithms for the computation of the cumulants of the state vector and the output process of a multiple input multiple output time varying state space model are derived using a kronecker product representation for the cumulants of vector processes the noise processes are not assumed to be stationary symmetry relations for the cumulants of vector processes are discussed computational aspects are examined in detail it is conjectured that the recursive equations derived will be useful in developing state estimators and even optimal controllers based on higher order statistics

improved estimates via exponential sums for the minimum distance of z sub 4 linear trace codes an upper hound for weil type exponential sums over galois rings was derived by kumar helleseth and calderbank see ibid vol 41 no 3 p 456 1995 this bound leads directly to an estimate for the minimum distance of z sub 4 linear trace codes an improved minimum distance estimate is presented first mceliece s result on the divisibility of the weights of binary cyclic codes is extended to z sub 4 trace codes the divisibility result is then combined with the techniques of serre 1983 and of moreno and moreno see ibid vol 40 no 11 p 1101 1994 to derive the improved minimum distance estimate the improved estimate is tight for the kerdock code as well as for the delsarte goethals codes

affinity learning with diffusion on tensor product graph in many applications we are given a finite set of data points sampled from a data manifold and represented as a graph with edge weights determined by pairwise similarities of the samples often the pairwise similarities which are also called affinities are unreliable due to noise or due to intrinsic difficulties in estimating similarity values of the samples as observed in several recent approaches more reliable similarities can be obtained if the original similarities are diffused in the context of other data points where the context of each point is a set of points most similar to it compared to the existing methods our approach differs in two main aspects first instead of diffusing the similarity information on the original graph we propose to utilize the tensor product graph tpg obtained by the tensor product of the original graph with itself since tpg takes into account higher order information it is not a surprise that we obtain more reliable similarities however it comes at the price of higher order computational complexity and storage requirement the key contribution of the proposed approach is that the information propagation on tpg can be computed with the same computational complexity and the same amount of storage as the propagation on the original graph we prove that a graph diffusion process on tpg is equivalent to a novel iterative algorithm on the original graph which is guaranteed to converge after its convergence we obtain new edge weights that can be interpreted as new learned affinities we stress that the affinities are learned in an unsupervised setting we illustrate the benefits of the proposed approach for data manifolds composed of shapes images and image patches on two very different tasks of image retrieval and image segmentation with learned affinities we achieve the bull s eye retrieval score of 99 99 percent on the mpeg 7 shape dataset which is much higher than the state of the art algorithms when the data points are image patches the ncut with the learned affinities not only significantly outperforms the ncut with the original affinities but it also outperforms state of the art image segmentation methods

further results on the asymptotic mutual information of rician fading mimo channels rician fading multiple input multiple output mimo channels with kronecker decomposable covariance structure are considered in this study to derive information measures characterizing their achievable rate analytic results are obtained in the asymptotic setting corresponding to infinitely many transmit and receive antennas whose numbers approach a finite limiting ratio more specifically the asymptotic mean and variance of the mutual information are derived in this asymptotic regime and it is shown that the corresponding probability distribution converges to a gaussian distribution all these results are based on the replica method whose mathematical principles are discussed in this study the ergodic capacity is also addressed by deriving an iterative water filling algorithm yielding the capacity achieving input covariance matrix under an average power constraint resorting to the asymptotic gaussianity of the mutual information distribution an analytic expression of the outage capacity is derived based on the mean and variance of the mutual information finally numerical results are reported to assess the level of approximation of the analytic results in comparison with those obtained by monte carlo simulation

z4 linear codes obtained as projections of kerdock and delsarte goethals codes abstract the kerdock and delsarte goethals codes can be very simply constructed as binary images under a certain natural map called the gray map of linear codes over z 4 the integers modulo 4 we consider the gray images of linear codes over z 4 obtained from the kerdock and delsarte goethals codes by projection on a hyperplane for m odd certain gray images have the same weight distribution as duals of extended binary bch codes of length 2 m but are not equivalent to these codes inequivalence follows from a general theorem identifying binary linear codes that are not gray images of linear codes over z 4