on error saturation nonlinearities in nlms adaptation the effect of a saturation type error nonlinearity in the weight update equation in normalized least mean square nlms adaptation is investigated for system identification for a white gaussian data model nonlinear recursions are derived for the weight mean error and mean square deviation msd that include the effect of an error function erf saturation type nonlinearity on the error sequence driving the algorithm the nonlinear recursion for the msd is solved numerically and shown in excellent agreement with monte carlo simulations supporting the theoretical model assumptions the theory is extended to tracking a markov channel and accurately predicts the tracking behavior as well the saturation behavior of the algorithm is easily studied by varying a single parameter in the error function varying from a linear device to a hard limiter for the white data case the excess mean square error emse is simply related to the msd the tradeoff between the extent of error saturation steady state emse and algorithm convergence rate is studied using these results

another generalization of abelian equivalence the binomial coefficient of two words u and v is the number of times v occurs as a subsequence of u based on this classical notion we introduce the m binomial equivalence of two words refining the abelian equivalence two words x and y are m binomially equivalent if for all words v of length at most m the binomial coefficients of x and v and respectively y and v are equal the m binomial complexity of an infinite word x maps an integer n to the number of m binomial equivalence classes of factors of length n occurring in x we study the first properties of m binomial equivalence we compute the m binomial complexity of two classes of words sturmian words and pure morphic words that are fixed points of parikh constant morphisms like the thue morse word i e images by the morphism of all the letters have the same parikh vector we prove that the frequency of each symbol of an infinite recurrent word with bounded 2 binomial complexity is rational

the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

stochastic analysis of the lms algorithm with a saturation nonlinearity following the adaptive filter output this paper presents a statistical analysis of the least mean square lms algorithm with a zero memory scaled error function nonlinearity following the adaptive filter output this structure models saturation effects in active noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals the problem is first defined as a nonlinear signal estimation problem and the mean square error mse performance surface is studied analytical expressions are obtained for the optimum weight vector and the minimum achievable mse as functions of the saturation these results are useful for adaptive algorithm design and evaluation the lms algorithm behavior with saturation is analyzed for gaussian inputs and slow adaptation deterministic nonlinear recursions are obtained for the time varying mean weight and mse behavior simplified results are derived for white inputs and small step sizes monte carlo simulations display excellent agreement with the theoretical predictions even for relatively large step sizes the new analytical results accurately predict the effect of saturation on the lms adaptive filter behavior

adaptive filter performance with nonlinearities in the correlation multiplier it is well known that the correlation multiplier used to form the update signal in adaptive transversal filters need not be a true multiplier by using an approximate multiplier such as a sign bit only multiplier implementation can be made less costly in this paper we derive expressions for the convergence rate and misalignment noise of adaptive transversal filters with arbitrary correlation multiplier non linearities the analysis shows any such nonlinearities to always harm performance and quantifies the penalty suffered sign bit only multiplication suffers a significant penalty and power of two multiplication is suggested as a better alternative its performance is almost equal to that of a true multiplier and in a digital implementation at least it is only slightly more difficult to realize than a sign bit only multiplier the theory is developed in a way emphasizing physical understanding and has been verified to be in good agreement with simulation

saturation effects in lms adaptive echo cancellation for binary data the effect of a saturation type error nonlinearity in the weight update equation in least mean square lms adaptive echo cancellation is investigated for an independent binary data model a nonlinear difference equation is derived for the mean norm of the difference between the estimate and the unknown filter to be estimated by the algorithm the difference equation is evaluated numerically it is shown that far end binary data interference is much more deleterious to algorithm transient behavior than far end gaussian data interference the number of additional bits for the same cancellation convergence rates for binary versus gaussian interference of the same power is studied as a function of various system parameters algorithm convergence rates are studied as a function of an arbitrary probability density function pdf for the far end data it is shown that a binary pdf causes the worst degradation and a gaussian shaped pdf causes the least degradation

counting and equidistribution for quaternion algebras we aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting we prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure it leads to answering the sato tate conjectures in this case and contains the counting law of the universal family with a power savings error term

a compactness property of the k abelian monoids abstract the k abelian equivalence of words counting the numbers of occurrences of factors of length at most k has been analyzed in recent years from several different directions we continue this analysis the k abelian equivalence classes are known to constitute a monoid hence equations over these monoids are well defined we show that these monoids satisfy a compactness property each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems we give two proofs for this compactness result one is based the fact that the monoid can be embedded into the multiplicative monoid of matrices and the other directly on linear algebra the former method allows the application of hilbert s basis theorem the latter one in turn allows to conclude an upper bound for the size of the finite subsystem

asymptotic trace formula for the hecke operators given integers m n and k we give an explicit formula with an optimal error term with square root cancelation for the petersson trace formula involving the mth and nth fourier coefficients of an orthonormal basis of s k left n right the weight k newforms with fixed square free level n provided that 4 pi sqrt mn k o left k frac 1 3 right moreover we establish an explicit formula with a power saving error term for the trace of the hecke operator mathcal t n on s k left n right averaged over k in a short interval by bounding the second moment of the trace of mathcal t n over a larger interval we show that the trace of mathcal t n is unusually large in the range 4 pi sqrt n k o left n frac 1 6 right as an application for any fixed prime p coprime to n we show that there exists a sequence k n of weights such that the error term of weyl s law for mathcal t p is unusually large and violates the prediction of arithmetic quantum chaos in particular this generalizes the result of gamburd et al j eur math soc 1 1 51 85 1999 theorem 1 4 with an improved exponent

on error saturation nonlinearities in lms adaptation the effect of a saturation type error nonlinearity in the weight update equation in least mean squares lms adaptation is investigated for a white gaussian data model nonlinear difference equations are derived for the eight first and second moments which include the effect of an error function erf saturation type nonlinearity on the error sequence driving the algorithm a nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures the steady state second moment weight behavior is evaluated exactly for the erf nonlinearity using the above results the tradeoff between the extent of error saturation steady state excess mean square error and rate of algorithm convergence is studied the tradeoff shows that 1 starting with a sign detector the convergence rate is increased by nearly a factor of two for each additional bit and 2 as the number of bits is increased further the additional bit by very little in convergence speed asymptotically approaching the behavior of the linear algorithm

central limit theorems for elliptic curves and modular forms with smooth weight functions abstract in 11 the second and third named authors established a central limit theorem for the error term in the sato tate law for families of modular forms this method was adapted to families of elliptic curves in 3 by the first and second named authors in this context a central limit theorem was established only under a strong hypothesis going beyond the riemann hypothesis in the present paper we consider a smoothed version of the sato tate conjecture which allows us to overcome several limitations in particular for the smoothed version we are able to establish a central limit theorem for much smaller families of modular forms and we succeed in proving a theorem of this type for families of elliptic curves under the riemann hypothesis for l functions associated to hecke eigenforms for the full modular group

limit of the smallest eigenvalue of a large dimensional sample covariance matrix in this paper the authors show that the smallest if p leq n or the p n 1 th smallest if p n eigenvalue of a sample covariance matrix of the form 1 n xx tends almost surely to the limit 1 sqrt y 2 as n rightarrow infty and p n rightarrow y in 0 infty where x is a p times n matrix with iid entries with mean zero variance 1 and fourth moment finite also as a by product it is shown that the almost sure limit of the largest eigenvalue is 1 sqrt y 2 a known result obtained by yin bai and krishnaiah the present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices

nonlinear quantization effects in the lms and block lms adaptive algorithms a comparison digital implementations of the least mean square lms and block lms blms algorithms are compared with respect to finite word effects the algorithm stalling phenomenon is studied using gaussian data models and conditional expectation arguments it is shown that the blms algorithm requires 1 2 log sub 2 l k fewer bits for the same stalling behavior l block length and k lies between 0 2 and 1 0 depending on the precise definition of algorithm stalling on the other hand the lms algorithm requires log sub 2 l fewer bits than blms for the same level of saturation behavior transient response at algorithm initialization hence the lms algorithm requires 1 2 log sub 2 l k fewer bits than the blms algorithm for the same saturation and stalling effects

eigenvalues of graphs and a simple proof of a theorem of greenberg abstract in his ph d thesis greenberg proved that if x is the spectral radius of the universal cover x of a finite graph x then for each 0 a positive proportion depending only on x and of the eigenvalues of x have absolute value at least x in this paper we show that the same result holds true if we remove absolute from the previous result we also prove an analogue result for the smallest eigenvalues of x

transcendence of hecke operators in the big hecke algebra we prove a transcendence result of hecke operator t l over the ring generated by diamond operators over in a non cm component of the cyclotomic ordinary big p adic hecke algebra and we discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic invariant 2014

concentration of norms and eigenvalues of random matrices we prove concentration results for lpn operator norms of rectangular random matrices and eigenvalues of self adjoint random matrices the random matrices we consider have bounded entries which are independent up to a possible self adjointness constraint our results are based on an isoperimetric inequality for product spaces due to talagrand

transient and tracking performance analysis of the quantized lms algorithm for time varying system identification this paper investigates the statistical behavior of the finite precision lms adaptive filter in the identification of an unknown time varying stochastic system nonlinear recursions are derived for the mean and mean square behavior of the adaptive weights transient and tracking algorithm performance curves are generated from the recursions and shown to be in excellent agreement with monte carlo simulations our results demonstrate that linear models are inappropriate for analyzing the transient and the steady state algorithm behavior the performance curves indicate that the transient and tracking capabilities cannot be determined from perturbations about the infinite precision case it is shown that the transient phase of the algorithm increases as the digital wordlength or the speed of variation of the unknown system decrease design examples illustrate how the theory can be used to select the algorithm step size and the number of bits in the quantizer

on error saturation nonlinearities for lms adaptation in impulsive noise this correspondence extends the analytic results in n j bershad ldquoon error nonlinearities in lms adaptation rdquo ieee transactions on acoustics speech and signal processing vol assp 36 no 4 pp 440 452 april 1988 to least mean square lms adaptation in impulsive observation noise a scalar recursion for the weight misadjustment is derived for the white input data case monte carlo simulations verify the accuracy of the theoretical model the theoretical recursion is then used to study the effects of the impulse noise on algorithm convergence speed and steady state weight misadjustment for a wide variety of parameter values

an explicit trace formula of jacquet zagier type for hilbert modular forms abstract we give an exact formula of the average of adjoint l functions of holomorphic hilbert cusp forms with a fixed weight and a square free level which is a generalization of zagier s formula known for the case of elliptic cusp forms on sl 2 z as an application we prove that the satake parameters of hilbert cusp forms with a fixed weight and with growing square free levels are equidistributed in an ensemble constructed by values of the adjoint l functions

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

a nonlinear analytical model for the quantized lms algorithm the power of two step size case presents a study of the quantization effects in the finite precision lms algorithm with power of two step sizes deterministic nonlinear recursions are presented for the mean and second moment matrix of the weight vector about the wiener weight for white gaussian data models and small algorithm step size spl mu the numerical solutions of these recursions are shown to agree very closely with the monte carlo simulations during all phases of the adaptation process design examples demonstrate the selection of the number of quantizer bits and the adaptation step size spl mu to yield a desired transient behavior and cancellation performance the results obtained indicate that previous models are too conservative in predicting the converged mse for a given number of bits

elliptic curves over a finite field and the trace formula we prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k points of the curves contain a chosen subgroup these formulas express the moments in terms of traces of hecke operators for certain congruence subgroups of sl 2 z as our main technical input we prove an eichler selberg trace formula for a family of congruence subgroups of sl 2 z which include as special cases the groups 1 n and n our formulas generalize results of birch and ihara the case of the trivial subgroup and the full modular group and previous work of the authors the subgroups z 2z and z 2z 2 and congruence subgroups 0 2 0 4 we use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field generalizing results of vladu gekeler howe and others

on weight update saturation nonlinearities in lms adaptation the effect of a saturation type nonlinearity in the weight update equation in least mean square lms adaptation is investigated for a white gaussian data model nonlinear difference equations that include the effect of a 1 e sup ax saturation type nonlinearity on the update term driving the algorithm are derived for the weight first and second moments a nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures the steady state second moment weight behavior is evaluated approximately for the nonlinearity using these results the tradeoff between the extent of weight update saturation steady state excess mean square error and rate of algorithm convergence is studied for the same steady state misadjustment error the tradeoff results are discussed

an estimate for the average spectral measure of random band matrices for a class of random band matrices of band width w we prove regularity of the average spectral measure at scales w 0 99 and find its asymptotics at these scales

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

walk generating functions and spectral measures of infinite graphs let g be a locally finite graph with bounded vertex degrees its adjacency matrix a gives rise to a bounded self adjoint operator on l2 v g let e be the resolution of the identity for a and ev the vector in l2 v g having the v coordinate 1 and all other coordinates 0 the function uv e eu ev is called the spectral measure of g corresponding to vertices u v of g it is related to the generating function wuv z for the number of walks of given length between u and v the spectral measures are used to show the following the pairing theorem characterizing finite bipartite graphs in terms of their spectrum is generalized from finite to infinite graphs the average closed walk generating function can be defined in some cases generalizing the plancherel measure to not necessarily vertex transitive graphs some theorems about the convergence of graph measures are given with applications to the calculation of the expected eigenvalue distribution of a large finite random regular or bipartite semiregular graph and the calculation of spectral measures of several examples

on the lms algorithm with constant and variable leakage factor in a nonlinear environment this paper studies the application of the linear least mean square lms algorithm with leakage to a system having a memoryless saturation type nonlinearity this approach represents an interesting alternative to the nonlinear lms nllms algorithm the major drawback to implement the latter is that the model parameters of the system nonlinearity must be known in contrast the linear lms algorithm with leakage does not require such knowledge it permits to approximate the performance of the nllms by properly selecting a constant leakage factor value to cope with an eventual change of the environment a strategy for a variable leakage is also proposed several numerical simulations are presented with the aim of ratifying the feasibility of the proposed approach

the tracy widom law for some sparse random matrices consider the random matrix obtained from the adjacency matrix of a random d regular graph by multiplying every entry by a random sign the largest eigenvalue converges after proper scaling to the tracy widom distribution

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

the composite regressor algorithm for iir adaptive systems an adaptive iir algorithm called the composite regressor algorithm cra is developed the algorithm is a generalization of the common equation error a priori output error and a posteriori output error adaptive iir algorithms the cra is analyzed for convergence in a noiseless environment and for bias in a stochastic setting it is determined that by using a parameter called the regressor composition parameter a tradeoff can be obtained between the automatic convergence but large bias results of the equation error algorithm and the difficult convergence condition but small bias results of the output error algorithms in proving results for the cra it is shown that the a posteriori output error algorithm produces estimates with nonzero bias when the adaptive gain is small but bounded away from zero a convergence condition for the a priori output error algorithm is derived for the first time

statistical analysis of the single layer backpropagation algorithm i mean weight behavior the single layer backpropagation algorithm is a gradient descent method that adjusts the connection weights of a single layer perceptron to minimize the mean square error at the output it is similar to the standard least mean square al gorithm except the output of the linear combiner contains a differentiable nonlinearity in this paper we present a statis tical analysis of the mean weight behavior of the single layer backpropagation algorithm for gaussian input signals it is based on a nonlinear system identification model of the desired response which is capable of generating an arbitrary hyper plane decision boundary it is demonstrated that although the weights grow unbounded the algorithm on average quickly learns the correct hyperplane associated with the system iden tification model

a nonlinear analytical model for the quantized lms algorithm the arbitrary step size case this paper extends conditional moment techniques previously developed for the study of nonlinear versions of the lms algorithm to the study of the effects of quantizers in the finite precision case deterministic nonlinear recursions are derived for the mean and second moment matrix of the weight vector about the wiener weight for white gaussian data models and small algorithm step sizes spl mu these recursions are solved numerically and shown to be in very close agreement with the monte carlo simulations during all phases of the adaptation process a design example is presented that demonstrates how the theory can be used to select the number of quantizer bits and the adaptation step size spl mu to yield a desired transient behavior and cancellation performance

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

equidistribution theorems for holomorphic siegel modular forms for gsp_4 hecke fields and n level density this paper is a continuation of kim et al j inst math jussieu 2018 we supplement four results on a family of holomorphic siegel cusp forms for gsp 4 mathbb q first we improve the result on hecke fields namely we prove that the degree of hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of gsp 4 second we prove simultaneous vertical sato tate theorem namely we prove simultaneous equidistribution of hecke eigenvalues at finitely many primes third we compute the n level density of degree 4 spinor l functions and thus we can distinguish the symmetry type depending on the root numbers this is conditional on certain conjecture on root numbers fourth we consider equidistribution of paramodular forms in this case we can prove the conjecture on root numbers main tools are the equidistribution theorem in our previous work and shin templier s compos math 150 12 2003 2053 2014 work

the spectral edge of some random band matrices we study the asymptotic distribution of the eigenvalues of random hermitian periodic band matrices focusing on the spectral edges the eigenvalues close to the edges converge in distribution to the airy point process if and only if the band is sufficiently wide w n 5 6 otherwise a different limiting distribution appears

rate of convergence to the semi circular law a stochastic bound of order op n 1 2 for the kolmogorov distance between the spectral distribution function of an n n matrix from wigner ensemble and the distribution function of the semi circular law is obtained the result holds assuming that the twelfth moment of the entries of the matrix is uniformly bounded

eigenvalue spacings for quantized cat maps according to one of the basic conjectures in quantum chaos the eigenvalues of a quantized chaotic hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices we study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms cat maps our numerical experiments indicate that for generic choices of cat maps the unfolded consecutive spacing distribution in the irreducible components of the nth quantization given by the n dimensional weil representation approaches the goe gse law of random matrix theory for certain special arithmetic transformations related to the ramanujan graphs of lubotzky phillips and sarnak the experiments indicate that the unfolded consecutive spacing distribution follows poisson statistics we provide a sharp estimate in that direction

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

functional limit theorems for random regular graphs consider d uniformly random permutation matrices on n labels consider the sum of these matrices along with their transposes the total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices we consider limit theorems for various combinatorial and analytical properties of this graph or the matrix as n grows to infinity either when d is kept fixed or grows slowly with n in a suitable weak convergence framework we prove that the finite but growing in length sequences of the number of short cycles and of cyclically non backtracking walks converge to distributional limits we estimate the total variation distance from the limit using stein s method as an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix a key step in this latter derivation is an extension of the kahn szemeredi argument for estimating the second largest eigenvalue for all values of d and n

balances for fixed points of primitive substitutions an infinite word defined over a finite alphabet a is balanced if for any pair of factors of the same length and for any letter a in the alphabet a a 1 where a denotes the number of occurrences of the letter a in the word in this paper we generalize this notion and introduce a measure of balance for an infinite sequence in the case of fixed points of primitive substitutions we show that the asymptotic behaviour of this measure is in part ruled by the spectrum of the incidence matrix associated with the substitution connections with frequencies of letters and other balance properties are also discussed

on the optimum data nonlinearity in lms adaptation the effect of an arbitrary nonlinear operation on the data input to the weight update equation in the lms adaptive algorithm is investigated for a gaussian data model exact difference equations are derived for the weight first and second moments that include the effects of an arbitrary nonlinear operation on the data sequence the difference equations are used to obtain expressions for the transient behavior of the mean square error the mean square error is minimized over the choice of nonlinearity for a fixed transient behavior the best choice of nonlinearity is shown exactly to be of the form x 1 bx2 when the input data are white for an arbitrary data covariance matrix the optimum nonlinearity is shown to be linear when the product of the algorithm step size and input power is much less than unity

when kloosterman sums meet hecke eigenvalues by elaborating a two dimensional selberg sieve with asymptotics and equidistributions of kloosterman sums from ell adic cohomology as well as a bombieri vinogradov type mean value theorem for kloosterman sums in arithmetic progressions it is proved that for any given primitive hecke maass cusp form of trivial nebentypus the eigenvalue of the n th hecke operator does not coincide with the kloosterman sum mathrm kl 1 n for infinitely many squarefree n with at most 100 prime factors this provides a partial negative answer to a problem of katz on modular structures of kloosterman sums

localization of eigenvector centrality in networks with a cut vertex we show that eigenvector centrality exhibits localization phenomena on networks that can be easily partitioned by the removal of a vertex cut set the most extreme example being networks with a cut vertex three distinct types of localization are identified in these structures one is related to the well established hub node localization phenomenon and the other two are introduced and characterized here we gain insights into these problems by deriving the relationship between eigenvector centrality and katz centrality this leads to an interpretation of the principal eigenvector as an approximation to more robust centrality measures which exist in the full span of an eigenbasis of the adjacency matrix

statistical analysis of the single layer backpropagation algorithm ii mse and classification performance for pt i see ibid p 583 91 1993 the analysis of pt i is extended to the evaluation of the mean square error and the probability of correct classification it is shown that the mean square error and the corresponding performance surface are such that the single layer perceptron is prevented from correctly classifying with probability one until the weights converge at infinity

on the exponential generating function for non backtracking walks we derive an explicit formula for the exponential generating function associated with non backtracking walks around a graph we study both undirected and directed graphs our results allow us to derive computable expressions for non backtracking versions of network centrality measures based on the matrix exponential we find that eliminating backtracking walks in this context does not significantly increase the computational expense we show how the new measures may be interpreted in terms of standard exponential centrality computation on a certain multilayer network insights from this block matrix interpretation also allow us to characterize centrality measures arising from general matrix functions rigorous analysis on the star graph illustrates the effect of non backtracking and shows that unwanted localization effects can be eliminated when we restrict to non backtracking walks we also investigate the localization issue on synthetic networks

avoiding 2 binomial squares and cubes two finite words u v are 2 binomially equivalent if for all words x of length at most 2 the number of occurrences of x as a scattered subword of u is equal to the number of occurrences of x in v this notion is a refinement of the usual abelian equivalence a 2 binomial square is a word uv where u and v are 2 binomially equivalent in this paper considering pure morphic words we prove that 2 binomial squares resp cubes are avoidable over a 3 letter resp 2 letter alphabet the sizes of the alphabets are optimal