on differentiability of convex operators abstract the main known results on differentiability of continuous convex operators f from a banach space x to an ordered banach space y are due to j m borwein and n k kirov our aim is to prove some supergeneric results i e to show that sometimes the set of g teaux or frechet nondifferentiability points is not only a first category set but also smaller in a stronger sense for example we prove that if y is countably daniell and the space l x y of bounded linear operators is separable then each continuous convex operator f x y is frechet differentiable except for a null angle small set some applications of such supergeneric results are shown

on the pointwise limits of bivariate lagrange projectors a linear algebra proof is given of the fact that the nullspace of a finite rank linear projector on polynomials in two complex variables is an ideal if and only if the projector is the bounded pointwise limit of lagrange projectors i e projectors whose nullspace is a radical ideal i e the set of all polynomials that vanish on a certain given finite set a characterization of such projectors is also given in the real case more generally a characterization is given of those finite rank linear projectors on polynomials in d complex variables with nullspace an ideal that are the bounded pointwise limit of lagrange projectors the characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences

on frechet differentiability of lipschitz maps between banach spaces a well known open question is whether every countable collection of lipschitz functions on a banach space x with separable dual has a common point of frechet differentiability we show that the answer is positive for some infinite dimensional x previously even for collections consisting of two functions this has been known for finite dimensional x only although for one function the answer is known to be affirmative in full generality our aims are achieved by introducing a new class of null sets in banach spaces called f null sets whose definition involves both the notions of category and measure and showing that the required differentiability holds almost everywhere with respect to it we even obtain existence of frechet derivatives of lipschitz functions between certain infinite dimensional banach spaces no such results have been known previously our main result states that a lipschitz map between separable banach spaces is frechet differentiable f almost everywhere provided that it is regularly gateaux differentiable f almost everywhere and the gateaux derivatives stay within a norm separable space of operators it is easy to see that lipschitz maps of x to spaces with the radon nikodym property are gateaux differentiable f almost everywhere moreover gateaux differentiability implies regular gateaux differentiability with exception of another kind of negligible sets so called a porous sets the answer to the question is therefore positive in every space in which every r porous set is f null we show that this holds for c k with k countable compact the tsirelson space and for all subspaces of co but that it fails for hilbert spaces

commuting pairs and triples of matrices and related varieties abstract in this note we show that the set of all commuting d tuples of commuting n n matrices that are contained in an n dimensional commutative algebra is a closed set and therefore gerstenhaber s theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs we show that the variety of commuting triples of 4 4 matrices is irreducible we also study the variety of n dimensional commutative subalgebras of m n f and show that it is irreducible of dimension n 2 n for n 4 but reducible of dimension greater than n 2 n for n 7

fast coupled adaptation for sparse impulse responses using a partial haar transform this work presents a novel scheme for identifying the impulse response of a sparse channel the scheme consists of two adaptive filters operating sequentially the first adaptive filter adapts using a partial haar transform of the input and yields an estimate of the location of the peak of the sparse impulse response the second adaptive filter is then centered about this estimate both filters are short in comparison to the delay uncertainty of the unknown channel the principle advantage of this scheme is that two short adaptive filters can be used instead of one long adaptive filter resulting in faster overall convergence and reduced computational complexity and storage the scheme is analyzed in detail for a least mean squares lms lms lms type of structure although it can be implemented using any combination of adaptive algorithms monte carlo simulations are shown to be in good agreement with the theoretical model for the behavior of the peak estimating filter as well as for the mean square error mse behavior of the second filter

stochastic modeling of the transform domain varepsilon rm lms algorithm this paper presents a statistical analysis of the transform domain least mean square tdlms algorithm resulting in a more accurate model than those discussed in the current open literature the motivation to analyze such an algorithm comes from the fact that the tdlms presents a higher convergence speed for correlated input signals as compared with other adaptive algorithms possessing a similar computational complexity such a fact makes it a highly competitive alternative to some applications approximate analytical models for the first and second moments of the filter weight vector are obtained the tdlms algorithm has an orthonormal transformation stage accomplishing a decomposition of the input signal into distinct frequency bands in which the interband samples are practically uncorrelated on the other hand the intraband samples are correlated the larger the number of bands the higher their correlation the model is then derived taking into account such a correlation requiring that a high order hyperelliptic integral be computed in addition to the proposed model an approximate procedure for computing high order hyperelliptic integrals is presented a regularization parameter is also considered in the model expressions permitting to assess its impact on the adaptive algorithm behavior an upper bound for the step size control parameter is also obtained through simulation results the accuracy of the proposed model is assessed

adaptive filtering in subbands with critical sampling analysis experiments and application to acoustic echo cancellation an exact analysis of the critically subsampled two band modelization scheme is given and it is demonstrated that adaptive cross filters between the subbands are necessary for modelization with small output errors it is shown that perfect reconstruction filter banks can yield exact modelization these results are extended to the critically subsampled multiband schemes and important computational savings are seen to be achieved by using good quality filter banks the problem of adaptive identification in critically subsampled subbands is considered and an appropriate adaptation algorithm is derived the authors give a detailed analysis of the computational complexity of all the discussed schemes and experimentally verify the theoretical results that are obtained the adaptive behavior of the subband schemes that were tested is discussed

v subdifferentials of convex operators abstract for any convex operator from a convex set c of a topological vector space e into another one f endowed with a convex cone f a notion of v subdifferential v a of at a c is introduced although it is equivalent to the notion of e sub differential when f r it enjoys many important properties which are not satisfied by the e subdifferential whenever int f o the nonvacuity of v a is proved whenever v is a neighbourhood of zero in f and belongs to a large class of mappings analogous to the class of lower semicontinuous real valued convex functions other properties of v subdifferentials are studied and applications to differentiability of are made

rapid and brief communication wavelet feature domain adaptive noise reduction using learning algorithm for text independent speaker recognition in this paper a type of thresholding method is developed for adaptive noise reduction here we propose a new type thresholding method unlike the standard thresholding functions the new thresholding functions are infinitely differentiable gradient based adaptive learning algorithms are presented to seek the optimal solution for noise reduction furthermore the learning algorithm can be used for any speaker data derived from discrete wavelet transform it is demonstrated that 94 correct classification rates can be achieved by the use of the first 32 variation features in talung database

svd compression unitary transforms and computational complexity the search for fast unitary transforms and the need for data compression in linear systems are complementary issues compression requires the definition of a threshold dependent on the condition number which is invariant over the unitary group with respect to this threshold it is shown that the svd is the optimal tool considerations in connection with the kronecker product and direct sum of unitary matrices show that the computational complexity of unitary transforms is entropy like in nature thereby indicating that the o n log n complexity unitary transforms are dense over the unitary group

approximating commuting operators the problem of approximating m tuples of commuting n n complex matrices by commuting m tuples of generic matrices is studied we narrow the gap for commuting triples by showing that they can be perturbed if n 29

on approximately simultaneously diagonalizable matrices abstract a collection a1 a2 ak of n n matrices over the complex numbers c has the asd property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable such a collection must perforce be commuting we show by a direct matrix proof that the asd property holds for three commuting matrices when one of them is 2 regular dimension of eigenspaces is at most 2 corollaries include results of gerstenhaber and neubauer sethuraman on bounds for the dimension of the algebra generated by a1 a2 ak even when the asd property fails our techniques can produce a good bound on the dimension of this subalgebra for example we establish dim c a 1 a k 5 n 4 for commuting matrices a1 ak when one of them is 2 regular this bound is sharp one offshoot of our work is the introduction of a new canonical form the h form for matrices over an algebraically closed field the h form of a matrix is a sparse jordan like upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular the jordan form itself does not accommodate this

a primal dual proximal algorithm for sparse template based adaptive filtering application to seismic multiple removal unveiling meaningful geophysical information from seismic data requires to deal with both random and structured noises as their amplitude may be greater than signals of interest primaries additional prior information is especially important in performing efficient signal separation we address here the problem of multiple reflections caused by wave field bouncing between layers since only approximate models of these phenomena are available we propose a flexible framework for time varying adaptive filtering of seismic signals using sparse representations based on inaccurate templates we recast the joint estimation of adaptive filters and primaries in a new convex variational formulation this approach allows us to incorporate plausible knowledge about noise statistics data sparsity and slow filter variation in parsimony promoting wavelet frames the designed primal dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters the approach demonstrates significantly good performance in low signal to noise ratio conditions both for simulated and real field data

convex analysis in groups and semigroups a sampler we define convexity canonically in the setting of monoids we show that many classical results from convex analysis hold for functions defined on such groups and semigroups rather than only vector spaces some examples and counter examples are also discussed

similarity of matrices over local rings abstract wasow investigated the problem of when for a pair of matrices of analytic functions pointwise similarity implies analytic similarity by using the topology of spec r this is studied for an arbitrary commutative ring results of wasow ohm schneider and friedland are generalized in this setting a canonical form is determined for matrices satisfying the wasow condition

super efficiency in vector optimization we introduce a new concept of efficiency in vector optimization this concept super efficiency is shown to have many desirable properties in particular we show that in reasonable settings the super efficient points of a set are norm dense in the efficient frontier we also provide a chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex

polynomials in a matrix and its commutant abstract given any norm u on the space of linear transformations matrices over a finite dimensional complex vector space and given commuting transformations t s and any e 0 there is a transformation t e such that u t e t p e such that p e t e s this result has implications for the theory of the numerical radius and other operator radii its proof involves the following fact of interest in itself if the jordan canonical form of s has k blocks then the transformations with k distinct eigenvalues are dense in the commutant of s

two generated commutative subalgebras of mn f abstract for two commuting matrices a b m n k k a field it has been known that dim k k a b n we show here that dim k cent m n k a b n and that equality for one implies equality for the other we also give both a module theoretic and a geometric characterization of those algebras for which equality is obtained furthermore we present a new proof of a result of gerstenhaber

on the uniqueness and stability of an inverse problem in photo acoustic tomography this article deals with the uniqueness and stability of the solution of a problem of optimal control related to the photo acoustic tomography process we prove stability results of the optimal solution with respect to the source and to the observation data and we compute the corresponding derivatives

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

wavelet transform domain adaptive fir filtering this paper presents and studies two new wavelet transform domain least mean square lms algorithms the algorithms exploit the special sparse structure of the wavelet transform of wide classes of correlation matrices and their cholesky factors in order to compute a whitening transformation of the input data in the wavelet domain and minimize computational complexity the procedures differ in the exact estimates they use and in the way they identify the data dependent whitening transformation the first approach explicitly computes a sparse estimate of the wavelet domain correlation matrix of the input process it then computes the cholesky factor of that matrix and uses its inverse to whiten the input the complexity of this first approach is o n log sup 2 n in contrast the second approach computes a sparse estimate of the cholesky factor of the wavelet domain correlation matrix of the input process directly this second approach has a computational complexity of o n log n floating point operations however it requires a more complex bookkeeping procedure both algorithms have a convergence rate that is faster than that of time domain lms and discrete fourier transform dft or discrete cosine transform dct based lms procedures the paper compares the two procedures and analyzes their mean and mean square performance

the minimal dimension of maximal commutative subalgebras of full matrix algebras abstract let f be a field and a a maximal commutative subalgebra of the full matrix algebra mn f it is shown that dim a 2n 2 3 1 it is also shown that if the radical of a has cube zero then dim a 3n 2 3 4 and that this result is best possible for infinitely many natural numbers n

on varieties of commuting nilpotent matrices abstract let n d n be the variety of all d tuples of commuting nilpotent n n matrices it is well known that n d n is irreducible if d 2 if n 3 or if d 3 and n 4 on the other hand n 3 n is known to be reducible for n 13 we study in this paper the reducibility of n d n for various values of d and n in particular we prove that n d n is reducible for all d n 4 in the case d 3 we show that it is irreducible for n 6

a variety of commuting triples abstract the problem of approximating triples of commuting n n matrices by generic matrices is equivalent to the problem of whether the variety c 3 n of commuting triples is irreducible the answer to the problem is known to be positive for dimensions no greater than 5 and negative for dimensions no smaller than 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimension 6 as well all the proofs are given over an arbitrary algebraically closed field of characteristic zero

region based wavelet packet adaptive algorithm for identification of sparse impulse responses identification of systems with sparse impulse response encounters large applicability numerous techniques have been proposed to identify such systems efficiently one strategy that leads to specially good results is to detect the active nonzero response samples and update only the corresponding adaptive coefficients wavelet based approaches have been shown to be specially effective to this end this paper proposes a new region based wavelet packet rbwp algorithm for efficient identification of systems with sparse impulse responses with arbitrary frequency spectra and with any delay of the effective response the discrete wavelet packet transform dwpt is adaptively tailored to the energy distribution of the unknown system s response spectrum the new algorithm leads to a reduced number of active weights and to a reduced computational complexity when compared with competing wavelet based algorithms monte carlo simulation results show good performances regarding convergence speed and robustness to design parameter choice

jordan forms for mutually annihilating nilpotent pairs abstract in this paper we completely characterize all possible pairs of jordan canonical forms for mutually annihilating nilpotent pairs i e pairs a b of nilpotent matrices such that ab ba 0

orthogonal time varying filter banks and wavelet packets considers the construction of orthogonal time varying filter banks by examining the time domain description of the two channel orthogonal filter bank the authors find it possible to construct a set of orthogonal boundary filters which allows to apply the filter bank to one sided or finite length signals without redundancy or distortion the method is constructive and complete there is a whole space of orthogonal boundary solutions and there is considerable freedom for optimization this may be used to generate subband tree structures where the tree varies over time and to change between different filter sets the authors also show that the iteration of discrete time time varying filter banks gives continuous time bases just as in the stationary case this gives rise to wavelet or wavelet packet bases for half line and interval regions

on tensors of border rank l in cm n l notice this is the author s version of a work that was accepted for publication in linear algebra and its applications changes resulting from the publishing process such as peer review editing corrections structural formatting and other quality control mechanisms may not be reflected in this document changes may have been made to this work since it was submitted for publication a definitive version was subsequently published in linear algebra and its applications june 23 2011 doi 10 1016 j laa 2011 05 013 the original publication is available at www elsevier com

signal modeling with filtered discrete fractional noise processes filtered versions of fractionally differenced gaussian noise fdgn processes are studied fractionally differenced gaussian noise is a discrete time equivalent of fractional brownian motion filtered versions of such processes are ideally suited for modeling signals with different short term and long term correlation structure two iterative algorithms for estimating the parameters of filtered fdgn processes are described the first technique is based on the expectation maximization algorithm it converges to a stationary point of the log likelihood function corresponding to the parameters of the model the second technique is a computationally efficient approximate approach it is found to converge experimentally but no proof of its convergence is given the usefulness of filtered fdgn models and the performance of the proposed iterative algorithms are illustrated by fitting filtered fdgn models to speech waveforms and other data corresponding to natural phenomena

performance of transform domain lms adaptive digital filters in this paper we analyze the performance particularly the convergence behavior of the transform domain least mean square lms adaptive digital filter adf using the discrete fourier transform and discrete orthogonal transforms such as discrete cosine and sine transforms we first obtain the optimum wiener solution and the minimum mean squared error mse in the transform domain it is shown that the two minimum mse s in the time and transform domains are identical independently of the transforms used we then study the convergence conditions and the steady state excess mse s of the transform domain lms trlms algorithms both for the cases of having a constant and a time varying convergence factors when a constant convergence factor is used the convergence behaviors of the lms and trlms adf s appear to be almost identical provided that each has an appropriate value of the convergence factor depending on the transform used also based on the concept of a self orthogonalizing algorithm in the transform domain it is shown that the convergence speed of the trlms adf can be improved significantly for the same excess mse as that of the lms adf in addition we compare the computational complexities of the lms and trlms adf s finally we investigate by computer simulation the effects of system parameter values and different transforms on the convergence behavior of the trlms adf

transform domain lms algorithm the concept of transform domain adaptive filtering is introduced in certain applications filtering in the transform domain results in great improvements in convergence rate over the conventional time domain adaptive filtering the relationship between several existing frequency domain adaptive filtering algorithms is established applications of the discrete fourier transform dft and the discrete cosine transform dct domain adaptive filtering algorithms in the areas of speech processing and adaptive line enhancers are discussed

a compact null set containing a differentiability point of every lipschitz function we prove that in a euclidean space of dimension at least two there exists a compact set of lebesgue measure zero such that any real valued lipschitz function defined on the space is differentiable at some point in the set such a set is constructed explicitly

stability and convergence analysis of transform domain lms adaptive filters with second order autoregressive process in this paper the stability and convergence properties of the class of transform domain least mean square lms adaptive filters with second order autoregressive ar process are investigated it is well known that this class of adaptive filters improve convergence property of the standard lms adaptive filters by applying the fixed data independent orthogonal transforms and power normalization however the convergence performance of this class of adaptive filters can be quite different for various input processes and it has not been fully explored in this paper we first discuss the mean square stability and steady state performance of this class of adaptive filters we then analyze the effects of the transforms and power normalization performed in the various adaptive filters for both first order and second order ar processes we derive the input asymptotic eigenvalue distributions and make comparisons on their convergence performance finally computer simulations on ar process as well as moving average ma process and autoregressive moving average arma process are demonstrated for the support of the analytical results

on varieties of commuting triples abstract the problem of approximating triples of commuting n n matrices by triples of generic matrices is equivalent to that whether the variety c 3 n of triples of commuting matrices is irreducible it is known that the variety is irreducible for n 7 and reducible for n 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we prove that c 3 8 is irreducible

pairs of commuting nilpotent matrices and hilbert function abstract let k be an infinite field there has been recent study of the family h n k of pairs of commuting nilpotent n n matrices relating this family to the fibre h n of the punctual hilbert scheme a n hilb n a 2 over the point np of the symmetric product sym n a 2 where p is a point of the affine plane a 2 v baranovsky the variety of pairs of commuting nilpotent matrices is irreducible transform groups 6 1 2001 3 8 r basili on the irreducibility of commuting varieties of nilpotent matrices j algebra 268 1 2003 56 80 a premet nilpotent commuting varieties of reductive lie algebras invent math 154 3 2003 653 683 in this study a pair of commuting nilpotent matrices a b is related to an artinian algebra k a b there has also been substantial study of the stratification of the local punctual hilbert scheme h n by the hilbert function as j briancon description de hilb n c x y invent math 41 1 1977 45 89 and others however these studies have been hitherto separate we first determine the stable partitions i e those for which p itself is the partition q p of a generic nilpotent element of the centralizer of the jordan nilpotent matrix j p we then explore the relation between h n k and its stratification by the hilbert function of k a b suppose that dim k k a b n and that k is algebraically closed of characteristic 0 or large enough p we show that a generic element of the pencil a b k has jordan partition the maximum partition p h whose diagonal lengths are the hilbert function h of k a b

differentiability properties of convex operators abstract in this paper we show that a convex operator between a weak asplund space and a suitable ordered locally convex space is gateaux differentiable on a dense g set of its domain

a universal differentiability set in banach spaces with separable dual abstract we show that any non zero banach space with a separable dual contains a totally disconnected closed and bounded subset s of hausdorff dimension 1 such that every lipschitz function on the space is frechet differentiable somewhere in s

an elementary method to compute the algebra generated by some given matrices and its dimension abstract we give an efficient solution to the following problem given x 1 x d and y some n by n matrices can we determine if y is in the unital algebra generated by x 1 x d as a subalgebra of all n by n matrices the solution also gives an easy method for computing the dimension of this algebra

nilpotent matrices having a given jordan type as maximum commuting nilpotent orbit abstract the jordan type of a nilpotent matrix is the partition giving the sizes of its jordan blocks we study pairs of partitions p q where q q p is the jordan type of a generic nilpotent matrix a commuting with a nilpotent matrix b of jordan type p t kosir and p oblak have shown that q has parts that differ pairwise by at least two such partitions which are also known as super distinct or rogers ramanujan are exactly those that are stable or self large in the sense that q q q in 2012 p oblak formulated a conjecture concerning the cardinality of q 1 q when q has two parts and proved some special cases r zhao refined this to posit that the partitions in q 1 q for q u u r with u r 1 could be arranged in an r 1 u r table t q where the entry in the k th row and l th column has k l parts we prove this table theorem and then generalize the statement to propose a box conjecture for the set of partitions q 1 q for an arbitrary partition q whose parts differ pairwise by at least two

thresholding neural network for adaptive noise reduction in the paper a type of thresholding neural network tnn is developed for adaptive noise reduction new types of soft and hard thresholding functions are created to serve as the activation function of the tnn unlike the standard thresholding functions the new thresholding functions are infinitely differentiable by using the new thresholding functions some gradient based learning algorithms become possible or more effective the optimal solution of the tnn in a mean square error mse sense is discussed it is proved that there is at most one optimal solution for the soft thresholding tnn general optimal performances of both soft and hard thresholding tnns are analyzed and compared to the linear noise reduction method gradient based adaptive learning algorithms are presented to seek the optimal solution for noise reduction the algorithms include supervised and unsupervised batch learning as well as supervised and unsupervised stochastic learning it is indicated that the tnn with the stochastic learning algorithms can be used as a novel nonlinear adaptive filter it is proved that the stochastic learning algorithm is convergent in certain statistical sense in ideal conditions numerical results show that the tnn is very effective in finding the optimal solutions of thresholding methods in an mse sense and usually outperforms other noise reduction methods especially it is shown that the tnn based nonlinear adaptive filtering outperforms the conventional linear adaptive filtering in both optimal solution and learning performance

conjugate convex operators convex mappings from a locally convex space x into f f are considered where f is an ordered topological vector space and an arbitrary greatest element adjoined to f in view of applications to the polarity theory of convex operators the possibility is investigated of representing a convex mapping taking values in f as a supremum of continuous affine mappings

on the structure of commutative matrices ii a finite set a of n n nilpotent commutative matrices that have one dimensional joint kernel is considered the theorem due to suprunenko and tyshkevich that the algebra a generated by a and the identity matrix has dimension equal to n is proved a canonical basis for a is given and related structure constants are discussed

wavelet based representations for a class of self similar signals with application to fractal modulation a potentially important family of self similar signals based upon a deterministic scale invariance characterization is introduced these signals which are referred to as dy homogeneous signals because they generalize the well known homogeneous functions have highly convenient representations in terms of orthonormal wavelet bases in particular wavelet representations can be exploited to construct orthonormal self similar bases for these signals the spectral and fractal characteristics of dy homogeneous signals make them appealing candidates for use in a number of applications as one potential example their use in a communications based context is considered specifically a strategy for embedding information into a dy homogeneous waveform on multiple time scales is developed this multirate modulation strategy called fractal modulation is potentially well suited for use with noisy channels of simultaneously unknown duration and bandwidth

commuting pairs in the centralizers of 2 regular matrices inmn k kan algebraically closed field we call a matrixl regular if each eigenspace is at mostl dimensional we prove that the variety of commuting pairs in the centralizer of a 2 regular matrix is the direct product of various affine spaces and various determinantal varieties zl mobtained from matrices over truncated polynomial rings we prove that these varieties zl mare irreducible and apply this to the case of thek algebra generated by three commuting matrices we show that if one of the three matrices is 2 regular then the algebra has dimension at mostn we also show that such an algebra is always contained in a commutative subalgebra ofmn k of dimension exactlyn

on stability of invariant subspaces of commuting matrices we study the stability of joint invariant subspaces of a finite set of commuting matrices we generalize some of the results of gohberg lancaster and rodman for the single matrix case for sets of two or more commuting matrices we exhibit some phenomena different from the single matrix case we show that each root subspace is a stable invariant subspace that each invariant subspace of a root subspace of a nonderogatory eigenvalue is stable and that even in the derogatory case the eigenspace is stable if it is one dimensional we prove that a pair of commuting matrices has only finitely many stable invariant subspaces at the end we discuss the stability of invariant subspaces of an algebraic multiparameter eigenvalue problem

on the dimension of the algebra generated by two positive semi commuting matrices abstract gerstenhaber s theorem states that the dimension of the unital algebra generated by two commuting n n matrices is at most n we study the analog of this question for positive matrices with a positive commutator we show that the dimension of the unital algebra generated by the matrices is at most n n 1 2 and that this bound can be attained we also consider the corresponding question if one of the matrices is a permutation or a companion matrix or both of them are idempotents in these cases the upper bound for the dimension can be reduced significantly in particular the unital algebra generated by two semi commuting positive idempotent matrices is at most 9 dimensional this upper bound can be attained

the use of the modified escalator algorithm to improve the performance of transform domain lms adaptive filters this paper describes a new algorithm that improves the convergence performance of the transform domain least mean square trlms algorithm the algorithm exploits the sparse structure of the correlation matrix of the transformed input process to derive a data dependent gram schmidt orthogonalization type transform of the process we show its faster convergence compared with the time domain least mean square lms algorithm and the dct or the dwt based trlms algorithm the gram schmidt orthogonalization is realized using a modified adaptive escalator algorithm the modification significantly reduces the computational complexity of the adaptive escalator algorithm and determines the computational complexity of the proposed algorithm

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

a groebner basis for the 2 2 determinantal ideal mod t2 abstract in an earlier paper t kosir b a sethuraman determinantal varieties over truncated polynomial rings j pure appl algebra 195 2005 75 95 we had begun a study of the components and dimensions of the spaces of k 1 th order jets of the classical determinantal varieties these are the varieties z r k m n obtained by considering generic m n m n matrices over rings of the form f t t k and for some fixed r setting the coefficients of powers of t of all r r minors to zero in this paper we consider the case where r k 2 and provide a groebner basis for the ideal i 2 2 m n which defines the tangent bundle to the classical 2 2 determinantal variety we use the results of these groebner basis calculations to describe the components of the varieties z r 4 m n where r is arbitrary the components of z r 2 m n and z r 3 m n were already described in the above cited paper

continuity and differentiability properties of convex operators this paper provides some theorems on the generic differentiability of convex operators between topological vector spaces these results are applied to give new conditions for subgradients of convex operators to be non empty and well behaved limiting examples are given

rapid identification of a sparse impulse response using an adaptive algorithm in the haar domain this paper proposes a fast convergence adaptive algorithm for identifying a sparse impulse response that is rich in spectral content a sparse impulse response is referred here as a discrete time impulse response that has a large number of zero or near zero coefficients the basic idea for rapid identification is to automatically determine the locations of the nonzero impulse response coefficients for their adaptation and eliminate the unnecessary adaptation of zero coefficients the proposed method which is called the haar basis algorithm employs a transform approach by modeling the sparse impulse response in the haar domain the haar transform has many basis sets and each of them contains basis vectors that span the entire time domain range this special nature of the haar transform allows for the selection of one small subset of adaptive filter coefficients whose basis vectors span the entire range of the impulse response these coefficients are adapted at the beginning and are then used subsequently to identify from the hierarchical structure of the haar transform the rest of the filter coefficients that must be adapted to correctly model the unknown sparse impulse response the consequence is avoiding adaptation of many zero coefficients leading to a dramatic improvement in either convergence speed or steady state excess mean square error ease while requiring no a priori knowledge such as the number of nonzero coefficients in the unknown sparse impulse response the proposed algorithm has been tested with a variety of unknown sparse systems using both white noise input and colored input whose spectrum closely resembles that of speech simulation results show that the new approach provides promising results

centralizers in endomorphism rings we prove that the centralizer cen endr m of a nilpotent endomorphism of a finitely generated semisimple left r module mr over an arbitrary ring r is the homomorphic image of the opposite of a certain z r subalgebra of the full m m matrix algebra mm r z where m is the dimension of ker if r is a local ring then we give a complete characterization of cen and of the containment cen cen where is a not necessarily nilpotent element of endr m for a k linear map a of a finite dimensional vector space over a field k we determine the pi degree of cen a

directional upper derivatives and the chain rule formula for locally lipschitz functions on banach spaces motivated by an attempt to find a general chain rule formula for differentiating the composition f g of lipschitz functions f and g that would be as close as possible to the standard formula f g x f g x g x we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments the idea behind these assignments is that the derivative of f at y is understood as defined only in the direction of a suitable tangent space u f y and so it exists at every point but these tangent spaces are chosen in such a way that for any g they contain the range of g x for almost every x showing the existence of such assignments leads us to detailed study of derived sets and the ways in which they describe pointwise behavior of lipschitz functions

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

the tropical commuting variety we study tropical commuting matrices from two viewpoints linear algebra and algebraic geometry in classical linear algebra there exist various criteria to test whether two square matrices commut

commutative matrix subalgebras and length function it is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion the examples are given showing that the converse does not hold to establish this result we prove several fundamental properties of the length function

new classes of examples satisfying the three matrix analog of gerstenhaber s theorem abstract in 1961 gerstenhaber proved the following theorem if k is a field and x and y are commuting d d matrices with entries in k then the unital k algebra generated by these matrices has dimension at most d the analog of this statement for four or more commuting matrices is false the three matrix version remains open we use commutative algebraic techniques to prove that the three matrix analog of gerstenhaber s theorem is true for some new classes of examples in particular we translate this three commuting matrix statement into an equivalent statement about certain maps between modules and prove that this commutative algebraic reformulation is true in special cases we end with ideas for an inductive approach intended to handle the three matrix analog of gerstenhaber s theorem more generally

a topological and geometric approach to fixed points results for sum of operators and applications abstract in this paper we establish a fixed point result of krasnoselskii type for the sum a b where a and b are continuous maps acting on locally convex spaces our results extend previous ones we apply such results to obtain strong solutions for some quasi linear elliptic equations with lack of compactness we also provide an application to the existence and regularity theory of solutions to a nonlinear integral equation modeled in a banach space in the last section we develop a sequentially weak continuity result for a class of operators acting on vector valued lebesgue spaces such a result is used together with a geometric condition as the main tool to provide an existence theory for nonlinear integral equations in l p e

on the separation property of orbits in representation spaces a subset x of a vector space v is said to have the separation property if it separates linear forms in the following sense given a pair of linearly independent linear forms on v there is a point x on x such that x 0 and x 0 a more geometric way to express this is the following every linear subspace h v of codimension 1 is linearly spanned by its intersection with x the separation property was first asked for conjugacy classes in simple lie algebras in connection with some classification problems we give a general answer for orbits in representation spaces of algebraic groups and discuss in detail some special cases we also introduce a strong and a weak separation property which come up very naturally in our setting it turns out that these separation properties have a number of very nice features for example we discovered the surprising fact that in an irreducible representation of a connected semisimple group every linear hyperplane meets every orbit and we show that a generic orbit always has the separation property

fast positive definite linear system solvers the authors show that the m band wavelet transforms of a wide class of covariance matrices consist of subblocks that are essentially banded furthermore they prove that the cholesky factors of the transformed covariance matrices also consist of subblocks that are essentially banded they combine these two observations to construct a fast o n sup 2 algorithm for solving the n spl times n linear positive definite systems of equations that arise in statistical signal processing finally they provide an error analysis of the proposed linear positive definite system solver

differentiability of lipschitz functions in lebesgue null sets we show that if n 1 then there exists a lebesgue null set in mathbb r n containing a point of differentiability of each lipschitz function f mathbb r n rightarrow mathbb r n 1 in combination with the work of others this completes the investigation of when the classical rademacher theorem admits a converse avoidance of sigma porous sets arising as irregular points of lipschitz functions plays a key role in the proof

adjoint process duality we study convex processes between topological vector spaces with particular emphasis on their adjoints this study is then applied to produce general duality results for classes of convex programs involving processes the use of processes allows one to exploit the symmetry of linear programming and to obtain significantly broader and stronger results

two generated commutative matrix subalgebras abstract let f be a field and let mn f be the algebra of n n matrices over f let a b mn f with ab ba and let a be the algebra generated by a b over f a theorem of gerstenhaber ann math 73 324 348 1991 states that the dimension of a is at most n gerstenhaber s proof uses the methods of algebraic geometry in this paper we obtain a purely matrix theoretic proof of the result we also examine the case when equality occurs the case where f is algebraically closed and a is indecomposable under similarity holds the key to the general situation and in the indecomposable case we obtain a cayley hamilton like theorem expressing bk as a polynomial in i b bk 1 with coefficients in f a where k denotes the number of blocks in the jordan form of a if all jordan blocks of a have the same size we obtain a nonderogatory like condition on b which is equivalent to dim f a n we also show that in this case dim f a n is equivalent to the maximality of a as a commutative subalgebra of mn f

the number of monodromy representations of abelian varieties of low p rank abstract let a g be an abelian variety of dimension g and p rank 1 over an algebraically closed field of characteristic p 0 we compute the number of homomorphisms from 1 et a g a to g l n f q where q is any power of p we show that for fixed g and n the number of such representations is polynomial in q and give an explicit formula for this polynomial we show that the set of such homomorphisms forms a constructible set and use the geometry of this space to deduce information about the coefficients and degree of the polynomial in the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups as a corollary we deduce that when 0 hom 1 et a g a g l n f q g l n f q is a laurent polynomial in q

some thoughts on gerstenhaber s theorem abstract we weigh the current evidence for and against an extension of gerstenhaber s 1961 theorem to three commuting matrices over a field