local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

decentralized fuzzy observer based output feedback control for nonlinear large scale systems an lmi approach this paper presents a decentralized fuzzy control problem for asymptotic stabilization of a class of nonlinear large scale systems that use an observer based output feedback scheme a takagi sugeno t s fuzzy model is adopted for the nonlinear large scale system which has unknown interconnection terms and a fuzzy controller is separately considered for measurable and nonmeasurable premise variable cases sufficient conditions are derived for both asymptotic stabilization and optimization of a maximum bound of interconnection and are formulated in terms of linear matrix inequalities finally numerical examples are provided to verify the effectiveness of the proposed techniques

a bayesian approach to geometric subspace estimation this paper presents a geometric approach to estimating subspaces as elements of the complex grassmann manifold with each subspace represented by its unique complex projection matrix variation between the subspaces is modeled by rotating their projection matrices via the action of unitary matrices elements of the unitary group u n subspace estimation or tracking then corresponds to inferences on u n taking a bayesian approach a posterior density is derived on u n and certain expectations under this posterior are empirically generated for the choice of the hilbert schmidt norm on u n to define estimation errors an optimal mmse estimator is derived it is shown that this estimator achieves a lower bound on the expected squared errors associated with all possible estimators the estimator and the bound are computed using metropolis adjusted langevin s diffusion algorithm for sampling from the posterior for use in subspace tracking a prior model on subspace rotation that utilizes newtonian dynamics is suggested

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

multiple emitter location and signal parameter estimation processing the signals received on an array of sensors for the location of the emitter is of great enough interest to have been treated under many special case assumptions the general problem considers sensors with arbitrary locations and arbitrary directional characteristics gain phase polarization in a noise interference environment of arbitrary covariance matrix this report is concerned first with the multiple emitter aspect of this problem and second with the generality of solution a description is given of the multiple signal classification music algorithm which provides asymptotically unbiased estimates of 1 number of incident wavefronts present 2 directions of arrival doa or emitter locations 3 strengths and cross correlations among the incident waveforms 4 noise interference strength examples and comparisons with methods based on maximum likelihood ml and maximum entropy me as well as conventional beamforming are included an example of its use as a multiple frequency estimator operating on time series is included

h spl infin decentralized fuzzy model reference tracking control design for nonlinear interconnected systems in general due to the interactions among subsystems it is difficult to design an h sub spl infin decentralized controller for nonlinear interconnected systems the model reference tracking control problem of nonlinear interconnected systems is studied via h sub spl infin decentralized fuzzy control method first the nonlinear interconnected system is represented by an equivalent takagi sugeno type fuzzy model a state feedback decentralized fuzzy control scheme is developed to override the external disturbances such that the h spl infin model reference tracking performance is achieved furthermore the stability of the nonlinear interconnected systems is also guaranteed if states are not all available a decentralized fuzzy observer is proposed to estimate the states of each subsystem for decentralized control consequently a fuzzy observer based state feedback decentralized fuzzy controller is proposed to solve the h sub spl infin tracking control design problem for nonlinear interconnected systems the problem of h sub spl infin decentralized fuzzy tracking control design for nonlinear interconnected systems is characterized in terms of solving an eigenvalue problem evp the evp can be solved very efficiently using convex optimization techniques finally simulation examples are given to illustrate the tracking performance of the proposed methods

a product theorem in free groups if a is a nite subset of a free group with at least two noncommuting elements thenja a aj jaj 2 logjaj o 1 more generally the same conclusion holds in an arbitrary virtually free group unless a generates a virtually cyclic subgroup the central part of the proof of this result is carried on by estimating the number of collisions in multiple products a1 ak we include a few simple observations showing that in this statistical context

random generation of finite and profinite groups and group enumeration the following article appeared in annals of mathematics 173 2 2011 769 814 and may be found at http annals math princeton edu 2011 173 2 p04

efficient subspace based algorithm for adaptive bearing estimation and tracking in some practical applications of array processing the directions of the incident signals should be estimated adaptively and or the time varying directions should be tracked promptly in this paper an adaptive bearing estimation and tracking abest algorithm is investigated for estimating and tracking the uncorrelated and correlated narrow band signals impinging on a uniform linear array ula based on the subspace based method without eigendecomposition sumwe where a linear operator is obtained from the array data to form a basis for the null space by exploiting the array geometry and its shift invariance property specifically the null space is estimated using the least mean square lms or normalized lms nlms algorithm and the directions are updated using the approximate newton method the transient analyses of the lms and nlms algorithms are studied where the weight i e the linear operator is in the form of a matrix and there is a correlation between the additive noise and input data that involve the instantaneous correlations of the received array data in the updating equation and the step size stability conditions are derived explicitly in addition the analytical expressions for the mean square error mse and mean square deviation msd learning curves of the lms algorithm are clarified the effectiveness of the abest algorithm is verified and the theoretical analyses are corroborated through numerical examples simulation results show that the abest algorithm is computationally simple and has good adaptation and tracking abilities

h_ infty fuzzy control synthesis for a large scale system with a reduced number of lmis this paper introduces an h infty fuzzy control synthesis method for a nonlinear large scale system with a reduced number of linear matrix inequalities lmis it is well known that a nonlinear large scale system can be transformed to a takagi sugeno t s fuzzy system by using sector nonlinearity or local approximation in fuzzy partition spaces methods next in order to achieve the fuzzy control design for this t s fuzzy system we solve the stabilization conditions that are represented by the lmis however if the number of lmis is large the control design process may become very complicated in this study based on the lyapunov method and s procedure several theorems are proposed for the synthesis of parallel distributed compensation pdc type fuzzy control such that the nonlinear large scale system achieves h infty control performance and the number of lmis to be solved is reduced explicitly as a result the control design process will become much easier furthermore if the modeling error between the nonlinear system and t s fuzzy system exists the robust h infty control performance and the number reduction of lmis are also achieved by the proposed theorem several examples are presented in this paper to show the number reduction effect of lmis and the effectiveness of the proposed controller synthesis

nonlinear decentralized state feedback controller for uncertain fuzzy time delay interconnected systems abstract in this paper the perturbed continuous time interconnected system with time delay is represented by an equivalent takagi sugeno type fuzzy model having r i rules for each subsystem based on lyapunov stability theorem and razumikhin theorem the nonlinear decentralized state feedback fuzzy controllers are proposed to stabilize the whole perturbed fuzzy time delay interconnected system asymptotically under using this nonlinear controller the design methods only consider the r i fuzzy rules rather than r i r i rules for each subsystem that is we can remove the negative effect of the coupled terms therefore these approaches can be less conservative moreover if all the time delays of each subsystem are the same for all rules we shall propose less conservative criteria these criteria do not need the solution of a lyapunov equation or a riccati equation we also do not need to find a common positive matrix p to satisfy any inequality the so called matching condition for the interconnection matrices and perturbations are not needed finally a numerical example is given to illustrate the control design and its effectiveness

adaptive eigensubspace algorithms for direction or frequency estimation and tracking the authors present an adaptive estimator of the complete noise or signal subspace of a sample covariance matrix as well as the estimator s practical implementations the general formulation of the proposed estimator results from an asymptotic argument which shows the signal or noise subspace computation to be equivalent to a constrained gradient search procedure a highly parallel algorithm denoted the inflation method is introduced for the estimation of the noise subspace the simulation results of these adaptive estimators show that the adaptive subspace algorithms perform substantially better than p a thompson s 1980 adaptive version of v f pisarenko s technique 1973 in estimating frequencies or directions of arrival doa of plane waves for tracking nonstationary parameters the simulation results also show that the adaptive subspace algorithms are better than direct eigendecomposition methods for which computational complexity is much higher than the adaptive versions

an approach to fuzzy control of nonlinear systems stability and design issues presents a design methodology for stabilization of a class of nonlinear systems first the authors represent a nonlinear plant with a takagi sugeno fuzzy model then a model based fuzzy controller design utilizing the concept of the so called parallel distributed compensation is employed the main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system the design procedure is conceptually simple and natural moreover the stability analysis and control design problems can be reduced to linear matrix inequality lmi problems therefore they can be solved efficiently in practice by convex programming techniques for lmis the design methodology is illustrated by application to the problem of balancing and swing up of an inverted pendulum on a cart

expansion in sl d z q z q arbitrary let s be a fixed finite symmetric subset of sl d z and assume that it generates a zariski dense subgroup g we show that the cayley graphs of q g with respect to the generating set q s form a family of expanders where q is the projection map z z q z

nilprogressions and groups with moderate growth abstract we show that doubling at some large scale in a cayley graph implies uniform doubling at all subsequent scales the proof is based on the structure theorem for approximate subgroups proved by green tao and the first author we also give a number of applications to the geometry and spectrum of finite cayley graphs for example we show that a finite group has moderate growth in the sense of diaconis and saloff coste if and only if its diameter is larger than a fixed power of the cardinality of the group we call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter we also give bounds on the cheeger constant first eigenvalue of the laplacian and mixing time this can be seen as a finite group version of gromov s theorem on groups with polynomial growth it also improves on a result of lackenby regarding property in towers of coverings another consequence is a universal upper bound on the diameter of all finite simple groups independent of the cfsg

uniform expansion bounds for cayley graphs of sl2 fp we prove that cayley graphs of sl2 fp are expanders with respect to the projection of any fixed elements in sl 2 z generating a non elementary subgroup and with respect to generators chosen at random in sl2 fp

decentralized sampled data h fuzzy filter for nonlinear large scale systems this paper presents a decentralized sampled data h fuzzy filter design method for nonlinear large scale systems which are represented by a takagi sugeno t s fuzzy model based on the t s fuzzy model the error system between the nonlinear large scale system and the filter is obtained the discretization process of the error system is accomplished with the exact discrete time approach to eliminate the exact approximate mismatch by using the discrete time lyapunov sense the sufficient condition of the asymptotic stability for the error system is given and a prescribed level of the h norm is ensured to guarantee the h fuzzy filter performance finally numerical examples are given to show the effectiveness of the proposed methods

the statistical performance of the music and the minimum norm algorithms in resolving plane waves in noise this paper presents an asymptotic statistical analysis of the null spectra of two eigen assisted methods music 1 and minimum norm 2 for resolving independent closely spaced plane waves in noise particular attention is paid to the average deviation of the null spectra from zero at the true angles of arrival for the plane waves these deviations are expressed as functions of signal to noise ratios number of array elements angular separation of emitters and the number of snapshots in the case of music an approximate expression is derived for the resolution threshold of two plane waves with equal power in noise this result is validated by monte carlo simulations

decentralized adaptive tracking control of nonaffine nonlinear large scale systems with time delays this paper addresses the problem of decentralized tracking control of large scale systems with uncertain nonaffine nonlinear isolated subsystems and nonlinear interconnections with time varying delays based on lyapunov krasovskii functional approach and implicit function theorem a delay independent decentralized tracking controller is proposed due to functional approximation capability of fuzzy logic systems fls neither strict structure restrictions on the isolated subsystems nor a priori knowledge of the strong interconnections with time varying delays is required in our control design furthermore transient performance of the resulting closed loop system is also addressed under an analytical framework finally two numerical examples are provided to show the effectiveness of the proposed controller

on the local global conjecture for integral apollonian gaskets we prove that a set of density one satisfies the local global conjecture for integral apollonian gaskets that is for a fixed integral primitive apollonian gasket almost every in the sense of density admissible passing local obstructions integer is the curvature of some circle in the gasket

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

asymptotic performance analysis of direction finding algorithms based on fourth order cumulants in the narrow band array processing context the use of higher order statistics has been often advocated because consistent and asymptotically unbiased parameter estimates can be obtained without it being necessary to know to model or to estimate the spatial covariance of the noise as long as it is normally distributed however experimentation shows that this noise insensitivity is traded for increased variability of the parameter estimates the main purpose of this contribution is to derive and work out closed form expressions of the asymptotic covariance of music like direction of arrival estimates based on two fourth order cumulant matrices the diagonal slice and the contracted quadricovariance this is compared with the standard covariance based music estimate establishing on a rational basis the domain of applicability of higher order statistics for doa estimation in particular the actual impact of the noise variance and of the dynamic range of the sources is investigated this asymptotic performance analysis is achieved within a general framework which we believe to be of general interest

the affine sieve we establish the main saturation conjecture connected with executing a brun sieve in the setting of an orbit of a group of affine linear transformations this is carried out under the condition that the zariski closure of the group is levi semisimple it is likely that this condition is also necessary for such saturation to hold 2013 american mathematical society

beyond expansion iii reciprocal geodesics we prove the existence of infinitely many low lying and fundamental closed geodesics on the modular surface which are reciprocal that is invariant under time reversal the method combines ideas from parts i and ii of this series namely the dispersion method in bilinear forms as applied to thin semigroups coming from restricted continued fractions

on the number of markoff numbers below a given bound according to a famous theorem of markoff the indefinite quadratic forms with exceptionally large minima greater than f of the square root of the discriminant are in 1 1 correspondence with the solutions of the diophantine equation p2 q2 r1 ipqr by relating markoffs algorithm for finding solutions of this equation to a problem of count ing lattice points in triangles it is shown that the number of solutions less than x equals clog2 3x 0 log x log log2 x with an explicitly computable constant c 0 18071704711507 numerical data up to 101300 is presented which suggests that the true error term is considerably smaller

word maps conjugacy classes and a noncommutative waring type theorem let w w x xd 1 be a nontrivial group word we show that if g is a sufficiently large finite simple group then every element g e g can be expressed as a product of three values of w in g this improves many known results for powers commutators as well as a theorem on general words obtained in 19 the proof relies on probabilistic ideas algebraic geometry and character theory our methods which apply the zeta function g s x eirrg xo s giye rise to various additional results of independent interest including applications to conjectures of ore and thompson

prime and almost prime integral points on principal homogeneous spaces we develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space in particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors in many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers when the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number and thus establish the zariski density of matrices all of whose entries are prime numbers among the key tools used are explicit approximations to the generalized ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms

asymptotic performance analysis of subspace adaptive algorithms introduced in the neural network literature in the neural network literature many algorithms have been proposed for estimating the eigenstructure of covariance matrices we first show that many of these algorithms when presented in a common framework show great similitudes with the gradient like stochastic algorithms usually encountered in the signal processing literature we derive the asymptotic distribution of these different recursive subspace estimators a closed form expression of the covariances in distribution of eigenvectors and associated projection matrix estimators are given and analyzed in particular closed form expressions of the mean square error of these estimators are given it is found that these covariance matrices have a structure very similar to those describing batch estimation techniques the accuracy of our asymptotic analysis is checked by numerical simulations and it is found to be valid not only for a small step size but in a very large domain finally convergence speed and deviation from orthonormality of the different algorithms are compared and several tradeoffs are analyzed

stability analysis and design of fuzzy control systems abstract the stability analysis and the design technique of fuzzy control systems using fuzzy block diagrams are discussed first we show the concept of fuzzy blocks and consider the connection problems of fuzzy blocks diagrams we derive some theorems and corollaries with respect to two basic types of connections of fuzzy blocks in order to preserve some properties in a connection of fuzzy blocks continuous piecewise polynomial membership functions are defined secondly a sufficient condition which guarantees the stability of fuzzy systems is obtained in terms of lyapunov s direct method we give an important fact based on this condition thirdly we propose a new design technique of a fuzzy controller the fuzzy block diagrams and the stability analysis are applied to the design problems of a model based fuzzy controller

an efficient pastd algorithm implementation for multiple direction of arrival tracking we propose a simple direction of arrival doa tracking scheme involving the projection approximation subspace tracking with deflation pastd algorithm and a kalman filter the doas are obtained via a newton type method initialized with the doas predicted by the kalman filter the data association is then avoided the tracking capability of the proposed algorithm is demonstrated by computer simulations in scenarios involving targets with crossing trajectories

on the diameter of permutation groups given a finite group g and a set a of generators the diameter diam g g a of the cayley graph g g a is the smallest l such that every element of g can be expressed as a word of length at most l in a a 1 we are concerned with bounding diam g max a diam g g a it has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n but the best previously known upper bound was exponential in nlogn v we give a quasipolynomial upper bound namely diam g exp o logn 4 loglogn exp loglog g o 1 for g sym n or g alt n where the implied constants are absolute this addresses a key open case of babai s conjecture on diameters of simple groups by a result of babai and seress 1992 our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n

growth and generation in mathrm sl _2 mathbb z p mathbb z we show that every subset of sl2 z pz grows rapidly when it acts on itself by the group operation it follows readily that for every set of generators a of sl2 z pz every element of sl2 z pz can be expressed as a product of at most o log p c elements of a a 1 where c and the implied constant are absolute

an explicit upper bound for the helfgott delta in sl 2 p abstract helfgott proved that there exists a 0 such that if s is a symmetric generating subset of sl 2 p containing 1 then either s 3 sl 2 p or s 3 s 1 it is known that 1 3024 here we show that log 2 7 1 6 0 3012 and we present evidence suggesting that this might be the true value of

growth in solvable subgroups of mathrm gl _r mathbb z p mathbb z let k mathbb z p mathbb z and let a be a subset of mathrm gl r k such that langle a rangle is solvable we reduce the study of the growth of a under the group operation to the nilpotent setting fix a positive number c ge 1 we prove that either a grows meaning a 3 ge c a or else there are groups u r and s with u r unlhd s unlhd langle a rangle such that s u r is nilpotent a k cap s is large and u r subseteq a k where k depends only on the rank r of mathrm gl r k here a k x 1 x 2 cdots x k x i in a cup a 1 cup 1 when combined with recent work by pyber and szabo the main result of this paper implies that it is possible to draw the same conclusions without supposing that langle a rangle is solvable

stabilization and estimation for perturbed discrete time delay large scale systems based on a derived algebraic inequality a criterion to guarantee the robust stabilization and the state estimation for perturbed discrete time delay large scale systems is proposed this criterion is independent of time delay and does not need the solution of a lyapunov equation or riccati equation furthermore the same problem for the continuous time systems is also solved by a much simpler derivation

control laws for takagi sugeno fuzzy models abstract this work presents control laws for fuzzy models of takagi sugeno ts sugeno and kang fuzzy sets and systems 28 1988 15 33 takagi and sugeno ieee trans systems man cybernet 15 1 1985 116 132 these laws are not directly put in the classical form called pdc for parallel distributed compensation wang et al ieee trans fuzzy systems 4 1 1996 14 23 on a particular family of ts fuzzy models the approach allows to reduce the conservatism of results found in the literature tanaka et al ieee trans fuzzy systems 6 2 1998 1 16 this approach utilizes the state space representation according to that remark observers are also considered and a separation principle is also given though results are provided mainly for the continuous case they are also provided for the discrete case an example in both cases illustrates the results obtained

growth in finite simple groups of lie type we prove that if l is a finite simple group of lie type and a a set of generators of l then either a grows i e vert a 3 vert vert a vert 1 varepsilon where varepsilon depends only on the lie rank of l or a 3 l this implies that for simple groups of lie type of bounded rank a well known conjecture of babai holds i e the diameter of any cayley graph is polylogarithmic we also obtain new families of expanders a generalization of our proof yields the following let a be a finite subset of sl n mathbb f mathbb f an arbitrary field satisfying big vert a 3 big vert le mathcal k vert a vert then a can be covered by mathcal k m i e polynomially many cosets of a virtually soluble subgroup of sl n mathbb f which is normalized by a where m depends on n see more at http www ams org journals jams 0000 000 00 s0894 0347 2014 00821 3 home html sthash lp65mzge dpuf

congruence properties of zariski dense subgroups i this paper deals with the following general situation we are given an algebraic group g defined over a number field k and a subgroup f of the group g k of k rational points of g then what should it mean for f to be a large subgroup we might require f to be a lattice in g to be arithmetic to contain many elements of a specific kind to have a large closure in some natural topology et cetera there are many theorems proving implications between conditions of size of this kind we shall consider the case of g a k simple group usually q simple the zariski topology of g for which the closed sets are those defined by the vanishing of polynomial functions is very coarse on the other hand the various valuations v of k each give rise to a strong topology on g k if v is non archimedean and kv is the completion of k with respect to v then g kv is a non archimedean lie group and if is the ring of integers of kv the open subgroup g 5 j of g kv of integral points is defined for all but finitely many v the zariski closure f of a subgroup f of g k is a c algebraic subgroup of g while the strong closure fy is a lie subgroup of g kv for elementary reasons the dimension of tv is at most the dimension of f our results are in the opposite direction we show under suitable conditions on g that if f g then for almost all v we have that tv contains g ov to illustrate this with a specific example if ml 5 mk are matrices in sl2 q generating a group f which is zariskidense in sl2 then for all sufficiently large prime numbers p we have f p sl2 zp further this result is effective in the sense that we could in principle check the hypothesis for given mx mk and derive a bound on p beyond which the conclusion holds it is perhaps worth remarking that this special case which is the simplest application of our results may be proved with much less effort than the general theorem

affine linear sieve expanders and sum product let mathcal o be an orbit in n of a finitely generated subgroup of gl n whose zariski closure zcl is suitably large e g isomorphic to sl2 we develop a brun combinatorial sieve for estimating the number of points on mathcal o at which a fixed integral polynomial is prime or has few prime factors and discuss applications to classical problems including pythagorean triangles and integral apollonian packings a fundamental role is played by the expansion property of the congruence graphs that we associate with mathcal o this expansion property is established when zcl sl2 using crucially sum product theorem in q for q square free

diophantine equations in the primes let mathfrak p mathfrak p 1 ldots mathfrak p r be a system of r polynomials with integer coefficients of degree d in n variables mathbf x x 1 ldots x n for a given r tuple of integers say mathbf s a general local to global type statement is shown via classical hardy littlewood type methods which provides sufficient conditions for the solubility of mathfrak p mathbf x mathbf s under the condition that each of the x i s is prime

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

performance analysis of an adaptive algorithm for tracking dominant subspaces this paper provides a performance analysis of a least mean square lms dominant invariant subspace algorithm based on an unconstrained minimization problem this algorithm is a stochastic gradient algorithm driving the columns of a matrix w to an orthonormal basis of a dominant invariant subspace of a correlation matrix we consider the stochastic algorithm governing the evolution of ww sup h to the projection matrix onto this dominant invariant subspace and study its asymptotic distribution a closed form expression of its asymptotic covariance is given in the case of independent observations and is further analyzed to provide some insights into the behavior of this lms type algorithm in particular it is shown that even though the algorithm does not constrain w to have orthonormal columns there is deviation from orthonormality of the first order we also give a closed form expression of the asymptotic covariance of the doas estimated by the music algorithm applied to the adaptive estimate of the projector it is found that the asymptotic distributions have a structure that is very similar to those describing batch estimation techniques because both algorithms are obtained from the minimization of the same criterion finally the accuracy of the asymptotic analysis is checked by numerical simulations and is found to be valid not only for a small step size but in a very large domain

asymptotic distributions associated to oja s learning equation for neural networks we perform a complete asymptotic performance analysis of the stochastic approximation algorithm denoted subspace network learning algorithm derived from oja s learning equation in the case where the learning rate is constant and a large number of patterns is available this algorithm drives the connection weight matrix w to an orthonormal basis of a dominant invariant subspace of a covariance matrix our approach consists in associating to this algorithm a second stochastic approximation algorithm that governs the evolution of ww sup t to the projection matrix onto this dominant invariant subspace then using a general result of gaussian approximation theory we derive the asymptotic distribution of the estimated projection matrix closed form expressions of the asymptotic covariance of the projection matrix estimated by the snl algorithm and by the smoothed snl algorithm that we introduce are given in the case of independent or correlated learning patterns and are further analyzed it is found that the structures of these asymptotic covariance matrices are similar to those describing batch estimation techniques the accuracy or our asymptotic analysis is checked by numerical simulations and it is found to be valid not only for a small learning rate but in a very large domain finally improvements brought by our smoothed snl algorithm are shown such as the learning speed misadjustment tradeoff and the deviation from orthonormality

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

analysis and design for a class of complex control systems part i fuzzy modelling and identification this paper is the first of two dealing with the analysis and design of a class of complex control systems this part deals with fuzzy modelling and identification the second paper is concerned with fuzzy controller design we believe that these two papers show a way to analyze and design complex control systems using a combination of fuzzy logic and modern control theory the complex systems studied in these two papers can be represented by a fuzzy aggregation of a set of local linear models the identification can be divided into two procedures the first procedure is the identification of the membership functions including the determination of the number of fuzzy rules and the estimation of the parameters in the membership functions the second is the identification of the local linear models including the determination of the structure and the parameters of the local rule maps and the global rule interpolation two examples are used to demonstrate the application of the modelling method

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

sieve methods in group theory i powers in linear groups a general sieve method for groups is formulated it enables one to measure subsets of a finitely generated group as an application we show that if is a finitely generated non virtually solvable linear group of characteristic zero then the set of proper powers in is exponentially small this is a far reaching strengthening of the main result of hkls

blind channel identification subspace tracking method without rank estimation subspace ss methods are an effective approach for blind channel identification however these methods also have two major disadvantages 1 they require accurate channel length estimation and or rank estimation of the correlation matrix which is difficult with noisy channels and 2 they require a large amount of computation for the singular value decomposition svd which makes it inconvenient for adaptive implementation although many adaptive subspace tracking algorithms can be applied the computational complexity is still o m sup 3 where m is the data vector length in this paper we introduce new recursive subspace algorithms using ulv updating and successive cancellation techniques the new algorithms do not need to estimate the rank of the correlation matrix furthermore the channel length can be overestimated initially and be recovered at the end by a successive cancellation procedure which leads to more convenient implementations the adaptive algorithm has computations of o m sup 2 in each recursion the new methods can be applied to either the single user or the multiuser cases simulations demonstrate their good performance

explicit helfgott type growth in free products and in limit groups abstract we adapt safin s result on powers of sets in free groups to obtain helfgott type growth in free products if a is any finite subset of a free product of two arbitrary groups then either a is conjugate into one of the factors or the triple product a 3 of a satisfies a 3 1 7776 a 2 or a generates an infinite cyclic or infinite dihedral group we also point out that if a is any finite subset of a limit group then a 3 satisfies the above inequality unless a generates a free abelian group this gives rise to many infinite groups g where there exist c 0 and 1 such that any finite subset a of g either satisfies a 3 c a 1 or generates a virtually nilpotent group

approximate subgroups of residually nilpotent groups we show that a k approximate subgroup a of a residually nilpotent group g is contained in boundedly many cosets of a finite by nilpotent subgroup the nilpotent factor of which is of bounded step combined with an earlier result of the author this implies that a is contained in boundedly many translates of a coset nilprogression of bounded rank and step the bounds are effective and depend only on k in particular if g is nilpotent they do not depend on the step of g as an application we show that there is some absolute constant c such that if g is a residually nilpotent group and if there is an integer n 1 such that the ball of radius n in some cayley graph of g has cardinality bounded by n c log log n then g is virtually log n step nilpotent

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed

on uniform exponential growth for linear groups let be a finitely generated group given a finite set of generators s of the word length ls for an element is defined to be the smallest positive integer for which there exist s1 sn s s 1 such that s1 sn for each n n denote by bs n the set of elements in whose word length with respect to s is at most n it follows from the subadditive property of ls that limn bs n 1 n exists which we denote by s a finitely generated group is said to be of exponential growth if s 1 of polynomial growth if for some c 0 and d n bs n c n d for all n 1 and of intermediate growth otherwise for some finite generating set s of observe that the growth type of does not depend on the choice of generating set s if a finitely generated group is linear it is known that is either of polynomial growth in which case is virtually nilpotent or of exponential growth otherwise tit72 mil68 wol68