factorizing a finite group into conjugates of a subgroup abstract for every non nilpotent finite group g there exists at least one proper subgroup m such that g is the setwise product of a finite number of conjugates of m we define cp g to be the smallest number k such that g is a product in some order of k pairwise conjugated proper subgroups of g we prove that if g is non solvable then cp g 36 while if g is solvable then cp g can attain any integer value bigger than 2 while on the other hand cp g 4 log 2 g

on efficient computation and asymptotic sharpness of kalantari s bounds for zeros of polynomials we study an infinite family of lower and upper bounds on the modulus of zeros of complex polynomials derived by kalantari we first give a simple characterization of these bounds which leads to an efficient algorithm for their computation for a polynomial of degree n our algorithm computes the first m bounds in kalantari s family in o mn operations we further prove that for every complex polynomial these lower and upper bounds converge to the tightest annulus containing the roots and thus settle a problem raised in kalantari s paper

new algorithms for finding irreducible polynomials over finite fields we present a new algorithm for finding an irreducible polynomial of specified degree over a finite field our algorithm is deterministic and it runs in polynomial time for fields of small characteristic we in fact prove the stronger result that the problem of finding irreducible polynomials of specified degree over a finite field is deterministic polynomial time reducible to the problem of factoring polynomials over the prime field

adaptive estimation of eigensubspace in a recent work we recast the problem of estimating the minimum eigenvector eigenvector corresponding to the minimum eigenvalue of a symmetric positive definite matrix into a neural network framework we now extend this work using an inflation technique to estimate all or some of the orthogonal eigenvectors of the given matrix based on these results we form a cost function for the finite data case and derive a newton based adaptive algorithm the inflation technique leads to a highly modular and parallel structure for implementation the computational requirement of the algorithm is o n sup 2 n being the size of the covariance matrix we also present a rigorous convergence analysis of this adaptive algorithm the algorithm is locally convergent and the undesired stationary points are unstable computer simulation results are provided to compare its performance with that of two adaptive subspace estimation methods proposed by yang and kaveh 1988 and an improved version of one of them for stationary and nonstationary signal scenarios the results show that the proposed approach performs identically to one of them and is significantly superior to the remaining two

self similar groups and the zig zag and replacement products of graphs abstract every finitely generated self similar group naturally produces an infinite sequence of finite d regular graphs n we construct self similar groups whose graphs n can be represented as an iterated zig zag product and graph powering n 1 n k z k 1 we also construct self similar groups whose graphs n can be represented as an iterated replacement product and graph powering n 1 n k r k 1 this gives simple explicit examples of self similar groups whose graphs n form an expanding family and examples of automaton groups whose graphs n have linear diameters diam n o n and bounded girth

global convergence of oja s pca learning algorithm with a non zero approaching adaptive learning rate a non zero approaching adaptive learning rate is proposed to guarantee the global convergence of oja s principal component analysis pca learning algorithm most of the existing adaptive learning rates for oja s pca learning algorithm are required to approach zero as the learning step increases however this is not practical in many applications due to the computational round off limitations and tracking requirements the proposed adaptive learning rate overcomes this shortcoming the learning rate converges to a positive constant thus it increases the evolution rate as the learning step increases this is different from learning rates which approach zero which slow the convergence considerably and increasingly with time rigorous mathematical proofs for global convergence of oja s algorithm with the proposed learning rate are given in detail via studying the convergence of an equivalent deterministic discrete time ddt system extensive simulations are carried out to illustrate and verify the theory derived simulation results show that this adaptive learning rate is more suitable for oja s pca algorithm to be used in an online learning situation

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

on the conjectures of j thompson and o ore if g is a finite simple group of lie type over a field containing more than 8 elements for twisted groups xn ql we require q 8 except for 2b2 q2 2g2 q2 and 2 f4 q2 where we assume q2 8 then g is the square of some conjugacy class and consequently every element in g is a commutator

property t for noncommutative universal lattices we establish a new spectral criterion for kazhdan s property t which is applicable to a large class of discrete groups defined by generators and relations as the main application we prove property t for the groups el n r where n 3 and r is an arbitrary finitely generated associative ring we also strengthen some of the results on property t for kac moody groups from dymara and januszkiewicz in invent math 150 3 579 627 2002

uniformly exhaustive submeasures and nearly additive set functions every uniformly exhaustive submeasure is equivalent to a measure from this we deduce that every vector measure with compact range in an f space has a control measure we also show that co or any e space is a t space i e cannot be realized as the quotient of a nonlocally convex f space by a one dimensional subspace

a dual purpose principal and minor subspace gradient flow the dual purpose principal and minor subspace gradient flow can be used to track principal subspace ps and if altered simply by the sign it can also serve as a minor subspace ms trackor this is of practical significance in the implementations of algorithms in this paper a unified information criterion is proposed and a dual purpose principal and minor subspace gradient flow is derived based on the information criterion in this dual purpose gradient flow the weight matrix length is self stabilizing i e moving towards unit length in each learning step the energy function associated with the dual purpose gradient flow for tracking ps and ms is given and it exhibits a unique global minimum attained if and only if its state matrices span the ps or ms of the autocorrelation matrix of a vector data stream the other stationary points of its energy function are unstable saddle points the proposed dual purpose gradient flow can efficiently track an orthonormal basis of the ps or ms which is illustrated through simulation experiments

conjugate gradient eigenstructure tracking for adaptive spectral estimation a conjugate gradient iteration is derived that converges to the set of r dominant subdominant eigenpairs this iteration is used to construct two eigenstructure tracking algorithms that track the r dimensional dominant or subdominant subspaces of time varying data or data covariance matrices the two eigenstructure tracking algorithms have update complexities o m sup 2 r and the other o mr sup 2 where m is the data dimension the algorithms are customized to solve high resolution temporal and spatial frequency tracking problems they are compared with existing techniques by tying into published simulation based performance tests the algorithms demonstrate rapid convergence and tracking characteristics at a competitive cost

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

entropy waves the zig zag graph product and new constant degree expanders the main contribution of this work is a new type of graph product which we call the zig zag product taking a product of a large graph with a small graph the resulting graph inherits roughly its size from the large one its degree from the small one and its expansion properties from both iteration yields simple explicit constructions of constant degree expanders of arbitrary size starting from one constant size expander crucial to our intuition and simple analysis of the properties of this graph product is the view of expanders as functions which act as entropy wave propagators they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated in these terms the graph product affords the constructive interference of two such waves

finite simple groups as expanders abstract we prove that there exist k and 0 e such that every non abelian finite simple group g which is not a suzuki group has a set of k generators for which the cayley graph cay g s is an e expander expander graphs ramanujan complexes

universal lattices and property tau we prove that the universal lattices the groups g sl d r where r x 1 x k have property for d 3 this provides the first example of linear groups with which do not come from arithmetic groups we also give a lower bound for the constant with respect to the natural generating set of g our methods are based on bounded elementary generation of the finite congruence images of g a generalization of a result by dennis and stein on k 2 of some finite commutative rings and a relative property t of mathrm sl 2 r ltimes r 2 r 2

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

a neural network learning for adaptively extracting cross correlation features between two high dimensional data streams this paper proposes a novel cross correlation neural network cnn model for finding the principal singular subspace of a cross correlation matrix between two high dimensional data streams we introduce a novel nonquadratic criterion nqc for searching the optimum weights of two linear neural networks lnn the nqc exhibits a single global minimum attained if and only if the weight matrices of the left and right neural networks span the left and right principal singular subspace of a cross correlation matrix respectively the other stationary points of the nqc are unstable saddle points we develop an adaptive algorithm based on the nqc for tracking the principal singular subspace of a cross correlation matrix between two high dimensional vector sequences the nqc algorithm provides a fast online learning of the optimum weights for two lnn the global asymptotic stability of the nqc algorithm is analyzed the nqc algorithm has several key advantages such as faster convergence which is illustrated through simulations

golod shafarevich groups with property t and kac moody groups we construct golod shafarevich groups with property t and thus provide counterexamples to a conjecture stated in a recent paper of zelmanov ze2 explicit examples of such groups are given by lattices in certain topological kac moody groups over finite fields we provide several applications of this result including examples of residually finite torsion non amenable groups

adelic profinite groups let us call a profinite group d adelic if d is a subgroup of the product sl z for some n g 2 one of the main cases of interest is when d p n p is the profinite completion of an s arithmetic subgroup of a semisimple algebraic group let k be an algebraic number field s be a finite set of places of k containing the infinite ones g be a semisimple connected algebraic group over k and g be an s arithmetic group we say that g satisfies the congruence subgroup property if the s congruence kernel s w x 4 c g is a finite group see pr2 chap 9 if n g 3 and s s the s s congruence kernel c sl is trivial and so we can identify the profinite n completion sl z of sl z with the product sl z n n p n p the purpose of this note is twofold firstly we prove some results on bounded generation for an arbitrary finitely generated adelic profinite group then we apply these results to the case of arithmetic groups to prove the following conjecture of a lubotzky conjecture if the profinite completion g of an s arithmetic group g is adelic then g satisfies the congruence subgroup property

on relative convergence properties of principal component analysis algorithms we investigate the convergence properties of two different stochastic approximation algorithms for principal component analysis and analytically explain some commonly observed experimental results in our analysis we use the theory of stochastic approximation and in particular the results of fabian 1968 to explore the asymptotic mean square errors amses of the algorithms this study reveals the conditions under which the algorithms produce smaller amses and also the conditions under which one algorithm has a smaller amse than the other experimental study with multidimensional gaussian data corroborate our analytical findings we next explore the convergence rates of the two algorithms our experiments and an analytical explanation reveals the conditions under which the algorithms converge faster to the solution and also the conditions under which one algorithm converges faster than the other

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

kazhdan groups have cost 1 we prove that every countably infinite group with kazhdan s property t has cost 1 answering a well known question of gaboriau it remains open if they have fixed price 1

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

principal component extraction using recursive least squares learning a new neural network based approach is introduced for recursive computation of the principal components of a stationary vector stochastic process the neurons of a single layer network are sequentially trained using a recursive least squares squares rls type algorithm to extract the principal components of the input process the optimality criterion is based on retaining the maximum information contained in the input sequence so as to be able to reconstruct the network inputs from the corresponding outputs with minimum mean squared error the proof of the convergence of the weight vectors to the principal eigenvectors is also established a simulation example is given to show the accuracy and speed advantages of this algorithm in comparison with the existing methods finally the application of this learning algorithm to image data reduction and filtering of images degraded by additive and or multiplicative noise is considered

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

groups of lie type as products of sl2 subgroups abstract we prove that apart from the suzuki groups every finite simple group of lie type of rank r over a field of q elements can be written as a product of c r subgroups isomorphic to sl 2 q or psl 2 q where c r is a quadratic function this has an application to the theory of expander graphs

property t for kac moody groups over rings abstract let r be a finitely generated commutative ring with 1 let a be an indecomposable 2 spherical generalized cartan matrix of size at least 2 and m m a the largest absolute value of a non diagonal entry of a we prove that there exists an integer n n a such that the kac moody group g a r has property t whenever r has no proper ideals of index less than n and all positive integers less than or equal to m are invertible in r

universal lattices and unbounded rank expanders we study the representations of non commutative universal lattices and use them to compute lower bounds of the constant for the commutative universal lattices g d k sl d x 1 x k for d 3 with respect to several generating sets as an application we show that the cayley graphs of the finite groups text sl 3k mathbb f p can be made expanders with a suitable choice of generators this provides the first example of expander families of groups of lie type where the rank is not bounded and provides counter examples to two conjectures of a lubotzky and b weiss

small bias probability spaces efficient constructions and applications it is shown how to efficiently construct a small probability space on n binary random variables such that for every subset its parity is either zero or one with almost equal probability they are called epsilon biased random variables the number of random bits needed to generate the random variables is o log n log frac 1 epsilon thus if epsilon is polynomially small then the size of the sample space is also polynomial random variables that are epsilon biased can be used to construct almost k wise independent random variables where epsilon is a function of k these probability spaces have various applications l derandomization of algorithms many randomized algorithms that require only k wise independence of their random bits where k is bounded by o log n can be derandomized by using epsilon biased random variables 2 reducing the number of random bits required by certain randomized algorithms e g verification of matrix multiplication 3 exhaustive tes

symmetric groups and expander graphs we construct explicit generating sets sn and tilde s n of the alternating and the symmetric groups which turn the cayley graphs mathcal c text alt n s n and mathcal c text sym n tilde s n into a family of bounded degree expanders for all n

a survey of conjugate gradient algorithms for solution of extreme eigen problems of a symmetric matrix a survey of various conjugate gradient cg algorithms is presented for the minimum maximum eigen problems of a fixed symmetric matrix the cg algorithms are compared to a commonly used conventional method found in imsl it is concluded that the cg algorithms are more flexible and efficient than some of the conventional methods used in adaptive spectrum analysis and signal processing

on the infinite families of zero bounds of complex polynomials we derive several classes of zero bounds for univariate polynomials with complex coefficients the derivations are based on the iterative algorithms of graffe and jin and the approximation methods of newton and kalantari all of the proposed bounds are shown to be asymptotically sharp and the convergence speeds are compared in particular we derive an improving factor to tighten kalantari s bounds this factor and kalantari s bound can both be readily computed with a precalculated lookup table we also propose a modified version of kalantari s bound that is independent of the table and we prove that under some conditions our bound is tighter numerical examples are provided to illustrate the effectiveness of our results

post classification version of jordan s theorem on finite linear groups abstract using classification of finite simple groups i show that a finite subgroup g of gln c where c the complex numbers contains a commutative normal subgroup m of index at most n 1 nalogn b moreover if g is primitive and does not contain normal subgroups that are direct products of large alternating groups then the factor n 1 can be dropped i further show that similar statements hold also in characteristics p 2 if one takes m to be an extension of a group of lie type of characteristic p by a solvable group that has a normal p subgroup with commutative p quotient these results improve the celebrated theorems of jordan and of brauer and feit

an optimal orthonormal system for discriminant analysis abstract this paper proposes a new discriminant analysis with orthonormal coordinate axes of the feature space in general the number of coordinate axes of the feature space in the traditional discriminant analysis depends on the number of pattern classes therefore the discriminatory capability of the feature space is limited considerably the new discriminant analysis solves this problem completely in addition it is more powerful than the traditional one in so far as the discriminatory power and the mean error probability for coordinate axes are concerned this is also shown by a numerical example

convergence analysis of the ojan mca learning algorithm by the deterministic discrete time method minor component analysis mca is a statistical method of extracting the eigenvector associated with the smallest eigenvalue of the covariance matrix of input signals convergence is essential for mca algorithms towards practical applications traditionally the convergence of mca algorithms is indirectly analyzed via their corresponding deterministic continuous time dct systems however the dct method requires the learning rate to approach zero which is not reasonable in many applications due to the round off limitation and tracking requirements this paper studies the convergence of the deterministic discrete time ddt system associated with the ojan mca learning algorithm unlike the dct method the ddt method does not require the learning rate to approach zero in this paper some important convergence results are obtained for the ojan mca learning algorithm via the ddt method simulations are carried out to illustrate the theoretical results achieved

the product replacement algorithm and kazhdan s property t a problem of great importance in computational group theory is to generate nearly uniformly distributed random elements in a finite groupg a good example of such an algorithm should start at any given set of generators use no prior knowledge of the structure of g and in a polynomial number of group operations return the desired random group elements see bb2 then these random elements can be used further to determine the structure of g in a pioneer paper bb1 babai found such an algorithm which provably generates nearly uniformly distributed random elements in o log g group multiplications too many for practical applications a different heuristic the product replacement algorithm was designed by leedham green and soicher lg and studied by celler et al in clmno in spite of the fact that very little theoretical justification was known practical experiments showed excellent performance so it quickly became the most popular practical algorithm to generate random group elements and was included in the two most frequently used group algebra packages gap sc and magma bcp a systematic and quantitative approach was carried out by diaconis and saloffcoste ds1 ds2 see also bb2 cg but their results did not reveal the mystery of the truly outstanding performance of the algorithm the aim of this paper is to propose a conceptual explanation based on kazhdan s property t from representation theory of lie groups and to improve some of the previous estimates on the running time the product replacement algorithm works as follows clmno given a finite group g let k g be the set of k tuples g g1 gk of elements of g such that g1 gk g we call elements of k g the generating k tuples given a generating k tuple g1 gk define a move on it in the following way choose uniformly a pair i j such that 1 i 6 j k then apply one of the following four operations with equal probability

an infinite family of bounds on zeros of analytic functions and relationship to smale s bound smale s analysis of newton s iteration function induce a lower bound on the gap between two distinct zeros of a given complex valued analytic function f z in this paper we make use of a fundamental family of iteration functions b m z m 2 to derive an infinite family of lower bounds on the above gap however even for m 2 where b 2 z coincides with newton s our lower bound is more than twice as good as smale s bound or its improved version given by blum cucker shub and smale when f z is a complex polynomial of degree n for small m the corresponding bound is computable in o n log n arithmetic operations for quadratic polynomials as m increases the lower bounds converge to the actual gap we show how to use these bounds to compute lower bounds on the distance between an arbitrary point and the nearest root of f z in particular using the latter result we show that given a complex polynomial f z a n z n a 0 a n a o 0 for each m 2 we can compute upper and lower bounds u m and l m such that the roots of f z lie in the annulus z l m z u m in particular l 2 1 2 max a k a 0 1 k k 1 n u 2 2 max a n k a n 1 k k 1 n and l3 5 1 2 max a 1 a k 1 a 0 a k a 2 0 1 k k 2 n 1 u 3 5 1 2 max a n 1 a n k 1 a n o n k a 2 n 1 k k 2 n 1 where a 1 a n 1 0 an application of the latter bounds is within weyl s classical quad tree algorithm for computing all roots of a given complex polynomial

simulating independence new constructions of condensers ramsey graphs dispersers and extractors we present new explicit constructions of deterministic randomness extractors dispersers and related objects we say that a distribution x on binary strings of length n is a source if x assigns probability at most 2 n to any string of length n for every 0 we construct the following poly n time computable functions 2 source disperser d 0 1 n 2 0 1 such that for any two independent sources x1 x2 we have that the support of d x1 x2 is 0 1 bipartite ramsey graph let ne2n a corollary is that the function d is a 2 coloring of the edges of kn n the complete bipartite graph over two sets of n vertices such that any induced subgraph of size n by n is not monochromatic 3 source extractor e 0 1 n 3 0 1 such that for any three independent sources x1 x2 x3 we have that e x1 x2 x3 is o 1 close to being an unbiased random bit no previous explicit construction was known for either of these for any a component in these results is a new construction of condensers that may be of independent interest this is a function c 0 1 n 0 1 n c d where c and d are constants that depend only on such that for every source x one of the output blocks of c x is exponentially close to a 0 9 source this result was obtained independently by ran raz the constructions are quite involved and use as building blocks other new and known objects a recurring theme in these constructions is that objects that were designed to work with independent inputs sometimes perform well enough with correlated high entropy inputs the construction of the disperser is based on a new technique which we call the challenge response mechanism that in some sense allows identifying high entropy regions in a given pair of sources using only one sample from the two sources

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

fast adaptive algorithms and networks for class separability features abstract in this article we introduce accelerated algorithms for linear discriminant analysis lda and feature extraction from unimodal multiclass gaussian data current adaptive methods based on the gradient descent optimization technique use a fixed or a monotonically decreasing step size in each iteration which results in a slow convergence rate here we use a variable step size optimally computed in each iteration using the steepest descent method in order to accelerate the convergence of the algorithm based on the new adaptive algorithm we present a self organizing neural network for adaptive computation of the square root of the inverse covariance matrix 1 2 and use it i in a network for optimal feature extraction from gaussian data and ii in cascaded form with a principal component analysis network for lda experimental results demonstrate fast convergence and high stability of the algorithm and justify its advantages for on line pattern recognition applications with stationary and non stationary input data

adaptive multiple minor directions extraction in parallel using a pca neural network a principal component analysis pca neural network is developed for online extraction of the multiple minor directions of an input signal the neural network can extract the multiple minor directions in parallel by computing the principal directions of the transformed input signal so that the stability speed problem of directly computing the minor directions can be avoided to a certain extent on the other hand the learning algorithms for updating the net weights use constant learning rates this overcomes the shortcoming of the learning rates approaching zero in addition the proposed algorithms are globally convergent so that it is very simple to choose the initial values of the learning parameters this paper presents the convergence analysis of the proposed algorithms by studying the corresponding deterministic discrete time ddt equations rigorous mathematical proof is given to prove the global convergence the theoretical results are further confirmed via simulations

counting primes groups and manifolds let lambda sl 2 z be the modular group and let c n lambda be the number of congruence subgroups of lambda of index at most n we prove that lim n infinity log c n lambda log n 2 log log n 3 2 sqrt 2 4 the proof is based on the bombieri vinogradov riemann hypothesis on the average and on the solution of a new type of extremal problem in combinatorial number theory similar surprisingly sharp estimates are obtained for the subgroup growth of lattices in higher rank semisimple lie groups if g is such a lie group and gamma is an irreducible lattice of g it turns out that the subgroup growth of gamma is independent of the lattice and depends only on the lie type of the direct factors of g it can be calculated easily from the root system the most general case of this result relies on the generalized riemann hypothesis but many special cases are unconditional the proofs use techniques from number theory algebraic groups finite group theory and combinatorics

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

a family of fuzzy learning algorithms for robust principal component analysis neural networks in this paper we analyze xu and yuille s robust principal component analysis rpca learning algorithms by means of the distance measurement in space based on the analysis a family of fuzzy rpca learning algorithms is proposed which is robust against outliers these algorithms can explicitly be understood from the viewpoint of fuzzy set theory though xu and yuille s algorithms were proposed based on a statistical physics approach in the proposed algorithms an adaptive learning procedure overcomes the difficulty of selection of learning parameters in xu and yuille s algorithms furthermore the robustness of proposed algorithms is investigated by using the theory of influence functions simulations are carried out to illustrate the robustness of these algorithms

on finitely generated profinite groups ii products in quasisimple groups we prove two results 1 there is an absolute constant d such that for any finite quasisimple group s given 2d arbitrary automorphisms of s every element of s is equal to a product of d twisted commutators defined by the given automorphisms 2 given a natural number q there exist c c q and m m q such that if s is a finite quasisimple group with s z s c j j 1 m are any automorphisms of s and qj j 1 m are any divisors of q then there exist inner automorphisms j of s such that s m s j j q j these results which rely on the classification of finite simple groups are needed to complete the proofs of the main theorems of part i

algorithms and networks for accelerated convergence of adaptive lda we introduce and discuss new accelerated algorithms for linear discriminant analysis lda in unimodal multiclass gaussian data these algorithms use a variable step size optimally computed in each iteration using i the steepest descent ii conjugate direction and iii newton raphson methods in order to accelerate the convergence of the algorithm current adaptive methods based on the gradient descent optimization technique use a fixed or a monotonically decreasing step size in each iteration which results in a slow convergence rate furthermore the convergence of these algorithms depends on appropriate choices of the step sizes the new algorithms have the advantage of automatic optimal selection of the step size using the current data samples based on the new adaptive algorithms we present self organizing neural networks for adaptive computation of s 1 2 and use them in cascaded form with a pca network for lda experimental results demonstrate fast convergence and robustness of the new algorithms and justify their advantages for on line pattern recognition applications with stationary and non stationary multidimensional input data

hard examples for resolution exponential lower bounds are proved for the length of resolution refutations of sets of disjunctions constructed from expander graphs using the method of tseitin since these sets of clauses encode biconditionals they have short polynomial length refutations in a standard axiomatic formulation of propositional calculus

on a question of m newman on the number of commutators abstract m newman asked whether there is an absolute constant c such that every matrix in slnr is the product of at most c commutators where r ranges over euclidean commutative rings and n 3 we give here a negative answer however if for the ring r every matrix in slmr is the product of a bounded number of commutators for some fixed m 3 then for all sufficiently large n every matrix in slnr is the product of six commutators

on finitely generated profinite groups i strong completeness and uniform bounds we prove that in every finitely generated profinite group every subgroup of finite index is open this implies that the topology on such groups is determined by the algebraic structure this is deduced from the main result about finite groups let w be a locally finite group word and d n then there exists f f w d such that in every d generator finite group g every element of the verbal subgroup w g is equal to a product of f w values an analogous theorem is proved for commutators this implies that in every finitely generated profinite group each term of the lower central series is closed the proofs rely on some properties of the finite simple groups to be established in part ii

algorithms for accelerated convergence of adaptive pca we derive and discuss adaptive algorithms for principal component analysis pca that are shown to converge faster than the traditional pca algorithms due to oja and karhunen 1985 sanger 1989 and xu 1993 it is well known that traditional pca algorithms that are derived by using gradient descent on an objective function are slow to converge furthermore the convergence of these algorithms depends on appropriate choices of the gain sequences since online applications demand faster convergence and an automatic selection of gains we present new adaptive algorithms to solve these problems we first present an unconstrained objective function which can be minimized to obtain the principal components we derive adaptive algorithms from this objective function by using 1 gradient descent 2 steepest descent 3 conjugate direction and 4 newton raphson methods although gradient descent produces xu s lmser algorithm the steepest descent conjugate direction and newton raphson methods produce new adaptive algorithms for pca we also provide a discussion on the landscape of the objective function and present a global convergence proof of the adaptive gradient descent pca algorithm using stochastic approximation theory extensive experiments with stationary and nonstationary multidimensional gaussian sequences show faster convergence of the new algorithms over the traditional gradient descent methods we also compare the steepest descent adaptive algorithm with state of the art methods on stationary and nonstationary sequences

reduced 1 cohomology and relative property t shalom characterized property t in terms of the vanishing of all reduced first cohomology for compactly generated groups we characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes we show that in a strong sense this is inequivalent to relative property t even for compactly generated group pairs

uniform kazhdan constant for some families of linear groups abstract let r be a ring generated by l elements with stable range r assume that the group el d r has kazhdan constant 0 0 for some d r 1 we prove that there exist 0 l 0 and k n s t for every n d el n r has a generating set of order k and a kazhdan constant larger than as a consequence we obtain for sl n z where n 3 a kazhdan constant which is independent of n w r t generating set of a fixed size

fast subspace tracking and neural network learning by a novel information criterion we introduce a novel information criterion nic for searching for the optimum weights of a two layer linear neural network nn the nic exhibits a single global maximum attained if and only if the weights span the desired principal subspace of a covariance matrix the other stationary points of the nic are unstable saddle points we develop an adaptive algorithm based on the nic for estimating and tracking the principal subspace of a vector sequence the nic algorithm provides a fast on line learning of the optimum weights for the two layer linear nn we establish the connections between the nic algorithm and the conventional mean square error mse based algorithms such as oja s algorithm oja 1989 lmser past apex and gha the nic algorithm has several key advantages such as faster convergence which is illustrated through analysis and simulation

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