decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

structure of chevalley groups over rings via universal localization abstract in the current article we study the structure of a chevalley group g r over a commutative ring r we generalize and improve the following results on the standard relative and multi relative commutator formulas the nilpotent structure of relative k 1 the bounded word length of commutators to this end we enlarge the elementary group construct a generic element for the extended elementary group and use localization in the universal ring the key step is a construction of a generic element for the principal congruence subgroup corresponding to a principal ideal

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

products of two unipotent matrices of index 2 abstract necessary and sufficient conditions are presented for a square matrix a over a general field f to be the product of two unipotent matrices of index 2 this generalizes a result established by wang and wu 1991 4 for the case where f is the complex field

divisibility of reduction in groups of rational numbers given a multiplicative group of nonzero rational numbers and a positive integer m we consider the problem of determining the density of the set of primes p for which the order of the reduction modulo p of the group is divisible by m in the case when the group is finitely generated the density is explicitly computed some examples of groups with infinite rank are considered

sums and products of two quadratic matrices abstract necessary and sufficient conditions are given for a complex matrix t to be the sum of an idempotent and a square zero matrix these conditions are expressed in terms of 1 the block structure of t under similarity as well as 2 the elementary divisor structure of t this leads to a complete characterization of which complex matrices are a sum of two quadratic matrices also using a similar scheme we characterize when a matrix is the product of an involution and a unipotent matrix of index 2

approximation by unitaries with finite spectrum in purely infinite c algebras abstract we prove that a unital purely infinite simple c algebra has c exponential rank at most 1 and therefore has the property weak fu every element of the identity component of the unitary group is norm limit of unitaries with finite spectrum we also prove that the appropriate modification of this statement for nonunital algebras

analysis of a finite pml approximation to the three dimensional elastic wave scattering problem we consider the application of a perfectly matched layer pml technique to approximate solutions to the elastic wave scattering problem in the frequency domain the pml is viewed as a complex coordinate shift in spherical coordinates which leads to a variable complex coefficient equation for the displacement vector posed on an infinite domain the complement of the scatterer the rapid decay of the pml solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation for the truncated problem we prove existence and uniqueness of the solutions to the infinite domain and truncated domain pml equations provided that the truncated domain is sufficiently large we also show exponential convergence of the solution of the truncated pml problem to the solution of the original scattering problem in the region of interest we then analyze a galerkin numerical approximation to the truncated pml problem and prove that it is well posed provided that the pml damping parameter and mesh size are small enough finally computational results illustrating the efficiency of the finite element pml approximation are presented

the lower central series of subgroups of the vershik kerov group abstract we describe the derived and the lower central series of certain subgroups of the vershik kerov group we are concerned with the subgroups consisting of infinite matrices having finite number of nonzero entries in each row we consider the group of matrices over rings which are associative commutative of stable rank at most one and such that the identity can be written as a sum of two units for this case we give a complete description of the derived and the lower central series moreover we prove that every element of discussed commutator subgroups can be written as a product of a finite number of commutators

dimension reduction with extreme learning machine data may often contain noise or irrelevant information which negatively affect the generalization capability of machine learning algorithms the objective of dimension reduction algorithms such as principal component analysis pca non negative matrix factorization nmf random projection rp and auto encoder ae is to reduce the noise or irrelevant information of the data the features of pca eigenvectors and linear ae are not able to represent data as parts e g nose in a face image on the other hand nmf and non linear ae are maimed by slow learning speed and rp only represents a subspace of original data this paper introduces a dimension reduction framework which to some extend represents data as parts has fast learning speed and learns the between class scatter subspace to this end this paper investigates a linear and non linear dimension reduction framework referred to as extreme learning machine ae elm ae and sparse elm ae selm ae in contrast to tied weight ae the hidden neurons in elm ae and selm ae need not be tuned and their parameters e g input weights in additive neurons are initialized using orthogonal and sparse random weights respectively experimental results on usps handwritten digit recognition data set cifar 10 object recognition and norb object recognition data set show the efficacy of linear and non linear elm ae and selm ae in terms of discriminative capability sparsity training time and normalized mean square error

on euclidean ideal classes in certain abelian extensions in this article we show that certain abelian extensions k with unit rank greater than or equal to three have cyclic class group if and only if it has a euclidean ideal class this result improves an earlier result of murty and graves one can improve this result up to unit rank 2 if one assumes the elliott and halberstam conjecture see conjecture 1 in preliminaries these results are known under generalized riemann hypothesis by the work of lenstra j lond math soc 10 457 465 see also weinberger proc symp pure math 24 321 332

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

products of involutory matrices over rings abstract let gln a be a matrix over a commutative ring a with 1 such that det 2 1 if is cyclic it can be written as a product of at most three involutions when a satisfies the first bass stable range condition then can be written as a product of at most five involutions if in addition either n 3 or n 4 and det 1 then can be written as a product of at most four involutions when a is a dedekind ring of arithmetic type the number of involutions needed to express is uniformly bounded for any n 3 when a c x the number of involutions is unbounded for any n 2

products of elementary and idempotent matrices over integral domains abstract a ring r such that invertible matrices over r are products of elementary matrices is called after cohn generalized euclidean extending results proved by ruitenburg for bezout domains characterizations of generalized euclidean commutative domains are obtained that relate them with the property that singular matrices are products of idempotent matrices this latter property is investigated focusing on 2 2 matrices which is not restrictive in the context of bezout domains it is proved that domains r that satisfy a suitable property of ideals called princ are necessarily bezout domains if 2 2 singular matrices over r are products of idempotent matrices the class of rings satisfying princ includes factorial and projective free domains the connection with the existence of a weak euclidean algorithm a notion introduced by o meara for dedekind domains is also investigated

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

a unified framework for semi supervised dimensionality reduction in practice many applications require a dimensionality reduction method to deal with the partially labeled problem in this paper we propose a semi supervised dimensionality reduction framework which can efficiently handle the unlabeled data under the framework several classical methods such as principal component analysis pca linear discriminant analysis lda maximum margin criterion mmc locality preserving projections lpp and their corresponding kernel versions can be seen as special cases for high dimensional data we can give a low dimensional embedding result for both discriminating multi class sub manifolds and preserving local manifold structure experiments show that our algorithms can significantly improve the accuracy rates of the corresponding supervised and unsupervised approaches

products of commutators of dilatations abstract we consider the group sl n f of all n n matrices with determinant 1 over a field let res a denote rank a i for any a in gl n f a matrix a gl n f is called a dilatation if a is similar to a matrix i n 1 a a 1 we prove that if f 4 then every matrix a in sl n f is the product of at most res a 2 1 commutators of dilatations for n 2

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

pml method for electromagnetic scattering problem in a two layer medium the perfectly matched layer pml method is well studied for acoustic scattering problems electromagnetic scattering problems and more recently elastic scattering problems with homogeneous background media the purpose of this paper is to present the stability and exponential convergence of the pml method for a three dimensional electromagnetic scattering problem in a two layer medium the main contributions of this paper are threefold first we establish the well posedness of the original scattering problem for any dirichlet boundary value in boldsymbol h 1 2 rm div gamma d where gamma d stands for the boundary of the scatterer second we propose a new weak formulation for the original problem where the dirichlet to neumann operator is proposed on a truncation boundary inside the pml this argument is favorable in the analysis of the pml dirichlet to neumann operator the inf sup condition is proved for the bilinear form third we establish the well posedness of the pml problem a

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

on the r rank artin conjecture we assume the generalized riemann hypothesis and prove an asymptotic formula for the number of primes for which f p can be generated by r given multiplicatively independent numbers in the case when the r given numbers are primes we express the density as an euler product and apply this to a conjecture of brown zassenhaus j number theory 3 1971 306 309 finally in some examples we compare the densities approximated with the natural densities calculated with primes up to 9 10 4

on decomposing any matrix as a linear combination of three idempotents abstract in a recent article we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients in this one we use those results to improve on a recent theorem of rabanovich we establish that every square matrix is a linear combination of three idempotents for an arbitrary coefficient field rather than just one of characteristic 0

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

word length in elementary matrices let r be an associative ring with identity recall that the group e n r is generated by those n x n matrices over r differing from the identity matrix z in at most one entry which must be off the main diagonal defining v a to be the minimal number of nontrivial generators necessary to obtain the product a one may ask whether there is an absolute bound on v a as a varies in e n r if such a bound exists denote by e r the maximum attained by v on e n r it is then of interest to estimate e r as a by product of their results on commutator products r k dennis and l n vaserstein 2 give the following relative bound which holds for n m1 sr r where w r is the stable rank in s

absorbing boundary condition for the elastic wave equation velocity stress formulation this paper describes an absorbing boundary condition for finite difference modeling of elastic wave propagation in two and three dimensions the boundary condition is particularly effective for obliquely incident waves typically quite troublesome for absorbing boundaries analytical predictions of the boundary reflection coefficients of a few percent or less for angles of incidence up to 89 are verified in example finite difference applications the algorithm is appropriate for use in a velocity stress finite difference vs fd formulation it is computationally simpler than a similar absorbing boundary given previously for the standard displacement formulation a second algorithm is presented which may be advantageous when the boundary of interest is exposed to strong evanescent waves both algorithms require that the adjacent elastic medium be locally homogeneous

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

joint low rank and sparse principal feature coding for enhanced robust representation and visual classification recovering low rank and sparse subspaces jointly for enhanced robust representation and classification is discussed technically we first propose a transductive low rank and sparse principal feature coding lspfc formulation that decomposes given data into a component part that encodes low rank sparse principal features and a noise fitting error part to well handle the outside data we then present an inductive lspfc i lspfc i lspfc incorporates embedded low rank and sparse principal features by a projection into one problem for direct minimization so that the projection can effectively map both inside and outside data into the underlying subspaces to learn more powerful and informative features for representation to ensure that the learned features by i lspfc are optimal for classification we further combine the classification error with the feature coding error to form a unified model discriminative lspfc d lspfc to boost performance the model of d lspfc seamlessly integrates feature coding and discriminative classification so the representation and classification powers can be enhanced the proposed approaches are more general and several recent existing low rank or sparse coding algorithms can be embedded into our problems as special cases visual and numerical results demonstrate the effectiveness of our methods for representation and classification

nonlinear dimensionality reduction by topologically constrained isometric embedding many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space the embedding process usually relies on distances between neighboring features mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to its non convexity distortions resulting from using long geodesics indiscriminately lead to a known limitation of the isomap algorithm when used to map non convex manifolds presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of flat manifolds with boundaries the resulting algorithm filters out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the boundary of the feature manifold thus avoiding an inherent limitation of the isomap algorithm since the proposed algorithm matches non local structures it is robust to strong noise we show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques both global and local in nature

products of conjugacy classes of two by two matrices abstract we compute the covering number and the extended covering number for the group psl 2 f with a field f similar numbers are defined for any group and they are computed for the groups gl 2 f pgl 2 f and sl 2 f

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

supervised optimal locality preserving projection in the past few years the computer vision and pattern recognition community has witnessed a rapid growth of a new kind of feature extraction method the manifold learning methods which attempt to project the original data into a lower dimensional feature space by preserving the local neighborhood structure among these methods locality preserving projection lpp is one of the most promising feature extraction techniques unlike the unsupervised learning scheme of lpp this paper follows the supervised learning scheme i e it uses both local information and class information to model the similarity of the data based on novel similarity we propose two feature extraction algorithms supervised optimal locality preserving projection solpp and normalized laplacian based supervised optimal locality preserving projection nl solpp optimal here means that the extracted features via solpp or nl solpp are statistically uncorrelated and orthogonal we compare the proposed solpp and nl solpp with lpp orthogonal locality preserving projection olpp and uncorrelated locality preserving projection ulpp on publicly available data sets experimental results show that the proposed solpp and nl solpp achieve much higher recognition accuracy

bagging constraint score for feature selection with pairwise constraints constraint score is a recently proposed method for feature selection by using pairwise constraints which specify whether a pair of instances belongs to the same class or not it has been shown that the constraint score with only a small amount of pairwise constraints achieves comparable performance to those fully supervised feature selection methods such as fisher score however one major disadvantage of the constraint score is that its performance is dependent on a good selection on the composition and cardinality of constraint set which is very challenging in practice in this work we address the problem by importing bagging into constraint score and a new method called bagging constraint score bcs is proposed instead of seeking one appropriate constraint set for single constraint score in bcs we perform multiple constraint score each of which uses a bootstrapped subset of original given constraint set diversity analysis on individuals of ensemble shows that resampling pairwise constraints is helpful for simultaneously improving accuracy and diversity of individuals we conduct extensive experiments on a series of high dimensional datasets from uci repository and gene databases and the experimental results validate the effectiveness of the proposed method

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

regularized generalized eigen decomposition with applications to sparse supervised feature extraction and sparse discriminant analysis we propose a general technique for obtaining sparse solutions to generalized eigenvalue problems and call it regularized generalized eigen decomposition rged for decades fisher s discriminant criterion has been applied in supervised feature extraction and discriminant analysis and it is formulated as a generalized eigenvalue problem thus rged can be applied to effectively extract sparse features and calculate sparse discriminant directions for all variants of fisher discriminant criterion based models particularly rged can be applied to matrix based and even tensor based discriminant techniques for instance 2d linear discriminant analysis 2d lda furthermore an iterative algorithm based on the alternating direction method of multipliers is developed the algorithm approximately solves rged with monotonically decreasing convergence and at an acceptable speed for results of modest accuracy numerical experiments based on four data sets of different types of images show that rged has competitive classification performance with existing multidimensional and sparse techniques of discriminant analysis highlightswe propose a new technique called regularized generalized eigen decomposition rged rged solves generalized eigenvalue problems and obtains sparse solutions it is easy and straightforward applying rged to sparse discriminant analysis and feature extraction an algorithm is developed to solve it with monotonically decreasing convergence rged has competitive classification performance comparing with other methods

new anisotropic perfectly matched layer boundary condition for elastic waves in solids the perfectly matched layer pml is currently a popular absorbing boundary condition for numerical modeling of electromagnetic acoustic and elastic wave problems existing pml formulation for elastic waves uses a field splitting scheme which increases the number of variables in the computation the new anisotropic elastic wave pml modifies the elasticity or compliance tensors to be anisotropic it utilizes fewer variables for both isotropic and anisotropic media so that computing memory and time can be reduced significantly its formulations for cartesian and curvilinear coordinate systems will be discussed moreover this anisotropic pml can be directly used in the finite element method and easily be implemented in the finite difference time domain method as well work supported by nasa nag3 2147 and by nsf cts 9870015

multi manifold discriminant isomap for visualization and classification isomap aims to recover the intrinsic geometric structure of manifold by preserving geodesic distances between all pairs of data points however it is an unsupervised dimensionality reduction method usually using class label information can increase the discriminating capability hence a new supervised isomap is proposed in this paper dubbed multi manifold discriminant isomap mmd isomap first data points are partitioned into different manifolds according to their class label information then mmd isomap aims at seeking an optimal nonlinear subspace to preserve the geometrical structure of each manifold according to the isomap criterion meanwhile to enhance the discriminating capability by maximizing the distances between data points of different manifolds finally the corresponding optimization problem is solved by using a majorization algorithm furthermore two new numerical metrics are designed to measure the performance of dimensionality reduction method in both visualization and classification experiments mmd isomap achieves improved performance over many state of the art methods preserve the geometrical structure and enhance the discriminating capability smacof algorithm is introduced to solve the optimization problem two metrics are designed to evaluate dimensionality reduction methods experiments show the effectiveness of the method in visualization and classification

products of involutions in the stable general linear group abstract in the stable general linear group over an arbitrary field we prove that every element with determinant 1 is the product of three involutions and of no less in general we also obtain several results of the same flavor with applications to decompositions of automorphisms of an infinite dimensional vector space that are scalar multiples of finite rank perturbations of the identity

trace ratio criterion based generalized discriminative learning for semi supervised dimensionality reduction dealing with high dimensional data has always been a major problem in many pattern recognition and machine learning applications trace ratio criterion is a criterion that can be applicable to many dimensionality reduction methods as it directly reflects euclidean distance between data points of within or between classes in this paper we analyze the trace ratio problem and propose a new efficient algorithm to find the optimal solution based on the proposed algorithm we are able to derive an orthogonal constrained semi supervised learning framework the new algorithm incorporates unlabeled data into training procedure so that it is able to preserve the discriminative structure as well as geometrical structure embedded in the original dataset under such a framework many existing semi supervised dimensionality reduction methods such as sda lap lda ssdr ssmmc can be improved using our proposed framework which can also be used to formulate a corresponding kernel framework for handling nonlinear problems theoretical analysis indicates that there are certain relationships between linear and nonlinear methods finally extensive simulations on synthetic dataset and real world dataset are presented to show the effectiveness of our algorithms the results demonstrate that our proposed algorithm can achieve great superiority to other state of art algorithms

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

semi supervised local multi manifold isomap by linear embedding for feature extraction we explore the discriminative feature extraction problem a semisupervised local multimanifold isomap by linear embedding is proposed our model can use labeled and unlabeled data to deliver manifold features our model aims to minimize pairwise intraclass distances in the same manifold our model aims to maximize the distances between different manifolds in this paper we mainly propose a semi supervised local multi manifold isomap learning framework by linear embedding termed ssmm isomap that can apply the labeled and unlabeled training samples to perform the joint learning of neighborhood preserving local nonlinear manifold features and a linear feature extractor the formulation of ssmm isomap aims at minimizing pairwise distances of intra class points in the same manifold and maximizing the distances over different manifolds to enhance the performance of nonlinear manifold feature learning we also incorporate the neighborhood reconstruction error to preserve local topology structures between both labeled and unlabeled samples to enable our ssmm isomap to extract local manifold features from the outside new data we also add a feature approximation error that correlates manifold features with embedded features by the jointly learnt feature extractor thus the learnt linear extractor can extract the local manifold features from the new data efficiently by direct embedding to optimize the proposed objective function two effective schemes are presented i e scaling by majorizing a complicated function and eigen decomposition notice that the comparison of the proposed two solvers is also described we mainly evaluate ssmm isomap for manifold feature learning data clustering and classification extensive simulation results verify the effectiveness of our ssmm isomap algorithm compared with other related feature learning techniques

products of unipotent matrices of index 2 abstract we show that an n n complex matrix t is the product of two unipotent matrices of index 2 if and only if t is similar to a matrix of the form d d 1 i n i mi 1 ji where 0 and 1 are not eigenvalues of d n is nilpotent and each ji is a nilpotent jordan block of even size on the other hand t is the product of finitely many unipotent matrices of index 2 if and only if det t 1 in this case the minimal number of required unipotents is 1 if n 1 3 if n 2 and 4 if n 3

analysis of a cartesian pml approximation to acoustic scattering problems in r2 and r3 we consider the application of a perfectly matched layer pml technique applied in cartesian geometry to approximate solutions of the acoustic scattering problem in the frequency domain the pml is viewed as a complex coordinate shift stretching and leads to a variable complex coefficient equation for the acoustic wave posed on an infinite domain the complement of the bounded scatterer the use of cartesian geometry leads to a pml operator with simple coefficients although still complex symmetric non hermitian the pml reformulation results in a problem whose solution coincides with the original solution inside the pml layer while decaying exponentially outside the rapid decay of the pml solution suggests truncation to a bounded domain with a convenient outer boundary condition and subsequent finite element approximation for the truncated problem this paper provides new stability estimates for the cartesian pml approximations both on the infinite and the truncated domain we first investigate the stability of the infinite pml approximation as a function of the pml strength s 0 this is done for pml methods which involve continuous piecewise smooth stretching as well as piecewise constant stretching functions we next introduce a truncation parameter m which determines the size of the pml layer our analysis shows that the truncated pml problem is stable provided that the product of email protected 0 is sufficiently large in which case the solution of the problem on the truncated domain converges exponentially to that of the original problem in the domain of interest near the scatterer this justifies the simple computational strategy of selecting a fixed pml layer and increasing s 0 to obtain the desired accuracy the results of numerical experiments varying m and s 0 are given which illustrate the theoretically predicted behavior

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

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

a note on the existence and uniqueness of solutions of frequency domain elastic wave problems a priori estimates in h1 abstract in this note we provide existence and uniqueness results for frequency domain elastic wave problems these problems are posed on the complement of a bounded domain r 3 the scatterer the boundary condition at infinity is given by the kupradze sommerfeld radiation condition and involves different sommerfeld conditions on different components of the field our results are obtained by setting up the problem as a variational problem in the sobolev space h 1 on a bounded domain we use a nonlocal boundary condition which is related to the dirichlet to neumann conditions used for acoustic and electromagnetic scattering problems we obtain stability results for the source problem a necessary ingredient for the analysis of numerical methods for this problem based on finite elements or finite differences

analysis of a cartesian pml approximation to acoustic scattering problems in r2 abstract in this paper we consider a cartesian pml approximation to solutions of acoustic scattering problems on an unbounded domain in r 2 the perfectly matched layer pml technique in a curvilinear coordinate system has been researched for acoustic scattering applications both in theory and computation our goal will be to extend the results of spherical cylindrical pml to pml in cartesian coordinates that is the well posedness of cartesian pml approximation on both the unbounded and truncated domains the exponential convergence of approximate solutions as a function of domain size is also shown we note that once the stability and convergence of the continuous truncated problem has been achieved the analysis of the resulting finite element approximations is then classical finally the results of numerical computations illustrating the theory and efficiency of the cartesian pml approach will be given

primes in intervals of bounded length in april 2013 yitang zhang proved the existence of a finite bound b such that there are infinitely many pairs of distinct primes which differ by no more than b this is a massive breakthrough makes the twin prime conjecture look highly plausible which can be re interpreted as the conjecture that one can take b 2 and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable the original purpose of this talk was to discuss zhang s extraordinary work putting it in its context in analytic number theory and to sketch a proof of his theorem zhang had even proved the result with b 70 000 000 moreover a co operative team polymath8 collaborating only on line had been able to lower the value of b to 4680 not only had they been more careful in several difficult arguments in zhang s original paper they had also developed zhang s techniques to be both more powerful and to allow a much simpler proof indeed the proof of zhang s theorem that will be given in the write up of this talk is based on these developments in november inspired by zhang s extraordinary breakthrough james maynard dramatically slashed this bound to 600 by a substantially easier method both maynard and terry tao who had independently developed the same idea were able to extend their proofs to show that for any given integer m 1 there exists a bound bm such that there are infinitely many intervals of length bm containing at least m distinct primes we will also prove this much stronger result herein even showing that one can take bm e 8m if zhang s method is combined with the maynard tao set up then it appears that the bound can be further reduced to 576 if all of these techniques could be pushed to their limit then we would obtain b b2 12 so new ideas are still needed to have a feasible plan for proving the twin prime conjecture the article will be split into two parts the first half which appears here we will introduce the work of zhang polymath8 maynard and tao and explain their arguments that allow them to prove their spectacular results as we will discuss zhang s main novel contribution is an estimate for primes in relatively short arithmetic progressions the second half of this article sketches a proof of this result the bulletin article will contain full details of this extraordinary work part 1 primes in short intervals