principal component analysis in the local differential privacy model abstract in this paper we study the principal component analysis pca problem under the distributed non interactive local differential privacy model for the low dimensional case i e p n we show the optimal rate of k p n 2 omitting the eigenvalue terms for the private minimax risk of the k dimensional pca using the squared subspace distance as the measurement where n is the sample size and is the privacy parameter for the high dimensional i e p n row sparse case we first give a lower bound of k s log p n 2 on the private minimax risk where s is the underlying sparsity parameter then we provide an efficient algorithm to achieve the upper bound of o s 2 log p n 2 experiments on both synthetic and real world datasets confirm our theoretical guarantees

influential features pca for high dimensional clustering we consider a clustering problem where we observe feature vectors x i in r p i 1 2 ldots n from k possible classes the class labels are unknown and the main interest is to estimate them we are primarily interested in the modern regime of p gg n where classical clustering methods face challenges we propose influential features pca if pca as a new clustering procedure in if pca we select a small fraction of features with the largest kolmogorov smirnov ks scores obtain the first k 1 left singular vectors of the post selection normalized data matrix and then estimate the labels by applying the classical k means procedure to these singular vectors in this procedure the only tuning parameter is the threshold in the feature selection step we set the threshold in a data driven fashion by adapting the recent notion of higher criticism as a result if pca is a tuning free clustering method we apply if pca to 10 gene microarray data sets the method has competitive performance in clustering especially in three of the data sets the error rates of if pca are only 29 or less of the error rates by other methods we have also rediscovered a phenomenon on empirical null by efron j amer statist assoc 99 2004 96 104 on microarray data with delicate analysis especially post selection eigen analysis we derive tight probability bounds on the kolmogorov smirnov statistics and show that if pca yields clustering consistency in a broad context the clustering problem is connected to the problems of sparse pca and low rank matrix recovery but it is different in important ways we reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each

cofree quiver settings abstract we give a complete classification of all quivers q and dimension vectors for which the representation space rep q is cofree as a representation of its natural symmetry group gl

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

some algorithms for semi invariants of quivers abstract we present some theorems and algorithms for calculating perpendicular categories and locally semi simple decompositions we implemented a computer program tetiva based on these algorithms and we offer this program for everybody s use

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

on the equations and classification of toric quiver varieties toric quiver varieties moduli spaces of quiver representations are studied given a quiver and a weight there is an associated quasi projective toric variety together with a canonical embedding into projective space it is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3 in each fixed dimension d up to isomorphism there are only finitely many d dimensional toric quiver varieties a procedure for their classification is outlined

solving large scale linear multicommodity flow problems with an active set strategy and proximal accpm in this paper we propose to solve the linear multicommodity flow problem using a partial lagrangian relaxation the relaxation is restricted to the set of arcs that are likely to be saturated at the optimum this set is itself approximated by an active set strategy the partial lagrangian dual is solved with proximal accpm a variant of the analytic center cutting plane method the new approach makes it possible to solve huge problems when few arcs are saturated at the optimum as appears to be the case in many practical problems

normality of marsden weinstein reductions for representations of quivers we prove that the marsden weinstein reductions for the moment map associated to representations of a quiver are normal varieties we give an application to conjugacy classes of matrices

qurves and quivers abstract in this paper we associate to a k qurve a formerly known as a quasi free algebra j cuntz d quillen algebra extensions and nonsingularity j amer math soc 8 1995 251 289 or formally smooth algebra m kontsevich a rosenberg noncommutative smooth spaces math ag 9812158 1998 the one quiver q 1 a and dimension vector 1 a this pair contains enough information to reconstruct for all n n the gl n etale local structure of the representation scheme rep n a in an appendix we indicate how one might extend this to qurves over non algebraically closed fields further we classify all finitely generated groups g such that the group algebra kg is a k qurve if char k 0 these are exactly the virtually free groups we determine the one quiver setting in this case and indicate how it can be used to study the finite dimensional representations of virtually free groups as this approach also applies to fundamental algebras of graphs of separable k algebras we state the results in this more general setting

asymptotic expansions for trace functionals abstract we obtain taylor approximations for functionals v tr f h 0 v defined on the bounded self adjoint operators where h 0 is a self adjoint operator with compact resolvent and f is a sufficiently nice scalar function relaxing assumptions on the operators made in 17 and derive estimates and representations for the remainders of these approximations

poisson structures on moduli spaces of representations abstract we show that a poisson structure can be induced on the affine moduli space of semisimple representations of an associative algebra from a suitable lie algebra structure on the zeroth hochschild homology of the algebra in particular this applies to necklace lie algebra for path algebras of doubled quivers and preprojective algebras we call such structures h 0 poisson structures and show that they are well behaved for azumaya algebras and under morita equivalence

quiver varieties and lusztig s algebra abstract we study preprojective algebras of graphs and their relationship to module categories over representations of quantum sl 2 as an application ade quiver varieties of nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by lusztig

classes of degeneracy loci for quivers the thom polynomial point of view the chern classes for degeneracy loci of quivers are natural generalizations of the thom porteous giambelli formula suppose that e f are vector bundles over a manifold m and that s e f is a vector bundle homomorphism the question is which cohomology class is defined by the set6k s m consisting of points m where the linear map s m has corank k the answer due to i porteous is a determinant in terms of chern classes of the bundles e f we can generalize the question by giving more bundles over m and bundle maps among them the situation can be conveniently coded by an oriented graph called a quiver assigning vertices for bundles and arrows for maps we give a new method for calculating chern class formulae for degeneracy loci of quivers we show that for representation finite quivers this is a special case of the problem of calculating thom polynomials for group actions this allows us to apply a method for calculating thom polynomials developed by the authors the method reducing the calculations to solving a system of linear equations is quite different from the method of a buch and w fulton developed for calculating chern class formulae for degeneracy loci of a n quivers and it is more general can be applied to an dn e6 e7 and e8 quivers we provide sample calculations for a 3 and d4 quivers

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

double poisson algebras in this paper we develop poisson geometry for non commutative algebras this generalizes the bi symplectic geometry which was recently and independently introduced by crawley boevey etingof and ginzburg our quasi poisson brackets induce classical quasi poisson brackets on representation spaces as an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by crawley boevey and shaw carry a natural poisson structure

fibers of generic brauer severi schemes abstract in this note we investigate the fibers of van den bergh s brauer severi scheme bs m n v m n of the trace ring of m generic n n matrices over the variety of matrix invariants if v m n is a point of representation type m 1 n 1 m r n r then the reduced variety of 1 has m i m i irreducible components each of dimension equation

quantum harmonic analysis and geometric invariants we develop two topics in parallel and show their inter relation the first centers on the notion of a fractional differentiable structure on a commutative or a non commutative space we call this studyquantum harmonic analysis the second concerns homotopy invariants for these spaces and is an aspect of non commutative geometry we study an algebra a which will be a banach algebra with unit represented as an algebra of operators on a hilbert space h in order to obtain a geometric interpretation of a we define a derivative on elements of a we do this in a hilbert space context takingdaas a commutatorda q a hereqis a basic self adjoint operator with discrete spectrum increasing sufficiently rapidly that exp q 2has a trace whenever 0 we can define fractional differentiability of order with 0 0 we concentrate on subalgebras a of j which have certain additional covariance properties under a group z2 g acting on h by a unitary representation u g in addition the derivativeqis assumed to be g invariant the geometric interpretation flows from the assumption that elements of a possess an arbitrarily small fractional derivative we study homotopy invariants of a in terms of equivariant entire cyclic cohomology in fact the existence of a fractional derivative on a allows the construction of the cochain jlo which plays the role of the integral of differential forms we give a simple expression for a homotopy invariant zq a g determined by pairing jlo with a g invariant elementa a such thatais a square root of the identity this invariant is zq a g 1 e t2tr u g ae q2 it da dt this representation of the pairing is reminiscent of the heat kernel representation for an index in fact this quantity is an invariant in the following sense we isolate a simple condition on a familyq of differentiations that yields a continuously differentiable family jlo of cochains since zq a g need not be an integer continuity of jlo in is insufficient to prove the constancy of the pairing however the existence of the derivative leads to the existence of the homotopy as a vanishes for a coboundary and asd jlo d is a coboundary our condition onq ensures that zq a g is independent of hence it is a homotopy invariant the theory of zq a g reduces to the study of the radon transform of sequences of certain functions the fractional differentiability properties of elements of a translate into properties of the asymptotics of the sequences of radon transforms the condition that jlofit into the framework of entire cyclic cohomology translates to the existence of some fractional derivative for functions in the algebra under study and in particular the assumption 1 thus the study of fractionally differentiable structures dove tails naturally with the theory of homotopy invariants in our study of quantum harmonic analysis we introduce spaces t of operator valued distributions these spaces are bounded linear operators between sobolev spaces the elements of the interpolation spaces the banach algebras j have derivativesdawhich belong to the spaces t for a certain range of and we extend the theory of the radon transform from products of regularized bounded operators to products of regularized operator valued distributions we sometimes wish to evaluate such an invariant at the endpoint of an interval such as 0 1 where zq a g becomes singular as 0 we discuss in brief a procedure to regularize the endpoint and a method to recover zq a g fully from certain partial information at the endpoint finally we generalize this approach to cover the case whenqcan be split into the sum of independent partsq1 q2 such that q1 q2 2 q1 2 q2 2 here we assume thatq1and q2 2are g invariant but not necessarilyq2 with further assumptions ona the most important being that q1 2 q2 2commutes witha we obtain a modified formula for an invariant namely z qj a g 1 e t2tr u g ae q21 q22 2 itd1a dt

finding a sparse vector in a subspace linear sparsity using alternating directions is it possible to find the sparsest vector direction in a generic subspace mathcal s subseteq mathbb r p with text dim left mathcal s right n this problem can be considered a homogeneous variant of the sparse recovery problem and finds connections to sparse dictionary learning sparse pca and many other problems in signal processing and machine learning in this paper we focus on a planted sparse model for the subspace the target sparse vector is embedded in an otherwise random subspace simple convex heuristics for this planted recovery problem provably break down when the fraction of nonzero entries in the target sparse vector substantially exceeds o 1 sqrt n in contrast we exhibit a relatively simple nonconvex approach based on alternating directions which provably succeeds even when the fraction of nonzero entries is omega 1 to the best of our knowledge this is the first practical algorithm to achieve linear scaling under the planted sparse model empirically our proposed algorithm also succeeds in more challenging data models e g sparse dictionary learning

the flat locus of brauer severi fibrations of smooth orders given a cayley hamilton smooth order a in a central simple algebra we determine the flat locus of the brauer severi fibration of a moreover we give a classification of all reduced central singularities where the flat locus differs from the azumaya locus and show that the fibers over the flat non azumaya points near these central singularities can be described as fibered products of graphs of projection maps this generalizes an old result of artin on the fibers of the brauer severi fibration of a maximal order over a ramified point finally we show these fibers are also toric quiver varieties and use this fact to compute their cohomology

a proof of the popov conjecture for quivers abstract let q be a quiver with dimension vector we show that if the space of isomorphism classes of semisimple representations iss q of q of dimension vector is not smooth then the quotient map rep q iss q is not equidimensional in other words we prove the popov conjecture for the natural action of the linear reductive group gl on the space rep q of dimensional representations of the quiver q

polynomial identity rings as rings of functions ii abstract in characteristic zero zinovy reichstein and the author generalized the usual relationship between irreducible zariski closed subsets of the affine space their defining ideals coordinate rings and function fields to a non commutative setting where varieties carry a pgl n action regular and rational functions on them are matrix valued coordinate rings are prime polynomial identity algebras and function fields are central simple algebras of degree n in the present paper much of this is extended to prime characteristic in addition a mistake in the earlier paper is corrected one of the results is that the finitely generated prime pi algebras of degree n are precisely the rings that arise as coordinate rings of n varieties in this setting for n 1 the definitions and results reduce to those of classical affine algebraic geometry

local structure of schelter procesi smooth orders in this paper we give the etale local classification of schelterprocesi smooth orders in central simple algebras in particular we prove that if is a central simple k algebra of dimension n2 where k is a field of trancendence degree d then there are only finitely many etale local classes of smooth orders in this result is a non commutative generalization of the fact that a smooth variety is analytically a manifold and so has only one type of etale local behaviour

exact sequences hochschild cohomology and the lie module structure over the m relative center abstract in this article we present actions by central elements on hochschild cohomology groups with arbitrary bimodule coefficients as well as an interpretation of these actions in terms of exact sequences since our construction utilizes the monoidal structure that the category of bimodules possesses we will further recognize that these actions are compatible with monoidal functors and thus as a consequence are invariant under morita equivalences by specializing the bimodule coefficients to the underlying algebra itself our efforts in particular yield a description of the degree n 0 part of the lie bracket in hochschild cohomology and thereby close a gap in earlier work by s schwede

derived representation schemes and cyclic homology we describe the derived functor drepv a of the affine representation scheme repv a parametrizing the representations of an associative k algebra a on a finite dimensional vector space v we construct the characteristic maps trv a n hcn a hn drepv a extending the canonical trace trv a hc0 a k repv a to the higher cyclic homology of the algebra a and describe a related derived version of the representation functor introduced recently by m van den bergh we study various operations on the homology of drepv a induced by known operations on cyclic and hochschild homology of a

noncommutative geometry and quiver algebras abstract we develop a new framework for noncommutative differential geometry based on double derivations this leads to the notion of moment map and of hamiltonian reduction in noncommutative symplectic geometry for any smooth associative algebra b we define its noncommutative cotangent bundle t b which is a basic example of noncommutative symplectic manifold applying hamiltonian reduction to noncommutative cotangent bundles gives an interesting class of associative algebras b that includes preprojective algebras associated with quivers our formalism of noncommutative hamiltonian reduction provides the space with a lie algebra structure analogous to the poisson bracket on the zero fiber of the moment map in the special case where is the preprojective algebra associated with a quiver of non dynkin type we give a complete description of the gerstenhaber algebra structure on the hochschild cohomology of in terms of the lie algebra

on the lower central series of an associative algebra abstract for an associative algebra a define its lower central series l 0 a a l i a a l i 1 a and the corresponding quotients b i a l i a l i 1 a in this paper we study the structure of b i a n for a free algebra a n we construct a basis for b 2 a n and determine the structure of b 3 a 2 and b 4 a 2 in appendix a we study the structure of b 2 a for any associative algebra a over c

on quiver grassmannians and orbit closures for representation finite algebras we show that auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective injective its endomorphism ring is called the projective quotient algebra for any representation finite algebra we use the projective quotient algebra to construct desingularizations of quiver grassmannians orbit closures in representation varieties and their desingularizations this generalizes results of cerulli irelli feigin and reineke

a formal inverse to the cayley hamilton theorem abstract we prove that a q algebra r with formal trace can be realized as n n matrices if and only if it satisfies the cayley hamilton identity of degree n

necklace lie algebras and noncommutative symplectic geometry recently v ginzburg proved that calogero phase space is a coadjoint orbit for some infinite dimensional lie algebra coming from noncommutative symplectic geometry 12 in this note we generalize his argument to specific quotient varieties of representations of deformed preprojective algebras this result was also obtained independently by v ginzburg 13 using results of w crawley boevey and m holland 10 8 and 9 we give a combinatorial description of all the relevant couples alpha lambda which are coadjoint orbits we give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

accpm with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems this paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems the new approach exploits the property that the objective of the lagrangian dual problem has a smooth component with second order derivatives readily available in closed form the cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self concordant augmented barrier our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component the paper also proposes an approximation scheme for the original objective an active set strategy can be applied to the transformed problem it reduces the dimension of the dual space and accelerates computations the new approach solves huge instances with high accuracy the method is compared to alternative approaches proposed in the literature

quasi homomorphismes cohomologie cyclique et positivite we show that cyclic cohomology of an algebraa is obtained from traces with suitable domains on the algebraqa of the second author whena is ac algebra so isqa and the notion of positive trace makes sense we hence get a notion of positivity for cyclic cocycles we prove that a positive trace onqa defines a type i or ii fredholm module ona

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

smooth quiver representation spaces abstract in this paper we classify all quivers and corresponding dimension vectors having a smooth space of semisimple representation classes the result is that these quiver settings can be reduced via some specific reduction steps to 3 simple types

quiver representations with an irreducible null cone abstract let d be a prehomogeneous dimension vector for a quiver q there is an action of the product gl d of linear groups on the vector space rep q d of representations of q with dimension vector d and there is a representation t with a dense gl d orbit in rep q d we give a construction for a dense subset f q d of the variety z q d of common zeros of all semi invariants in k rep q d of positive degree and we show that this set is stable for big dimension vectors i e f q n d x t n 1 x f q d moreover we show that the existence of a dense orbit in z q d depends on a quiver q such that the category of representations of q is equivalent to the right perpendicular category t

stratifications of marsden weinstein reductions for representations of quivers and deformations of symplectic quotient singularities we investigate the poisson geometry of the marsden weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver we show that the stratification by representation type equals the stratification by symplectic leaves the deformed symplectic quotient singularities spectra of centres of symplectic reflection algebras associated to a wreath product are shown to be isomorphic to reductions of a certain quiver this establishes a method to calculate when these deformations are smooth furthermore the isomorphism identifies symplectic leaves and so one can give a description of their symplectic leaves in terms of roots of the quiver

perpendicular categories null cones and dense orbits abstract let q be a quiver k an algebraically closed field and d a prehomogeneous dimension vector for q with canonical decomposition d i 1 r i dim t i we denote by z q d the set of common zeros of all semi invariants in k rep q d of positive degree we study the relations between the existence of a dense gl d orbit in the null cone z q d and the position of the representations t 1 t r in the auslander reiten quiver associated to q

the spectral norm of random inner product kernel matrices we study an inner product kernel random matrix model whose empirical spectral distribution was shown by xiuyuan cheng and amit singer to converge to a deterministic measure in the large n and p limit we provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a marchenko pastur law by comparing the tracial moments of this matrix to an additive deformation of a wigner matrix we establish that for odd kernel functions the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting

recollements and hochschild theory it is shown that a recollement of derived categories of algebras induces those of tensor product algebras and opposite algebras respectively which is applied to clarify the relations between recollements of derived categories of algebras and smoothness and hochschild cohomology of algebras

multiplicative preprojective algebras middle convolution and the deligne simpson problem abstract we introduce a family of algebras which are multiplicative analogues of preprojective algebras and their deformations as introduced by m p holland and the first author we show that these algebras provide a natural setting for the middle convolution operation introduced by n m katz in his book rigid local systems and put in an algebraic setting by m dettweiler and s reiter and h volklein we prove a homological formula relating the dimensions of hom and ext spaces study varieties of representations of multiplicative preprojective algebras and use these results to study simple representations we apply this work to the deligne simpson problem obtaining a sufficient and conjecturally necessary condition for the existence of an irreducible solution to the equation a 1 a 2 a k 1 with the a i in prescribed conjugacy classes in gl n c

finitely presented simple modules over leavitt path algebras abstract let e be an arbitrary graph and k be any field we construct various classes of non isomorphic simple modules over the leavitt path algebra l k e induced by vertices which are infinite emitters closed paths which are exclusive cycles and paths which are infinite and call these simple modules chen modules it is shown that every primitive ideal of l k e can be realized as the annihilator ideal of some chen module our main result establishes the equivalence between a graph theoretic condition and various conditions concerning the structure of simple modules over l k e

on curvature in noncommutative geometry a general definition of a bimodule connection in noncommutative geometry has been recently proposed for a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra the corresponding curvatures are also compared

localized functional principal component analysis we propose localized functional principal component analysis lfpca looking for orthogonal basis functions with localized support regions that explain most of the variability of a random process the lfpca is formulated as a convex optimization problem through a novel deflated fantope localization method and is implemented through an efficient algorithm to obtain the global optimum we prove that the proposed lfpca converges to the original functional principal component analysis fpca when the tuning parameters are chosen appropriately simulation shows that the proposed lfpca with tuning parameters chosen by cross validation can almost perfectly recover the true eigenfunctions and significantly improve the estimation accuracy when the eigenfunctions are truly supported on some subdomains in the scenario that the original eigenfunctions are not localized the proposed lfpca also serves as a nice tool in finding orthogonal basis functions that balance between interpretability and the capability of exp

test instances for the multicommodity flow problem an erratum this note is an erratum to babonneau et al 2006 in which some data were incorrectly described in babonneau et al 2006 the accpm method is used to solve the linear multicommodity flow problem on four sets of instances i e the planar and the grid instances some telecommunication problems and some transportation problems from bar gera 2007 http www bgu ac il bargera tntp sioux falls winnipeg barcelona chicago region and philadelphia the mistakes concern the last category of instances the new data and the associated results can now be found in the electronic companion at http or journal informs org this site also provides the full information on test instances in the multicommodity flow problem mcf and the traffic assignment problem to facilitate benchmarking it gives the data and the results using three specific objective functions used in the literature e g bureau of public roads bpr kleinrock or linear for more details on the optimization problem we refer the reader to babonneau et al 2006 and babonneau and vial 2007

sparse principal component analysis based on least trimmed squares sparse principal component analysis pca is used to obtain stable and interpretable principal components pcs from high dimensional data a robust sparse pca method is proposed to handle potentia

unbounded bivariant k theory and correspondences in noncommutative geometry by adapting the algebraic notion of universal connection to the setting of unbounded kk cycles we show that the kasparov product of such cycles can be defined directly by an algebraic formula in order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable c module the theory of operator spaces provides the required tools finally the above mentioned kk cycles with connection can be viewed as the morphisms in a category whose objects are spectral triples

invariants of quivers under the action of classical groups abstract we consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical linear groups instead of a product of the general linear groups and by considering the dual action of groups on vertex vector spaces together with the usual action a generating system for the corresponding algebra of invariants is found in particular semi invariants of supermixed representations of quivers are established as a consequence a generating system for the algebra of so n invariants of several matrices is constructed over a field of characteristic different from 2 the proof uses the reduction to semi invariants of mixed representations and the decomposition formula that generalizes amitsur s formula for the determinant

symmetric quiver settings with a regular ring of invariants in this paper we classify all the symmetric quivers and corresponding dimension vectors having a smooth space of semisimple representation classes the result we obtain is that such quivers can be decomposed as a connected sum of a few number of basic quivers

on the symplectic structure of instanton moduli spaces abstract we study the complex symplectic structure of the quiver varieties corresponding to the moduli spaces of su 2 instantons on both commutative and non commutative r 4 we identify global darboux coordinates and quadratic hamiltonians on classical phase spaces for which these quiver varieties are natural completions we also show that the group of non commutative symplectomorphisms of the corresponding path algebra acts transitively on the moduli spaces of non commutative instantons this paper should be viewed as a step towards extending known results for calogero moser spaces to the instanton moduli spaces

isomorphisms of quantizations via quantization of resolutions abstract in this paper we prove the existence of isomorphisms between certain non commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain poisson algebras we show that quantizations of kleinian singularities obtained by three different constructions are isomorphic to each other the constructions are via symplectic reflection algebras quantum hamiltonian reduction and w algebras next we prove that parabolic w algebras in type a are isomorphic to quantum hamiltonian reductions associated to quivers of type a finally we show that the symplectic reflection algebras for wreath products of the symmetric group and a kleinian group are isomorphic to certain quantum hamiltonian reductions our results involving w algebras are new while for those dealing with symplectic reflection algebras we just find new proofs a key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate poisson varieties

generating graded central simple algebras abstract we solve the generator problem for z graded central simple algebras applications are given to automorphisms of trace rings of generic matrices and to periodic fat point modules

free products cyclic homology and the gauss manin connection abstract we use the techniques of cuntz and quillen to present a new approach to periodic cyclic homology our construction is based on a t d t i a noncommutative equivariant de rham complex of an associative algebra a here d is the karoubi de rham differential and i is an operation analogous to contraction with a vector field as a byproduct we give a simple explicit construction of the gauss manin connection introduced earlier by e getzler on the relative periodic cyclic homology of a flat family of associative algebras over a central base ring we introduce and study free product deformations of an associative algebra a new type of deformation over a not necessarily commutative base ring natural examples of free product deformations arise from preprojective algebras and group algebras for compact surface groups

noncommutative smoothness and coadjoint orbits abstract in math ag 0010030 math z to appear r bocklandt and the author proved that certain quotient varieties of representations of deformed preprojective algebras are coadjoint orbits for the necklace lie algebra n q of the corresponding quiver q a conjectural ring theoretical explanation of these results was given in terms of noncommutative smoothness in the sense of c procesi j algebra 107 1987 63 74 in this paper we prove these conjectures the main tool in the proof is the etale local description due to w crawley boevey math ag 0105247 along the way we determine the smooth locus of the marsden weinstein reductions for quiver representations

formal structures and representation spaces abstract m kapranov introduced and studied in math ag 9802041 the noncommutative formal structure of a smooth affine variety in this note we show that his construction is a particular case of micro localization and extend the construction functorially to representation schemes of affine algebras we describe explicitly the formal completions in the case of path algebras of quivers and initiate the study of their finite dimensional representations

minimal generating set for semi invariants of quivers of dimension two abstract a minimal by inclusion generating set for the algebra of semi invariants of a quiver of dimension 2 2 is established over an infinite field of arbitrary characteristic the mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees where in the case of characteristic different from two we take only admissible paths as a consequence we describe relations modulo decomposable semi invariants

generating invariant rings of quivers in arbitrary characteristic abstract it is well known that the ring of polynomial invariants of a reductive group is finitely generated however it is difficult to give strong upper bounds on the degrees of the generators especially over fields of positive characteristic in this paper we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi invariants in arbitrary characteristic and consequently for matrix invariants our results generalize to invariants and semi invariants of quivers

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1

quivers geometric invariant theory and moduli of linear dynamical systems we use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems a general approach to this problem is presented and applied to two well known cases we show how both lomadze s and helmke s compactification arises naturally as a geometric invariant theory quotient both moduli spaces are proven to be smooth projective manifolds furthermore a description of lomadze s compactification as a quot scheme is given whereas helmke s compactification is shown to be an algebraic grassmann bundle over a quot scheme this gives an algebro geometric description of both compactifications as an application we determine the cohomology ring of helmke s compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive