some groups of transformations defined by jordan algebras ii groups of type f4 the principal objective of this paper is the study of one of the types of linear groups defined in i 15 the groups of automorphisms of central simple exceptional jordan algebras this can also be considered as the second step in a program of studying exceptional linear groups analogous to the exceptional simple lie groups via non associative algebras the first step in this program has been taken in another paper 13 in which we studied the counterparts of the lie groups of type g 2 as groups of automorphisms of cayley algebras in an analogous fashion the present paper deals with the counterparts of the lie group f 4 as groups of automorphisms of exceptional jordan algebras

image denoising in steerable pyramid domain based on a local laplace prior this paper presents a new image denoising algorithm based on the modeling of coefficients in each subband of steerable pyramid employing a laplacian probability density function pdf with local variance this pdf is able to model the heavy tailed nature of steerable pyramid coefficients and the empirically observed correlation between the coefficient amplitudes within this framework we describe a novel method for image denoising based on designing both maximum a posteriori map and minimum mean squared error mmse estimators which relies on the zero mean laplacian random variables with high local correlation despite the simplicity of our spatially adaptive denoising method both in its concern and implementation our denoising results achieves better performance than several published methods such as bayes least squared gaussian scale mixture bls gsm technique that is a state of the art denoising technique

technical note a generalized bounding method for multifacility location models this paper develops a new lower bound for multifacility location problems with lp distances we prove that the method produces superior results to other known procedures the new bound is also computationally efficient

on triality and automorphisms and derivations of composition algebras abstract the principle of triality for orthogonal transformations on eight dimensional spaces is used to obtain different expressions for the automorphisms derivations and orthogonal transformations of cayley dickson algebras and symmetric composition algebras over fields of arbitrary characteristic

theory and design of signal adapted fir paraunitary filter banks we study the design of signal adapted fir paraunitary filter banks using energy compaction as the adaptation criterion we present some important properties that globally optimal solutions to this optimization problem satisfy in particular we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi infinite programming sip problem the remaining filters are related to the first through a matrix eigenvector decomposition we discuss uniqueness and sensitivity issues the sip problem is solved using a discretization method and a standard simplex algorithm we also show how regularity constraints may be incorporated into the design problem to obtain globally optimal in the energy compaction sense filter banks with specified regularity we also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be dct based such constraints may also be incorporated into our optimization algorithm therefore we are able to obtain globally optimal filter banks subject to regularity and or computational complexity constraints numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters as well as the effects of the various constraints the conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for fir filter banks

poisson gerstenhaber brackets in representation algebras abstract we introduce cyclic bilinear forms on coalgebras and use them to generalize van den bergh s poisson brackets in representation algebras

azumaya loci and discriminant ideals of pi algebras abstract we prove that under mild assumptions for all positive integers l the zero set of the discriminant ideal d l r z r tr of a prime affine polynomial identity pi algebra r coincides with the zero set of the modified discriminant ideal m d l r z r tr of r and give an explicit description of this set in terms of the dimensions of the irreducible representations of r furthermore we prove that when l is the square of the pi degree of r this zero set is precisely the complement of the azumaya locus of r this description is used to classify the azumaya loci of the multiparameter quantized weyl algebras at roots of unity as another application we prove that the zero set of the top discriminant ideal of a prime affine pi algebra r coincides with the singular locus of the center of r provided that the discriminant ideal has height at least 2 r has finite global dimension and r is a cohen macaulay module over its center

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

calabi yau algebras and weighted quiver polyhedra dimer models have been used in string theory to construct path algebras with relations that are 3 dimensional calabi yau algebras these constructions result in algebras that share some specific properties they are finitely generated modules over their centers and their representation spaces are toric varieties in order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3 dimensional calabi yau algebras can be constructed from dimer models on a torus toric orders are examples of a much broader class of algebras positively graded cancellation algebras for these algebras the cy 3 condition implies the existence of a weighted quiver polyhedron which is an extension of dimer models obtained by replacing the torus with any two dimensional compact orientable orbifold

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 modified weiszfeld algorithm for the fermat weber location problem this paper gives a new simple monotonically convergent algorithm for the fermat weber location problem with extensions covering more general cost functions

vector bundles whose restriction to a linear section is ulrich an ulrich sheaf on an n dimensional projective variety x subseteq mathbb p n is an initialized acm sheaf which has the maximum possible number of global sections using a construction based on the representation theory of roby clifford algebras we prove that every normal acm variety admits a reflexive sheaf whose restriction to a general 1 dimensional linear section is ulrich we call such sheaves delta ulrich in the case n 2 where delta ulrich sheaves satisfy the property that their direct image under a general finite linear projection to mathbb p 2 is a semistable instanton bundle on mathbb p 2 we show that some high veronese embedding of x admits a delta ulrich sheaf with a global section

spectroscopy of gauge theories based on exceptional lie groups we generate by computer a basis of invariants for the fundamental representations of the exceptional lie groups e6 and e7 up to degree 18 we discuss the relevance of this calculation for the study of supersymmetric gauge theories and revisit the self dual exceptional models we study the chiral ring of g2 to degree 13 as well as a few classical groups the homological dimension of a ring is a natural estimator of its complexity and provides a guideline for identifying theories that have a good chance to be amenable to a solution

open questions concerning weiszfeld s algorithm for the fermat weber location problem the fermat weber location problem is to find a point in n that minimizes the sum of the weighted euclidean distances fromm given points in n a popular iterative solution method for this problem was first introduced by weiszfeld in 1937 in 1973 kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by weiszfeld s scheme converges to the unique optimal solution we demonstrate that kuhn s convergence theorem is not always correct we then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points the convergence is ensured for all but a denumerable number of starting points

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

footnotes to a paper of domokos i abstract we use regev s double centralizer theorem from 7 to study invariants of matrices with group actions this theory then has applications to trace identities and poincare series

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

robust guided image filtering using nonconvex potentials filtering images using a guidance signal a process called guided or joint image filtering has been used in various tasks in computer vision and computational photography particularly for noise reduction and joint upsampling this uses an additional guidance signal as a structure prior and transfers the structure of the guidance signal to an input image restoring noisy or altered image structure the main drawbacks of such a data dependent framework are that it does not consider structural differences between guidance and input images and that it is not robust to outliers we propose a novel sd for static dynamic filter to address these problems in a unified framework and jointly leverage structural information from guidance and input images guided image filtering is formulated as a nonconvex optimization problem which is solved by the majorize minimization algorithm the proposed algorithm converges quickly while guaranteeing a local minimum the sd filter effectively controls the underlying image structure at different scales and can handle a variety of types of data from different sensors it is robust to outliers and other artifacts such as gradient reversal and global intensity shift and has good edge preserving smoothing properties we demonstrate the flexibility and effectiveness of the proposed sd filter in a variety of applications including depth upsampling scale space filtering texture removal flash non flash denoising and rgb nir denoising

entropy based algorithms for best basis selection adapted waveform analysis uses a library of orthonormal bases and an efficiency functional to match a basis to a given signal or family of signals it permits efficient compression of a variety of signals such as sound and images the predefined libraries of modulated waveforms include orthogonal wavelet packets and localized trigonometric functions and have reasonably well controlled time frequency localization properties the idea is to build out of the library functions an orthonormal basis relative to which the given signal or collection of signals has the lowest information cost the method relies heavily on the remarkable orthogonality properties of the new libraries all expansions in a given library conserve energy and are thus comparable several cost functionals are useful one of the most attractive is shannon entropy which has a geometric interpretation in this context

on the convergence of miehle s algorithm for the euclidean multifacility location problem for the euclidean single facility location problem e weiszfeld proposed a simple iterative algorithm in 1937 later it was proved by numerous authors that it is a convergent descent algorithm w miehle extended weiszfeld s algorithm to solve the euclidean multifacility location problem then l m ostresh proved that miehle s algorithm is a descent algorithm recently f rado modified miehle s algorithm and provided several sets of sufficient conditions for the modified algorithm to converge he also indicated that the convergence of miehle s algorithm was an open problem in this paper the relationship between miehle s multifacility location algorithm and weiszfeld s single facility location algorithm is analyzed counterexamples show that miehle s algorithm may converge to nonoptimal points for well structured problems

geometric methods for spectral analysis this paper explores a geometric framework for modeling nonstationary but slowly varying time series based on the assumption that short windowed power spectra capture their spectral character and that energy transference in the frequency domain has a physical significance the framework relies on certain notions of transportation distance and their respective geodesics to model possible nonparametric changes in the power spectral density with respect to time we discuss the relevance of this framework to applications in spectral tracking spectral averaging and speech morphing

signal analysis using a multiresolution form of the singular value decomposition this paper proposes a multiresolution form of the singular value decomposition svd and shows how it may be used for signal analysis and approximation it is well known that the svd has optimal decorrelation and subrank approximation properties the multiresolution form of svd proposed here retains those properties and moreover has linear computational complexity by using the multiresolution svd the following important characteristics of a signal may he measured at each of several levels of resolution isotropy sphericity of principal components self similarity under scaling and resolution of mean squared error into meaningful components theoretical calculations are provided for simple statistical models to show what might be expected results are provided with real images to show the usefulness of the svd decomposition

on the cocharacter sequence of 3 3 matrices abstract let m 3 f be the algebra of 3 3 matrices with involution over a field f of characteristic zero we study the polynomial identities of m 3 f where t is the transpose involution through the representation theory of the hyperoctahedral group b n after decomposing the space of multilinear polynomial identities of degree n under the b n action we determine which irreducible b n modules appear with non zero multiplicity in symbols we write the nth cocharacter n m 3 f r 0 n r h 6 n r h 3 m where and are partitions of r and n r respectively is the irreducible b n character associated to the pair and m 0 is the corresponding multiplicity we prove that for any n the multiplicities m are always non zero except the trivial case 1 6 and

on the global convergence of a generalized iterative procedure for the minisum location problem with l p distances for p 2 this paper presents a procedure to solve the classical location median problem where the distances are measured with lp norms with p 2 in order to do that we consider an approximated problem the global convergence of the sequence generated by this iterative scheme is proved therefore this paper closes the still open question of giving a modification of the weiszfeld algorithm that converges to an optimal solution of the median problem with lp norms and p in 2 infty the paper ends with a computational analysis of the different provided iterative schemes

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

analysis and extension of multiresolution singular value decomposition abstract this paper presents an analysis of the multiresolution form of singular value decomposition svd mr svd and generalizes the analysis into a random multiresolution form of singular value decomposition r mr svd with an intrinsic randomness the core idea behind the proposed transform is to introduce randomness in the computing process based on parameters without which one can neither decompose nor reconstruct the data correctly the proposed transform inherits the excellent properties of mr svd along with its own unique features which can be useful in many research areas image encryption lossy image compression and face recognition are the primary applications used to illustrate the practical usage of the proposed transform

isotopes of octonion algebras g2 torsors and triality octonion algebras over rings are in contrast to those over fields not determined by their norm forms octonion algebras whose norm is isometric to the norm q of a given algebra c are twisted forms of c by means of the aut c torsor o q o q aut c we show that over any commutative unital ring these twisted forms are precisely the isotopes c a b of c with multiplication given by x y xa by for unit norm octonions a b of c the link is provided by the triality phenomenon which we study from new and classical perspectives we then study these twisted forms using the interplay thus obtained between torsor geometry and isotope computations thus obtaining new results on octonion algebras over e g rings of laurent polynomials

matrix algebras with involution and standard polynomial identities in symmetric variables abstract let m be a positive integer and m 2 m f be the algebra of 2 m 2 m matrices over an algebraically closed field of characteristic zero f endowed with the transpose or the symplectic involution in this paper we construct a basis b of m 2 m f over f such that b is a group whose elements are symmetric or skew with respect to the given involution moreover all elements of this basis commute or anti commute among themselves the construction is based on a specific irreducible representation of b an extra special 2 group as an application this basis solves the problem on finding the minimal degree of a standard polynomial identity in symmetric variables of m 2 m f s where s is the symplectic involution

a quadratically convergent method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum nolimits i 1 m r i x here the residuals r i x are affine functions fromr n tor 1 n 1 2 m 2 this arises in a number of applications including single and multi facility location problems the functionf is in general not differentiable atx if at least oner i x is zero

computing the shortest network under a fixed topology we show that in any given uniform orientation metric plane the shortest network interconnecting a given set of points under a fixed topology can be computed by solving a linear programming problem whose size is bounded by a polynomial in the number of terminals and the number of legal orientations when the given topology is restricted to a steiner topology our result implies that the steiner minimum tree under a given steiner topology can be computed in polynomial time in any given uniform orientation metric with spl lambda legal orientations for any fixed integer spl lambda spl ges 2 this settles an open problem posed by brazil thomas and weng 2000

a description of the brauer severi scheme of trace rings of generic matrices abstract here we use invariant theory to describe the brauer severi scheme of the fibers of trace rings of generic matrices with an algebraically closed base field of characteristic zero when the trace ring is viewed as a sheaf of algebras over the variety of matrix invariants using this approach we first prove that the brauer severi scheme of a trace ring is isomorphic to proj q for a graded ring q whose generators we describe in the first section this description also has a relevant interpretation over base fields of arbitrary characteristic in the second section of this paper we show that the brauer severi scheme of the fiber of a trace ring over a point that is not too degenerate will have smooth irreducible components meeting transversally and describe these irreducible components as brauer severi schemes of certain algebras

a projected weiszfeld algorithm for the box constrained weber location problem abstract the weber problem consists of finding a point in r n that minimizes the weighted sum of distances from m points in r n that are not collinear an application that motivated this problem is the optimal location of facilities in the 2 dimensional case a classical method to solve the weber problem proposed by weiszfeld in 1937 is based on a fixed point iteration in this work we generalize the weber location problem considering box constraints we propose a fixed point iteration with projections on the constraints and demonstrate descending properties it is also proved that the limit of the sequence generated by the method is a feasible point and satisfies the kkt optimality conditions numerical experiments are presented to validate the theoretical results

a globally convergent algorithm for facility location on a sphere abstract in this paper we study the problem of facility location on a sphere this is a generalization of the planar euclidean facility location problem this problem was first studied by katz and cooper and by drezner and wesolowsky where weiszfeld like algorithms were proposed however convergence has never been proved in this paper we first prove a hull property of the problem i e every global minimizer is in the spherical convex hull of the existing facilities we then study the relationship between the spherical facility location problem and a planar euclidean facility location problem corresponding to each approximate solution to the spherical facility location problem optimality conditions for the spherical facility location problem are established in terms of optimality conditions for the corresponding planar euclidean facility location problem and a gradient algorithm is proposed to solve the spherical facility location problem we prove that our algorithm always converges to a global minimizer of the spherical facility location problem computational results are also given

a heuristic algorithm for constrained multi source weber problem the variational inequality approach abstract for solving the well known multi source weber problem mwp each iteration of the heuristic alternate location allocation algorithm consists of a location phase and an allocation phase the task of the location phase is to solve finitely many single source weber problems swp which are reduced by the heuristic of nearest center reclassification for the customers in the previous allocation phase this paper considers the more general and practical case the mwp with constraints cmwp in particular a variational inequality approach is contributed to solving the involved constrained swp cswp and thus a new heuristic algorithm for cmwp is presented the involved cswp in the location phases are reformulated into some linear variational inequalities whose special structures lead to a new projection contraction pc method global convergence of the pc method is proved under mild assumptions the new heuristic algorithm using the pc method in the location phases approaches to the heuristic solution of cmwp efficiently which is verified by the preliminary numerical results reported in this paper

embeddings for the jordan algebra of a bilinear form abstract let k be a field of characteristic zero and let j be a jordan algebra with a formal trace we prove that the algebra j can be embedded into a jordan algebra of a non degenerate symmetric bilinear form over some associative and commutative k algebra c if and only if j satisfies all trace identities of the jordan algebra of a non degenerate symmetric bilinear form over the field k this is an extension of results of procesi and berele concerning the analogous problem for the associative matrix algebras and the matrix algebras with involution as a consequence of these results we also prove that the ideal of all trace identities of the jordan algebra of a non degenerate symmetric bilinear form over k satisfies the specht property

theory and design of optimum fir compaction filters the problem of optimum fir energy compaction filter design for a given number of channels m and a filter order n is considered the special cases where n m and n spl infin have analytical solutions that involve eigenvector decomposition of the autocorrelation matrix and the power spectrum matrix respectively we deal with the more difficult case of m n spl infin for the two channel case and for a restricted but important class of random processes we give an analytical solution for the compaction filter that is characterized by its zeros on the unit circle this also corresponds to the optimal two channel fir filter bank that maximizes the coding gain under the traditional quantization noise assumptions with a minor extension this can also be used to generate optimal wavelets for the arbitrary m channel case we provide a very efficient suboptimal design method called the window method the method involves two stages that are associated with the above two special cases as the order increases the suboptimality becomes negligible and the filter converges to the ideal optimal solution we compare the window method with a previously introduced technique based on linear programming

a smoothing newton method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum i 1 m b i a i tx this problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero in this paper we present a smoothing newton method for this problem by applying the smoothing newton method proposed by qi sun and zhou math programming 87 2000 pp 1 35 directly to a system of strongly semismooth equations derived from primal and dual feasibility and a complementarity condition this method is globally and quadratically convergent as applications to this problem smoothing newton methods are presented for the euclidean facilities location problem and the steiner minimal tree problem under a given topology preliminary numerical results indicate that this method is extremely promising

on the convergence of a hyperboloid approximation procedure for the perturbed euclidean multifacility location problem for the euclidean single facility location problem e weiszfeld proposed a simple closed form iterative algorithm in 1937 later numerous authors proved that it is a convergent descent algorithm in 1973 j eyster j white and w wierwille extended weiszfeld s idea and proposed a hyperboloid approximation procedure hap for solving the euclidean multifacility location problem they believed based on considerable computational experience that the hap always converges in 1977 ostresh proved that the hap is a descent algorithm under certain conditions in 1981 morris proved that a variant of the hap always converges however no convergence proof for the original hap has ever been given in this paper we prove that the hap is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point

asymptotics in cocharacters of simple algebras with involution abstract we compute the asymptotics of the colength and maximal multiplicity sequences of simple algebras with involution

convergence of the weiszfeld algorithm for weber problems using a generalized distance function this paper considers a generalization of the single and multisource weber problem for the case when the distance function is some power k of the usual ip distance properties of the generalized problem are established and an appropriate generalization of the weiszfeld iterative approach is given a convergence proof is supplied for an approximation to the original problem under certain restrictions on p and k

laplace operator and polynomial invariants abstract let a be a finite dimensional simple algebra not necessarily associative over the field of complex numbers c and let g denote the automorphism group aut a suppose that a has a symmetric nondegenerate associative g invariant bilinear form x y and a compact real form i e a subalgebra b over r of dimension dim r b dim c a where a is equal to the span of b over c and the restriction of x y to b is positive definite we describe generators of the algebra of polynomial g invariants of a system of several vectors from a in terms of x y and laplace operators in particular we give generators of the algebra of polynomial invariants of the adjoint representation of a simple linear algebraic group of any exceptional type e 6 as a consequence we get the first main theorem on matrix invariants invariants of minimal representation of g 2 and f 4

the euclidean multifacility location problem to solve the euclidean multifacility location problem w miehle extended the weiszfeld algorithm from one to several facilities l m ostresh established that the objective function for the iterates of miehle s algorithm decreases in value but his results do not give a convergence proof in this paper we give a new extension of the weiszfeld algorithm and by generalizing the kuhn ostresh convergence theorem prove separate results for convergence and for optimality both under mild hypotheses

the fermat weber location problem revisited the fermat weber location problem requires finding a point in n that minimizes the sum of weighted euclidean distances tom given points a one point iterative method was first introduced by weiszfeld in 1937 to solve this problem since then several research articles have been published on the method and generalizations thereof global convergence of weiszfeld s algorithm was proven in a seminal paper by kuhn in 1973 however since them given points are singular points of the iteration functions convergence is conditional on none of the iterates coinciding with one of the given points in addressing this problem kuhn concluded that whenever them given points are not collinear weiszfeld s algorithm will converge to the unique optimal solution except for a denumerable set of starting points as late as 1989 chandrasekaran and tamir demonstrated with counter examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of n we resolve this open question by proving that weiszfeld s algorithm converges to the unique optimal solution for all but a denumerable set of starting points if and only if the convex hull of the given points is of dimensionn

a conjecture of regev about the capelli polynomial abstract let x1 xn2 y1 yn2 be generic n n matrices over a field k of characteristic zero if x 1 x n is a multilinear invariant of x1 xn then s z 2 sign f x 1 y 1 x 2 y 2 y 3 y 4 x 3 y 5 y 9 x n y n 2 2n 2 y n 2 f x 1 x n y where sn2 is the symmetric group of degree n2 y is the discriminant of y1 yn2 and tf x 1 x n is a uniquely defined multilinear invariant of x1 xn thus tf defines a function from the vector space of multilinear invariants of x1 xn to itself an analysis of this function is used to prove regev s conjecture that s z 2 sign x n 1 y 1 x 2 x 3 x 4 y 2 y 3 y 4 x 5 x n 2 2 2 x n 2 y n 2 2n 2 y n 2 is nonzero in addition a variant of the above function is used to evaluate the capelli polynomial

on the convergence of alternating minimization for convex programming with applications to iteratively reweighted least squares and decomposition schemes this paper is concerned with the alternating minimization am method for solving convex minimization problems where the decision variables vector is split into two blocks the objective function is a sum of a differentiable convex function and a separable possibly nonsmooth extended real valued convex function and consequently constraints can be incorporated we analyze the convergence rate of the method and establish a nonasymptotic sublinear rate of convergence where the multiplicative constant depends on the minimal block lipschitz constant we then analyze the iteratively reweighted least squares irls method for solving convex problems involving sums of norms based on the results derived for the am method we establish a nonasymptotic sublinear rate of convergence of the irls method in addition we show an asymptotic rate of convergence whose efficiency estimate does not depend on the data of the problem finally we study the convergence properties of a decomposition based approach designed t

non levi closed conjugacy classes of sp q 2 n we construct an explicit quantization of semi simple conjugacy classes of the complex symplectic group sp 2n with non levi isotropy subgroups through an operator realization on highest weight modules over the quantum group u q bigl mathfrak sp 2n bigr

theory of optimal orthonormal subband coders the theory of the orthogonal transform coder and methods for its optimal design have been known for a long time we derive a set of necessary and sufficient conditions for the coding gain optimality of an orthonormal subband coder for given input statistics we also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time several theoretical properties of optimal compaction filters and optimal subband coders are then derived especially pertaining to behavior as the number of subbands increases significant theoretical differences between optimum subband coders transform coders and predictive coders are summarized finally conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

an efficient algorithm for minimizing a sum of euclidean norms with applications in recent years rich theories on polynomial time interior point algorithms have been developed these theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications in this paper the problem of minimizing a sum of euclidean norms is studied this problem is convex but not everywhere differentiable by transforming the problem into a standard convex programming problem in conic form we show that an epsilon optimal solution can be computed efficiently using interior point algorithms as applications to this problem polynomial time algorithms are derived for the euclidean single facility location problem the euclidean multifacility location problem and the shortest network under a given tree topology in particular by solving the newton equation in linear time using em gaussian elimination on leaves of a tree we present an algorithm which computes an epsilon optimal solution to the shortest network under a given full steiner topology interconnecting n regular points in o n sqrt n log bar c epsilon log n arithmetic operations where bar c is the largest pairwise distance among the given points the previous best known result on this problem is a graphical algorithm which requires o n2 arithmetic operations under certain conditions

rederivation of the fuzzypossibilistic clustering objective function through bayesian inference unsupervised clustering of a set of datums into homogeneous groups is a primitive operation required in many signal and image processing applications in fact different incarnations and hybrids of fuzzy c means fcm and possibilistic c means pcm have been suggested which address additional requirements such as accepting weighted sets and being robust to the presence of outliers nevertheless arriving at a general framework which is independent of the datum model and the notion of homogeneity of a particular problem class is a challenge however this process has not been followed organically and clustering algorithms are generally based on exogenous objective functions which are heuristically engineered and are believed to lead to the satisfaction of a required behavior these techniques also commonly depend on regularization coefficients which are to be set prudently by the user or through separate processes in contrast in this work we utilize bayesian inference and derive a robustified objective function for a fuzzypossibilistic clustering algorithm by assuming a generic datum model and a generic notion of cluster homogeneity we utilize this model for the purpose of cluster validity assessment as well we emphasize the epistemological importance of the theoretical basis on which the developed methodology rests at the end of this paper we present experimental results to exhibit the utilization of the developed framework in the context of four different problem classes

quasi identities on matrices and the cayley hamilton polynomial abstract we consider certain functional identities on the matrix algebra m n that are defined similarly as the trace identities except that the coefficients are arbitrary polynomials not necessarily those expressible by the traces the main issue is the question of whether such an identity is a consequence of the cayley hamilton identity we show that the answer is affirmative in several special cases and moreover for every such an identity p and every central polynomial c with zero constant term there exists m n such that the affirmative answer holds for c m p in general however the answer is negative we prove that there exist antisymmetric identities that do not follow from the cayley hamilton identity and give a complete description of a certain family of such identities

on the convergence of the weiszfeld algorithm in this work we analyze the paper brimberg j 1995 the fermat weber location problem revisited mathematical programming 71 71 76 which claims to close the question on the conjecture posed by chandrasekaran and tamir in 1989 on the convergence of the weiszfeld algorithm some counterexamples are shown to the proofs showed in brimberg s paper

an o nm algorithm for a special case of the multimedian location problem on a tree abstract the multimedian location problem on a connected network g is concerned with locating m new facilities on the network the network g has one existing facility customer at each of the n nodes of g the objective is to locate the new facilities so as to minimize total cost which is the sum of a weighted distances between pairs of new and existing facilities and b weighted distances between certain pairs of new facilities we show that this problem can be solved in o nm when g is a tree and the set of pairs of new facilities which interact has special structure

a note on convergence in the single facility minisum location problem abstract the single facility minisum location problem requires finding a point in rn that minimizes a sum of weighted distances to m given points the distance measure is typically assumed in the literature to be either euclidean or rectangular or the more general lp norm global convergence of a well known iterative solution method named the weiszfeld procedure has been proven under the proviso that none of the iterates coincide with a singular point of the iteration functions the purpose of this paper is to examine the corresponding set of bad starting points which result in failure of the algorithm for a general lp norm an important outcome of this analysis is that the set of bad starting points will always have a measure zero in the solution space rn thereby validating the global convergence properties of the weiszfeld procedure for any lp norm p 1 2

on the relationships between svd klt and pca abstract in recent literature on digital image processing much attention is devoted to the singular value decomposition svd of a matrix many authors refer to the karhunen loeve transform klt and principal components analysis pca while treating the svd in this paper we give definitions of the three transforms and investigate their relationships it is shown that in the context of multivariate statistical analysis and statistical pattern recognition the three transforms are very similar if a specific estimate of the column covariance matrix is used in the context of two dimensional image processing this similarity still holds if one single matrix is considered in that approach the use of the names klt and pca is rather inappropriate and confusing if the matrix is considered to be a realization of a two dimensional random process the svd and the two statistically defined transforms differ substantially

isotropic trialitarian algebraic groups abstract by modifying a construction from knus et al we construct all isotropic algebraic groups of type 3 d 4 and 6 d 4 over an arbitrary field of characteristic 2 we also provide a nice isomorphism criterion for such groups the results of this paper extend the main results of allison using entirely different methods to fields of nonzero characteristic and algebraic groups

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

non commutative resolutions of quotient singularities for reductive groups in this paper we generalize standard results about non commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups we show in particular that quotient singularities for reductive groups always have non commutative resolutions in an appropriate sense moreover we exhibit a large class of such singularities which have twisted non commutative crepant resolutions we discuss a number of examples both new and old that can be treated using our methods notably we prove that twisted non commutative crepant resolutions exist in previously unknown cases for determinantal varieties of symmetric and skew symmetric matrices in contrast to almost all prior results in this area our techniques are algebraic and do not depend on knowing a commutative resolution of the singularity

the characteristic polynomial of an algebra and representations we pose two new linear preserver problems first given a field k and a finite dimensional k algebra b we show that the only linear maps k d b k d b which preserve the unit and the n th roots of unity for some n 2n 2 coprime to char k char k are the algebra homomorphisms second we consider linear maps a b a b between finite dimensional k algebras which preserve the cayley hamilton relations we show that if preservation of the cayley hamilton property is understood in a certain non commutative sense and a k da k d then the only such linear mappings are the algebra homomorphisms

models of the lie algebra f4 abstract we describe the models of the exceptional lie algebra f 4 which are based on its semisimple subalgebras of rank 4 the underlying fact is that any reductive subalgebra of maximal rank of a simple lie algebra induces a grading on this algebra by means of an abelian group in such a way that the nontrivial components of the grading are irreducible modules

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

some remarkable degenerations of quantum groups we show that an irreducible representation of a quantized enveloping algebraue at a lth root of 1 has maximal dimension l n if the corresponding symplectic leaf has maximal dimension 2n the method of the proof consists of a construction of a sequence of degenerations ofue the last one being aq commutative algebrau e 2n this allows us to reduce many problems concerningue to that concerningu e 2n

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

invariants mod 2 and subgroups of g2 and f4 abstract let k be a field of characteristic different from 2 and 3 let g be a simple group of type f 4 or g 2 defined over k in this paper we discuss embeddings of connected simple algebraic groups of type a 1 and a 2 in g in terms of the mod 2 galois cohomological invariants attached to these groups we prove that k groups of type f 4 resp g 2 arising from division algebras are generated by k subgroups of type a 2 resp a 1 see theorem 3 11 theorem 4 1 we derive a necessary and sufficient condition for an albert algebra to have zero f 5 invariant theorem 3 4 further for a k group g of type f 4 we derive a condition necessary for a k group h of type a 1 or a 2 to embed in g over k we prove that in order to embed h in g over k the mod 2 invariant of h must divide f 5 g see section 2 and remark 2 8 remark 2 3 for definition of invariants along similar lines we derive a condition for a k group h of type a 2 to embed in a k group g of type g 2 we prove that h embeds in g over k if and only if f 3 h f 3 g theorem 4 4 next we derive a necessary and sufficient condition for a k group h of type a 1 to embed in a k group g of type g 2 if g aut c is a group of type g 2 over k for an octonion algebra c over k then this condition provides a natural bijection between k conjugacy classes of involutions in g k and isometry classes of 2 fold pfister divisors of the pfister form n c the norm form of the octonion algebra c section 4 proposition 4 2

algebraic families of galois representations and potentially semi stable pseudodeformation rings we construct and study the moduli of continuous representations of a profinite group with integral p adic coefficients we present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable when this profinite group is the absolute galois group of a p adic local field we show that these moduli spaces admit zariski closed loci cutting out galois representations that are potentially semi stable with bounded hodge tate weights and a given hodge and galois type as a consequence we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation

svd based modeling for image texture classification using wavelet transformation this paper introduces a new model for image texture classification based on wavelet transformation and singular value decomposition the probability density function of the singular values of wavelet transformation coefficients of image textures is modeled as an exponential function the model parameter of the exponential function is estimated using maximum likelihood estimation technique truncation of lower singular values is employed to classify textures in the presence of noise kullback leibler distance kld between estimated model parameters of image textures is used as a similarity metric to perform the classification using minimum distance classifier the exponential function permits us to have closed form expressions for the estimate of the model parameter and computation of the kld these closed form expressions reduce the computational complexity of the proposed approach experimental results are presented to demonstrate the effectiveness of this approach on the entire 111 textures from brodatz database the experimental results demonstrate that the proposed approach improves recognition rates using a lower number of parameters on large databases the proposed approach achieves higher recognition rates compared to the traditional subband energy based approach the hybrid imm svm approach and the ggd based approach

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