lie theory for nilpotent l algebras the deligne groupoid is a functor from nilpotent differential graded lie algebras concentrated in positive degrees to groupoids in the special case of lie algebras over a field of characteristic zero it gives the associated simply connected lie group we generalize the deligne groupoid to a functor from l algebras concentrated in degree n to n groupoids we actually construct the nerve of the n groupoid which is an enriched kan complex the construction of gamma is quite explicit it is based on dupont s proof of the de rham theorem and yields higher dimensional analogues of holonomy and of the campbell hausdorff formula in the case of abelian l algebras i e chain complexes the functor is the dold kan simplicial set

the structured sensitivity of vandermonde like systems we consider a general class of structured matrices that includes possibly confluent vandermonde and vandermonde like matrices here the entries in the matrix depend nonlinearly upon a vector of parameters we define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right hand side convenient expressions are derived for the infinity norm based condition numbers and order of magnitude estimates are given for condition numbers defined in terms of a general vector norm we then discuss the computation of the corresponding backward errors after linearising the constraints we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity norm solution of an underdetermined linear system exact componentwise condition numbers are also derived for matrix inversion and the least squares problem and the linearised least squares backward error is characterised

condition numbers for the outer inverse and constrained singular linear system in this paper we investigate the condition number of the outer inverse at s 2 and outer inverse at s 2 solution of constrained linear system ax b x t where a is a real m n matrix b and x are real vectors t is a subspace let and be two positive real numbers when we consider the weighted frobenius norm a b qp f on the data we get the formula of condition number for the outer inverse at s 2 solution of constrained linear system for the normwise condition number the sensitivity of the relative condition number itself is studied and the componentwise perturbation is also investigated

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

displacement rank of the drazin inverse in this paper we study the displacement rank of the drazin inverse both sylvester displacement and the generalized displacement are discussed we present upper bounds for the ranks of the displacements of the drazin inverse the general results are applied to the group inverse of a structured matrix such as close to toeplitz generalized cauchy toeplitz plus hankel and bezoutians

a flexible condition number for weighted linear least squares problem and its statistical estimation in this paper we mainly focus on the derivation of a flexible condition number for weighted linear least squares problem and its estimation with frechet derivative and kronecker product the explicit expression of the condition number is obtained when the coefficient matrices are large and dense considering the difficulties of explicitly forming the expression of condition number in computer we also present its simplified form through a simple and easy to operate method two different condition estimation methods are introduced and the numerical experiments are also performed to show the efficiency of the condition estimators

comment on condition number of drazin inverse and their condition numbers of singular linear systems abstract in this paper we establish the condition number of drazin inverse of a singular matrix a where r a k r a k and k index a by the schur decomposition based on this form the spectral norm and frobenius norm of relative condition number for the drazin inverse and level 2 condition number of the drazin inverse are characterized the sensitivity for the drazin inverse solution of singular systems is also presented we obtain several formulas for the condition number of the drazin inverse by spectral norm instead of the p norm where p is a transformation matrix of the jordan canonical form of a thereby improving the earlier work y wei g wang d wang condition number of drazin inverse and their condition numbers of singular linear systems appl math comput 146 2003 455 467

the kronecker product and stochastic automata networks this paper can be thought of as a companion paper to van loan s the ubiquitous kronecker product paper j comput appl math 123 2000 85 we collect and catalog the most useful properties of the kronecker product and present them in one place we prove several new properties that we discovered in our search for a stochastic automata network preconditioner we conclude by describing one application of the kronecker product omitted from van loan s list of applications namely stochastic automata networks

mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation abstract in this paper the mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation are considered the explicit expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed some simple numerical experiments show that the mixed and componentwise condition numbers can be sharp

kodaira spencer theory of gravity and exact results for quantum string amplitudes we develop techniques to compute higher loop string amplitudes for twistedn 2 theories with 3 i e the critical case an important ingredient is the discovery of an anomaly at every genus in decoupling of brst trivial states captured to all orders by a master anomaly equation in a particular realization of then 2 theories the resulting string field theory is equivalent to a topological theory in six dimensions the kodaira spencer theory which may be viewed as the closed string analog of the chern simons theory using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in calabi yau manifolds it is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingn 2 theory relations withc 1 strings are also pointed out

on level 2 condition number for the weighted moore penrose inverse in this paper we present characterizations for the level 2 condition number of the weighted moore penrose inverse i e cond m n a cond m n 2 a cond m n a 1 where cond m n a is the condition number of the weighted moore penrose inverse of a rectangular matrix and cond m n 2 a is the level 2 condition number of this problem this paper extends the result by cucker diao and wei f cucker h diao y wei on the level 2 condition number for moore penrose inversion 2005 unpublished report and improves the results by wei and wang y wei d wang condition numbers and perturbation of weighted moore penrose inverse and weighted linear least squares problem appl math comput 145 2003 45 58

on the b twisted topological sigma model and calabi yau geometry we provide a rigorous perturbative quantization of the b twisted topological sigma model via a first order quantum field theory on derived mapping space in the formal neighborhood of constant maps we prove that the first chern class of the target manifold is the obstruction to the quantization via batalin vilkovisky formalism when the first chern class vanishes i e on calabiyau manifolds the factorization algebra of observables gives rise to the expected topological correlation functions in the b model we explain a twisting procedure to generalize to the landau ginzburg case and show that the resulting topological correlations coincide with vafa s residue formula

condition number of bott duffin inverse and their condition numbers various normwise relative condition numbers that measure the sensitivity of bott duffin inverse and the solution of constrained linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of componentwise condition numbers

some topics on weighted moore penrose inverse weighted least squares and weighted regularized tikhonov problems it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of linear systems the classical normwise relative condition number formulas are given by higham linear algebra appl 214 1995 193 here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b moreover we consider the nearness to rank deficiency finally we give some decomposition tools and their application in the regularized weighted tikhonov problems

condition number of drazin inverse and their condition numbers of singular linear systems various normwise relative condition numbers that measure the sensitivity of drazin inverse and the solution of singular linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of structured perturbation

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

condition numbers and structured perturbation of the w weighted drazin inverse we give the explicit condition number formulas for the w weighted drazin inverse of a rectangular matrix we also present the structured perturbation of the w weighted drazin inverse and consider the perturbation for the w weighted drazin inverse of a rectangular matrix and give an explicit expression for the w weighted drazin inverse of a perturbed matrix under some certain conditions finally we discuss the minimum quality of the condition number

on mixed and componentwise condition numbers for moore penrose inverse and linear least squares problems classical condition numbers are normwise they measure the size of both input perturbations and output errors using some norms to take into account the relative of each data component and in particular a possible data sparseness componentwise condition numbers have been increasingly considered these are mostly of two kinds mixed and componentwise in this paper we give explicit expressions computable from the data for the mixed and componentwise condition numbers for the computation of the moore penrose inverse as well as for the computation of solutions and residues of linear least squares problems in both cases the data matrices have full column row rank

a note on the componentwise perturbation bounds of matrix inverse and linear systems in this note we present sharp bounds for the componentwise perturbation of matrix inverse and linear systems especially for nonsingular m matrix we also use matrix derivatives to deduce the matrix componentwise condition number and present some numerical tests to show our results

structured condition numbers for some matrix factorizations of structured matrices abstract using the modified matrix vector approach and the differential calculus we study the structured condition numbers for lu cholesky and qr factorizations of some structured matrices that can be represented by sets of parameters the obtained explicit expressions of these structured condition numbers are very general which are applicable to most of linear and non linear structured matrices and include the popular normwise mixed and componentwise condition numbers as special cases more specific explicit expressions of the structured condition numbers for linear structured matrices are also provided we compare the structured condition numbers with the corresponding unstructured ones in theory and experiment numerical results show that for non linear structured matrices the structured condition numbers can be much smaller than the unstructured ones in addition we also test the applications of structured condition numbers in estimating the first order perturbation bounds of matrix factorizations using numerical examples

two kinds of condition numbers for the quadratic matrix equation in the situation that the quadratic matrix equation has the dominant or minimal solvents the mixed and componentwise condition numbers of the quadratic matrix equation are considered the explicit approximate expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed the theoretical results are illustrated by using some simple numerical examples

condition number for the drazin inverse and the drazin inverse solution of singular linear system with their condition numbers in this paper we investigate the condition number of drazin inverse and drazin inverse solution of singular linear system ax b where a is a nxn rank deficient matrix and b a real vector of size n x a real vector let a and b be two positive real numbers when we consider the weighted frobenius norm aa bb p q f on the data we get the formula of condition number of the drazin inverse solution of singular linear system for the normwise condition number the sensitivity of the relative condition number itself is studied the componentwise perturbation is also investigated

on structured componentwise condition numbers for hamiltonian eigenvalue problems abstract in this paper we introduce the structured componentwise perturbation analysis to hamiltonian eigenvalue problems the explicit expressions for the relative structured componentwise condition number for hamiltonian eigenvalue problems are derived also we will compare the relative structured componentwise condition number with the relative unstructured componentwise condition number and the relative structured normwise condition number respectively we show the superiority of the relative structured componentwise condition number through numerical examples

a note on condition numbers for generalized inverse at s 2 and constrained linear systems abstract in this paper we use the schur decomposition to measure the sensitivity of the general inverse a t s 2 and the constrained singular linear system ax b with regard to 2 norm and f norm rather than pq norm in 12 where p and q are nonsingular matrices the explicit forms of the condition numbers approximate the sensitivity of a t s 2 and the constrained singular linear system pretty well furthermore since schur decomposition is well posed the evaluation of the sensitivity can be numerically easy and stable

equivariant cobordism for torus actions abstract we study the equivariant cobordism groups for the action of a split torus t on varieties over a field k of characteristic zero we show that for t acting on a variety x there is an isomorphism t x b t l x as applications we show that for a connected linear algebraic group g acting on a k variety x the forgetful map g x x is surjective with rational coefficients as a consequence we describe the rational algebraic cobordism ring of algebraic groups and flag varieties we prove a structure theorem for the equivariant cobordism of smooth projective varieties with torus action using this we prove various localization theorems and a form of bott residue formula for such varieties as an application we show that the equivariant cobordism of a smooth variety x with torus action is generated by the invariant cobordism cycles in x as b t module

condition number related with w weighted drazin inverse and singular linear systems in this paper we introduce a pq norm and discuss the condition number of the w weighted drazin inverse we get some results on the condition number with respect to a d w and singular linear systems wawx b b r wa k 2 x r aw k 1 which are similar to the regular inverse a 1 and nonsingular linear systems ax b

structured perturbations of group inverse and singular linear system with index one in this paper we study the normwise perturbation theory of singular linear structured system with index one the structures under investigation are toeplitz circulant hankel matrices we show that the structured condition number is smaller than unstructured condition number for specific right hand side we present the condition number for structured system

the condition numbers for weighted moore penrose inverse and weighted linear least squares problem condition numbers play an important role in numerical analysis classical normise condition numbers are used to measure the size of both input perturbations and output errors in this paper we study the weighted normwise relative condition numbers for the weighted moore penrose inverse and the weighted linear least squares wls problems in the case of the full column rank matrix the bounds or formulas for the weighted condition numbers are presented the obtained results can be viewed as extensions of the earlier works studied by others

motivic decompositions of twisted flag varieties and representations of hecke type algebras motivated by the motivic galois group and the kostant kumar results on equivariant cohomology of flag varieties we provide a uniform de scription of motivic direct sum decompositions with integer coefficients of versal flag varieties in terms of integer representations of the associated affine nil hecke algebra h more generally we establish an equivalence between the h motivic subcat egory generated by the motive of e b and the category of projective modules of the associated rational algebra d of push pull operators where e is a tor sor for a split semisimple linear algebraic group g over a field k b is a borel subgroup of g h is an algebraic oriented cohomology theory in the sense of levine morel e g chow ring ch or an algebraic cobordism the algebra d can be think of as an integer analogue of the hopf algebra of the h motivic galois group of e b as an application taking h ch and specializing the coefficients to the finite field fp we obtain that p modular projective representations of d h are generated by an irreducible h module corresponding to the generalized rost voevodsky motive for g p

perturbation bounds for the generalized inverses at s 2 with application to constrained linear system in this paper the more general perturbation problem of the generalized inverse a t s 2 is considered we derive its perturbation bounds when the subspaces t and s also have perturbation and in special case when t and s have no perturbation we get some results on the condition numbers with respect to a t s 2 and constrained linear system ax b x t which are similar to the regular inverse a 1 and nonsingular linear system ax b

on perturbation bounds of kronecker product linear systems and their level 2 condition numbers the classical perturbation theory of linear systems ax b is extended to kronecker product linear systems email protected b x d upper bounds are derived for the normwise and componentwise condition number the nearness to singularity and the sensitivity of the condition numbers are analyzed

on mixed and componentwise condition numbers for indefinite least squares problem in this paper we mainly consider the mixed and componentwise condition numbers for the indefinite least squares problem without constraints or with least squares constraints the explicit expressions and the easily computable upper bounds for these condition numbers are presented two numerical examples are given to test these results and illustrate their superiority compared with the corresponding normwise condition numbers furthermore we also provide the mixed and componentwise condition numbers for the moore penrose inverse in indefinite inner product spaces

renormalization for holomorphic field theories we introduce the concept of a holomorphic field theory on any complex manifold in the language of the batalin vilkovisky formalism when the complex dimension is one this setting agrees with that of chiral conformal field theory our main result concerns the behavior of holomorphic theories under renormalization group flow namely we show that holomorphic theories are one loop finite we use this to completely characterize holomorphic anomalies in any dimension throughout we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories

condition numbers and perturbation of the weighted moore penrose inverse and weighted linear least squares problem it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b these explicit expressions extend the earlier work of several authors as we know the computed weighted least squares solution x of min email protected email protected m in the presence of the round off error satisfies the perturbed equation min email protected a e y b f m finally an upper bound of email protected n is derived for the case where the weighted matrix and vector norms and the assumption rank a e rank a are used

condition numbers and their condition numbers abstract various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized new results are derived for the cases where two common noninduced matrix norms are used and where different vector norms are used for the domain and range of the matrix condition numbers that respect the structure of symmetric problems are also analyzed the sensitivity of the condition number itself is then investigated and we obtain sharp examples of demmel s general result that for certain problems in numerical analysis the condition number of the condition number is the condition number finally upper bounds are derived for the sensitivity of componentwise condition numbers

structured condition numbers of structured tikhonov regularization problem and their estimations both structured componentwise and structured normwise perturbation analysis of the tikhonov regularization are presented the structured matrices under consideration include toeplitz hankel vandermonde and cauchy matrices structured normwise mixed and componentwise condition numbers for the tikhonov regularization are introduced and their explicit expressions are derived for the general linear structure based on the derived expressions we prove structured condition numbers are smaller than their corresponding unstructured counterparts by means of the power method and small sample statistical condition estimation sce fast condition estimation algorithms are proposed our estimation methods can be integrated into tikhonov regularization algorithms that use the generalized singular value decomposition gsvd for large scale linear structured tikhonov regularization problems we show how to incorporate the sce into the preconditioned conjugate gradient pcg method to get the posterior error estimations the structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones and the proposed condition estimation algorithms are reliable also an image restoration example is tested to show the effectiveness of the sce for large scale linear structured tikhonov regularization problems

condition number related to the outer inverse of a complex matrix in this paper we obtain the formula for computing the condition number of a complex matrix which is related to the outer generalized inverse of a given matrix we use the schur decomposition of a matrix we characterize the spectral norm and the frobenius norm of the relative condition number of the generalized inverse and the level 2 condition number of the generalized inverse the sensitivity for the generalized drazin inverse solution of linear systems is presented we also present the structured perturbation of the generalized inverse

a contribution to the theory of condition in this paper the use of the condition number of a problem as defined by rice in 1966 is discussed for the eigenvalue eigenvector and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived

on transversely holomorphic flows ii this paper is a complement to the preceding one by marco brunella 2 our purpose is to complete the classi cation of transversely holomorphic ows on closed 3 manifolds avoiding the rationality assumption made in 2 the main tool is still the existence of a harmonic atlas established by m brunella and we shall only add some simple ideas coming from the theory of complex surfaces we follow the notations of 2 as a matter of fact our main result is the following

optimal a priori clipping estimation for wavelet based method of moments matrices wavelet bases are mainly used in the methods of moments mom to render the system matrix sparse clipping entries below a given threshold is an essential operation to obtain the desired sparse matrix in this paper we present a novel a priori way to estimate the clipping threshold that can be used if one wants an error on the solution below a desired accuracy the procedure has a very low computational complexity typically of o n

perturbation analysis for a class of fuzzy linear systems in this paper the perturbation analysis is presented for a class of fuzzy linear systems which can be solved by an embedding method we deduce the nonlinear upper perturbation bounds for the solutions and discuss the normwise mixed and componentwise condition numbers the results show how the perturbations of the coefficient matrix and the right hand vector impact the solutions to fuzzy linear systems

optimal two sided diagonal scaling for filter bank frames two sided diagonal scaling for transfer matrices is formulated an efficient algorithm is proposed to compute globally optimal solutions using the iterated bi section and linear matrix inequality lmi solver it is shown that the two sided scaling of filter bank fb frames can be implemented by the adjustment of channel gains and the periodic precoding of source signal and that the frame bound ratio of fb frames can be effectively improved by such scaling explicit formulas are established for both uniform and nonuniform fb frames including detail formulas for discrete weyl heisenberg frames and tree structured fbs discrete wavelets different examples show the effectiveness of the obtained results

structured eigenvalue condition numbers for parameterized quasiseparable matrices the development of fast algorithms for performing computations with n times n n n low rank structured matrices has been a very active area of research during the last two decades as a consequence of the numerous applications where these matrices arise the key ideas behind these fast algorithms are that low rank structured matrices can be described in terms of o n parameters and that these algorithms operate on the parameters instead on the matrix entries therefore the sensitivity of any computed quantity should be measured with respect to the possible variations that the parameters defining these matrices may suffer since this determines the maximum accuracy of a given fast computation in other words it is necessary to develop condition numbers with respect to parameters for different magnitudes and classes of low rank structured matrices but as far as we know this has not yet been accomplished in any case in this paper we derive structured relative eigenvalue condition numbers for the important class of low rank structured matrices known as 1 1 1 1 quasiseparable matrices with respect to relative perturbations of the parameters in the quasiseparable and in the givens vector representations of these matrices and we provide fast algorithms for computing them comparisons among the new structured condition numbers and the unstructured one are also presented as well as numerical experiments showing that the structured condition numbers can be small in situations where the unstructured one is huge in addition the approach presented in this paper is general and may be extended to other problems and classes of low rank structured matrices

componentwise perturbation theory for linear systems with multiple right hand sides existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right hand sides and to a general class of componentwise measure of perturbations involving holder p norms it is shown that for a system of order n with r right hand sides the componentwise backward error can be computed by finding the minimum p norm solutions to n underdetermined linear systems and an explicit expression is obtained in the case r 1 a perturbation bound is derived and from this the componentwise condition number is obtained to within a multiplicative constant applications of the results are discussed to invariant subspace computations quasi newton methods based on multiple secant equations and an inverse ode problem

structured perturbations and symmetric matrices abstract for a given n n matrix the ratio between the componentwise distance to the nearest singular matrix and the inverse of the optimal bauer skeel condition number cannot be larger than 3 2 2 n in this note a symmetric matrix is presented where the described ratio is equal to n for the choice of most interest in numerical computation for relative perturbations of the individual matrix components it is shown that a symmetric linear system can be arbitrarily ill conditioned while any symmetric and entrywise relative perturbation of the matrix of less than 100 does not produce a singular matrix that means that the inverse of the condition number and the distance to the nearest ill posed problem can be arbitrarily far apart finally we prove that restricting structured perturbations to symmetric entrywise perturbations cannot change the condition number by more than a factor 3 2 2 n

geometric representations of the formal affine hecke algebra abstract for any formal group law there is a formal affine hecke algebra defined by hoffnung malagon lopez savage zainoulline coming from this formal group law there is also an oriented cohomology theory we identify the formal affine hecke algebra with a convolution algebra coming from the oriented cohomology theory applied to the steinberg variety as a consequence this algebra acts on the corresponding cohomology of the springer fibers this generalizes the action of classical affine hecke algebra on the k theory of the springer fibers constructed by lusztig we also give a residue interpretation of the formal affine hecke algebra which generalizes the residue construction of ginzburg kapranov vasserot when the formal group law comes from a 1 dimensional algebraic group

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems