minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

automatic numerical differentiation by discrete mollification abstract a new very simple totally automated and powerful technique for numerical differentiation based on the computation of the derivative of a suitable filtered version of the noisy data by discrete mollification is presented several numerical examples of interest are also analyzed

matrix coefficient realization theory of noncommutative rational functions abstract noncommutative rational functions i e elements of the universal skew field of fractions of a free algebra can be defined through evaluations of noncommutative rational expressions on tuples of matrices this interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas from free real algebraic geometry to systems and control theory if a noncommutative rational function is regular at the origin it can be described by a linear object called a realization in this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rational functions and is well adapted for studying matrix evaluations of special interest are the minimal realizations which compensate the absence of a canonical form for noncommutative rational functions the non minimality of a realization is assessed by obstruction modules associated with it they enable us to devise an efficient method for obtaining minimal realizations with them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity using these results we determine concrete size bounds for rational identity testing construct minimal symmetric realizations and prove an effective local global principle for linear dependence of noncommutative rational functions

numerical identification of forcing terms by discrete mollification abstract a new and totally automated technique for the approximate reconstruction of the unknown forcing terms in a system of ordinary differential equations when the experimental information is obtained through measured data on a discrete set of points is presented the numerical method is based on the computation of the derivative of a filtered version of the noisy data by discrete mollification the unknown forcing terms are reconstructed in a compact subset of the interval where the solution is observed the compact subset being automatically determined by the amount of noise in the data rigorous stability bounds are derived and several numerical examples of interest are analyzed

comment on identification in the linear errors in variables model on considere un modele etudie par kapteyn et wansbeek 1983 et on presente des conditions necessaires et suffisantes d identification

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

whittaker coinvariants for gl m n abstract let w m n be the finite w algebra attached to the principal nilpotent orbit in the general linear lie superalgebra gl m n c in this paper we study the whittaker coinvariants functor which is an exact functor from category o for gl m n c to a certain category of finite dimensional modules over w m n we show that this functor has properties similar to soergel s functor v in the setting of category o for a semisimple lie algebra we also use it to compute the center of w m n explicitly and deduce consequences for the classification of blocks of o up to morita derived equivalence

consistent sets of estimates for regressions with correlated or uncorrelated measurement errors in arbitrary subsets of all variables over the last decade the problem of measurement errors in the independent variables of a regression equation has attracted renewed interest among econometricians in the fifties and sixties the problem was considered to be more or less hopeless due to its inherent underidentification e g theil 1971 apart from instrumental variables the most frequently cited textbook solution was wald s method of grouping wald 1940 recent insight into the properties of the method of grouping can be interpreted as making this method worthless in most practical cases pakes 1982 since about 1970 new approaches to the problem have been explored basically along three lines viz embedding the error ridden equation into a set of multiple equations e g zellner 1970 goldberger 1972 into a set of simultaneous equations e g hsiao 1976 geraci 1976 and using the dynamics of the equation if present e g maravall and aigner 1977 in view of the underidentification of the basic model it is clear that all these methods invoke additional information of some kind if this information takes the form of exact or stochastic knowledge of certain parameters in the model the construction of consistent estimators is fairly straightforward e g fuller 1980 kapteyn and wansbeek 1984 for an overview of the state of the art see aigner et al 1984 an approach somewhat orthogonal to the ones described above has been to take the model as it is and to use prior ideas about the size of the measurement errors to diagnose how serious the probem is examples are blomqvist 1972 hodges and moore 1972 and davies and hutton 1975 leamer 1983 starts from the opposite direction by asking how serious the measurement error problem has to be in order to render the data useless for inference that is to say when measurement error is large enough to make it impossible to put bounds on regression parameters in an empirical example he shows that even very small measurement errors in some explanatory variables would open up the possibility of perfectly collinear explanatory variables and hence make the data useless for statistical inference at least without additional prior information the most systematic analysis of the information loss caused by measurement error is due to klepper and leamer 1984 they start out by invoking a minimal amount of prior information and then ask the question under what conditions it is still possible to make some inferences regarding the vector of unknown regression parameters p in the special case where the measurement errors are assumed uncorrelated and the k 1 estimates of 3 obtained by regressing each of the k 1 variables involved i e the one dependent variable and the k independent variables on the remaining k variables are all in the same orthant one can bound the ml estimates of p in that case the convex hull of the k 1 regressions contains all possible ml estimates and any point in the hull is a possible ml estimate if the k 1 regressions are not all in the same orthant then the set of ml estimates is unbounded in that case klepper and leamer 1984 introduce extra prior information which allows them to bound the set of maximum likelihood estimates the prior information comes in two forms firstly a researcher is supposed to be able to specify a maximum value of r2 if all exogenous variables were measured accurately it is shown that if this maximum is low enough one can again bound the set of ml estimates by a convex hull secondly if

discrete stability analysis of the mollification method for numerical differentiation abstract the method of numerical differentiation by discrete mollification is presented in a fully discretized environment new and rigorous results for the numerical stability and error analysis of the algorithm in the presence of noisy data are derived to illustrate the accuracy and flexibility of the numerical procedure the application of the method to a function identification problem is also investigated

kleene stars of the plane polylogarithms and symmetries we extend the definition and construct several bases for polylogarithms li t where t are some series recognizable by a finite state multiplicity automaton of alphabet 4 x x 0 x 1 the kernel of this new polylogarithmic map li is also characterized and provides a rewriting process which terminates to a normal form we concentrate on algebraic and analytic aspects of this extension allowing index polylogarithms at non positive multi indices by rational series and regularize polyzetas at non positive multi indices

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

lie idempotent algebras abstract besides the inner product inherited from each symmetric group algebra ksn on the direct sum ks nksn a second convolution product arises from the hopf algebra structure on ks for any subalgebra a of ks for both the inner and the convolution product it is shown that the lie idempotent algebra l a of a generated by all lie idempotents in a with respect to the convolution product is also closed under inner products its nth homogeneous component l n a is thus a subalgebra of ksn for all n provided that a contains a lie idempotent in ksk for all k n solomon s descent algebra d n is a subalgebra of l n a in this case a number of results on the structure of l n a is derived including a description of its jacobson radical its principal indecomposables and its cartan invariants furthermore solomon s epimorphism from d n onto the ring of class functions clk sn of sn extends to an epimorphism of algebras c n l n a cl k s n additional results are obtained concerning a bialgebra structure of l a in the particular case of a n d n this amounts to a new approach to several well known results on d n

vertex operators arising from jacobi trudi identities we give an interpretation of the boson fermion correspondence as a direct consequence of the jacobi trudi identity this viewpoint enables us to construct from a generalized version of the jacobi trudi identity the action of a clifford algebra on the polynomial algebras that arrive as analogues of the algebra of symmetric functions a generalized giambelli identity is also proved to follow from that identity as applications we obtain explicit formulas for vertex operators corresponding to characters of the classical lie algebras shifted schur functions and generalized schur symmetric functions associated to linear recurrence relations

the positive semidefiniteness of partitioned matrices abstract the positive semidefiniteness of a partitioned matrix is characterized in terms of its submatrices the result is applied to a variety of problems concerning lowner ordered matrices which need not be partitioned themselves

inversion des matrices de hankel abstract hankel determinants can be viewed as special schur symmetric functions this provides without computations of determinants or matrices most of the algebra of hankel determinants and explains their links with other classical fields

malcev neumann series on the free group and questions of rationality let g be an ordered free group b a basis contained in g the sub monoid spanned by b and k a field we construct a family q n g n n of sub division rings in k m g the division ring consisting of the so called malcev neumann series describe their intersections with k and give some applications concerning rational series in k

proxies versus omitted variables in regression analysis when some variables to be used in a regression analysis contain measurement error the ols estimator will be inconsistent when these variables are left out ols will also be inconsistent in an appropriate metric the first type of inconsistency is smaller than the second type

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

new identities in dendriform algebras abstract dendriform structures arise naturally in algebraic combinatorics where they allow for example the splitting of the shuffle product into two pieces and through rota baxter algebra structures the latter appear among others in differential systems and in the renormalization process of pqft we prove new combinatorial identities in dendriform algebras that appear to be strongly related to classical phenomena such as the combinatorics of lyndon words rewriting rules in lie algebras or the fine structure of the malvenuto reutenauer algebra one of these identities is an abstract noncommutative dendriform generalization of the bohnenblust spitzer identity and of an identity involving iterated chen integrals due to c s lam

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

three dimensional gaussian fluctuations of non commutative random surfaces along time like paths abstract we construct a continuous time non commutative markov operator on u gl n with corresponding morphisms u gl n end l 2 u n at any fixed times 0 t 1 t 2 this is an analog of a continuous time non commutative markov operator on the group von neumann algebra v n u n constructed in 16 and is a variant of discrete time non commutative random walks on u gl n 2 9 it is also shown that when restricting to the gelfand tsetlin subalgebra of u gl n the non commutative markov operator matches a 2 1 dimensional random surface model introduced in 7 as an application it is then proved that the moments converge to an explicit gaussian field along time like paths combining with 7 which showed convergence to the gaussian free field along space like paths this computes the entire three dimensional gaussian field in particular it matches a gaussian field from eigenvalues of random matrices 5

ideals of quasi symmetric functions and super covariant polynomials for sn the aim of this work is to study the quotient ring r n of the ring q x 1 x n over the ideal j n generated by non constant homogeneous quasi symmetric functions we prove here that the dimension of r n is given by c n the n th catalan number this is also the dimension of the space sh n of super covariant polynomials that is defined as the orthogonal complement of j n with respect to a given scalar product we construct a basis for r n whose elements are naturally indexed by dyck paths this allows us to understand the hilbert series of sh n in terms of number of dyck paths with a given number of factors

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

usefulness of proxy variables in linear models with stochastic regressors abstract this paper considers a linear model with two stochastic regressors where one regressor is not observed directly but may be observed with error proxy variable the interest is focussed upon the estimation accuracy of the regression coefficient of the observable regressor the possibility of estimating this coefficient more accurately using a proxy variable in regression instead of omitting the non observable variable is considered and an approximate condition for this to occur is given and discussed

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

a hopf algebra approach to inner plethysm abstract we use the hopf algebra structure of the algebra of symmetric functions to study the adams operators of the complex representation rings of symmetric groups and we give new proofs of all of littlewood s formulas for inner plethysm we also study the adams operations for orthogonal and symplectic group characters

the algebra of quasi symmetric functions is free over the integers let z denote the leibniz hopf algebra which also turns up as the solomon descent algebra and the algebra of noncommutative symmetric functions as an algebra z z z1 z2 the free associative algebra over the integers in countably many indeterminates the coalgebra structure is given by zn ni 0 zi zn i z0 1 let m be the graded dual of z this is the algebra of quasi symmetric functions the ditters conjecture says that this algebra is a free commutative algebra over the integers in this paper the ditters conjecture is proved

algebraic structures on grothendieck groups of a tower of algebras abstract the grothendieck group of the tower of symmetric group algebras has a self dual graded hopf algebra structure inspired by this we introduce by way of axioms a general notion of a tower of algebras and study two grothendieck groups on this tower linked by a natural paring using representation theory we show that our axioms give a structure of graded hopf algebras on each grothendieck groups and these structures are dual to each other we give some examples to indicate why these axioms are necessary we also give auxiliary results that are helpful to verify the axioms we conclude with some remarks on generalized towers of algebras leading to a structure of generalized bialgebras in the sense of loday on their grothendieck groups

a generating set of solomon s descent algebra abstract solomon s descent algebra is generated by sums of descent classes corresponding to certain hook shapes this particularly implies that the ring of class functions of any finite symmetric group s n is generated by the irreducible characters corresponding to certain hook partitions of n as another consequence a second generating set of solomon s descent algebra and of the ring of class functions of s n is obtained related to the major index of permutations

quantum and quasi plucker coordinates abstract we demonstrate a passage from the quasi plucker coordinates of gelfand and retakh to the quantum plucker coordinates built from q generic matrices in the process we rediscover the defining relations of the quantum grassmannian of taft and towber and provide that algebra with more concrete geometric origins

renormalization of multiple zeta values abstract multiple zeta values mzvs in the usual sense are the special values of multiple variable zeta functions at positive integers their extensive studies are important in both mathematics and physics with broad connections and applications in contrast very little is known about the special values of multiple zeta functions at non positive integers since the values are usually undefined we define and study multiple zeta functions at integer values by adapting methods of renormalization from quantum field theory and following the hopf algebra approach of connes and kreimer this definition of renormalized mzvs agrees with the convergent mzvs and extends the work of ihara kaneko zagier on renormalization of mzvs with positive arguments we further show that the important quasi shuffle stuffle relation for usual mzvs remains true for the renormalized mzvs

vandermonde and wronskian matrices over division rings in l the first author initiated the study of vandermonde matrices with entries over a division ring k even in the case when k is a field the definition of a vandermonde matrix in l is already more general than that in the classical sense as it takes into account a given endomorphism s of k under the epilogue of l it was further pointed out that the results of that paper remain valid if in the definition of a vandermonde matrix one allows for a given s derivation d and uses the appropriate definition of the power functions in the s d setting such a generalization is worthwhile because in this setting the study of vandermonde matrices over k also encompasses the study of wronskian matrices with respect to d so results on vandermonde matrices may be used to study linear differential equations arising from the operator d more generally vandermonde matrices seem to be an important tool in studying the skew polynomial ring k t s d so a deeper understanding of vandermonde matrices should be important to the study of the arithmetic of a division ring the purpose of this paper is threefold first we shall develop under sec tion 2 the basic facts on skew polynomials in the s d setting define the evaluation of such polynomials and prove the all important product theorem 2 7 second we shall give a general computation of the rank of a vandermonde matrix with respect to s d as in l the computation is first reduced to the case when the elements a a used to build the vandermonde matrix are pairwise s d conjugate in this case the rank of the vandermonde matrix is computed by the dimension of a vector space over a certain division subring of k cf theorem 4 5 last under