computation of rational interpolants with prescribed poles a constructive proof for existence and unicity of the rational r belonging to gm v m 0 n 0 having prescribed n poles and interpolating m 1 hermite data is given it is based upon explicit computation of the confluent cauchy vandermonde determinant in terms of the nodes and the poles an algorithm to compute r z is presented it calculates this value from a triangular field of values of rational interpolants of lowest possible orders the algorithm is based upon a neville aitken interpolation formula and has arithmetical complexity 0 l l max m 1 n

branching rules for conformal embeddings we give explicit formulas for the branching rules of the conformal embeddingssu n n 1 2 1 su n n 2 su n n 1 2 1 su n n 2 sp n 1 so n 4 su 2 n andso m n 1 so m 1 so n 1 withm andn odd

some schur complement identities and applications to matrix extrapolation methods the schur complement has many applications in linear algebra and in numerical analysis schur complement identities have been used for implementing some scalar and vector extrapolation methods in this paper we define some matrix sequence transformations and give two new schur complement identities that will be used to derive algorithms for implementing these transformations

ect b splines defined by generalized divided differences ect spline curves are generated from different local ect systems via connection matrices if they are nonsingular lower triangular and totally positive there is a basis of the space of ect splines consisting of functions having minimal compact supports normalized either to form a nonnegative partition of unity or to have integral one in this paper such ect b splines are defined by generalized divided differences this definition reduces to the classical one in case of a schoenberg space under suitable assumptions it leads to a recursive method for computing the ect b splines that reduces to the de boor mansion cox recursion in case of ordinary polynomial splines and to lyche s recursion in case of tchebycheff splines muhlbach and tang calculation of ect b splines and of ect spline curves recursively in preparation there is an ect spline space naturally adjoint to every ect spline space we also construct b splines via generalized divided differences for this space and study relations between the two adjoint spaces

linear extrapolation by rational functions exponentials and logarithmic functions abstract in this paper linear extrapolation by rational functions with given poles is considered from an arithmetical point of view it is shown that the classical interpolation algorithms of lagrange neville aitken and newton which are well known for polynomial interpolation can be extended in a natural way to this problem yielding recursive methods of nearly the same complexity the proofs are based upon explicit representations of generalized vandermonde determinants which are calculated by the elimination method combined with analytical considerations as an application a regularity criterion for certain linear sequence transformations is given also by the same method simplified recurrence relations for linear extrapolation by exponentials and logarithmic functions at special knots are derived

a construction of the c 1 modular invariant partition functions decomposition theorems for certain representations of kac moody algebras which are needed for the construction of modular invariant unitary conformal models are proved it is shown that allc 1 modular invariant models can then be recovered from gauged free fermionic models including the exceptional cases

can sylvester s determinantal identity equivalently muir s law of extensible minors be generalized abstract we show that sylvester s classical determinantal identity is equivalent to its generalizations given by the muhlbach gasca lopez carmona ramirez identity and the beckermann muhlbach identity we also show that the generating extension principles associated with each of these three identities muir s law of extensible minors muhlbach s extension principle and the extension principle given in camargo 2013 3 are equivalent and we use this fact to conclude that all these six statements and also jacobi s identity on the minors of the adjugate and cayley s law of complementaries are actually equivalent these findings lead naturally to enquire whether there would exist a generalization of sylvester s identity or an extension principle which could not be derived from sylvester s identity itself

elimination technique from extrapolation to totally positive matrices and cagd in this survey we will show some connections between several mathematical problems such as extrapolation linear systems totally positive matrices and computer aided geometric design with elimination techniques as the common tool to deal with all of them

a generalization of sylvester s identity on determinants and some applications abstract a generalization of sylvester s identity on determinants is proved by elimination techniques as an application some inequalities for determinants of totally positive or positive definite matrices are derived

conformal embeddings rank level duality and exceptional modular invariants we compute the branching rules of the conformal embeddingsso 4nk 1 sp 2n k sp 2k n andso rq 1 so r q so q r forrq even using this we prove that the affine algebrassp 2n k andsp 2k n have the sames matrix and modular invariants as a second application we show how the triality ofso 8 leads to an exceptional modular invariant forsu 2 at level 16 and for allso q 4 at level 8

z nz conformal field theories in this paper we give a brief introduction to conformal field theory cft following the presentation of g segal we explain how to reconstruct part of a cft from its fusion rules the possible choices of s matrices are indexed by some automorphisms of the fusion algebra we illustrate this procedure by computing the modular properties of the possible genus one characters when the fusion algebra is the representation algebra of a finite group we also classify the modular invariant partition functions of these theories we recover as special cases the a n 1 wzw theories and the rational gaussian model

convergence properties of some block krylov subspace methods for multiple linear systems in the present paper we give some convergence results of the global minimal residual methods and the global orthogonal residual methods for multiple linear systems using the schur complement formulae and a new matrix product we give expressions of the approximate solutions and the corresponding residuals we also derive some useful relations between the norm of the residuals

decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of z2 graded lie algebras we give uniform formulas for the branching rules of level 1 modules over orthogonal affine lie algebras for all conformal pairs associated to symmetric spaces we also provide a combinatorial interpretation of these formulas in terms of certain abelian subalgebras of simple lie algebras

other manifestations of the schur complement abstract many series and sequence transformations used in numerical analysis are defined as ratios of determinants they are implemented by recursive algorithms based on determinantal identities the aim of this paper is to study the connections of these transformations and algorithms with the schur complement of a matrix the schur s formula is extended to the vector case thus providing the same treatment for vector sequence transformations and the corresponding recursive algorithms thanks to these connections particular rules for avoiding division by zero or numerical instability are obtained for these algorithms some fixed point methods and continued fractions also fit in this framework

chern classes for singular hypersurfaces we prove a formula expressing the chern schwartz macpherson class of a hypersurface in a nonsingular variety as a variation on another definition of the homology chern class of singular varieties introduced by w fulton and we discuss the relation between these classes and others such as mather s chern class and the l class we introduced in previous work 0 introduction there are several candidates for a notion of homology chern class of a possibly singular algebraic variety x agreeing with the class obtained as dual of the total chern class of the tangent bundle of x if x is nonsingular one such notion was introduced by r macpherson macpherson agreeing as it was later understood with a class introduced earlier by m ij schwartz and enjoying good functorial properties a different class was defined by w fulton fulton 4 2 6 in a different and somewhat more general setting the main purpose of this paper is to prove a precise relation between the schwartz macpherson and the fulton class of a hypersurface x of a nonsingular variety we denote these two classes by csm x cf x respectively a summary of the context and essential definitions is given in 1 where we present several different statements of the result the version which expresses most directly the link between the two classes mentioned above goes as follows fulton s class for a subscheme x of a nonsingular variety m is defined by cf x m c tmix n s x m that is by capping the total chern class of the ambient space against the segre class of the subscheme it is proved in fulton 4 2 6 that this definition does not depend on the choice of the ambient variety m for a hypersurface x of a nonsingular variety m let y be the singular subscheme of x locally defined by the partial derivatives of an equation of x for an integer k 0 we can consider the k th thickening x k of x along y and then consider its fulton class cf x k it is easily seen that cf x k is a polynomial in k with coefficients in the chow group a x so it can be formally evaluated at negative k s we can then define a class c x cf x 1 the main result of this paper is theorem c x is equal to the chern schwartz macpherson class of x received by the editors june 3 1997 1991 mathematics subject classification prinmary 14c17 32s60 supported in part by nsf grant dms 9500843 1999 american mathbemiatical society 3989 this content downloaded from 157 55 39 136 on thu 01 dec 2016 05 45 16 utc all use subject to http about jstor org terms

level rank duality in nonunitary coset theories it is shown that the conformal field theories su n k su n 1 su n k 1 and su m l su m 1 su m 1 1 are equivalent where k n n m l m m n for a pair m n of relatively prime positive integers

meromorphic c 24 conformal field theories modular invariant conformal field theories with just one primary field and central charge c 24 are considered it has been shown previously that if the chiral algebra of such a theory contains spin 1 currents it is either the leech lattice cft or it contains a kac moody sub algebra with total central charge 24 in this paper all meromorphic modular invariant combinations of the allowed kac moody combinations are obtained the result suggests the existence of 71 meromorphic c 24 theories including the 41 that were already known

a general determinantal identity of sylvester type and some applications abstract a general determinantal identity of sylvester type over arbitrary commutative fields is derived while its proof is rather short and conceptually simpler than earlier attempts the result contains sylvester s classical determinantal identity and more recent extensions as well as some old and some new determinantal identities an application to the e algorithm is added

matrix extrapolation algorithms abstract extrapolation algorithms are used for accelerating scalar or vector sequences they are also used for solving systems of linear and nonlinear equations these algorithms are expressed in terms of a ratio of two determinants as the e algorithm and the general recursive projection algorithm grpa in this paper we define the matrix extrapolation problem and we use the schur complement and the sylvester identity for solving this problem we give two transformations equivalent to the e algorithm and the grpa for the matrix case we give also a characterization of their kernels

on extending determinantal identities abstract a new principle for extending determinantal identities is established which generalizes muir s classical law of extensible minors the proof makes use of general elimination strategies and of generalized schur complements this principle allows either the list of columns or the list of rows extending the corresponding list in the given identity to depend on the latter as applications of this technique karlin s and a generalization of sylvester s identity are derived

two composition methods for solving certain systems of linear equations this note is concerned with the following problem given a systema x b of linear equations and knowing that certains of its subsystemsa 1 x 1 b 1 a m x m b m can be solved uniquely what can be said about the regularity ofa and how to find the solutionx fromx 1 x m this question is of particular interest for establishing methods computing certain linear or quasilinear sequence transformations recursively 7 13 15 in dieser arbeit wird die folgende fragestellung behandelt wenn man wei da gewisse teilsystemea 1 x 1 b 1 a m x m b m eines linearen gleichungssystemsa x b eindeutig losbar sind was kann man uber die losbarkeit des systemsa x b sagen und wie kann man gegebenenfallsx ausx 1 x m berechnen die losung dieses problems erlaubt die rekursive berechnung gewisser linearer oder quasilinearer folgentransformationen 7 13 15

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

the branching rules of conformal embeddings after recalling the main properties of a conformal embedding of lie algebras g id p which is defined by the equality of the sugawara central charges on both sides we launch a systematic study of their branching rules the bulk of the paper is devoted to the proof of a general formula in the case suim su m n su m at the end we give some applications to the con struction of modular invariant partition functions

a residue formula for the index of a holomorphic flow let p be a normal isolated complex analytic singularity of dimension n 1 and consider germs of holomorphic vector fields on at p one would like to be able to distinguish different germs if p is a regular point of y there are a number of invariants that sometimes allow us to distinguish germs of vector fields at p however for p singular little is known if x is a holomorphic vector field on singular only at p then a natural way for studying the geometry the topology and the dynamics of x near p is to go up to a resolution of p and see what happens there see for instance hh more precisely let

quasi galois symmetries of the modular s matrix the recently introduced galois symmetries of rational conformal field theory are generalized for the case of wzw theories to quasi galois symmetries these symmetries can be used to derive a large number of equalities and sum rules for entries of the modular matrixs including some that previously had been observed empirically in addition quasi galois symmetries allow us to construct modular invariants and to relates matrices as well as modular invariants at different levels they also lead us to a convenient closed expression for the branching rules of the conformal embeddings open image in new window