a formula for iterated derivatives along trajectories of nonlinear systems abstract the paper presents a formula for the n th order total derivative of the output function of a nonlinear systems in terms of derivatives of the inputs the formula is given in terms of noncommuting variables and the shuffle product and has potential applications to input output decoupling and inverses of nonlinear systems

a combinatorial approach to nonlinear functional expansions an introduction with an example in this paper we present a new approach to causal functionals we introduce combinatorial interpretations of the solutions of nonlinear differential equations with forcing terms this theory parallels the algebraic approach with formal power series in noncommutative variables developed by fliess lamnabhi and lamnabhi lagarrigue this theory makes use of certain combinatorial objects called weighted increasing trees weighted paths and histories we can deduce very efficient algorithms for the computation of the corresponding volterra kernels we present an introduction to our combinatorial theory an example with a nonlinear circuit gives the flavor of our approach the complete proofs and general theory will be discussed in a further paper

on rational solutions of yang baxter equations maximal orders in loop algebra in 1982 belavin and drinfeld listed all elliptic and trigonometric solutionsx u v of the classical yang baxter equation cybe wherex takes values in a simple complex lie algebrag and left the classification problem of the rational one open in 1984 drinfeld conjectured that if a rational solution is equivalent to a solution of the formx u v c2 u v r u v wherec2 is the quadratic casimir element andr is a polynomial inu v then deg u r deg v r 1 in another paper i proved this conjecture forg sl n and reduced the problem of listing nontrivial i e nonequivalent toc2 u v solutions of cybe to classification of quasi frobenius subalgebras of g they in turn are related with the so called maximal orders in the loop algebra of g corresponding to the vertices of the extended dynkin diagramd e g in this paper i give an algorithm which enables one to list all solutions and illustrate it with solutions corresponding to vertices ofd e g with coefficient 2 or 3 in particular i will find all solutions forg o 5 and some solutions forg o 7 o 10 o 14 andg2

formality theorem for lie bialgebras and quantization of twists and coboundary r matrices abstract let g be a lie bialgebra let u g a quantization of g through etingof kazhdan functor we prove the existence of a l morphism between the lie algebra c g g and the tensor algebra without unit t u t u g 1 with lie algebra structure given by the gerstenhaber bracket when s is a twist for g we deduce from the formality morphism the existence of a quantum twist f when g r is a coboundary lie bialgebra we get the existence of a quantization r of r

event history graphs for censored survival data a compact graphical device for combining survival and time varying covariate information is proposed the proposed graph contains the kaplan meier estimator for right censored data and a simultaneous display of the behaviour of time dependent covariate s and the lifetime for each subject in the sample the observed levels of time dependent covariates are possibly subjected to an initial dimension reduction or smoothing step to produce a continuous covariate function values of this function are plotted on a horizontal bar for the length of the lifetime of the subject covariate information for censored data is also incorporated the union of the horizontal bars forms the kaplan meier estimator of the survival function our graphical method is implemented with a new s plus function and demonstrated in several applications copyright 2001 john wiley sons ltd

survival probabilities with time dependent treatment indicator quantities and non parametric estimators the landmark and simon and makuch non parametric estimators of the survival function are commonly used to contrast the survival experience of time dependent treatment groups in applications such as stem cell transplant versus chemotherapy in leukemia however the theoretical survival functions corresponding to the second approach were not clearly defined in the literature and the use of the simon and makuch estimator was criticized in the biostatistical community here we review the landmark approach showing that it focuses on the average survival of patients conditional on being failure free and on the treatment status assessed at the landmark time we argue that the simon and makuch approach represents counterfactual survival probabilities where treatment status is forced to be fixed the patient is thought as under chemotherapy without possibility to switch treatment or as under transplant since the beginning of the follow up we argue that the simon and makuch estimator leads to valid estimates only under the markov assumption which is however less likely to occur in practical applications this motivates the development of a novel approach based on time rescaling which leads to suitable estimates of the counterfactual probabilities in a semi markov process the method is also extended to deal with a fixed landmark time of interest

the representation type of segre varieties abstract the goal of this work is to prove that all segre varieties n 1 n s p n n i 1 s n i 1 1 unless the quadric surface in p 3 support families of arbitrarily large dimension and rank of simple ulrich and hence acm vector bundles therefore they are of wild representation type

on the representation dimension of finite dimensional algebras the importance of the representation finite artin algebras for the whole representation theory of artin algebras is well understood and there is much literature on the subject see 1 6 12 these are artin algebras such that every indecomposable module is finitely generated and every module is a direct sum of indecomposable modules moreover there is a bijective correspondence between the class of representation finite artin algebras and that of artin algebras with global dimension of at most 2 and with dominant dimension of at least 2 this result was established by auslander in 1 motivated by this correspondence auslander introduced the concept of representation dimension for artin algebras to study the connection of arbitrary artin algebras with representation finite artin algebras it is hoped that this notion gives a reasonable way of measuring how far an artin algebra is from being representation finite type 1 p 134 representation dimension is a morita invariant of artin algebras and its definition involves homological dimensions and modules which are both generators and cogenerators unfortunately little seems to be known about representation dimension so there are a lot of essential questions on this invariant dealing with the computation and estimation of the representation dimension of an artin algebra there are a few cases known auslander proved that the representation dimension of an artin algebra is two if and

mckay correspondence for the poincare series of kleinian and fuchsian singularities we give a uniform and to a large extent geometrical proof that the poincare series of the coordinate algebra of a kleinian singularity and of a fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two coxeter elements these coxeter elements are interpreted geometrically using 2 calabi yau triangulated categories and spherical twist functors

classification des triples de manin pour les algebres de lie reductives complexes resume we study real and complex manin triples for a complex reductive lie algebra g first we generalize results of e karolinsky 1996 math phys anal geom3 545 563 1999 preprint math qa 9901073 on the classification of lagrangian subalgebras then we show that if g is noncommutative one can attach to each manin triple in g another one for a strictly smaller reductive complex lie subalgebra of g this gives a powerful tool for induction then we classify complex manin triples in terms of what we call generalized belavin drinfeld data this generalizes by other methods the classification of a belavin and v g drinfeld of certain r matrices i e the solutions of modified triangle equations for constants cf a belavin and v g drinfeld triangle equations and simple lie algebras mathematical physics reviews vol 4 pp 93 165 harwood academic chur 1984 theorem 6 1 we get also results for real manin triples in passing we retrieve a result of a panov 1999 preprint math qa 9904156 which classifies certain lie bialgebra structures on a real simple lie algebra

application of a new functional expansion to the cubic anharmonic oscillator a new representation of causal functionals is introduced which makes use of noncommutative generating power series and iterated integrals this technique allows the solutions of nonlinear differential equations with forcing terms to be obtained in a simple and natural way it generalizes some properties of fourier and laplace transforms to nonlinear systems and leads to effective computations of various perturbative expansions illustrations by means of the cubic anharmonic oscillator are given in both the deterministic and the stochastic cases

equivariant quantization of poisson homogeneous spaces and kostant s problem we find a partial solution to the longstanding problem of kostant concerning description of the so called locally finite endomorphisms of highest weight irreducible modules the solution is obtained by means of its reduction to a far reaching extension of the quantization problem while the classical quantization problem consists in finding product deformations of the commutative algebras of functions we consider the case when the initial object is already a noncommutative algebra the algebra of functions within q calculus

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

almost split sequences in dimension two a does not split b if h x r c is a morphism in c which is not a splittable epimorphism in c then there is a t x c such that gt h c if j ci y is a morphism in c which is not a splittable monomorphism in c then there is an s c y in c such that j sf almost split sequences were first introduced by us in the case a is the category of finitely generated modules over an artin algebra i and c a see 6 71 the reader is referred to the expository articles 11 121 for an account of the role almost split sequences have played in the theory of the representation theory of artin algebras that almost split sequences exist for various subcategories of module categories over a much wider class of rings than artin algebras was first shown in 2 31 where various existence theorems for almost split sequences were given these results seemed to indicate that there might be connections between the structure and existence of almost split sequences and algebraic geometry recent developments show that this is indeed the case in 4 it is shown that the structure of the almost split sequences in the category of reflexive modules of a complete rational double point over the complex numbers determines the desingularization graph of the singularity also it has been shown in s that a complete cohen macaulay local ring s is an isolated singularity if and only if the category of cohen macaulay

explicit quantization of dynamical r matrices for finite dimensional semisimple lie algebras 1 1 classical r matrices in the early eighties belavin and drinfeld bd classified nonskewsymmetric classical r matrices for simple lie algebras it turned out that such r matrices up to isomorphism and twisting by elements from the exterior square of the cartan subalgebra are classified by combinatorial objects which are now called belavin drinfeld triples by definition a belavin drinfeld triple for a simple lie algebra g is a triple 1 2 t where 1 2 are subsets of the dynkin diagram of g and t 1 2 is an isomorphism which preserves the inner product and satisfies the nilpotency condition if 1 then there exists k such that t k 1 1 but t 1 the r matrix corresponding to such a triple is given by a certain explicit formula these results generalize in a straightforward way to semisimple lie algebras in s the third author generalized the work of belavin and drinfeld and classified classical nonskewsymmetric dynamical r matrices for simple lie algebras it turns out that they have an even simpler classification up to gauge transformations they are classified by generalized belavin drinfeld triples which are defined as the usual belavin drinfeld triples but without any nilpotency condition the dynamical rmatrix corresponding to such a triple is given by a certain explicit formula as before these results can be generalized to semisimple lie algebras

simple vector bundles on a nodal weierstrass cubic and quasi trigonometric solutions of the classical yang baxter equation in this paper we study the combinatorics of quasi trigonometric solutions of the classical yang baxter equation arising from simple vector bundles on a nodal weierstrab cubic

on some lie bialgebra structures on polynomial algebras and their quantization we study classical twists of lie bialgebra structures on the polynomial current algebra mathfrak g u where mathfrak g is a simple complex finite dimensional lie algebra we focus on the structures induced by the so called quasi trigonometric solutions of the classical yang baxter equation it turns out that quasi trigonometric r matrices fall into classes labelled by the vertices of the extended dynkin diagram of mathfrak g we give the complete classification of quasi trigonometric r matrices belonging to multiplicity free simple roots which have coefficient 1 in the decomposition of the maximal root we quantize solutions corresponding to the first root of mathfrak sl n

the quantum group structure associated with non linearly extended virasoro algebras recently an infinite family of chiral virasoro vertex operators whose exchange algebra is given by the universalr matrix ofsl 2 q has been constructed in the present paper the case of non linearly w extended virasoro symmetries related to the algebrasan n 1 is considered along the same line contrary to the previous case a1 the standardr matrix ofsl n 1 q does not come out and a different solution of yang and baxter s equations is derived the associated quantum group structure is displayed

bilinear system interconnections and generating series of weighted petri nets this paper has three objectives the first objective is to provide a simpler proof concerning a sufficient condition due to ferfera under which bilinearity is preserved for cascade interconnections the next objective is to provide a specific counterexample to show that this condition does not apply to the feedback connection the final objective is to show that the well known correspondence between rational series and formal power series recognized by weighted finite state automata can be generalized to produce a correspondence between the generating series of cascaded and feedback connected bilinear systems and a class of weighted petri nets

quantization of coboundary lie bialgebras we show that any coboundary lie bialgebra can be quantized for this we prove that a etingof kazhdan quantization functors are compatible with lie bialgebra twists and b if such a quantization functor corresponds to an even associator then it is also compatible with the operation of taking coopposites we also use the relation between the etingof kazhdan construction of quantization functors and the alternative approach to this problem which was established in a previous work

tame and wild projective curves and classification of vector bundles abstract we propose a new method of classifying vector bundles on projective curves especially singular ones according to their representation type in particular we prove that the classification problem of vector bundles respectively of torsion free sheaves on projective curves is always finite tame or wild we completely classify curves which are of finite respectively tame vector bundle type by their dual graph moreover our methods yield a geometric description of all indecomposable vector bundles and torsion free sheaves on finite and tame curves

robust inference for event probabilities with non markov event data summary multistate event data in which a single subject is at risk for multiple events is common in biomedical applications this article considers nonparametric estimation of the vector of probabilities of state membership at time t estimators derived under the markov assumption have been shown datta and satten 2001 statistics and probability letters55 403 411 to be consistent for data that is non markov inference however must take into account possibly non markov transitions when constructing confidence bands for event curves we develop robust confidence bands for these curves evaluate them via simulation and illustrate the method on two datasets

classical yang baxter equation and the a constraint abstract we show that elliptic solutions of classical yang baxter equation cybe can be obtained from triple massey products on an elliptic curve we introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions we also study some degenerations of these solutions

tame biserial algebras a similar result is suspected if we drop the property representationfinite in classes 1 and 2 and we will start an investigation of biserial algebras of infinite representation type the key to the result for representation finite algebras was to observe that each representation finite biserial algebra is special 13 see also the basic notations below which means that it can be presented by a quiver with nice relations special algebras play an important role in the modular representation theory of finite groups namely each representation finite block of a group algebra and some of the tame blocks they occur only in characteristic 2 are special 7 10 4 moreover in the complex representation theory of the lorentz group the so called harish chandra modules are defined over tame special algebras 6 in this paper we show that l a 62 for any special algebra a and that special algebras of infinite representation type are tame our paper consists of four parts first we prove that each special algebra is a factor of a special symmetric algebra theorem 1 5 for the latter the methods of gelfand and ponomarev used in the classification of the indecomposable harish chandra modules of the lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras proposition 2 3 presented in section 2 from this list we 480 0021 8693 85 3 00