a hypergeometric system of the heun equation and middle convolution a new method of obtaining integral transformation of the heun equation is described the heun equation is reduced to the so called hypergeometric system a linear fuchsian system of rank 3 and a detweiler and reiter algebraic analogue of the katz middle convolution functor is applied a linear fuchsian system of rank 2 associated with the heun equation is also studied

quantized knizhnik zamolodchikov equations quantum yang baxter equation and difference equations for q hypergeometric functions thesl2 quantized knizhnik zamolodchikov equations are solved inq hypergeometric functions new difference equations are derived for generalq hypergeometric functions the equations are given in terms of quantum yang baxter matrices and have the form similar to quantum knizhnik zamolodchikov equations for quantum affine algebras introduced by frenkel and reshetikhin

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

rigid fuchsian systems in 2 dimensional conformal field theories we investigate fuchsian equations arising in the context of 2 dimensional conformal field theory cft and we apply the katz theory of fucshian rigid systems to solve some of these equations we show that the katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions we focus on the case of mathcal w 3 toda cft we argue that the differential equations arising for four point conformal blocks with one nth level semi degenerate field and a fully degenerate one in the fundamental sl3 representation are associated to fuchsian rigid systems we show how to apply katz theory to determine the explicit form of the differential equations the integral expression of solutions and the monodromy group representation the theory of twisted homology is also used in the analysis of the integral expression the computation of the connection coefficients is done for the first time in the case of a katz system with multiplicities thus extending the work done by oshima in the multiplicity free case this approach allows us to construct the corresponding fusion matrices and to perform the whole bootstrap program new explicit factorization of mathcal w 3 correlation functions as well as shift relations between structure constants for general toda theories are also provided

on globally nilpotent differential equations abstract in a previous work of the authors a middle convolution operation on the category of fuchsian differential systems was introduced in this note we show that the middle convolution of fuchsian systems preserves the property of global nilpotence this leads to a globally nilpotent fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems

linear differential equations on the riemann sphere and representations of quivers our interest in this article is a generalization of the additive deligne simpson problem which was originally defined for fuchsian differential equations on the riemann sphere we extend this problem to differential equations having an arbitrary number of unramified irregular singular points and we determine the existence of solutions of the generalized additive deligne simpson problems moreover we apply this result to the geometry of the moduli spaces of stable meromorphic connections of trivial bundles on the riemann sphere namely open embedding of the moduli spaces into quiver varieties and the nonemptiness condition of the moduli spaces furthermore we detail the connectedness of the moduli spaces

lattice filters for rls estimation of a delta operator based model a lattice filter algorithm is derived for the recursive least squares rls estimation of the parameters of the input output model of a plant based on the delta operator formulation the lattice derivation presented allows a numerically efficient implementation particularly for large orders the order recursive property of the lattices is ideally suited for order determination techniques and applications where the order of the plant may change during operation an example concerning parameter identification for a fourth order plant with measurement noise is given

improved consistent estimation of the order of stochastic control systems a more consistent method of estimating the order of feedback control systems is developed the algorithm is of the recursive type strong consistency is derived under less stringent conditions than those of h f chen and l guo ibid vol ac 32 p 531 5 1987

discretized stability and error growth of the nonautonomous pantograph equation this paper is concerned with the stability properties of runge kutta methods for the pantograph equation a functional differential equation with a proportional delay the focus is on nonautonomous equations both linear and nonlinear cases are considered sufficient and necessary conditions for the asymptotic stability of the numerical solution of general neutral pantograph equations are given an upper bound for the error growth is investigated for algebraically stable methods applied to nonneutral equations finally some stability results are extended to the case of a more general class of equations

local variable selection of nonlinear nonparametric systems by first order expansion local variable selection by first order expansion for nonlinear nonparametric systems is investigated in the paper by substantially modifying the algorithms developed in our earlier work bai et al 2014 the previous results have been considerably strengthened under much less restrictive conditions firstly the estimates generated by the modified algorithms are shown to have both the set and parameter convergence with probability one rather than only the set convergence in probability given in our earlier work secondly several technical assumptions e g the lower and upper bounds on the growth of some random sequences which practically are uncheckable have been removed thirdly not only the consistency but also the convergence rate of estimates have been established besides a generalization of the proposed algorithms is also introduced

on the t method for delay differential equations with infinite lag abstract this paper discusses the method for the numerical solution of delay differential equations with infinite lag our analysis is based on the test equation y t ay t by t where a b c and 0 1 in order to solve the storage problem involved in studying the long time behaviour of the solution we use a grid with increased stepsizes monotonicity uniform boundedness asymptotic stability algebraic decay and global discretisation error of the numerical solution are investigated

werte hypergeometrischer funktionen a partial result on a problem of siegel is given hypergeometric functions with rational parameters have transcendental values in almost all algebraic points up to some natural exceptions these exceptions are the well known algebraic functions and an unexpected second class of examples related to certain shimura curves

topological theory for selberg type integral associated with rigid fuchsian systems using the analytic realization of middle convolution due to dettweiler and reiter we show that any rigid fuchsian system can be obtained as a subsystem of some generating system which has an integral representation of solutions of selberg type twisted homology groups and twisted cohomology groups associated with such integrals are studied in particular contiguity relations and twisted cycles which realize local exponents are obtained

consistent order estimation for linear stochastic feedback control systems carma model abstract a new criterion cic is introduced to estimate orders p0 q0 r0 of the linear stochastic feedback control system with correlated noise described by a carma model it is proved that the estimate is strongly consistent when the upper bounds for p0 q0 and r0 are available but neither the stability condition nor the ergodicity of the input and output are imposed on the system

subvarieties of shimura varieties 7 proof of the main result 19 7 3 the case where is bounded 20 7 4 the case where is not bounded 22 both authors were partially supported by the european research training network contract hprn ct 200000120 arithmetic algebraic geometry adresses irmar campus de beaulieu 35042 rennes cedex france edix maths univ rennes1 fr ayafaev maths univ rennes1 fr

recursive identification of overparametrized systems a recursive identification algorithm based on extended least squares is proposed to deal with the contingency of overparametrization the algorithm is relatively simple compared to those involving online order determination being based on adaptively introducing suitable excitation into the algorithm to avoid ill conditioning in the case of extended least squares based adaptive estimation then the regressors are appropriately stochastically perturbed the algorithm is shown to converge to a uniquely defined signal model with any pole zero cancellations at the origin

braid groups are linear in a previous work 11 the author considered a representation of the braid group p bn glm z ql1 t l m n n 1 2 and proved it to be faithful for n 4 bigelow 3 then proved the same representation to be faithful for all n by a beautiful topological argument the present paper gives a different proof of the faithfulness for all n we establish a relation between the charney length in the braid group and exponents of t a certain bn invariant subset of the module is constructed whose properties resemble those of convex cones we relate line segments in this set with the thurston normal form of a braid

hypergeometric integrals and arrangements abstract this is an outline of the aomoto gelfand theory of multivariable hypergeometric integrals and varchenko s formula for the determinant of the period matrix of the hypergeometric pairing a significant feature of this work is the use of the theory of arrangements of hyperplanes to transform a problem in analysis into one in combinatorics

identification and adaptive control for systems with unknown orders delay and coefficients an analysis that gives recursive estimates for time delay system orders and coefficients of single input single output linear discrete time deterministic and stochastic systems with uncorrelated noise is presented it is assumed that a lower bound for the time delay and upper bounds for system order are known the optimal adaptive control is designed for both tracking and linear quadratic regulation when the system parameters including time delay orders and coefficients are unknown the rates of convergence are derived the coefficient estimates to their true values and the loss functions to their minima

an application of aomoto gelfand hypergeometric functions to the su n knizhnik zamolodchikov equation solutions to the knizhnik zamolodchikov equation for verma modules of the lie algebra mathfrak s mathfrak l n 1 mathbb c are explicitly given by certain integrals called aomoto gelfand hypergeometric functions

integral representations of solutions of differential equations free from accessory parameters abstract we show that every accessory parameter free system of differential equations of okubo normal form has integral representation of solutions the proof is constructive we study the change of solutions under the operations the extension and the restriction which have been introduced by yokoyama construction of systems of differential equations of okubo normal form with rigid monodromy preprint in order to construct every such system of differential equations several examples are given

nonstationary time series identification abstract in this paper strongly consistent estimates are given for both unknown parameters and orders of the nonstationary time series where the nonstationarity arises because 1 at any time n a feedback control is added to the usual arma process which is successfully applied to modelling economic systems and 2 det a z may have zeros on the unit circle in addition to those located outside the unit disk where a z is the matrix polynomial corresponding to the ar part of the arma process

regular flat structure and generalized okubo system we study a relationship between regular flat structures and generalized okubo systems we show that the space of variables of isomonodromic deformations of a regular generalized okubo system can be equipped with a flat structure as its consequence we introduce flat structures on the spaces of independent variables of generic solutions to classical painleve equations except for pi in our framework the painleve equations pvi pii can be treated uniformly as just one system of differential equations called the four dimensional extended wdvv equation then the well known coalescence cascade of the painleve equations corresponds to the degeneration scheme of the jordan normal forms of a square matrix of rank four

stokes geometry of higher order linear ordinary differential equations and middle convolution abstract the middle convolution introduced by katz and developed by dettweiler reiter oshima and others defines an operation of reduction of linear ordinary differential equations with polynomial coefficients in this paper employing an idea of the exact steepest descent method proposed by aoki kawai takei we study how to determine the complete stokes geometry and how to obtain the borel summability of wkb solutions of a higher order linear ordinary differential equation with a large parameter when it is reduced to a second order equation via middle convolution to show the practical usefulness of the method we also investigate some concrete examples numerically

consistent estimation of the order of stochastic control systems a consistent estimate of the order of feedback control systems with unknown matrix coefficients estimated by the least squares method is derived by minimizing a modified version of the bayesian information criterion

strongly consistent estimation of the order of stochastic control systems carma model abstract in this paper a variant of the plc predictive least squares criterion for order estimation is combined with an adaptive control strategy and applied to multidimensional carma systems it is shown that with this combination we can estimate recursively and in a strongly consistent way both the order and coefficients of the controlled system while achieving asymptotically optimal cost

the b and m shapiro conjecture in real algebraic geometry and the bethe ansatz we prove the b and m shapiro conjecture that if the wronskian of a set of polynomials has real roots only then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients this in particular implies the following result if all ramification points of a parametrized rational curve 1 r lie on a circle in the riemann sphere 1 then maps this circle into a suitable real subspace r r the proof is based on the bethe ansatz method in the gaudin model the key observation is that a symmetric linear operator on a euclidean space has real spectrum in appendix a we discuss properties of differential operators associated with bethe vectors in the gaudin model in particular we prove a statement which may be useful in complex algebraic geometry it claims that certain schubert cycles in a grassmannian intersect transversally if the spectrum of the corresponding gaudin hamiltonians is simple in appendix b we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with lie algebras of types a r b r and c r

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

multiplicative middle convolution for kz equations we extend the middle convolution for monodromy of fuchsian ordinary differential equations due to katz to an operation for monodromy of kz equations this operation also called the middle convolution gives a monodromy of kz equation obtained by the additive middle convolution since the fundamental group of the domain of definition of kz equation is the pure braid group our middle convolution also gives a recursive way of constructing irreducible representations of the pure braid group several applications are considered

stability of the discretized pantograph differential equation in this paper we study discretizations of the general pantograph equation y t ay t by 6 t cy o t t 0 y o yo where a b c and yo are complex numbers and where 0 and 0 are strictly increasing functions on the nonnegative reals with 6 0 q o 0 and 0 t t 0 t t for positive t our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations and we will identify conditions on a b c and the stepsize which imply that the solution sequence yn i is bounded or that it tends to zero algebraically as a negative power of n

convergence rate of least squares identification and adaptive control for stochastic systems the strong consistency and the convergence rate of least squares identification for the multidimensional armax model are established under some decaying excitation conditions which are satisfied if both input and output do not grow too fast and the attenuating excitation technique is applied the parameter identification results are applied to adaptive control systems with a quadratic loss function the rate of convergence of the loss function to its minimum is also obtained

middle convolution and the hypergeometric equation the addition and the detweiler and reiter algebraic analogue of katz s middle convolution are certain transformations of fuchsian systems their compositions are useful to get nontrivial relations between solutions of these systems the aim of this paper is to study the result of these transformations for a 2 2 linear fuchsian system with triangular matrices and four singularities the isomonodromy deformations of which are given by the hypergeometric functions

on the asymptotics of the trapezoidal rule for the pantograph equation the paper deals with the trapezoidal rule discretization of a class of linear delay differential equations with a special emphasis on equations with a proportional delay our purpose is to analyse the asymptotic properties of the numerical solutions and formulate their upper bounds we also survey the known results and show that our formulae improve and generalize these results in particular we set up conditions under which the numerical solution of the scalar pantograph equation has the same decay rate as the exact solution

on algebraic equations satisfied by hypergeometric correlators in wzw models ii the paper contains an explicit description of genus 0 conformal block bundles for wess zumino witten models of conformal field theory we prove that an earlier construction due to the second and the third authors gives a map of these bundles to certain de rham cohomology bundles

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

stability of runge kutta methods for the generalized pantograph equation this paper deals with stability properties of runge kutta rk methods applied to a non autonomous delay differential equation dde with a constant delay which is obtained from the so called generalized pantograph equation an autonomous dde with a variable delay by a change of the independent variable it is shown that in the case where the rk matrix is regular stability properties of the rk method for the dde are derived from those for a difference equation which are examined by similar techniques to those in the case of autonomous ddes with a constant delay as a result it is shown that some rk methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation stability of algebraically stable natural rk methods is also considered

the stability of modified runge kutta methods for the pantograph equation in the present paper the modified runge kutta method is constructed and it is proved that the modified runge kutta method preserves the order of accuracy of the original one the necessary and sufficient conditions under which the modified runge kutta methods with the variable mesh are asymptotically stable are given as a result the 0 methods with 1 2 1 the odd stage gauss legendre methods and the even stage lobatto iiia and iiib methods are asymptotically stable some experiments are given

order determination theorems and system identification theorems for order determination without a priori knowledge of upper bounds on the order in mimo dynamic systems are developed also deterministic procedures are introduced to determine orders and estimate parameters simultaneously by recursively computing the order determining quantity s sub n a b k which plays a crucial role in order determination procedures and the least squares estimate theta sub n a b of theta p q with p and q denoting the true orders

kernel based local order estimation of nonlinear nonparametric systems we consider the local order estimation of nonlinear autoregressive systems with exogenous inputs narx which may have different local dimensions at different points by minimizing the kernel based local information criterion introduced in this paper the strongly consistent estimates for the local orders of the narx system at points of interest are obtained the modification of the criterion and a simple procedure of searching the minimum of the criterion are also discussed the theoretical results derived here are tested by simulation examples

brief on line order estimation and parameter identification for linear stochastic feedback control systems carma model this paper presents a new recursive estimate method for orders and coefficients of linear stochastic feedback control systems carma model under the assumption that the upper bounds of system orders are known the strong consistency of the estimates for orders and coefficients is proved and the convergence rate of coefficient estimates to their true values is also obtained the estimate algorithm is applied to adaptive tracking of the systems with unknown orders and unknown coefficients the resulting closed loop systems are then globally stable and the tracking sample mean square error is minimized as well simultaneously the estimates of the adaptive tracking for orders and coefficients are also strongly consistent the simulation results given here show that the new developed algorithms of both system identification and adaptive tracking are effective

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

recursive estimation of mixed autoregressive moving average order summary the order p q of an autoregressive moving average sequence y t may be estimated by minimizing a criterion log 2 p q log t t with respect to p and q where 2 is the maximum likelihood estimate of the variance of the innovations t it is suggested that instead a2 be estimated from a series of regressions of y t on y t 1 y t 2 y t p 5 t 1 t q where the t are obtained by fitting a long autoregression to the data it is shown how the sequence of regressions may for p q be economically recursively calculated by embedding them in a sequence of bivariate autoregressions asymptotic properties of the procedure are established under very general conditions

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

a taylor method for numerical solution of generalized pantograph equations with linear functional argument this paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument in this paper we introduce a numerical method based on the taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case the method is illustrated by studying the initial value problems the results obtained are compared by the known results

a differential equation with monodromy group 2 j2 abstract we construct a sixth order differential equation having the central extension of c 2 by the hall janko group j 2 as monodromy group moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation