special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

matching procedure for the sixth painleve equation in the framework of the isomonodromy deformation method we present a constructive procedure to obtain the critical behaviour of painleve vi transcendents and solve the connection problem this procedure yields two and one parameter families of solutions including trigonometric and logarithmic behaviours and three classes of solutions with taylor expansion at a critical point

euler and algebraic geometry euler s work on elliptic integrals is a milestone in the history of algebraic geometry the founders of calculus understood that some algebraic functions could be integrated using elementary functions logarithms and inverse trigonometric functions euler realized that integrating other algebraic functions leads to genuinely different functions elliptic integrals these functions are not something ugly as abel discovered their inverses are doubly periodic functions on the complex plane what we now call elliptic curves algebraic curves of genus 1 take their name from elliptic integrals although these curves had been studied earlier indeed in great depth by fermat it is euler s analysis that clarifies the key points elliptic curves are fundamentally different from rational curves and not only in a negative way they have a richer symmetry the famous group structure possessed by an elliptic curve this paper considers two main themes in algebraic geometry descended from euler s work integrals of algebraic functions in fancier terms hodge theory and birational geometry in section 1 we reach a major open problem of algebraic geometry which representations of the fundamental group are summands of the cohomology of some family of algebraic varieties or equivalently which linear differential equations can be solved by integrals of algebraic functions one might not expect any good answer to these questions but in fact there are two promising approaches the simpson and bombieri dwork conjectures section 2 more elementary gives an introduction to birational geometry i hope to explain the significance of the problem of finite generation of the canonical ring which has just been solved thanks to carlos simpson for his comments on an earlier version

a ga based fuzzy modeling approach for generating tsk models this paper proposes a new genetic based modeling method for building simple and well defined tsk models with scatter type input partitions our approach manages all attributes characterizing the structure of a tsk model simultaneously particularly it determines the number of rules the input partition the participating inputs in each rule and the consequent parameters the model building process is divided into two phases in phase one the structure learning task is formulated as a multi objective optimization problem which is resolved using a novel genetic based structure learning gbsl scheme apart from the mean square error mse and the number of rules three additional criteria are introduced in the fitness function for measuring the quality of the partitions optimization of these measures leads to models with representative rules small overlapping and efficient data cover in order to obtain models with accurate data fitting and good local performance the consequent parameters are determined using a local mse function while the overall model is evaluated on the basis of a global mse function the search capabilities of the suggested structure learning scheme are significantly enhanced by including a highly effective local search operator implemented by a micro genetic algorithm and four problem specific operators finally a genetic based parameter learning gbpl scheme is suggested in phase two which performs fine tuning of the initial models obtained after structure learning the performance of the proposed modeling approach is evaluated using a static example and a well known dynamic benchmark problem simulation results demonstrate that our models outperform those suggested by other methods with regard to simplicity model structure and accuracy

on the interpretation and identification of dynamic takagi sugeno fuzzy models dynamic takagi sugeno fuzzy models are not always easy to interpret in particular when they are identified from experimental data it is shown that there exists a close relationship between dynamic takagi sugeno fuzzy models and dynamic linearization when using affine local model structures which suggests that a solution to the multiobjective identification problem exists however it is also shown that the affine local model structure is a highly sensitive parametrization when applied in transient operating regimes due to the multiobjective nature of the identification problem studied here special considerations must be made during model structure selection experiment design and identification in order to meet both objectives some guidelines for experiment design are suggested and some robust nonlinear identification algorithms are studied these include constrained and regularized identification and locally weighted identification their usefulness in the present context is illustrated by examples

algebraic isomonodromic deformations of logarithmic connections on the riemann sphere and finite braid group orbits on character varieties we study algebraic isomonodromic deformations of flat logarithmic connections on the riemann sphere with n ge 4 poles for arbitrary rank we introduce a natural property of algebraizability for the germ of universal deformation of such a connection we relate this property to a peculiarity of the corresponding monodromy representation to yield a finite braid group orbit on the appropriate character variety under reasonable assumptions on the deformed connection we may actually establish an equivalence between both properties we apply this result in the rank two case to relate finite branching and algebraicity for solutions of garnier systems for general rank a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension

gromov witten classes quantum cohomology and enumerative geometry the paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry in particular it contains an axiomatic treatment of gromov witten classes and a discussion of their properties for fano varieties cohomological field theories are defined and it is proved that tree level theories are determined by their correlation functions application to counting rational curves on del pezzo surfaces and projective spaces are given

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

shear coordinate description of the quantized versal unfolding of a d4 singularity in this communication by using teichmuller theory of a sphere with four holes orbifold points we obtain a system of flat coordinates on the general affine cubic surface having a d4 singularity at the origin we show that the goldman bracket on the geodesic functions on the four holed orbifold sphere coincides with the etingof ginzburg poisson bracket on the affine d4 cubic we prove that this bracket is the image under the riemann hilbert map of the poisson lie bracket on we realize the action of the mapping class group by the action of the braid group on the geodesic functions this action coincides with the procedure of analytic continuation of solutions of the sixth painleve equation finally we produce the explicit quantization of the goldman bracket on the geodesic functions on the four holed orbifold sphere and of the braid group action

canonical structure and symmetries of the schlesinger equations the schlesinger equations s n m describe monodromy preserving deformations of order m fuchsian systems with n 1 poles they can be considered as a family of commuting time dependent hamiltonian systems on the direct product of n copies of m m matrix algebras equipped with the standard linear poisson bracket in this paper we present a new canonical hamiltonian formulation of the general schlesinger equations s n m for all n m and we compute the action of the symmetries of the schlesinger equations in these coordinates

fuzzy instrumental variable approach for nonlinear discrete time systems identification in a noisy environment nonlinear discrete time systems identification based on instrumental variable iv method and takagi sugeno ts fuzzy model is proposed in this paper the ivs statistically uncorrelated with noise are adequately chosen and mapped to fuzzy sets partitioning the input space of the nonlinear plant in subregions for unbiased estimates of the ts fuzzy model consequent parameters in a noisy environment the fuzzy instrumental variable fiv concept is proposed so that consistency and unbias properties of the fiv algorithm are derived from lemmas and theorem simulation results show the efficiency of the fiv algorithm for nonlinear system identification in a noisy environment and its application for modeling of an aluminium beam with free free configuration which is commonly used for active vibration control design in aircrafts and aerospace vehicles

green function painleve vi equation and eisenstein series of weight one we study the problem how many singular points of a solution t to the painleve vi equation with parameter 8 1 8 1 8 3 8 might have in c 0 1 here t0 c 0 1 is called a singular point of t if t0 0 1 t0 based on hitchin s formula we explore the connection of this problem with green function and the eisenstein series of weight one among other things we prove i there are only three solutions which have no singular points in c 0 1 ii for a special type of solutions called real solutions here any branch of a solution has at most two singular points in particular at most one pole in c 0 1 iii any riccati solution has singular points in c 0 1 iv for each n 5 and n 6 6 we calculate the number of the real j values of zeros of the eisenstein series en 1 k1 k2 of weight one where k1 k2 runs over 0 n 1 2 with gcd k1 k2 n 1 the geometry of the critical points of the green function on a flat torus e as varies in the modulim1 plays a fundamental role in our analysis of the painleve iv equation in particular the conjectures raised in 22 on the shape of the domain 5 m1 which consists of tori whose green function has extra pair of critical points are completely solved here

isomonodromic deformations and twisted yangians arising in teichmuller theory abstract in this paper we build a link between the teichmuller theory of hyperbolic riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the fuchsian group associated to the given hyperbolic riemann surface by the poincare uniformization in the case of a one sheeted hyperboloid with n orbifold points we show that the poisson algebra d n of geodesic length functions is the semiclassical limit of the twisted q yangian y q o n for the orthogonal lie algebra o n defined by molev ragoucy and sorba we give a representation of the braid group action on d n in terms of an adjoint matrix action we characterize two types of finite dimensional poissonian reductions and give an explicit expression for the generating function of their central elements finally we interpret the algebra d n as the poisson algebra of monodromy data of a frobenius manifold in the vicinity of a non semi simple point

the strong macdonald conjecture and hodge theory on the loop grassmannian we prove the strong macdonald conjecture of hanlon and feigin for reductive groups g in a geometric reformulation we show that the dolbeault cohomology hq x op of the loop grassmannian x is freely generated by de rham s forms on the disk coupled to the indecomposables of h bg equating the two euler characteristics gives an identity independently known to macdonald m which generalises ramanujan s 1 1 sum for simply laced root systems at level 1 we also find a strong form of bailey s 4 4 sum failure of hodge decomposition implies the singularity of x and of the algebraic loop groups some of our results were announced in t2

picard and chazy solutions to the painleve vi equation we study the solutions of a particular family of painleve vi equations with parameters beta gamma 0 delta frac 1 2 and 2 alpha 2 mu 1 2 for 2 mu in mathbb z we show that in the case of half integer mu all solutions can be written in terms of known functions and they are of two types a two parameter family of solutions found by picard and a new one parameter family of classical solutions which we call chazy solutions we give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0 1 infty and their nonlinear monodromy we study the structure of analytic continuation of the solutions to the pvi mu equation for any mu such that 2 mu in mathbb z as an application we classify all the algebraic solutions for mu half integer we show that they are in one to one correspondence with regular polygons or star polygons in the plane for mu integer we show that all algebraic solutions belong to a one parameter family of rational solutions

hurwitz action on tuples of euclidean reflections we show that if a tuple of euclidean reflections has a finite orbit under the hurwitz action of the artin braid group then the group generated by these reflections is finite humphries has published a similar statement but his proof is irremediably flawed at the same time as correcting his proof our proof is much simpler that dubrovin and mazocco s proof for triples of reflections

pole distribution of pvi transcendents close to a critical point abstract the distribution of the poles of painleve vi transcendents associated to semi simple frobenius manifolds is determined close to a critical point it is shown that the poles accumulate at the critical point asymptotically along two rays as an example the frobenius manifold given by the quantum cohomology of c p 2 is considered the general pvi is also considered

recursive fuzzy instrumental variable based evolving neuro fuzzy identification for non stationary dynamic system in a noisy environment abstract in this paper an online identification algorithm for instrumental variable based evolving neuro fuzzy modeling applied to dynamic systems in noisy environment is proposed the adopted methodology is based an online neuro fuzzy inference system with takagi sugeno evolving structure which employs an adaptive distance norm based on the maximum likelihood criterion with instrumental variable recursive parameter estimation the application and performance analysis of the proposed algorithm is based on black box modeling of a 2dof helicopter with errors in the variables

fuzzy regulators and fuzzy observers relaxed stability conditions and lmi based designs this paper presents new relaxed stability conditions and lmi linear matrix inequality based designs for both continuous and discrete fuzzy control systems they are applied to design problems of fuzzy regulators and fuzzy observers first takagi and sugeno s fuzzy models and some stability results are recalled to design fuzzy regulators and fuzzy observers nonlinear systems are represented by takagi sugeno s ts fuzzy models the concept of parallel distributed compensation is employed to design fuzzy regulators and fuzzy observers from the ts fuzzy models new stability conditions are obtained by relaxing the stability conditions derived in previous papers lmi based design procedures for fuzzy regulators and fuzzy observers are constructed using the parallel distributed compensation and the relaxed stability conditions other lmi s with respect to decay rate and constraints on control input and output are also derived and utilized in the design procedures design examples for nonlinear systems demonstrate the utility of the relaxed stability conditions and the lmi based design procedures

on the existence of monodromies for the rabi model we discuss the existence of monodromies associated with the singular points of the eigenvalue problem for the rabi model the complete control of the full monodromy data requires the taming of the stokes phenomenon associated with the unique irregular singular point the monodromy data in particular the composite monodromy are written in terms of the parameters of the model via the isomonodromy method and the function of the painleve v these data provide a systematic way to obtain the quantized spectrum of the rabi model

a generalization of determinant formulae for the solutions of painleve ii and xxxiv equations a generalization of determinant formulae for the classical solutions of painleve xxxiv and painleve ii equations are constructed using the technique of darboux transformation and hirota s bilinear formalism it is shown that the solutions admit determinant formulae even for the transcendental case

exact solutions of a q discrete second painleve equation from its iso monodromy deformation problem i rational solutions in this paper we present a new method of deducing infinite sequences of exact solutions of q discrete painlev equations by using their associated linear problems the specific equation we consider

quantum painleve calogero correspondence for painleve vi this paper is a continuation of our previous paper where the painleve calogero correspondence has been extended to auxiliary linear problems associated with painleve equations we have proved for the first five equations from the painleve list that one of the linear problems can be recast in the form of the non stationary schrodinger equation whose hamiltonian is a natural quantization of the classical calogero like hamiltonian for the corresponding painleve equation in the present paper we establish the quantum painleve calogero correspondence for the most general case the painleve vi equation we also show how the desired special gauge and the needed choice of variables can be derived starting from the corresponding schlesinger system with rational spectral parameter

a brylinski filtration for affine kac moody algebras abstract braverman and finkelberg have recently proposed a conjectural analogue of the geometric satake isomorphism for untwisted affine kac moody groups as part of their model they conjecture that at dominant weights lusztig s q analog of weight multiplicity is equal to the poincare series of the principal nilpotent filtration of the weight space as occurs in the finite dimensional case we show that the conjectured equality holds for all affine kac moody algebras if the principal nilpotent filtration is replaced by the principal heisenberg filtration the main body of the proof is a lie algebra cohomology vanishing result we also give an example to show that the poincare series of the principal nilpotent filtration is not always equal to the q analog of weight multiplicity finally we give some partial results for indefinite kac moody algebras

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

4 dimensional frobenius manifolds and painleve vi a frobenius manifold has tri hamiltonian structure if it is even dimensional and its spectrum is maximally degenerate we study the case of the lowest nontrivial dimension n 4 and show that under the assumption of semisimplicity the corresponding isomonodromic fuchsian system is described by the painleve hbox vi mu equation since the solutions of this equation are known to parametrize semisimple frobenius manifolds of dimension n 3 this leads to an explicit procedure mapping 3 dimensional frobenius structures of 4 dimensional ones and giving all tri hamiltonian structures in four dimensions we illustrate the construction by computing two examples in the framework of frobenius structures on hurwitz spaces

on linear systems and τ functions associated with lame s equation and painleve s equation vi abstract painleve s transcendental differential equation p vi may be expressed as the consistency condition for a pair of linear differential equations with 2 2 matrix coefficients with rational entries by a construction due to tracy and widom this linear system is associated with certain kernels which give trace class operators on hilbert space this paper expresses such operators in terms of hankel operators of linear systems which are realised in terms of the laurent coefficients of the solutions of the differential equations let p t l 2 0 l 2 t be the orthogonal projection then the fredholm determinant t det i p t defines the function which is here expressed in terms of the solution of a matrix gelfand levitan equation for suitable values of the parameters solutions of the hypergeometric equation give a linear system with similar properties for meromorphic transfer functions that have poles on an arithmetic progression the corresponding hankel operator has a simple form with respect to an exponential basis in l 2 0 so det i p t can be expressed as a series of finite determinants this applies to elliptic functions of the second kind such as satisfy lame s equation with l 1

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

teichmuller spaces as degenerated symplectic leaves in dubrovin ugaglia poisson manifolds abstract in this paper we study the goldman bracket between geodesic length functions both on a riemann surface g s 0 of genus g with s 1 2 holes and on a riemann sphere 0 1 n with one hole and n orbifold points of order two we show that the corresponding teichmuller spaces t g s 0 and t 0 1 n are realised as real slices of degenerated symplectic leaves in the dubrovin ugaglia poisson algebra of upper triangular matrices s with 1 on the diagonal

rational solutions of the painleve vi equation in this paper we classify all values of the parameters and of the painleve vi equation such that there are rational solutions we give a formula for them up to the birational canonical transformations and the symmetries of the painleve vi equation

observer based robust fuzzy control of nonlinear systems with parametric uncertainties this paper addresses the robust fuzzy control problem for nonlinear systems in the presence of parametric uncertainties and the state variables unavailable for measurement the takagi sugeno t s fuzzy model system with parametric uncertainties is adopted for modeling the nonlinear system and establishing fuzzy state observer sufficient conditions are derived for robust stabilization in the sense of lyapunov asymptotic stability and are formulated in the format of linear matrix inequalities lmis the effectiveness of the proposed fuzzy controller and fuzzy observer design methodology is finally demonstrated through numerical simulations on an inverted pendulum system

real root property of the spectral polynomial of the treibich verdier potential and related problems abstract we study the spectral polynomial of the treibich verdier potential such spectral polynomial which is a generalization of the classical lame polynomial plays fundamental roles in both the finite gap theory and the ode theory of heun s equation in this paper we prove that all the roots of such spectral polynomial are real and distinct under some assumptions the proof uses the classical concept of sturm sequence and isomonodromic theories we also prove an analogous result for a polynomial associated with a generalized lame equation where we apply a new approach based on the viewpoint of the monodromy data

holonomy of the ising model form factors we study the ising model two point diagonal correlation function c n n by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of wu mccoy tracy and barouch we extend this expansion weighting by powers of a variable the j particle contributions f j n n in the form factor expansion the corresponding extension of the two point diagonal correlation function c n n is shown for arbitrary to be a solution of the sigma form of the painleve vi equation introduced by jimbo and miwa in their isomonodromic approach to the ising model fuchsian linear differential equations for the form factors f j n n are obtained for j 9 and shown to have both a russian doll nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second order linear differential operator associated with the elliptic integral e from this we show that each f j n n is unexpectedly simple being expressed polynomially in terms of the elliptic integrals e and k in contrast we exhibit some mathematical objects built from these form factors f j n n which break the direct sum of symmetric powers decomposition with its associated polynomial expressions first we show that the scaling limit of these differential operators and form factors breaks the direct sum structure but not the russian doll structure secondly we show that the previous extension of two point diagonal correlation functions c n n is for singled out values cos m n m n integers also solutions of fuchsian linear differential equations these solutions of painleve vi are not polynomial in e and k but are actually algebraic functions being associated with modular curves

an iv qr algorithm for neuro fuzzy multivariable online identification in this paper a new algorithm for neuro fuzzy identification of multivariable discrete time nonlinear dynamic systems more specifically applied to consequent parameters estimation of the neuro fuzzy inference system is proposed based on a decomposed form as a set of coupled multiple input and single output miso takagi sugeno ts neuro fuzzy networks an on line scheme is formulated for modeling a nonlinear autoregressive with exogenous input narx recurrent neuro fuzzy structure from input output samples of a multivariable nonlinear dynamic system in a noisy environment the adaptive weighted instrumental variable wiv algorithm by qr factorization based on the numerically robust orthogonal householder transformation is developed to modify the consequent parameters of the ts multivariable neuro fuzzy network

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

lax pairs of discrete painleve equations a2 a1 1 case in this paper we provide a comprehensive method for constructing lax pairs of discrete painleve equations by using a reduced hypercube structure in particular we consider the a 5 1 surface q painleve system which has the affine weyl group symmetry of type a 2 a 1 1 two new lax pairs are found

an ergodic study of painleve vi an ergodic study of painleve vi is developed the chaotic nature of its poincare return map is established for almost all loops the exponential growth of the numbers of periodic solutions is also shown principal ingredients of the arguments are a moduli theoretical formulation of painleve vi a riemann hilbert correspondence the dynamical system of a birational map on a cubic surface and the lefschetz fixed point formula

building sugeno type models using fuzzy discretization and orthogonal parameter estimation techniques this paper develops a new approach to building sugeno type models the essential idea is to separate the premise identification from the consequence identification while these are mutually related in the previous methods a fuzzy discretization technique is suggested to determine the premise of the model and an orthogonal estimator is provided to identify the consequence of the model the orthogonal estimator can provide information about the model structure or which terms to include in the model and final parameter estimates in a very simple and efficient manner the well known gas furnace data of box and jenkins is used to illustrate the proposed modeling approach and to compare its performance with other statistical and fuzzy modeling approaches it shows that the performance of the new approach compares favorably with these existing techniques

the logarithmic asymptotics of the sixth painleve equation we compute the monodromy group associated with the solutions of the sixth painleve equation with a logarithmic asymptotic behavior at a critical point and we characterize the asymptotic behavior in terms of the monodromy itself

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

computation of highly ramified coverings an almost belyi covering is an algebraic covering of the projective line such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line in general there are 1 dimensional families of these coverings with a fixed ramification pattern that is hurwitz spaces for these coverings are curves in this paper three almost belyi coverings of degrees 11 12 and 20 are explicitly constructed we demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth painleve equation

stability conditions on a non compact calabi yau threefold we study the space of stability conditions stab x on the non compact calabi yau threefold x which is the total space of the canonical bundle of mathbb p 2 we give a combinatorial description of an open subset of stab x and state a conjecture relating stab x to the frobenius manifold obtained from the quantum cohomology of mathbb p 2 we give some evidence from mirror symmetry for this conjecture

stokes matrices and monodromy of the quantum cohomology of projective spaces in this paper we compute stokes matrices and monodromy of the quantum cohomology of projective spaces this problem can be formulated in a classical framework as the problem of computation of stokes matrices and monodromy of differential equations with regular and irregular singularities we prove that the stokes matrix of the quantum cohomology coincides with the gram matrix in the theory of derived categories of coherent sheaves we also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups

determinant structure of the rational solutions for the painleve ii equation two types of determinant representations of the rational solutions for the painleve ii equation are discussed by using the bilinear formalism one of them is a representation by the devisme polynomials and another one is hankel determinant representation they are derived from the determinant solutions of the kp hierarchy and toda lattice respectively

correction factors for kac moody groups and t deformed root multiplicities we study a correction factor for kac moody root systems which arises in the theory of p adic kac moody groups in affine type this factor is known and its explicit computation is the content of the macdonald constant term conjecture the data of the correction factor can be encoded as a collection of polynomials m lambda in mathbb z t indexed by positive imaginary roots lambda at t 0 these polynomials evaluate to the root multiplicities so we consider m lambda to be a t deformation of mathrm mult lambda we generalize the peterson algorithm and the berman moody formula for root multiplicities to compute m lambda as a consequence we deduce fundamental properties of m lambda

schlesinger transformations for painleve vi equation a method to obtain the schlesinger transformations for painleve vi equation is given the procedure involves formulating a riemann hilbert problem for a transformation matrix which transforms the solution of the linear problem but leaves the associated monodromy data the same

a family of solutions of a higher order pvi equation near a regular singularity restriction of the n dimensional garnier system to a complex line yields a system of second order nonlinear differential equations which may be regarded as a higher order version of the sixth painleve equation near a regular singularity of the system we present a 2n parameter family of solutions expanded into convergent series these solutions are constructed by iteration and their convergence is proved by using a kind of majorant series for simplicity we describe the proof in the case n 2

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

the t analog of the basic string function for twisted affine kac moody algebras abstract we study lusztig s t analog of weight multiplicities associated to level one representations of twisted affine kac moody algebras an explicit closed form expression is obtained for the corresponding t string function using constant term identities of macdonald and cherednik the closed form involves the generalized exponents of the graded pieces of the twisted affine algebra considered as modules for the underlying finite dimensional simple lie algebra this extends previous work on level 1 t string functions for the untwisted simply laced affine kac moody algebras

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection