exponential stabilization of nonholonomic chained systems this paper presents a feedback control scheme for the stabilization of two input driftless chained nonholonomic systems also called chained form these systems are controllable but not asymptotically stabilizable by a smooth static state feedback control law in addition exponential stability cannot be obtained with a smooth time varying feedback control law here global asymptotical stability with exponential convergence is achieved about any desired configuration by using a nonsmooth time varying feedback control law the control law depends in addition to the state and time on a function which is constant except at predefined instants of time where the function is recomputed as a nonsmooth function of the state the inputs are differentiable with respect to time and tend exponentially toward zero for use in the analysis a lemma on the exponential convergence of a stable time varying nonlinear system perturbed by an exponentially decaying signal is presented simulation results are also shown

quasifinite highest weight modules over the super w 1 algebra we study quasifinite highest weight modules over the supersymmetric extension of thew1 algebra on the basis of the analysis by kac and radul we find that the quasifiniteness of the modules is again characterized by polynomials and obtain the differential equations for highest weights the spectral flow free field realization over the b c system and the embedding into widehat gl are also presented

further results on incremental input to state stability incremental stability notions defined by means of comparison functions in a state space framework are further investigated in particular we provide a continuous converse lyapunov theorem for incremental integral iss and show that the incremental integral property is at least as strong as incremental iss

formal power series solutions of supersymmetric n 3 yang mills equations formal power series solutions of the linear system with one spectral parameter associated to the constraint equations for yang mills superconnections on n extended super minkowski space are considered for n 3 the integrability equations reduce to the supersymmetric field equations the method of approach is identical to that used by takasaki for the self dual equations based upon formal power series in a spectral parameter and in spatial variables the problem is reduced to a linear system of equations for a superfield with values in an dimensional grassmann manifold the formal solution is expressed in terms of data on a 3 2n dimensional superhypersurface however a difficulty arises with respect to the cauchy problem which becomes formally solvable only for an extended system breaking the relativistic invariance through introduction of additional superfields

the cr geometry of weighted extremal kahler and sasaki metrics we establish an equivalence between conformally einstein maxwell kahler 4 manifolds recently studied in apostolov et al j reine angew math 721 109 147 2016 apostolov and maschler j eur math soc 21 1319 1360 2019 futaki and ono j math soc jpn 70 1493 1521 2018 koca et al ann glob anal geom 50 29 46 2016 lebrun einstein maxwell equations extremal kahler metrics and seiberg witten theory in the many facets of geometry a tribute to nigel hitchin oxford university press oxford pp 17 33 2009 lebrun j geom phys 91 171 2015 and lebrun commun math phys 344 621 653 2016 and extremal kahler 4 manifolds in the sense of calabi extremal kahler metrics seminar on differential geometry annals of mathematics studies vol 102 pp 259 290 princeton university press princeton 1982 with nowhere vanishing scalar curvature the corresponding pairs of kahler metrics arise as transversal kahler structures of sasaki metrics compatible with the same cr structure and having commuting sasaki reeb vector fields this correspondence extends to higher dimensions using the notion of a weighted extremal kahler metric apostolov et al in levi kahler reduction of cr structures product of spheres and toric geometry arxiv 1708 05253 apostolov et al in weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 lahdili in j geom anal 29 542 568 2019 lahdili in int math res not arxiv 1710 00235 lahdili in proc lond math soc 119 1065 1114 2019 illuminating and uniting several explicit constructions in kahler and sasaki geometry it also leads to new existence and non existence results for extremal sasaki metrics suggesting a link between the notions of relative weighted k stability of a polarized variety introduced in apostolov et al weighted extremal kahler metrics and the einstein maxwell geometry of projective bundles arxiv 1808 02813 and lahdili proc lond math soc 119 1065 1114 2019 and relative k stability of the kahler cone corresponding to a sasaki polarization studied in boyer and van coevering math res lett 25 1 19 2018 and collins and szekelyhidi j differ geom 109 81 109 2018

on the τ function of the painleve equations abstract concerning the hamiltonian structure associated with painleve equations we define the function related to a hamiltonian and show that the correlation function given in the study of the two dimensional ising model is a function for the third painleve equation the list of the hamiltonians which are polynomials in the two canonical variables is given and we prove that the function related to each of them is holomorphic

transformation group acting on a self dual yang mills hierarchy a gl n c self dual yang mills hierarchy is introduced it is an infinite system of self dual yang mills equations having an infinite number of independent variables cauchy problems for the hierarchy are formally solved by using lie transforms of a wave matrix a relationship between the kadomtsev petviashvili hierarchy and the self dual yang mills hierarchy is discussed furthermore it is shown that an infinite dimensional transformation group acts on a solution space to the n 2 self dual yang mills hierarchy a parametric solution to the hierarchy is also given as a representation of the transformation group

symmetries of hyper kahler or poisson gauge field hierarchy symmetry properties of the space of complex or formal hyper kahler metrics are studied in the language of hyper kahler hierarchies the construction of finite symmetries is analogous to the theory of riemann hilbert transformations loop group elements now taking values in a pseudo group of canonical transformations of a simplectic manifold in spite of their highly nonlinear and involved nature infinitesimal expressions of these symmetries are shown to have a rather simple form these infinitesimal transformations are extended to the plebanski key functions to give rise to a nonlinear realization of a poisson loop algebra the poisson algebra structure turns out to originate in a contact structure behind a set of symplectic structures inherent in the hyper kahler hierarchy possible relations to membrane theory are briefly discussed

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

confluent kz equations for mathfrak sl _n with poincare rank 2 at infinity we construct confluent kz equations with poincar rank 2 at infinity for the case of and the integral representation for the solutions hamiltonians of these confluent kz equations are derived from suitable quantization of dlog constructed in the theory of monodromy preserving deformation mpd in jimbo et al 1981 physica d 2 306 52 our confluent kz equations can be viewed as a quantization of mpd with poincar rank 2 at infinity

periodic reduction of self dual yang mills equations a class of gl invariant self dual yang mills sdym equations is considered it is shown that the gl sdym equations are reduced to the gl n c sdym hierarchy by imposing a periodic condition this reduction procedure makes clear a relationship between our gl sdym equations and the usual infinite matrix representation of a single gl n c sdym equation

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

a non quadratic criterion for stability of forced oscillations abstract a new test for stability of forced oscillations in nonlinear systems is proposed the result is illustrated by an example of a saturated pi control of an integrator with a harmonic reference signal the range of frequencies and amplitudes that guarantee stability of the corresponding oscillations is found

subcomplexes in curved bgg sequences bgg sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries for locally flat geometries the resulting sequences are complexes but in general the compositions of the operators in such a sequence are nonzero in this paper we show that under appropriate torsion freeness and or semi flatness assumptions certain parts of all bgg sequences are complexes several examples of structures including quaternionic structures hypersurface type cr structures and quaternionic contact structures are discussed in detail in the case of quaternionic structures we show that several families of complexes obtained in this way are elliptic

the group theoretical aspects of infinitesimal riemann hilbert transform and hidden symmetry we obtain explicit expressions for infinitesimal regular riemann hilbert rh transforms using them the group theoretical aspects of infinitesimal rh transforms are discussed with an eye to the comparison with the hidden symmetry transformations proposed by us before we find that the rh transforms have very rich group structure e g in the 2 d principal chiral models their group contains two kac moody algebras as subalgebras but not all of them are nontrivial hidden symmetries of the theory

the kontsevich matrix integral convergence to the painleve hierarchy and stokes phenomenon we show that the kontsevich integral on n times n matrices n infty is the isomonodromic tau function associated to a 2 times 2 riemann hilbert problem the approach allows us to gain control of the analysis of the convergence as n to infty by an appropriate choice of the external source matrix in kontsevich s integral we show that the limit produces the isomonodromic tau function of a special tronquee solution of the first painleve hierarchy and we identify the solution in terms of the stokes data of the associated linear problem we also show that there are several tau functions that are analytic in appropriate sectors of the space of parameters and that the formal witten kontsevich tau function is the asymptotic expansion of each of them in their respective sectors thus providing an analytic tool to analyze its nonlinear stokes phenomenon

convergence conditions for some classes of nonlinear systems abstract systems of differential equations with nonlinearities of a sector type and almost periodic perturbations are studied with the aid of the lyapunov direct method conditions are found under which the considered systems admit globally asymptotically stable almost periodic solutions moreover it is shown that the proposed approach permits us to derive new convergence conditions for some models of neural networks and generalized lotka volterra models of population dynamics an example is presented to demonstrate the effectiveness of the obtained results

antiquantization isomonodromy and integrability an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear an extended analysis of links between linear differential equations and the nonlinear painleve equation pv i is given for linear equations second order equations in different forms as well as various first order systems are chosen the role of an accessory parameter is explained the relationship to the schlesinger system is made clear

linear integral transformations and hierarchies of integrable nonlinear evolution equations abstract integrable hierarchies of nonlinear evolution equations are investigated on the basis of linear integral equations these are riemann hilbert type of integral transformations which leave invariant an infinite sequence of ordinary differential matrix equations of increasing order in an indefinite parameter k the potential matrices in these equations obey a set of nonlinear recursion relations leading to a heirarchy of nonlinear partial differential equations in decreasing order the same equations give rise to a reciprocal hierarchy associated with heisenberg ferromagnet type of equations central in the treatment is an embedding of the hierarchy into an infinite matrix structure which is constructed on the basis of the integral equations in terms of this infinite matrix structure the equations governing the hierarchies become quite simple furthermore it leads in a straightforward way to various generalizations such as to other types of linear spectral problems multicomponent system and lattice equations generalizations to equations associated with noncommuting flows follow as a direct consequence of the treatment finally some results on conserved densities and the hamiltonian structure are briefly discussed

projective compactness and conformal boundaries for complete affine manifolds we treat a recently introduced general notion of compactification based on the projective differential geometry i e geodesic path data of the given connection this involves a real parameter called the order of projective compactness for volume preserving connections this order is a measure of asymptotic volume growth via the levi civita connection this concept applies to complete pseudo riemannian metrics so projective compactness provides a geometrically motivated alternative to conformal compactification we study the fundamental links between interior geometry and the embedding of the boundary hypersurface at infinity and develop the tools that describe the induced geometry on this boundary we prove that a pseudo riemannian metric which is projectively compact of order two admits a certain asymptotic form this form was known to be sufficient for projective compactness so the result establishes that it provides an equivalent characterization from a projectively compact connection on m one obtains a projective structure on the compactified manifold with boundary overline m we show this induces a possibly degenerate conformal structure on the boundary hypersurface partial m we prove that in the case of metrics which are projectively compact of order two this boundary structure is in fact always non degenerate we also prove that in this case the metric is necessarily asymptotically einstein in a natural sense finally a non degenerate conformal boundary geometry gives rise to a conformal standard tractor bundle on the boundary endowed with a canonical linear connection and we explicitly describe these in terms of the projective data of the interior geometry

extreme superposition rogue waves of infinite order and the painleve iii hierarchy we study the fundamental rogue wave solutions of the focusing nonlinear schrodinger equation in the limit of large order using a recently proposed riemann hilbert representation of the rogue wave solution of arbitrary order k we establish the existence of a limiting profile of the rogue wave in the large k limit when the solution is viewed in appropriate rescaled variables capturing the near field region where the solution has the largest amplitude the limiting profile is a new particular solution of the focusing nonlinear schrodinger equation in the rescaled variables the rogue wave of infinite order which also satisfies ordinary differential equations with respect to space and time the spatial differential equations are identified with certain members of the painleve iii hierarchy we compute the far field asymptotic behavior of the near field limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving riemann hilbert problems in a certain transitional region for the asymptotics the near field limit function is described by a specific globally defined tritronquee solution of the painleve ii equation these properties lead us to regard the rogue wave of infinite order as a new special function

on the initial condition for instanton solutions to each gauge equivalence class of both local and global framed in the sense of donaldson self dual solutions with the gauge group u r there is related the unique canonical initial condition in the sense of takasaki and in this way the gauge freedom is eliminated a geometric interpretation is given and consequently the complete transcription of the adhm construction into the inverse scattering formalism is derived as an application an injection holomorphic mapping of the instanton moduli space into a finite dimensional complex vector space is described and the loop group action on the transition functions is discussed the results suggest the possibility of a new description of the framed instanton moduli spaces directly as algebraic sets

on the solvability of the discrete second painleve equation the inverse monodromy method for studying the riemann hilbert problem associated with classical painlev e equations is applied to the discrete second painlev e equation

linear integral equations and multicomponent nonlinear integrable systems i abstract a systematic method for obtaining multicomponent generalizations of integrable nonlinear partial differential equations pde s is developed the method starts from a general type of linear integral equations containing integrations over an arbitrary contour with an arbitrary measure in the complex plane of k the spectral parameter special k dependent factors in the integrand are shown to induce an extra coupling between solutions of these integral equations with a different source term in this way the direct linearization is obtained of multicomponent generalizations of various nonlinear pde s such as the nonlinear schrodinger equation the derivative nonlinear schrodinger equations the isotropic heisenberg spin chain equation the complex sine gordon equation and the massive thirring model equations in the present paper i we present the general framework to derive finite matrix pde s and we also discuss backlund transformations and multicomponent lattice versions in a subsequent paper ii we treat a variety of examples of multicomponent pde s and discuss miura transformations and gauge equivalences

stability analysis via averaging functions a new class of lyapunov functions is proposed for analysis of incremental stability for nonlinear systems this class of lyapunov functions allows to establish input dependent incremental stability criteria two substantially different situations are considered when incremental stability is guaranteed by the inputs of sufficiently small amplitude and when similar to the excited van der pol oscillator the stability is induced by sufficiently large inputs

complete integrability of the kadomtsev petviashvili equation abstract the total hierarchy of the kadomtsev petviashvili kp equation is transformed to the system of linear partial differential equations with constant coefficients the complete integrability of the kp equation is proved by using this linear system the existence and uniqueness theorem of the cauchy problem of the kp hierarchy is obtained

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

approximating the steady state periodic solutions of contractive systems we consider contractive systems whose trajectories evolve on a compact and convex state space it is well known that if the time varying vector field of the system is periodic then the system admits a unique globally asymptotically stable periodic solution obtaining explicit information on this periodic solution and its dependence on various parameters is important both theoretically and in numerous applications we develop an approach for approximating such a periodic trajectory using the periodic trajectory of a simpler system e g an lti system the approximation includes an error bound that is based on the input to state stability property of contractive systems we show that in some cases this error bound can be computed explicitly we also use the bound to derive a new theoretical result namely that a contractive system with an additive periodic input behaves like a low pass filter we demonstrate our results using several examples from systems biology

painleve transcendent evaluations of finite system density matrices for 1d impenetrable bosons the recent experimental realisation of a one dimensional bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable bose gas of primary concern is the ground state occupation of effective single particle states in the finite system and thus the tendency for bose einstein condensation this requires the computation of the density matrix for the impenetrable bose gas on a circle we evaluate the density matrix in terms of a particular painleve vi transcendent in form and furthermore show that the density matrix satisfies a recurrence relation in the number of particles for the impenetrable bose gas in a harmonic trap and with dirichlet or neumann boundary conditions we give a determinant form for the density matrix a form as an average over the eigenvalues of an ensemble of random matrices and in special cases an evaluation in terms of a transcendent related to painleve v and vi we discuss how our results can be used to compute the ground state occupations

convergent dynamics a tribute to boris pavlovich demidovich we review and pay tribute to a result on convergent systems by the russian mathematician boris pavlovich demidovich in a sense demidovich s approach forms a prelude to a field which is now called incremental stability of dynamical systems developments on incremental stability are reviewed from a historical perspective

self dual yang mills hierarchy a nonlinear integrable dynamical system in higher dimensions a self dual yang mills sdym hierarchy is presented that has some properties similar to those of the celebrated kadomtsev petviashvili kp hierarchy having one spacial variable the sdym hierarchy is an infinite system of gl invariant compatible sdym equations depending on an infinite number of spacial variables on which an exponential operator acts as a time evolution operator a nontrivial gl symmetry of the sdym hierarchy called the bruhat transformation is found the existence of the gl sdym hierarchy supports the complete integrability of the single gl sdym equation with a reduction condition however our observation suggests that the sdym equation without the condition may not be completely integrable in sharp contrast with the kp equation

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

a new approach to the self dual yang mills equations inspired by sato s new theory for soliton equations we find a new approach to the self dual yang mills equations we first establish a correspondence of solutions between the self dual yang mills equations and a new system of equations with infinitely many unknown functions it then turns out that the latter equations can be easily solved by a simple explicit procedure this leads to an explicit description of a very broad class of solutions to the self dual yang mills equations and also to a construction of transformations acting on these solutions

character and determinant formulae of quasifinite representation of thew1 algebra we diagonalize the hilbert space of some subclass of the quasifinite module of thew1 algebra states are classified according to their eigenvalues for infinitely many commuting charges and the young diagrams the parameter dependence of their norms is explicitly derived the full character formulae of the degenerate representations are given as summation of the bilinear combinations of the schur polynomials

hierarchy structure in integrable systems of gauge fields and underlying lie algebras an improved version of nakamura s self dual yang mills hierarchy is presentd and its symmetry contents are studied the new hierarchy as well as the previous one represents a set of commuting dynamical flows in an infinite dimensional manifolds of loop type but includes a large set of dependent variables because of new degrees of freedom the theory acquires a more symmetric form with richer structures for example it allows a large symmetry algebra of riemann hilbert type which is actually a direct sum of two subalgebras left and right this phenomenon is basically the same as observed recently by avan and bellon on the case of principal chiral models in addition to these rather familiar symmeties a new type of symmetries referred to as coordinate transformation type are also introduced generators of the above dynamical flows are all included therein these two types of symmetries altogether form a big lie algebra which lead to more satisfactory understanding of symmetry properties of integrable systems of guage fields

scattering of plane waves in self dual yang mills theory we consider the classical self dual yang mills equation in 3 1 dimensional minkowski space we have found a new solution it describes the scattering of n plane waves the construction which we use is similar to the quantum inverse scattering method we introduce a monodromy matrix it acts in the direct product of the universal enveloping of su n algebra and an auxiliary linear space in order to obtain the solution of the self dual yang mills equation we take a special matrix element of in the auxiliary space

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

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

the hamiltonian structure of yang mills theories and instantons ii abstract the formalism of constraints reviewed in paper i is applied to yang mills theory to determine the physical phase space this turns out to be the cotangent bundle of orbit space the space of gauge inequivalent potentials self dual configurations are not hamiltonian with respect to the symplectic structure inherited from the general system

factorizations for self dual gauge fields for a particular class of patching matrices onp3 including those for the complex instanton bundles with structure group sp k oro 2k we show that the associated riemann hilbert problemg x g x g 1 x can be generically solved in the factored formg 1 2 n if n is the potential generated in the usual way fromg and we set i 1 i with n g then each ialso generates a selfdual gauge potential i the potentials are connected via the dressing transformations gamma iota phi i 1 cdot gamma iota 1 cdot phi i phi i 1 d phi i of zakharov shabat the factorization is not unique it depends on the arbitrary ordering of the poles of the patching matrix

the partition function of the two matrix model as an isomonodromic τ function we consider the itzykson zuber eynard mehta two matrix model and prove that the partition function is an isomonodromic function in a sense that generalizes that of jimbo et al monodromy preserving deformation of linear ordinary differential equations with rational coefficients physica d 2 306 1981 in order to achieve the generalization we need to define a notion of function for isomonodromic systems where the adregularity of the leading coefficient is not a necessary requirement

the cauchy problem for the second member of a rm p_ iv hierarchy a rigorous method the inverse monodromy transform for studying the riemann hilbert rh problem associated with the classical painleve equations is applied to the second member of a fourth painleve hierarchy we show that the cauchy problem for the second member of this hierarchy admits in general a global meromorphic solution in x moreover for a particular choice of the monodromy data the associated rh problem can be reduced to a set of scalar rh problems and a special solution which can be written in terms of the airy function is obtained

infinite dimensional lie algebras acting on the solution space of various sigma models infinite dimensional lie algebras of infinitesimal transformations acting on the solution space of various two dimensional models are investigated the main tools are i takasaki s interpretation commun math phys 94 35 1984 of the solutions of the associated linear system in terms of points in an infinite dimensional grassmann manifold and ii mikhailov s reduction procedure physica d 3 73 1981 for linear systems takasaki s approach leads for the models with values in a lie group g to a set of transformations that has the structure of the loop algebra g r t t 1 where g is the lie algebra of g this algebra has already been encountered by dolan phys rev lett 47 1371 1981 and by wu nucl phys b 211 160 1983 among others the models with a wess zumino term are also considered the algebraic structure is found to be the same finally mikhailov s procedure is used to study the models with values in a riemannian symmetric space rss g h which is not a lie group th

witten s gauge field equations and an infinite dimensional grassmann manifold witten s gauge fields are interpreted as motions on an infinite dimensional grassmann manifold unlike the case of self dual yang mills equations in takasaki s work the initial data must satisfy a system of differential equations since witten s equations comprise a pair of spectral parameters solutions corresponding to anti self dual yang mills fields are characterized in the space of initial data and in application some yang mills fields which are not self dual anti self dual nor abelian can be constructed

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

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 indefinite anti self dual metrics and the painleve equations we classify the su 2 invariant anti self dual metrics with a signature the metrics are specified by a solution of painleve vi v iii or ii moreover we show the geometric meaning of the metrics specified by each type of painleve function

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

on infinite dimensional algebras of symmetries of the self dual yang mills equations infinite dimensional algebras of hidden symmetries of the self dual yang mills equations are considered a current type algebra of symmetries and an affine extension of conformal symmetries introduced recently are discussed using the twistor picture it is shown that the extended conformal symmetries of the self dual yang mills equations have a simple description in terms of ward s twistor construction

quantum painleve calogero correspondence the painleve calogero correspondence is extended to auxiliary linear problems associated with painleve equations the linear problems are represented in a new form which has a suggestive interpretation as a quantized version of the painleve calogero correspondence namely the linear problem responsible for the time evolution is brought into the form of non stationary schrodinger equation in imaginary time t 12 x2 v x t whose hamiltonian is a natural quantization of the classical calogero like hamiltonian h 12p2 v x t for the corresponding painleve equation in present paper we present explicit constructions for the first five equations from the painleve list

the conformal factor in the sas einstein maxwell field equations and a central extension of a formal loop group we consider a relation between the conformal factor in the stationary axisymmetric sas einstein maxwell field equations and a central extension of a formal loop group which is described by a group 2 cocycle on the formal loop group the corresponding 2 cocycle on the lie algebra of the formal loop group is the one which describes an affine lie algebra as a result we see that the space of formal solutions with conformal factors is a homogeneous space of a central extension of the hauser group

the k tacnode process the tacnode process is a universal behavior arising in nonintersecting particle systems and tiling problems for dyson brownian bridges the tacnode process describes the grazing collision of two packets of walkers we consider such a dyson sea on the unit circle with drift for any k in mathbb z we show that an appropriate double scaling of the drift and return time leads to a generalization of the tacnode process in which k particles are expected to wrap around the circle we derive winding number probabilities and an expression for the correlation kernel in terms of functions related to the generalized hastings mcleod solutions to the inhomogeneous painleve ii equation the method of proof is asymptotic analysis of discrete orthogonal polynomials with a complex weight

the ernst equation as a motion on a universal grassmann manifold we find a linearization of the ernst equation by means of universal grassmann manifold ugm techniques all local analytic solutions defined at the origin are obtained by solving an initial value problem for a linear differential equation on a ugm we give an explicit formula which represents solutions of the ernst equation by using this formula we generate several special solutions