a feynman kac ito formula for magnetic schrodinger operators on graphs in this paper we prove a feynman kac ito formula for magnetic schrodinger operators on arbitrary weighted graphs to do so we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation on the operator side we identify a very general class of potentials that allows the definition of magnetic schrodinger operators on the probabilistic side we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials apart from linking the world of discrete magnetic operators with the probabilistic world through the feynman kac ito formula the insights from this paper gained on both sides should be of an independent interest as applications of the feynman kac ito formula we prove a kato inequality a golden thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

on twistor solutions of the dkp equation the factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered a method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dkp equation

kato class measures of symmetric markov processes under heat kernel estimates abstract we establish the coincidence of two classes of kato class measures in the framework of symmetric markov processes admitting upper and lower estimates of heat kernel under mild conditions one class of kato class measures is defined by way of the heat kernel another is defined in terms of the green kernel depending on some exponents related to the heat kernel estimates we also prove that p th integrable functions on balls with radius 1 having a uniformity of its norm with respect to centers are of kato class if p is greater than a constant related to the estimate under the same conditions these are complete extensions of some results for the brownian motion on euclidean space by aizenman and simon our result can be applicable to many examples for instance symmetric relativistic stable processes jump processes on d sets brownian motions on riemannian manifolds diffusions on fractals and so on

coisotropic deformations of associative algebras and dispersionless integrable hierarchies the paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies like the dispersionless kp and modified kp hierarchies and the universal whitham hierarchy of genus zero it stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras it discusses the link between the structure constants and hirota s tau function and shows that the dispersionless hirota bilinear equations are within this approach a way of writing the associativity conditions for the structure constants in terms of the tau function it also suggests a simple interpretation of the algebro geometric construction of the universal whitham equations of genus zero due to krichever

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

twistor theory of hyper kahler metrics with hidden symmetries we review the hierarchy for the hyper kahler equations and define a notion of symmetry for solutions of this hierarchy a four dimensional hyper kahler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry it is shown that a hyper kahler metric admits a hidden symmetry if it admits a certain killing spinor we show that if the hidden symmetry is tri holomorphic then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a twistor group action as introduced by bielawski twistor quotients of hyper kahler manifolds world scientific river edge nj 2001 this leads to a construction for the solution to the hierarchy in terms of linear equations and variants of the generalized legendre transform for the hyper kahler metric itself given by ivanov and rocek commun math phys 182 291 1996 we show that the ale spaces are examples of hyper kahler metrics admitting three tri holomorphic killing spinors these metrics are in this sense

path integrals and the essential self adjointness of differential operators on noncompact manifolds we consider schrodinger operators on possibly noncompact riemannian manifolds acting on sections in vector bundles with locally square integrable potentials whose negative part is in the underlying kato class using path integral methods we prove that under geodesic completeness these differential operators are essentially self adjoint on mathsf c infty 0 and that the corresponding operator closures are semibounded from below these results apply to nonrelativistic pauli dirac operators that describe the energy of hydrogen type atoms on riemannian 3 manifolds

on the solutions of the second heavenly and pavlov equations we have recently solved the inverse scattering problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields such as the heavenly equation of plebanski the dispersionless kadomtsev petviashvili dkp equation and the two dimensional dispersionless toda 2ddt equation as well as with the commutation of one dimensional vector fields such as the pavlov equation we also showed that the associated riemann hilbert inverse problems are powerful tools to establish if the solutions of the cauchy problem break at finite time to construct their long time behaviour and characterize classes of implicit solutions in this paper using the above theory we concentrate on the heavenly and pavlov equations i establishing that their localized solutions evolve without breaking unlike the cases of dkp and 2ddt ii constructing the long time behaviour of the solutions of their cauchy problems iii characterizing a distinguished class of implicit solutions of the heavenly equation

self adjointness of schrodinger type operators with locally integrable potentials on manifolds of bounded geometry abstract we consider a schrodinger type differential expression h v v where is a c bounded hermitian connection on a hermitian vector bundle e of bounded geometry over a manifold of bounded geometry m g with metric g and positive c bounded measure d and v lloc1 ende is a linear self adjoint bundle map we define the maximal operator hv max associated to hv as an operator in l2 e given by hv maxu hvu for all u dom h v max u l 2 e vu l loc 1 e h v u l 2 e where u in h v u u vu is understood in distributional sense we give a sufficient condition for the self adjointness of hv max the proof adopts kato s technique to our setting but it requires a more general version of kato s inequality for bochner laplacian operator as well as a result on the positivity of u l2 m satisfying the equation m b u where m is the scalar laplacian on m b 0 is a constant and 0 is a positive distribution on m for local estimates we use a family of cut off functions constructed with the help of regularized distance on manifolds of bounded geometry

on generalized schrodinger semigroups abstract we prove a feynman kac formula for schrodinger type operators on vector bundles over arbitrary riemannian manifolds where the potentials are allowed to have strong singularities like those that typically appear in atomic quantum mechanical problems this path integral formula is then used to prove several l p type results like bounds on the ground state energy and l 2 l p smoothing properties of the corresponding schrodinger semigroups as another main result we will prove that with a little control on the riemannian structure the latter semigroups are also l 2 bounded continuous smoothing for kato decomposable potentials

essential self adjointness for semi bounded magnetic schrodinger operators on non compact manifolds abstract we prove essential self adjointness for semi bounded below magnetic schrodinger operators on complete riemannian manifolds with a given positive smooth measure which is fixed independently of the metric some singularities of the scalar potential are allowed this is an extension of the povzner wienholtz simader theorem the proof uses the scheme of wienholtz but requires a refined invariant integration by parts technique as well as the use of a family of cut off functions which are constructed by a non trivial smoothing procedure due to karcher

the approach to the dispersionless kp hierarchy the dispersionless limit of the scalar nonlocal problem is derived it is given by a special class of nonlinear first order equations a quasiclassical version of the dressing method is presented it is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of beltrami type equations the universal whitham hierarchy and in particular the dispersionless kp hierarchy turn out to be rings of symmetries for the quasiclassical problem pacs number 02 30 uk

the dispersionless 2d toda equation dressing cauchy problem longtime behaviour implicit solutions and wave breaking we have recently solved the inverse spectral problem for one parameter families of vector fields and used this result to construct the formal solution of the cauchy problem for a class of integrable nonlinear partial differential equations in multidimensions including the second heavenly equation of plebanski and the dispersionless kadomtsev petviashvili dkp equation arising as commutation of vector fields in this paper we make use of the above theory i to construct the nonlinear riemann hilbert dressing for the so called two dimensional dispersionless toda equation elucidating the spectral mechanism responsible for wave breaking ii we present the formal solution of the cauchy problem for the wave form of it exp tt xx yy iii we obtain the longtime behaviour of the solutions of such a cauchy problem showing that it is essentially described by the longtime breaking formulae of the dkp solutions confirming the expected universal character of the dkp equation as prototype model in the description of the gradient catastrophe of two dimensional waves iv we finally characterize a class of spectral data allowing one to linearize the rh problem corresponding to a class of implicit solutions of the pde

functions with bounded variation on a class of riemannian manifolds with ricci curvature unbounded from below after establishing some new global facts like a measure theoretic structure theorem and a new invariant characterization of sobolev functions about complex valued functions with bounded variation on arbitrary noncompact riemannian manifolds we extend results of miranda the second author paronetto preunkert and of carbonaro mauceri on the heat semigroup characterization of the variation of mathsf l 1 functions to a class of riemannian manifolds with possibly unbounded from below ricci curvature

multi hamiltonian structure of plebanski s second heavenly equation we show that plebanski s second heavenly equation when written as a first order nonlinear evolutionary system admits multi hamiltonian structure therefore by magri s theorem it is a completely integrable system thus it is an example of a completely integrable system in four dimensions

estimation of a simple linear regression model for fuzzy random variables a generalized simple linear regression statistical probabilistic model in which both input and output data can be fuzzy subsets of r p is dealt with the regression model is based on a fuzzy arithmetic approach and it considers the possibility of fuzzy valued random errors specifically the least squares estimation problem in terms of a versatile metric is addressed the solutions are established in terms of the moments of the involved random elements by employing the concept of support function of a fuzzy set some considerations concerning the applicability of the model are made

essential selfadjointness of singular magnetic schrodinger operators on riemannian manifolds in this paper we extend the well known leinfelder simader theorem on the essential selfadjointness of singular schrodinger operators to arbitrary complete riemannian manifolds this improves some earlier results of shubin milatovic and others

integrable structure of the dirichlet boundary problem in multiply connected domains we study the integrable structure of the dirichlet boundary problem in two dimensions and extend the approach to the case of planar multiply connected domains the solution to the dirichlet boundary problem in the multiply connected case is given through a quasiclassical tau function which generalizes the tau function of the dispersionless toda hierarchy it is shown to obey an infinite hierarchy of hirota like equations which directly follow from properties of the dirichlet green function and from the fay identities the relation to multi support solutions of matrix models is briefly discussed

bubble break off in hele shaw flows singularities and integrable structures abstract bubbles of inviscid fluid surrounded by a viscous fluid in a hele shaw cell can merge and break off during the process of break off a thinning neck pinches off to a universal self similar singularity we describe this process and reveal its integrable structure it is a solution of the dispersionless limit of the akns hierarchy the singular break off patterns are universal not sensitive to details of the process and can be seen experimentally we briefly discuss the dispersive regularization of the hele shaw problem and the emergence of the painleve ii equation at the break off

heat equation derivative formulas for vector bundles we use martingale methods to give bismut type derivative formulas for differentials and co differentials of heat semigroups on forms and more generally for sections of vector bundles the formulas are mainly in terms of weitzenbock curvature terms in most cases derivatives of the curvature are not involved in particular our results improve b k driver s formula in 1997 j math pures appl 9 76 703 737 for logarithmic derivatives of the heat kernel measure on a riemannian manifold our formulas also include the formulas of k d elworthy and x m li 1998 c r acad sci paris ser i math 327 87 92

on the geometry of semiclassical limits on dirichlet spaces abstract this paper is a contribution to semiclassical analysis for abstract schrodinger type operators on locally compact spaces let x be a metrizable separable locally compact space let be a radon measure on x with a full support let t x y p t x y be a strictly positive pointwise consistent heat kernel and assume that the generator h p 0 of the corresponding self adjoint contraction semigroup in l 2 x induces a regular dirichlet form then given a function 0 1 0 such that the limit lim t 0 p t x x t exists for all x x we prove that for every potential w x r one has lim t 0 t tr e t h p w e w x lim t 0 p t x x t d x for the schrodinger type operator h p w provided w satisfies very mild conditions at that are essentially only made to guarantee that the sum of quadratic forms h p w t is self adjoint and bounded from below for small t and to guarantee that e w x lim t 0 p t x x t d x the proof is probabilistic and relies on a principle of not feeling the boundary for p t x x in particular this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete riemannian manifolds and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs