on the analyticity and the almost periodicity of the solution to the euler equations with non decaying initial velocity abstract the cauchy problem of the euler equations in the whole space is considered with non decaying initial velocity in the frame work of b 1 1 it is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables furthermore a new estimate for the size of the radius of convergence of taylor s expansion is established the key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms it is also shown the propagation of the almost periodicity in spatial variables

nonparametric tests of the markov model for survival data summary jones crowley 1989 introduced a general class of nonparametric statistics for right censored survival data that includes many of the existing procedures involving a single covariate this paper considers this class in the framework of a markov model and suggests some tests especially appropriate for the markov setting application to left truncated and right censored data is also given small sample properties and asymptotic relative efficiencies for several statistics of this class are investigated for both relative and excess risk models for the hazard

localization and diagonalization a review of functional integral techniques for low dimensional gauge theories and topological field theories localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low dimensional gauge theories are reviewed these are the functional integral counterparts of the mathai quillen formalism the duistermaat heckman theorem and the weyl integral formula respectively in each case the necessary mathematical background euler classes of vector bundles equivariant cohomology topology of lie groups is introduced and the finite dimensional integration formulae described then some applications to path integrals are discussed and an overview of the relevant literature is given the applications include supersymmetric quantum mechanics cohomological field theories phase space path integrals and two dimensional yang mills theory

asymptotically simple solutions of the vacuum einstein equations in even dimensions we show that a set of conformally invariant equations derived from the fefferman graham tensor can be used to construct global solutions of vacuum einstein equations in all even dimensions this gives in particular a new simple proof of friedrich s result on the future hyperboloidal stability of minkowski space time and extends its validity to even dimensions

radius of analyticity for the camassa holm equation on the line abstract using estimates in sobolev spaces we prove that the solution to the cauchy problem for the camassa holm equation on the line with analytic initial data u 0 x and satisfying the mckean condition that is the quantity m 0 x 1 x 2 u 0 x does not change sign is analytic in the spatial variable for all time furthermore we obtain explicit lower bounds for the radius of spatial analyticity r t given by r t a 1 1 c 1 b t 1 exp c 0 u 0 h 1 t where a b c 1 and c 0 are suitable positive constants

representation of the propagator and schwinger functions of dirac fields in terms of brownian motions a mathematical path integral representation of the propagator and schwinger functions of the dirac fields on r4 interacting with a smooth non abelian gauge field is presented the representation is essentially in terms of expectations with respect to the brownian motion on r4

propagation de l analyticite des solutions de systemes hyperboliques non lineaires on montre que si l on se place dans un domaine ou la solution est a priori assez reguliere la regularite analytique s etend des donnees de cauchy a la solution selon la geometrie des domaines d influence de l equation comme dans le cas lineaire en particulier si les donnees de cauchy sont analytiques une solution est analytique jusqu au premier choc

conditioning and initial enlargement of filtration on a riemannian manifold we extend to riemannian manifolds the theory of conditioned stochastic differential equations we also provide some enlargement formulas for the brownian filtration in this nonflat setting

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

on the domain of analyticity of solutions to semilinear klein gordon equations abstract we consider initial value problems for semilinear klein gordon equations u t t x u u f u 0 with periodic boundary conditions assuming that both the initial data and the nonlinear forcing term f u are analytic we provide explicit lower bounds on the decay of the radius t of analyticity of the solutions as a function of time in particular in one space dimension with u real valued and f u u 2 k 1 we prove that the decay of t is not worse than 1 t the results are given in a general framework including gevrey class solutions

the atiyah singer theorems a probabilistic approach ii the lefschetz fixed point formulas abstract in this paper the lefschetz fixed point formulas of m f atiyah and i m singer ann of math 87 1968 484 530 546 604 for classical elliptic complexes are proved by using probabilistic methods

analytic regularity of uniformly stable shock fronts with analytic data on considere un systeme hyperbolique de lois de conservation u t j 1 f j u x j o ou x x 1 x n appartient a un sous ensemble ouvert de r n t o et u t u 1 u m f u f 1 u f n u est une application c d un sous ensemble ouvert g de r m dans r m n

pseudodifferential operators on supermanifolds and the atiyah singer index theorem fermionic quantization or clifford algebra is combined with pseudodifferential operators to simplify the proof of the atiyah singer index theorem for the dirac operator on a spin manifold

the infinitesimal lefschetz formulas a heat equation proof abstract in this paper the lefschetz formulas of atiyah and bott ann of math 86 1967 374 407 88 1968 451 491 and atiyah and singer ann of math 87 1968 546 604 93 1971 119 138 are proved by heat equation methods in infinitesimal form i e without using the usual localization on the fixed point set the proof is obtained by a homotopic deformation of the dirac operator and uses probabilistic methods the main applications are potentially in noncompact situations where fixed point formulas do not always hold while their infinitesimal analogues are still valid

on the lagrangian and eulerian analyticity for the euler equations abstract we address preservation of the lagrangian analyticity radius of solutions to the euler equations belonging to natural analytic space based on the size of taylor or gevrey coefficients we prove that if the solution belongs to such space then the solution also belongs to it for a positive amount of time we also prove the local analog of this result for a sufficiently large gevrey parameter however we show that the preservation holds independently of the size of the radius finally we construct a solution which shows that the first result does not hold in the eulerian setting

well posedness of the good boussinesq equation in analytic gevrey spaces and time regularity abstract the cauchy problem for the good boussinesq equation with data in analytic gevrey spaces on the line and the circle is considered and its local well posedness in these spaces is proved the proof is based on bilinear estimates in bourgain type spaces incorporating the symbol of the linear part of the equation and an exponential weight expressing the analytic gevrey regularity of the solution in the spatial variable also gevrey regularity of the solution in time variable is provided

on the radius of spatial analyticity for semilinear symmetric hyperbolic systems abstract we study the problem of propagation of analytic regularity for semi linear symmetric hyperbolic systems we adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius width e 0 the same happens for the solution u t for a certain radius e t as long as the solution exists our focus is on precise lower bounds on the spatial radius of analyticity e t as t grows we also get similar results for the schrodinger equation with a real analytic electromagnetic potential

covariate adjustment for non parametric tests for censored survival data we develop a method for covariate adjustment for a general class of non parametric tests for censored survival data which includes the widely used logrank and wilcoxon tests the covariate adjusted tests improve the power of the unadjusted counterparts and have advantages over the covariate adjusted cox score test when there are outliers in the covariable space we investigate the small sample properties of such test statistics through monte carlo simulations examples are given to illustrate the proposed procedures

stochastic multiplicative measures generalized markov semigroups and group valued stochastic processes and fields abstract we extend the well known relation between markov semigroups and processes with independent increments to the case of generalized group valued semigroups indexed by some measurable space and stochastic multiplicative measures we give a kolmogorov type theorem and a classification theorem connecting the stochastic multiplicative measures with vector valued levy khinchine fields the results have applications to the study of stochastic connections and differential equations for forms on lie groups

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

a higher dispersion kdv equation on the line abstract the cauchy problem for a korteweg devries equation with dispersion of order m 2 j 1 where j is a positive integer kdvm is studied with data in sobolev and analytic spaces first optimal bilinear estimates in bourgain spaces are proved and using them well posedness in sobolev spaces h s s j 1 4 is established then well posedness in analytic gevrey spaces g s 0 is proved by using an analytic version of the bilinear estimates this implies that the uniform radius of analyticity persist for some time for the later times a lower bound for the radius of spacial analyticity is derived which is given by t c t with 4 3 e for any e 0 when j 1 and 1 when j 2 finally it is shown that the regularity of the solution in the time variable is gevrey of order m and this is optimal

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

the versatility of function indexed weighted log rank statistics abstract two sample weighted log rank statistics are used in the presence of right censoring to test whether failure times from two populations have different survival distributions kosorok has showed that large families of these statistics form stochastic processes indexed by weight functions and that these function indexed statistics can be used to construct versatile test procedures simultaneously sensitive to a wide array of both ordered hazards and stochastic ordering alternatives the complexity of the asymptotic distribution of these statistics precludes obtaining p values through analytical means in this article we develop a monte carlo method for accurately obtaining these p values and we evaluate the moderate sample size properties of this method and compare the power of function indexed statistics with previously developed weighted log rank tests these statistics are also examined in a data analysis of the beta blocker heart attack trial bhat the results of this article demonstrate that

well posedness and regularity of the generalized burgers equation in periodic gevrey spaces we consider the generalized burgers equation in a class of gevrey functions g s t we show that the generalized burgers equation is well posed in this space furthermore we show that the solution is gevrey in the spacial variable and gevrey 2 in the time variable

stochastic path ordered exponentials we prove convergence of an approximation of the stochastic product integral for conditional wiener paths to the solution of a certain stochastic integral equation this is used to establish the wiener ito representation for the kernel of the semigroup exp t a where a 1 a 2 for functions a with values in the space of anti hermitian matrices

a stochastic approach to the poincare hopf theorem in this paper the poincare hopf theorem hopf 8 is proved by a probabilistic method the proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution we use the malliavian calculus to estimate the fundametal solution our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions

asymptotic properties of a general class of nonparametric tests for survival analysis many of the popular nonparametric test statistics for censored survival data used in two sample s sample trend and single continuous covariate situations are special cases of a general statistic differing only in the choice of covariate based label and weight function formulated in terms of counting processes and martingales this general statistic standardized by the square root of its consistent variance estimator is shown to be asymptotically normal under the null hypothesis and under a sequence of contiguous hazard alternatives that includes both relative and excess risk models

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

generalized martingale residual processes for goodness of fit inference in cox s type regression models in the paper a general class of stochastic processes based on the sums of weighted martingale transform residuals for goodness of fit inference in general cox s type regression models is studied their form makes the inference robust to covariate outliers a weak convergence result for such processes is obtained giving the possibility of establishing the randomness of their graphs together with the construction of the formal x 2 type goodness of fit tests by using the khmaladze innovation approach a modified version of the initial class of processes is also defined weak convergence results for the processes are derived this leads to the main application which concerns the formal construction of the kolmogorov smirnov and cramer von mises type goodness of fit tests this is done within the general situation considered

demailly s asymptotic morse inequalities a heat equation proof on donne une demonstration qui s appuie sur l equation de la chaleur pour les inegalites asymptotiques de demailly pour le complexe

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

stochastic areas winding numbers and hopf fibrations we define and study stochastic areas processes associated with brownian motions on the complex symmetric spaces mathbb cp n and mathbb ch n the characteristic functions of those processes are computed and limit theorems are obtained in the case n 1 we also study windings of the brownian motion on those spaces and compute the limit distributions for mathbb cp n the geometry of the hopf fibration plays a central role whereas for mathbb ch n it is the anti de sitter fibration

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

on the radius of spatial analyticity for the 1d dirac klein gordon equations abstract we study the well posedness of the dirac klein gordon system in one space dimension with initial data that have an analytic extension to a strip around the real axis it is proved that the radius of analyticity t of the solutions at time t cannot decay faster than 1 t 4 as t

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

the entropy production of diffusion processes on manifolds and its circulation decompositions in non equilibrium statistical mechanics the entropy production is used to describe flowing in or pumping out of the entropy of a time dependent system even if a system is in a steady state invariant in time prigogine suggested that there should be a positive entropy production if it is open in 1979 the first author of this paper and qian min ping discovered that the entropy production describes the irreversibility of stationary markov chains and proved the circulation decomposition formula of the entropy production they also obtained the entropy production formula for drifted brownian motions on euclidean space r n see a report without proof in the proc 1st world congr bernoulli soc by the topological triviality of r n there is no discrete circulation associated to the diffusion processes on r n in this paper the entropy production formula for stationary drifted brownian motions on a compact riemannian manifold m is proved furthermore the entropy production is decomposed into two parts in addition to the first part analogous to that of a diffusion process on r n some discrete circulations intrinsic to the topology of m appear the first part is called the hidden circulation and is then explained as the circulation of a lifted process on m s 1 around the circle s 1 the main result of this paper is the circulation decomposition formula which states that the entropy production of a stationary drifted brownian motion on m is a linear sum of its circulations around the generators of the fundamental group of m and the hidden circulation

gevrey regularity of the periodic gkdv equation abstract using the multilinear estimates which were derived for proving well posedness of the generalized korteweg de vries gkdv equation it is shown that if the initial data belongs to gevrey space g 1 in the space variable then the solution to the corresponding cauchy problem for gkdv belongs also to g in the space variable moreover the solution is not necessarily g in the time variable however it belongs to g 3 near 0 when 1 these are analytic regularity results for gkdv

a feynman kac formula for the quantum heisenberg ferromagnet i this article continues the analysis of the first arcticle under the same title using methods of stochastic analysis we prove feynman kac formulas for the relevant heat kernels we also present classical limit theorems

supersymmetric path integrals the supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the wiener integral it is then pushed forward to a compact riemannian manifold by means of a malliavin type construction the relation to index theory is discussed

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

localization formulas superconnections and the index theorem for families in this paper we give a new proof of the localization formulas of berline and vergne 9 and duistermaat and heckman 18 when interpreted in the framework of atiyah 2 the probabilistic heat equation proof of the index theorem given in our paper 12 appears as the rigorous infinite dimensional version of this new proof of the localization formulas in finite dimensions the results of quillen 25 on superconnections are briefly presented the heat equation proofs 15 of the index theorem for families are described it is shown that in this framework the superconnections formalism is the operator theoretic description of integration along the fiber in the loop space

a generalized formula of ito and some other properties of stochastic flows a stochastic differential equation with smooth coefficients is considered which defines a continuous flow t of c 8 mappings of rd in rd if zt is a continuous semi martingale t zt s is proved to be a semi martingale for which an ito type formula is established it is shown that a s for any t t is an onto diffeomorphism if zt is a continuous semi martingale t 1 zt is proved to be a semi martingale whose ito decomposition is explicitly found