a tree approach to p variation and to integration we consider a real valued path it is possible to associate a tree to this path and we explore the relations between the tree the properties of p variation of the path and integration with respect to the path in particular the fractal dimension of the tree is estimated from the variations of the path and young integrals with respect to the path as well as integrals from the rough paths theory are written as integrals on the tree examples include some stochastic paths such as martingales levy processes and fractional brownian motions for which an estimator of the hurst parameter is given

weighted variation inequalities for differential operators and singular integrals abstract we prove weighted strong q variation inequalities with 2 q for differential and singular integral operators for the first family of operators the weights used can be either sawyer s one sided a p weights or muckenhoupt s a p weights according to whether the differential operators in consideration are one sided or symmetric we use only muckenhoupt s a p weights for the second family all these inequalities hold equally in the vector valued case that is for functions with values in l for 1 as application we show variation inequalities for mean bounded positive invertible operators on l p with positive inverses

ell p left mathbb z d right estimates for discrete operators of radon type variational estimates we prove ell p left mathbb z d right bounds for p in 1 infty of r variations r in 2 infty for discrete averaging operators and truncated singular integrals of radon type we shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of p and r which coincide with the ranges of their continuous counterparts

on bernstein approximants and the f variation of a fuzzy random variable abstract in this paper we study different properties of bernstein approximants of a fuzzy random variable as those in relation with the approximation of fuzzy random variables by means of its n bernstein approximant moreover we define and study the concept of variation for a fuzzy random variable

the strong p variation of martingales and orthogonal series let 1 p and letx x n n 0 be a sequence of scalars the strongp variation ofx denoted byw p x is defined as w p x sup left left x 0 p sum limits k 1 infty x n k x n k 1 p right 1 p right where the supremum runs over all increasing sequences of integers 0 n 0 n 1 n 2 let 1 p 2 and letm m n n 0 be a martingale inl p our main results are as follows if sigma mathbb e m n m n 1 p infty thenw p m is finite a s and we have mathbb e w p m p leqq c mathbb e m 0 p sum limits n geqq 1 mathbb e m n m n 1 p for some constantc depending only onp on the other hand let n be an arbitrary orthonormal system of functions inl 2 considerx x n n 0 inl 2 and lets n 0 x i i ands s n n 0 we prove that if x n p 1 p 2 thenw p s t for a e t and w p s 2 c x n p 1 p for some constantc each of these results is an extension of a result proved by bretagnolle for sums of independent mean zero r v s the casep 2 in also discussed our proofs use the real interpolation method of lions peetre they admit extensions in the banach space valued case provided suitable assumptions are imposed on the banach space

a new criterion and a special class of k positive maps abstract we study k positive maps on operators we obtain a new criterion on k positivity in terms of the k numerical range and use it to improve and refine some earlier results on k positive maps related to the study of quantum information science we also consider a special class of positive maps extending the construction of choi on positive maps that are not completely positive some open questions on the decomposability of such maps are answered

frechet differentiability p variation and uniform donsker classes classically a stastistical functional is defined on a space of distribution f on the real line with the supremum norn the values may be real or themselves functions such as the quantile function f 1 nonlinear functionals are studied via their derivatives this paper and a related one dudley 1991a will show that the differentiability properties originally proved by reeds 1976 also treated in fernholz 1983 can be improved substantially

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

the p variation of partial sum processes and the empirical process the p variation of a function f is the supremum of the sums of the p th powers of absolute increments of f over nonoverlapping intervals let f be a continuous probability distribution function dudley has shown that the p variation of the empirical process is bounded in probability as n to infty if and only if p 2 and for 1 leq p leq 2 the p variation of the empirical process is at least n 1 p 2 and is at most of the order n 1 p 2 log log n p 2 in probability in this paper we prove that the exact order of the 2 variation of the empirical process is log log n in probability and for 1 leq p 2 the p variation of the empirical process is of exact order n 1 p 2 in expectation and almost surely let s j x 1 x 2 dots x j then the p variation of the partial sum process for x 1 x 2 dots x n is defined as that of f on 0 n where f t s j for j 1 t leq j j 1 2 dots n bretagnolle has shown that the expectation of the p variation for independent centered random variables x i with bounded p th moments is of order n for 1 leq p 2 we prove that for p 2 the 2 variation of the partial sum process of i i d centered nonconstant random variables with finite 2 delta moment for some delta 0 is of exact order n log log n in probability

weighted jump and variational inequalities for rough operators abstract in this paper we systematically study weighted jump and variational inequalities for rough operators more precisely we show some weighted jump and variational inequalities for the families t t e e 0 of truncated singular integrals and m m t t 0 of averaging operators with rough kernels which are defined respectively by t e f x y e y y n f x y d y and m t f x 1 t n y t y f x y d y where the kernel belongs to l q s n 1 for q 1

a unified approach to operator monotone functions abstract the notion of operator monotonicity dates back to a work by lowner in 1934 a map f s n s m is called operator monotone if a b implies f a f b here s n is the space of symmetric matrices with the semidefinite partial order often the function f is defined in terms of an underlying simpler function f of main interest is to find the properties of f that characterize operator monotonicity of f in that case it is said that f is also operator monotone classical examples are the lowner operators and the spectral scalar valued isotropic functions operator monotonicity for these two classes of functions is characterized in seemingly very different ways this work extends the notion of operator monotonicity to symmetric functions f on k arguments the latter is used to define generated k isotropic maps f s n s n k for any n k necessary and sufficient conditions are given for f to generate an operator monotone k isotropic map f when k 1 the k isotropic map becomes a lowner operator and when k n it becomes a spectral function this allows us to reconcile and explain the differences between the conditions for monotonicity for the lowner operators and the spectral functions

vector valued q variation for ergodic averages and analytic semigroups abstract in this paper we establish umd lattice valued variational inequalities for ergodic averages of contractively regular operators and analytic semigroups these results generalize on the one hand some scalar valued variational inequalities in ergodic theory on the other hand xu s very recent result on umd lattice valued maximal inequality as an application we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence for semigroups acting on umd lattice valued functions

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

superdiffusive limits for deterministic fast slow dynamical systems we consider deterministic fast slow dynamical systems on mathbb r m times y of the form begin aligned left begin array ll x k 1 n x k n n 1 a big x k n big n 1 alpha b big x k n big v y k y k 1 f y k end array right end aligned where alpha in 1 2 under certain assumptions we prove convergence of the m dimensional process x n t x lfloor nt rfloor n to the solution of the stochastic differential equation begin aligned mathrm d x a x mathrm d t b x diamond mathrm d l alpha end aligned where l alpha is an alpha stable levy process and diamond indicates that the stochastic integral is in the marcus sense in addition we show that our assumptions are satisfied for intermittent maps f of pomeau manneville type

vector valued q variation for differential operators and semigroups i in this paper we establish mathcal b valued variational inequalities for differential operators ergodic averages and symmetric diffusion semigroups under the condition that banach space mathcal b has martingale cotype property these results generalize on the one hand pisier and xu s result on the variational inequalities for mathcal b valued martingales on the other hand many classical variational inequalities in harmonic analysis and ergodic theory moreover we show that rademacher cotype q is necessary for the mathcal b valued q variational inequalities as applications of the variational inequalities we deduce the jump estimates and obtain quantitative information on the rate of convergence it turns out the rate of convergence depends upon the geometry of the banach space under consideration while cowling and leinert have shown that convergence holds for all separable banach spaces

jump inequalities via real interpolation jump inequalities are the r 2 endpoint of lepingle s inequality for r variation of martingales extending earlier work by pisier and xu probab theory relat fields 77 4 497 514 1988 we interpret these inequalities in terms of banach spaces which are real interpolation spaces this interpretation is used to prove endpoint jump estimates for vector valued martingales and doubly stochastic operators as well as to pass via sampling from mathbb r d to mathbb z d for jump estimates for fourier multipliers

variational estimates for the bilinear iterated fourier integral abstract we prove pointwise variational l p bounds for a bilinear fourier integral operator in a large but not necessarily sharp range of exponents this result is a joint strengthening of the corresponding bounds for the classical carleson operator the bilinear hilbert transform the variation norm carleson operator and the bi carleson operator terry lyon s rough path theory allows for extension of our result to multilinear estimates we consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations

a perturbation of a periodic hamiltonian system in this paper we examine a hamiltonian system with a superquadratic term that is of the form u u f t v u where v u behaves like a power greater than 2 of u we look for solutions homoclinic to zero or just homoclinic that is solutions u with u t u t 0 as t such solutions can be difficult to find due to the lack of compactness of the domain and the subsequent failure of the palaissmale condition in variational formulations of the problem see 1 for an example of such a problem with no homoclinic solutions the existence of such solutions depends delicately on f and v it is known that they exist when f is periodic we will examine a system in which f is a perturbation of a periodic function assuming reasonably weak conditions on the periodic problem we show that the perturbed problem has a solution if the perturbation is small enough the conditions required of the periodic problem can be verified for some specific examples we employ variational mountain pass techniques the result almost extends to an analogous elliptic partial differential equation but subtle differences in the topology of r and rn n 2 prevent a complete extension nevertheless we find that for a large class of nearly autonomous elliptic pde there exists a nonzero solution decaying at infinity consider the following system containing a periodic term

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

derivatives of compound matrix valued functions abstract primary matrix functions and spectral functions are two classes of orthogonally invariant functions on a symmetric matrix argument many of their properties have been investigated thoroughly and find numerous applications both theoretical and applied in areas ranging from engineering image processing optimization and physics we propose a family of maps that provide a natural connection and generalization of these two classes of functions the family of maps also contains the well known multiplicative and additive compound matrices we explain when each member of this family is a differentiable function and exhibit a formula for its derivative

jump estimates for operators associated with polynomials abstract in this study we prove the jump inequalities for singular integrals and averages of radon type with rough kernels