Quadratic variation of functionals of two-parameter Wiener process

The quadratic variation of functionals of the two-parameter Wiener process of the form f(W(s, t)) is investigated, where W(s, t) is the standard two-parameter Wiener process and f is a function on the reals. The existence of the quadratic variation is obtained under the condition that f′ is locally absolutely continuous and f^N is locally square integrable.     

Invarianti e ipoellitticità per una classe di operatori pseudodifferenziali

Riassunto In questa nota si introduce un invariante, mediante il quale si danno condizioni sufficienti di ipoellitticità e propagazione delle singolarità per operatori pseudo-differenziali, degeneri di certi ordini ≥2 su una varietà chiusa conica involutiva di*R ^R. Si utilizzano risultati ottenuti per operatori quasi-omogenei [5], [7], [8], [9], provando che, almeno microlocalmente, gli operatori considerati sono di questo tipo.

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Summary In this paper we shall introduce a class of pseudodifferential operators for which we define an invariant and we obtain sufficient conditions for the hypoellipticity and propagation of singularities. Our approach here is to consider the operators as P.D.O. which are anisotropic (quasi-homogeneous) with respect to some weight: for the definitions and the relevant properties of quasi-homogeneous P.D.O. we refer to [5] or [7].

     

Quadratic Covariation and Itô's Formula for Smooth Nondegenerate Martingales

In this paper we prove the existence of the quadratic covariation [f(X),X], where f is a locally square integrable function and X _t = ^t ₀ u _s dW _s is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus.     

Extensions of Meyers-Ziemer results

Letp∈(1, +∞) ands ∈ (0, +∞) be two real numbers, and letH _p ^s (ℝ ⁿ ) denote the Sobolev space defined with Bessel potentials. We give a classA of operators, such thatB _s,p -almost all points ℝ ⁿ are Lebesgue points ofT(f), for allf ∈H _p ^s (ℝ ⁿ ) and allT ∈A (B _s,p denotes the Bessel capacity); this extends the result of Bagby and Ziemer (cf. [2], [15]) and Bojarski-Hajlasz [4], valid wheneverT is the identity operator. Furthermore, we describe an interesting special subclassC ofA (C contains the Hardy-Littlewood maximal operator, Littlewood-Paley square functions and the absolute value operatorT: f→|f|) such that, for everyf ∈H _p ^s (ℝ ⁿ ) and everyT ∈C, T(f) is quasiuniformly continuous in ℝ ⁿ ; this yields an improvement of the Meyers result [10] which asserts that everyf ∈H _p ^s (ℝ ⁿ ) is quasicontinuous. However,T (f) does not belong, in general, toH _p ^s (ℝ ⁿ ) wheneverT ∈C ands≥1+1/p (cf. Bourdaud-Kateb [5] or Korry [7]).     

A remark on random and equidistributed sequences

The average of the values of a function f on the points of an equidistributed sequence in [0, 1] ^s converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions.     

An extension of Alexandrov?s theorem on second derivatives of convex functions

If f is a function of n variables that is locally L¹ approximable by a sequence of smooth functions satisfying local L¹ bounds on the determinants of the minors of the Hessian, then f admits a second order Taylor expansion almost everywhere. This extends a classical theorem of A.D. Alexandrov, covering the special case in which f is locally convex.     

Subconvexity for Rankin-Selberg L-Functions of Maass Forms

In this paper we prove a subconvexity bound for Rankin–Selberg L-functions associated with a Maass cusp form f and a fixed cusp form g in the aspect of the Laplace eigenvalue 1/4 + k² of f, on the critical line Re s = 1/2. Using this subconvexity bound, we prove the equidistribution conjecture of Rudnick and Sarnak [RS] on quantum unique ergodicity for dihedral Maass forms, following the work of Sarnak [S2] and Watson [W]. Also proved here is that the generalized Lindelöf hypothesis for the central value of our L-function is true on average.     

An Error Expansion for some Gauss–Turán Quadratures and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-Estimates of the Remainder Term

Our aim in this paper is to obtain error expansions in the Gauss–Turán quadrature formula ∫₋₁¹f(t)w(t) dt=∑_ν=1ⁿ∑_i=0^2sA_i,νf⁽ⁱ⁾(τ_ν)+R_n,s(f), in the case when f is an analytic function in some region of the complex plane containing the interval [−1,1] in its interior. Using a representation of the remainder term R_n,s(f) in the form of contour integral over confocal ellipses, we obtain R_n,1(f) for the four Chebyshev weights and R_n,2(f) for the Chebyshev weight of the first kind. Also, we get a few new L¹-estimates of the remainder term, which are stronger than the previous ones. Some numerical results, illustrations and comparisons are also given. AMS subject classification (2000) 41A55, 65D30, 65D32.Received January 2004. Accepted October 2004. Communicated by Lothar Reichel.M. M. Spalević: This work was supported in part by the Serbian Ministry of Science and Environmental Protection (Project: Applied Orthogonal Systems, Constructive Approximation and Numerical Methods, grant number 2002).     

On the existence of convex decompositions of partially separable functions

The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsf _i whose Hessians have nontrivial nullspacesN _i , Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionf _i is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf _i such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureN _i ¹ have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachN _i is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDL^T factorization without fill-in.     

Sur les équations fonctionnelles aux itérées

Summary We study solutions of functional equationsP(f ^[10] ,,f ^[s] ) = 0, whereP is a non zero polynomial ins + 1 variables andf ^[k] denotes thekth iterate of a functionf. We deal with three distinct cases: first,f is an entire function of a complex variable, we show then thatf is a polynomial. Second, we also prove thatf is a polynomial if it is an entire function of ap-adic variable. Third, we considerf a formal power series with coefficients in a number fieldK; subject to some apparently natural restrictions onf and onP, we find thatf is an algebraic power series over the ring of polynomials inK[x].

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Sur les équations fonctionnelles aux itérées

     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

An Extension Of The Ruzsa-Szemerédi Theorem

  We let G ^(r)(n,m) denote the set of r-uniform hypergraphs with n vertices and m edges, and f ^(r)(n,p,s) is the smallest m such that every member of G ^(r)(n,m) contains amember of G ^(r)(p,s). In this paper we are interested in fixed values r,p and s for which f ^(r)(n,p,s) grows quadratically with n. A probabilistic construction of Brown, Erds and T. Sós ([2]) implies that f ^(r)(n,s(r-2)+2,s)=(n ²). In the other direction the most interesting question they could not settle was whether f ⁽³⁾(n,6, 3) = o(n ²). This was proved by Ruzsa and Szemerédi [11]. Then Erds, Frankl and Rödl [6] extended this result to any r: f ^(r)(n, 3(r-2)+3, 3)=o(n ²), and they conjectured ([4], [6]) that the Brown, Erds and T. Sós bound is best possible in the sense that f ^(r)(n,s(r-2)+3,s)=o(n ²).In this paper by giving an extension of the Erds, Frankl, Rödl Theorem (and thus the Ruzsa–Szemerédi Theorem) we show that indeed the Brown, Erds, T. Sós Theorem is not far from being best possible. Our main result is

Globalwell-posedness and I method for the fifth order Korteweg-de Vries equation

The Kawahara equation has fewer symmetries than the KdV equation; in particular, it has no invariant scaling transform and is not completely integrable. Thus its analysis requires different methods. We prove that the Kawahara equation is locally well posed in H ^−7/4, using the ideas of an [`(F)] ^s{\overline F ^s}-type space [8]. Then we show that the equation is globally well posed in H ^s for s ≥ −7/4, using the ideas of the “I-method” [7].     

Separating sets by Darboux Baire 1 and approximately continuous functions

We characterize sets A₀, A₁ for which there is a DB1 function f with [f = 0] = A₀ and [f = 1] = A₁. This characterization is a conjunction of necessary conditions for Darboux and for Baire 1 functions. We also characterize sets A^?, A⁺ for which there is a DB₁ function with [f < 0] = A^? and [f > 0] = A⁺. The same characterzations are provided for approximately continuous functions.     

Construction of Continuous Functions with Prescribed Local Regularity

In this paper we investigate from both a theoretical and a practical point of view the following problem: Let s be a function from [0;1] to [0;1] . Under which conditions does there exist a continuous function f from [0;1] to R such that the regularity of f at x , measured in terms of H?lder exponent, is exactly s(x) , for all x ∈ [0;1] ? We obtain a necessary and sufficient condition on s and give three constructions of the associated function f . We also examine some extensions regarding, for instance, the box or Tricot dimension or the multifractal spectrum. Finally, we present a result on the ``size' of the set of functions with prescribed local regularity. November 30, 1995. Date revised: September 30, 1996.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

Locally Compact Path Spaces

It is shown that the space X^[0,1], of continuous maps [0,1]X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T₁-space X, it follows that X^[0,1] is locally compact if and only if X is locally compact and totally path-disconnected. Mathematics Subject Classifications (2000) 54C35, 54E45, 55P35, 18B30, 18D15.     

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f _s,t (n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K _s }}, where the minimum is taken over all K _t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n ^1/2+o(1)) ≤ f _s,s+1(n) ≤ O(n ^1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f _s,s+1(n) ≤ O(n ^2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n ^1/2−ɛ ) ≤ f _s,s+k (n) ≤ O(n ^1/2+ɛ . In addition, we also discuss some connections between the function f _s,t and vertex Folkman numbers.     

Algebraic Independence by Approximation Method (II)

Let f (x) be a continued fraction with elements a _n x, where coefficients a _n are positive algebraic numbers. Using the criterion of [l] for any nonzero real algebraic numbers α₁,...,α_s with distinct absolute values the algebraic independence of the values f(α₁), ..., f(α_s) is proved under certain assumption concerning only with a _n . For some transcendental numbers ξ the algebraic independence of values f(ξ^j)(j∈ℤ) is also established. Received March 27, 1998, Accepted September 28, 1998