Formule d'Itô pour des Diffusions Uniformément Elliptiques,et Processus de Dirichlet

Soit X une diffusion uniformément elliptique sur R ^d ,F une fonction dans H _loc ¹(R ^d ) et la loi initiale de la diffusion. On montre que si l'intégrale |F|²(x)U(x)dx est finie, oùU désigne le potentiel de la mesure , alors F(X) est un processus de Dirichlet. Si de plus, F appartient àH ² _loc(R ^d ) et si les intégrales |F|²(x)U(x)dx et |f _k |²(x)U(x)dx sont finies, pour les dérivées faibles f _k de F, alors on peut écrire une formule d'Itô. En particulier, on définit l'intégrale progressive F(X)dX et on prouve l'existence des covariations quadratiques [f _k (X),X ^k ].     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .     

Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf: ^p × ^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y ^q }. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx ₀ in terms of those off at (x ₀,y ₀), wherey ₀ is any element for which (x ₀)=f(x ₀,y ₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.     

The intuitionistic propositional calculus with quantifiers

Let L be the language of the intuitionistic propositional calculus J completed by the quantifiers and , and let calculus 2J in language L contain, besides the axioms of J, the axioms xB (x) B(y) and B(y) xB (x). A Kripke semantics is constructed for 2J and a completeness theorem is proven. A result of D. Gabbay is generalized concerning the undecidability of C2J⁺-extension of 2J by schemes x (x B) and x(A B(x))A xB (x) specificially: the undecidability is proven of each T theory in language L such that [2J]T [C2J+] ([2J] ([2J] denotes the set of all theorems of calculus 2J).Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 69–76, July, 1977.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

A remark on generalized iterates

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf ^|n| denote then'th iterate off. Clearlyf ^|m|(f ^|n|(x))=f ^|m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F ^||:G} on the real line with the properties (i)F ^||(F ^||(x))=F ^|+| (x) for allxR and (ii)F ^|n|(x)=f ^|n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday     

Positive definite dot product kernels in learning theory

In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=_l=0^Na_l(xy)^l generating polynomials of degree N, and dot product kernels K(x,y)=_l=0⁺a_l(xy)^l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xⁱx^j)]_i,j=1^m is positive semi-definite for any x¹,x²,...,x^mRⁿ, n2. Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

Some analytical properties of γ-convex functions on the real line

This paper deals with the analytical properties of -convex functions, which are defined as those functions satisfying the inequalityf(x ₁ )+f(x ₂ )f(x ₁)+f(x ₂), forx _i [x ₁,x ₂], |x _i –x _i |=, i=1,2, whenever |x ₁–x ₂|>, for some given positive . This class contains all convex functions and all periodic functions with period . In general, -convex functions do not have ideal properties as convex functions. For instance, there exist -convex functions which are totally discontinuous or not locally bounded. But -convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length . It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of -convex functions on the real line. However, -convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of -convexity are given. Ways for generating and representing -convex functions are described.This research was supported by the Deutsche Forschungsgemeinschaft. The first author thanks Prof. Dr. E. Zeidler and Prof. Dr. H. G. Bock for their hospitality and valuable support.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

A Bernstein result for energy minimizing hypersurfaces

We show that there are no entire, positive, stable solutions in ⁿ of the Euler equation corresponding to the singular variational integral ,>0, if+n<5.236.... Furthermore we prove a related result for smooth boundaries of least-energy |x _n+1||D _U | in ⁿ⁺¹.     

An Investigation of the Fields of Bounded Formal Power Series by Means of Theory of Cuts

We consider a class K of real closed fields F, |F|=|G|=₁, where G is a group of Archimedean classes of F, and cofinality of each symmetric gap of F is ₁. We will show that this class is exactly a class of all bounded formal power series RG,₁, where G is a divisible Abelian group, card(G)=₁, under CH. A nonstandard real line ^*R, which is ₁-set belongs to this class; we will also consider a construction RG(L,P),₁ of fields from this class, where L is a totally ordered set, P is a totally ordered field, G(L,P) is a group of finite words. It will be describes symmetric gaps of such two fields in K, which are not ₁-set. Mathematics Subject Classifications (2000) 03E04, 12J15, 12J25.The work was supported by grant of Ministry of Education PD02-1.1-386.     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Convergence of random products of compact contractions in Hilbert space

Let {T₁, ..., T_N} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. Roughly, the S_n converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): x_n1, Tx_n1 implies (I-T)x_n0 weakly, (S): x_n1, Tx_n1 implies (I-T)x_n0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x²K(x²–Tx²).We have three main results in the event that the T_i's are compact contractions. First, if r assumes each value infinitely often, then S_n converges uniformly to the projection Q on the subspace _i= ₁ ^N [x|T_ix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T₁, ..., T_N) denotes the algebraic semigroup generated by the T_i's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.     

Asymptotic behavior of general M-estimates for regression and scale with random carriers

Summary Let (xini, y _i be a sequence of independent identically distributed random variables, where x _i R ^p and y _i R, and let R ^p be an unknown vector such that y _i =x _i +u _i (^*), where u _i is independent of x _i and has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, ( ^*,^*), defined as solutions of the system: , where r= (y _i –x _i ^1*/)^*, with R ^p ×RR and RR. This class contains estimators of (, ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on and and assuming the joint distribution of (x _i , y _i ) to fulfill the model (^*) only approximately.     

Application of Topological Degree to the Study of Bifurcation in von Kármán Equations

The aim of this paper is to illustrate the use of topological degree for the study of bifurcation in von Kármán equations with two real positive parameters and for a thin elastic disk lying on the elastic base under the action of a compressing force, which may be written in the form of an operator equation F(x, , ) = 0 in some real Banach spaces X and Y. The bifurcation problem that we study is a mathematical model for a certain physical phenomenon and it is very important in the mechanics of elastic constructions. We reduce the bifurcation problem in the solution set of equation F(x, , ) = 0 at a point (0, ₀, ₀) X × IR ₊ ² to the bifurcation problem in the solution set of a certain equation in IR ⁿ at a point (0, ₀, ₀) IR ⁿ × IR ₊ ², where n = dim Ker F _x (0, ₀, ₀) and F _x (0, ₀, ₀): X Y is a Fréchet derivative of F with respect to x at (0, ₀, ₀). To solve the bifurcation problem obtained as a result of reduction, we apply homotopy and degree theory.     

A White Noise Approach to Stochastic Neumann Boundary-Value Problems

We illustrate the use of white noise analysis in the solution of stochastic partial differential equations by explicitly solving the stochastic Neumann boundary-value problem LU(x)–c(x)U(x)=0, xDR ^d ,(x)U(x)=–W(x), xD, where L is a uniformly elliptic linear partial differential operator and W(x), xR ^d , is d-parameter white noise.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.