Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

Convergence of diagonally stationary sequences in convex optimization

LetX be a real normed linear space,f, f ⁿ, n , be extended real-valued proper closed convex functions onX. A sequence {x _n} inX is called diagonally stationary for {f ⁿ} if for alln there existsx* _n f ⁿ (x _n) such that x* _{n *} 0. Such sequences arise in approximation methods for the problem of minimizingf. Some general convergence results based upon variational convergence theory and appropriate equi-well-posedness are presented.     

On the Ranked Excursion Heights of a Kiefer Process

Let (K(s,t), 0s1, t1) be a Kiefer process, i.e., a continuous two-parameter centered Gaussian process indexed by [0,1]×₊ whose covariance function is given by (K(s₁,t₁) K(s₂,t₂))=(s₁s₂-s₁s₂)t₁t₂, 0s₁, s₂1, t₁, t₂ 0. For each t>0, the process K(·,t) is a Brownian bridge on the scale of . Let M ₁ ^* (t) M ₂ ^* (t) M _j ^* (t) 0 be the ranked excursion heights of K(,t). In this paper, we study the path properties of the process tM _j ^* (t). Two laws of the iterated logarithm are established to describe the asymptotic behaviors of M _j ^* (t) as t goes to infinity.     

On unconditional bases in certain Banach function spaces

Let X be a Banach space, L ([0,1])XL ¹([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f _n} _m=1 , where f _n in C([0,1]), nN. In this paper, we introduce the notion that X ^*has the singularity property of X ^*at a point t ₀[0,1]. It is proved that if X ^*has the singularity property at a point t ₀ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.     

Convexity of Nonlinear Image of a Small Ball with Applications to Optimization

Let f: XY be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f (x) is Lipschitz in B(a,r) with Lipschitz constant L and f (a) is a surjection: f (a)X=Y; this implies the existence of >0 such that f (a)^* yy, yY. Then, if r,/(2L), the image F=f(B(a,)) of the ball B(a,) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a. Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control.     

Mappings that preserve cones in Lobachevskii space

Let ⁿ be n-dimensional Lobachevskii space, and {l_x:x ⁿ} be a family of lines, parallel to a linel ₀, 0ⁿ (in a given direction). Let {c_x:Xⁿ} be a family of circular cones in ⁿ of opening with axes l_X and vertex X. Then, iff:ⁿⁿ(n>2) is a bijective mapping andf(C_x)=C _f(x), it follows thatf is a motion in the space ⁿ.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 687–694, May, 1973.     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

Generalised Holley-Preston inequalities on measure spaces and their products

Summary It is shown that if (X, ) is a product of totally ordered measure spaces andf _j (j=1,2,3,4) are measurable non-negative functions onX satisfyingf ₁(x)f₂(y)f₃(xy)f₄(xy), where (, ) are the lattice operations onX, then (f ₁ d)(f ₂ d)(f ₃ d)(f ₄ d). This generalises results of Ahlswede and Daykin (for counting measure on finite sets) and Preston (for special choices off _j).     

Minty Variational Inequalities,Increase-Along-Rays Property and Optimization<Superscript>1</Superscript>

Let E be a linear space, let K E and f:K . We formulate in terms of the lower Dini directional derivative problem GMVI (f ^,K ), which can be considered as a generalization of MVI (f ^,K ), the Minty variational inequality of differential type. We investigate, in the case of K star-shaped (SS), the existence of a solution x ^* of GMVI (f ^K ) and the property of f to increase-along-rays starting at x ^*, fIAR (K,x ^*). We prove that the GMVI (f ^,K ) with radially l.s.c. function f has a solution x ^* ker K if and only if fIAR (K,x ^*). Further, we prove that the solution set of the GMVI (f ^,K ) is a convex and radially closed subset of ker K. We show also that, if the GMVI (f ^,K ) has a solution x ^*K, then x ^* is a global minimizer of the problem min f(x), xK. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove that, in the case of a quasiconvex function f, these sets coincide.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

γ-Subdifferential and γ-convexity of functions on a normed space

In this paper, the -subdifferential is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: DX attains its global minimum onD atx ^*, then 0 f(x ^*). This necessary condition always holds, even iff is not continuous orx ^* is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable ₊, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called -convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a -convex function. There are -convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two -convex functions. For all that, -convex functions still have properties similar to those of convex functions. For instance, each -local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its -boundary.This research was supported by the Alexander von Humboldt Foundation. The author thanks Professors R. Bulirsch, K. H. Hoffmann, and H. G. Bock for inviting him to Munich and Augsburg where this research was done.     

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 ].     

Orbits, weak orbits and local capacity of operators

LetT be an operator on a Banach spaceX. We give a survey of results concerning orbits {T ⁿ x:n=0,1,...} and weak orbits {T ⁿ x,x ^*:n=0,1,...} ofT wherexX andx ^*X ^*. Further we study the local capacity of operators and prove that there is a residual set of pointsxX with the property that the local capacity cap(T, x) is equal to the global capacity capT. This is an analogy to the corresponding result for the local spectral radius.The research was supported by the grant No. A1019801 of AV R.     

D'Alembert's functional equation on restricted domains

Summary In the class of functionalsf:X , whereX is an inner product space with dimX 3, we study the D'Alembert functional equationf(x + y) + f(x – y) = 2f(x)f(y) (1) on the restricted domainsX ₁ = {(x, y) X ²/x, y = 0} andX ₂ = {(x, y) X ²/x = y}. In this paper we prove that the equation (1) restricted toX ₁ is not equivalent to (1) on the whole spaceX. We also succeed in characterizing all common solutions if we add the conditionf(2x) = 2f²(x) – 1. Using this result, we prove the equivalence between (1) restricted toX ₂ and (1) on the whole spaceX. This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.     

Surjectivity for Hamiltonian loop group spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,), and assume that the moment map :XL ^* is proper. We consider the function ||²:X, and use a version of Morse theory to show that the inclusion map j:^-1(0)X induces a surjection j ^*:H _G ^*(X)H _G ^*(^-1(0)), in analogy with Kirwans surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.     

On the central limit theorem in R k

Summary Let F ^*n denote the n th convolution of a distribution function F on R ^k and suppose that F has zero moments of the first order and finite second order moment matrix. It is well-known that F ^*n (n·) converges to a Gaussian d.f. as n + t8. These d.f.s determine measures F ^*n (nA) and (A) for Borelsets A, We present a method that admits the estimation of the remainder-term F ^*n (n A)- (A) when A belongs to a certain class of Borelsets. This class contains all convex sets. If F has finite absolute third order moments then the remainder-term is of the order n ^–1/2. Also the remainder term's dependence on the dimension k is given. These results strengthen and generalize earlier results in the same direction.This paper was first communicated at the Scandinavian mathematical congress in Oslo, August 1968.     

Density functions for prime and relatively prime numbers

Letr ^*(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ^*(x) denote the maximum number ofprime integers in any interval (y,y+x] whereyx. Throughout this paper we assume the primek-tuples hypothesis. (This hypothesis could be avoided by using an alternative sievetheoretic definition of ^*(x); cf. the beginning of Section 1.) We investigate the differencer ^*(x)—^*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr ^*(x)—^*(x)<x ^c for somec>0 (whenx). This improves the previous lower bound of logx. As an upper bound we getr ^*(x)—^*(x)=o[x/(logx)²]. It is known that ^*(x)—(x)>const.[x/(logx)²];.; thus the difference betweenr ^*(x) and ^*(x) is negligible compared to ^*(x)—(x). The results mentioned so far involve the upper bound or maximizing sieve. In Section 2, similar comparisons are made between two types of minimum sieves. One of these is the erasing sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of minimax for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.     

On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth