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1.
Coulbois Thierry; Hilion Arnaud; Lustig Martin 《Journal London Mathematical Society》2008,78(3):755-766
We study the map which associates to a current its support (whichis a lamination). We show that this map is Out(FN)-equivariant,not injective, not surjective and not continuous. However itis semi-continuous and almost surjective in a suitable sense.Given an -tree T (with dense orbits) in the boundary of outerspace and a current µ carried by the dual lamination ofT, we define a dual pseudo-distance dµ on T. When thetree and the current come from a measured geodesic laminationon a surface with boundary, the dual distance is the originaldistance of the tree T. In general, such a good correspondencedoes not occur. We prove that when the tree T is the attractivefixed point of a non-geometric irreducible, with irreduciblepowers, outer automorphism, the dual lamination of T is uniquelyergodic and the dual distance dµ is either zero or infinitethroughout T. 相似文献
2.
3.
Let T : X X be a continuous surjection of a topologicalspace, and let f : X be upper semi-continuous. Wewish to identify those T-invariant measures µ which maximize f dµ. We call such measures f-maximizing, and denotethe maximum by ß(f). The study of such measures andtheir properties has recently been dubbed ergodic optimization.A first step to understanding the structure of a function'smaximizing measures is to establish the following subordinationprinciple defined by T. Bousch: if µ and are T-invariantmeasures such that supp supp µ and µ is f-maximizing,then is also f-maximizing. Previous authors have approachedthis result by constructing a continuous function g : X such that f ß(f) g T g. We providea sufficient condition for the subordination principle whichhas advantages when the space X is noncompact. 相似文献
4.
In 1903 Minkowski showed that, given pairwise different unitvectors µ1, ..., µm in Euclidean n-space Rn whichspan Rn, and positive reals µ1, ..., µm such thatmi=1µiµi = 0, there exists a polytope P in Rn, uniqueup to translation, with outer unit facet normals µ1, ...,µm and corresponding facet volumes µ1, ..., µm.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of Minkowski reconstruction has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity. 相似文献
5.
Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T2, T3, ...} of iteratesof a single operator T on the Hilbert space L2(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more natural than Ruppert's. We also showthat there are enough functions of our type to distinguish 2cidempotentsin wN. 相似文献
6.
A Radon measure µ on Rn is said to be k-monotone if is a non-decreasing function on (0,) for every x Rn. (If µ is the k-dimensional Hausdorffmeasure restricted to a k-dimensional minimal surface then thisimportant property is expressed by the monotonicity formula.)We give an example of a 1-monotone measure µ in R2 withnon-unique and non-conical tangent measures at a point. Furthermore,we show that µ can be the one-dimensional Hausdorff measurerestricted to a closed set A R2. 2000 Mathematics Subject Classification49Q05, 49Q20 (primary), 28A75, 53A10 (secondary). 相似文献
7.
Received on 21 November 1995. Revised on 12 July 1996. This article is concerned with the numerical computation ofhomoclinic solutions converging to a hyperbolic or semi-hyperbolicequilibrium of a system u = f(u, µ). The approximationis done by replacing the original problem with a boundary valueproblem on a finite interval and introducing an additional phasecondition to make the solution unique. Numerical experimentshave indicated that the parameter µ is much better approximatedthan the homoclinic solution. This was proved in Schecter (1995IMA J. Numer Anal. 15, 23–60) for phase conditions satisfyingan additional niceness assumption, which is unfortunatelynot satisfied for the phase condition most commonly used innumerical experiments and which actually suggested the super-convergenceresult. Here, this result is proved for arbitrary phase conditions.Moreover, it is shown that it suffices to approximate the originalboundary value problem to first order when considering semi-hyperbolicequilibria, extending a result of Schecter (1993 SIAM J. NumerAnal. 30, 1155–78).
Permanent address: WIAS, MohrenstraBe 39, 10117 Berlin, Germany 相似文献
8.
Lifshits Mikhail A.; Linde Werner; Shi Zhan 《Proceedings London Mathematical Society》2006,92(1):224-250
We investigate RiemannLiouville processes RH, with H> 0, and fractional Brownian motions BH, for 0 < H <1, and study their small deviation properties in the spacesLq([0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of RiemannLiouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary). 相似文献
9.
Franchi Bruno; Perez Carlos; Wheeden Richard L. 《Proceedings London Mathematical Society》2000,80(3):665-689
We derive SobolevPoincaré inequalities that estimatethe Lq(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL1(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25. 相似文献
10.
Akkouchi Mohamed; Bakali Allal; Khalil Idriss 《Journal London Mathematical Society》1998,57(3):694-705
Let G be a locally compact group not necessarily unimodular.Let µ be a regular and bounded measure on G. We study,in this paper, the following integral equation,
E(µ) This equation generalizes the functional equation for sphericalfunctions on a Gel'fand pair. We seek solutions in the spaceof continuous and bounded functions on G. If is a continuousunitary representation of G such that (µ) is of rank one,then tr((µ)(x)) is a solution of E(µ). (Here, trmeans trace). We give some conditions under which all solutionsare of that form. We show that E(µ) has (bounded and)integrable solutions if and only if G admits integrable, irreducibleand continuous unitary representations. We solve completelythe problem when G is compact. This paper contains also a listof results dealing with general aspects of E(µ) and propertiesof its solutions. We treat examples and give some applications. 相似文献
11.
Bogachev V. I.; Da Prato G.; Rockner M.; Stannat W. 《Bulletin London Mathematical Society》2007,39(4):631-640
We study uniqueness of solutions of parabolic equations formeasures µ(dt dx) = µt(dx)dt of the type L*µ = 0, satisfying µt as t 0, where each µtis a probability measure on d, L = t + aij(t, x)xixj + bi(t,x)xj is a differential operator on (0, T) x d and is a giveninitial measure. One main result is that uniqueness holds underuniform ellipticity and Lipschitz conditions on aij but forbi merely local integrability and coercivity conditions aresufficient. 相似文献
12.
In this note, the maximal Large Sieve inequality has been usedto obtain following estimate:
where D(s)=, s=+it, is a Dirichletseries, dµ is a nonnegative Borel measure on R and U =[Clog1N] (say T1). In addition, the higher-dimensionalversion of (*) has been obtained. 相似文献
13.
On the Centred Hausdorff Measure 总被引:2,自引:0,他引:2
Let v be a measure on a separable metric space. For t, q R,the centred Hausdorff measures µh with the gauge functionh(x, r) = rt(vB(x, r))q is studied. The dimension defined bythese measures plays an important role in the study of multifractals.It is shown that if v is a doubling measure, then µh isequivalent to the usual spherical measure, and thus they definethe same dimension. Moreover, it is shown that this is trueeven without the doubling condition, if q 1 and t 0 or ifq 0. An example in R2 is also given to show the surprisingfact that the above assertion is not necessarily true if 0 <q < 1. Another interesting question, which has been askedseveral times about the centred Hausdorff measure, is whetherit is Borel regular. A positive answer is given, using the aboveequivalence for all gauge functions mentioned above. 相似文献
14.
Let µ be the self-similar measure for a linear functionsystem Sjx=x+bj (j=1,2,...,m) on the real line with the probabilityweight . Under the condition that satisfies the finite type condition, the Lq-spectrum (q) of µ is shown to bedifferentiable on (0,); as an application, µ is exactdimensional and satisfies the multifractal formalism. 相似文献
15.
The stability of the solutions of Hill's equation+[Bµø(t)]x= 0 is studied through the use of a fundamental matrix of solutionsevaluated after one period. This fundamental matrix is expandedin a power series in µ, but its determinant is computedhere by using series expansion rather than the Jacobi-Liouvilleformula. Using this procedure, it is shown that successive approximationsof boundaries of stability regions in the (B, µ) planecan be obtained simply, in particular without expansion of Bin power series of µ and Ø(t) in a Fourier series.A simple expression is obtained for the first approximation.Two examples are treated, with stability curves drawn up tothe second approximation. 相似文献
16.
This paper introduces the concept of bounded-norm matrix-inversemappings, i.e. mappings µ : RmxnRnxm such that, for allnonzero mxn matrices A, the matrix µ(A) is a generalizedinverse of A and ||µ(A)||> k/s(A), where K < 0 isa constant and s(A) is the nonzero singular value of A havingsmallest absolute value. It is shown how the definition of suchmappings is motivated by the need to ensure finite terminationof the inner-iterations of generalized elimination methods forthe solution of nonlinearly constrained optimization problems.The main result of the paper is that the mapping defined byµ(A) = Ab is a bounded-norm matrix-inverse mapping, providedthat the basic inverse Ab is calculated using Gaussian eliminationwith complete pivoting. The concept of bounded-norm matrix-inversemappings is then extended to that of boundednorm least-squaresmatrix-inverse mappings. It is proved that the mapping definedby µ(A) = Aß is a bounded-norm least-squaresmatrix-inverse mapping, provided that the basic least-squaresinverse Aß is calculated using the QR decompositionwith column pivoting. 相似文献
17.
Let µ be a positive bounded Borel measure on a subsetI of the real line and = {1, ..., n} a sequence of arbitrary complexpoles outside I. Suppose {1, ..., n} is the sequence of rationalfunctions with poles in orthonormal on I with respect to µ. First, we are concernedwith reducing the number of different coefficients in the three-termrecurrence relation satisfied by these orthonormal rationalfunctions. Next, we consider the case in which I = [–1, 1] and µ satisfies the Erdos–Turán conditionµ' > 0 a.e. on I (where µ' is the Radon–Nikodymderivative of the measure µ with respect to the Lebesguemeasure) to discuss the convergence of n+1(x)/n(x) as n tendsto infinity and to derive asymptotic formulas for the recurrencecoefficients in the three-term recurrence relation. Finally,we give a strong convergence result for n(x) under the morerestrictive condition that µ satisfies the Szeg condition(1 – x2)–1/2 log µ'(x) L1([– 1, 1]). 相似文献
18.
Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves. 相似文献
19.
Let X and Y be reflexive Banach spaces with strictly convexduals, and let T be a compact linear map from X to Y. It isshown that a certain nonlinear equation, involving T and itsadjoint, has a normalised solution (an eigenvector)corresponding to an eigenvalue, and that the sameis true for each member of a countable family of similar equationsinvolving the restrictions of T to certain subspaces of X. Theaction of T can be described in terms of these eigenvectors.There are applications to the p-Laplacian, the p-biharmonicoperator and integral operators of Hardy type. 相似文献
20.
Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a symplecticblowup of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X. 相似文献