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35
Area and Hausdorff Dimension of Julia Sets of Entire Functions
 Trans. Amer. Math. Soc
, 1987
"... Introduction There is no known rational function R whose Julia set J has positive area and yet is not the whole Riemann sphere. (Indeed there is as yet no example for which J is proved to be a proper subset of the sphere with Hausdorff dimension two.) On the other hand, when the forward orbits of th ..."
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Introduction There is no known rational function R whose Julia set J has positive area and yet is not the whole Riemann sphere. (Indeed there is as yet no example for which J is proved to be a proper subset of the sphere with Hausdorff dimension two.) On the other hand, when the forward orbits of the critical points do not accumulate on J, the map R is expanding and Sullivan shows dim(J) < 2 (in fact J has positive finite measure in its dimension [S]). So disregarding the case J = C , it is fairly hard to find large Julia sets. The purpose of this note is to describe a contrasting situation for some entire functions. We consider two families of functions: {f(z) = le z : l 0} (the "exponential family") {f(z) = sin(az+b) : a 0} (the "sine family") Each of these families is "t
Fractals in Noncommutative Geometry
 in the Proceedings of the Conference ”Mathematical Physics in Mathematics and Physics”, Siena 2000, Edited by R. Longo, Fields Institute Communications
, 2001
"... Abstract. To any spectral triple (A, D, H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that D  −d has non trivial logarithmic Dixmier trace. Moreover, when d ∈ (0, ∞), there always exists a singular trace which ..."
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Abstract. To any spectral triple (A, D, H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that D  −d has non trivial logarithmic Dixmier trace. Moreover, when d ∈ (0, ∞), there always exists a singular trace which is finite nonzero on D  −d, giving rise to a noncommutative integration on A. Such results are applied to fractals in R, using Connes ’ spectral triple, and to limit fractals in R n, a class which generalises selfsimilar fractals, using a new spectral triple. The noncommutative dimension or measure can be computed in some cases. They are shown to coincide with the (classical) Hausdorff dimension and measure in the case of selfsimilar fractals. 1 Introduction. This paper is both a survey and an announcement of results concerning singular traces on B(H), and their application to the study of fractals in the framework of
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Absolute continuity for some onedimensional processes
, 2008
"... We introduce an elementary method for proving the absolute continuity of the time marginals of onedimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the onestep Euler approximation of the underlying process. We obtain some absolute ..."
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We introduce an elementary method for proving the absolute continuity of the time marginals of onedimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the onestep Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations, and to some Lévydriven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavindifferentiable.
On Hausdorff's moment problem in higher dimensions
"... Denote by n = R x n 1 1 . . . x n d d d(x) the n'th moment of a Borel probability measure on the unit cube I d = [0, 1] d in R d . Generalizing results of Hausdor#, Hildebrandt and Schoenberg, we give a su#cient condition in terms of moments, that is absolutely continuous with respect ..."
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Denote by n = R x n 1 1 . . . x n d d d(x) the n'th moment of a Borel probability measure on the unit cube I d = [0, 1] d in R d . Generalizing results of Hausdor#, Hildebrandt and Schoenberg, we give a su#cient condition in terms of moments, that is absolutely continuous with respect to a second Borel measure # on I d . We also review a constructive approximation of measures by atomic measures using finitely many moments. Keywords : Moment problems Abreviated title : Hausdor#'s Moment problem. AMS subject Classification: 44A60 1 Introduction Moment problems occur in di#erent mathematical contexts like in probability theory, mathematical physics, statistical mechanics, potential theory, constructive analysis or dynamical systems. In one dimension, the problem is to get information about a measure , if the moments R x n d(x) are known. For example, in probability theory, the question can occur to find properties of the law of a bounded random variable X , if the mom...
Subdiffusive quantum transport for 3DHamiltonians with absolutely continuous spectra
"... Both in the 3D Anderson model at low disorder and in 3D quasicrystals, the local density of states is expected to be absolutely continuous, although the quantum transport is diusive or subdiusive respectively. By studying sums of 1D models with wellunderstood spectral and transport properties, we e ..."
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Both in the 3D Anderson model at low disorder and in 3D quasicrystals, the local density of states is expected to be absolutely continuous, although the quantum transport is diusive or subdiusive respectively. By studying sums of 1D models with wellunderstood spectral and transport properties, we exhibit a 3D model with absolutely continuous spectrum for which the diusion exponent characterizing the growth of the mean square displacement is only slightly bigger than imposed by Guarneri's lower bound. Keywords: anomalous quantum transport, fractal measures Quantum diusion in aperiodic media is slowed down due to destructive interference phenomena. The most prominent example is the Anderson model describing an electron in a disordered potential. In low dimension or for a strong disordered potential, the motion is completely localized and the spectrum is known to be purepoint (see [1] and references therein). On the other hand, scaling arguments and numerical calculations [14] show ...
ANALYTIC REPRESENTATION OF FUNCTIONS AND A NEW TYPE OF QUASIANALYTICITY
, 2004
"... ABSTRACT. We characterize precisely the possible rate of decay of the antianalytic half of a trigonometric series converging to zero almost everywhere. 1. ..."
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ABSTRACT. We characterize precisely the possible rate of decay of the antianalytic half of a trigonometric series converging to zero almost everywhere. 1.
The Sidon constant of sets with three elements. arxiv.org/ math/0102145
, 2001
"... We solve an elementary minimax problem and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is sec(π/2n) for n = max ni − nj/gcd(n1 − n0, n2 − n0). 1 ..."
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We solve an elementary minimax problem and obtain the exact value of the Sidon constant for sets with three elements {n0, n1, n2}: it is sec(π/2n) for n = max ni − nj/gcd(n1 − n0, n2 − n0). 1
Szegö’s theorem and its probabilistic descendants
, 2011
"... Abstract: The theoryoforthogonal polynomialsontheunitcircle(OPUC) dates back to Szegö’s work of 191521, and has been given a great impetus by the recent work of Simon, in particular his survey paper and three recent books; we allude to the title of the third of these, Szegö’s theorem and its descen ..."
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Abstract: The theoryoforthogonal polynomialsontheunitcircle(OPUC) dates back to Szegö’s work of 191521, and has been given a great impetus by the recent work of Simon, in particular his survey paper and three recent books; we allude to the title of the third of these, Szegö’s theorem and its descendants, in ours. Simon’s motivation comes from spectral theory and analysis. Another major area of application of OPUC comes from probability, statistics, time series and prediction theory; see for instance the classic book by Grenander and Szegö, Toeplitz forms and their applications. Coming to the subject from this background, our aim here is to complement this recent work by giving some probabilistically motivated