Kahane using brownian local times of intersection

We show that if a set E in the positive real line has Hausdorff dimension greater than d/2 m, then the m-fold algebraic sum of the image of E by d-dimensional Brownian motion has an interior point. This extends a result of Kahane. The proof uses techniques found in Rosen (1983) and Geman, Horowitz and Rosen. We then show that the results do not hold for random sets and demonstrate that the above condition on the Hausdorff dimension of E is not close to being necessary     

Brownian sheet images and Bessel-Riesz capacity

We show that the image of a 2-dimensional set under -dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ()-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.

     

On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

We prove that the set of exceptional \({\lambda\in (1/2,1)}\) such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.     

Lognormal {\star} -scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.     

On exact scaling log-infinitely divisible cascades

In this paper we extend some classical results valid for canonical multiplicative cascades to exact scaling log-infinitely divisible cascades. We present an alternative construction of exact scaling infinitely divisible cascades based on a family of cones whose geometry naturally induces the exact scaling property. We complete previous results on non-degeneracy and moments of positive orders obtained by Barral and Mandelbrot, and Bacry and Muzy: we provide a necessary and sufficient condition for the non-degeneracy of the limit measures of these cascades, as well as for the finiteness of moments of positive orders of their total mass, extending Kahane’s result for canonical cascades. Our main results are analogues to the results by Kahane and Guivarc’h regarding the asymptotic behavior of the right tail of the total mass. They come from a “non-independent” random difference equation satisfied by the total mass of the measures. The non-independent structure brings new difficulties to study the random difference equation, which we overcome thanks to Dirichlet’s multiple integral formula and Goldie’s implicit renewal theory. We also discuss the finiteness of moments of negative orders of the total mass, and some geometric properties of the support of the measure.     

Notions of boundaries for function spaces

Sibony and the author independently defined a higher order generalization of the usual Shilov boundary of a function algebra which yielded extensions of results about analytic structure from one dimension to several dimensions. Tonev later obtained an alternative characterization of this generalized Shilov boundary by looking at closed subsets of the spectrum whose image under the spectral mapping contains the topological boundary of the joint spectrum. In this note we define two related notions of what it means to be a higher order/higher dimensional boundary for a space of functions without requiring that the boundary be a closed set. We look at the relationships between these two boundaries, and in the process we obtain an alternative proof of Tonev's result. We look at some examples, and we show how the same concepts apply to convex sets and linear functions.

     

From restoration by topological gradient to medical image segmentation via an asymptotic expansion

The aim of this article is to present an application of the topological asymptotic expansion to the medical image segmentation problem. We first recall the classical variational of the image restoration problem, and its resolution by topological asymptotic analysis in which the identification of the diffusion coefficient can be seen as an inverse conductivity problem. The conductivity is set either to a small positive coefficient (on the edge set), or to its inverse (elsewhere). In this paper a technique based on a power series expansion of the solution to the image restoration problem with respect to this small coefficient is introduced. By considering the limit when this coefficient goes to zero, we obtain a segmented image, but some numerical issues do not allow a too small coefficient. The idea is to use the series expansion to approximate the asymptotic solution with several solutions corresponding to positive (larger than a threshold) conductivity coefficients via a quadrature formula. We illustrate this approach with some numerical results on medical images.     

Théorèmes du type Ingham et application à la théorie du contrôle

Ingham [4] improved a previous result of Wiener [10] on nonharmonic Fourier series. Modifying his weight function we obtain optimal results improving several earlier theorems of Kahane [7], Castro and Zuazua [2] and of Jaffard, Tucsnak, and Zuazua [5]. Then we apply these results to simultaneous observability problems.     

GENERALIZED BROWNIAN SHEET IMAGES AND BESSEL-RIESZ CAPACITY

Let W be a two-parameter, Rd-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under W. He also presents the connections between the Lebesgue measure of the image of W and Bessel-Riesz capacity. His conclusions also solve a problem proposed by J.-P.Kahane.     

Distance sets,orthogonal projections and passing to weak tangents

We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of ‘passing to weak tangents’. First, we solve the analogue of Falconer’s distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.     

Hitting Times in Markov Chains with Restart and their Application to Network Centrality

Motivated by applications in telecommunications, computer science and physics, we consider a discrete-time Markov process with restart. At each step the process either with a positive probability restarts from a given distribution, or with the complementary probability continues according to a Markov transition kernel. The main contribution of the present work is that we obtain an explicit expression for the expectation of the hitting time (to a given target set) of the process with restart. The formula is convenient when considering the problem of optimization of the expected hitting time with respect to the restart probability. We illustrate our results with two examples in uncountable and countable state spaces and with an application to network centrality.     

The multicriteria p-facility median location problem on networks

In this paper we discuss the multicriteria p-facility median location problem on networks with positive and negative weights. We assume that the demand is located at the nodes and can be different for each criterion under consideration. The goal is to obtain the set of Pareto-optimal locations in the graph and the corresponding set of non-dominated objective values. To that end, we first characterize the linearity domains of the distance functions on the graph and compute the image of each linearity domain in the objective space. The lower envelope of a transformation of all these images then gives us the set of all non-dominated points in the objective space and its preimage corresponds to the set of all Pareto-optimal solutions on the graph. For the bicriteria 2-facility case we present a low order polynomial time algorithm. Also for the general case we propose an efficient algorithm, which is polynomial if the number of facilities and criteria is fixed.     

The loci of abelian varieties with points of high multiplicity on the theta divisor

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to the singular set of the universal theta divisor. We obtain bounds on the dimensions of these loci and relations among their dimensions, and make further conjectures about their structure. Research of the first author is supported in part by National Science Foundation under the grant DMS-05-55867.     

The role of Billingsley dimensions in computing fractal dimensions on Cantor-like spaces

We consider a Cantor-like set as a geometric projection of a Bernoulli process. P. Billingsley (1960) and C. Dai and S.J. Taylor (1994) introduced dimension-like indices in the probability space of a stochastic process. Under suitable regularity conditions we find closed formulae linking the Hausdorff, box and packing metric dimensions of the subsets of the Cantor-like set, to the corresponding Billingsley dimensions associated with a suitable Gibbs measure. In particular, these formulae imply that computing dimensions in a number of well-known fractal spaces boils down to computing dimensions in the unit interval endowed with a suitable metric. We use these results to generalize density theorems in Cantor-like spaces. We also give some examples to illustrate the application of our results.

     

Old Friends Revisited: the Multifractal Nature of Some Classical Functions

We study some explicit functions introduced by Riemann, Jordan, Lévy, Kahane… These functions share the property of having a dense set of discontinuities. We prove that they are examples of multifractal functions.     

Structure of positive solution sets of sub-linear semi-positone operator equations

In this paper, first we obtain some results on the structure of the positive solution set of a nonlinear operator equation. Then using these results, we obtain some existence results for positive solutions of nonlinear operator equations. We use global bifurcation theories to show our main results.     

On Markov–Mandelbrot martingales

We consider the actions on ℓ-Markov measures of multiplicative chaos operators associated to Mandelbrot cascades. Necessary and sufficient conditions are obtained for the singularity and the regularity and for the existence of moments. The Hausdorff dimension of the image measures is determined. These results generalize those of Kahane–Peyrière for the action on the Lebesgue measure. We also consider the actions on Gibbs measures.     

Complete controllability and contractibility in multimodal systems

The concept of a strictly positive definite set of Hermitian matrices is introduced. It is shown that a strictly positive definite set is always a positive definite set, and conditions are found under which a positive definite set is strictly positive definite. We also show that a set of Hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix. For state dimension two, we use this concept to find necessary and sufficient conditions for a two-mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm. For general state dimensions, we give necessary and sufficient conditions for a special-type two-mode completely controllable irreducible system to be contractible relative to a weakly monotone norm. Applying the above results, we show that, for state dimension two, there exists a completely controllable two-mode system which is not contractible relative to either an elliptic or a weakly monotone norm. We leave open the question whether or not complete controllability implies contractibility, relative to some norm, for multimodal systems of two or more modes.     

The Hausdorff dimension of chaotic sets for interval self-maps

We obtain two sufficient conditions for an interval self-map to have a chaotic set with positive Hausdorff dimension. Furthermore, we point out that for any interval Lipschitz maps with positive topological entropy there is a chaotic set with positive Hausdorff dimension.     

The Lazer-McKenna conjecture and a free boundary problem in two dimensions

We prove that certain super-linear elliptic equations in twodimensions have many solutions when the diffusion is small.We find these solutions by constructing solutions with manysharp peaks. In three or more dimensions, this has already beenproved by the authors in Comm. Partial Differential Equations30 (2005) 1331–1358. However, in two dimensions, the problemis much more difficult because there is no limit problem inthe whole space. Therefore, the proof is quite different, thoughstill a reduction argument. A direct consequence of this resultis that we give a positive answer to the Lazer–McKennaconjecture for some typical nonlinearities in two dimensions.