Uniform Random Covering Problems

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega =(\omega _n)_{n\geq 1}$ uniformly distributed on the unit circle $\mathbb{T}$ and a sequence $(r_n)_{n\geq 1}$ of positive real numbers with limit $0$. We investigate the size of the random set $$\begin{align*} & {\operatorname{{{\mathcal{U}}}}} (\omega):=\{y\in \mathbb{T}: \ \forall N\gg 1, \ \exists n \leq N, \ \text{s.t.} \ | \omega_n -y | < r_N \}. \end{align*}$$Some sufficient conditions for ${\operatorname{{{\mathcal{U}}}}}(\omega )$ to be almost surely the whole space, of full Lebesgue measure, or countable, are given. In the case that ${\operatorname{{{\mathcal{U}}}}}(\omega )$ is a Lebesgue null measure set, we provide some estimations for the upper and lower bounds of Hausdorff dimension.

Boundary value problems associated with singular strongly nonlinear equations with functional terms

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}}_x(t))\,\rho(t, x'(t)) = 0,$$ on a compact interval [a, b]. These equations are quite general due to the presence of a strictly increasing homeomorphism Φ, the so-called Φ-Laplace operator, of a non-negative function k, which may vanish on a set of null measure, and moreover of a functional term Gx. We look for solutions, in a suitable weak sense, which belong to the Sobolev space W1,1([a, b]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.

FAT AND THIN MORAN SETS FOR DOUBLING MEASURES

For a large class of Moran sets, we find sufficient and necessary conditions implying that a set has positive or null measure for doubling measures.

Implicit Vector Integral Equations Associated with Discontinuous Operators

LetI∶=[0,1]. We consider the vector integral equationh(u(t))=ft,∫Ig(t,z),u(z),dzfor a.e.t∈I,wheref:I×J→R, g:I×I→ [0,+∞[,andh:X→Rare given functions andX,Jare suitable subsets ofRn. We prove an existence result for solutionsu∈Ls(I, Rn), where the continuity offwith respect to the second variable is not assumed. More precisely,fis assumed to be a.e. equal (with respect to second variable) to a functionf*:I×J→Rwhich is almost everywhere continuous, where the involved null-measure sets should have a suitable geometry. It is easily seen that such a functionfcan be discontinuous at each pointx∈J. Our result, based on a very recent selection theorem, extends a previous result, valid for scalar casen=1.

Finite blocking property versus pure periodicity

A translation surface 𝒮 is said to have the finite blocking property if for every pair (O,A) of points in 𝒮 there exists a finite number of ‘blocking’ points B1,…,Bn such that every geodesic from O to A meets one of the Bis. 𝒮 is said to be purely periodic if the directional flow is periodic in each direction whose directional flow contains a periodic trajectory (this implies that 𝒮 admits a cylinder decomposition in such directions). We will prove that the finite blocking property implies pure periodicity. We will also classify the surfaces that have the finite blocking property in genus two: such surfaces are exactly the torus branched coverings. Moreover, we prove that in every stratum such surfaces form a set of null measure. In Appendix A, we prove that completely periodic translation surfaces form a set of null measure in every stratum.

DISCRETE AND CONTINUOUS AMBIGUITIES IN THE ISOSPIN ANALYSIS OF THE B→πK DECAYS

A polynomial equation of the fifth degree, which solves completely the theoretical problem of the isospin analysis of the B→πK decays, is derived. It is shown that the problem has in general an even number (two or four) of non-trivially-related solutions. There are six particular cases of null measure, but which might be relevant due to the experimental errors, in which the problem has a continuous set of solutions. It is concluded that the isospin analysis, even if experimentally feasible, has a large probability of being inconclusive.