We show that there exist $(d-1)$ -Ahlfors regular compact sets $E \subset\mathbb{R}^{d}$ , $d\geq2$ such that for any $t< d-1$ , we have $$\sup_{T} \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^{t}}< \infty $$ where the supremum is over all tubes $T$ with width $w(T) >0$ . This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.

A general framework for constructions of endoscopy correspondences via automorphic integral transforms for classical groups is formulated in terms of the Arthur classification of the discrete spectrum of square-integrable automorphic forms, which is called the Principle of Endoscopy Correspondences, extending the well-known Howe Principle of Theta Correspondences. This suggests another principle, called the (τ, b)-theory of automorphic forms of classical groups, to reorganize and extend the series of work of Piatetski- Shapiro, Rallis, Kudla and others on standard L-functions of classical groups and theta correspondence.

This paper studies a special class of population-size-dependent branching processes, in which the offspring distribution is supercritical when the population size does not exceed a given threshold 𝐾𝐾, and is subcritical or critical when the population size exceeds 𝐾𝐾. Up to now, the author has found no paper concerning continuous time threshold processes, and study of the discrete case is also limited to the extinction time 𝑇𝑇 and 𝔼𝔼𝑇𝑇. In this paper, both of the two cases were studied more thoroughly: for the discrete case, a limit theorem of the process has been proved; for the continuous case, the existence of Markovian threshold processes has been proved, and an equivalent condition for the process to extinct almost surely was also given. This paper also further generalized discrete time threshold processes by giving a random delay to the threshold’s influence of offspring distribution. The properties of these delayed threshold processes were studied, and a limit theorem was obtained.

The case of dimension two was proved by Andreotti-Frankel [-7] and the case of dimension three by Mabuchi [18] using the result of Kobayashi-Ochiai [12]. Our method of proof uses harmonic maps and the characterization of the complex projective space obtained by Kobayashi-Ochiai [15]. According to the result of Kobayashi-Ochiai the complex projective space of dimension n is characterized by the fact that its first Chern class equals 2c1 (F) for some 2&gt; n+ 1 and some positive holomorphic line bundle F over it. Since by the result of Bishop-Goldberg [-2] the second Betti number of a compact Kiihler manifold M of positive bisectional curvature is 1, for the Main Theorem it suffices to show that ca (M) is 2 times a generator of H2 (M, 7Z.) for some 2&gt; 1+ dim M. This can be done by proving that a generator of the free part of H2 (M, Z) can be represented by a rational curve, because the tangent bundle of M restricted to the rational curve splits into a direct sum of holomorphic line bundles over the rational curve according to the result of Grothendieck [-11]. The existence of the rational curve is obtained in the following way. According to the result of Sacks-Uhlenbeck [22] and its improved formulation by Meeks-Yau [-19], the infimum of the energies of maps from S 2 to M representing the generator of rcz (M) can be achieved by a sum of stable harmonic maps f~ from S 2 to M (1&lt; i&lt; m). The key step in our proof is to show that each f~ is either holomorphic or conjugate holomorphic. The known methods of proving the complex-analyticity of a harmonic map use the formula for the Laplacian of the energy function [23, 25, 26] or a variation of it [24]. Here we use

No abstract uploaded!