Lu :“ ́ε4u` fpx, y;uq “ 0 for px, yq P Ω, u “ 0 on BΩ, (1.1) posed in a, possibly non-Lipschitz, polygonal domain Ω Ă R. Here 0 ă ε ď 1. We also assume that f is continuous on ΩˆR and satisfies fp ̈; sq P L8pΩq for all s P R, and the one-sided Lipschitz condition fpx, y;uq ́ fpx, y; vq ě Cf ru ́ vs whenever u ě v, with some constant Cf ě 0. Then there is a unique solution u PW 2 ` pΩq ĎW 1 q Ă CpΩ̄q...

Let K be a compact convex body in R, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fipKnq denote the number of i-dimensional faces of Kn. We show that for planar convex sets, Erf0pKnqs is increasing in n. In dimension d ě 3 we prove that if limnÑ8 Erfd ́1pKnqs An “ 1 for some constants A and c ą 0 then the function n ÞÑ Erfd ́1pKnqs is increasing for n lar...

Let pX,μq be a standard probability space. An automorphism T of this space has the weak Pinsker property if for every ε ą 0 it is isomorphic to a direct product of a Bernoulli shift and an automorphism of entropy less than ε. This property was introduced by Thouvenot, who conjectured that it holds for all measure-preserving systems. This paper proves Thouvenot’s conjecture. The proof applies wi...

We give improved algorithmic time bounds for two fundamental problems, and establish a new complexity connection between them. The first is computing dominance product: given a set of n points p1, . . . , pn in R , compute a matrix D, such that Dri, js “ ˇ ˇtk | pirks ď pjrksu ˇ ; this is the number of coordinates at which pj dominates pi. Dominance product computation has often been applied in...

This book is fundamentally about the estimation of risk. At an intuitive level, risk is easy to understand: given an asset with a current price of say $100, what is the likelihood that at some future time—the risk horizon—its price will be less than $90? Or more than $120? And how does the probability of observing a price below $90 change as a function of the risk horizon? Such heuristic notion...

Let Σ`` d be the set of positive definite matrices with determinant 1 in dimension d ě 2. Identifying any two SLdpZq–congruent elements in Σ `` d gives rise to the space of reduced quadratic forms of determinant one, which in turn can be identified with the locally symmetric space Xd :“ SLdpZqzSLdpRq{SOdpRq. Equip the latter space with its natural probability measure coming from a Haar measure ...

7 VP T J üú û üŗ 4 J W VN»»S » û þĀŗ VS O ". . *Ć Ğ Ğ /,(Ą ðĞ Ć Ć ăāĂĚ ăÿüĚ Āú Ě þăüû Ě üŗ Preface " Jokainen tsäänssi on mahdollisuus! " – Matti Nykänen (Every chance is an opportunity) The work presented in this thesis was carried out in Optoelectronics Laboratory , of Electrical Engineering between 2003 and 2012. I want to express my sincere gratitude to Professor Harri Lipsanen for supervis...

We study the following basic problem called Bi-Covering. Given a graph GpV,Eq, find two (not necessarily disjoint) sets A Ď V and B Ď V such that A Y B “ V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize maxt|A|, |B|u. This is the most simple case of the Channel Allocation problem [13]. A solution that outputs V,H gives ratio ...

Background. Interventional endoscopy has largely reduced mortality in patients with peptic ulcer hemorrhage. Study aims. To evaluate the efficacy and safety of endoscopic hemostasis with argon plasma coagulation and injection sclerotherapy in bleeding peptic ulcer and determine the mortality of patients in a prospective, controlled study. Patients and methods. The study includes 100 patients wi...

Every collection of n (arbitrary-oriented) unit squares can be packed translatively into any equilateral triangle of side length 2.3755 ̈ ? n. Let the coordinate system in the Euclidean plane be given. For 0 ≤ αi ă π{2, denote by Spαiq a square in the plane with sides of unit length and with an angle between the x-axis and a side of Spαiq equal to αi. Furthermore, let T psq be an equilateral tr...