Probability and Real Trees
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On Two Dimensional Markov Processes with Branching Property^)
ON TWO DIMENSIONAL MARKOV PROCESSES WITH BRANCHING PROPERTY^) BY SHINZO WATANABE Introduction. A continuous state branching Markov process (C.B.P.) was introduced by Jirina [8] and recently Lamperti [10] determined all such processes on the half line. (A quite similar result was obtained independently by the author.) This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller [2]. The main objective of the present paper is to extend Lamperti's result to multi-dimensional case. For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2). In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them. Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe [5], through backward equations (or in the terminology of [5] through S-equations) for a simpler case and then in the general case by a limiting procedure. A special attention will be paid to the case of diffusions. We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3). This fact may be of some interest since the solutions of a stochastic equation with coefficients Holder continuous of exponent 1/2 (which is our case) are not known to be unique in general. Next we shall examine the behavior of sample functions near the boundaries (xj-axis or x2-axis). -
Acylindrical Accessibility for Groups Acting on R-Trees
ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES ILYA KAPOVICH AND RICHARD WEIDMANN Abstract. We prove an acylindrical accessibility theorem for finitely generated groups acting on R-trees. Namely, we show that if G is a freely indecomposable non-cyclic k-generated group acting minimally and M- acylindrically on an R-tree X then there is a finite subtree Y ⊆ X of measure at most 2M(k − 1) + such that GY = X. This generalizes theorems of Z.Sela and T.Delzant about actions on simplicial trees. 1. Introduction An isometric action of a group G on an R-tree X is said to be M- acylindrical (where M ≥ 0) if for any g ∈ G, g 6= 1 we have diam Fix(g) ≤ M, that is any segment fixed point-wise by g has length at most M. For example the action of an amalgamated free product G = A ∗C B on the corresponding Bass-Serre tree is 2-acylindrical if C is malnormal in A and 1-acylindrical if C is malnormal in both A and B. In fact the notion of acylindricity seems to have first appeared in this context in the work of Karras and Solitar [32], who termed it being r-malnormal. Sela [37] proved an important acylindrical accessibility result for finitely generated groups which, when applied to one-ended groups, can be restated as follows: for any one-ended finitely generated group G and any M ≥ 0 there is a constant c(G, M) > 0 such that for any minimal M-acylindrical action of G on a simplicial tree X the quotient graph X/G has at most c(G, M) edges. -
Antwerpen, Belgium
10th European Congress on Tropical Medicine and International Health Antwerpen, Belgium Preliminary Programme www.ECTMIH2017.be Table of Contents Legend ....................................................................................................... 4 Programme Monday Opening Ceremony ................................................................. 7 Tuesday Programme at a Glance .......................................................... 8 Programme S and OS ............................................................. 10 Wednesday Programme at a Glance .......................................................... 28 Programme S and OS ............................................................. 30 Thursday Programme at a Glance .......................................................... 44 Programme S and OS ............................................................. 46 Friday Programme at a Glance .......................................................... 65 Programme S and OS ............................................................. 66 Posters Poster List Tuesday............................................................................ 71 Poster List Wednesday ...................................................................... 92 Poster List Thursday .......................................................................... 114 2 www.ectmih2017.be www.ectmih2017.be 3 Legend Colour Codes The programme is organised in 8 tracks. These 8 tracks are listed on page 5. Track 1. Breakthroughs and innovations in tropical biomedical -
Texas Public Safety Threat Overview 2017 (PDF)
UNCLASSIFIED LAW ENFORCEMENT SEN Texas Public Safety Threat Overview A State Intelligence Estimate Produced by the Texas Department of Public Safety In collaboration with other law enforcement and homeland security agencies January 2017 1 UNCLASSIFIED UNCLASSIFIED Executive Summary (U) Texas faces the full spectrum of threats, and the state’s vast size, geography, and large population present unique challenges to public safety and homeland security. Texas employs a systematic approach to detect, assess, and prioritize public safety threats within seven categories: terrorism, crime, motor vehicle crashes, natural disasters, public health threats, industrial accidents, and cyber threats. (U) Due to the recent actions of lone offenders or small groups affiliated with or inspired by the Islamic State of Iraq and Syria (ISIS) and other foreign terrorist organizations, we assess that the current terrorism threat to Texas is elevated. We recognize that ISIS has had considerable success in inspiring and inciting lone offenders to attack targets in the United States and other Western countries using simple yet effective tactics that are difficult to detect and disrupt. We expect this heightened threat to persist over at least the next year, due in part to the relatively high number of recent terrorism-related arrests and thwarted plots inside the US, and the prevalence and effectiveness of ISIS’s online recruitment and incitement messaging, as the organization is slowly defeated on the battlefield. We are especially concerned about the potential for terrorist infiltration across the US-Mexico border, particularly as foreign terrorist fighters depart Syria and Iraq and enter global migration flows. We are concerned about the challenges associated with the security vetting of Syrian war refugees or asylum seekers who are resettled in Texas – namely, that derogatory security information about individuals is inaccessible or nonexistent. -
Ergodic Currents Dual to a Real Tree Thierry Coulbois, Arnaud Hilion
Ergodic currents dual to a real tree Thierry Coulbois, Arnaud Hilion To cite this version: Thierry Coulbois, Arnaud Hilion. Ergodic currents dual to a real tree. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2016, 36 (3), pp.745-766. 10.1017/etds.2014.78. hal- 01481866 HAL Id: hal-01481866 https://hal.archives-ouvertes.fr/hal-01481866 Submitted on 3 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ERGODIC CURRENTS DUAL TO A REAL TREE THIERRY COULBOIS, ARNAUD HILION Abstract. Let T be an R-tree with dense orbits in the boundary of Outer space. When the free group FN acts freely on T , we prove that the number of projective classes of ergodic currents dual to T is bounded above by 3N − 5. We combine Rips induction and splitting induction to define unfolding induction for such an R-tree T . Given a current µ dual to T , the unfolding induction produces a sequence of approximations converging towards µ. We also give a unique ergodicity criterion. 1. Introduction 1.1. Main results. Let FN be the free group with N generators. -
Long Term Risk: a Martingale Approach
Long Term Risk: A Martingale Approach Likuan Qin∗ and Vadim Linetskyy Department of Industrial Engineering and Management Sciences McCormick School of Engineering and Applied Sciences Northwestern University Abstract This paper extends the long-term factorization of the pricing kernel due to Alvarez and Jermann (2005) in discrete time ergodic environments and Hansen and Scheinkman (2009) in continuous ergodic Markovian environments to general semimartingale environments, with- out assuming the Markov property. An explicit and easy to verify sufficient condition is given that guarantees convergence in Emery's semimartingale topology of the trading strate- gies that invest in T -maturity zero-coupon bonds to the long bond and convergence in total variation of T -maturity forward measures to the long forward measure. As applications, we explicitly construct long-term factorizations in generally non-Markovian Heath-Jarrow- Morton (1992) models evolving the forward curve in a suitable Hilbert space and in the non-Markovian model of social discount of rates of Brody and Hughston (2013). As a fur- ther application, we extend Hansen and Jagannathan (1991), Alvarez and Jermann (2005) and Bakshi and Chabi-Yo (2012) bounds to general semimartingale environments. When Markovian and ergodicity assumptions are added to our framework, we recover the long- term factorization of Hansen and Scheinkman (2009) and explicitly identify their distorted probability measure with the long forward measure. Finally, we give an economic interpre- tation of the recovery theorem of Ross (2013) in non-Markovian economies as a structural restriction on the pricing kernel leading to the growth optimality of the long bond and iden- tification of the physical measure with the long forward measure. -
Beyond Serre's" Trees" in Two Directions: $\Lambda $--Trees And
Beyond Serre’s “Trees” in two directions: Λ–trees and products of trees Olga Kharlampovich,∗ Alina Vdovina† October 31, 2017 Abstract Serre [125] laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from [125]: the theory of isometric group actions on Λ-trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees. 1 Introduction Serre [125] laid down the fundamentals of the theory of groups acting on sim- plicial trees. The book [125] consists of two parts. The first part describes the basics of what is now called Bass-Serre theory. This theory makes it possi- ble to extract information about the structure of a group from its action on a simplicial tree. Serre’s original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. These groups are considered in the second part of the book. Bass-Serre theory states that a group acting on a tree can be decomposed arXiv:1710.10306v1 [math.GR] 27 Oct 2017 (splits) as a free product with amalgamation or an HNN extension. Such a group can be represented as a fundamental group of a graph of groups. -
A Short History of Stochastic Integration and Mathematical Finance
A Festschrift for Herman Rubin Institute of Mathematical Statistics Lecture Notes – Monograph Series Vol. 45 (2004) 75–91 c Institute of Mathematical Statistics, 2004 A short history of stochastic integration and mathematical finance: The early years, 1880–1970 Robert Jarrow1 and Philip Protter∗1 Cornell University Abstract: We present a history of the development of the theory of Stochastic Integration, starting from its roots with Brownian motion, up to the introduc- tion of semimartingales and the independence of the theory from an underlying Markov process framework. We show how the development has influenced and in turn been influenced by the development of Mathematical Finance Theory. The calendar period is from 1880 to 1970. The history of stochastic integration and the modelling of risky asset prices both begin with Brownian motion, so let us begin there too. The earliest attempts to model Brownian motion mathematically can be traced to three sources, each of which knew nothing about the others: the first was that of T. N. Thiele of Copen- hagen, who effectively created a model of Brownian motion while studying time series in 1880 [81].2; the second was that of L. Bachelier of Paris, who created a model of Brownian motion while deriving the dynamic behavior of the Paris stock market, in 1900 (see, [1, 2, 11]); and the third was that of A. Einstein, who proposed a model of the motion of small particles suspended in a liquid, in an attempt to convince other physicists of the molecular nature of matter, in 1905 [21](See [64] for a discussion of Einstein’s model and his motivations.) Of these three models, those of Thiele and Bachelier had little impact for a long time, while that of Einstein was immediately influential. -
Lecture 4, Geometric and Asymptotic Group Theory
Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Quasi-isometry Limit groups Free actions Lecture 4, Geometric and asymptotic group theory Olga Kharlampovich NYC, Sep 16 1 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The universal property of free groups allows one to describe arbitrary groups in terms of generators and relators. Let G be a group with a generating set S. By the universal property of free groups there exists a homomorphism ': F (S) ! G such that '(s) = s for s 2 S. It follows that ' is onto, so by the first isomorphism theorem G ' F (S)=ker('): In this event ker(') is viewed as the set of relators of G, and a group word w 2 ker(') is called a relator of G in generators S. If a subset R ⊂ ker(') generates ker(') as a normal subgroup of F (S) then it is termed a set of defining relations of G relative to S. 2 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The pair hS j Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS j Ri is finite if both sets S and R are finite. A group is finitely presented if it has at least one finite presentation. Presentations provide a universal method to describe groups. Example of finite presentations 1 G = hs1;:::; sn j [si ; sj ]; 81 ≤ i < j ≤ ni is the free abelian group of rank n. -
Llogl Condition for Supercritical Branching Hunt Processes
J Theor Probab (2011) 24: 170–193 DOI 10.1007/s10959-010-0322-7 L log L Condition for Supercritical Branching Hunt Processes Rong-Li Liu · Yan-Xia Ren · Renming Song Received: 17 March 2009 / Revised: 17 July 2010 / Published online: 19 October 2010 © Springer Science+Business Media, LLC 2010 Abstract In this paper we use the spine decomposition and martingale change of measure to establish a Kesten–Stigum L log L theorem for branching Hunt processes. This result is a generalization of the results in Asmussen and Hering (Z. Wahrschein- lichkeitstheor. Verw. Geb. 36:195–212, 1976) and Hering (Branching Processes, pp. 177–217, 1978) for branching diffusions. Keywords Hunt processes · Branching Hunt processes · Kesten–Stigum theorem · Martingales · Martingale change of measure Mathematics Subject Classification (2000) Primary 60J80 · 60F15 · Secondary 60J25 1 Introduction Suppose that {Zn; n ≥ 1} is a Galton–Watson process with each particle having prob- ability pn of giving birth to n offspring. Let L stand for a random variable with this := ∞ offspring distribution. Let m n=1 npn be the mean number of offspring per par- n n ticle. Then Zn/m is a non-negative martingale. Let W be the limit of Zn/m as The research of Y.-X. Ren was supported by NSFC (Grant No. 10871103 and 10971003). R.-L. Liu · Y.-X. Ren LMAM School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China R.-L. Liu e-mail: [email protected] Y.-X. Ren e-mail: [email protected] R. Song () Department of Mathematics, The University of Illinois, Urbana, IL 61801, USA e-mail: [email protected] J Theor Probab (2011) 24: 170–193 171 n →∞. -
Fractal Geometry and Applications in Forest Science
ACKNOWLEDGMENTS Egolfs V. Bakuzis, Professor Emeritus at the University of Minnesota, College of Natural Resources, collected most of the information upon which this review is based. We express our sincere appreciation for his investment of time and energy in collecting these articles and books, in organizing the diverse material collected, and in sacrificing his personal research time to have weekly meetings with one of us (N.L.) to discuss the relevance and importance of each refer- enced paper and many not included here. Besides his interdisciplinary ap- proach to the scientific literature, his extensive knowledge of forest ecosystems and his early interest in nonlinear dynamics have helped us greatly. We express appreciation to Kevin Nimerfro for generating Diagrams 1, 3, 4, 5, and the cover using the programming package Mathematica. Craig Loehle and Boris Zeide provided review comments that significantly improved the paper. Funded by cooperative agreement #23-91-21, USDA Forest Service, North Central Forest Experiment Station, St. Paul, Minnesota. Yg._. t NAVE A THREE--PART QUE_.gTION,, F_-ACHPARToF:WHICH HA# "THREEPAP,T_.<.,EACFi PART" Of:: F_.AC.HPART oF wHIct4 HA.5 __ "1t4REE MORE PARTS... t_! c_4a EL o. EP-.ACTAL G EOPAgTI_YCoh_FERENCE I G;:_.4-A.-Ti_E AT THB Reprinted courtesy of Omni magazine, June 1994. VoL 16, No. 9. CONTENTS i_ Introduction ....................................................................................................... I 2° Description of Fractals .................................................................................... -
Damit Die Gorillas in Itombwe Auf Lange Sicht Vor Dem Aussterben
Zeitschrift der Berggorilla & Regenwald Direkthilfe Nr. 54 – Juni 2017 gorilla 4 Fährtenleser in Sarambwe 4 Projekte am Tshiaberimu Damit die Gorillas in Itombwe auf lange 6 Cantsbee 6 Ndahura und Isabukuru 7 Gorillas in Tofala Sicht vor dem Aussterben bewahrt werden 7 Superhighway-Neuigkeiten 8 Colo können, muss sich die Population erholen. 8 Torfmoor im Kongobecken 9 Tierhändler verhaftet Die größte Gefahr ist die Wilderei; Men- 10 Treffen in Krefeld schenaffen sind als Wildfleisch begehrt. 5 Studie in Itombwe Mgahinga Gorilla National Park Mt. Tshiaberimu (Kyavirimu) Uganda, Nationalpark Die von uns geförderte Fischzucht hat sich Teil der Virunga Conservation Area als Erfolg für viele Teilnehmer des Projekts Fläche: 33,7 km2, 2600–4127 m erwiesen (S. 4). Wir unterstützen in die- Gorillabesuche zeitweise möglich sem Jahr weitere Gemeindeprojekte in verschiedenen Bereichen. Bwindi Impenetrable National Park Uganda, Nationalpark Fläche: 310 km2, 1190–2607 m Gorillas: ca. 400 Berggorillas Gorillabesuche möglich Réserve Naturelle de Sarambwe Demokratische Republik Kongo Fläche: 9 km2 Gorillas: zeitweise, aus Bwindi Maiko-Nationalpark Parc National des Volcans In einem Teil des Parks soll demnächst Ruanda, Nationalpark eine Bestandsaufnahme stattfinden. Teil der Virunga Conservation Area (dort ca. 480 Berggorillas) Fläche: 120 km2, bis 4507 m Gorillabesuche möglich Parc National des Virunga (Mikeno- Sektor) Demokratische Republik Kongo Nationalpark, Weltnaturerbe Teil der Virunga Conservation Area Gorillabesuche möglich Mt. Tshiaberimu (3100 m) Demokratische Republik Kongo Kahuzi-Biega-Nationalpark Teil des Parc National des Virunga Ende April besuchte ein Team der Gorillas: 7 Grauergorillas UNESCO den Park, um zu prüfen, ob Gorillas sind habituiert, Besuche aus der Status „Weltnaturerbe in Gefahr“ Sicherheitsgründen nicht möglich noch berechtigt ist.