scholarly journals Statistical behaviour of the leaves of Riccati foliations

2009 ◽  
Vol 30 (1) ◽  
pp. 67-96 ◽  
Author(s):  
CH. BONATTI ◽  
X. GÓMEZ-MONT ◽  
R. VILA-FREYER
Keyword(s):  
Lebesgue Measure ◽  
Geodesic Flow ◽  
Conformal Structure ◽  
Holomorphic Foliation ◽  
Horocycle Flow ◽  
Linear Ordinary Differential Equations ◽  
Positive Time ◽  
Measure Class ◽  
Dimension One

AbstractWe introduce the geodesic flow on the leaves of a holomorphic foliation with leaves of dimension one and hyperbolic, corresponding to the unique complete metric of curvature −1 compatible with its conformal structure. We do these for the foliations associated to Riccati equations, which are the projectivization of the solutions of linear ordinary differential equations over a finite Riemann surface of hyperbolic type S, and may be described by a representation ρ:π1(S)→GL(n,ℂ). We give conditions under which the foliated geodesic flow has a generic repeller–attractor statistical dynamics. That is, there are measures μ− and μ+ such that for almost any initial condition with respect to the Lebesgue measure class the statistical average of the foliated geodesic flow converges for negative time to μ− and for positive time to μ+ (i.e. μ+ is the unique Sinaï, Ruelle and Bowen (SRB)-measure and its basin has total Lebesgue measure). These measures are ergodic with respect to the foliated geodesic flow. These measures are also invariant under a foliated horocycle flow and they project to a harmonic measure for the Riccati foliation, which plays the role of an attractor for the statistical behaviour of the leaves of the foliation.

scholarly journals Relative bifurcation sets and the local dimension of univoque bases

2020 ◽  
pp. 1-33
Author(s):  
PIETER ALLAART ◽  
DERONG KONG
Keyword(s):  
Hausdorff Dimension ◽  
Explicit Expression ◽  
Lebesgue Measure ◽  
Pairwise Disjoint ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Local Dimension ◽  
Dimension Zero ◽  
Image Position ◽  
Dimension One

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$ . We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$ , where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.

Eigenvalues, and instabilities of solitary waves

1992 ◽  
Vol 340 (1656) ◽  
pp. 47-94 ◽  
Keyword(s):  
Solitary Waves ◽  
Wave Speed ◽  
Linear Ordinary Differential Equations ◽  
Finite Dimensional ◽  
Constant Coefficients ◽  
The Stability ◽  
Invariant Functional ◽  
Time Invariant ◽  
Derivatives Of

We study a type of ‘eigenvalue’ problem for systems of linear ordinary differential equations with asymptotically constant coefficients by using the analytic function D (λ) introduced by J. W. Evans (1975) in his study of the stability of nerve impulses. We develop a general theory of D (λ) that clarifies the role of the essential spectrum in applications. New formulae for derivatives of D (λ) are used to study linear exponential instabilities of solitary waves for generalizations of: (1) the Korteweg-de Vries equation (KdV); (2) the Benjamin-Bona-Mahoney equation (BBM); and (3) the regularized Boussinesq equation. A pair of real eigenvalues exists, indicating a non-oscillatory instability, when the ‘momentum’ of the wave (a time-invariant functional associated with the hamiltonian structure of the equation) is a decreasing function of wave speed. Also we explain the mechanism of the transition to instability. Unexpectedly, these transitions are unlike typical transitions to instability in finite-dimensional hamiltonian systems. Instead they can be understood in terms of the motion of poles of the resolvent formula extended to a multi-sheeted Riemann surface. Finally, for a generalization of the KdV-Burgers equation (a model for bores), we show that a conjectured transition to instability does not involve real eigenvalues emerging from the origin, suggesting an oscillatory type of instability.

scholarly journals Role of Maternal Autologous Neutralizing Antibody in Selective Perinatal Transmission of Human Immunodeficiency Virus Type 1 Escape Variants

2006 ◽  
Vol 80 (13) ◽  
pp. 6525-6533 ◽  
Author(s):  
Ruth Dickover ◽  
Eileen Garratty ◽  
Karina Yusim ◽  
Catherine Miller ◽  
Bette Korber ◽  
...  
Keyword(s):  
Neutralizing Antibody ◽  
Perinatal Transmission ◽  
Infected Infant ◽  
Immunodeficiency Virus ◽  
Positive Time ◽  
Cross Neutralization ◽  
Hiv 1 ◽  
Time Point

ABSTRACT Perinatal human immunodeficiency virus type 1 (HIV-1) transmission is characterized by acquisition of a homogeneous viral quasispecies, yet the selective factors responsible for this genetic bottleneck are unclear. We examined the role of maternal autologous neutralizing antibody (aNAB) in selective transmission of HIV-1 escape variants to infants. Maternal sera from 38 infected mothers at the time of delivery were assayed for autologous neutralizing antibody activity against maternal time-of-delivery HIV-1 isolates in vitro. Maternal sera were also tested for cross-neutralization of infected-infant-first-positive-time-point viral isolates. Heteroduplex and DNA sequence analyses were then performed to identify the initial infecting virus as a neutralization-sensitive or escape HIV-1 variant. In utero transmitters (n = 14) were significantly less likely to have aNAB to their own HIV-1 strains at delivery than nontransmitting mothers (n = 17, 14.3% versus 76.5%, P = 0.003). Cross-neutralization assays of infected-infant-first-positive-time-point HIV-1 isolates indicated that while 14/21 HIV-1-infected infant first positive time point isolates were resistant to their own mother's aNAB, no infant isolate was inherently resistant to antibody neutralization by all sera tested. Furthermore, both heteroduplex (n = 21) and phylogenetic (n = 9) analyses showed that selective perinatal transmission and/or outgrowth of maternal autologous neutralization escape HIV-1 variants occurs in utero and intrapartum. These data indicate that maternal autologous neutralizing antibody can exert powerful protective and selective effects in perinatal HIV-1 transmission and therefore has important implications for vaccine development.

ON THE GROWTH OF HOLOMORPHIC PROJECTIVE FOLIATIONS

2002 ◽  
Vol 13 (07) ◽  
pp. 695-726 ◽  
Author(s):  
B. C. AZEVEDO SCÁRDUA ◽  
J. C. CANILLE MARTINS
Keyword(s):  
Algebraic Curves ◽  
Hermitian Manifold ◽  
Holomorphic Foliation ◽  
Global Dynamics ◽  
Subexponential Growth ◽  
Codimension One ◽  
Singular Foliations ◽  
Invariant Algebraic Curves ◽  
Dimension One ◽  
Compact Hermitian Manifold

In the theory of real (non-singular) foliations, the study of the growth of the leaves has proved to be useful in the comprehension of the global dynamics as the existence of compact leaves and exceptional minimal sets. In this paper we are interested in the complex version of some of these basic results. A natural question is the following: What can be said of a codimension one (possibly singular) holomorphic foliation on a compact hermitian manifold M exhibiting subexponential growth for the leaves? One of the first examples comes when we consider the Fubini–Study metric on [Formula: see text] and dimension one foliations. In this case, under some non-degeneracy hypothesis on the singularities, we may classify the foliation as a linear logarithmic foliation. In particular, the limit set of ℱ is a union of singularities and invariant algebraic curves. Applications of this and other results we prove are given to the general problem of uniformization of the leaves of projective foliations.

scholarly journals COVID-19 – Bewährungsprobe 2.0 für die deutschen Förderbanken

2020 ◽  
Vol 89 (2) ◽  
pp. 81-105
Author(s):  
Iris Bethge-Krauß
Keyword(s):  
Financial Crisis ◽  
Federal Government ◽  
Economic Losses ◽  
Development Banks ◽  
Dimension One ◽  
Acute Crisis

Zusammenfassung: Die Corona-Pandemie und die ergriffenen Maßnahmen zur Eindämmung bedeuten erhebliche wirtschaftliche Einbußen für viele Unternehmen in Deutschland. Die durch die Corona-Krise ausgelöste Rezession dürfte mindestens so schlimm wie in der letzten Finanzkrise ausfallen. Die Politik handelte sehr entschlossen und konnte dabei auf leistungsstarke Förderbanken setzen. Denn zur Abmilderung akuter Krisenerscheinungen leisten Förderbanken einen entscheidenden Beitrag. Als Krise ganz neuer Dimension kann man dabei von einer Bewährungsprobe 2.0 sprechen. Im vorliegenden Aufsatz wird die Rolle der Förderbanken des Bundes und der Länder als Stabilitätsanker in Zeiten der Corona-Krise dargestellt. Summary: The corona pandemic and the measures taken to contain it mean considerable economic losses for many companies in Germany. The recession triggered by the corona crisis is likely to be at least as severe as during the last financial crisis. Politicians acted very decisively and were able to rely on powerful development banks. This is because development banks make a decisive contribution to mitigating acute crisis symptoms. As a crisis of a completely new dimension, one can speak of a test of endurance 2.0. This paper describes the role of the development banks of the Federal Government and the Länder as an anchor of stability in times of the corona crisis.

Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form

1999 ◽  
Vol 19 (4) ◽  
pp. 1093-1109 ◽  
Author(s):  
WILLIAM A. VEECH
Keyword(s):  
Riemann Surface ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Quadratic Differential ◽  
Probability Measures ◽  
Locally Finite ◽  
Horocycle Flow ◽  
Closed Riemann Surface ◽  
Borel Probability ◽  
Uniquely Ergodic

We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.

23. Criminal justice principles

2021 ◽  
pp. 693-717
Author(s):  
Steve Case ◽  
Phil Johnson ◽  
David Manlow ◽  
Roger Smith ◽  
Kate Williams
Keyword(s):  
Criminal Justice ◽  
Restorative Justice ◽  
Rule Of Law ◽  
Due Process ◽  
The Rule Of Law ◽  
Independent Judiciary ◽  
The World ◽  
Democratic Systems ◽  
Dimension One

This chapter describes the key principles of the criminal justice system. These key principles behind the abstract aims of criminal justice include the rule of law, adversarial justice, and restorative justice. The chapter particularly focuses on the rule of law doctrine to illustrate its status as the ultimate authority for democratic systems of justice around the world, but it also reflects on three of its supplementary concepts: an independent judiciary, due process, and human rights. Meanwhile, the traditional adversarial contest in a courtroom between two opposing sides means such hearings can lack impartiality as the role of the judge is limited to ensuring that the rules are followed. The restorative justice principle offers a different dimension, one that prioritises repairing the harms suffered by the injured parties.

A generalization of the Mariot theorem on congruences of null geodesics

1986 ◽  
Vol 405 (1828) ◽  
pp. 41-48 ◽  
Keyword(s):  
Vector Field ◽  
Conformal Structure ◽  
Dimensional Manifold ◽  
Hopf Fibration ◽  
Null Geodesics ◽  
Null Congruence ◽  
Dimension One

It is shown that the property of a congruence of curves to consist of null geodesics can be defined in terms of a distribution of a co-dimension one, without reference to the conformal structure of the underlying differen­tiable manifold: if k is the vector field tangent to the congruence and k is a 1-form characterizing the distribution, then the congruence is said to be null if k ˩ k = 0 and geodesic if, and only if, k ∧£ k = 0. The geodesic property of the congruence, on an n -dimensional manifold, means that if F is an ( n —2)-form such that k ˩ F = 0 and k ∧ F = 0, then k ∧ d F = 0. A twisting geodesic null congruence on S 1 ᵡ S 2 l + 1 , associated with the Hopf fibration S 2 l + 1 → CP l , is constructed as an illustration.

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