Basic Facts about the Geodesic Flow

1978 ◽  
pp. 13-52
Author(s):  
Arthur L. Besse
Keyword(s):  
Geodesic Flow ◽  
Basic Facts

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Dramatis Institutiones

2017 ◽  
Author(s):  
C. Randall Henning
Keyword(s):  
European Union ◽  
Central Bank ◽  
International Monetary Fund ◽  
Euro Area ◽  
European Central Bank ◽  
The European Union ◽  
European Central ◽  
Regime Complex ◽  
Basic Facts ◽  
Institutional Conflict

The regime complex for crisis finance in the euro area included the European Council, Council of the European Union, and Eurogroup in addition to the three institutions of the troika. As the member states acted largely, though not exclusively, through the council system, these bodies stood at the center of the institutional mix. This chapter reviews the institutions as a prelude to examining the dilemmas that confronted them over the course of the crises. It presents a brief review of some of the basic facts about their origins, membership, and organization. Each section then delves more deeply into these institutions’ governance and principles to understand their capabilities and strategic challenges. As a consequence of different mandates and design, the European Commission, European Central Bank, and International Monetary Fund diverged with respect to their approach to financing, adjustment, conditionality, and debt sustainability. This divergence set the stage for institutional conflict in the country programs.

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

Processing subject focus across two Spanish varieties

Probus ◽  
2020 ◽  
Vol 32 (1) ◽  
pp. 93-127
Author(s):  
Bradley Hoot ◽  
Tania Leal
Keyword(s):  
Language Processing ◽  
Sentence Processing ◽  
Word Order ◽  
Judgment Task ◽  
Spanish Speakers ◽  
Experimental Results ◽  
Final Position ◽  
Reading Task ◽  
New Information ◽  
Basic Facts

AbstractLinguists have keenly studied the realization of focus – the part of the sentence introducing new information – because it involves the interaction of different linguistic modules. Syntacticians have argued that Spanish uses word order for information-structural purposes, marking focused constituents via rightmost movement. However, recent studies have challenged this claim. To contribute sentence-processing evidence, we conducted a self-paced reading task and a judgment task with Mexican and Catalonian Spanish speakers. We found that movement to final position can signal focus in Spanish, in contrast to the aforementioned work. We contextualize our results within the literature, identifying three basic facts that theories of Spanish focus and theories of language processing should explain, and advance a fourth: that mismatches in information-structural expectations can induce processing delays. Finally, we propose that some differences in the existing experimental results may stem from methodological differences.

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

The algebraic integrability of geodesic flow onSO(4)

1982 ◽  
Vol 67 (2) ◽  
pp. 297-331 ◽  
Author(s):  
M. Adler ◽  
P. van Moerbeke
Keyword(s):  
Geodesic Flow ◽  
Algebraic Integrability

scholarly journals Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces

2015 ◽  
Vol 37 (3) ◽  
pp. 939-970 ◽  
Author(s):  
RUSSELL RICKS
Keyword(s):  
Geodesic Flow ◽  
Isometric Action ◽  
Structural Result ◽  
Curved Manifolds ◽  
Rank One ◽  
Unit Speed ◽  
Flat Strip ◽  
Geodesically Complete ◽  
Positively Curved ◽  
General Technical

Let$X$be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group$\unicode[STIX]{x1D6E4}$with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of$X$modulo the$\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in$\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when$X$has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when$X$is a tree with all edge lengths in$c\mathbb{Z}$for some$c>0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

scholarly journals Basic Facts of Body Water and Ions

1960 ◽  
Vol 60 (11) ◽  
pp. 1661
Author(s):  
Lutie C. Leavell ◽  
Stewart M. Brooks
Keyword(s):  
Body Water ◽  
Basic Facts
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