Completeness in affine and statistical geometry

The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple (g,∇, ∇*) of a Riemannian metric g and two affine connections ∇ and ∇*. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning ∇- and ∇*-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.

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THE STATISTICAL GEOMETRY OF THE STRUCTURE OF THE MOLECULAR DYNAMIC MODEL OF LIQUID AND AMORPHOUS ALUMINIUM

Le Journal de Physique Colloques ◽

10.1051/jphyscol:1980872 ◽

1980 ◽

Vol 41 (C8) ◽

pp. C8-284-C8-288 ◽

Cited By ~ 3

Author(s):

V. A. Poluchin ◽

M. M. Dzugutov ◽

V. F. Uchov ◽

R. A. Vatolin

Keyword(s):

Molecular Dynamic ◽

Dynamic Model ◽

Statistical Geometry ◽

Molecular Dynamic Model

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Dramatis Institutiones

10.1093/oso/9780198801801.003.0003 ◽

2017 ◽

Cited By ~ 1

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.

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Probability and Random Processes

10.1093/oso/9780198821939.003.0002 ◽

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.

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Processing subject focus across two Spanish varieties

Probus ◽

10.1515/probus-2019-0004 ◽

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.

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Affine connections and frames of reference in classical mechanics

Reports on Mathematical Physics ◽

10.1016/0034-4877(91)90036-m ◽

1991 ◽

Vol 30 (1) ◽

pp. 21-32 ◽

Cited By ~ 4

Author(s):

N. Kadianakis

Keyword(s):

Classical Mechanics ◽

Frames Of Reference ◽

Affine Connections

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Statistical geometry of hard-sphere systems

Journal of the Chemical Society Faraday Transactions 2 Molecular and Chemical Physics ◽

10.1039/f29807600693 ◽

1980 ◽

Vol 76 ◽

pp. 693 ◽

Cited By ~ 50

Author(s):

Robin J. Speedy

Keyword(s):

Hard Sphere ◽

Statistical Geometry

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CHOICE-FREE STONE DUALITY

Journal of Symbolic Logic ◽

10.1017/jsl.2019.11 ◽

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.

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