Basic Facts in Euclidean and Minkowski Plane Geometry

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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Spacetime Geometry

10.1093/oso/9780199658633.003.0011 ◽

2018 ◽

Author(s):

David M. Wittman

Keyword(s):

Coordinate System ◽

Special Relativity ◽

Displacement Vector ◽

Proper Time ◽

Plane Geometry ◽

Spatial Displacement ◽

Spacetime Geometry ◽

Space And Time ◽

Displacement Vectors

This chapter shows that the counterintuitive aspects of special relativity are due to the geometry of spacetime. We begin by showing, in the familiar context of plane geometry, how a metric equation separates frame‐dependent quantities from invariant ones. The components of a displacement vector depend on the coordinate system you choose, but its magnitude (the distance between two points, which is more physically meaningful) is invariant. Similarly, space and time components of a spacetime displacement are frame‐dependent, but the magnitude (proper time) is invariant and more physically meaningful. In plane geometry displacements in both x and y contribute positively to the distance, but in spacetime geometry the spatial displacement contributes negatively to the proper time. This is the source of counterintuitive aspects of special relativity. We develop spacetime intuition by practicing with a graphic stretching‐triangle representation of spacetime displacement vectors.

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Three-point perspective and plane geometry

Journal of Mathematics and the Arts ◽

10.1080/17513470701884180 ◽

2007 ◽

Vol 1 (4) ◽

pp. 213-223 ◽

Cited By ~ 1

Author(s):

Marc Frantz ◽

Annalisa Crannell

Keyword(s):

Plane Geometry

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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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Static and dynamic response of sandwich beams with lattice and pantographic cores

Journal of Sandwich Structures & Materials ◽

10.1177/10996362211033896 ◽

2021 ◽

pp. 109963622110338

Author(s):

Yury Solyaev ◽

Arseniy Babaytsev ◽

Anastasia Ustenko ◽

Andrey Ripetskiy ◽

Alexander Volkov

Keyword(s):

Mechanical Performance ◽

Lattice Structure ◽

Unit Length ◽

Experimental Tests ◽

Plane Geometry ◽

Sandwich Beams ◽

Static Tests ◽

Three Point Bending ◽

The Core ◽

The Impact

Mechanical performance of 3d-printed polyamide sandwich beams with different type of the lattice cores is investigated. Four variants of the beams are considered, which differ in the type of connections between the elements in the lattice structure of the core. We consider the pantographic-type lattices formed by the two families of inclined beams placed with small offset and connected by stiff joints (variant 1), by hinges (variant 2) and made without joints (variant 3). The fourth type of the core has the standard plane geometry formed by the intersected beams lying in the same plane (variant 4). Experimental tests were performed for the localized indentation loading according to the three-point bending scheme with small span-to-thickness ratio. From the experiments we found that the plane geometry of variant 4 has the highest rigidity and the highest load bearing capacity in the static tests. However, other three variants of the pantographic-type cores (1–3) demonstrate the better performance under the impact loading. The impact strength of such structures are in 3.5–5 times higher than those one of variant 4 with almost the same mass per unit length. This result is validated by using numerical simulations and explained by the decrease of the stress concentration and the stress state triaxiality and also by the delocalization effects that arise in the pantographic-type cores.

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Evolutoids and Pedaloids of Minkowski Plane Curves

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-021-01091-1 ◽

2021 ◽

Author(s):

Gülşah Aydın Şekerci ◽

Shyuichi Izumiya

Keyword(s):

Minkowski Plane ◽

Plane Curves

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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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