scholarly journals MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES

2020 ◽  
Vol 8 ◽  
Author(s):  
CLINTON T. CONLEY ◽  
ANDREW S. MARKS ◽  
SPENCER T. UNGER
Keyword(s):  
General Question ◽  
Abstract System ◽  
Congruence Relations ◽  
Paradoxical Decompositions

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$ -divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$ -sphere. This answers a question by Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_{2}(\mathbb{Z})$ on $\mathsf{P}^{1}(\mathbb{R})$ . The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.

Identifying Community Interests in International Law

2018 ◽  
Author(s):  
Rüdiger Wolfrum
Keyword(s):  
International Law ◽  
Dispute Settlement ◽  
International Community ◽  
Available P ◽  
General Question ◽  
Global Commons ◽  
Settlement Systems ◽  
Community Interest ◽  
National Jurisdiction ◽  
Advisory Opinions

This chapter explores the general question of how to establish that the regulation of a certain matter constitutes a matter of community-wide concern, which is the necessary step for the recognition of community obligation. The hypothesis is that such a qualification must, first, be well founded factually and, secondly, accepted as such in a legal or political legitimizing process. On this basis, the chapter suggests that the governance of spaces beyond national jurisdiction constitutes a community interest and has to be guided by the interests of the international community. Exploring this question with respect to key common spaces and particular issues, the chapter notes the difficulty of most of the dispute settlement systems, which, being bilateral, are not fully adequate to address questions related to the management of global commons as well as for the protection of the environment. To avoid this difficulty, the chapter suggests greater reliance on advisory opinions where available.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals A New Methodological Approach to Correlate Protective and Microscopic Properties by Soft X-ray Microscopy and Solid State NMR Spectroscopy: The Case of Cusa’s Stone

2021 ◽  
Vol 11 (13) ◽  
pp. 5767
Author(s):  
Veronica Ciaramitaro ◽  
Alberto Spinella ◽  
Francesco Armetta ◽  
Roberto Scaffaro ◽  
Emmanuel Fortunato Gulino ◽  
...  
Keyword(s):  
Solid State ◽  
Nmr Spectroscopy ◽  
Solid State Nmr ◽  
Methodological Approach ◽  
Polymer Chains ◽  
General Question ◽  
Protective Agent ◽  
Vapour Permeability ◽  
X Ray ◽  
Solid State Nmr Spectroscopy

Hydrophobic treatment is one of the most important interventions usually carried out for the conservation of stone artefacts and monuments. The study here reported aims to answer a general question about how two polymers confer different protective performance. Two fluorinated-based polymer formulates applied on samples of Cusa’s stone confer a different level of water repellence and water vapour permeability. The observed protection action is here explained on the basis of chemico-physical interactions. The distribution of the polymer in the pore network was investigated using scanning electron microscopy and X-ray microscopy. The interactions between the stone substrate and the protective agents were investigated by means of solid state NMR spectroscopy. The ss-NMR findings reveal no significant changes in the chemical neighbourhood of the observed nuclei of each protective agent when applied onto the stone surface and provide information on the changes in the organization and dynamics of the studied systems, as well as on the mobility of polymer chains. This allowed us to explain the different macroscopic behaviours provided by each protective agent to the stone substrate.

The pace of “the good life”: Connecting past, present, and future in the context of a housing affordability crisis

2021 ◽  
pp. 0961463X2098781
Author(s):  
Petr Kubala ◽  
Tomáš Hoření Samec
Keyword(s):  
Life Course ◽  
Good Life ◽  
Social Inequalities ◽  
Housing Affordability ◽  
General Question ◽  
The Good Life ◽  
The Czech Republic ◽  
Ideal Types ◽  
Depth Interviews ◽  
The Ideal

This article focuses on the topic of the young adult’s cleft habitus influenced by a housing affordability crisis in the Czech Republic and examines how this situation affects the young adult’s relation to the imagination of a temporally structured life course and synchronization of life spheres (housing, family, and work). This article is based on qualitative in-depth interviews conducted in the four cities most affected by the house and rent price increase. The general question addresses if and how social inequalities, sharpened by the current housing affordability crisis, affect the process of narrative life course coherence creation (the connection of past, present, and future) in relation to an orientation toward a vision of “the good life.” We furthermore complement the already existing ideal types of the young adult’s relation toward time— confident continuity and cautious contingency—with two other two types— cautious continuity and total contingency—defined on the basis of our data. We argue that the ability of young adults to envision a coherent future is related to the feeling of secured housing and that the idea of the good life is depicted to a large extent through the ideal of homeownership, although the precarity of the housing market makes homeownership harder to reach for those from unprivileged backgrounds.

Congruence relations on almost distributive lattices-II

2019 ◽  
pp. 2150005
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
Distributive Lattice ◽  
Congruence Relation ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Ideal Formula ◽  
Congruence Relations

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.

scholarly journals The action of anæsthetics on living tissues. Part I.— The action on isolated nerve

1906 ◽  
Vol 77 (517) ◽  
pp. 267-283 ◽  
Keyword(s):  
Carbon Dioxide ◽  
General Question ◽  
Nerve Action ◽  
Physical Action ◽  
The Subject ◽  
Living Tissues

The action of anaesthetics on isolated nerve has already been studied by Waller* as regards the effect on the negative variation in the sciatic of the frog, and by myself in mammalian nerves.f In the course of these researches it became evident that the anaesthetics used (chloroform, ether, carbon dioxide) affected not only the negative variation, but also the injury current, and as this action has not been studied before, as far as I am aware, it seemed desirable to investigate the matter not only in nerve, but also in other tissues. The inquiry falls naturally under two heads: first, in how far the phenomena throw light on the processes of nerve action ; and, secondly, as regards the chemical and physical action of anaesthetics on the animal protoplasm generally. These are obviously only parts of the same story, but for convenience I have considered the subject mainly under the first heading in this part, leaving the more general question to a future occasion.

Industry-Sponsored Medical Education — In the Quest for Professional Integrity and Legal Certainty

2007 ◽  
Vol 14 (4) ◽  
pp. 313-319
Author(s):  
Benedikt Buchner
Keyword(s):  
Medical Education ◽  
Self Regulation ◽  
State Regulation ◽  
General Question ◽  
Legal Certainty ◽  
European Directive ◽  
Starting Point ◽  
Good Starting Point ◽  
Advertising Law ◽  
The Right

AbstractIndustry-sponsored medical education is a much disputed issue. So far, there has been no regulatory framework which provides clear and definite rules as to whether and under what circumstances the sponsorship of medical education is acceptable. State regulation does not exist, or confines itself to a very general principle. Professional regulation, even though applied frequently, is rather vague and indefinite, raising the general question as to whether self-regulation is the right approach at all. Certainly, self-regulation by industry cannot and should not replace other regulatory approaches. Ultimately, advertising law in general and the European Directive 2001/83/EC specifically, might be a good starting point in providing legal certainty and ensuring the independence of medical education. Swiss advertising law illustrates how the principles of the European Directive could be implemented clearly and unambiguously.

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