scholarly journals A Note on Topological Properties of Non-Hausdorff Manifolds

2009 ◽  
Vol 2009 ◽  
pp. 1-4
Author(s):  
Steven L. Kent ◽  
Roy A. Mimna ◽  
Jamal K. Tartir
Keyword(s):  
Topological Space ◽  
The Other ◽  
Topological Manifold ◽  
Topological Properties ◽  
Countable Base ◽  
Standard Definition ◽  
Hausdorff Topological Space ◽  
Definition Of

The notion of compatible apparition points is introduced for non-Hausdorff manifolds, and properties of these points are studied. It is well known that the Hausdorff property is independent of the other conditions given in the standard definition of a topological manifold. In much of literature, a topological manifold of dimension is a Hausdorff topological space which has a countable base of open sets and is locally Euclidean of dimension . We begin with the definition of a non-Hausdorff topological manifold.

scholarly journals Enlightenment Views of Hope

2020 ◽  
pp. 61-76
Author(s):  
Claudia Blöser
Keyword(s):  
Immanuel Kant ◽  
David Hume ◽  
Thomas Hobbes ◽  
The Other ◽  
Probability Estimate ◽  
Standard Definition ◽  
General Account ◽  
The Enlightenment ◽  
Definition Of

Abstract This chapter discusses accounts of hope found in the works of important Enlightenment thinkers: René Descartes, Thomas Hobbes, Baruch de Spinoza, David Hume, and Immanuel Kant. The paper’s guiding questions are: Where are discussions of hope located within these thinkers’ works? Do the authors provide an account of what hope is? Do they ascribe a certain function to hope? Most authors of the Enlightenment, with the exception of Kant, write about hope in the context of a general account of the passions. Their characterization of hope closely resembles the “standard definition” of hope in contemporary debates. According to this definition, hope consists of a desire and a belief in the possibility, but not the certainty, of the desired outcome. It turns out, however, that Descartes, Hobbes, and Hume advocate a stronger evidential condition for hope than is common today: According to their view, we do not hope for what we take to be merely possible, no matter how unlikely it is; we hope for what we take to be more likely. Kant’s account differs from the other ones in important respects: He does not treat hope as an affect and he does not require a probability estimate, but grounds hope in faith.

scholarly journals Some fuzzy SP-topological properties

2004 ◽  
Vol 2004 (28) ◽  
pp. 1487-1501
Author(s):  
S. E. Abbas
Keyword(s):  
Topological Space ◽  
Topological Properties ◽  
Fuzzy Topological Space ◽  
Definition Of ◽  
Closed Mappings

We introducer-fuzzy strongly preopen andr-fuzzy strongly preclosed sets in fuzzy topological space in view of the definition of Šostak (1985). We investigate some properties of them. Moreover, the concepts of fuzzy SP-continuous, fuzzy SP-irresolute continuous, fuzzy SP-irresolute open (closed) mappings, and a fuzzy SP-irresolute homeomorphism are introduced and studied. Some of their characteristic properties are considered.

scholarly journals A Case Study of Transporting Urysohn’s Lemma from Topology via Open Sets into Topology via Neighborhoods

2020 ◽  
Vol 28 (3) ◽  
pp. 227-237
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Space ◽  
Type Theory ◽  
The Other ◽  
Logical Framework ◽  
Link Type ◽  
Homotopy Type Theory ◽  
Mizar Mathematical Library ◽  
Abstract Structure ◽  
Definition Of

Summary Józef Białas and Yatsuka Nakamura has completely formalized a proof of Urysohn’s lemma in the article [4], in the context of a topological space defined via open sets. In the Mizar Mathematical Library (MML), the topological space is defined in this way by Beata Padlewska and Agata Darmochwał in the article [18]. In [7] the topological space is defined via neighborhoods. It is well known that these definitions are equivalent [5, 6]. In the definitions, an abstract structure (i.e. the article [17, STRUCT 0] and its descendants, all of them directly or indirectly using Mizar structures [3]) have been used (see [10], [9]). The first topological definition is based on the Mizar structure TopStruct and the topological space defined via neighborhoods with the Mizar structure: FMT Space Str. To emphasize the notion of a neighborhood, we rename FMT TopSpace (topology from neighbourhoods) to NTopSpace (a neighborhood topological space). Using Mizar [2], we transport the Urysohn’s lemma from TopSpace to NTop-Space. In some cases, Mizar allows certain techniques for transporting proofs, definitions or theorems. Generally speaking, there is no such automatic translating. In Coq, Isabelle/HOL or homotopy type theory transport is also studied, sometimes with a more systematic aim [14], [21], [11], [12], [8], [19]. In [1], two co-existing Isabelle libraries: Isabelle/HOL and Isabelle/Mizar, have been aligned in a single foundation in the Isabelle logical framework. In the MML, they have been used since the beginning: reconsider, registration, cluster, others were later implemented [13]: identify. In some proofs, it is possible to define particular functors between different structures, mainly useful when results are already obtained in a given structure. This technique is used, for example, in [15] to define two functors MXR2MXF and MXF2MXF between Matrix of REAL and Matrix of F-Real and to transport the definition of the addition from one structure to the other: [...] A + B -> Matrix of REAL equals MXF2MXR ((MXR2MXF A) + (MXR2MXF B)) [...]. In this paper, first we align the necessary topological concepts. For the formalization, we were inspired by the works of Claude Wagschal [20]. It allows us to transport more naturally the Urysohn’s lemma ([4, URYSOHN3:20]) to the topological space defined via neighborhoods. Nakasho and Shidama have developed a solution to explore the notions introduced in various ways https://mimosa-project.github.io/mmlreference/current/ [16]. The definitions can be directly linked in the HTML version of the Mizar library (example: Urysohn’s lemma http://mizar.org/version/current/html/urysohn3.html#T20).

scholarly journals How directed is a directed network?

2020 ◽  
Vol 7 (9) ◽  
pp. 201138
Author(s):  
R. S. MacKay ◽  
S. Johnson ◽  
B. Sansom
Keyword(s):  
Gene Expression ◽  
Trophic Levels ◽  
Structure Stability ◽  
Directed Network ◽  
Topological Properties ◽  
Cycle Structure ◽  
Standard Definition ◽  
Supply Chain Networks ◽  
Global Language ◽  
Definition Of

The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and percolation. The standard definition of trophic levels, however, borrowed from ecology, suffers from drawbacks such as requiring basal nodes, which limit its applicability. Here we propose simple improved definitions of trophic levels and coherence that can be computed on any directed network. We demonstrate how the method can identify node function in examples including ecosystems, supply chain networks, gene expression and global language networks. We also explore how trophic levels and coherence relate to other topological properties, such as non-normality and cycle structure, and show that our method reveals the extent to which the edges in a directed network are aligned in a global direction.

scholarly journals Λ-Ib-Sets in Topological Space

2019 ◽  
pp. 71-74
Author(s):  
Carlos Andres Granados Ortiz
Keyword(s):  
Topological Space ◽  
The Other ◽  
Other Hand ◽  
New Type ◽  
Definition Of

In this paper we studied a new type of sets that called Λ-Ib-sets, for planting this sets it was necessary to know about b-opens sets were introduced by [4], b-I-opens sets were introduced by [2] and the notion of ideal in topological space was defined by [5], thus we used those concepts and created the sets which were studied, furthermore we studied relationship between Λ-I sets and Λ-Is sets, in fact some properties were planted, on the other hand, the definition of continuity from Λ-Ib-sets was added and some examples are shown.

Documents as Timed Abstract Objects

2012 ◽  
Author(s):  
Claus Huitfeldt ◽  
Fabio Vitali ◽  
Silvio Peroni
Keyword(s):  
Space Time ◽  
The Other ◽  
Abstract Objects ◽  
Abstract Object ◽  
Ontological Status ◽  
Standard Definition ◽  
Definition Of

At Balisage 2009 and 2010 Renear and Wickett discussed problems in reconciling the view that documents are abstract objects with the fact that they can undergo change. In this paper we present an account of documents which we believe is quite common, but which was not discussed by Renear and Wickett. According to this account documents are indeed abstract objects, but this is easily reconciled with the fact that they are created and can undergo change. We then point to a similarity between this account and the notion of so-called space-time slices. We argue that the proposed account of documents as timed abstract objects may be subject to the same kind of criticism that has been raised against the notion of space-time slices. We believe that our account fares no worse than the other accounts given of documents as abstract objects. But it still fails, and we remain agnostic about the ontological status of documents and their relation to abstract objects, as well as about the nature of abstract objects. We conclude that either documents are not (or not related to) abstract objects, or they are (or are related to) abstract objects of a kind which does not correspond to the standard definition of what an abstract object is.

Immunoferritin localization of intracellular antigens in positively stained frozen sections

1978 ◽  
Vol 36 (2) ◽  
pp. 168-169
Author(s):  
K. T. Tokuyasu
Keyword(s):  
Surface Tension ◽  
Water Surface ◽  
Thin Layers ◽  
The Other ◽  
Positive Staining ◽  
Frozen Sections ◽  
The Past ◽  
Ultrathin Frozen Sections ◽  
Definition Of

During the past investigations of immunoferritin localization of intracellular antigens in ultrathin frozen sections, we found that the degree of negative staining required to delineate u1trastructural details was often too dense for the recognition of ferritin particles. The quality of positive staining of ultrathin frozen sections, on the other hand, has generally been far inferior to that attainable in conventional plastic embedded sections, particularly in the definition of membranes. As we discussed before, a main cause of this difficulty seemed to be the vulnerability of frozen sections to the damaging effects of air-water surface tension at the time of drying of the sections.Indeed, we found that the quality of positive staining is greatly improved when positively stained frozen sections are protected against the effects of surface tension by embedding them in thin layers of mechanically stable materials at the time of drying (unpublished).

scholarly journals Language Family Engineering with Product Lines of Multi-level Models

2021 ◽  
Author(s):  
Juan de Lara ◽  
Esther Guerra
Keyword(s):  
Software Engineering ◽  
Arbitrary Number ◽  
Formal Theory ◽  
The Other ◽  
Language Family ◽  
Top Down ◽  
Product Lines ◽  
Modelling Languages ◽  
Multi Level ◽  
Definition Of

AbstractModelling is an essential activity in software engineering. It typically involves two meta-levels: one includes meta-models that describe modelling languages, and the other contains models built by instantiating those meta-models. Multi-level modelling generalizes this approach by allowing models to span an arbitrary number of meta-levels. A scenario that profits from multi-level modelling is the definition of language families that can be specialized (e.g., for different domains) by successive refinements at subsequent meta-levels, hence promoting language reuse. This enables an open set of variability options given by all possible specializations of the language family. However, multi-level modelling lacks the ability to express closed variability regarding the availability of language primitives or the possibility to opt between alternative primitive realizations. This limits the reuse opportunities of a language family. To improve this situation, we propose a novel combination of product lines with multi-level modelling to cover both open and closed variability. Our proposal is backed by a formal theory that guarantees correctness, enables top-down and bottom-up language variability design, and is implemented atop the MetaDepth multi-level modelling tool.

A practical metric for melanopic metrology

2019 ◽  
Vol 51 (8) ◽  
pp. 1178-1191 ◽  
Author(s):  
SM Berman ◽  
RD Clear
Keyword(s):  
Ganglion Cell ◽  
Retinal Ganglion Cell ◽  
Vision Research ◽  
Spectral Response ◽  
Retinal Ganglion ◽  
Standard Definition ◽  
Standard Unit ◽  
Vision Science ◽  
The Past ◽  
Definition Of

Over the past decade, there has been a growing interest in lighting research on the effects of the recently discovered melanopsin receptor (also referred to as the intrinsically photosensitive retinal ganglion cell) and its impacts on health and vision. Presently, there is not a generally accepted metrology for dealing with the spectral response of the melanopsin receptor as applied to both lighting and vision research. A proposition to handle this issue from a vision science perspective has been presented in 2014 in the journal Trends in Neurosciences and from a more lighting perspective in 2017 in Lighting Research and Technology. These propositions are complex, and do not retain the CIE standard definition of a lumen. In this paper, we propose an approach based on effective watts and melanopic/photopic ratios that is both simpler and more closely aligned with CIE standard unit definitions. In addition, we include some practical examples of how such ratios are accessible now, and can be used for both lighting and vision research as well as applications.

Identity, Community and Belonging in gcc States

2018 ◽  
Vol 6 (4) ◽  
pp. 401-428
Author(s):  
Miriam R. Lowi
Keyword(s):  
Religious Affiliation ◽  
Protective Role ◽  
The Other ◽  
National Project ◽  
Scholarly Literature ◽  
National Community ◽  
The Difference ◽  
Definition Of

Studies of identity and belonging in Gulf monarchies tend to privilege tribal or religious affiliation, if not the protective role of the ruler as paterfamilias. I focus instead on the ubiquitous foreigner and explore ways in which s/he contributes to the definition of national community in contemporary gcc states. Building upon and moving beyond the scholarly literature on imported labor in the Gulf, I suggest that the different ‘categories’ of foreigners impact identity and the consolidation of a community of privilege, in keeping with the national project of ruling families. Furthermore, I argue that the ‘European,’ the non-gcc Arab, and the predominantly Asian (and increasingly African) laborer play similar, but also distinct roles in the delineation of national community: while they are differentially incorporated in ways that protect the ‘nation’ and appease the citizen-subject, varying degrees of marginality reflect Gulf society’s perceptions or aspirations of the difference between itself and ‘the other(s).’

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