Spaces with a Unique Uniformity

1983 ◽  
Vol 35 (6) ◽  
pp. 1001-1009
Author(s):  
Richard H. Warren
Keyword(s):  
Companion Paper ◽  
The Other ◽  
Completely Regular ◽  
Continuous Maps ◽  
Uniformly Continuous ◽  
Compact Subsets

The major results in this paper are nine characterizations of completely regular spaces with a unique compatible uniformity. All prior results of this type assumed that the space is Tychonoff (i.e., completely regular and Hausdorff) until the appearance of a companion paper [9] which began this study. The more important characterizations use quasi-uniqueness of R1-compactifications which relate to uniqueness of T2-comPactifications. The features of the other characterizations are: (i) compact subsets linked to Cauchy filters, (ii) C- and C*-embeddings, and (iii) lifting continuous maps to uniformly continuous maps.Section 2 contains information on T0-identification spaces which we will use later in the paper. In Section 3 several properties of uniform identification spaces are developed so that they can be used later. The nine characterizations are established in Section 4. Also it is shown that a space with a unique compatible uniformity is normal if and only if each of its closed subspaces has a unique compatible uniformity.

LIFTING THE FUNCTOR ITO THE CATEGORY OF UNIFORM SPACES

2020 ◽  
Vol 4 (1) ◽  
pp. 29-39
Author(s):  
Dilrabo Eshkobilova ◽  
Keyword(s):  
Compact Support ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Continuous Maps ◽  
Uniformly Continuous

Uniform properties of the functor Iof idempotent probability measures with compact support are studied. It is proved that this functor can be lifted to the category Unif of uniform spaces and uniformly continuous maps

scholarly journals Compactifications of totally bounded quasi-uniform spaces

1986 ◽  
Vol 28 (1) ◽  
pp. 31-36 ◽  
Author(s):  
P. Fletcher ◽  
W. F. Lindgren
Keyword(s):  
Hausdorff Space ◽  
Uniform Space ◽  
Continuous Extension ◽  
Sufficient Condition ◽  
Uniform Spaces ◽  
Completely Regular ◽  
Totally Bounded ◽  
Continuous Map ◽  
Total Boundedness ◽  
Uniformly Continuous

The notation and terminology of this paper coincide with that of reference [4], except that here the term, compactification, refers to a T1-space. It is known that a completely regular totally bounded Hausdorff quasi-uniform space (X, ) has a Hausdorff compactification if and only if contains a uniformity compatible with ℱ() [4, Theorem 3.47]. The use of regular filters by E. M. Alfsen and J. E. Fenstad [1] and O. Njåstad [5], suggests a construction of a compactification, which differs markedly from the construction obtained in [4]. We use this construction to show that a totally bounded T1 quasi-uniform space has a compactification if and only if it is point symmetric. While it is pleasant to have a characterization that obtains for all T1-spaces, the present construction has several further attributes. Unlike the compactification obtained in [4], the compactification given here preserves both total boundedness and uniform weight, and coincides with the uniform completion when the quasi-uniformity under consideration is a uniformity. Moreover, any quasi-uniformly continuous map from the underlying quasi-uniform space of the compactification onto any totally bounded compact T1-space has a quasi-uniformly continuous extension to the compactification. If is the Pervin quasi-uniformity of a T1-space X, the compactification we obtain is the Wallman compactification of (X, ℱ ()). It follows that our construction need not provide a Hausdorff compactification, even when such a compactification exists; but we obtain a sufficient condition in order that our compactification be a Hausdorff space and note that this condition is satisfied by all uniform spaces and all normal equinormal quasi-uniform spaces. Finally, we note that our construction is reminiscent of the completion obtained by Á. Császár for an arbitrary quasi-uniform space [2, Section 3]; in particular our Theorem 3.7 is comparable with the result of [2, Theorem 3.5].

scholarly journals OLCI A/B Tandem Phase Analysis, Part 1: Level 1 Homogenisation and Harmonisation

2020 ◽  
Vol 12 (11) ◽  
pp. 1804 ◽  
Author(s):  
Nicolas Lamquin ◽  
Sébastien Clerc ◽  
Ludovic Bourg ◽  
Craig Donlon
Keyword(s):  
European Space Agency ◽  
Companion Paper ◽  
The Other ◽  
Satellite Mission ◽  
Space Agency ◽  
The Future ◽  
Level 1 ◽  
Level 2 ◽  
Very High ◽  
Sentinel 2

Copernicus is a European system for monitoring the Earth in support of European policy. It includes the Sentinel-3 satellite mission which provides reliable and up-to-date measurements of the ocean, atmosphere, cryosphere, and land. To fulfil mission requirements, two Sentinel-3 satellites are required on-orbit at the same time to meet revisit and coverage requirements in support of Copernicus Services. The inter-unit consistency is critical for the mission as more S3 platforms are planned in the future. A few weeks after its launch in April 2018, the Sentinel-3B satellite was manoeuvred into a tandem configuration with its operational twin Sentinel-3A already in orbit. Both satellites were flown only thirty seconds apart on the same orbit ground track to optimise cross-comparisons. This tandem phase lasted from early June to mid October 2018 and was followed by a short drift phase during which the Sentinel-3B satellite was progressively moved to a specific orbit phasing of 140° separation from the sentinel-3A satellite. In this paper, an output of the European Space Agency (ESA) Sentinel-3 Tandem for Climate study (S3TC), we provide a full methodology for the homogenisation and harmonisation of the two Ocean and Land Colour Instruments (OLCI) based on the tandem phase. Homogenisation adjusts for unavoidable slight spatial and spectral differences between the two sensors and provide a basis for the comparison of the radiometry. Persistent radiometric biases of 1–2% across the OLCI spectrum are found with very high confidence. Harmonisation then consists of adjusting one instrument on the other based on these findings. Validation of the approach shows that such harmonisation then procures an excellent radiometric alignment. Performed on L1 calibrated radiances, the benefits of harmonisation are fully appreciated on Level 2 products as reported in a companion paper. Whereas our methodology aligns one sensor to behave radiometrically as the other, discussions consider the choice of the reference to be used within the operational framework. Further exploitation of the measurements indeed provides evidence of the need to perform flat-fielding on both payloads, prior to any harmonisation. Such flat-fielding notably removes inter-camera differences in the harmonisation coefficients. We conclude on the extreme usefulness of performing a tandem phase for the OLCI mission continuity as well as for any optical mission to which the methodology presented in this paper applies (e.g., Sentinel-2). To maintain the climate record, it is highly recommended that the future Sentinel-3C and Sentinel-3D satellites perform tandem flights when injected into the Sentinel-3 time series.

scholarly journals Action minimization and macroscopic interface motion under forced displacement

2018 ◽  
Vol 24 (2) ◽  
pp. 765-792 ◽  
Author(s):  
Panagiota Birmpa ◽  
Dimitrios Tsagkarogiannis
Keyword(s):  
Stochastic Process ◽  
Evolution Equation ◽  
Fixed Time ◽  
Companion Paper ◽  
The Other ◽  
Final Position ◽  
Dimensional Model ◽  
Interface Motion ◽  
One Dimensional ◽  
Non Local

We study an one dimensional model where an interface is the stationary solution of a mesoscopic non local evolution equation which has been derived by a microscopic stochastic spin system. Deviations from this evolution equation can be quantified by obtaining the large deviations cost functional from the underlying stochastic process. For such a functional, derived in a companion paper, we investigate the optimal way for a macroscopic interface to move from an initial to a final position distant by R within fixed time T. We find that for small values of R∕T the interface moves with a constant speed, while for larger values there appear nucleations of the other phase ahead of the front.

Real-valued models with metric equality and uniformly continuous predicates

1982 ◽  
Vol 47 (4) ◽  
pp. 772-792 ◽  
Author(s):  
Michael Katz
Keyword(s):  
Metric Spaces ◽  
Uniform Continuity ◽  
The Other ◽  
Inference Rules ◽  
Finite Sets ◽  
Uniformly Continuous

AbstractTwo real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.

Inflection and Derivation

2019 ◽  
pp. 241-261
Author(s):  
Andrew Spencer
Keyword(s):  
The Other ◽  
Past Tense ◽  
Personal Pronoun ◽  
Past Participle ◽  
Double Object ◽  
Completely Regular ◽  
Auxiliary Verbs ◽  
The Past ◽  
Irregular Verbs ◽  
Categorial Status

The chapter presents an overview of phenomena which pose important problems of description and analysis. I focus on the inflectional system, which has undergone severe attrition and shows idiosyncrasies typical of such systems. For nominals I describe the personal pronoun paradigm and the ‘possessive -s’ clitic/phrasal affix. The controversial categorial status of adverbs in -ly is discussed, while for verbs, all the subcategories prove to be highly problematical. For instance, only 50 irregular verbs distinguish past tense from past participle (e.g. wrote/written), so it is not even clear whether the past participle category is a highly restricted subcategory, with the vast majority of verbs showing past tense/past participle syncretism, or whether this is a case of ‘overdifferentiation’, like the forms am, are, were of the verb BE. On the other hand, the polyfunctionality of the completely regular -ing suffix, which derives verb, noun, and adjective forms, also poses serious unresolved problems. Auxiliary verbs and related phenomena alternate between periphrastic, clitic, and genuinely morphological (affixal) constructions. The chapter concludes with consideration of those aspects of derivational morphology which seem to be indisputably productive, hence part of the grammar, including (certain types of) event nominalization, some cases of double object alternation, and morphosemantic mismatches of the kind electrical engineering ⇒ electrical engineer.

scholarly journals On the Nemytskii Operator in the Space of Functions of Bounded (p, 2, α)-Variation with Respect to the Weight Function

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Wadie Aziz
Keyword(s):  
Weight Function ◽  
Bounded Variation ◽  
The Other ◽  
Affine Function ◽  
Nemytskii Operator ◽  
Other Hand ◽  
Generator Function ◽  
Uniformly Continuous

AbstractIn this paper, we consider the Nemytskii operator (Hf)(t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p,2,α)-variation (with respect to a weight function α) into the space of functions of bounded (q,2,α)-variation (with respect to α) 1<q<p, then H is of the form (Hf)(t) = A(t)f(t)+B(t). On the other hand, if 1<p<q then H is constant. It generalize several earlier results of this type due to Matkowski-Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

scholarly journals Digital products with $ PN_k $-adjacencies and the almost fixed point property in $ DTC_k^\blacktriangle $

2021 ◽  
Vol 6 (10) ◽  
pp. 11550-11567
Author(s):  
Jeong Min Kang ◽  
◽  
Sang-Eon Han ◽  
Sik Lee ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Digital Images ◽  
Fixed Point Property ◽  
The Other ◽  
Point Property ◽  
Digital Products ◽  
Adjacency Relation ◽  
Digital Product ◽  
New Concepts ◽  
Continuous Maps

<abstract><p>Given two digital images $ (X_i, k_i), i \in \{1, 2\} $, first of all we establish a new $ PN_k $-adjacency relation in a digital product $ X_1 \times X_2 $ to obtain a relation set $ (X_1 \times X_2, PN_k) $, where the term $ ''$$ PN $" means $ ''$pseudo-normal". Indeed, a $ PN $-$ k $-adjacency is softer or broader than a normal $ k $-adjacency. Next, the present paper initially develops both the notion of $ PN $-$ k $-continuity and a $ PN $-$ k $-isomorphism. Furthermore, it proves that these new concepts, the $ PN $-$ k $-continuity and a $ PN $-$ k $-isomorphism, need not be equal to the typical $ k $-continuity and a $ k $-isomorphism, respectively. Precisely, we prove that none of the typical $ k $-continuity (<italic>resp.</italic> typical $ k $-isomorphism) and the $ PN $-$ k $-continuity (<italic>resp.</italic> $ PN $-$ k $-isomorphism) implies the other. Then we prove that for each $ i \in \{1, 2\} $, the typical projection map $ P_i: X_1 \times X_2 \to X_i $ preserves a $ PN_k $-adjacency relation in $ X_1 \times X_2 $ to the $ k_i $-adjacency relation in $ (X_i, k_i) $. In particular, using a $ PN $-$ k $-isomorphism, we can classify digital products with $ PN_k $-adjacencies. Furthermore, in the category of digital products with $ PN_k $-adjacencies and $ PN $-$ k $-continuous maps between two digital products with $ PN_k $-adjacencies, denoted by $ DTC_k^\blacktriangle $, we finally study the (almost) fixed point property of $ (X_1 \times X_2, PN_k) $.</p></abstract>

scholarly journals Etude Histopathologique et Neurochimique Suite a des Lesions Unilaterales du Locus Coeruleus Chez le Rat et de la Region Postlocus Chez le Chat (Part 2)

1979 ◽  
Vol 6 (2) ◽  
pp. 113-122 ◽  
Author(s):  
R. Marchand ◽  
P. Chauvel ◽  
M. Fantino ◽  
L.J. Poirier
Keyword(s):  
Spinal Cord ◽  
Cerebral Cortex ◽  
Locus Coeruleus ◽  
Marked Decrease ◽  
Cerebellar Nuclei ◽  
Companion Paper ◽  
Important Change ◽  
The Other ◽  
Contralateral Cortex

SummaryUnilateral lesions involving the area immediately caudal to the locus coeruleus in the cat are associated with slight decreases of noradrenaline (NA) in both sides of the spinal cord without any important change of serotonin (5-HT) concentration in the spinal cord and of N A and 5-HT concentrations in the cerebral cortex of both sides. In other cats with similar lesions involving the same area bilaterally (postlocus lesions) NA is markedly decreased and 5-H T slightly decreased in the hypothalamus and spinal cord but the concentrations of NA and 5-HT of the cerebral cortex, striât urn and thalamus are not conspicuously modified by such postlocus lesions.Unilateral lesions in the area of the loci coeruleus and subcoeruleus in the rat are associated with a marked decrease of noradrenaline (NA) in the ipsilateral cerebral cortex without any important change of NA in the contralateral cortex and in the spinal cord of both sides. Under such conditions the concentrations of serotonin (5-HT) are not significantly modified in the cerebral cortex and spinal cord of both sides. NA and 5-HT concentrations of the cerebral cortex of both sides are unmodified by unilateral lesions of the cerebellar nuclei in the rat. In the latter group of animals slight increases of NA and 5-HT in the spinal cord are significant in comparison to the values obtained in the control animals but not from one side of the spinal cord to the other.These results further support the suggestion that the noradrenergic coeruleo-cortical pathway originating in the loci coeruleus and subcoeruleus ascends on the same side and ends in the ipsilateral cerebral cortex. In the light of the results of this investigation and of those reported in a companion paper (Marchand et ai, 1979) NA fibers reaching the hypothalamus and spinal cord most likely arise from neurons located caudally in respect to the locus coeruleus. Moreover NA fibers ending in the hypothalamus do not ascend in the dorsolateral part of the isthmic area and, therefore, have a different course than the N A fibers reaching the cerebral cortex and thalamus.

The half-wetted bearing. Part 2: Potential application in low load contacts

2003 ◽  
Vol 217 (1) ◽  
pp. 15-26 ◽  
Author(s):  
H. A. Spikes
Keyword(s):  
Microelectromechanical Systems ◽  
Hydrodynamic Lubrication ◽  
Hard Disk Drives ◽  
Critical Shear Stress ◽  
Slip Boundary Condition ◽  
Companion Paper ◽  
The Other ◽  
Current Evidence ◽  
Slip Boundary ◽  
Low Load

In a companion paper (Part 1) a hydrodynamic lubrication model was developed to describe the flow and pressure in a bearing in which the lubricant has a no-slip boundary condition with one bearing surface but can slip against the other surface at a critical shear stress. This paper examines the current evidence for liquid slip at solid surfaces and outlines the problems inherent in the lubrication of low load contacts. The paper then shows that a bearing design in which one surface is very smooth and oleophobic, thereby allowing lubricant slip to occur, while the other surface is conventionally oleophilic, may provide a very low friction coefficient in low load conditions such as are present in microelectromechanical systems and hard disk drives.

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