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Topological properties of real number representations

2002
*
Theoretical Computer Science
*

We prove three results about

doi:10.1016/s0304-3975(01)00066-4
fatcat:vxxivq7a5raarnhjwzktmzhvg4
*representations**of**real**numbers*(or elements*of*other*topological*spaces) by inÿnite strings. ... Third, we show that a rather interesting*property**of*admissible*real**number**representations*holds true also for a certain non-admissible*representation*, namely for the naive Cauchy*representation*: the ... The third result is related to another*property**of*admissible*representations**of**real**numbers*: for any admissible*representation**of*R, a function f mapping*real**numbers*to*real**numbers*is continuous i ...##
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Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets

2007
*
Order
*

We will show that although the set

doi:10.1007/s11083-006-9052-y
fatcat:6b4sqeyiuzb2vchgss4sssjce4
*of*irrational*numbers*does not satisfy the Debreu*property*it is still nonetheless true that any lower (respectively, upper) semicontinuous total preorder*representable*... The objective*of*this paper is to investigate the role*of*the set*of*irrational*numbers*as the codomain*of*order-preserving functions defined on*topological*totally preordered sets. ... Acknowledgements We want to express our gratitude to an anonymous referee, to whom we are indebted for the detailed reading*of*the manuscript and his/her very helpful suggestions and ideas. ...##
###
Representations of the real numbers and of the open subsets of the set of real numbers

1987
*
Annals of Pure and Applied Logic
*

In this paper the concepts developed so far are used for the foundation

doi:10.1016/0168-0072(87)90065-0
fatcat:4wdyn5fmcvg25pgdzqiesk7owu
*of*a new kind*of*constructive analysis. Different standard*representations**of*the*real**numbers*are compared. ... In the second part some effective*representations**of*the open subsets*of*the*real**numbers*are introduced and compared. 0168-0072/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland) ...*Representations**of*the*real**numbers*By R we denote the set*of**real**numbers*. The standard*topology*rR on R can be defined by the basis {(x;y)~_R Ix, y eR, x<y}*of*open intervals on R. ...##
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Order representability in groups and vector spaces

2012
*
Expositiones mathematicae
*

Then, we consider groups and

doi:10.1016/j.exmath.2012.01.007
fatcat:lvjh5hv43zcnzojkaqrmbm6ezm
*real*vector spaces equipped with a*topology*, and analyze the fulfillment*of*the so-called continuous*representability**property*; the latter meaning that every continuous total ... preorder defined on the given*topological*space admits a continuous*real*-valued order-preserving function. ... [12, 14] : The weak*topology**of*a locally convex*topological**real*vector space satisfies the continuous*representability**property*. ...##
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Relative Property (T) for Topological Groups

2014
*
Applied Mathematics
*

In this paper, we investigate Kazhdan's relative

doi:10.4236/am.2014.519285
fatcat:hnjer3dzs5fg3acio5aetbc2b4
*Property*(T) for pairs ( ) G X , , where G is a*topological*group and X is any subset*of*G . ... We show that the pair ( ) G X , has*Property*(FH) and every function conditionally*of*negative type on G is X-bounded if the pair ( ) G X , has relative*Property*(T). ... . [1] Let π be an orthogonal*representation**of*the*topological*group G on a*real*Hilbert space Definition 2.6. ...##
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Admissibly Represented Spaces and Qcb-Spaces
[article]

2020
*
arXiv
*
pre-print

The latter form a well-behaved subclass

arXiv:2004.09450v1
fatcat:dquy6j77kneategcryn563v46a
*of**topological*spaces. ... A basic concept*of*Type Two Theory*of*Effectivity (TTE) is the notion*of*an admissibly represented space. Admissibly represented spaces are closely related to qcb-spaces. ... The Cauchy*Representation**of*the*Real**Numbers*We present a*representation*for the*real**numbers*which induces a reasonable notion*of*computability. ...##
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Modes of interaction between individuals dominate the topologies of real world networks
[article]

2014
*
biorxiv/medrxiv
*
pre-print

These dependencies explain contrasting observations among previous

doi:10.1101/011288
fatcat:tklu2boghnev5lyhqthl2fbwki
*topological*analyses*of**real*world complex systems. ... We also observe this trend in systems with natural hierarchies, in which alternate*representations**of*the same networks, but which capture different levels*of*the hierarchy, manifest these signature*topological*... We also observed a dichotomy among the*real*world networks for additional*topological**properties*. ...##
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Modes of Interaction between Individuals Dominate the Topologies of Real World Networks

2015
*
PLoS ONE
*

These dependencies explain contrasting observations among previous

doi:10.1371/journal.pone.0121248
pmid:25793969
pmcid:PMC4368763
fatcat:bqapeeg55ffbjdtkbzand6iiwu
*topological*analyses*of**real*world complex systems. ... We also observe this trend in systems with natural hierarchies, in which alternate*representations**of*the same networks, but which capture different levels*of*the hierarchy, manifest these signature*topological*... We also observed a dichotomy among the*real*world networks for additional*topological**properties*. ...##
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Page 807 of Mathematical Reviews Vol. 26, Issue 4
[page]

1963
*
Mathematical Reviews
*

(m an integer, r

*real*), (ii) the supple- mentary series*of**representations*7'* (0 < s < 1), and (iii) the identity*representation*I. ... The*topologies*are not Hausdorff, though their deviations from this*property*are rather weak (see Corollaries 2 and 6*of*Theorem 3.1). ...##
###
Exact computation over topological spaces
[article]

2018
*
arXiv
*
pre-print

We retrieve existing theory and derive strong new results on the efficient

arXiv:1806.01636v1
fatcat:4olgukx46ncvdeqq7jb4xgny3q
*representation**of*continuous*real*-valued functions defined on a general class*of**topological*spaces (called natural spaces). ... The NToP-definition*of*the*real**numbers*(and continuous*real*functions) matches recent expert recommendations for exact*real*computation (see [Bauer&Kavkler2008] and [Bauer&Kavkler2009]). ... These morphisms reveal the*topology*behind different*representations**of*the*real**numbers*on a computer, and transitions between these*representations*. ...##
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Pointwise Debreu Lexicographic Powers

2009
*
Order
*

The most typical example

doi:10.1007/s11083-009-9132-x
fatcat:77wfkhmitjabxn3dnp2sje7stu
*of*a Debreu chain is the linearly ordered*topological*space (R, <, τ < )*of**real**numbers*endowed with its usual order < and the order*topology*τ < . ... Traditionally, the literature has been dealing with utility*representations*such that the base chain (Y, <) is the set R*of**real**numbers*, endowed with its natural order <. ... This additional*property**of*the set*of**real**numbers*might be a further justification to employ only R-valued utility*representations*. ...##
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Topological Analysis of Representations
[chapter]

2018
*
Lecture Notes in Computer Science
*

Examples

doi:10.1007/978-3-319-94418-0_22
fatcat:p4dtwyn5greslor2ejuax2xvvi
*of*countably-based spaces are: -The*real**numbers*with the Euclidean*topology*, generated by the rational open intervals: the standard*representation*is equivalent to the Cauchy*representation*. ... Countably-based spaces The standard way*of*representing a*real**number*x is by giving a sequence*of*rational*numbers*converging at a certain fixed rate to x. ...##
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Interval Representations

2004
*
TEMA
*

This paper presents the concept

doi:10.5540/tema.2004.05.02.0317
fatcat:nqk5w3nirjhjbekue5od5c63wa
*of*interval*representation*and shows some*of*its*properties*. ... The idea*of*interval objects as*representation**of**real*objects is defined and its relation with some aspects*of*interval analysis is showed. ... Aspects*of*Continuity Metrics and Moore*topology*On the set*of**real**numbers*and on the set*of*intervals the notion*of*distance between two*real**numbers*and two intervals, is given, respectively, by the ...##
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A Multiview Representation Learning Framework for Large-Scale Urban Road Networks

2022
*
Applied Sciences
*

Moreover,

doi:10.3390/app12136301
fatcat:5kx24pyt5jfeplajkrbpvevuvu
*real*-world road networks usually contain millions*of*segments, which poses a great challenge regarding the memory usage and computational efficiency*of*existing methods. ... In the experiment, two*real*-world datasets were used to demonstrate the superior performance*of*our framework through a comparative analysis. ... Acknowledgments: This work was conducted in part using computing resources at the High-Performance Computing Platform*of*the Central South University. ...##
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A unified approach to constructive and recursive analysis
[chapter]

1984
*
Lecture notes in mathematics
*

Futhermore xw(p) = IR\ U {I .11. n x (p) = ¢} and J J w I j n X w (p) = ¢ <=> (3p E dom X ) I. n U C () = ¢ P J q, X p q (where I. denates the closure

doi:10.1007/bfb0099489
fatcat:nuara6a5ubffjggejazwh2bcb4
*of*the interval I.). ... The definition*of*the*real**numbers*as cuts*of*rational*numbers*induces several*representations*. ... the clear*topological**properties**of*6 are concealed by the composition with the*numbering*the*topological**properties**of*which are not easily to understand. ...
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