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Topological properties of real number representations

Vasco Brattka, Peter Hertling
2002 Theoretical Computer Science  
We prove three results about 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  ... 
doi:10.1016/s0304-3975(01)00066-4 fatcat:vxxivq7a5raarnhjwzktmzhvg4

Order Embeddings with Irrational Codomain: Debreu Properties of Real Subsets

M. J. Campión, J. C. Candeal, E. Induráin, G. B. Mehta
2007 Order  
We will show that although the set 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.  ... 
doi:10.1007/s11083-006-9052-y fatcat:6b4sqeyiuzb2vchgss4sssjce4

Representations of the real numbers and of the open subsets of the set of real numbers

Klaus Weihrauch, Christoph Kreitz
1987 Annals of Pure and Applied Logic  
In this paper the concepts developed so far are used for the foundation 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.  ... 
doi:10.1016/0168-0072(87)90065-0 fatcat:4wdyn5fmcvg25pgdzqiesk7owu

Order representability in groups and vector spaces

Juan C. Candeal, Esteban Induráin, Manuel Sanchis
2012 Expositiones mathematicae  
Then, we consider groups and 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.  ... 
doi:10.1016/j.exmath.2012.01.007 fatcat:lvjh5hv43zcnzojkaqrmbm6ezm

Relative Property (T) for Topological Groups

Jicheng Tao, Wen Yan
2014 Applied Mathematics  
In this paper, we investigate Kazhdan's relative 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.  ... 
doi:10.4236/am.2014.519285 fatcat:hnjer3dzs5fg3acio5aetbc2b4

Admissibly Represented Spaces and Qcb-Spaces [article]

Matthias Schröder
2020 arXiv   pre-print
The latter form a well-behaved subclass 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.  ... 
arXiv:2004.09450v1 fatcat:dquy6j77kneategcryn563v46a

Modes of interaction between individuals dominate the topologies of real world networks [article]

Insuk Lee, Eiru Kim, Edward M. Marcotte
2014 biorxiv/medrxiv   pre-print
These dependencies explain contrasting observations among previous 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.  ... 
doi:10.1101/011288 fatcat:tklu2boghnev5lyhqthl2fbwki

Modes of Interaction between Individuals Dominate the Topologies of Real World Networks

Insuk Lee, Eiru Kim, Edward M. Marcotte, Philip M. Kim
2015 PLoS ONE  
These dependencies explain contrasting observations among previous 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.  ... 
doi:10.1371/journal.pone.0121248 pmid:25793969 pmcid:PMC4368763 fatcat:bqapeeg55ffbjdtkbzand6iiwu

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]

Frank Waaldijk
2018 arXiv   pre-print
We retrieve existing theory and derive strong new results on the efficient 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.  ... 
arXiv:1806.01636v1 fatcat:4olgukx46ncvdeqq7jb4xgny3q

Pointwise Debreu Lexicographic Powers

Alfio Giarlotta, Stephen Watson
2009 Order  
The most typical example 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.  ... 
doi:10.1007/s11083-009-9132-x fatcat:77wfkhmitjabxn3dnp2sje7stu

Topological Analysis of Representations [chapter]

Mathieu Hoyrup
2018 Lecture Notes in Computer Science  
Examples 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.  ... 
doi:10.1007/978-3-319-94418-0_22 fatcat:p4dtwyn5greslor2ejuax2xvvi

Interval Representations

R.H.N. Santiago, B.R.C Bedregal, B.M. Acióly
2004 TEMA  
This paper presents the concept 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  ... 
doi:10.5540/tema.2004.05.02.0317 fatcat:nqk5w3nirjhjbekue5od5c63wa

A Multiview Representation Learning Framework for Large-Scale Urban Road Networks

Kaiqi Chen, Guowei Chu, Kaiyuan Lei, Yan Shi, Min Deng
2022 Applied Sciences  
Moreover, 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.  ... 
doi:10.3390/app12136301 fatcat:5kx24pyt5jfeplajkrbpvevuvu

A unified approach to constructive and recursive analysis [chapter]

Christoph Kreitz, Klaus Weihrauch
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 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.  ... 
doi:10.1007/bfb0099489 fatcat:nuara6a5ubffjggejazwh2bcb4
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