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Topological properties of real number representations
2002
Theoretical Computer Science
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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
2007
Order
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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
1987
Annals of Pure and Applied Logic
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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
2012
Expositiones mathematicae
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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
2014
Applied Mathematics
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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]
2020
arXiv
pre-print
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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]
2014
biorxiv/medrxiv
pre-print
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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
2015
PLoS ONE
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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
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(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
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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
2009
Order
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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]
2018
Lecture Notes in Computer Science
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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
2004
TEMA
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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
2022
Applied Sciences
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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]
1984
Lecture notes in mathematics
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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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