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Computability on subsets of Euclidean space I: closed and compact subsets

Vasco Brattka, Klaus Weihrauch
1999 Theoretical Computer Science  
In this paper we introduce and compare computability concepts on the set of closed subsets of Euclidean space.  ...  The resultant canonical computability concepts induce computability of objects as well as computability of operators on the space of closed and compact subsets.  ...  Conclusion In this paper we have discussed computability concepts on the closed and on the compact subsets of Euclidean space.  ... 
doi:10.1016/s0304-3975(98)00284-9 fatcat:3hnwpzi5m5bilj2vxh4iluukd4

Volume distortion for subsets of Euclidean spaces

James R. Lee
2006 Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06  
We show that this holds without any assumption on the aspect ratio, and give an improved bound of O( √ log n(log k) 1/4 ).  ...  Our main result is an upper bound of O( √ log n log log n) independent of the value of k, nearly resolving the main open questions of [Dunagan-Vempala 2001] and [Krauthgamer-Linial-Magen 2004] .  ...  We thank Sanjeev Arora and Assaf Naor for relevant discussions during the work of [ALN08] , and Satish Rao for related conversations over the years.  ... 
doi:10.1145/1137856.1137888 dblp:conf/compgeom/Lee06 fatcat:wdrysy632fe5rhu23wcxtzewne

Diophantine approximation for negatively curved manifold, I [article]

Sa'ar Hersonsky, Frederic Paulin
1999 arXiv   pre-print
In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood.  ...  Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to  ...  Parts of this paper were written during a stay of the first author at the IHES in August-September 1998, and during a one week stay of the second author at the Warwick University under an Alliance project  ... 
arXiv:math/9909131v1 fatcat:ubtv6oxug5eihjizf5jwtp4y6m

Generalized sums over histories for quantum gravity (I). Smooth conifolds

Kristin Schleich, Donald M. Witt
1993 Nuclear Physics B  
This paper proposes to generalize the histories included in Euclidean functional integrals from manifolds to a more general set of compact topological spaces.  ...  Consequently, generalized Euclidean functional integrals based on these conifold histories yield semiclassical amplitudes for sequences of both manifold and conifold histories that approach a stationary  ...  Acknowledgments This work is the outgrowth of results presented by the authors at the Fifth Marcel  ... 
doi:10.1016/0550-3213(93)90649-a fatcat:leabetfvpbfxra7khjgox6soay

Hilbert's fourth problem, I

Z.I Szabó
1986 Advances in Mathematics  
Stacho, and V. Totik for their valuable discussions.  ...  '(x) on any compact subset of W.  ...  tend uniformly to d(x, y) on any compact subset of the AE coordinate system.  ... 
doi:10.1016/0001-8708(86)90056-3 fatcat:5d562foknjfstbezhdlov6kgui

Polyhedral compactifications, I [article]

Corina Ciobotaru, Linus Kramer, Petra Schwer
2020 arXiv   pre-print
In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods  ...  , that is, ultrapowers of the spaces at hand.  ...  Since X is a metric space, a function on X is continuous if and only if its restriction to every compact subset of X is continuous, see [Dug66, VI.8.3 and XI.9.3].  ... 
arXiv:2002.12422v3 fatcat:jmxtxjg3ebeajklgofi4s4kjpy

Morse Quasiflats I [article]

Jingyin Huang, Bruce Kleiner, Stephan Stadler
2021 arXiv   pre-print
In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety  ...  of examples.  ...  on parameters of super-Euclidean divergence and X).  ... 
arXiv:1911.04656v4 fatcat:i6ekp6b7wfabtewncxj6aiaxpu

Linear foliations of complex spheres I. Chains [article]

Laurent Dufloux
2018 arXiv   pre-print
We provide coordinate-free versions of the classical projection Theorem of Marstrand-Kaufman-Mattila.  ...  fact that the singular values of g belong to a compact subset of ]0, +∞[) and the last inequality is an easy computation.  ...  the inverse image of any compact subset is a compact subset.)  ... 
arXiv:1704.08010v2 fatcat:34plhqwehree5f3yq55y7uuxt4

Algebraic geometry of topological spaces I

Guillermo Cortiñas, Andreas Thom
2012 Acta Mathematica  
Weibel for a useful email discussion on Beȋlinson-Soulé's conjecture.  ...  Part of the research for this article was carried out during a visit of the first author to Universität Göttingen. He is indebted to this institution for their hospitality.  ...  from N to a precompact subset of the space Y an of closed points of Y equipped with the topology inherited by the euclidean topology on C n .  ... 
doi:10.1007/s11511-012-0082-6 fatcat:jtaecjxarzdizjxpsbvnpxndsi

Algebraic Geometry of Topological Spaces I [article]

Guillermo Cortiñas, Andreas Thom
2011 arXiv   pre-print
We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological  ...  space X.  ...  Part of the research for this article was carried out during a visit of the first named author to Universität Göttingen. He is indebted to this institution for their hospitality.  ... 
arXiv:0912.3635v3 fatcat:mtluc5ibjvgq7ivtcv2hbvfgn4

Topological ordered C- (resp. I-)spaces and generalized metric spaces

Hans-Peter A. Künzi, Zechariah Mushaandja
2009 Topology and its Applications  
The following result due to Hanai, Morita, and Stone is well known: Let f be a closed continuous map of a metric space X onto a topological space Y .  ...  While each fiber of a map is open or compact under the hypotheses mentioned in Balachandran's result (compare [25, p. 700]), the sets i(x) and d(x) (x ∈ X) of a metrizable topological ordered C -and I  ...  The closure of a subset D of ]0, 1[ with respect to the Euclidean topology on ]0, 1[ will be denoted by D.  ... 
doi:10.1016/j.topol.2008.12.040 fatcat:otxmox6sqfbvliavcgpacgwidu

Distance Geometry in Quasihypermetric Spaces. I [article]

Peter Nickolas, Reinhard Wolf
2008 arXiv   pre-print
Let (X, d) be a compact metric space and let M(X) denote the space of all finite signed Borel measures on X.  ...  The metric space (X, d) is quasihypermetric if for all n ∈, all α_1, ..., α_n ∈ satisfying ∑_i=1^n α_i = 0 and all x_1, ..., x_n ∈ X, one has ∑_i,j=1^n α_i α_j d(x_i, x_j) ≤ 0.  ...  The authors are grateful for the financial support and hospitality of the University of Salzburg and the Centre for Pure Mathematics in the School of Mathematics and Applied Statistics at the University  ... 
arXiv:0809.0740v1 fatcat:veupkzjwsbfudglofgp75yty2i


2009 Bulletin of the Australian Mathematical Society  
Let (X, d) be a compact metric space and let M(X ) denote the space of all finite signed Borel measures on X .  ...  of this topology to the weak- * topology and the measure-norm topology on M 0 (X ); and the functional-analytic properties of M 0 (X ) as a semi-inner product space, including the question of its completeness  ...  For n ≥ 2, let S n−1 denote the Euclidean unit sphere in R n , let X be a compact subset of S n−1 and let d(x, y) = x − y 2 for all x, y ∈ X , where · is the Euclidean norm.  ... 
doi:10.1017/s0004972708000932 fatcat:ncyso4dmwrfzblyboj5apu3tvm

Entropy for hyperbolic Riemann surface laminations I [article]

Tien-Cuong Dinh, Viet-Anh Nguyen, Nessim Sibony
2011 arXiv   pre-print
When the lamination is compact and transversally smooth, we show that the entropy is finite and the Poincare metric on leaves is transversally Holder continuous.  ...  A notion of metric entropy is also introduced for harmonic measures.  ...  When X is not compact, we can also consider the supremum of the entropies on compact subsets of X. Note that if Y and Y ′ are two subsets of X, then h(Y ∪ Y ′ ) = max(h(Y ), h(Y ′ )).  ... 
arXiv:1105.2307v2 fatcat:ol5afetelfa6vnmzd2xzk7pdla

Constructions of $$H_r$$ H r -hypersurfaces, barriers and Alexandrov theorem in $$\mathrm{I\!H}^n\times \mathrm{I\!R}$$ I H n × I R

Maria Fernanda Elbert, Ricardo Sa Earp
2014 Annali di Matematica Pura ed Applicata  
The dependence on r and n is a distinguishing point from the theory of H r -hypersurfaces in IH n × IR (H r > 0) from that of the euclidean or hyperbolic spaces, where the critical points are 0 and 1,  ...  Namely, we prove that an embedded compact H r -hypersurface in IH n × IR is rotational (H r > 0). rotational H-hypersurface, H > 0, depends on the value of H and we distinguish the cases of H greater or  ...  when H r tends to n−r n . iii) λ(ρ) → 0 uniformly in compacts subsets of [0, ∞] when H r → 0.  ... 
doi:10.1007/s10231-014-0446-y fatcat:ynsixgrbnbh53mwqklfcozyezi
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