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Definable one dimensional topologies in o-minimal structures [article]

Ya'acov Peterzil, Ayala Rosel
2019 arXiv   pre-print
We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space (X,τ) is definably homeomorphic to an affine  ...  definable space (namely, a definable subset of M^n with the induced subspace topology).  ...  Introduction The goal of this article is to study definable one-dimensional Hausdorff topologies in o-minimal structures, and to understand when they are definably homeomorphic to a definable set in some  ... 
arXiv:1807.08086v2 fatcat:cj6af5pk7zauhplz2y2cslcyju

Topology of definable abelian groups in o-minimal structures

Elías Baro, Alessandro Berarducci
2011 Bulletin of the London Mathematical Society  
In this note we show that every definably connected, definably compact abelian definable group in an o-minimal expansion of a real closed field of dimension not 4 is definably homeomorphic to a torus of  ...  Moreover, in the semialgebraic case the result holds for all dimensions.  ...  Wall for their comments concerning low dimensional topology and classification of manifolds.  ... 
doi:10.1112/blms/bdr108 fatcat:yfadeen2mvac3a53zr2tvfguw4

Taming the Landscape of Effective Theories [article]

Thomas W. Grimm
2021 arXiv   pre-print
are definable using the tame geometry built from an o-minimal structure.  ...  In particular, we will exploit the fact that coset spaces and period mappings are definable in an o-minimal structure and argue for non-trivial tameness results in higher-supersymmetric theories and in  ...  The basic strategy in defining a tame topology based on o-minimal structures [7] is to specify the space of allowed subsets of Rn , for every n.  ... 
arXiv:2112.08383v1 fatcat:knzm6u3bzzfldm6j6ugf4pytne

Page 575 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
In the case of a two-dimensional definably connected group definable in an o-minimal structure, the structure theorem goes via the facts that such a group is solvable and has a normal one- dimensional  ...  Let G be a three-dimensional nonsolvable definably connected group which is definable in an o-minimal structure M.  ... 


2002 International journal of shape modeling  
The main contribution of the paper is an existence proof for least stratifications (of sets in O-minimal structures).  ...  Results from the study of O-minimal structures are used to formalise fundamental objects and operations for geometric modelling kernels.  ...  The proof depends on cellular decomposition results for O-minimal structures. The next section formally defines O-minimal structures, Icells and the decomposition of definable sets.  ... 
doi:10.1142/s0218654302000054 fatcat:3axydiciizf4xn5mkdbhmvvsce

Tame Complex Analysis and o-minimality

Ya'acov Peterzil, Sergei Starchenko
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
O-minimal structures offer a tame setting for various areas of mathematics. Some features: Topology: Order topology on R and product topology on R n .  ...  From now on: We work in the structure R an,exp . A torus means a one-dimensional torus. In the structure R an,exp : Let A t , t ∈ T , be a definable family of g-dimensional abelian varieties over C.  ... 
doi:10.1142/9789814324359_0040 fatcat:egznsme3dnh4ljajhkn2qlaruq

Zero-groups and maximal tori [article]

Alessandro Berarducci
2005 arXiv   pre-print
We give a presentation of various results on zero-groups in o-minimal structures together with some new observations.  ...  In particular we prove that if G is a definably connected definably compact group in an o-minimal expansion of a real closed field, then for any maximal definably connected abelian subgroup T of G, G is  ...  in the o-minimal structure (R, <, +, ·).  ... 
arXiv:math/0511162v1 fatcat:7o3r6zxshfdsxjb5nc3akrgsiq

Three-dimensional stress-based topology optimization using SIMP method

Hailu Shimels Gebremedhen, Dereje Engida Woldemicahel, Fakheruldin M. Hashim
2019 International Journal for Simulation and Multidisciplinary Design Optimization  
Structural topology optimization problems have been formulated and solved to minimize either compliance or weight of a design domain under volume or stress constraints.  ...  The introduction of threedimensional analysis is a more realistic approach to many applications in industry and research, but most of the developments in stress-based topology optimization are two-dimensional  ...  The developments in structural optimization that uses SIMP method focus on two-dimensional problems and those that are formulated for three-dimensional problems consider a compliance minimization problem  ... 
doi:10.1051/smdo/2019005 fatcat:lshyb4oj2jabrlbznlknqzfcfu

One Dimensional T.T.T Structures [article]

Daniel Lowengrub
2012 arXiv   pre-print
In this paper we analyze the relationship between o-minimal structures and the notion of \omega -saturated one dimensional t.t.t structures.  ...  As a corollary we obtain the result that if an \omega -saturated one dimensional t.t.t structure admits a topological group structure then it is locally o-minimal.  ...  I'd like to thank Ehud Hrushovski for providing valuable insights and guidance throughout the time spent working on this paper.  ... 
arXiv:1210.5603v1 fatcat:g4m6jjjmyrghrhel3ar5phj4v4

Aspects in topology-based geometric modeling Possible tools for discrete geometry? [chapter]

Pascal Lienhardt
1997 Lecture Notes in Computer Science  
Topology-based Geometric Modeling is concerned with modeling subdivisions of geometric spaces. Methods are close to that of combinatorial topology, but for different purposes.  ...  construction operations can be easily defined on these structures.  ...  Topological properties are defined on semi-simplicial sets and can be extended on generalized maps.  ... 
doi:10.1007/bfb0024828 fatcat:c3knhescqvdafc6edvqyv35zgi

Groups and rings definable in d-minimal structures [article]

Antongiulio Fornasiero
2021 arXiv   pre-print
We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality.  ...  In particular, we prove that we can endow a definable group with a definable topology making it a topological group, and that a definable ring of dimension at least 1 and without zero divisors is a skew  ...  Pillay ([Pil88] ) proved that every group definable in an o-minimal structure can be endowed with a definable topology that makes it both a topological group and a definable manifold.  ... 
arXiv:1205.4177v2 fatcat:7kvkr4sxybfbxoomrnwdlhc2vq

Compactness and connectedness in topological groups

Dikran Dikranjan
1998 Topology and its Applications  
(b) q(G) = 1, (c) G is zero-dimensional; in particular, always c(G) = q(G) for such a group G.  ...  We also discuss the role of connectedness for the question when topology or algebra alone determine the topological group structure of a compact-like group. 0 1998 Elsevier Science B.V.  ...  Wiegandt, Connectedness and disconnectedness in topology, Gen. Topology Appl. 5 (197.5) 137-154. [2] A. Arkhangel'skii, Mappings connected with topological groups, Dokl. Akad. Nauk SSSR  ... 
doi:10.1016/s0166-8641(97)00094-1 fatcat:77rbdmpbvzfnhpzmt5jkc223ke

Affiness of topological space definable in a definably complete uniformly locally o-minimal structure of the second kind [article]

Masato Fujita, Tomohiro Kawakami
2021 arXiv   pre-print
We give a necessary and sufficient condition for a one-dimensional regular and Hausdorff topological space definable in a definably complete uniformly locally o-minimal structure of the second kind having  ...  definable bounded multiplication being affine.  ...  Rosel gave a necessary and sufficient condition for a one-dimensional topological space with a topology definable in an o-minimal structure being affine in [9] when the definable set in consideration  ... 
arXiv:2110.15616v1 fatcat:e3zynktrlnhhrowm3d3aupr24m

Book Review: Tame topology and o-minimal structures

David Marker
2000 Bulletin of the American Mathematical Society  
Tame topology and o-minimal structures, by Lou van den Dries, Cambridge Univ.  ...  Speissegger [18] extended this by showing that the "Pfaffian closure" of an o-minimal structure is o-minimal. §4 The book Van den Dries' Tame topology and O-minimal structures can be viewed as a compelling  ... 
doi:10.1090/s0273-0979-00-00866-1 fatcat:p5tpkcjbfbay5pzyu6ieajpqmq

Spacetime granularity from finite-dimensionality of local observable algebras [article]

Matti Raasakka
2019 arXiv   pre-print
Motivated by these ideas, we show that operational spacetime topology is described by an atomistic Boolean algebra if (i) local observable algebras are finite-dimensional factors, (ii) the intersection  ...  Thus, in this case, spacetime has a point-free granular behavior at small scales.  ...  Acknowledgments I thank Paolo Bertozzini, Philipp Höhn and Ted Jacobson for enlightening discussions and helpful comments on an earlier version of the manuscript.  ... 
arXiv:1908.09293v3 fatcat:fvfbvvfumnhnbbj6nuxqdnhchq
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