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Applications of cohomology to questions in set theory i: hausdorff gaps [article]

Daniel Talayco
1993 arXiv   pre-print
We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps.  ...  The cohomology theory is introduced with enough generality to be applicable to other questions in set theory.  ...  Section 1 Introduction Hausdorff gaps appear in a wide scope of applications in the literature on set theory and a fitting and voluminous tribute to the importance of these objects has recently been published  ... 
arXiv:math/9311205v1 fatcat:tmwtogf3rzapdmc2w3ol3krfsy

Applications of cohomology to set theory I: Hausdorff gaps

Daniel E. Talayco
1995 Annals of Pure and Applied Logic  
We explore an application of homological algebra to set theoretic objects by developing a cohomology theory for Hausdorff gaps.  ...  The cohomology theory is introduced with enough generality to be applicable to other questions in set theory.  ...  Introduction Hausdorff gaps appear in a wide scope of applications in the literature on set theory and a fitting and voluminous tribute to the importance of these objects has recently been published by  ... 
doi:10.1016/0168-0072(94)00020-4 fatcat:fsr3npihsnfd5kh5ewoxp35hku

Pathology in dynamical systems I: General theory

Richard C Churchill, David L Rod
1976 Journal of Differential Equations  
This paper presents definitions of "transversal" and "nondegenerate" which are more readily verified in applications, and which still lead to topological results on the pathology of orbits analogous to  ...  Part I is done completely without smoothness assumptions.  ...  The theory can also be shown to be continuous, and in fact, for compact Hausdorff pairs, is identical with the usual Tech theory.  ... 
doi:10.1016/0022-0396(76)90018-8 fatcat:qfmrxukuknayzdliogopytzibq

Spectral Continuity for Aperiodic Quantum Systems I. General Theory [article]

Siegfried Beckus, Jean Bellissard, Giuseppe De Nittis
2018 arXiv   pre-print
The use of the Hausdorff topology turns out to be fundamental in understanding why and how these approximations work.  ...  In this work a positive answer is provided using the rather general setting of groupoid C^∗-algebras.  ...  Some of these results are slightly extended to cover all potential applications.  ... 
arXiv:1709.00975v2 fatcat:cjspnkubjrfpbphxmqjfujlusq

Twisted index theory on good orbifolds, I: noncommutative Bloch theory [article]

Matilde Marcolli
1999 arXiv   pre-print
In Part I we study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group.  ...  We apply these results to obtain qualitative results on real and complex hyperbolic spaces in 2 and 4 dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and the Ten Martini Problem  ...  More precisely, an orbifold M of dimension m is a Hausdorff, second countable topological space with a Satake atlas V = {U i , φ i } which covers M , consisting of open sets U i and homeomorphisms φ i  ... 
arXiv:math/9911102v1 fatcat:v55y5mgn5jaipdkjd5cfxq5n74

Page 8840 of Mathematical Reviews Vol. , Issue 2002M [page]

2002 Mathematical Reviews  
Denoting by £; the set of reals where the asymptotic frequency of some digit in the f-expansion differs by at least 6 from the frequency prescribed by wv, they prove that E; has Hausdorff dimension less  ...  certain applications.”  ... 

Page 638 of Mathematical Reviews Vol. , Issue 85b [page]

1985 Mathematical Reviews  
theory provides cohomology theories for these categories as well.  ...  It seems still to be open whether the gap between cmp X amd def X can be arbitrarily large.  ... 

Page 5887 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
Kalnins (NZ-WAIK) 90j:58119 58F19 14K20 35P99 35Q20 47F0S 58G25 Krichever, I. M. (2-MPI) Spectral theory of two-dimensional periodic operators and its applications. (Russian) Uspekhi Mat.  ...  Each cross section to the suspension flow is dual to a cohomology class and each of these cohomology classes determines an infinite cyclic cover of M.  ... 

Three-manifolds, Foliations and Circles, I [article]

William P. Thurston
1997 arXiv   pre-print
The skew R-covered Anosov foliations of Sergio Fenley are examples. We hope later to use this structure for geometrization of slithered 3-manifolds.  ...  We show that all such foliations admit transverse pseudo-Anosov flows, and that in the universal cover of the hyperbolic cases, the leaves limit to sphere-filling Peano curves.  ...  Then J ∩ I ν is a Cantor set, since gaps are dense and there are no atoms to L + (ν).  ... 
arXiv:math/9712268v1 fatcat:uiozlkod2rbcxhlz34wsqcdlse

Book Review: Lie theory and special functions

Louis Weisner
1969 Bulletin of the American Mathematical Society  
Rectifiable sets have become objects of fundamental importance to measure theoretic geometry since they are essential in describing the structure of sets of finite ^-dimensional Hausdorff measure.  ...  The irreducible representations of the associate Lie algebra are obtained but, because of computational difficulties, detailed applications are omitted. We expect this gap to be filled in the future.  ... 
doi:10.1090/s0002-9904-1969-12292-5 fatcat:c4vvcmogs5dw7pdctczubgqdra

Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem [article]

Eric Akkermans, Yaroslav Don, Jonathan Rosenberg, Claude L. Schochet
2020 arXiv   pre-print
properties (of Hamiltonians defined on such tilings) defined by K-theory, and to show their equivalence in dimensions ≤ 3.  ...  The purpose of this paper is to show the relationship in all dimensions between the structural (diffraction pattern) aspect of tilings (described by Čech cohomology of the tiling space) and the spectral  ...  The transversal N in our applications is a Cantor set, zero-dimensional, so its cohomology vanishes in positive dimensions.  ... 
arXiv:2007.15961v1 fatcat:6wkmi3apajhibii2zv73l46fvi

Equivariant Cartan-Eilenberg supergerbes for the Green-Schwarz superbranes I. The super-Minkowskian case [article]

Rafał R. Suszek
2020 arXiv   pre-print
cohomology of its Lie superalgebra, the gap between the two cohomologies being bridged by a super-variant of the classic Chevalley-Eilenberg construction.  ...  straightforward identification of the supergeometric mechanisms and unobstructed development of formal tools to be employed in more complex circumstances.  ...  [Mur96, MS00, Ste00, BCM+02, CJM02, CJM+05] -ample structural applications in the study of σ-models and the associated conformal field theories and string theories, and in particular in a neat cohomological  ... 
arXiv:1706.05682v5 fatcat:hnutu5j2kbdk5okwhmliqojpoe

A spectral sequence for the K-theory of tiling spaces

JEAN SAVINIEN, JEAN BELLISSARD
2009 Ergodic Theory and Dynamical Systems  
. , e d } is the canonical basis of Z d , α i = α e i is the restriction of α to the ith component of Z d , whereas x∧ is the exterior multiplication by x ∈ Z d .  ...  There is a spectral sequence converging to the K -theory of A E r s 2 ⇒ K r +s+d (A α Z d ), with page-2 isomorphic to the cohomology of the PV complex. † J.B. is indebted to M.  ...  (for cohomology or K -theory in particular).  ... 
doi:10.1017/s0143385708000539 fatcat:zk2zaxzlznchzkxa2y7ypv7moi

Page 389 of Mathematical Reviews Vol. 56, Issue 2 [page]

1978 Mathematical Reviews  
The independence of the results from Zermelo- Frankel set theory remains open. Paul C. Eklof (Irvine, Calif.) Manin, Ju. I. 2821 The continuum problem.  ...  ., 1971] tried to prove that the assumption | A?=A was equivalent to being a homomorphic image of a Suslin algebra of sets (with the real operation A), but there was a gap.  ... 

Page 2271 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
To do this, he starts by introducing a number of equivalence relations, most on the “setof all met- ric spaces—among them coarse equivalence, Gromov-Hausdorff equivalence, Lipschitz (homotopy) equivalence  ...  Part I of the paper under review is devoted to equivari- ant elliptic cohomology with respect to a finite group G of odd order n and an elliptic curve Eyniy given by y?  ... 
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