scholarly journals On a topological version of Pach's overlap theorem

2019 ◽  
Vol 52 (2) ◽  
pp. 275-282
Author(s):  
Boris Bukh ◽  
Alfredo Hubard
Keyword(s):  
Topological Version

scholarly journals Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

2021 ◽  
Vol 51 (3) ◽  
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Uncertainty Principle ◽  
Convex Geometry ◽  
Geometric Approach ◽  
Point Of View ◽  
Reconstruction Problem ◽  
Orthogonal Projections ◽  
Topological Version ◽  
Mahler Conjecture ◽  
Quantum Indeterminacy ◽  
Dual Quantum

AbstractWe define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

scholarly journals Photonic implementation of Majorana-based Berry phases

2018 ◽  
Vol 4 (10) ◽  
pp. eaat6533 ◽  
Author(s):  
Jin-Shi Xu ◽  
Kai Sun ◽  
Jiannis K. Pachos ◽  
Yong-Jian Han ◽  
Chuan-Feng Li ◽  
...  
Keyword(s):  
Quantum Systems ◽  
Majorana Fermions ◽  
Topological Characteristic ◽  
Fundamental Physics ◽  
Geometric Phases ◽  
Physical Systems ◽  
All Optical ◽  
Topological Version ◽  
Berry Phases ◽  
Statistical Evolution

Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies alike. The exotic statistics of anyons realized in physical systems can be interpreted as a topological version of geometric phases. However, non-Abelian statistics has not yet been demonstrated in the laboratory. Here, we use an all-optical quantum system that simulates the statistical evolution of Majorana fermions. As a result, we experimentally realize non-Abelian Berry phases with the topological characteristic that they are invariant under continuous deformations of their control parameters. We implement a universal set of Majorana-inspired gates by performing topological and nontopological evolutions and investigate their resilience against perturbative errors. Our photonic experiment, though not scalable, suggests the intriguing possibility of experimentally simulating Majorana statistics with scalable technologies.

scholarly journals A topological version of the Ambrosetti-Prodi theorem

1996 ◽  
Vol 64 (2) ◽  
pp. 121-130
Author(s):  
Bogdan Przeradzki
Keyword(s):  
Topological Version

scholarly journals ON THE TOPOLOGY OF T-DUALITY

2005 ◽  
Vol 17 (01) ◽  
pp. 77-112 ◽  
Author(s):  
ULRICH BUNKE ◽  
THOMAS SCHICK
Keyword(s):  
Homotopy Type ◽  
Cohomology Class ◽  
Dimensional Torus ◽  
Duality Transformation ◽  
Classifying Space ◽  
Simple Derivation ◽  
Torus Bundles ◽  
Twisted Cohomology ◽  
Higher Dimensional ◽  
Topological Version

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T-duality for higher-dimensional torus bundles.

scholarly journals Banach algebras of topologically bounded index

1997 ◽  
Vol 56 (1) ◽  
pp. 51-62
Author(s):  
J.J. Green
Keyword(s):  
Banach Algebras ◽  
Stability Properties ◽  
Topological Version ◽  
Finiteness Properties

We consider normed and Banach algebras satisfying a condition topologically analogous to bounded index for rings. We investigate stability properties, prove a topological version of a theorem of Jacobson, and find in many cases co-incidence with well-known finiteness properties.

scholarly journals The Combinatorics of Hyperbolized Manifolds

2015 ◽  
Vol 117 (1) ◽  
pp. 31
Author(s):  
Allan L. Edmonds ◽  
Steven Klee
Keyword(s):  
Euler Characteristic ◽  
Generating Functions ◽  
Characteristic Sign ◽  
Combinatorial Properties ◽  
Topological Version ◽  
Aspherical Manifold

A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$. Gromov defined several hyperbolization functors which produce an aspherical manifold from a given simplicial or cubical manifold. We investigate the combinatorics of several of these hyperbolizations and verify the Euler Characteristic Sign Conjecture for each of them. In addition, we explore further combinatorial properties of these hyperbolizations as they relate to several well-studied generating functions.

scholarly journals ON THE sl(2) FOAM COHOMOLOGY COMPUTATIONS

2009 ◽  
Vol 18 (09) ◽  
pp. 1313-1328 ◽  
Author(s):  
CARMEN CAPRAU
Keyword(s):  
Divide And Conquer ◽  
Cohomology Groups ◽  
Topological Version ◽  
Knots And Links

We show how to use Bar-Natan's "divide and conquer" approach to computation to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.

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