Geometry and dynamics II: geodesic flow on a surface

Berger, Marcel

doi:10.1007/978-3-540-70997-8_12

Cited by 4 publications

(5 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It can be seen that  P is a convex polytope embedded in  P [87]; in other words, it is a convex body, convex hull of its vertices that are the pure points of this convex (i.e., the points that cannot be written as convex combination of several points of the set) [88,89].…”

Section: Discussionmentioning

confidence: 99%

General entropy-like uncertainty relations in finite dimensions

Zozor¹,

Bosyk²,

Portesi³

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicrú generalized (h, φ)-entropies, including Rényi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (c A , c B , c A,B ) with c A (resp. c B ) being the overlap between the elements of the POVM A (resp. B) and c A,B the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sánchez-Ruiz [Phys. Rev. A 77, 042110 (2008)] and consists in a minimization of the entropy sum subject to the Landau-Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Rényi or Tsallis entropic formulations of the uncertainty principle, we overcome the Hölder conjugacy constraint imposed on the entropic indices by the Riesz-Thorin theorem. In the case of nondegenerate observables, we show that for given c A,B > 1 √ 2 , the bound obtained is optimal; and that, for Rényi entropies, our bound improves Deutsch one, but Maassen-Uffink bound prevails when c A,B ≤ 1 2 . Finally, we illustrate by comparing our bound with known previous results in particular cases of Rényi and Tsallis entropies.

show abstract

Section: Discussionmentioning

confidence: 99%

General entropy-like uncertainty relations in finite dimensions

Zozor¹,

Bosyk²,

Portesi³

2014

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

show abstract

“…The assumption that ξ and η can be identified with these coordinate vector fields for all t is again a restriction on the coordinate gauge which we choose later. According to the definition of Euler coordinates, we see that the two Killing fields generate two families of 'conjugate circles' in S 3 which yield a foliation of a dense subset of S 3 in terms of 2-tori; this is related to the Clifford parallelism discussed in [7,8]. Our symmetry action (and hence the foliation in terms of 2-tori) degenerates, in the sense that the group orbits become one-dimensional, precisely at the 'poles' θ = 0 and at θ = π .…”

Section: Symmetry Reduction For Spacetimes Of Spatial 3-sphere Topologymentioning

confidence: 99%

Smooth Gowdy-symmetric generalized Taub–NUT solutions

Beyer¹,

Hennig²

2012

Class. Quantum Grav.

View full text Add to dashboard Cite

Abstract. We study a class of S 3 -Gowdy vacuum models with a regular past Cauchy horizon which we call smooth Gowdy symmetric generalized Taub-NUT solutions. In particular, we prove existence of such solutions by formulating a singular initial value problem with asymptotic data on the past Cauchy horizon. We prove that also a future Cauchy horizon exists for generic asymptotic data, and derive an explicit expression for the metric on the future Cauchy horizon in terms of the asymptotic data on the past horizon. This complements earlier results about S 1 × S 2 -Gowdy models.

show abstract

“…This question is interestingly hard to answer simply! As it is stated in [2], a convex polyhedron in X which is a d−dimensional real affine space is a subset of X obtained a finite intersection of closed haf-spaces. A polytope is a compact convex polyhedron with non-empty interior.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. (for more details see to [1,2] and [5,18]) There are many thinkers that worked on polyhedra among the ancient Greeks. Early civilizations worked out mathematics as problems and their solutions.…”

Section: Introductionmentioning

confidence: 99%

Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces

Gelişgen

Yavuz

2019

Mathematical Sciences and Applications E-Notes

View full text Add to dashboard Cite

Polyhedra have interesting symmetries. Therefore they have attracted the attention of scientists and artists from past to present. Thus polyhedra are discussed in a lot of scientific and artistic works. There are only five regular convex polyhedra known as the platonic solids. There are many relationships between metrics and polyhedra. Some of them are given in previous studies. In this study, we introduce two new metrics, and show that the spheres of the 3-dimensional analytical space furnished by these metrics are chamfered cube and chamfered octahedron. Also we give some properties about these metrics. We show that the group of isometries of the 3-dimesional space covered by CC−metric and CO−metric are the semi-direct product of O h and T (3), where octahedral group O h is the (Euclidean) symmetry group of the octahedron and T (3) is the group of all translations of the 3-dimensional space.

show abstract

Geometry and dynamics II: geodesic flow on a surface

Cited by 4 publications

References 41 publications

General entropy-like uncertainty relations in finite dimensions

General entropy-like uncertainty relations in finite dimensions

Smooth Gowdy-symmetric generalized Taub–NUT solutions

Isometry Groups of Chamfered Cube and Chamfered Octahedron Spaces

Contact Info

Product

Resources

About