Geometric study of the beta-integers for a Perron number and mathematical quasicrystals

AbstractWe contribute to the algebraic-geometric study of discrete integrable systems generated by planar birational maps: (a) we find geometric description of Manin involutions for elliptic pencils consisting of curves of higher degree, birationally equivalent to cubic pencils (Halphen pencils of index 1), and (b) we characterize special geometry of base points ensuring that certain compositions of Manin involutions are integrable maps of low degree (quadratic Cremona maps). In particular, we identify some integrable Kahan discretizations as compositions of Manin involutions for elliptic pencils of higher degree.

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A Geometric Study of Wasserstein Spaces: Isometric Rigidity in Negative Curvature

International Mathematics Research Notices ◽

10.1093/imrn/rnv177 ◽

2015 ◽

Vol 2016 (5) ◽

pp. 1368-1386 ◽

Cited By ~ 2

Author(s):

Jérôme Bertrand ◽

Benoît R. Kloeckner

Keyword(s):

Negative Curvature ◽

Geometric Study

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Geometric Study of Various Chimney Graft Configurations in an In Vitro Juxtarenal Aneurysm Model

Journal of Endovascular Therapy ◽

10.1583/1545-1550-20.2.184 ◽

2013 ◽

Vol 20 (2) ◽

pp. 184-190 ◽

Cited By ~ 33

Author(s):

Jorg L. de Bruin ◽

Kak K. Yeung ◽

Wouter W. Niepoth ◽

Rutger J. Lely ◽

Qingfeng Cheung ◽

...

Keyword(s):

Juxtarenal Aneurysm ◽

Chimney Graft ◽

Aneurysm Model ◽

Geometric Study

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A Geometric Study of Wasserstein Spaces: An Addendum on the Boundary

Lecture Notes in Computer Science - Geometric Science of Information ◽

10.1007/978-3-642-40020-9_44 ◽

2013 ◽

pp. 405-412

Author(s):

Jérôme Bertrand ◽

Benoît R Kloeckner

Keyword(s):

Geometric Study

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Hydrodynamics Around a Deep-Draft Semi-Submersible With Various Corner Shapes

Volume 9: Offshore Geotechnics; Honoring Symposium for Professor Bernard Molin on Marine and Offshore Hydrodynamics ◽

10.1115/omae2018-77135 ◽

2018 ◽

Author(s):

Yibo Liang ◽

Longbin Tao

Keyword(s):

Numerical Model ◽

Experimental Test ◽

Numerical Study ◽

Water Channel ◽

Hydrodynamic Characteristics ◽

Shape Design ◽

Circulating Water ◽

Fluid Physics ◽

Shape Effects ◽

Geometric Study

A numerical study on flow over a stationary deep-draft semi-submersible (DDS) with various corner shapes was carried out to investigate the corner shape effects on the overall hydrodynamics. Three models based on a typical DDS design with different corner shapes were numerically investigated under 45° incidence. The present numerical model has been validated by an experimental test carried out in a circulating water channel. It is demonstrated that, as the corner shape design changed, the hydrodynamic characteristics alter drastically. In addition, the flow patterns were examined to reveal some insights of the fluid physics due to the changing of different corner shape designs. The detailed numerical results from the geometric study will provide a good guidance for future practical designs.

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Non-negative integral matrices with given spectral radius and controlled dimension

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.93 ◽

2021 ◽

pp. 1-24

Author(s):

MEHDI YAZDI

Keyword(s):

Spectral Radius ◽

Positive Real Number ◽

Algebraic Integer ◽

Log P ◽

Irreducible Matrix ◽

Positive Real ◽

Shift Of Finite Type ◽

Primitive Matrix ◽

Integral Matrices ◽

Perron Number

Abstract A celebrated theorem of Douglas Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number p, we prove that there is an integral irreducible matrix with spectral radius p, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number p, there is an irreducible shift of finite type with entropy $\log (p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.

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Negative curvature in graph braid groups

International Journal of Algebra and Computation ◽

10.1142/s0218196721500053 ◽

2020 ◽

pp. 1-36

Author(s):

Anthony Genevois

Keyword(s):

Braid Group ◽

Negative Curvature ◽

Braid Groups ◽

Gromov Hyperbolic ◽

Relatively Hyperbolic ◽

Geometric Study

In this paper, we initiate a geometric study of graph braid groups. More precisely, by applying the formalism of special colorings introduced in a previous paper, we determine precisely when a graph braid group is Gromov-hyperbolic, toral relatively hyperbolic and acylindrically hyperbolic.

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Dynamical degrees of affine-triangular automorphisms of affine spaces

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.90 ◽

2021 ◽

pp. 1-42

Author(s):

JÉRÉMY BLANC ◽

IMMANUEL VAN SANTEN

Keyword(s):

Affine Space ◽

Affine Spaces ◽

Dynamical Degree ◽

Perron Number ◽

Dynamical Degrees ◽

Affine Automorphism

Abstract We study the possible dynamical degrees of automorphisms of the affine space $\mathbb {A}^n$ . In dimension $n=3$ , we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalizes the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb {A}^n$ for some n, and we give the best possible n for quadratic integers, which is either $3$ or $4$ .

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