This article develops a method to compute the action induced on the fundamental group by an automorphism of a real rational surface. Then, it uses this method to compute the induced actions for a ...certain family of basic quadratic real automorphisms. By utilizing an invariant set in the fundamental group, we introduce a method to estimate a lower bound of the action’s growth rate on the fundamental group. The growth rate of this induced action provides a lower bound for the entropy of the real surface automorphism. These calculations are carried out for an important family of real surface automorphisms, and new lower bounds are obtained for the entropy of these automorphisms.
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For a stratified group
G
, we construct a class of Lie groups endowed with a left-invariant distribution locally diffeomorphic to the flat distribution of
G
. Vice versa, we show that all Lie groups ...with a left-invariant distribution that is locally diffeomorphic to the flat distribution of
G
belong to the class we constructed, if the Lie algebra of
G
has finite Tanaka prolongation.
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Let
M
be an oriented three-dimensional Riemannian manifold of constant sectional curvature
k
=
0
,
1
,
-
1
and let
S
O
M
be its direct orthonormal frame bundle (direct refers to positive ...orientation), which may be thought of as the set of all positions of a small body in
M
. Given
λ
∈
R
, there is a three-dimensional distribution
D
λ
on
S
O
M
accounting for infinitesimal rototranslations of constant pitch
λ
. When
λ
≠
k
2
, there is a canonical sub-Riemannian structure on
D
λ
. We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For
k
=
0
,
-
1
, we compute the sub-Riemannian length spectrum of
S
O
M
,
D
λ
in terms of the complex length spectrum of
M
(given by the lengths and the holonomies of the periodic geodesics) when
M
has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral.
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494.
String theory and string Newton–Cartan geometry Bergshoeff, Eric A; Gomis, Jaume; Rosseel, Jan ...
Journal of physics. A, Mathematical and theoretical,
01/2020, Volume:
53, Issue:
1
Journal Article
This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. ...Tropical geometry is a new and rapidly developing field of mathematics which has deep connections with various areas of mathematics and physics, such as algebraic geometry, symplectic geometry, complex analysis, dynamical systems, combinatorics, statistical physics, and string theory. As reflected by the content of this volume, this meeting was mainly focused on the geometric side of the tropical world with an emphasis on relations between tropical geometry, algebraic geometry, and combinatorics. This volume provides an overview of current trends concerning algebraic and combinatorial aspects of tropical geometry through eleven papers combining expository parts and development of modern techniques and tools.
In this paper, we construct cataclysm deformations for
θ
-Anosov representations into a semisimple non-compact connected real Lie group
G
with finite center, where
θ
⊂
Δ
is a subset of the simple ...roots that is invariant under the opposition involution. These generalize Thurston’s cataclysms on Teichmüller space and Dreyer’s cataclysms for Borel-Anosov representations into
PSL
(
n
,
R
)
. We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for
θ
-Anosov representations.
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In wireless networks assisted by intelligent reflecting surfaces (IRSs), jointly modeling the signal received over the direct and indirect (reflected) paths is a difficult problem. In this work, we ...show that the network geometry (locations of serving base station, IRS, and user) can be captured using the so-called triangle parameter <inline-formula><tex-math notation="LaTeX">\Delta</tex-math></inline-formula>. We introduce a decomposition of the effect of the combined link into a signal amplification factor and an effective channel power coefficient <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula>. The amplification factor is monotonically increasing with both the number of IRS elements <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">\Delta</tex-math></inline-formula>. For <inline-formula><tex-math notation="LaTeX">G</tex-math></inline-formula>, since an exact characterization of the distribution seems unfeasible, we propose three approximations depending on the value of the product <inline-formula><tex-math notation="LaTeX">N\Delta</tex-math></inline-formula> for Nakagami fading and the special case of Rayleigh fading. For two relevant models of IRS placement, we prove that their performance is identical if <inline-formula><tex-math notation="LaTeX">\Delta</tex-math></inline-formula> is the same given an <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula>. We also show that no gains are achieved from IRS deployment if <inline-formula><tex-math notation="LaTeX">N</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">\Delta</tex-math></inline-formula> are both small. We further compute bounds on the diversity gain to quantify the channel hardening effect of IRSs. Hence only with a judicious selection of IRS placement and other network parameters, non-trivial gains can be obtained.
We prove that a three-dimensional smooth complete intersection of two quadrics over a field k is k-rational if and only if it contains a line defined over k. To do so, we develop a theory of ...intermediate Jacobians for geometrically rational threefolds over arbitrary, not necessarily perfect, fields. As a consequence, we obtain the first examples of smooth projective varieties over a field k which have a k-point, and are rational over a purely inseparable field extension of k, but not over k.
In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass ...transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).
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