arXiv subjects

Hassan Attarchi

Publications and source records attributed to Hassan Attarchi.

10 recordsLinked to original sources

arXiv2023

HyperKahler Contact Distributions

Let $(φ_α,ξ_α,g)$ for $α=1,2$, and $3$ be a contact metric $3$-structure on the manifold $M^{4n+3}$. We show that the $3$-contact distribution of this structure admits a HyperKahler structure whenever $(M^{4n+3},φ_α,ξ_α,g)$ is a $3$-Sasakian manifold. In this case, we call it HyperKahler contact distribution. To analyze the curvature properties of this distribution, we define a special metric connection that is completely determined by the HyperKahler contact distribution. We prove that the $3$-Sasakian manifold is of constant $φ_α$-sectional curvatures if and only if its HyperKahler contact distribution has constant holomorphic sectional curvatures.

math.DG↗

arXiv2023

$3$-Kenmotsu Manifolds

In this paper, a $3$-Kenmotsu structure is defined on a $4n+1$ dimensional manifold where such structure seems to be never studied before.

math.DG↗

arXiv2021

Unexpected Linearly Stable Orbits in $3$-Dimensional Billiards

In this work, we construct linearly stable periodic orbits in $3$-dimensional domains with boundaries containing focusing components (small pieces of a sphere) where we place these components arbitrarily far apart. It demonstrates that we cannot directly implement the construction methods of planar billiards with focusing boundaries and chaotic properties into $3$-dimensional billiards.

math.DS↗

arXiv2021

Contact Structure on the Indicatrix Bundle of Finslerian Warped Product Manifolds

In this paper, a comprehensive study of contact and Sasakian structures on the indicatrix bundle of Finslerian warped product manifolds is reconstructed. In addition, the Kahler structure on the tangent bundle of these manifolds is studied for some different metrics. Throughout the paper, the contact structure of indicatrix bundle in warped product Finsler manifolds is presented. It is shown that indicatrix bundle cannot be a Sasakian manifold.

math.MG↗

arXiv2021

Cartan Spaces and Natural Foliations on the Cotangent Bundle

In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M,K) are studied. It is shown that the geometry of these foliations is closely related to the geometry of the Cartan space (M,K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.

math.DS↗

arXiv2020

Bridge to Hyperbolic Polygonal Billiards

It is well-known that billiards in polygons cannot be chaotic (hyperbolic). Particularly Kolmogorov-Sinai entropy of any polygonal billiard is zero. We consider physical polygonal billiards where a moving particle is a hard disc rather than a point (mathematical) particle and show that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov- Sinai entropy for any positive radius of the moving particle (provided that the particle is not so big that it cannot move within a polygon). This happens because a typical physical polygonal billiard is equivalent to a mathematical (point particle) semi-dispersing billiard. We also conjecture that in fact typical physical billiard in polygon is ergodic under the same conditions.

math.DS↗

arXiv2020

Ehrenfest Wind-Tree Model is Dynamically Richer Than the Lorentz Gas

We consider a physical Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. We demonstrate that a physical periodic Wind-Tree model is dynamically richer than a physical or mathematical periodic Lorentz gas. Namely, the physical Wind-Tree model may have diffusive behavior as the Lorentz gas does, but it has more superdiffusive regimes than the Lorentz gas. The new superdiffusive regime where the diffusion coefficient D(t)~(ln t)^2 of dynamics seems to be never observed before in any model.

math.DS↗

arXiv2020

Collision of a Hard Ball with Singular Points of the Boundary

Recently were introduced physical billiards where a moving particle is a hard sphere rather than a point as in standard mathematical billiards. It has been shown that in the same billiard tables the physical billiards may have totally different dynamics than mathematical billiards. This difference appears if the boundary of a billiard table has visible singularities (internal corners if the billiard table is two-dimensional), i.e. the particle may collide with these singular points. Here, we consider the collision of a hard ball with a visible singular point and demonstrate that the motion of the smooth ball after collision with a visible singular point is indeed the one that was used in the studies of physical billiards. So such collision is equivalent to the elastic reflection of hard ball's center off a sphere with the center at the singular point and the same radius as the radius of the moving particle.

math.DS↗

arXiv2020

Why Escape Is Faster Than Expected

We consider chaotic (hyperbolic) dynamical systems which have a generating Markov partition. Then, open dynamical systems are built by making one element of a Markov partition a hole through which orbits escape. We compare various estimates of the escape rate which correspond to a physical picture of leaking in the entire phase space. Moreover, we uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Exact computations are present for the skewed tent map and Arnold's cat map.

math.DS↗

arXiv2012

3D-Algorithms of Composed Pursuit Navigation

The problem of pursuing a moving target is always one of the main topics in navigation. In the literatures, there are two well-known algorithms called Pure Pursuit and Pure Rendezvous navigation in the 3-dimensional space $\mathbb{R}^3$. In this paper, these two methods are combined to introduce a novel family of pursuing algorithms called Composed Pursuit Navigation. The Kinematic and geometric properties of this navigation is studied. The trajectories of this new family of algorithms benefit the advantages of two known methods and its prominence is demonstrated in two real examples. Moreover, it is shown that the metric related to the algorithms are given by Matsumoto metrics.

math.DG↗