Dense Set Of Points Research Articles

Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms

We discuss different regularities on stable/unstable holonomies of cw-hyperbolic homeomorphisms and prove that if a cw-hyperbolic homeomorphism has continuous joint stable/unstable holonomies, then it has a dense set of periodic points in its non-wandering set. For that, we prove that the hyperbolic cw-metric (introduced in Artigue et al (2024 J. Differ. Equ. 378 512–38)) can be adapted to be self-similar (as in Artigue (2018 Ergodic Theory Dyn. Syst. 38 2422–46)) and, in this case, continuous joint stable/unstable holonomies are pseudo-isometric. We also prove transitivity of cw-hyperbolic homeomorphisms assuming that the stable/unstable holonomies are isometric. In the case the ambient space is a surface, we prove that a cw F -hyperbolic homeomorphism has continuous joint stable/unstable holonomies when every bi-asymptotic sector is regular.

Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system.

Synchronizing dynamical systems: Their groupoids and 𝐶*-algebras

Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic C ∗ C^\ast -algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various C ∗ C^\ast -algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space.

Norming and dense sets of extreme points of the unit ball in spaces of bounded Lipschitz functions

On spaces of finite signed Borel measures on a metric space one has introduced the Fortet-Mourier and Dudley norms, by embedding the measures into the dual space of the Banach space of bounded Lipschitz functions, equipped with different – but equivalent – norms: the FM-norm and the BL-norm, respectively. The norm of such a measure is then obtained by maximising the value of the measure when applied by integration to extremal functions of the unit ball. We introduce Lipschitz extension operators, essentially based on those defined by McShane, and investigate their properties. A remarkable one is that non-trivial extreme points are mapped to non-trivial extreme points of FM- and BL-norm unit balls. Using these extension operators, we define suitable ‘small’ subsets of extremal functions that are weak-star dense in the full set of extreme points of the unit ball, for any underlying metric space. For connected metric spaces, we additionally find a larger set of extremal functions for the BL-norm, similar to such a set that was defined previously by J. Johnson for the FM-norm. This set is then also weak-star dense in the extremal functions. These results may open an avenue to obtaining computational approaches for the Dudley norm on signed Borel measures.

SPECTRAL PROPERTIES AND INVERSE NODAL PROBLEMS FOR SINGULAR DIFFUSION EQUATION

In this study, some properties for the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues were learned, then the solutions of the inverse problem were given to determine the potential function and parameters of the boundary condition with the help of a dense set of nodal points and lastly we obtain a constructive solution to the inverse problems of this class.

Relatively pointwise recurrence on local dendrites

Let X be a local dendrite and let f : X → X be a continuous map. Denote by P ( f ) and R ( f ) the sets of periodic and recurrent points of f, respectively. We say that f is relatively pointwise recurrent if R ( f ) ¯ = X . We show that, if X is almost meshed (i.e the union of the interior of all free arcs is dense in X), then f is relatively pointwise recurrent if and only if either f is topologically conjugate to an irrational rotation on the circle or it has a dense set of periodic points.

Inverse nodal problem for singular Sturm–Liouville operator on a star graph

Abstract In this study, singular Sturm–Liouville operators on a star graph with edges are investigated. First, the behavior of sufficiently large eigenvalues is learned. Then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points. Lastly, a constructive solution to the inverse problems of this class is obtained.

RBF-Based Camera Model Based on a Ray Constraint to Compensate for Refraction Error.

A camera equipped with a transparent shield can be modeled using the pinhole camera model and residual error vectors defined by the difference between the estimated ray from the pinhole camera model and the actual three-dimensional (3D) point. To calculate the residual error vectors, we employ sparse calibration data consisting of 3D points and their corresponding 2D points on the image. However, the observation noise and sparsity of the 3D calibration points pose challenges in determining the residual error vectors. To address this, we first fit Gaussian Process Regression (GPR) operating robustly against data noise to the observed residual error vectors from the sparse calibration data to obtain dense residual error vectors. Subsequently, to improve performance in unobserved areas due to data sparsity, we use an additional constraint; the 3D points on the estimated ray should be projected to one 2D image point, called the ray constraint. Finally, we optimize the radial basis function (RBF)-based regression model to reduce the residual error vector differences with GPR at the predetermined dense set of 3D points while reflecting the ray constraint. The proposed RBF-based camera model reduces the error of the estimated rays by 6% on average and the reprojection error by 26% on average.

Densely branching trees as models for Hénon-like and Lozi-like attractors

Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For the Hénon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.

Decidability of CPC-irreducibility of subshifts of finite type over free groups

This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the ({textsf{d}},{textsf{c}})-reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided.

EFFICIENTLY FILLING SPACE

We show that for each n=3,4,… there is a space-filling curve f:[0,1]→[0,1]n such that f is at most (n+1)-to-1 at every point of [0,1]n. The fact that any such dimension raising continuous function is at least (n+1)-to-1 has been known since the 1930s, so the examples we provide here are, in that sense, the best possible. The classic space-filling curves due to first Peano and a year later, Hilbert, that map [0,1] onto [0,1]2 are both 4-to-1 at a dense set of points and their generalizations to [0,1]n are known to be 2n-to-1 at a dense set of points. Flaten, Humke, Olson and Vo (J. Math. Anal. Appl. 500:2 (2021), art. id. 125113) gave an example, f:[0,1]→[0,1]2 based on the Hilbert linear ordering of somewhat altered Hilbert partitions which is at most 3-to-1 at every point of [0,1]2, but there are technical difficulties with generalizing that example to higher dimensions. In a sense, this paper represents an overcoming of those difficulties.

Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges

Abstract In this study, a diffusion operator is investigated on a star graph with nonhomogeneous edges. First, the behaviors of sufficiently large eigenvalues are learned, and then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points and to obtain a constructive solution to the inverse problems of this class.

Topological equicontinuity and topological uniform rigidity for dynamical system

In this paper, we study topological equicontinuity, topological uniform rigidity and their properties. For a dynamical system, on a compact, T3 space, we study relations among the set of recurrent points of the map, the set of non-wandering points of the map and the intersection of the range sets of all iterations of the map. We define topological version of uniform rigidity and show that on a compact and T3 space any dynamical system is topologically uniformly rigid if it is first countable, almost topologically equicontinuous and transitive or it is second countable, topologically equicontinuous and has a dense set of periodic points. We show that a topologically uniformly rigid dynamical system, on a compact, Hausdorff space, has zero topological entropy. Moreover, we provide necessary examples and counterexamples.

Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions

In this paper, a Sturm–Liouville problem with some nonlocal boundary conditions of the Bitsadze-Samarskii type is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of the potential function and some other coefficients in the boundary conditions.

PU-MFA: Point Cloud Up-Sampling via Multi-Scale Features Attention

Recently, research using point clouds has been increasing with the development of 3D scanner technology. According to this trend, the demand for high-quality point clouds is increasing, but there is still a problem with the high cost of obtaining high-quality point clouds. Therefore, with the recent remarkable development of deep learning, point cloud up-sampling research, which uses deep learning to generate high-quality point clouds from low-quality point clouds, is one of the fields attracting considerable attention. This paper proposes a new point cloud up-sampling method called Point cloud Up-sampling via Multi-scale Features Attention (PU-MFA). Inspired by prior studies that reported good performance at generating high-quality dense point set using the multi-scale features or attention mechanisms, PU-MFA merges the two through a U-Net structure. In addition, PU-MFA adaptively uses multi-scale features to refine the global features effectively. The PU-MFA was compared with other state-of-the-art methods in various evaluation metrics through various experiments using the PU-GAN dataset, which is a synthetic point cloud dataset, and the KITTI dataset, which is the real-scanned point cloud dataset. In various experimental results, PU-MFA showed superior performance of generating high-quality dense point set in quantitative and qualitative evaluation compared to other state-of-the-art methods, proving the effectiveness of the proposed method. The attention map of PU-MFA was also visualized to show the effect of multi-scale features.

Failure of the Brauer–Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.

Local groups in Delone sets in the Euclidean space.

A Delone (Delaunay) set is a uniformly discrete and relatively dense set of points located in space, and is a natural mathematical model of the set of atomic positions of any solid, whether it is crystalline, quasi-crystalline or amorphous. A Delone set has two positive parameters: r is the packing radius and R is the covering radius. The value 2r can be interpreted as the minimum distance between points of the set. The covering radius R is the radius of the biggest `empty' ball, i.e. the radius of the biggest ball containing no points from the set. The central concept of this article is the so-called local group at a point of X which is defined as a group of a cluster (neighborhoods) around the point of radius 2R. This value 2R is notable because it is the minimum size of cluster that provides the finiteness of the cluster group at each point for any set X from the family of all Delone sets with the covering radius R. A few conjectures and theorems on the local groups for arbitrary Delone sets in the Euclidean plane and 3D space are discussed. Some of these statements significantly refine and generalize the famous Bravais theorem on the impossibility of fifth-order axes in 2D and 3D lattices. A complete proof is given that, in a Delone set X in the 3D Euclidean space, the subset \widetilde{X} of all points at which the local groups contain rotations of order at most 6 is also a Delone set with a certain covering radius \widetilde{R}, where \widetilde{R} < 3R and R is the covering radius for X.

Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions

In this paper, Dirac operator with some integral type nonlocal boundary conditions is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of some coefficients of the operator.

Torsion points on isogenous abelian varieties

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

Spatial information enhancement network for 3D object detection from point cloud

LiDAR-based 3D object detection pushes forward an immense influence on autonomous vehicles. Due to the limitation of the intrinsic properties of LiDAR, fewer points are collected at the objects farther away from the sensor. This imbalanced density of point clouds degrades the detection accuracy but is generally neglected by previous works. To address the challenge, we propose a novel two-stage 3D object detection framework, named SIENet. Specifically, we design the Spatial Information Enhancement (SIE) module to predict the spatial shapes of the foreground points within proposals, and extract the structure information to learn the representative features for further box refinement. The predicted spatial shapes are complete and dense point sets, thus the extracted structure information contains more semantic representation. Besides, we design the Hybrid-Paradigm Region Proposal Network (HP-RPN) which includes multiple branches to learn discriminate features and generate accurate proposals for the SIE module. Extensive experiments on the KITTI 3D object detection benchmark show that our elaborately designed SIENet outperforms the state-of-the-art methods by a large margin. Codes will be publicly available at https://github.com/Liz66666/SIENet.