Geodesic Research Articles

Spectrum of the Laplace operator on closed surfaces

AbstractA survey is given of classical and relatively recent results on the distribution of the eigenvalues of the Laplace operator on closed surfaces. For various classes of metrics the dependence of the behaviour of the second term in Weyl’s formula on the geometry of the geodesic flow is considered. Various versions of trace formulae are presented, along with ensuing identities for the spectrum. The case of a compact Riemann surface with the Poincaré metric is considered separately, with the use of Selberg’s formula. A number of results on the stochastic properties of the spectrum in connection with the theory of quantum chaos and the universality conjecture are presented.Bibliography: 51 titles.

Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type

Let \mathscr{L} be a smooth second-order real differential operator in divergence form on a manifold of dimension n . Under a bracket-generating condition, we show that the ranges of validity of spectral multiplier estimates of Mikhlin–Hörmander type and wave propagator estimates of Miyachi–Peral type for \mathscr{L} cannot be wider than the corresponding ranges for the Laplace operator on \mathbb{R}^n . The result applies to all sub-Laplacians on Carnot groups and more general sub-Riemannian manifolds, without restrictions on the step. The proof hinges on a Fourier integral representation for the wave propagator associated with \mathscr{L} and nondegeneracy properties of the sub-Riemannian geodesic flow.

On Differential Geometric Formulations of Slow Invariant Manifold Computation: Geodesic Stretching and Flow Curvature

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

Trajectory Grouping With Curvature Regularization for Tubular Structure Tracking.

Tubular structure tracking is a crucial task in the fields of computer vision and medical image analysis. The minimal paths-based approaches have exhibited their strong ability in tracing tubular structures, by which a tubular structure can be naturally modeled as a minimal geodesic path computed with a suitable geodesic metric. However, existing minimal paths-based tracing approaches still suffer from difficulties such as the shortcuts and short branches combination problems, especially when dealing with the images involving complicated tubular tree structures or background. In this paper, we introduce a new minimal paths-based model for minimally interactive tubular structure centerline extraction in conjunction with a perceptual grouping scheme. Basically, we take into account the prescribed tubular trajectories and curvature-penalized geodesic paths to seek suitable shortest paths. The proposed approach can benefit from the local smoothness prior on tubular structures and the global optimality of the used graph-based path searching scheme. Experimental results on both synthetic and real images prove that the proposed model indeed obtains outperformance comparing with the state-of-the-art minimal paths-based tubular structure tracing algorithms.

Realization of geodesic flows with a linear first integral by billiards with slipping

An arbitrary geodesic flow on the projective plane or Klein bottle with an additional, linear in the momentum, first integral is modelled using billiards with slipping on table complexes. The requisite table of a circular topological billiard with slipping is constructed algorithmically. Furthermore, linear integrals of geodesic flows can be reduced to the same canonical integral of a circular planar billiard. Bibliography: 36 titles.

Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape

Within the Kardar–Parisi–Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points (x,s) and (y,t) with s<t. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations x1 and x2, and consider geodesics traveling (x1,0)→(y,1) and (x2,0)→(y,1). We prove that the set of y∈R for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints (x,0) and (y,1) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such (x,y)∈R2 also has Hausdorff dimension one-half. The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in [10]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [40].

Multiple Borel–Cantelli Lemma in dynamics and MultiLog Law for recurrence

&lt;p style='text-indent:20px;'&gt;A classical Borel–Cantelli Lemma gives conditions for deciding whether an infinite number of rare events will happen almost surely. In this article, we propose an extension of Borel–Cantelli Lemma to characterize the multiple occurrence of events on the same time scale. Our results imply multiple Logarithm Laws for recurrence and hitting times, as well as Poisson Limit Laws for systems which are exponentially mixing of all orders. The applications include geodesic flows on compact negatively curved manifolds, geodesic excursions on finite volume hyperbolic manifolds, Diophantine approximations and extreme value theory for dynamical systems.&lt;/p&gt;

Generic properties of geodesic flows on analytic hypersurfaces of Euclidean space

&lt;p style='text-indent:20px;'&gt;Consider the geodesic flow on a real-analytic closed hypersurface &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathbb{R}^n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, equipped with the induced metric. How commonly can we expect such flows to have a transverse homoclinic orbit? In this paper, we give the following two partial answers to this question:&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;● If &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; is a real-analytic closed hypersurface in &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \mathbb{R}^n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (with &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ n \geq 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) on which the geodesic flow with respect to the induced metric has a nonhyperbolic periodic orbit, then &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ C^{\omega} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-generically the geodesic flow on &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;● There is a &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ C^{\omega} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-open and dense set of real-analytic, closed, and strictly convex surfaces &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ M $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \mathbb{R}^3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; on which the geodesic flow with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;These are among the first perturbation-theoretic results for real-analytic geodesic flows.&lt;/p&gt;

Realization of geodesic flows with a linear first integral by billiards with slipping

При помощи биллиардов с проскальзыванием на столах-комплексах удалось промоделировать произвольный геодезический поток на проективной плоскости и бутылке Клейна, имеющий линейный по компонентам импульса дополнительный первый интеграл. Требуемый стол кругового топологического биллиарда с проскальзыванием строится алгоритмически. При этом линейные интегралы геодезических потоков удается свести к одному каноническому интегралу плоского кругового биллиарда. Библиография: 36 названий.

Geodesic fields for Pontryagin type C0-Finsler manifolds

Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ TxM} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : TxM → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C0-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C0-Finsler structure where ℰ is a locally Lipschitz vector field.

A Geometric View on the Generalized Proudman–Johnson and r-Hunter–Saxton Equations

We show that two families of equations, the generalized inviscid Proudman-Johnson equation, and the $r$-Hunter-Saxton equation (recently introduced by Cotter et al.) coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman-Johnson equations as geodesic equations of right invariant homogeneous $W^{1,r}$-Finsler metrics on the diffeomorphism group. Generalizing a construction of Lenells for the Hunter-Saxton equation, we analyze these equations using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby we show that the periodic case is equivalent to the geodesic equations on the $L^r$-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior.

Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero.&#x0D; &#x0D; Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= &amp;lambda;(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function &amp;lambda; satisfies the equation R(&amp;part;/&amp;part;z（&amp;lambda;(c∆^-1&amp;lambda;_zz+a))= 0 with some complex constants a and c&amp;ne;0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function &amp;lambda;. If the functions &amp;lambda; and &amp;lambda; + &amp;lambda;_0 satisfy the equation for a real constant &amp;lambda;0, 0, then there exists a non-zero Killing vector field on the torus.

On the Lagrange and Markov Dynamical Spectra for Anosov Flows in Dimension 3

We consider the Lagrange and the Markov dynamical spectra associated with a conservative Anosov flow on a compact manifold of dimension 3 (including geodesic flows of negative curvature and suspension flows). We show that for a large set of real functions and typical conservative Anosov flows, both the Lagrange and Markov dynamical spectra have a non-empty interior.

A weighted planar stochastic lattice with scale-free, small-world and multifractal properties

We investigate a class of weighted planar stochastic lattice (WPSL1) created by random sequential nucleation of seed from which a crack is grown parallel to one of the sides of the chosen block and ceases to grow upon hitting another crack. It results in the partitioning of the square into contiguous and non-overlapping blocks. Interestingly, we find that the dynamics of WPSL1 is governed by infinitely many conservation laws and each of the conserved quantities, except the trivial conservation of total mass or area, is a multifractal measure. On the other hand, the dual of the lattice is a scale-free network as its degree distribution exhibits a power-law P(k)∼k−γ with γ=4. The network is also a small-world network as we find that (i) the total clustering coefficient C is high and independent of the network size and (ii) the mean geodesic path length grows logarithmically with N. Besides, the clustering coefficient Ck of the nodes which have degree k decreases exactly as 2/(k−1) revealing that it is also a nested hierarchical network.

Geometry and entropies in a fixed conformal class on surfaces

We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature.

Finiteness in polygonal billiards on hyperbolic plane

\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.

T-wave propagation from the Pacific to the Atlantic: The 2020 M7.4 Kermadec Trench earthquake case.

An Mw7.4 submarine earthquake occurred in the Kermadec Trench, northeast of New Zealand, on 18 June, 2020. This powerful earthquake triggered energetic tertiary waves (T-waves) that propagated through the South Pacific Ocean into the South Atlantic Ocean, where the T-waves were recorded by a hydrophone station near Ascension Island, 15 127 km away from the epicenter. Different T-wave arrivals were identified during the earthquake period with arrival angles deviating from the geodesic path. A three-dimensional sound propagation model has been utilized to investigate the cause of the deviation and confirm the horizontal diffraction of the T-waves at the Drake Passage.

Atrophy and partial volume related bias in cortical region of interest NODDI metrics

AbstractBackgroundNeurite Orientation Dispersion and Density Imaging (NODDI) provides in‐vivo indices of neurite density (NDI) and orientation dispersion (ODI) within the tissue compartment of each voxel. However, NDI and ODI are treated equally when calculating region of interest (ROI) means, despite tissue fraction (TF) varying within regions undergoing neurodegeneration. Covariation between TF and cortical NODDI measures bias these conventional means and we recommend using tissue‐weighted averages to address this.MethodIn this study, we included 22 healthy controls and 33 individuals affected by Young‐Onset Alzheimer’s disease (YOAD, see Table 1 for demographics) with suitable diffusion‐weighted and T1‐weighted 3T MR images. Diffusion data were corrected for eddy currents, motion and susceptibility artefacts before fitting the NODDI model to produce NDI, ODI, isotropic volume fraction (ISO) and TF maps (TF=1‐ISO). T1w images were parcellated into cortical ROIs using Geodesic Information Flow (Cardoso et al., IEEE Trans.Med.Im.; 34:1976‐1988, 2015). Five bilateral ROIs expected to undergo neurodegeneration were analysed (Precuneus, Fusiform Gyrus, Superior Parietal, Middle and Inferior Temporal cortex). ROIs were resampled into native diffusion space and two regional measures calculated: 1) conventional, unweighted NDI and ODI averages and 2) tissue‐weighted averages using voxel TF as weights. Within‐participant differences between conventional and tissue‐weighted measures were calculated. Spearman’s rank tested correlations and Wilcoxon tests evaluated within‐ and between‐participant differences (Bonferroni adjusted for multiple comparisons).ResultTF positively correlated with GM volume (rs range=0.33‐0.68,p&lt;0.05) in all ROIs except the left fusiform gyrus (rs=0.31,p=0.06) (Figure 1). YOAD individuals had lower TF than healthy controls in all ROIs (Figure 2a), and lower volumes in all ROIs except the right fusiform gyrus (W=475,p=0.27) (Figure 2b). NDI showed small positive/negative biases in six of the ten ROIs (Figure 3a), while ODI showed significant positive biases across all ROIs (Figure 3b). Biases decreased as TF increased towards its maximum of one (Figure 4a‐4b).ConclusionLower cortical volumes in YOAD were associated with lower TF and higher bias, suggesting a greater risk for misestimation of cortical region NODDI metrics in studies involving neurodegenerative disease. We recommend tissue‐weighted averages to account for varying intra‐regional TF in NODDI measures.

Rolling Bearing Incipient Fault Diagnosis Method Based on Improved Transfer Learning with Hybrid Feature Extraction.

Data-driven based rolling bearing fault diagnosis has been widely investigated in recent years. However, in real-world industry scenarios, the collected labeled samples are normally in a different data distribution. Moreover, the features of bearing fault in the early stages are extremely inconspicuous. Due to the above mentioned problems, it is difficult to diagnose the incipient fault under different scenarios by adopting the conventional data-driven methods. Therefore, in this paper a new unsupervised rolling bearing incipient fault diagnosis approach based on transfer learning is proposed, with a novel feature extraction method based on a statistical algorithm, wavelet scattering network, and a stacked auto-encoder network. Then, the geodesic flow kernel algorithm is adopted to align the feature vectors on the Grassmann manifold, and the k-nearest neighbor classifier is used for fault classification. The experiment is conducted based on two bearing datasets, the bearing fault dataset of Case Western Reserve University and the bearing fault dataset of Xi’an Jiaotong University. The experiment results illustrate the effectiveness of the proposed approach on solving the different data distribution and incipient bearing fault diagnosis issues.

Superintegrable Bertrand Magnetic Geodesic Flows

The problem of description of superintegrable systems (i.e., systems with closed trajectories in a certain domain) in the class of rotationally symmetric natural mechanical systems goes back to Bertrand and Darboux. We describe all superintegrable (in a domain of slow motions) systems in the class of rotationally symmetric magnetic geodesic flows. We show that all sufficiently slow motions in a central magnetic field on a two-dimensional manifold of revolution are periodic if and only if the metric has a constant scalar curvature and the magnetic field is homogeneous, i.e. proportional to the area form.