2D Manifolds Research Articles

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Morphometric brain organization across the human lifespan reveals increased dispersion linked to cognitive performance.

The human brain is organized as segregation and integration units and follows complex developmental trajectories throughout life. The cortical manifold provides a new means of studying the brain's organization in a multidimensional connectivity gradient space. However, how the brain's morphometric organization changes across the human lifespan remains unclear. Here, leveraging structural magnetic resonance imaging scans from 1,790 healthy individuals aged 8 to 89 years, we investigated age-related global, within- and between-network dispersions to reveal the segregation and integration of brain networks from 3D manifolds based on morphometric similarity network (MSN), combining multiple features conceptualized as a "fingerprint" of an individual's brain. Developmental trajectories of global dispersion unfolded along patterns of molecular brain organization, such as acetylcholine receptor. Communities were increasingly dispersed with age, reflecting more disassortative morphometric similarity profiles within a community. Increasing within-network dispersion of primary motor and association cortices mediated the influence of age on the cognitive flexibility of executive functions. We also found that the secondary sensory cortices were decreasingly dispersed with the rest of the cortices during aging, possibly indicating a shift of secondary sensory cortices across the human lifespan from an extreme to a more central position in 3D manifolds. Together, our results reveal the age-related segregation and integration of MSN from the perspective of a multidimensional gradient space, providing new insights into lifespan changes in multiple morphometric features of the brain, as well as the influence of such changes on cognitive performance.

Disentanglement of the chemodynamical assembly: mapping the Milky Way discs

ABSTRACT The formation and structure of the Milky Way have a fundamental role in our understanding of the universe and its evolution, and thanks to the Gaia mission and large spectroscopic surveys, we live an exceptional moment of data availability, allowing us to trace the building blocks of the Galactic disc and their relations. In this sense, we propose here the exploration of a large data set in a top-down fashion, elaborating a similarity map of the local Galactic volume in order to segregate and characterize its main components, searching for hints about their relations. We have used the t-SNE algorithm with chemical, orbital and kinematic properties of the stars to produce 2D manifolds and dissect their structure by isolating populations to further analyse their behaviour. The young thin disc could be clearly separated from the older thick disc, also showing a puzzling transition zone with hints about the aftermath of the Gaia–Sausage–Enceladus merger. Moving groups and resonant features also appear prominently in the maps, splitting the disc into inner and outer portions as consequence of the resonances produced by the Galactic bar. The dynamical halo appears as an extreme end related to the heated portion of the thick disc, showing sub-structures corresponding to known accreted populations. Open and globular clusters also appear in their chemical/evolutionary context. We present details of the developed strategy, an overview of the different populations and their relations, as well as a discussion and insights of our results in the scenario of Galactic evolution.

Analyzing multi-step reaction mechanisms: Unveiling non-linear dynamics with advanced reduction techniques

This study aims to achieve the computational results for the oxidation of carbon monoxide using the spectral quasi-equilibrium manifold and the intrinsic low-dimensional manifold techniques, enabling the identification of the slow invariant manifold and subsequent simulation of the solutions. In addition, the application of Gibbs’ rule facilitates the reduction of the system, thereby effectively transforming higher dimensions into lower dimensions, ultimately revealing the underlying 1D manifold dynamics. This research serves to elucidate the intricacies of phase-shifting behaviors in the context of key species within the overarching and interconnected multi-step reaction mechanisms, contributing significantly to the advancement of understanding in this complex field.

Dynamic 3D Point Cloud Sequences as 2D Videos.

Dynamic 3D point cloud sequences serve as one of the most common and practical representation modalities of dynamic real-world environments. However, their unstructured nature in both spatial and temporal domains poses significant challenges to effective and efficient processing. Existing deep point cloud sequence modeling approaches imitate the mature 2D video learning mechanisms by developing complex spatio-temporal point neighbor grouping and feature aggregation schemes, often resulting in methods lacking effectiveness, efficiency, and expressive power. In this paper, we propose a novel generic representation called Structured Point Cloud Videos (SPCVs). Intuitively, by leveraging the fact that 3D geometric shapes are essentially 2D manifolds, SPCV re-organizes a point cloud sequence as a 2D video with spatial smoothness and temporal consistency, where the pixel values correspond to the 3D coordinates of points. The structured nature of our SPCV representation allows for the seamless adaptation of well-established 2D image/video techniques, enabling efficient and effective processing and analysis of 3D point cloud sequences. To achieve such re-organization, we design a self-supervised learning pipeline that is geometrically regularized and driven by self-reconstructive and deformation field learning objectives. Additionally, we construct SPCV-based frameworks for both low-level and high-level 3D point cloud sequence processing and analysis tasks, including action recognition, temporal interpolation, and compression. Extensive experiments demonstrate the versatility and superiority of the proposed SPCV, which has the potential to offer new possibilities for deep learning on unstructured 3D point cloud sequences. Code will be released at https://github.com/ZENGYIMING-EAMON/SPCV.

Fermi–Walker magnetic curves and Killing trajectories in 3D Riemannian manifolds

This paper deals with obtaining some new characterizations on the dynamics of the moving charged particle acting under particular forces in the three‐dimensional space. In this perspective, we have enlarged the research of magnetic curves and associated Killing magnetic fields by adding the Fermi–Walker parallelism and derivative. By doing this, we investigate the Hall–Killing–Fermi effect on the charged particle, then we define new types of Fermi–Walker magnetic curves corresponding to the Killing magnetic field. We carefully construct the Fermi–Walker character of the magnetic field to prove the parallelism of the magnetic trajectories and uniformly accelerated motion of the charged particle. It leads to defining a very special family of magnetic fields, that is, Fermi–Walker uniform magnetic fields. Moreover, we use the Killing feature of the magnetic field to show that the Frenet–Serret scalars and quasislopes of the magnetic curves must meet particular differential equations that lead to obtaining some geometric interpretations of the magnetic trajectories. Last but not the least, both of these two characters allow us to define Killing and Fermi–Walker electric vector fields influencing the motion of the charged particle experiencing the Lorentz force in the three‐dimensional setting.

Steiner and tube formulae in 3D contact sub-Riemannian geometry

We prove a Steiner formula for regular surfaces with no characteristic points in 3D contact sub-Riemannian manifolds endowed with an arbitrary smooth volume. The formula we obtain, which is equivalent to a half-tube formula, is of local nature. It can thus be applied to any surface in a region not containing characteristic points. We provide a geometrical interpretation of the coefficients appearing in the expansion, and compute them on some relevant examples in three-dimensional sub-Riemannian model spaces. These results generalize those obtained in [Z. M. Balogh, F. Ferrari, B. Franchi, E. Vecchi and K. Wildrick, Steiner’s formula in the Heisenberg group, Nonlinear Anal. 126 (2015) 201–217; M. Ritoré, Tubular neighborhoods in the sub-Riemannian Heisenberg groups, Adv. Calc. Var. 14(1) (2021) 1–36] for the Heisenberg group.

Statistical Evaluation of 3D Manifolds Shape Retention during Simplification Stages

Statistical Evaluation of 3D Manifolds Shape Retention during Simplification Stages

The number of rational points of two parameter Calabi-Yau manifolds as toric hypersurfaces

The number of rational points in toric data are given for two-parameter Calabi-Yau $n$-folds as toric hypersurfaces over finite fields $\mathbb F_p$ . We find that the fundamental period is equal to the number of rational points of the Calabi-Yau $n$-folds in zeroth order $p$-adic expansion. By analyzing the solution set of the GKZ-system given by the enhanced polyhedron, we deduce that under type II/F-theory duality the 3D and 4D Calabi-Yau manifolds have the same number of rational points in zeroth order. Taking the quintic and its duality as an example, the number of rational points in some specific complex moduli are given by numerical calculation to support our results.

Numerical investigations of stochastic Newell–Whitehead–Segel equation in (2+1) dimensions

The stochastic Newell–Whitehead–Segel in [Formula: see text] dimensions is under consideration. It represents the population density or dimensionless temperature and it discusses how stripes appear in temporal and spatial dimensional systems. The Newell–Whitehead–Segel equation (NWSE) has applications in different areas such as ecology, chemical, mechanical, biology and bio-engineering. The important thing is if we see the problem in the two-dimensional (2D) manifold, then the whole 3D picture can be included in the model. The 3D space is embedded compactly in the 2D manifolds. So, 2D problems for the Newell–White–Segel equation are very important because they consider the one, two and three dimensions in it. The numerical solutions of the underlying model have been extracted successfully by two schemes, namely stochastic forward Euler (SFE) and the proposed stochastic nonstandard finite difference (SNSFD) schemes. The existence of the solution is guaranteed by using the contraction mapping principle and Schauder’s fixed-point theorem. The consistency of each scheme is proved in the mean square sense. The stability of the schemes is shown by using von Neumann criteria. The SFE scheme is conditionally stable and the SNSFD scheme is unconditionally stable. The efficacy of the proposed methods is depicted through the simulations. The 2D and 3D graphs are plotted for various values of the parameters.

A Local Iterative Approach for the Extraction of 2D Manifolds from Strongly Curved and Folded Thin-Layer Structures.

Ridge surfaces represent important features for the analysis of 3-dimensional (3D) datasets in diverse applications and are often derived from varying underlying data including flow fields, geological fault data, and point data, but they can also be present in the original scalar images acquired using a plethora of imaging techniques. Our work is motivated by the analysis of image data acquired using micro-computed tomography ([Formula: see text]) of ancient, rolled and folded thin-layer structures such as papyrus, parchment, and paper as well as silver and lead sheets. From these documents we know that they are 2-dimensional (2D) in nature. Hence, we are particularly interested in reconstructing 2D manifolds that approximate the document's structure. The image data from which we want to reconstruct the 2D manifolds are often very noisy and represent folded, densely-layered structures with many artifacts, such as ruptures or layer splitting and merging. Previous ridge-surface extraction methods fail to extract the desired 2D manifold for such challenging data. We have therefore developed a novel method to extract 2D manifolds. The proposed method uses a local fast marching scheme in combination with a separation of the region covered by fast marching into two sub-regions. The 2D manifold of interest is then extracted as the surface separating the two sub-regions. The local scheme can be applied for both automatic propagation as well as interactive analysis. We demonstrate the applicability and robustness of our method on both artificial data as well as real-world data including folded silver and papyrus sheets.

Perceptually Uniform Construction of Illustrative Textures.

Illustrative textures, such as stippling or hatching, were predominantly used as an alternative to conventional Phong rendering. Recently, the potential of encoding information on surfaces or maps using different densities has also been recognized. This has the significant advantage that additional color can be used as another visual channel and the illustrative textures can then be overlaid. Effectively, it is thus possible to display multiple information, such as two different scalar fields on surfaces simultaneously. In previous work, these textures were manually generated and the choice of density was unempirically determined. Here, we first want to determine and understand the perceptual space of illustrative textures. We chose a succession of simplices with increasing dimensions as primitives for our textures: Dots, lines, and triangles. Thus, we explore the texture types of stippling, hatching, and triangles. We create a range of textures by sampling the density space uniformly. Then, we conduct three perceptual studies in which the participants performed pairwise comparisons for each texture type. We use multidimensional scaling (MDS) to analyze the perceptual spaces per category. The perception of stippling and triangles seems relatively similar. Both are adequately described by a 1D manifold in 2D space. The perceptual space of hatching consists of two main clusters: Crosshatched textures, and textures with only one hatching direction. However, the perception of hatching textures with only one hatching direction is similar to the perception of stippling and triangles. Based on our findings, we construct perceptually uniform illustrative textures. Afterwards, we provide concrete application examples for the constructed textures.

Classical and Quantum mechanics on 3D contact manifolds

Classical and Quantum mechanics on 3D contact manifolds

Surfaces of genus g ≥ 1 in 3D contact sub-Riemannian manifolds

We consider smooth embedded surfaces in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the fmiteness/not-finiteness of the distance.

TetCNN: Convolutional Neural Networks on Tetrahedral Meshes.

Convolutional neural networks (CNN) have been broadly studied on images, videos, graphs, and triangular meshes. However, it has seldom been studied on tetrahedral meshes. Given the merits of using volumetric meshes in applications like brain image analysis, we introduce a novel interpretable graph CNN framework for the tetrahedral mesh structure. Inspired by ChebyNet, our model exploits the volumetric Laplace-Beltrami Operator (LBO) to define filters over commonly used graph Laplacian which lacks the Riemannian metric information of 3D manifolds. For pooling adaptation, we introduce new objective functions for localized minimum cuts in the Graclus algorithm based on the LBO. We employ a piece-wise constant approximation scheme that uses the clustering assignment matrix to estimate the LBO on sampled meshes after each pooling. Finally, adapting the Gradient-weighted Class Activation Mapping algorithm for tetrahedral meshes, we use the obtained heatmaps to visualize discovered regions-of-interest as biomarkers. We demonstrate the effectiveness of our model on cortical tetrahedral meshes from patients with Alzheimer's disease, as there is scientific evidence showing the correlation of cortical thickness to neurodegenerative disease progression. Our results show the superiority of our LBO-based convolution layer and adapted pooling over the conventionally used unitary cortical thickness, graph Laplacian, and point cloud representation.

Canards Underlie Both Electrical and Ca$^{2+}$-Induced Early Afterdepolarizations in a Model for Cardiac Myocytes

Early afterdepolarizations (EADs) are voltage oscillations that can occur during the plateau phase of a cardiac action potential. EADs at the cellular level have been linked to potentially deadly tissue-level arrhythmias, and the mechanisms for their arisal are not fully understood. There is ongoing debate as to which is the predominant biophysical mechanism of EAD production: imbalanced interactions between voltage-gated transmembrane currents or overactive Ca$^{2+}$-dependent transmembrane currents brought about by pathological intracellular Ca$^{2+}$ release dynamics. In this article, we address this issue using a foundational 10-dimensional biophysical ventricular action potential model which contains both electrical and intracellular Ca$^{2+}$ components. Surprisingly, we find that the model can produce EADs through both biophysical mechanisms, which hints at a more fundamental dynamical mechanism for EAD production. Fast-slow analysis reveals EADs, in both cases, to be canard-induced mixed-mode oscillations. While the voltage-driven EADs arise from a fast-slow problem with two slow variables, the Ca$^{2+}$-driven EADs arise from the addition of a third slow variable. Hence, we adapt existing computational methods in order to compute 2D slow manifolds and 1D canard orbits in the reduced 7D model from which voltage-driven EADs arise. Further, we extend these computational methods in order to compute, for the first time, 2D sets of maximal canards which partition the 3D slow manifolds of the 8D problem from which Ca$^{2+}$-driven EADs arise. The canard viewpoint provides a unifying alternative to the voltage- or Ca$^{2+}$-driven viewpoints while also providing explanatory and predictive insights that cannot be obtained through the use of the traditional fast-slow approach.

Mass Production of 2D Manifolds of Graphene Oxide by Shear Flow

Mass Production of 2D Manifolds of Graphene Oxide by Shear Flow

Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds

We prove the pathwise uniqueness of solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov, [7], to the stochastic setting. The proof is based on the deterministic and new stochastic spectrally localized Strichartz estimates and the Littlewood-Paley decomposition.

Kimesurface Representation and Tensor Linear Modeling of Longitudinal Data.

Many modern techniques for analyzing time-varying longitudinal data rely on parametric models to interrogate the time-courses of univariate or multivariate processes. Typical analytic objectives include utilizing retrospective observations to model current trends, predict prospective trajectories, derive categorical traits, or characterize various relations. Among the many mathematical, statistical, and computational strategies for analyzing longitudinal data, tensor-based linear modeling offers a unique algebraic approach that encodes different characterizations of the observed measurements in terms of state indices. This paper introduces a new method of representing, modeling, and analyzing repeated-measurement longitudinal data using a generalization of event order from the positive reals to the complex plane. Using complex time (kime), we transform classical time-varying signals as 2D manifolds called kimesurfaces. This kime characterization extends the classical protocols for analyzing time-series data and offers unique opportunities to design novel inference, prediction, classification, and regression techniques based on the corresponding kimesurface manifolds. We define complex time and illustrate alternative time-series to kimesurface transformations. Using the Laplace transform and its inverse, we demonstrate the bijective mapping between time-series and kimesurfaces. A proposed general tensor regression based linear model is validated using functional Magnetic Resonance Imaging (fMRI) data. This kimesurface representation method can be used with a wide range of machine learning algorithms, artificial intelligence tools, analytical approaches, and inferential techniques to interrogate multivariate, complex-domain, and complex-range longitudinal processes.

On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient K̂ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.