2D Manifolds Research Articles

Embedding Functional Brain Networks in Low Dimensional Spaces Using Manifold Learning Techniques.

Background: fMRI data is inherently high-dimensional and difficult to visualize. A recent trend has been to find spaces of lower dimensionality where functional brain networks can be projected onto manifolds as individual data points, leading to new ways to analyze and interpret the data. Here, we investigate the potential of two powerful non-linear manifold learning techniques for functional brain networks representation: (1) T-stochastic neighbor embedding (t-SNE) and (2) Uniform Manifold Approximation Projection (UMAP) a recent breakthrough in manifold learning.Methods: fMRI data from the Human Connectome Project (HCP) and an independent study of aging were used to generate functional brain networks. We used fMRI data collected during resting state data and during a working memory task. The relative performance of t-SNE and UMAP were investigated by projecting the networks from each study onto 2D manifolds. The levels of discrimination between different tasks and the preservation of the topology were evaluated using different metrics.Results: Both methods effectively discriminated the resting state from the memory task in the embedding space. UMAP discriminated with a higher classification accuracy. However, t-SNE appeared to better preserve the topology of the high-dimensional space. When networks from the HCP and aging studies were combined, the resting state and memory networks in general aligned correctly.Discussion: Our results suggest that UMAP, a more recent development in manifold learning, is an excellent tool to visualize functional brain networks. Despite dramatic differences in data collection and protocols, networks from different studies aligned correctly in the embedding space.

A comparative study of flame-wall interaction and flame-cooling air interaction

This paper presents a three-dimensional (3D) direct numerical simulation (DNS) study of flame-wall interaction (FWI) and flame-cooling air interaction (FCAI). A preheated, methane/air mixture enters a channel with constant temperature walls, where the top wall is effusion cooled. An imposed vertical hot sheet near the inlet creates two flame branches interacting with the top and bottom walls. The flame is observed to be leaner in the region where it interacts with the effusion cooling jets. In this region, the flame is longer and features reduced CO mass fraction. The fluctuations in the heat release rate (HRR) and CO mass fraction are also relatively small near the top wall. Near the bottom wall, finger-like flame structures are formed due to the interaction of turbulent vortices with the flame surface. These flame structures initially move away from the wall as they propagate further downstream before eventually collapsing at the wall. This leads to the creation of regions of high wall heat flux and CO. While analysis of the CO thermochemical state shows a complex picture near the bottom wall, two-dimensional (2D) manifolds can be identified near the top wall. Therefore, a framework to estimate CO mass fraction due to FCAI based on 1D freely-propagating flame solutions is proposed showing a good agreement with the DNS results.

Comparison Theorems for 3D Manifolds With Scalar Curvature Bound

Abstract Two sharp comparison results are derived for 3D complete noncompact manifolds with scalar curvature bounded from below. The 1st one concerns the Green’s function. When the scalar curvature is nonnegative, it states that the rate of decay of an energy quantity over the level set is strictly less than that of the Euclidean space unless the manifold itself is isometric to the Euclidean space. The result is in turn converted into a sharp area comparison for the level set of the Green’s function when in addition the Ricci curvature of the manifold is assumed to be asymptotically nonnegative at infinity. The 2nd result provides a sharp upper bound of the bottom spectrum in terms of the scalar curvature lower bound, in contrast to the classical result of Cheng, which involves a Ricci curvature lower bound.

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

AbstractSelf‐organizing 2D sheets into a 3D structure with a well‐defined arrangement and tuned bandgap has garnered significant interest. However, there exist challenges such as scalable and feasible synthetic methods and preservation of 2D properties while preventing re‐stacking. Herein, a facile method for the mass production of 2D manifolds of graphene oxide (GO), which are 3D microtubes, while maintaining their 2D properties by applying shear to roll up GO sheets, is reported. The GO mesotubes are formed by shear flow in aluminum–glass parallel plates. Redox reaction plays a vital role during the formation of GO mesotubes by the vorticity alignment during slow shear flow. A high yield of 94% is achieved at an initial pH of 2.2. The maximum d‐spacing of the GO sheet in the tube wall is 21 nm, indicating no re‐stacking of the graphitic structure. It is demonstrated that the GO mesotubes behaves like a soft gel, rendering them a candidate material for soft robotics, e‐skins, and a visible light sensor owing to their reduced optical bandgap.

Probabilistic modelling of general noisy multi-manifold data sets

The intrinsic nature of noisy and complex data sets is often concealed in low-dimensional structures embedded in a higher dimensional space. Number of methodologies have been developed to extract and represent such structures in the form of manifolds (i.e. geometric structures that locally resemble continuously deformable intervals of Rj1). Usually a-priori knowledge of the manifold's intrinsic dimensionality is required. Additionally, their performance can often be hampered by the presence of a significant high-dimensional noise aligned along the low-dimensional core manifold. In real-world applications, the data can contain several low-dimensional structures of different dimensionalities. We propose a framework for dimensionality estimation and reconstruction of multiple noisy manifolds embedded in a noisy environment. To the best of our knowledge, this work represents the first attempt at detection and modelling of a set of coexisting general noisy manifolds by uniting two aspects of multi-manifold learning: the recovery and approximation of core noiseless manifolds and the construction of their probabilistic models. The easy-to-understand hyper-parameters can be manipulated to obtain an emerging picture of the multi-manifold structure of the data. We demonstrate the workings of the framework on two synthetic data sets, presenting challenging features for state-of-the-art techniques in Multi-Manifold learning. The first data set consists of multiple sampled noisy manifolds of different intrinsic dimensionalities, such as Möbius strip, toroid and spiral arm. The second one is a topologically complex set of three interlocked toroids. Given the absence of such unified methodologies in the literature, the comparison with existing techniques is organized along the two separate aspects of our approach mentioned above, namely manifold approximation and probabilistic modelling. The framework is then applied to a complex data set containing simulated gas volume particles from a particle simulation of a dwarf galaxy interacting with its host galaxy cluster. Detailed analysis of the recovered 1D and 2D manifolds can help us to understand the nature of Star Formation in such complex systems.

True Mechanism of Spontaneous Order from Turbulence in Two-Dimensional Superfluid Manifolds.

In a 2D turbulent fluid containing pointlike vortices, Lars Onsager predicted that adding energy to the fluid can lead to the formation of persistent clusters of like-signed vortices, i.e., Onsager vortex (OV) clusters. In the evolution of 2D superfluid turbulence in a uniform disk-shaped Bose-Einstein condensate (BEC), it was discovered that a pair of OV clusters with opposite signs can form without any energy input. This striking spontaneous order was explained as being due to a vortex evaporative-heating mechanism, i.e., annihilations of vortex-antivortex pairs which remove the lowest-energy vortices and thereby boost the mean energy per vortex. However, in our search for exotic OV states in a boundaryless 2D spherical BEC, we found that OV clusters never form despite the annihilations of vortex pairs. Our analysis reveals that contrary to the general belief, vortex-pair annihilation emits intense sound waves, which damp the motion of all vortices and hence suppress the formation of OV clusters. We also present unequivocal evidence showing that the true mechanism underlying the observed spontaneous OV state is the vortices exiting the BEC boundaries. Uncovering this mechanism paves the way for a comprehensive understanding of emergent vortex orders in 2D manifolds of superfluids driven far from equilibrium.

Random fields representation over manifolds via isometric feature mapping‐based dimension reduction

AbstractGenerating random fields over irregular geometries (e.g., two‐dimensional (2D) manifolds embedded in the three‐dimensional (3D) Euclidean space) is a great challenge because the geometry structure is complex and the correlation function can hardly be derived, thus the traditional methods, for example, spatial discretization methods or series expansion methods, cannot be directly adopted. To solve this issue, the present paper develops a two‐stage strategy to simulate random fields over manifolds. The core idea is to map the manifolds into the 2D Euclidean space through Isometric feature mapping (Isomap), with which the geodesic distance between points in the mapped 2D Euclidean space and the original manifold space is kept as the same. Therefore, the correlation function can be readily derived and the conventional methods can be directly employed to generate random fields. To validate the proposed method, several different types of manifolds are dimensionally reduced to 2D planar domains, and the stochastic harmonic function method is used to generate the random fields. Finally, case studies on the stochastic finite element analysis of two different structures are also performed to demonstrate the applicability and efficiency of the proposed method.

Random Assignment Problems on 2d Manifolds

We consider the assignment problem between two sets of N random points on a smooth, two-dimensional manifold Omega of unit area. It is known that the average cost scales as E_{Omega }(N)sim {1}/{2pi }ln N with a correction that is at most of order sqrt{ln Nln ln N}. In this paper, we show that, within the linearization approximation of the field-theoretical formulation of the problem, the first Omega -dependent correction is on the constant term, and can be exactly computed from the spectrum of the Laplace–Beltrami operator on Omega . We perform the explicit calculation of this constant for various families of surfaces, and compare our predictions with extensive numerics.

Large eddy simulation of the Cambridge/Sandia stratified flame with flamelet-generated manifolds: Effects of non-unity Lewis numbers and stretch

The Cambridge/Sandia turbulent stratified flame (SwB5) is simulated with the LES and Flamelet-Generated Manifolds (FGM) combustion model. Three 3D FGM manifolds are adopted. With the purpose to examine the influence of transport properties, unity and non-unity Lewis numbers (Le) are included in the first two manifolds, respectively. The combined effects of non-unity Le and stretch are investigated in the third manifold. Heat loss to the wall is also modeled. Good agreement is found between the simulation and experiment. The equivalence ratio, temperature and mass fractions of CO and H2 are all well reproduced in contrast with previous simulations. It is found that using non-unity Le can even deteriorate the near-wall temperature modeling. Non-unity Le is proposed to be crucial for the CO prediction as well, besides H2. The equivalence ratio modeling is observed to be very important, which accounts for several non-unity Le effects. Flame stretch shows almost no impact on the velocity fields, whereas its effects on the species, equivalence ratio and temperature are identified, although to a limited extent for the Cambridge/Sandia flame.

The Climatic Temporal Feature Space: Continuous and Discrete

Climatic zones, representing seasonal variations in temperature (T) and precipitation (P), are generally mapped geographically using discrete classifications with distinct boundaries. However, it is well known that global T and P vary continuously in space and time with steep gradients occurring infrequently. The objective of this analysis is to use complementary forms of dimensionality reduction to quantify the spatiotemporal dimensionality of the climate system and to produce a continuous representation of global climate based on the temporal feature space of historical T and P alone. We characterize the continuous global feature space using principal components (PCs) to identify a parsimonious set of temporal endmember T and P patterns bounding the feature space of all observed T and P patterns. These endmember T and P patterns provide the basis for a linear temporal mixture model that can represent decadal T and P patterns of any geographic location as fractions of the endmember T and P patterns. Inverting this linear mixture model for each geographic T+P time series gives an estimate of the fractional contribution of each endmember to the observed time series. The resulting temporal endmember fraction maps provide a continuous representation of the Euclidean proximity of T and P observations at every geographic location to each of the temporal endmember climates bounding the space. The spatiotemporal dimensionality implied by the variance partition of T + P time series for 67,420 land-based observations suggests that the T + P temporal feature space is effectively 3D, accounting for 92% of total variance. From the topology of the feature space, we identify 4 bounding temporal endmembers upon which to base the linear temporal mixture model. Inversion of the model for each normalized observed time series yields endmember fraction estimates and a model misfit distribution with 99% of misfit < 0.21. For comparison, we also render temporal feature spaces from ensembles of 2D manifolds within the T + P space derived from suites of t-distributed Stochastic Neighbor Embeddings (t-SNE) to identify discontinuities in the feature space. Comparison of spatial PC(t-SNE) across hyperparameter settings reveals consistent structure and little hyperparameter sensitivity to temporal feature spaces rendered by t-SNE. Combining the physically interpretable continuous global structure resolved by the PC feature space with the finer scale manifold structure resolved by the t-SNE feature space provides a continuous alternative to discrete classifications of climate that cannot represent the continuous character of its temporal feature space.

Parametric failure manifolds for laminated composites

The failure modes of laminated composites are complex and varied, being strongly dependant on layup and loading conditions. In this paper, a novel representation of damage initiation in laminated composites on the structural scale is presented. The proposed approach combines the knowledge of the composite design space with detailed meso-scale models to generate a complete representation of damage initiation under all possible combinations of bi-axial loading and for any layup in the composite design space. The 2D bi-axial failure envelopes are then combined with the knowledge of the composite’s physical behaviour and the surface compatibility conditions to generate 3D failure manifolds using Gauss surface equations. The resulting system is a complete parametric description of damage initiation in laminated composite for any layup under general loading conditions. The proposed manifolds are accurate and computationally efficient tools for structural (macro) scale analyses, design and optimisation.

Optimal Submarine Cable Path Planning and Trunk-and-Branch Tree Network Topology Design

We study the path planning of submarine cable systems with trunk-and-branch tree topology on the surface of the earth. Existing work on path planning represents the earth's surface by triangulated manifolds and takes account of laying cost of the cable including material, labor, alternative protection levels, terrain slope and survivability of the cable. Survivability issues include the risk of future cable break associated with laying the cable through sensitive and risky areas, such as, in particular, earthquake-prone regions. The key novelty of this paper is an examination and solution of the path planning of submarine cable systems with trunk-and-branch tree topology. We formulate the problem as a Steiner minimal tree problem on irregular 2D manifolds in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> . For a given Steiner topology, we propose a polynomial time computational complexity numerical method based on the dynamic programming principle. If the topology is unknown, a branch and bound algorithm is adopted. Simulations are performed on real-world three-dimensional geographical data.

Density Visualization Pipeline: A Tool for Cellular and Network Density Visualization and Analysis.

Neuron classification is an important component in analyzing network structure and quantifying the effect of neuron topology on signal processing. Current quantification and classification approaches rely on morphology projection onto lower-dimensional spaces. In this paper a 3D visualization and quantification tool is presented. The Density Visualization Pipeline (DVP) computes, visualizes and quantifies the density distribution, i.e., the “mass” of interneurons. We use the DVP to characterize and classify a set of GABAergic interneurons. Classification of GABAergic interneurons is of crucial importance to understand on the one hand their various functions and on the other hand their ubiquitous appearance in the neocortex. 3D density map visualization and projection to the one-dimensional x, y, z subspaces show a clear distinction between the studied cells, based on these metrics. The DVP can be coupled to computational studies of the behavior of neurons and networks, in which network topology information is derived from DVP information. The DVP reads common neuromorphological file formats, e.g., Neurolucida XML files, NeuroMorpho.org SWC files and plain ASCII files. Full 3D visualization and projections of the density to 1D and 2D manifolds are supported by the DVP. All routines are embedded within the visual programming IDE VRL-Studio for Java which allows the definition and rapid modification of analysis workflows.

On the Extended-Hamiltonian Structure of Certain Superintegrable Systems on Constant-Curvature Riemannian and Pseudo-Riemannian Surfaces

We prove the integrability and superintegrability of a family of natural Hamiltonians which includes and generalises those studied in some literature, originally defined on the 2D Minkowski space. Some of the new Hamiltonians are a perfect analogy of the well-known superintegrable system on the Euclidean plane proposed by Tremblay-Turbiner-Winternitz and they are defined on Minkowski space, as well as on all other 2D manifolds of constant curvature, Riemannian or pseudo-Riemannian. We show also how the application of the coupling-constant-metamorphosis technique allows us to obtain new superintegrable Hamiltonians from the previous ones. Moreover, for the Minkowski case, we show the quantum superintegrability of the corresponding quantum Hamiltonian operator.Our results are obtained by applying the theory of extended Hamiltonian systems, which is strictly connected with the geometry of warped manifolds.

Tensor network approach to two-dimensional bosonization

We present a 2D bosonization duality using the language of tensor networks. Specifically, we construct a tensor network operator (TNO) that implements an exact 2D bosonization duality. The primary benefit of the TNO is that it allows for bosonization at the level of quantum states. Thus, we use the TNO to provide an explicit algorithm for bosonizing fermionic projected entangled pair states (fPEPs). A key step in the algorithm is to account for a choice of spin-structure, encoded in a set of bonds of the bosonized fPEPS. This enables our tensor network approach to bosonization to be applied to systems on arbitrary triangulations of orientable 2D manifolds.

Discontinuous Galerkin solver for the shallow-water equations in covariant form on the sphere and the ellipsoid

A Discontinuous Galerkin (DG) method for the solution of the shallow-water equations (SWE) on arbitrary 2-dimensional (2D) manifolds is presented. To this purpose the SWE are formulated in covariant form using tensor notation. This allows to correctly transform the numerical fluxes between the local coordinate systems of any two neighboring grid cells. In particular, the covariant form of the numerical diffusion term in the Lax-Friedrichs numerical flux has been derived, too. This general approach has the advantage that it avoids any coordinate singularity. It is tested for the SWE on the sphere with several standard test setups. Beyond this, a recently published test case with an analytic solution for linear inertial-gravity wave expansion has been performed. The derived formalism on arbitrary 2D manifolds allows an easy extension from the sphere to the ellipsoid. The comparison of a barotropic instability test case for the earth shows a non-negligible difference between the solution on these two bodies. The presented approach may be a starting point for the development of a dynamical core for numerical weather and climate prediction models based on the DG method.

Magnetic Trajectories on Lightlike Surfaces

Divergence free vector fields are called magnetic vector fields in three-dimensional semi-Riemannian manifolds. When a charged particle enters the magnetic vector field, it traces a new trajectory called magnetic curve by the influenced of magnetic field. In the present paper, we investigate the magnetic curves on the lightlike surfaces corresponding to the Killing magnetic fields in 3D semi-Riemannian manifolds. Moreover, we give some characterizations of these curves. As an application, we determine all magnetic curves on the lightlike cone. Finally, we give various examples to confirm the main results.

Approximating lower-star persistence via 2D combinatorial map simplification

Filtration simplification consists of simplifying a given filtration while simultaneously controlling the perturbation in the associated persistence diagrams. In this paper, we propose a filtration simplification algorithm for orientable 2-dimensional (2D) manifolds with or without boundary (meshes) represented by 2D combinatorial maps. Given a lower-star filtration of the mesh, faces are added into contiguous clusters according to a “height” function and a parameter ϵ. Faces in the same cluster are merged into a single face, resulting in a lower resolution mesh and a simpler filtration. We prove that the parameter ϵ bounds the perturbation in the original persistence diagrams, and we provide experiments demonstrating the computational advantages of the simplification process.

Volume of small balls and sub-Riemannian curvature in 3D contact manifolds

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

Integrable Dynamical Systems with Dissipation on Tangent Bundles of 2D and 3D Manifolds

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.