Browse
Submit Data
Databases
API
Help

Dataset Information

0 Views

0 Connections

0 Citations

0 Reanalyses

0 Downloads

Omics score: 0

On Interpolating Sesqui-Harmonic Maps Between Riemannian Manifolds.

ABSTRACT: Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In this article we initiate a rigorous mathematical treatment of this functional and study various basic aspects of its critical points.

SUBMITTER: Branding V

PROVIDER: S-EPMC6994476 | biostudies-literature | 2020

REPOSITORIES: biostudies-literature

ACCESS DATA

Json Xml

Publications

On Interpolating Sesqui-Harmonic Maps Between Riemannian Manifolds.

Branding Volker V

Journal of geometric analysis 20190114 1

Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In this article we initiate a rigorous mathematical treatment of this functional and study various basic aspects of its critical points. ...[more]

PMID: 32063695

Similar Datasets

Some analytic results on interpolating sesqui-harmonic maps.

Project description:In this article, we study various analytic aspects of interpolating sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the case of a spherical target. The latter are critical points of an energy functional that interpolates between the functionals for harmonic and biharmonic maps. In the case of a spherical target, we will derive a conservation law and use it to show the smoothness of weak solutions. Moreover, we will obtain several classification results for interpolating sesqui-harmonic maps.

| S-EPMC7484910 | biostudies-literature

The obstacle problem for conformal metrics on compact Riemannian manifolds.

Project description:We prove a priori estimates up to their second order derivatives for solutions to the obstacle problem of curvature equations on Riemannian manifolds (Mn,g) arising from conformal deformation. With the a priori estimates the existence of a C1,1 solution to the obstacle problem with Dirichlet boundary value is obtained by approximation.

| S-EPMC6154054 | biostudies-literature

Local Riemannian geometry of model manifolds and its implications for practical parameter identifiability.

Project description:When non-linear models are fitted to experimental data, parameter estimates can be poorly constrained albeit being identifiable in principle. This means that along certain paths in parameter space, the log-likelihood does not exceed a given statistical threshold but remains bounded. This situation, denoted as practical non-identifiability, can be detected by Monte Carlo sampling or by systematic scanning using the profile likelihood method. In contrast, any method based on a Taylor expansion of the log-likelihood around the optimum, e.g., parameter uncertainty estimation by the Fisher Information Matrix, reveals no information about the boundedness at all. In this work, we present a geometric approach, approximating the original log-likelihood by geodesic coordinates of the model manifold. The Christoffel Symbols in the geodesic equation are fixed to those obtained from second order model sensitivities at the optimum. Based on three exemplary non-linear models we show that the information about the log-likelihood bounds and flat parameter directions can already be contained in this local information. Whereas the unbounded case represented by the Fisher Information Matrix is embedded in the geometric framework as vanishing Christoffel Symbols, non-vanishing constant Christoffel Symbols prove to define prototype non-linear models featuring boundedness and flat parameter directions of the log-likelihood. Finally, we investigate if those models could allow to approximate and replace computationally expensive objective functions originating from non-linear models by a surrogate objective function in parameter estimation problems.

| S-EPMC6546239 | biostudies-literature

Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images.

Project description:Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., (xi , yi ) pairs, are Euclidean is well studied. The setting where yi is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, xi ? R, against a manifold-valued variable, yi ? . We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -a manifold-valued dependent variable against multiple independent variables, i.e., f : Rn ? . Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL(n)/O(n) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.

| S-EPMC4288036 | biostudies-literature

Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry.

Project description:Most high-dimensional datasets are thought to be inherently low-dimensional-that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace-Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold's embedding to its curvature using the Second Fundamental Form and the Gauss-Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.

| S-EPMC8307776 | biostudies-literature

Global Weak Rigidity of the Gauss–Codazzi–Ricci Equations and Isometric Immersions of Riemannian Manifolds with Lower Regularity.

Project description:We are concerned with the global weak rigidity of the Gauss–Codazzi–Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with corresponding different metrics.

| S-EPMC6294193 | biostudies-other

Extended Theory of Harmonic Maps Connects General Relativity to Chaos and Quantum Mechanism.

Project description:General relativity and quantum mechanism are two separate rules of modern physics explaining how nature works. Both theories are accurate, but the direct connection between two theories was not yet clarified. Recently, researchers blur the line between classical and quantum physics by connecting chaos and entanglement equation. Here, we showed the Duan's extended HM theory, which has the solution of the general relativity, can also have the solutions of the classic chaos equations and even the solution of Schrödinger equation in quantum physics, suggesting the extended theory of harmonic maps may act as a universal theory of physics.

| S-EPMC6516492 | biostudies-literature

Using Low-Dimensional Manifolds to Map Relationships Between Dynamic Brain Networks.

Project description:As the field of dynamic brain networks continues to expand, new methods are needed to allow for optimal handling and understanding of this explosion in data. We propose here a novel approach that embeds dynamic brain networks onto a two-dimensional (2D) manifold based on similarities and differences in network organization. Each brain network is represented as a single point on the low dimensional manifold with networks of similar topology being located in close proximity. The rich spatio-temporal information has great potential for visualization, analysis, and interpretation of dynamic brain networks. The fact that each network is represented by a single point makes it possible to switch between the low-dimensional space and the full connectivity of any given brain network. Thus, networks in a specific region of the low-dimensional space can be examined to identify network features, such as the location of brain network hubs or the interconnectivity between brain circuits. In this proof-of-concept manuscript, we show that these low dimensional manifolds contain meaningful information, as they were able to successfully discriminate between cognitive tasks and study populations. This work provides evidence that embedding dynamic brain networks onto low dimensional manifolds has the potential to help us better visualize and understand dynamic brain networks with the hope of gaining a deeper understanding of normal and abnormal brain dynamics.

| S-EPMC6914694 | biostudies-literature

Regression Models on Riemannian Symmetric Spaces.

Project description:The aim of this paper is to develop a general regression framework for the analysis of manifold-valued response in a Riemannian symmetric space (RSS) and its association with multiple covariates of interest, such as age or gender, in Euclidean space. Such RSS-valued data arises frequently in medical imaging, surface modeling, and computer vision, among many others. We develop an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution in RSS. We propose various link functions to map from the Euclidean space of multiple covariates to the RSS of responses. We develop a two-stage procedure to calculate the parameter estimates and determine their asymptotic distributions. We construct the Wald and geodesic test statistics to test hypotheses of unknown parameters. We systematically investigate the geometric invariant property of these estimates and test statistics. Simulation studies and a real data analysis are used to evaluate the finite sample properties of our methods.

| S-EPMC5433528 | biostudies-literature

Flow through randomly curved manifolds.

Project description:We present a computational study of the transport properties of campylotic (intrinsically curved) media. It is found that the relation between the flow through a campylotic media, consisting of randomly located curvature perturbations, and the average Ricci scalar of the system, exhibits two distinct functional expressions, depending on whether the typical spatial extent of the curvature perturbation lies above or below the critical value maximizing the overall scalar of curvature. Furthermore, the flow through such systems as a function of the number of curvature perturbations is found to present a sublinear behavior for large concentrations, due to the interference between curvature perturbations leading to an overall less curved space. We have also characterized the flux through such media as a function of the local Reynolds number and the scale of interaction between impurities. For the purpose of this study, we have also developed and validated a new lattice Boltzmann model.

| S-EPMC3813940 | biostudies-literature

OmicsDI is part of the ELIXIR infrastructure

OmicsDI is an Elixir interoperability service. Learn more ›

Tweets

OmicsDI Databases

PRIDE
PeptideAtlas
MassIVE
JPOST Repository
Physiome Model Repository

EGA
EVA
ENA
LINCS
PAXDB
Cell Collective

MetaboLights
Metabolomics Workbench
MetabolomeExpress
GNPS
BioModels
FAIRDOMHub

ArrayExpress
dbGaP
ExpressionAtlas
GEO
NODE

Information

Databases
Help
API
Contact us
Code on GitHub
Terms of use
Submit Data