The obstacle problem for conformal metrics on compact Riemannian manifolds.
Ontology highlight
ABSTRACT:
SUBMITTER: Bao S
PROVIDER: S-EPMC6154054 | biostudies-literature | 2018
REPOSITORIES: biostudies-literature
Publications
The obstacle problem for conformal metrics on compact Riemannian manifolds.
Journal of inequalities and applications 20180912 1
We prove a priori estimates up to their second order derivatives for solutions to the obstacle problem of curvature equations on Riemannian manifolds ( M n , g ) arising from conformal deformation. With the a priori estimates the existence of a C 1 , 1 solution to the obstacle problem with Dirichlet boundary value is obtained by approximation. ...[more]
Similar Datasets
Project description: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.
| S-EPMC6994476 | biostudies-literature
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
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
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
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
Project description:Riboswitches are thought generally to function by modulating transcription elongation or translation initiation. In rare instances, ligand binding to a riboswitch has been found to alter the rate of RNA degradation by directly stimulating or inhibiting nearby cleavage. Here, we show that guanidine-induced pseudoknot formation by the aptamer domain of a guanidine III riboswitch from Legionella pneumophila has a different effect, stabilizing mRNA by protecting distal cleavage sites en masse from ribonuclease attack. It does so by creating a coaxially base-paired obstacle that impedes scanning from a monophosphorylated 5' end to those sites by the regulatory endonuclease RNase E. Ligand binding by other riboswitch aptamers peripheral to the path traveled by RNase E does not inhibit distal cleavage. These findings reveal that a riboswitch aptamer can function independently of any overlapping expression platform to regulate gene expression by acting directly to prolong mRNA longevity in response to ligand binding.
| S-EPMC7796952 | biostudies-literature
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
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
Project description:A community is a subset of a wider network where the members of that subset are more strongly connected to each other than they are to the rest of the network. In this paper, we consider the problem of identifying and tracking communities in graphs that change over time - dynamic community detection - and present a framework based on Riemannian geometry to aid in this task. Our framework currently supports several important operations such as interpolating between and averaging over graph snapshots. We compare these Riemannian methods with entry-wise linear interpolation and find that the Riemannian methods are generally better suited to dynamic community detection. Next steps with the Riemannian framework include producing a Riemannian least-squares regression method for working with noisy data and developing support methods, such as spectral sparsification, to improve the scalability of our current methods.
| S-EPMC6214282 | biostudies-literature
Project description:The use of unmanned underwater vehicles is steadily increasing for a variety of applications such as mapping, monitoring, inspection and intervention within several research fields and industries, e.g., oceanography, marine biology, military, and oil and gas. Particularly interesting types of unmanned underwater vehicles are bio-inspired robots such as underwater snake robots (USRs). Due to their flexible and slender body, these versatile robots are highly maneuverable and have better access capabilities than more conventional remotely operated vehicles (ROVs). Moreover, the long and slender body allows for energy-efficient transit over long distances similar to torpedo-shaped autonomous underwater vehicles (AUVs). In addition, USRs are capable of performing light intervention tasks, thereby providing intervention capabilities which exceed those of AUVs and inspection class ROVs. USRs may also propel themselves using energy-efficient motion patterns inspired by their biological counterparts. They can thereby increase the propulsion efficiency during transit and maneuvering, which is among the great challenges for autonomous underwater vehicles. In this paper, a control system for path following, and algorithms for obstacle detection and avoidance, are presented for a USR with thrusters attached at the tail module. The position of the obstacles is detected using a single camera in the head module of the USR and a developed computer vision algorithm. For the proposed control concept the robot joints are used for directional control while the thrusters are used for forward propulsion. The USR circumvents obstacles by following a circular path around them before converging back to the main straight line path when this is safe. Experimental results that validate the proposed methods are also presented.
| S-EPMC7805762 | biostudies-literature