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Xu Xu

Publications and source records attributed to Xu Xu.

At least 73 records · Page 4Linked to original sources
arXiv2020

Combinatorial Ricci flow on compact 3-manifolds with boundary

Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analog of Chow-Luo's combinatorial Ricci flow on a triangulated surface and conjectured to find algorithmically the complete hyperbolic metric on the compact 3-manifold with totally geodesic boundary. In this paper, we prove Luo's conjecture affirmatively by extending the combinatorial Ricci flow through the singularities of the flow if the ideally triangulated compact 3-manifold with boundary admits such a metric.

math.GT↗
arXiv2020

Chromodynamic multi-relaxation time lattice Boltzmann scheme for fluids with density difference

We develop, after Dellar (P. J. Dellar, Phys. Rev. E. 65, 036309 (2002), J. Comput. Phys. 190, pp351 (2003)), a multiple-relaxation time (MRT), chromodynamic, multi-component lattice Boltzmann equation (MCLBE) scheme for simulation of isothermal, immiscible fluid flow with a density contrast. It is based on Lishchuk's method (J. U. Brackbill, D. B. Kothe and C. Zemach, J. Comp. Phys. 100, 335-354 (1992), S. V. Lishchuk, C. M. Care and I. Halliday, Phys. Rev. E. 67(3), 036701(2), (2003)) and the segregation of d'Ortona et al. (U. D'Ortona, D. Salin, M. Cieplak, R. B. Rybka and J. R. Banavar Phys. Rev. E. 51, 3718, (1995)). We focus on fundamental model verifiability but do relate some of our data to that from previous approaches, due to Ba et al. (Y. Ba, H. Liu, Q. Li, Q. Kang and J. Sun, Phys. Rev. E 94, 023310 (2016)) and earlier Liu et al. (H. Liu, A. J. Valocchi and Q. Kang, Phys. Rev. E 85, 046309 (2012)), who pioneered large density difference chromodynamic MCLBE and showed the practical benefits of a MRT collision model. Specifically, we test the extent to which chromodynamic MCLBE MRT schemes comply with the kinematic condition of mutual impenetrability and the continuous traction condition by developing analytical benchmarking flows. We conclude that our data, taken with those of Ba et al., verify the utility of MRT chromodynamic MCLBE.

physics.flu-dyn↗
arXiv2020

Analytical expressions of variable specific yield for layered soils in shallow water table environments

This paper presents analytical expressions of variable specific yield for layered soils in shallow water table environments, with introducing two distinct concepts of point specific yield (Syp) and interval average specific yield (Syi). The Syp and Syi refer to the specific yield for the water table fluctuation approaching zero infinitely and that for an interval fluctuation of water table, respectively. On the basis of specific yield definition and van Genuchten model of soil water retention, the analytical and semi-analytical expressions were respectively proposed for Syp and Syi towards layered soils. The analytical expressions are evaluated and verified by experimental data and comparison with the previous expressions. Analyses indicate our expressions for Syp and Syi could effectively reflect the changes and nonlinear properties affected by soil hydraulic properties and soil layering under shallow water table conditions. The previously confused understanding of Syp and Syi are also distinguished. The practicality and applicability for the specific yield expressions are comprehensively analyzed for the potential applications related to the subsurface water modeling and management issues.

physics.geo-ph↗
arXiv2020

Combinatorial Calabi flow with surgery on surfaces

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo. To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu and Gu-Luo-Sun-Wu, we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

math.GT↗
arXiv2019

A new proof of Bowers-Stephenson conjecture

Inversive distance circle packing on surfaces was introduced by Bowers-Stephenson as a generalization of Thurston's circle packing and conjectured to be rigid. The infinitesimal and global rigidity of circle packing with nonnegative inversive distance were proved by Guo and Luo respectively. The author proved the global rigidity of circle packing with inversive distance in $(-1,+\infty)$. In this paper, we give a new variational proof of the Bowers-Stephenson conjecture for inversive distance in $(-1,+\infty)$, which simplifies the existing proofs and could be generalized to three dimensional case.

math.GT↗
arXiv2018

On the global rigidity of sphere packings on 3-dimensional manifolds

In this paper, we prove the global rigidity of sphere packings on 3-dimensional manifolds. This is a 3-dimensional analogue of the rigidity theorem of Andreev-Thurston and was conjectured by Cooper and Rivin. We also prove a global rigidity result using a combinatorial scalar curvature introduced by Ge and the author.

math.GT↗
arXiv2018

On a combinatorial curvature for surfaces with inversive distance circle packing metrics

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton \cite{Ham1}. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to $α$-curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture \cite{BS}. We also use the combinatorial Ricci flow to study the corresponding $α$-Yamabe problem.

math.GT↗
arXiv2018

Rigidity of inversive distance circle packings revisited

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity and then Luo \cite{L3} proved the global rigidity. In this paper, based on an observation of Zhou \cite{Z}, we prove this conjecture for inversive distance in $(-1, +\infty)$ by variational principles. We also study the global rigidity of a combinatorial curvature introduced in \cite{GJ4,GX4,GX6} with respect to the inversive distance circle packing metrics where the inversive distance is in $(-1, +\infty)$.

math.GT↗
arXiv2018

Toward Marker-free 3D Pose Estimation in Lifting: A Deep Multi-view Solution

Lifting is a common manual material handling task performed in the workplaces. It is considered as one of the main risk factors for Work-related Musculoskeletal Disorders. To improve work place safety, it is necessary to assess musculoskeletal and biomechanical risk exposures associated with these tasks, which requires very accurate 3D pose. Existing approaches mainly utilize marker-based sensors to collect 3D information. However, these methods are usually expensive to setup, time-consuming in process, and sensitive to the surrounding environment. In this study, we propose a multi-view based deep perceptron approach to address aforementioned limitations. Our approach consists of two modules: a "view-specific perceptron" network extracts rich information independently from the image of view, which includes both 2D shape and hierarchical texture information; while a "multi-view integration" network synthesizes information from all available views to predict accurate 3D pose. To fully evaluate our approach, we carried out comprehensive experiments to compare different variants of our design. The results prove that our approach achieves comparable performance with former marker-based methods, i.e. an average error of $14.72 \pm 2.96$ mm on the lifting dataset. The results are also compared with state-of-the-art methods on HumanEva-I dataset, which demonstrates the superior performance of our approach.

cs.CV↗
arXiv2016

Beam Search for Learning a Deep Convolutional Neural Network of 3D Shapes

This paper addresses 3D shape recognition. Recent work typically represents a 3D shape as a set of binary variables corresponding to 3D voxels of a uniform 3D grid centered on the shape, and resorts to deep convolutional neural networks(CNNs) for modeling these binary variables. Robust learning of such CNNs is currently limited by the small datasets of 3D shapes available, an order of magnitude smaller than other common datasets in computer vision. Related work typically deals with the small training datasets using a number of ad hoc, hand-tuning strategies. To address this issue, we formulate CNN learning as a beam search aimed at identifying an optimal CNN architecture, namely, the number of layers, nodes, and their connectivity in the network, as well as estimating parameters of such an optimal CNN. Each state of the beam search corresponds to a candidate CNN. Two types of actions are defined to add new convolutional filters or new convolutional layers to a parent CNN, and thus transition to children states. The utility function of each action is efficiently computed by transferring parameter values of the parent CNN to its children, thereby enabling an efficient beam search. Our experimental evaluation on the 3D ModelNet dataset demonstrates that our model pursuit using the beam search yields a CNN with superior performance on 3D shape classification than the state of the art.

cs.CV↗
arXiv2016

A combinatorial Yamabe problem on two and three dimensional manifolds

In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.

math.DG↗
arXiv2015

A Discrete Ricci Flow on Surfaces in Hyperbolic Background Geometry

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which could be taken as the area of the hyperbolic disk packed at each vertex. We prove that the corresponding discrete Ricci flow converges if and only if there exists a circle packing metric with zero curvature. We also prove that the flow converges if the initial curvatures are all negative. Note that, this result does not require the existence of zero curvature metric or Thurston's combinatorial-topological condition. We further generalize the definition of combinatorial curvature to any given area element and prove the equivalence between the existence of zero curvature metric and the convergence of the corresponding flow.

math.DG↗
arXiv2015

$α$-curvatures and $α$-flows on low dimensional triangulated manifolds

In this paper, we introduce two discrete curvature flows, which are called $α$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving $α$ order discrete Gauss curvature to a constant. When $αχ(M)\leq0$, we prove that the convergence of the flow is equivalent to the existence of constant $α$-curvature metric. We further get a necessary and sufficient combinatorial-topological-metric condition, which is a generalization of Thurston's combinatorial-topological condition, for the existence of constant $α$-curvature metric. For triangulated 3-manifolds, we generalize the combinatorial Yamabe functional and combinatorial Yamabe problem introduced by the authors in \cite{GX2,GX4} to $α$-order. We also study the $α$-order flow carefully, aiming at evolving $α$ order combinatorial scalar curvature, which is a generalization of Cooper and Rivin's combinatorial scalar curvature, to a constant.

math.DG↗
arXiv2013

3-Dimensional Discrete curvature flows and discrete Einstein metric

We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.

math.DG↗
arXiv2013

Discrete Quasi-Einstein Metrics and Combinatorial Curvature Flows in 3-Dimension

We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when the initial energy is small enough. The proof relies on a careful analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix of the curvature map. As a consequence, combinatorial curvature flow provides an algorithm to compute discrete sphere packing metrics with prescribed curvatures.

math.DG↗
arXiv2013

2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2π$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.

math.DG↗