arXiv subjects

Xu Xu

Publications and source records attributed to Xu Xu.

At least 37 records · Page 2Linked to original sources

arXiv2025

Maximal principles in discrete conformal geometry with application to the rigidity of infinite triangulations

In this paper, maximum principles for Euclidean and hyperbolic discrete conformal structures on polyhedral surfaces are established. These maximum principles unify and generalize the maximum principles for vertex scalings and different types of circle packings in the literature. As an application of the hyperbolic discrete maximum principle, a discrete Schwarz-Ahlfors lemma is established. As another application, an infinite rigidity theorem for small Delaunay triangulations of the hyperbolic plane is proved.

math.MG↗

arXiv2025

Eu-doped CsSrCl$_3$ Large Nanocrystal Clusters with Self-Reduction Effect and Near-Unity Quantum Yield

Europium halide perovskites have emerged as promising candidates for environmental-friendly blue-emitting materials. However, their development is hindered by relative low photoluminescence quantum yields (PLQY, e.g. ~2-5% for intrinsic CsEuCl3) and poor stability against air. Here, we introduce a one-step-procedure for synthesizing Eu$^{2+}$-doped CsSrCl$_3$ large nanocrystal clusters (LNCs) with the effect of self-reduction, therefore eliminating the use of conventional reductant oleylamine (OAm) and ensuring phase purity. The CsSrCl$_3$:Eu LNCs shows photoluminescence emission centered at 430 nm with a full width at half-maximum (FWHM) of 25 nm and a PLQY of ~40%, which can be further enhanced to ~97% after passivating the surface defects by adding trioctylphosphine (TOP), the highest among all reported lead-free blue-emitting perovskite nanocrystals. The stability of CsSrCl$_3$:Eu can also be improved significantly by epitaxially growing ZnS shell on the surface. This work will shed more light on lanthanide and alkaline-earth metal (AEM)-based perovskites for nontoxic light-emitting materials.

cond-mat.mtrl-sci↗

arXiv2025

Generalized circle patterns on surfaces with cusps

Guo and Luo introduced generalized circle patterns on surfaces and proved their rigidity. In this paper, we prove the existence of Guo-Luo's generalized circle patterns with prescribed generalized intersection angles on surfaces with cusps, which partially answers a question raised by Guo-Luo and generalizes Bobenko-Springborn's hyperbolic circle patterns on closed surfaces to generalized hyperbolic circle patterns on surfaces with cusps. We further introduce the combinatorial Ricci flow and combinatorial Calabi flow for generalized circle patterns on surfaces with cusps, and prove the longtime existence and convergence of the solutions for these combinatorial curvature flows.

math.GT↗

arXiv2025

Rigidity of infinite inversive distance circle packings in the plane

In 2004, Bowers-Stephenson [2] introduced the inversive distance circle packings as a natural generalization of Thurston's circle packings. They further conjectured the rigidity of infinite inversive distance circle packings in the plane. Motivated by the recent work of Luo-Sun-Wu [22] on Luo's vertex scaling, we prove Bower-Stephenson's conjecture for inversive distance circle packings in the hexagonal triangulated plane. This generalizes Rodin-Sullivan's famous result [13] on the rigidity of infinite tangential circle packings in the hexagonal triangulated plane. The key tools include a maximal principle for generic weighted Delaunay inversive distance circle packings and a ring lemma for the inversive distance circle packings in the hexagonal triangulated plane.

math.GT↗

arXiv2024

Intermittency of bubble deformation in turbulence

The deformation of finite-sized bubbles in intense turbulence exhibits complex geometries beyond simple spheroids as the bubbles exchange energy with the surrounding eddies across a wide range of scales. This study investigates deformation via the velocity of the most stretched tip of the deformed bubble in 3D, as the tip extension results from the compression of the rest of the interface by surrounding eddies. The results show that the power spectrum based on the tip velocity exhibits a scaling akin to that of the Lagrangian statistics of fluid elements, but decays with a distinct timescale and magnitude modulated by the Weber number based on the bubble size. This indicates that the interfacial energy is primarily siphoned from eddies of similar sizes as the bubble. Moreover, the tip velocity appears much more intermittent than the velocity increment, and its distribution near the extreme tails can be explained by the proposed model that accounts for the fact that small eddies with sufficient energy can contribute to extreme deformation. These findings provide a framework for understanding the energy transfer between deformable objects and multiscale eddies in intense turbulence.

physics.flu-dyn↗

arXiv2024

Breaking Language Barriers: Cross-Lingual Continual Pre-Training at Scale

In recent years, Large Language Models (LLMs) have made significant strides towards Artificial General Intelligence. However, training these models from scratch requires substantial computational resources and vast amounts of text data. In this paper, we explore an alternative approach to constructing an LLM for a new language by continually pretraining (CPT) from existing pretrained LLMs, instead of using randomly initialized parameters. Based on parallel experiments on 40 model sizes ranging from 40M to 5B parameters, we find that 1) CPT converges faster and saves significant resources in a scalable manner; 2) CPT adheres to an extended scaling law derived from Hoffmann et al. (2022) with a joint data-parameter scaling term; 3) The compute-optimal data-parameter allocation for CPT markedly differs based on our estimated scaling factors; 4) The effectiveness of transfer at scale is influenced by training duration and linguistic properties, while robust to data replaying, a method that effectively mitigates catastrophic forgetting in CPT. We hope our findings provide deeper insights into the transferability of LLMs at scale for the research community.

cs.CL↗

arXiv2024

Discrete conformal structures on surfaces with boundary (I) -- Classification

In this paper, we introduce the discrete conformal structures on surfaces with boundary in an axiomatic approach, which ensures that the Poincaré dual of an ideally triangulated surface with boundary has a good geometric structure.Then we classify the discrete conformal structures on surfaces with boundary, which turns out to unify and generalize Guo-Luo's generalized circle packings \cite{GL2}, Guo's vertex scalings \cite{Guo} and Xu's partial discrete conformal structures \cite{Xu22} on surfaces with boundary.This generalizes the results of Glickenstein-Thomas \cite{GT} on closed surfaces to surfaces with boundary. Motivated by \cite{BPS, Zhang-Guo-Zeng-Luo-Yau-Gu}, we further study the relationships between the discrete conformal structures on surfaces with boundary and the hyperbolic trigonometry. Unexpectedly, we find that some subclasses of the discrete conformal structures on surfaces with boundary are closely related to the twisted generalized hyperbolic triangles introduced by Roger-Yang \cite{Roger-Yang}, which does not appear in the case of closed surfaces. Finally, we study the relationships between the discrete conformal structures on surfaces with boundary and the 3-dimensional hyperbolic geometry by constructing ten types of generalized hyperbolic tetrahedra. Some new generalized hyperbolic tetrahedra recently introduced by Belletti-Yang \cite{B-Y} naturally appears in the constructions.

math.DG↗

arXiv2024

On the classification of discrete conformal structures on surfaces

Glickenstein \cite{Glickenstein} and Glickenstein-Thomas \cite{GT} introduced the discrete conformal structures on surfaces in an axiomatic approach and studied its classification. In this paper, we give a full classification of the discrete conformal structures on surfaces, which completes Glickenstein-Thomas' classification. As a result, we find some new classes of discrete conformal structures on surfaces, including some of the generalized circle packing metrics introduced by Guo-Luo \cite{GL2}. The relationships between the discrete conformal structures on surfaces and the 3-dimensional hyperbolic geometry are also discussed.

math.DG↗

arXiv2024

Breaking bubbles across multiple timescales in turbulence

The familiar process of bubbles generated via breaking waves in the ocean is foundational to many natural and industrial applications. In this process, large pockets of entrained gas are successively fragmented by the ambient turbulence into smaller and smaller bubbles. The key question is how long it takes for the bubbles to reach terminal sizes for a given system. Despite decades of effort, the reported breakup time from multiple experiments differs significantly. Here, to reconcile those results, rather than focusing on one scale, we measure multiple timescales associated with the process through a unique experiment that resolves bubbles' local deformation and curvature. The results emphasize that the scale separation among various timescales is controlled by the Weber number, similar to how the Reynolds number determines the scale separation in single-phase turbulence, but shows a distinct transition at a critical Weber number.

physics.flu-dyn↗

arXiv2024

A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces. A discrete uniformization theorem is established for this discrete Gaussian curvature. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete Gaussian curvature, which is called the combinatorial $α$-curvature. To find decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures, we introduce the combinatorial $α$-Ricci flow for decorated piecewise hyperbolic metrics. To handle the potential singularities along the combinatorial $α$-Ricci flow, we do surgery along the flow by edge flipping under the weighted Delaunay condition. Then we prove the longtime existence and convergence of the combinatorial $α$-Ricci flow with surgery. As an application of the combinatorial $α$-Ricci flow with surgery, we give the existence of decorated piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. We further introduce the combinatorial $α$-Calabi flow with surgery and study its longtime behavior.

math.DG↗

arXiv2023

Deformation of discrete conformal structures on surfaces

Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. It includes Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. In this paper, we study the deformation of Glickenstein's discrete conformal structures by combinatorial curvature flows. The combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scalings. We prove that the solution of the combinatorial Ricci flow for Glickenstein's discrete conformal structures on triangulated surfaces can be uniquely extended. Furthermore, under some necessary conditions, we prove that the solution of the extended combinatorial Ricci flow on a triangulated surface exists for all time and converges exponentially fast for any initial value. We further introduce the combinatorial Calabi flow for Glickenstein's discrete conformal structures on triangulated surfaces and study the basic properties of the flow. These combinatorial curvature flows provide effective algorithms for finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.

math.DG↗

arXiv2023

A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces, II

In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of a geodesic disk at a vertex of a polyhedral surface. It is proved that each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a decorated piecewise Euclidean metric with this discrete curvature constant. We further investigate the prescribing combinatorial curvature problem for a parametrization of this discrete curvature and prove some Kazdan-Warner type results. The main tools are Bobenko-Lutz's discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints.

math.DG↗

arXiv2023

A discrete uniformization theorem for decorated piecewise Euclidean metrics on surfaces

In this paper, we introduce a new discretization of the Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of some dual cell of a weighted triangulation at the conic singularity. A discrete uniformization theorem for this discrete Gaussian curvature is established on surfaces with non-positive Euler number. The main tools are Bobenko-Lutz's discrete conformal theory for decorated piecewise Euclidean metrics on surfaces and variational principles with constraints.

math.DG↗

arXiv2023

Rigidity and deformation of generalized sphere packings on 3-dimensional manifolds with boundary

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized sphere packing metrics. We prove that the generalized sphere packing metric is determined by the combinatorial scalar curvature. To find the hyper-ideal polyhedral metrics on 3-dimensional manifolds with prescribed combinatorial scalar curvature, we introduce the combinatorial Ricci flow and combinatorial Calabi flow for the generalized sphere packings on 3-dimensional manifolds with boundary. Then we study the longtime existence and convergence for the solutions of these combinatorial curvature flows.

math.DG↗

arXiv2023

Rigidity of generalized Thurston's sphere packings on 3-dimensional manifolds with boundary

Motivated by Guo-Luo's generalized circle packings on surfaces with boundary \cite{GL2}, we introduce the generalized Thurston's sphere packings on 3-dimensional manifolds with boundary. Then we investigate the rigidity of the generalized Thurston's sphere packings. We prove that the generalized Thurston's sphere packings are locally determined by the combinatorial scalar curvatures. We further prove the infinitesimal rigidity that the generalized Thurston's sphere packings can not be deformed while keeping the combinatorial Ricci curvatures fixed.

math.GT↗

arXiv2023

Rigidity of discrete conformal structures on surfaces

In \cite{G3}, Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. Glickenstein's discrete conformal structures include Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. Glickenstein \cite{G5} further conjectured the rigidity of the discrete conformal structures on polyhedral surfaces. Glickenstein's conjecture includes Luo's conjecture on the rigidity of vertex scalings \cite{L1} and Bowers-Stephenson's conjecture on the rigidity of inversive distance circle packings \cite{BSt} as special cases. In this paper, we prove Glickenstein's conjecture using variational principles. This unifies and generalizes the well-known results of Luo \cite{L4} and Bobenko-Pinkall-Springborn \cite{BPS}. Our method provides a unified approach to similar problems. We further discuss the relationships of Glickenstein's discrete conformal structures on polyhedral surfaces and $3$-dimensional hyperbolic geometry. As a result, we obtain some new results on the convexities of the co-volume functions of some generalized $3$-dimensional hyperbolic tetrahedra.

math.DG↗

arXiv2023

Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces

Inversive distance circle packings introduced by Bowers-Stephenson are natural generalizations of Thurston's circle packings on surfaces. To find piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial $p$-th Calabi flow for the Euclidean inversive distance circle packings. Due to the singularities possibly developed by these combinatorial curvature flows, the longtime existence and convergence of these combinatorial curvature flows have been a difficult problem for a long time. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Using the discrete conformal theory recently established by Bobenko-Lutz for decorated piecewise Euclidean metrics on surfaces, we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. This provides effective algorithms for finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.

math.DG↗

arXiv2023

New Perspectives on Sensitivity and Identifiability Analysis using the Unscented Kalman Filter

Detailed dynamical systems' models used in the life sciences may include hundreds of state variables and many input parameters, often with physical meaning. Therefore, efficient and unique input parameter identification, from experimental data, is an essential but challenging task for this class of model. To clarify our understating of the process (which within a clinical context amounts to a personalisation), we utilise the computational methods of Unscented Kalman filtration (UKF), sensitivity and orthogonality analysis. We have applied these three techniques to a test-bench model of a single ventricle, coupled, via Ohmic valves, to a Compliance-Resistor-Compliance (CRC) Windkessel electrical analogue model of the systemic circulation, chosen in view of its relative simplicity, interpretability and prior art. Utilising an efficient, novel and real-time implementation of the UKF (Code available at https://github.com/H-Sax/CMSB-2023), we show how, counter-intuitively, input parameters are efficiently recovered from experimental data \emph{even if they are not sensitive parameters in the currently accepted sense}. This result (i) exposes potential limitations in the standard interpretation of what it means for an input parameter to be designated identifiable and (ii) suggests that the concepts of sensitivity and identifiability may have a weaker relationship than commonly thought - at least in the presence of an appropriate data set. We rationalise these observations. Practically, we present results which show the UKF to be an efficient method for assigning personalised input parameters from experimental data in real-time, which enhances the clinical significance of our approach.

q-bio.QM↗