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Xu Xu

Publications and source records attributed to Xu Xu.

At least 55 records · Page 3Linked to original sources

Thurston's sphere packings on 3-dimensional manifolds, I

Thurston's sphere packing on a 3-dimensional manifold is a generalization of Thusrton's circle packing on a surface, the rigidity of which has been open for many years. In this paper, we prove that Thurston's Euclidean sphere packing is locally determined by combinatorial scalar curvature up to scaling, which generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity that Thurston's Euclidean sphere packing can not be deformed (except by scaling) while keeping the combinatorial Ricci curvature fixed.

math.GT↗

Absolute continuity of the integrated density of states in the localized regime

We establish the absolute continuity of the integrated density of states (IDS) for quasi-periodic Schrödinger operators with a large trigonometric potential and Diophantine frequency. This partially solves Eliasson's open problem in 2002. Furthermore, this result can be extended to a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$. Our proof is based on stratified quantitative almost reducibility results of dual cocycles. Specifically, we prove that a generic analytic one-parameter family of cocycles, sufficiently close to constant coefficients, is reducible except for a zero Hausdorff dimension set of parameters. This result affirms Eliasson's conjecture in 2017.

math.SP↗

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I

In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterized discrete Yamabe flow for piecewise linear metrics on surfaces. To handle the possible singularities along the flow, we do surgery on the flow by flipping. Then we prove that the flow with surgery converges to a piecewise linear metric with constant discrete $α$-curvature, which confirms another generalized Luo conjecture on convergence of discrete Yamabe flow with surgery. We also introduce a parameterized discrete Calabi flow and prove the convergence of the flow with surgery, which generalizes the convergence proved by Zhu and the author.

math.GT↗

Combinatorial curvature flows for generalized circle packings on surfaces with boundary

In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$ type generalized circle packing metric introduced by Guo-Luo \cite{GL2}. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.

math.DG↗

Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

Discrete conformal structure on polyhedral surfaces is a discrete analogue of the smooth conformal structure on surfaces that assigns discrete metrics by scalar functions defined on vertices. In this paper, we introduce combinatorial $α$-curvature for discrete conformal structures on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. Then we prove the local and global rigidity of combinatorial $α$-curvature with respect to discrete conformal structures on polyhedral surfaces, which confirms parameterized Glickenstein rigidity conjecture. To study the Yamabe problem for combinatorial $α$-curvature, we introduce combinatorial $α$-Ricci flow for discrete conformal structures on polyhedral surfaces, which is a generalization of Chow-Luo's combinatorial Ricci flow for Thurston's circle packings and Luo's combinatorial Yamabe flow for vertex scaling on polyhedral surfaces. To handle the potential singularities of the combinatorial $α$-Ricci flow, we extend the flow through the singularities by extending the inner angles in triangles by constants. Under the existence of a discrete conformal structure with prescribed combinatorial curvature, the solution of extended combinatorial $α$-Ricci flow is proved to exist for all time and converge exponentially fast for any initial value. This confirms a parameterized generalization of another conjecture of Glickenstein on the convergence of combinatorial Ricci flow, gives an almost equivalent characterization of the solvability of Yamabe problem for combinatorial $α$-curvature in terms of combinatorial $α$-Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial $α$-curvatures.

math.GT↗

Combinatorial Calabi flows on surfaces with boundary

Motivated by Luo's combinatorial Yamabe flow on closed surfaces \cite{L1} and Guo's combinatorial Yamabe flow on surfaces with boundary \cite{Guo}, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.

math.DG↗

A new class of discrete conformal structures on surfaces with boundary

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using variational principles, which is a complement of Guo-Luo's rigidity of the discrete conformal structures and Guo's rigidity of vertex scaling on surface with boundary. As a result, new convexities of the volume of generalized hyperbolic pyramids with right-angled hyperbolic hexagonal bases are obtained. Motivated by Chow-Luo's combinatorial Ricci flow and Luo's combinatorial Yamabe flow on closed surfaces, we further introduce combinatorial Ricci flow and combinatorial Calabi flows to deform the new discrete conformal structures on surfaces with boundary. The basic properties of these combinatorial curvature flows are established. These combinatorial curvature flows provide effective algorithms for constructing hyperbolic metrics on surfaces with totally geodesic boundary components of prescribed lengths.

math.GT↗

A new proof for global rigidity of vertex scaling on polyhedral surfaces

The vertex scaling for piecewise linear metrics on polyhedral surfaces was introduced by Luo, who proved the local rigidity by establishing a variational principle and conjectured the global rigidity. Luo's conjecture was solved by Bobenko-Pinkall-Springborn, who also introduced the vertex scaling for piecewise hyperbolic metrics and proved its global rigidity. Bobenko-Pinkall-Spingborn's proof is based on their observation of the connection of vertex scaling and the geometry of polyhedra in $3$-dimensional hyperbolic space and the concavity of the volume of ideal and hyper-ideal tetrahedra. In this paper, we give an elementary and short variational proof of the global rigidity of vertex scaling without involving $3$-dimensional hyperbolic geometry. The method is based on continuity of eigenvalues of matrices and the extension of convex functions.

math.GT↗

Combinatorial Yamabe flow on hyperbolic bordered surfaces

This paper studies the combinatorial Yamabe flow on hyperbolic bordered surfaces. We show that the flow exists for all time and converges exponentially fast to conformal factor which produces a hyperbolic surface whose lengths of boundary components are equal to prescribed positive numbers. This provides an algorithm to such problems.

math.DG↗

Prescribing discrete Gaussian curvature on polyhedral surfaces

Vertex scaling of piecewise linear metrics on surfaces introduced by Luo is a straightforward discretization of smooth conformal structures on surfaces. Combinatorial $α$-curvature for vertex scaling of piecewise linear metrics on surfaces is a discretization of Gaussian curvature on surfaces. In this paper, we investigate the prescribing combinatorial $α$-curvature problem on polyhedral surfaces. Using Gu-Luo-Sun-Wu's discrete conformal theory for piecewise linear metrics on surfaces and variational principles with constraints, we prove some Kazdan-Warner type theorems for prescribing combinatorial $α$-curvature problem, which generalize the results obtained by Gu-Luo-Sun-Wu on prescribing combinatorial curvatures on surfaces. Gu-Luo-Sun-Wu conjectured that one can prove Kazdan-Warner's theorems via approximating smooth surfaces by polyhedral surfaces. This paper takes the first step in this direction.

math.DG↗

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

This paper investigates the combinatorial $α$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $α$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu, we prove the longtime existence and convergence of combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures.

math.GT↗

Fractional combinatorial Calabi flow on surfaces

Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established by Gu et al., we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.

math.GT↗

A Systematic Collection of Medical Image Datasets for Deep Learning

The astounding success made by artificial intelligence (AI) in healthcare and other fields proves that AI can achieve human-like performance. However, success always comes with challenges. Deep learning algorithms are data-dependent and require large datasets for training. The lack of data in the medical imaging field creates a bottleneck for the application of deep learning to medical image analysis. Medical image acquisition, annotation, and analysis are costly, and their usage is constrained by ethical restrictions. They also require many resources, such as human expertise and funding. That makes it difficult for non-medical researchers to have access to useful and large medical data. Thus, as comprehensive as possible, this paper provides a collection of medical image datasets with their associated challenges for deep learning research. We have collected information of around three hundred datasets and challenges mainly reported between 2013 and 2020 and categorized them into four categories: head & neck, chest & abdomen, pathology & blood, and ``others''. Our paper has three purposes: 1) to provide a most up to date and complete list that can be used as a universal reference to easily find the datasets for clinical image analysis, 2) to guide researchers on the methodology to test and evaluate their methods' performance and robustness on relevant datasets, 3) to provide a ``route'' to relevant algorithms for the relevant medical topics, and challenge leaderboards.

eess.IV↗

Influence of microstructural and crystallographic inhomogeneity on tensile anisotropy in thick-section Al-Li-Cu-Mg plates

Thick-section plates made from a recently developed Al-Cu-Mg-Li alloy have been evaluated to understand the influence of microstructure on the anisotropy of tensile strengths after natural and artificial ageing treatment. Pancake-shaped grains with a coarse substructure and strong crystallographic texture with a beta-fibre orientation at the mid-thickness position are observed. In addition, an inhomogeneous distribution of T1 precipitates through the plate thickness has been revealed with the volume fraction of intragranular precipitates ~ 40% higher at the plate centre than the thickness position. Altogether these microstructural features contribute to the in-plane anisotropy of tensile strengths that is ~ 5% higher at the mid-thickness position than the 1/4 thickness position. The variation of ageing-induced T1 precipitates through the plate thickness further contributes to the through-thickness anisotropy that is ~ 3% higher in T8 temper as compared to T3 temper.

cond-mat.mtrl-sci↗

Damage accumulation during high temperature fatigue of Ti/SiC$_f$ metal matrix composites under different stress amplitudes

The damage mechanisms and load redistribution of high strength TC17 titanium alloy/unidirectional SiC fibre composite (fibre diameter = 100 $μ$m) under high temperature (350 °C) fatigue cycling have been investigated in situ using synchrotron X-ray computed tomography (CT) and X-ray diffraction (XRD) for high cycle fatigue (HCF) under different stress amplitudes. The three-dimensional morphology of the crack and fibre fractures has been mapped by CT. During stable growth, matrix cracking dominates with the crack deflecting (by 50-100 $μ$m in height) when bypassing bridging fibres. A small number of bridging fibres have fractured close to the matrix crack plane especially under relatively high stress amplitude cycling. Loading to the peak stress led to rapid crack growth accompanied by a burst of fibre fractures. Many of the fibre fractures occurred 50-300 $μ$m from the matrix crack plane during rapid growth, in contrast to that in the stable growth stage, leading to extensive fibre pull-out on the fracture surface. The changes in fibre loading, interfacial stress, and the extent of fibre-matrix debonding in the vicinity of the crack have been mapped for the fatigue cycle and after the rapid growth by high spatial resolution XRD. The fibre/matrix interfacial sliding extends up to 600 $μ$m (in the stable growth zone) or 700 $μ$m (in the rapid growth zone) either side of the crack plane. The direction of interfacial shear stress reverses with the loading cycle, with the maximum frictional sliding stress reaching ~55 MPa in both the stable growth and rapid growth regimes.

cond-mat.mtrl-sci↗

Application of high-spatial-resolution secondary ion mass spectrometry for nanoscale chemical mapping of lithium in an Al-Li alloy

High-spatial-resolution secondary ion mass spectrometry offers a method for mapping lithium at nanoscale lateral resolution. Practical implementation of this technique offers significant potential for revealing the distribution of Li in many materials with exceptional lateral resolution and elemental sensitivity. Here, two state-of-the-art methods are demonstrated on an aluminium-lithium alloy to visualise nanoscale Li-rich phases by mapping the 7Li+ secondary ion. NanoSIMS 50L analysis with a radio frequency O- plasma ion source enabled visualisation of needle-shaped T1 (Al2CuLi) phases as small as 75 nm in width. A compact time-of-flight secondary ion mass spectrometry detector added to a focused ion beam scanning electron microscope facilitated mapping of the T1 phases down to 45 nm in width using a Ga+ ion beam. Correlation with high resolution electron microscopy confirms the identification of T1 precipitates, their sizes and distribution observed during SIMS mapping.

cond-mat.mtrl-sci↗

Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles

We prove that if the frequency of the quasi-periodic $\mathrm{SL}(2,\R)$ cocycle is Diophantine, then the following properties are dense in the subcritical regime: for any $\frac{1}{2}<κ<1$, the Lyapunov exponent is exactly $κ$-Hölder continuous; the extended eigenstates of the potential have optimal sub-linear growth; and the dual operator associated a subcritical potential has power-law decay eigenfunctions. The proof is based on fibered Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,\R)$ cocycles.

math.DS↗

Combinatorial Ricci flow on cusped 3-manifolds

Combinatorial Ricci flow on a cusped $3$-manifold is an analogue of Chow-Luo's combinatorial Ricci flow on surfaces and Luo's combinatorial Ricci flow on compact $3$-manifolds with boundary for finding complete hyperbolic metrics on cusped $3$-manifolds. Dual to Casson and Rivin's program of maximizing the volume of angle structures, combinatorial Ricci flow finds the complete hyperbolic metric on a cusped $3$-manifold by minimizing the co-volume of decorated hyperbolic polyhedral metrics. The combinatorial Ricci flow may develop singularities. We overcome this difficulty by extending the flow through the potential singularities using Luo-Yang's extension. It is shown that the existence of a complete hyperbolic metric on a cusped $3$-manifold is equivalent to the convergence of the extended combinatorial Ricci flow, which gives a new characterization of existence of a complete hyperbolic metric on a cusped $3$-manifold dual to Casson and Rivin's program. The extended combinatorial Ricci flow also provides an effective algorithm for finding complete hyperbolic metrics on cusped $3$-manifolds.

math.GT↗