On convergence to a football

We show that spheres of positive constant curvature with $n$ ($n\geq3$) conic points converge to a sphere of positive constant curvature with two conic points (or called an (American) football) in Gromov-Hausdorff topology when the corresponding singular divisors converge to a critical divisor in the sense of Troyanov. We prove this convergence in two different ways. Geometrically, the convergence follows from Luo-Tian's explicit description of conic spheres as boundaries of convex polytopes in $S^{3}$. Analytically, regarding the conformal factors as the singular solutions to the corresponding PDE, we derive the required a priori estimates and convergence result after proper reparametrization.Comment: 19 page

On the geometric flows solving K\"ahlerian inverse σk\sigma_k equations

In this note, we extend our previous work on the inverse $\sigma_k$ problem. Inverse $\sigma_{k}$ problem is a fully nonlinear geometric PDE on compact K\"ahler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.Comment: to appear in Pacific Journal of Mathematic

Volume bounds of conic 2-spheres

We obtain sharp volume bound for a conic 2-sphere in terms of its Gaussian curvature bound. We also give the geometric models realizing the extremal volume. In particular, when the curvature is bounded in absolute value by $1$, we compute the minimal volume of a conic sphere in the sense of Gromov. In order to apply the level set analysis and iso-perimetric inequality as in our previous works, we develop some new analytical tools to treat regions with vanishing curvature.Comment: 19 pages, 1 figur

Dependence in Propositional Logic: Formula-Formula Dependence and Formula Forgetting -- Application to Belief Update and Conservative Extension

Dependence is an important concept for many tasks in artificial intelligence. A task can be executed more efficiently by discarding something independent from the task. In this paper, we propose two novel notions of dependence in propositional logic: formula-formula dependence and formula forgetting. The first is a relation between formulas capturing whether a formula depends on another one, while the second is an operation that returns the strongest consequence independent of a formula. We also apply these two notions in two well-known issues: belief update and conservative extension. Firstly, we define a new update operator based on formula-formula dependence. Furthermore, we reduce conservative extension to formula forgetting.Comment: We find a mistake in this version and we need a period of time to fix i

Escobar-Yamabe compactifications for Poincare-Einstein manifolds and rigidity theorems

Let $(X^{n},g_+)$ $(n\geq 3)$ be a Poincar\'{e}-Einstein manifold which is $C^{3,\alpha}$ conformally compact with conformal infinity $(\partial X, [\hat{g}])$. On the conformal compactification $(\overline{X}, \bar g=\rho^2g_+)$ via some boundary defining function $\rho$, there are two types of Yamabe constants: $Y(\overline{X},\partial X,[\bar g])$ and $Q(\overline{X},\partial X,[\bar g])$. (See definitions (\ref{def.type1}) and (\ref{def.type2})). In \cite{GH}, Gursky and Han gave an inequality between $Y(\overline{X},\partial X,[\bar g])$ and $Y(\partial X,[\hat{g}])$. In this paper, we first show that the equality holds in Gursky-Han's theorem if and only if $(X^{n},g_+)$ is isometric to the standard hyperbolic space $(\mathbb{H}^{n}, g_{\mathbb{H}})$. Secondly, we derive an inequality between $Q(\overline{X},\partial X,[\bar g])$ and $Y(\partial X, [\hat g])$, and show that the equality holds if and only if $(X^{n},g_+)$ is isometric to $(\mathbb{H}^{n}, g_{\mathbb{H}})$. Based on this, we give a simple proof of the rigidity theorem for Poincar\'{e}-Einstein manifolds with conformal infinity being conformally equivalent to the standard sphere.Comment: 1

The Obata equation with Robin boundary condition

We study the Obata equation with Robin boundary condition $\frac{\partial f}{\partial \nu}+af=0$ on manifolds with boundary, where $a \in \mathbb{R}\setminus\{0\}$. Dirichlet and Neumann boundary conditions were previously studied by Reilly \cite{R}, Escobar \cite{Es} and Xia \cite{X}. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both $a>0$ and $a<0$. We also consider the Obata equation with non-vanishing Neumann condition $\frac{\partial f}{\partial \nu}=1$.Comment: 36 pages, 5 figure

On a class of fully nonlinear flow in K\"ahler geometry

In this paper, we study a class of fully nonlinear metric flow on K\"ahler manifolds, which includes the J-flow as a special case. We provide a sufficient and necessary condition for the long time convergence of the flow, generalizing the result of Song-Weinkove. As a consequence, under the given condition, we solved the corresponding Euler equation, which is fully nonlinear of Monge-Amp\`ere type. As an application, we also discuss a complex Monge-Amp\`ere type equation including terms of mixed degrees, which was first posed by Chen.Comment: Added the second Appendix, some minor mistakes corrected. To appear in Crelle's Journa

Content-Preserving Image Stitching with Regular Boundary Constraints

This paper proposes an approach to content-preserving stitching of images with regular boundary constraints, which aims to stitch multiple images to generate a panoramic image with regular boundary. Existing methods treat image stitching and rectangling as two separate steps, which may result in suboptimal results as the stitching process is not aware of the further warping needs for rectangling. We address these limitations by formulating image stitching with regular boundaries in a unified optimization. Starting from the initial stitching results produced by traditional warping-based optimization, we obtain the irregular boundary from the warped meshes by polygon Boolean operations which robustly handle arbitrary mesh compositions, and by analyzing the irregular boundary construct a piecewise rectangular boundary. Based on this, we further incorporate straight line preserving and regular boundary constraints into the image stitching framework, and conduct iterative optimization to obtain an optimal piecewise rectangular boundary, thus can make the panoramic boundary as close as possible to a rectangle, while reducing unwanted distortions. We further extend our method to panoramic videos and selfie photography, by integrating the temporal coherence and portrait preservation into the optimization. Experiments show that our method efficiently produces visually pleasing panoramas with regular boundaries and unnoticeable distortions.Comment: 12 figures, 13 page

An immersed boundary method for fluid--structure--acoustics interactions involving large deformations and complex geometries

This paper presents an immersed boundary (IB) method for fluid--structure--acoustics interactions involving large deformations and complex geometries. In this method, the fluid dynamics is solved by a finite difference method where the temporal, viscous and convective terms are respectively discretized by the third-order Runge-Kutta scheme, the fourth-order central difference scheme and a fifth-order W/TENO (Weighted/Targeted Essentially Non-oscillation) scheme. Without loss of generality, a nonlinear flexible plate is considered here, and is solved by a finite element method based on the absolute nodal coordinate formulation. The no-slip boundary condition at the fluid--structure interface is achieved by using a diffusion-interface penalty IB method. With the above proposed method, the aeroacoustics field generated by the moving boundaries and the associated flows are inherently solved. In order to validate and verify the current method, several benchmark cases are conducted: acoustic waves scattered from a stationary cylinder in a quiescent flow, sound generation by a stationary and a rotating cylinder in a uniform flow, sound generation by an insect in hovering flight, deformation of a red blood cell induced by acoustic waves and acoustic waves scattered by a stationary sphere. The comparison of the sound scattered by a cylinder shows that the present IB--WENO scheme, a simple approach, has an excellent performance which is even better than the implicit IB--lattice Boltzmann method. For the sound scattered by a sphere, the IB--TENO scheme has a lower dissipation compared with the IB--WENO scheme. Applications of this technique to model fluid-structure-acoustics interactions of flapping foils mimicking an insect wing section during forward flight and flapping foil energy harvester are also presented, considering the effects of foil shape and flexibility

Utility Optimal Thread Assignment and Resource Allocation in Distributed Systems

Achieving high performance in many distributed systems, such as a web hosting center or the cloud requires finding a good assignment of worker threads to servers and also effectively allocating each server's resources to its assigned threads. The assignment and allocation components of this problem have been studied extensively but largely separately in the literature. In this paper, we introduce the \emph{assign and allocate (AA)} problem, which seeks to simultaneously find an assignment and allocation that maximizes the total utility of the threads. Assigning and allocating the threads together can result in substantially better overall utility than performing the steps separately, as is traditionally done. We model each thread by a utility function giving its utility as a function of its assigned resources. We first prove that the AA problem is NP-hard. We then present a $2 (\sqrt{2}-1) > 0.828$ factor approximation algorithm for concave utility functions, which runs in $O(mn^2 + n (\log mC)^2)$ time for $n$ threads and $m$ servers with $C$ amount of resource each. We further present a faster algorithm with the same approximation ratio and lower time complexity of $O(n (\log mC)^2)$. We then extend our algorithms to solve AA problem with nonconcave utility functions and achieve an approximation ratio $\frac{1}{2}$. We conduct extensive experiments to test the performance of our algorithms on threads with both synthetic and realistic utility functions, and find that it achieves over 92\% of the optimal utility on average. We also compare our algorithm against several other assignment and allocation algorithms, and find that it achieves up to 9 times better total utility.Comment: 15 page