Gravitational instantons with faster than quadratic curvature decay (I)

In this paper, we study gravitational instantons (i.e., complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay). We prove three main theorems: 1.Any gravitational instanton must have known end----ALE, ALF, ALG or ALH. 2.In ALG and ALH-non-splitting cases, it must be biholomorphic to a compact complex elliptic surface minus a divisor. Thus, we confirm a long-standing question of Yau in ALG and ALH cases. 3.In ALF-D_k case, it must have an O(4)-multiplet

Gravitational instantons with faster than quadratic curvature decay (II)

This is our second paper in a series to study gravitational instantons, i.e. complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instanton must be the Cherkis-Hitchin-Ivanov-Kapustin-Lindstr\"om-Ro\v{c}ek metric.Comment: We add a corollary and the applications, correct the asymptotic rate of the multi-Taub-NUT metri

On J-equation

In this paper, we prove that for any K\"ahler metrics $\omega_0$ and $\chi$ on $M$, there exists $\omega_\varphi=\omega_0+\sqrt{-1}\partial\bar\partial\varphi>0$ satisfying the J-equation $\mathrm{tr}_{\omega_\varphi}\chi=c$ if and only if $(M,[\omega_0],[\chi])$ is uniformly J-stable. As a corollary, we can find many constant scalar curvature K\"ahler metrics with $c_1<0$. Using the same method, we also prove a similar result for the deformed Hermitian-Yang-Mills equation when the angle is in $(\frac{n\pi}{2}-\frac{\pi}{4},\frac{n\pi}{2})$

G2_2 manifolds with nodal singularities along circles

The goal of this paper is the construction of a compact manifold with G$_2$ holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by comparison to the untwisted connected sum case, it turns out that the obstruction space for the singular twisted connected sum construction is infinite dimensional. By analyzing the obstruction term, there are strong evidences that the obstruction may be resolved if a further gluing is performed in order to get a compact manifold with G$_2$ holonomy and isolated conical singularities with link $\mathbb{S}^3\times\mathbb{S}^3$.Comment: There is a simplification about homogenous harmonic forms on the nodal cone in this versio

Shi-type estimates and finite time singularities of flows of G2_2 structures

In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G$_2$ structures including the modified Laplacian co-flow. Then we prove a version of $\kappa$-non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G$_2$ structures.Comment: The previous version is too short. So it is merged into a new pape

Rate of asymptotic convergence near isolated singularity of G2_2 manifold

In this paper, a metric with G$_2$ holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold

On AP3AP_3 - covering sequences

Recently, motivated by Stanley sequences, Kiss, S\' andor and Yang introduced a new type sequence: a sequence $A$ of nonnegative integers is called an $AP_k$ - covering sequence if there exists an integer $n_0$ such that if $n > n_0$, then there exist $a_1\in A, \dots , a_{k-1}\in A$, $a_1<a_2<\cdots <a_{k-1}<n$ such that $a_1, \dots , a_{k-1}, n$ form a $k$-term arithmetic progression. They prove that there exists an $AP_3$ - covering sequence $A$ such that $\limsup\limits_{n\to\infty}{A(n)}/{\sqrt n}\le 34$. In this note, we prove that there exists an $AP_3$ - covering sequence $A$ such that $\limsup\limits_{n\to\infty}{A(n)}/{\sqrt n}=\sqrt{15}$.Comment: 5 page

Representations of bicircular lift matroids

Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid are the edge sets of those subgraphs of $G$ that contain at least two cycles, and are minimal with respect to this property. The main result of this paper is a characterization of when two graphs give rise to the same bicircular lift matoid, which answers a question proposed by Irene Pivotto. In particular, aside from some appropriately defined "small" graphs, two graphs have the same bicircular lift matroid if and only if they are $2$-isomorphic in the sense of Whitney

Asymptotic performance of regularized multi-task learning

This paper analyzes asymptotic performance of a regularized multi-task learning model where task parameters are optimized jointly. If tasks are closely related, empirical work suggests multi-task learning models to outperform single-task ones in finite sample cases. As data size grows indefinitely, we show the learned multi-classifier to optimize an average misclassification error function which depicts the risk of applying multi-task learning algorithm to making decisions. This technique conclusion demonstrates the regularized multi-task learning model to be able to produce reliable decision rule for each task in the sense that it will asymptotically converge to the corresponding Bayes rule. Also, we find the interaction effect between tasks vanishes as data size growing indefinitely, which is quite different from the behavior in finite sample cases

DenseImage Network: Video Spatial-Temporal Evolution Encoding and Understanding

Many of the leading approaches for video understanding are data-hungry and time-consuming, failing to capture the gist of spatial-temporal evolution in an efficient manner. The latest research shows that CNN network can reason about static relation of entities in images. To further exploit its capacity in dynamic evolution reasoning, we introduce a novel network module called DenseImage Network(DIN) with two main contributions. 1) A novel compact representation of video which distills its significant spatial-temporal evolution into a matrix called DenseImage, primed for efficient video encoding. 2) A simple yet powerful learning strategy based on DenseImage and a temporal-order-preserving CNN network is proposed for video understanding, which contains a local temporal correlation constraint capturing temporal evolution at multiple time scales with different filter widths. Extensive experiments on two recent challenging benchmarks demonstrate that our DenseImage Network can accurately capture the common spatial-temporal evolution between similar actions, even with enormous visual variations or different time scales. Moreover, we obtain the state-of-the-art results in action and gesture recognition with much less time-and-memory cost, indicating its immense potential in video representing and understanding.Comment: 7 page