Well-posedness and global existence of 2D viscous shallow water system in Besov spaces

In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by using the Littlewood-Paley theory, the Bony decomposition and the theories of transport equations and transport diffusion equations. Then we can prove the global existence of the system with small enough initial data in $B^s_{p,r}(\mathbb{R}^2)$, $1\leq p\leq2$, $1\leq r<\infty$ and $s>\frac{2}{p}$. Our obtained results generalize and cover the recent results in \cite{W}

Remarks on the Chern classes of Calabi-Yau moduli

We prove that the first Chern form of the moduli space of polarized Calabi-Yau manifolds, with the Hodge metric or the Weil-Petersson metric, represent the first Chern class of the canonical extensions of the tangent bundle to the compactification of the moduli space with normal crossing divisors.Comment: 14 pages, no figure

DeepSSM: Deep State-Space Model for 3D Human Motion Prediction

Predicting future human motion plays a significant role in human-machine interactions for a variety of real-life applications. In this paper, we build a deep state-space model, DeepSSM, to predict future human motion. Specifically, we formulate the human motion system as the state-space model of a dynamic system and model the motion system by the state-space theory, offering a unified formulation for diverse human motion systems. Moreover, a novel deep network is designed to build this system, enabling us to utilize both the advantages of deep network and state-space model. The deep network jointly models the process of both the state-state transition and the state-observation transition of the human motion system, and multiple future poses can be generated via the state-observation transition of the model recursively. To improve the modeling ability of the system, a unique loss function, ATPL (Attention Temporal Prediction Loss), is introduced to optimize the model, encouraging the system to achieve more accurate predictions by paying increasing attention to the early time-steps. The experiments on two benchmark datasets (i.e., Human3.6M and 3DPW) confirm that our method achieves state-of-the-art performance with improved effectiveness. The code will be available if the paper is accepted

Global existence for the two-component Camassa-Holm system and the modified two-component Camassa-Holm system

The present work is mainly concerned with global existence for the two-component Camassa-Holm system and the modified two-component Camassa-Holm system. By discovering new conservative quantities of these systems, we prove several new global existence results for these two-component shallow water systems.Comment: This paper has been withdrawn by the author due to a crucial erro

On the Cauchy problem of a two-component b-family equation

In this paper, we study the Cauchy problem of a two-component b-family equation. We first establish the local well-posedness for a two-component b-family equation by Kato's semigroup theory. Then, we derive precise blow-up scenarios for strong solutions to the equation. Moreover, we present several blow-up results for strong solutions to the equation

An Envelope for Davis-Yin Splitting and Strict Saddle Point Avoidance

It is known that operator splitting methods based on Forward Backward Splitting (FBS), Douglas-Rachford Splitting (DRS), and Davis-Yin Splitting (DYS) decompose a difficult optimization problems into simpler subproblems under proper convexity and smoothness assumptions. In this paper, we identify an envelope (an objective function) whose gradient descent iteration under a variable metric coincides with DYS iteration. This result generalizes the Moreau envelope for proximal-point iteration and the envelopes for FBS and DRS iterations identified by Patrinos, Stella, and Themelis. Based on the new envelope and the Stable-Center Manifold Theorem, we further show that, when FBS or DRS iterations start from random points, they avoid all strict saddle points with probability one. This results extends the similar results by Lee et al. from gradient descent to splitting methods

Global weak solutions to a weakly dissipative μ\muHS equation

This paper is concerned with global existence of weak solutions for a weakly dissipative $\mu$HS equation by using smooth approximate to initial data and Helly$^{,}$s theorem

On the Cauchy problem of a weakly dissipative μ\muHS equation

In this paper, we study the Cauchy problem of a weakly dissipative $\mu$HS equation. We first establish the local well-posedness for the weakly dissipative $\mu$HS equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation

Quantum Correction and the Moduli Spaces of Calabi-Yau Manifolds

We define the quantum correction of the Teichm\"uller space $\mathcal{T}$ of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichm\"uller space $\mathcal{T}$ is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image $\Phi(\mathcal{T})$ of the Teichm\"uller space $\mathcal{T}$ under the period map $\Phi$ is an open submanifold of a globally Hermitian symmetric space $W$ of the same dimension as $\mathcal{T}$. Finally, for Hyperk\"ahler manifold of dimension $2n \geq 4$, we find both locally and globally defined families of $(2,0)$ and $(2n,0)$-classes over the Teichm\"uller space of polarized Hyperk\"ahler manifolds.Comment: 36 page

CPM-sensitive AUC for CTR prediction

The prediction of click-through rate (CTR) is crucial for industrial applications, such as online advertising. AUC is a commonly used evaluation indicator for CTR models. For advertising platforms, online performance is generally evaluated by CPM. However, in practice, AUC often improves in offline evaluation, but online CPM does not. As a result, a huge waste of precious online traffic and human costs has been caused. This is because there is a gap between offline AUC and online CPM. AUC can only reflect the order on CTR, but it does not reflect the order of CTR*Bid. Moreover, the bids of different advertisements are different, so the loss of income caused by different reverse-order pair is also different. For this reason, we propose the CPM-sensitive AUC (csAUC) to solve all these problems. We also give the csAUC calculation method based on dynamic programming. It can fully support the calculation of csAUC on large-scale data in real-world applications