Positive curvature operator, projective manifold and rational connectedness

In his recent work \cite{Y1}, X. Yang proved a conjecture raised by Yau in 1982 (\cite{Yau82}), which states that any compact K\"{a}hler manifold with positive holomorphic sectional curvature must be projective. In this note, we prove that any compact Hermitian manifold $X$ with positive real bisectional curvature, its hodge number $h^{1,0}=h^{2,0}=h^{n-1,0}=h^{n,0}=0$. In particular, if in addition $X$ is K\"{a}hler, then $X$ is projective. Also, it is rationally connected manifold when $n=3$. This partially confirms the conjecture 1.11 \cite{Y1} which is proposed by X. Yang.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1708.06713, arXiv:1610.07165, arXiv:1802.08732 by other author

On the Dirichlet Problem for Backward Parabolic Stochastic Partial Differential Equations in General Smooth Domains

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the solutions to blow up near the boundary.Comment: 28 page

Strong Solution of Backward Stochastic Partial Differential Equations in C2C^2 Domains

This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the continuation method under fairly weak conditions on variable coefficients and $C^2$ domains. The problem is also considered in weighted Sobolev spaces which allow the derivatives of the solutions to blow up near the boundary. As applications, a comparison theorem is obtained and the semi-linear equation is discussed in the $C^2$ domain.Comment: 26 page

MacWilliams type identities on the Lee and Euclidean weights for linear codes over Zℓ\mathbb{Z}_{\ell}

Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}.$ Necessary and sufficient conditions for the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}$ are given. Some examples about such MacWilliams type identities are also presented

Wm,pW^{m,p}-Solution (p≥2p\geq2) of Linear Degenerate Backward Stochastic Partial Differential Equations in the Whole Space

In this paper, we consider the backward Cauchy problem of linear degenerate stochastic partial differential equations. We obtain the existence and uniqueness results in Sobolev space $L^p(\Omega; C([0,T];W^{m,p}))$ with both $m\geq 1$ and $p\geq 2$ being arbitrary, without imposing the symmetry condition for the coefficient $\sigma$ of the gradient of the second unknown---which was introduced by Ma and Yong [Prob. Theor. Relat. Fields 113 (1999)] in the case of $p=2$. To illustrate the application, we give a maximum principle for optimal control of degenerate stochastic partial differential equations.Comment: 29 page

LpL^{p} Theory for Super-parabolic Backward Stochastic Partial Differential Equations in the Whole Space

This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An $L^p$-theory is given for the Cauchy problem of BSPDEs, separately for the case of $p\in (1,2]$ and for the case of $p\in (2, \infty)$. A comparison theorem is also addressed

On the power of dominated players in team competitions

We investigate multi-round team competitions between two teams, where each team selects one of its players simultaneously in each round and each player can play at most once. The competition defines an extensive-form game with perfect recall and can be solved efficiently by standard methods. We are interested in the properties of the subgame perfect equilibria of this game. We first show that uniformly random strategy is a subgame perfect equilibrium strategy for both teams when there are no redundant players (i.e., the number of players in each team equals to the number of rounds of the competition). Secondly, a team can safely abandon its weak players if it has redundant players and the strength of players is transitive. We then focus on the more interesting case where there are redundant players and the strength of players is not transitive. In this case, we obtain several counterintuitive results. First of all, a player might help improve the payoff of its team, even if it is dominated by the entire other team. We give a necessary condition for a dominated player to be useful. We also study the extent to which the dominated players can increase the payoff. These results bring insights into playing and designing general team competitions.Comment: 8pages, AAMAS201

Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.Comment: 18 page

Decision Making with Machine Learning and ROC Curves

The Receiver Operating Characteristic (ROC) curve is a representation of the statistical information discovered in binary classification problems and is a key concept in machine learning and data science. This paper studies the statistical properties of ROC curves and its implication on model selection. We analyze the implications of different models of incentive heterogeneity and information asymmetry on the relation between human decisions and the ROC curves. Our theoretical discussion is illustrated in the context of a large data set of pregnancy outcomes and doctor diagnosis from the Pre-Pregnancy Checkups of reproductive age couples in Henan Province provided by the Chinese Ministry of Health

Effects of Weak Ties on Epidemic Predictability in Community Networks

Weak ties play a significant role in the structures and the dynamics of community networks. Based on the susceptible-infected model in contact process, we study numerically how weak ties influence the predictability of epidemic dynamics. We first investigate the effects of different kinds of weak ties on the variabilities of both the arrival time and the prevalence of disease, and find that the bridgeness with small degree can enhance the predictability of epidemic spreading. Once weak ties are settled, compared with the variability of arrival time, the variability of prevalence displays a diametrically opposed changing trend with both the distance of the initial seed to the bridgeness and the degree of the initial seed. More specifically, the further distance and the larger degree of the initial seed can induce the better predictability of arrival time and the worse predictability of prevalence. Moreover, we discuss the effects of weak tie number on the epidemic variability. As community strength becomes very strong, which is caused by the decrease of weak tie number, the epidemic variability will change dramatically. Compared with the case of hub seed and random seed, the bridgenss seed can result in the worst predictability of arrival time and the best predictability of prevalence. These results show that the variability of arrival time always marks a complete reversal trend of that of prevalence, which implies it is impossible to predict epidemic spreading in the early stage of outbreaks accurately.Comment: 8 pages, 6 figure