We introduce the notion of symplectic flatness for connections and fiber bundles over symplectic manifolds. Given an $A_\infty$-algebra, we present a flatness condition that enables the twisting of the differential complex associated with the $A_\infty$-algebra. The symplectic flatness condition arises from twisting the $A_\infty$-algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang-Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.
Jintai DingDepartment of Mathematical Science, University of Cincinnati, USAZheng ZhangDepartment of Mathematical Science, University of Cincinnati, USAJoshua DeatonDepartment of Mathematical Science, University of Cincinnati, USA
Advances in Mathematics of Communications, 15, (1), 65-72, 2021.2
We present a cryptanalysis of a signature scheme HIMQ-3 due to Kyung-Ah Shim et al , which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems in 2017. We will show that inherent to the signing process is a leakage of information of the private key. Using this information one can forge a signature.
Bobo HuaSchool of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, ChinaYong LinDepartment of Mathematics, Information School, Renmin University of China, Beijing 100872, ChinaYanhui SuCollege of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China
In this paper, we prove some analogues of Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet (discrete) Laplace eigenvalues on any subset in the integer lattice $\Z^n.$ This partially answers a question posed by Chung and Oden.
Yong LinDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaShuang LiuYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, ChinaHongye SongDepartment of Mathematics, Renmin University of China, Beijing 100872, China; Beijing International Studies University, Beijing 100024, China
We prove the equivalence between some functional inequalities and the ultracontractivity property of the heat semigroup on infinite graphs. These functional inequalities include Sobolev inequalities, Nash inequalities, Faber–Krahn inequalities, and log-Sobolev inequalities. We also show that, under the assumptions of volume growth and CDE(n, 0), which is regarded as the natural notion of curvature on graphs, these four functional inequalities and the ultracontractivity property of the heat semigroup are all true on graphs.
Yong LinDepartment of Mathematics, Renmin University of China, Beijing 100872, ChinaHongye SongSchool of General Education, Beijing International Studies University, Beijing 100024, China; Department of Mathematics, Renmin University of China, Beijing 100872, China
Analysis of PDEsDifferential Geometrymathscidoc:2207.03005
We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.