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Automated theorem Proving Practice with Null Geometric Algebra

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Journal of Systems Science & Complexity 2019年第1期32卷 95-123页

作者： LI Hongbo Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science University of Chinese Academy of Sciences Chinese Academy of Sciences

This paper presents the practice of automated theorem proving in Euclidean geometry with null geometric algebra, a combination of Conformal Geometric Algebra and Grassmann-Cayley algebra. This algebra helps generating extremely short and readable proofs: The proofs generated are mostly one-termed or two-termed. Besides, the theorems are naturally extended from qualitative description to quantitative characterization by removing one or more geometric constraints from the hypotheses.

关键词： Automated theorem discovering automated theorem extending automated theorem proving Clifford bracket algebra null geometric algebra

Interrogating imaginary optical force by the complex Maxwell stress tensor theorem

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Light(Science & Applications) 2023年第1期12卷 12-14页

作者： Jinwei Zeng Jian Wang Wuhan National Laboratory for Optoelectronics and School of Optical and Electronic Information Huazhong University of Science and TechnologyWuhan 430074 HubeiChina Optics Valley Laboratory Wuhan 430074 HubeiChina

The complex Maxwell stress tensor theorem has been developed to relate the imaginary optical force,reactive strength of canonical momentum and total optical force of a nanoparticle,which is essential to perfect optica... 详细信息

关键词： imaginary tensor theorem

Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem

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Chinese Physics B 2019年第10期28卷 253-256页

作者： Ji-Suo Wang Xiang-Guo Meng Hong-Yi Fan Shandong Provincial Key Laboratory of Laser Polarization and Information Technology College of Physics and EngineeringQufu Normal UniversityQufu 273165China Shandong Provincial Key Laboratory of Optical Communication Science and Technology School of Physical Science and Information EngineeringLiaocheng UniversityLiaocheng 252059China Department of Material Science and Engineering University of Science and Technology of ChinaHefei 230026China

We study how can an angular momentum coherent state |τ> keeps its form-invariant during time evolution governed by the Hamiltonian H = f(t)J++ f^*(t)J-+ g(t)Jz. We discuss this topic in the context of boson realization of |τ>. By employing the entangled state representation |ζ> and deriving a new binomial theorem involving two-subscript Hermite polynomials, we derive the wave function , which turns out to be a single-subscript Hermite polynomial. Based on this result the maintenance of angular momentum coherent state during time evolution is examined, and the value of τ(t) is totally determined by the parameters involved in the Hamiltonian.

关键词： invariance of angular momentum coherent state entangled state representation new binomial theorem time evolution

Generalized Ejiri’s Rigidity theorem for Submanifolds in Pinched Manifolds(In memory of Professor Chaohao Gu on his 90th birthday)

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Chinese Annals of Mathematics,Series B 2020年第2期41卷 285-302页

作者： Hongwei XU Li LEI Juanru GU Center of Mathematical Sciences Zhejiang UniversityHangzhou 310027China Department of Applied Mathematics Zhejiang University of TechnologyHangzhou 310023China

Let Mn(n ≥ 4) be an oriented compact submanifold with parallel mean curvature in an(n + p)-dimensional complete simply connected Riemannian manifold Nn+*** there exists a constant δ(n, p) ∈(0, 1) such that if the sectional curvature of N satisfies■ , and if M has a lower bound for Ricci curvature and an upper bound for scalar curvature, then N is isometric to Sn+p. Moreover, M is either a totally umbilic sphere■ , a Clifford hypersurface Sm■ in the totally umbilic sphere ■, or■ . This is a generalization of Ejiri’s rigidity theorem.

关键词： Minimal submanifold Ejiri rigidity theorem Ricci curvature Mean curvature

Three Series theorem for Independent Random Variables under Sub-linear Expectations with Applications

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Acta Mathematica Sinica,English Series 2019年第2期35卷 172-184页

作者： Jia Pan XU Li Xin ZHANG School of Mathematical Sciences Zhejiang University

In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng, we establish a three series theorem of independent random variables under the sub-linear expectations. As an application, we obtain the Marcinkiewicz's strong law of large numbers for independent and identically distributed random variables under the sub-linear expectations. The technical details are different from those for classical theorems because the sub-linear expectation and its related capacity are not additive.

关键词： Sub-linear expectation capacity Rosenthal's inequality Kolmogorov's three series theorem Marcinkiewicz's strong law of large numbers

Beurling Type theorem on the Hilbert Space Generated by a Positive Sequence

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Acta Mathematica Sinica,English Series 2019年第9期35卷 1511-1519页

作者： Chang Hui WU Zhi Jie WANG Tao YU School of Mathematical Sciences Dalian University of Technology Dalian 116024 P. R. China Department of Mathematics Physics and Information Engineering Jiaxing University Jiaxing 314001 P. R. China

Let H^2(γ) be the Hilbert space over the bidisk D^2 generated by a positive sequence γ={γnm}n,m≥0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H^2(γ) with γ={γnm}n,m≥0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk D^2.

关键词： Beurling type theorem invariant subspace wandering subspace

ON THE CONDITIONAL VERSION OFKOLMOGOROV'S THREE-SERIES theorem

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Acta Mathematica Scientia 1999年第3期19卷 300-306页

作者：刘国欣 Hebei Univ Technol Dept Math Tianjin 300130 Peoples R China

This paper studies the conditional version of Kolmogorov’s three-series theorem, and gets a new extention form of the conditional version. The results here present us an answer to the question when (or where) the conditional version also provide necessary conditions for convergence in dependent cases. Furthermore, some new sufficient conditions are obtained.

关键词： Three-series theorem dependent r.v. ’s predictable local absolute continuity martingale

LIOUVILLE TYPE theorem FOR THE STATIONARY EQUATIONS OF MAGNETO-HYDRODYNAMICS

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Acta Mathematica Scientia 2019年第2期39卷 491-497页

作者： Simon SCHULZ Mathematical Institute University of Oxford

We show that any smooth solution(u, H) to the stationary equations of magnetohydrodynamics belonging to both spaces L^6(R^3) and BMO^(-1)(R^3) must be identically *** is an extension of previous results, all of which systematically required stronger integrability and the additional assumption ▽u, ▽H∈L^2(R^3), i.e., finite Dirichlet integral.

关键词： Liouville theorem Caccioppoli inequality Navier-Stokes equations MHD

A mathematical aspect of Hohenberg-Kohn theorem

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Science China Mathematics 2019年第1期62卷 63-68页

作者： Aihui Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences

The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of *** this article, we study t... 详细信息

关键词： density functional theory electronic structure unique continuation principle Hohenberg-Kohn theorem