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Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

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Science China Mathematics 2024年第3期67卷 557-576页

作者： Jiaxiang Wang Bin Zhou Beijing International Center for mathematical Research Peking UniversityBeijing 100871China school of mathematical sciences Peking UniversityBeijing 100871China

In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

关键词： complex Monge-Ampère equations plurisubharmonic functions Sobolev inequality Moser-Trudinger inequality

The Weighted Basis for PHT-Splines

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工程与科学中的计算机建模(英文) 2024年第1期138卷 739-760页

作者： Zhiguo Yong Hongmei Kang Falai Chen school of mathematical sciences Soochow UniversitySuzhou215006China school of mathematical sciences University of Science and Technology of ChinaHefei230026China

PHT-splines are defined as polynomial splines over hierarchical T-meshes with very efficient local refinement *** original PHT-spline basis functions constructed by the truncation mechanism have a decay phenomenon,resulting in numerical *** non-decay basis functions are constructed as the B-splines that are defined on the 2×2 tensor product meshes associated with basis vertices in Kang et al.,but at the cost of losing the partition of *** the field of finite element analysis and topology optimization,forming the partition of unity is the default ingredient for constructing basis functions of approximate *** this paper,we will show that the non-decay PHT-spline basis functions proposed by Kang et *** be appropriately modified to form a partition of *** non-decay basis function is multiplied by a positive weight to form the weighted *** weights are solved such that the sum of weighted bases is equal to 1 on the *** provide two methods for calculatingweights,based on geometric information of basis functions and the subdivision of *** are given in the form of explicit formulas and can be efficiently *** also prove that the weights on the admissible hierarchical T-meshes are positive.

关键词： PHT-splines partition of unity weighted bases PHT-splines subdivision

Trotter–Kato Approximations of Impulsive Neutral SPDEs in Hilbert Spaces

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Acta Mathematica Sinica,English Series 2024年第5期40卷 1229-1243页

作者： Ming LIU Xia ZHANG Ling Fei DAI school of mathematical sciences Tiangong UniversityTianjin 300387P.R.China

This paper studies a class of impulsive neutral stochastic partial differential equations in real Hilbert *** main goal here is to consider the Trotter-Kato approximations of mild solutions of such equations in the pth-mean(p≥2).As an application,a classical limit theorem on the dependence of such equations on a parameter is *** novelty of this paper is that the combination of this approximating system and such equations has not been considered before.

关键词： Impulsive neutral stochastic partial diferential equation Trotter-Kato approximations classical limit theorem

P-symmetric Subharmonic Solutions for Nonlinear Hamiltonian Systems

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Acta Mathematica Sinica,English Series 2024年第6期40卷 1388-1408页

作者： Duan Zhi ZHANG Zhi Hao ZHAO school of mathematical sciences Nankai UniversityTianjin 300071P.R.China

In this paper,we prove that for each positive k≡1 mod m there exists a P-symmetric kmτ-periodic solution xk for asymptotically linear mτ-periodic Hamiltonian systems,which are nonautonomous and endowed with a *** the P-symmetric Hamiltonian function is semi-positive,one can prove,under a new iteration inequality of the Maslov-type P-index,that xk_(1) and xk_(2) are geometrically distinct for k_(1)/k_(2)≥(2n+1)m+1;and xk_(1),xk_(2) are geometrically distinct for k_(1)/k_(2)≥m+1 provided xk_(1) is non-degenerate.

关键词： P-symmetric subharmonic Hamiltonian systems Maslov-type index theory iteration inequality asymptotically linear

Quantitative Green's function estimates for lattice quasi-periodic Schrodinger operators

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Science China Mathematics 2024年第5期67卷 1011-1058页

作者： Hongyi Cao Yunfeng Shi Zhifei Zhang school of mathematical sciences Peking UniversityBeijing 100871China school of Mathematics Sichuan UniversityChengdu 610064China

In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger *** resonances in the estimates can be described via a pair of symmetric zeros of certain functions,and the estimates apply to the sub-exponential-type non-resonance *** the application of quantitative Green’s function estimates,we prove both the arithmetic version of Anderson localization and the finite volume version of(1/2-)-Holder continuity of the integrated density of states(IDS)for such QP Schrodinger *** gives an affirmative answer to Bourgain’s problem in Bourgain(2000).

关键词： Holder continuity of the IDS quantitative Green's function estimates quasi-periodic Schrodinger operators arithmetic Anderson localization multi-scale analysis

Topological Stability and Entropy for Certain Set-valued Maps

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Acta Mathematica Sinica,English Series 2024年第4期40卷 962-984页

作者： Yu ZHANG Yu Jun ZHU school of mathematical sciences Xiamen UniversityXiamen 361005P.R.China

In this paper,the dynamics(including shadowing property,expansiveness,topological stability and entropy)of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of *** is shown that(1)if f is a hyperbolic endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has theε-shadowing property,and moreover,if f is an expanding endomorphism then there exists a C^(1)-neighborhood U of f such that the induced set-valued map F_(f,U)has the Lipschitz shadowing property;(2)when a set-valued map F is generated by finite expanding endomorphisms,it has the shadowing property,and moreover,if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable;(3)if f is an expanding endomorphism then for eachε>0 there exists a C^(1)-neighborhood U of f such that h(F_(f,U,ε))=h(f);(4)when F is generated by finite expanding endomorphisms with no coincidence point,the entropy formula of F is ***,the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered.

关键词： Set-valued map orbit space hyperbolic endomorphism perturbation shadowing expansiveness topological stability entropy

MEAN SENSITIVITY AND BANACH MEAN SENSITIVITY FOR LINEAR OPERATORS

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Acta Mathematica Scientia 2024年第4期44卷 1200-1228页

作者： Quanquan YAO Peiyong ZHU school of mathematical sciences University of Electronic Science and Technology of ChinaChengdu 611731China

Let(X,T)be a linear dynamical system,where X is a Banach space and T:X→X is a bounded linear *** paper obtains that(X,T)is sensitive(Li-Yorke sensitive,mean sensitive,syndetically mean sensitive,respectively)if and only if(X,T)is Banach mean sensitive(Banach mean Li-Yorke sensitive,thickly multi-mean sensitive,thickly syndetically mean sensitive,respectively).Several examples are provided to distinguish between different notions of mean sensitivity,syndetic mean sensitivi`ty and mean Li-Yorke sensitivity.

关键词： mean sensitive Banach mean sensitive syndetically mean sensitive Li-Yorke sensitive mean Li-Yorke sensitive

Miyaoka-type inequalities for terminal threefolds with nef anti-canonical divisors

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Science China Mathematics 2024年

作者： Masataka Iwai Chen Jiang Haidong Liu Department of Mathematics Graduate School of Science Osaka University Shanghai Center for mathematical sciences & school of mathematical sciences Fudan University Department of mathematics Sun Yat-sen University

In this paper,we study Miyaoka-type inequalities on Chern classes of terminal projective 3-folds with nef anti-canonical *** X be a terminal projective 3-fold such that-KXis *** show that if c1（X）·c2（X）≠0,then c1（X）·c2（X）≥1/252;if further X is not rationally connected,then c1（X）·c2（X）≥4/5 and this inequality is *** order to prove this,we give a partial classification of such varieties along with many *** also study the nonvanishing of c1（X）dimX-2·c2（X） for terminal weak Fano varieties and prove a Miyaoka-Kawamata-type inequality.

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Investigation of Inside-Out Tracking Methods for Six Degrees of Freedom Pose Estimation of a Smartphone in Augmented Reality

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Computers, Materials & Continua 2024年第5期79卷 3047-3065页

作者： Chanho Park Takefumi Ogawa graduate school of Engineering The University of TokyoTokyo113-0033Japan Information Technology Center The University of TokyoChiba277-0882Japan

Six degrees of freedom(6DoF)input interfaces are essential formanipulating virtual objects through translation or rotation in three-dimensional(3D)space.A traditional outside-in tracking controller requires the installation of expensive hardware in *** inside-out tracking controllers have been proposed,they often suffer from limitations such as interaction limited to the tracking range of the sensor(e.g.,a sensor on the head-mounted display(HMD))or the need for pose value modification to function as an input interface(e.g.,a sensor on the controller).This study investigates 6DoF pose estimation methods without restricting the tracking range,using a smartphone as a controller in augmented reality(AR)*** approach involves proposing methods for estimating the initial pose of the controller and correcting the pose using an inside-out tracking *** addition,seven pose estimation algorithms were presented as candidates depending on the tracking range of the device sensor,the tracking method(e.g.,marker recognition,visual-inertial odometry(VIO)),and whether modification of the initial pose is *** two experiments(discrete and continuous data),the performance of the algorithms was *** results demonstrate enhanced final pose accuracy achieved by correcting the initial ***,the importance of selecting the tracking algorithm based on the tracking range of the devices and the actual input value of the 3D interaction was emphasized.

关键词： Smartphone inside-out tracking 6DoF pose 3D interaction augmented reality