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Simple Fourth-Degree Cubature Formulae with Few Nodes over General Product Regions

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Numerical Mathematics(Theory,Methods and Applications) 2014年第2期7卷 179-192页

作者： Ran Yu Zhaoliang Meng zhongxuan luo School of Mathematical Sciences Dalian University of TechnologyDalian116024P.R.China School of Software Dalian University of TechnologyDalian116620P.R.China

A simple method is proposed for constructing fourth-degree cubature formulae over general product regions with no symmetric assumptions. The cubatureformulae that are constructed contain at most n2 + 7n + 3 nodes and they are likelythe first kind of fourth-degree cubature formulae with roughly n2 nodes for nonsymmetric integrations. Moreover, two special cases are given to reduce the numberof nodes further. A theoretical upper bound for minimal number of cubature nodesis also obtained.

关键词： Fourth-degree cubature formula cubature formula product region non-symmetric region numerical integration

From a projective invariant to some new properties of algebraic hypersurfaces

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Science China Mathematics 2014年第11期57卷 2273-2284页

作者： luo zhongxuan ZHOU XinChen GU David XianFeng School of Mathematical Sciences Dalian University of Technology School of Software Dalian University of Technology Department of Computer Science State University of New York at Stony Brook

Projective invariants are not only important objects in mathematics especially in geometry,but also widely used in many practical applications such as in computer vision and object recognition. In this work,we show a projective invariant named as characteristic number,from which we obtain an intrinsic property of an algebraic hypersurface involving the intersections of the hypersurface and some lines that constitute a closed loop. From this property,two high-dimensional generalizations of Pascal's theorem are given,one establishing the connection of hypersurfaces of distinct degrees,and the other concerned with the intersections of a hypersurface and a simplex.

关键词： characteristic number algebraic hypersurfaces Pascal's theorem characteristic mapping simplex

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Feature Preserving Parameterization for Quadrilateral Mesh Generation Based on Ricci Flow and Cross Field

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Computer Modeling in Engineering & Sciences 2023年第10期137卷 843-857页

作者： Na Lei Ping Zhang Xiaopeng Zheng Yiming Zhu zhongxuan luo International School of Information Science and Engineering Dalian University of TechnologyDalian116000China School of Mathematical Sciences Dalian University of TechnologyDalian116000China School of Software Technology Dalian University of TechnologyDalian116000China

We propose a newmethod to generate surface quadrilateralmesh by calculating a globally defined parameterization with feature *** the field of quadrilateral generation with features,the cross field methods are wellknown because of their superior performance in feature *** methods based on metrics are popular due to their sound theoretical basis,especially the Ricci flow *** cross field methods’major part,the Poisson equation,is challenging to solve in three dimensions *** it comes to cases with a large number of elements,the computational costs are expensive while the methods based on metrics are on the *** addition,an appropriate initial value plays a positive role in the solution of the Poisson equation,and this initial value can be obtained from the Ricci flow *** we combine the methods based on metric with the cross field *** use the discrete dynamic Ricci flow algorithm to generate an initial value for the Poisson equation,which speeds up the solution of the equation and ensures the convergence of the *** experiments show that our method is effective in generating a quadrilateral mesh for models with features,and the quality of the quadrilateral mesh is reliable.

关键词： Quadrilateral mesh feature preserving Ricci flow cross field

The Nearest Complex Polynomial with a Prescribed Zero

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Journal of Mathematical Research with Applications 2015年第1期35卷 39-55页

作者： Wenyu HU zhongxuan luo College of Mathematics and Computer Science Gannan Normal University School of Mathematical Sciences Dalian University of Technology School of Software Dalian University of Technology

Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f(z) and a zero α, in this paper we explore the problem of computing a complex polynomial f(z) such that f(α) = 0 and the distance ∥f-f ∥ is minimal. Considering most of the existing works focus on either certain polynomial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of f(z), we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f(z) explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.

关键词： nearest polynomial explicit expression zero dual norm

Prismatic mesh generation based on anisotropic volume harmonic field

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Advances in Aerodynamics 2021年第1期3卷 228-243页

作者： Yiming Zhu Shengfa Wang Xiaopeng Zheng Na Lei zhongxuan luo Bo Chen Dalian University of Technology NO.2 Linggong RoadGanjingzi DistrictDalianChina Computational Aerodynamics Institute China Aerodynamics Research and Development CenterMianyangChina.

In this paper,we present an effective prismatic mesh generation method for viscous flow *** address the prismatic mesh collisions in recessed cavities or slit areas,we exploit 3D tensors controlled anisotropic volume harmonic field to generate prismatic ***,a well-fitting tetrahedral mesh is first constructed to serve as the discrete computation domain of volume harmonic ***,3D tensors are exploited to control the volume harmonic field that better fits the shape *** the topological perspective,the generation of the prismatic mesh can be treated as a topology-preserved morphing of the viscous ***,iso-surfaces in the volume harmonic field should be homeomorphic to the viscous wall while fitting its ***,a full prismatic mesh can be induced by estimating the forward directions and visible regions in the volume harmonic ***,to be compatible with different simulation situations,the thickness of prismatic meshes should be *** approach provides local adjustable thickness of prismatic meshes,which can be achieved by controlling local 3D *** proposed anisotropic volume harmonic field based prismatic meshes are efficient and robust,and a full prismatic mesh can be guaranteed without low quality *** experiments have shown that our proposed prismatic meshes have obvious advantages in terms of efficiency and effectiveness.

关键词： Mesh generation Prismatic mesh Volume harmonic field 3D tensor Minkouski sum

A Class of Multivariate Hermite Interpolation of Total Degree

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Journal of Mathematical Research with Applications 2017年第3期37卷 345-350页

作者： Zhongyong HU Zhaoliang MENG zhongxuan luo School of Mathematical Sciences Dalian University of Technology School of Mathematics and Statistics Taishan College School of Software Dalian University of Technology

In this paper we study a class of multivariate Hermite interpolation problem on 2;nodes with dimension d ≥ 2 which can be seen as a generalization of two classical Hermite interpolation problems of d = 2. Two combina... 详细信息

关键词： multivariate Hermite interpolation regularity total degree

A NODAL BASIS OF C~μ-RATIONAL SPLINE FUNCTIONS ON TRIANGULATIONS

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Analysis in Theory and Applications 1994年第4期10卷 13-24页

作者： zhongxuan luo Renhong Wang (Dalian University of Technology,China) 1. Institute of Mathematical Sciences Dalian University of Technology 116024 Dalian China

Let ι be a triangulation of a polygonal domain D R^2 with vertices V={v_1:1≤i≤N,}and RS^k(D,τ)={u∈C^k(D):T∈τ, u|_τis a rational function}.The purpose of this paper is to study the exist- ence and construction of C~~μ-rational spline functions on any triangulation τ for *** Hermite prob- lem H~μ(V,U)={find u∈U:D^au(v_1)=D^af(v_1),|x|≤μ}is solved by the generalized wedge function method in rational spline function family,i.e.U=RS“. this solution needs only the knowledge of partial derivatives of order≤μ at v_The explicit repesentations of all C”-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation are given.

关键词： RATIONAL SPLINE FUNCTIONS ON TRIANGULATIONS A NODAL BASIS OF C CAGD Math Rational 一刁 node

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A New Strategy to Construct Embedded Cubature Formulae over Two-Dimensional Regions

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Journal of Mathematical Research with Applications 2013年第2期33卷 142-154页

作者： Ran YU Zhaoliang MENG zhongxuan luo School of Mathematical Sciences Dalian University of Technology School of Software Technology Dalian University of Technology

The purpose of this paper is to study a new strategy to construct embedded cubature formulae over two-dimensional regions.A new kind of embedded cubature formulae with some nodes along the selected algebraic curve is ... 详细信息

关键词： cubature formulae embedded cubature formulae polynomial ideal moment.

A Geometric Understanding of Deep Learning

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Engineering 2020年第3期6卷 361-374页

作者： Na Lei Dongsheng An Yang Guo Kehua Su Shixia Liu zhongxuan luo Shing-Tung Yau Xianfeng Gu DUT-RU Co-Research Center of Advanced ICT for Active Life Dalian University of TechnologyDalian 116620China Department of Computer Science Stony Brook UniversityStony BrookNY 11794-2424USA School of Computer Science Wuhan UniversityWuhan 430072China School of Software Tsinghua UniversityBeijing 100084China Center of Mathematical Sciences and Applications Harvard UniversityCambridgeMA 02138USA

This work introduces an optimal transportation(OT)view of generative adversarial networks(GANs).Natural datasets have intrinsic patterns,which can be summarized as the manifold distribution principle:the distribution of a class of data is close to a low-dimensional *** mainly accomplish two tasks:manifold learning and probability distribution *** latter can be carried out using the classical OT *** the OT perspective,the generator computes the OT map,while the discriminator computes the Wasserstein distance between the generated data distribution and the real data distribution;both can be reduced to a convex geometric optimization ***,OT theory discovers the intrinsic collaborative-instead of competitive-relation between the generator and the discriminator,and the fundamental reason for mode *** also propose a novel generative model,which uses an autoencoder(AE)for manifold learning and OT map for probability distribution *** AE–OT model improves the theoretical rigor and transparency,as well as the computational stability and efficiency;in particular,it eliminates the mode *** experimental results validate our hypothesis,and demonstrate the advantages of our proposed model.

关键词： Generative Adversarial Deep learning Optimal transportation Mode collapse