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ON THE CLASSIFICATION OF POSITIVE DEFINITE unimodular HERMITIAN FORMS

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Chinese Science Bulletin 1991年第18期36卷 1506-1511页

作者：朱福祖 Department of Mathematics East China Normal University Shanghai 200062 PRC

1. Let F= Q (t^2=-1, m>0 and square free) be an imaginary quadratic field, D_m the ring of integers in F, H an n-ary positive definite Hermitian form over F(hereafter simply as H-form), and let(V, H), or simply V, denote a non-zero regular n-ary positive definite Hermitian space over F. L denotes a D_m-lattice on V, i. e. L is a finitely generated D_m-module of V and FL=V.

关键词： unimodular lattice (form) indecomposable lattice (form) normal genus reduced form opposite form even genus.

unimodular Gravity and Averaging

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Journal of Modern Physics 2012年第3期3卷 266-270页

作者： Alan Coley Johan Brannlund Joey Latta Department of Mathematics and Statistics Dalhousie University Halifax Canada

The question of the averaging of inhomogeneous spacetimes in cosmology is important for the correct interpretation of cosmological data. In this paper a conceptually simpler approach to averaging in cosmology is suggested, based on the averaging of scalars within unimodular gravity. As an illustration, the example of an exact spherically symmetric dust model is considered, and it is shown that within this approach averaging introduces correlations (corrections) to the effective dynamical evolution equation in the form of a spatial curvature term.

关键词： Inhomogeneous Cosmology Averaging unimodular Gravity

Almost split sequences for symmetric non-semisimple Hopf algebras

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2006年第6期7卷 1077-1083页

作者： SHI Mei-hua Department of Mathematics Zhejiang Education Institute Hangzhou 310012 China

We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be tmimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D（H）=（H^op）^＊∞ H） of any non-semisimple Hopf algebra.

关键词： Indecomposable unimodular Almost split sequences Symmetric non-semisimple Hopfalgebras

Matrix structures over connected rings

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Chinese Science Bulletin 1996年第7期41卷 549-551页

作者：陈焕艮 Department of Mathematics Hunan Normal University Changsha 410006 China

It is well known that the connected ring is one of the most useful rings. For example, in-tegral domain, local ring, PF ring and ring with torsion K0 group are all connected. LetX be a topological space. Then X is connected if and only if C（X） is a connected ring. Our main goal is to investigate the properties of connected rings by virtue of matrixtheories. Throughout this note, all rings are commutative with identity elements.

关键词： projective module idempotent matrix and unimodular vector.

Construction of indecomposable definite Hermitian forms

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Chinese Science Bulletin 1995年第3期40卷 177-181页

作者：朱福祖 Department of Mathematics East China Normal University Shanghai 200062 China

Let F=Q(-m1/2) （m>0 and square-free） be an imaginary quadratic field and Dm the ring of algebraic integers in F. The field F has a unique non-trivial involution （the complex conjugation） whose fixed field is Q. Let V be an n-dimensional non-degenerate Hermitian space over F equipped with a sesquilinear form φ on Ⅴ with respect to the involution and Hermitian form H associated with φ. Let L be a Dm-lattice on V, i.e. a finitely generated Dm-module in Ⅴ and FL=V. L is called even lattice if H（x）∈2Z for any x∈L; otherwise, it is odd. L is indecomposable if it cannot be expressed as an orthogonal sum of two non-zero sublattices; in other words, L=P （?） R implies P=0 or R=0. The minimum of a Hermitian Dm-lattice L With respect to its associated Hermitian form h

关键词： indecomposable lattice unimodular lattice genus even lattice.

POSITIVE DEFINITE HERMITIAN FORMS OVER IMAGINARY QUADRATIC FIELDS

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Chinese Science Bulletin 1990年第23期35卷 1946-1948页

作者：秦厚荣 Department of Mathematics East China Normal University Shanghai 200062 PRC Department of MathematicsNanjing UniversityNanjing210008PRC

The theory of integral Hermitian forms over algebraic number fields not only has connections with many branches of mathematics such as number theory, geometry of numbers, Lie theory and algebraic geometry but also has applications in some subjects such as code theory. Many papers dealt with such subject. Recently we generalize Kneser’s neighbor-lattice method to imaginary quadratic fields and develop a method to determine the class numbers and the representative forms of modular lattices (especially the unimodular lat-

关键词： unimodular lattice indecomposable lattice class number.

Using the Simplex Method for a Type of Allocation Problems

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American Journal of Computational Mathematics 2019年第2期9卷 25-31页

作者： Yoshihiro Tanaka Faculty of Economics and Business Hokkaido University Sapporo Japan

In this study we discuss the use of the simplex method to solve allocation problems whose flow matrices are doubly stochastic. Although these problems can be solved via a 0 - 1 integer programming method, H. W. Kuhn [1] suggested the use of linear programming in addition to the Hungarian method. Specifically, we use the existence theorem of the solution along with partially total unimodularity and nonnegativeness of the incidence matrix to prove that the simplex method facilitates solving these problems. We also provide insights as to how a partition including a particular unit may be obtained.

关键词： Allocation Problems Hall’s Theorem Totally unimodular Matrix Simplex Method

Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple

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American Journal of Computational Mathematics 2012年第1期2卷 21-26页

作者： Mohamed Salah Boudellioua Department of Mathematics and Statistics Sultan Qaboos University Muscat Oman

In this paper we show how the transformations associated with the reduction to the Smith form of some classes of mul-tivariate polynomial matrices are computed. Using a Maple implementation of a constructive version of the Quillen-Suslin Theorem, we present two algorithms for the reduction to a particular Smith form often associated with the simplification of linear systems of multidimensional equations.

关键词： Smith Form unimodular Equivalence Quillen-Suslin Theorem Maple