15 April 2002, Volume 18 Issue 2

    

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  Two-Dimensional Graphs Moving by Mean Curvature Flow

  CHEN Jing Yi, LI Jia Yu, TIAN Gang

  Acta Mathematica Sinica. 2002, 18(2): 209-224. https://doi.org/10.1007/s101140200163

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  A surface Σ is a graph in R⁴ if there is a unit constant 2-form ω on R⁴ such that <e₁∧e₂, ω≥v₀>0 where {e₁, e₂} is an orthonormal frame on Σ. We prove that, if on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M₁×M₂ where M₁ and M₂ are Riemann surfaces, if <e₁∧e₂, ω₁>≥v₀>0 where ω₁ is a Köhler form on M₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic.
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  Further Results on Finitely Generated Projective Modules

  WU Tong Suo

  Acta Mathematica Sinica. 2002, 18(2): 225-228. https://doi.org/10.1007/s101140000078

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  In this paper, the exchange rings R whose primitive factor rings are artinian are studied. The following results are proved: for any exchange ring R and any two-sided ideal I of R, K₀(π) : K₀(R)→K₀(R/I) is a group epimorphism with the kernel {[P]-[Q]|P = PI, Q = QI}; there is an isomorphism of ordered groups from K₀(R) to the gorup of all such functions f_P : X→Q(P∈p(R)), where X is the set of all primitive ideals of R and Q, the rational integers.
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  Abundant Semigroups with a Multiplicative Adequate Transversal

  GUO Xiao Jiang

  Acta Mathematica Sinica. 2002, 18(2): 229-244. https://doi.org/10.1007/s101140200170

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  The aim of this paper is to investigate abundant semigroups with a multiplicative adequate transversal. Some properties and characterizations for such semigroups are obtained. In particular, we establish the structure of this class of abundant semigroups in terms of left normal bands, right normal bands and adequate semigroups with some simple compatibility conditions. Finally, we apply this structure to some special cases.
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  Morita Duality for the Rings of Generalized Power Series

  LIU Zhong Kui

  Acta Mathematica Sinica. 2002, 18(2): 245-252. https://doi.org/10.1007/s101140200171

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  Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. For any bimodule _A M_B , we show that the bimodule defines a Morita duality if and only if _A M_B defines a Morita duality and A is left noetherian, B is right noetherian. As a corollary, it is shown that the ring[[A^S,≤]] of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _A M_B such that B is right noetherian.
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  The Monotone Iterative Technique and Periodic Boundary Value Problem for First Order Impulsive Functional Differential Equations

  HE Zhi Min, GE Wei Gao

  Acta Mathematica Sinica. 2002, 18(2): 253-262. https://doi.org/10.1007/s101140200155

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  This paper uses the method of upper and lower solutions and the monotone iterative technique to investigate the existence of maximal and minimal solutions of the periodic boundary value problem for first order impulsive functional differential equations.
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  Almost Periodic Solutions of Neutral Differential Difference Equations with Piecewise Constant Arguments

  PIAO Da Xiong

  Acta Mathematica Sinica. 2002, 18(2): 263-276. https://doi.org/10.1007/s101140100151

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  In this paper, we study the existence of almost periodic solutions of neutral differential difference equations with piecewise constant arguments via difference equation methods.
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  On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

  CAI Tian Xin, GRANVILLE Andrew

  Acta Mathematica Sinica. 2002, 18(2): 277-288. https://doi.org/10.1007/s101140100144

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  In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions.
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  A Short Proof of Krattenthaler Formulas

  MA Xin Rong

  Acta Mathematica Sinica. 2002, 18(2): 289-292. https://doi.org/10.1007/s101140200161

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  With an effort to investigate a unified approach to the Lagrange inverse Krattenthaler established operator method we finally found a general pair of inverse relations, called the Krattenthaler forumlas. The present paper presents a very short proof of this formula via Lagrange interpolation. Further, our method of proof declares that the Krattenthaler result is unique in the light of Lagrange interpolation.
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  Spectral Radius of Non-negative Matrices and Digraphs

  ZHANG Xiao Dong, LI Jiong Sheng

  Acta Mathematica Sinica. 2002, 18(2): 293-300. https://doi.org/10.1007/s101140200157

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  We present an upper and a lower bound for the spectral radius of non-negative matrices. Then we give the bounds for the spectral radius of digraphs.
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  Poisson Morphisms and Reduced Affine Poisson Group Actions

  YANG Qi Lin

  Acta Mathematica Sinica. 2002, 18(2): 301-310. https://doi.org/10.1007/s101140200158

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  We establish the concept of a quotient affine Poisson group, and study the reduced Poisson action of the quotient of an affine Poisson group G on the quotient of an affine Poisson-G-variety V. The Poisson morphisms (including equivariant cases) between Poisson affine varieties are also discussed.
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  Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

  ZHANG Li Xin

  Acta Mathematica Sinica. 2002, 18(2): 311-326. https://doi.org/10.1007/s101140200162

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  Let {X_n ;n≥1} be a sequence of i.i.d. random variables and let if |X_j | is the r-th maximum of |X₁|,..., |X_n |. Let S_n = X₁+…+X_n and Sufficient and necessary conditions for ^(r)S_n approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for ^(r)S_n are studied.
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  Non-Uniformity and Generalised Sacks Splitting

  COOPER S. Barry, LI Ang Sheng

  Acta Mathematica Sinica. 2002, 18(2): 327-334. https://doi.org/10.1007/s101140100150

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  We show that there do not exist computable functions f₁(e, i), f₂(e, i), g₁(e, i), g₂(e, i) such that for all e, i ∈ ω, (1)
  
  It follows that the splitting theorems of Sacks and Cooper cannot be combined uniformly.
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  On the Modular Version of the Gluck-Wolf Theorem

  LU Zi Qun

  Acta Mathematica Sinica. 2002, 18(2): 335-338. https://doi.org/10.1007/s101140100149

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  Let N be a normal subgroup of a finite group G. Let ψ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/φ(1) is a π'-number of any χ∈IBr_p (G/φ). If p'|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If pψπ then G/N has a metabelian Hall-π subgroup.
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  Dubrovin Valuation Skew Group Rings

  YI Zhong

  Acta Mathematica Sinica. 2002, 18(2): 339-346. https://doi.org/10.1007/s101140200172

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  Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given. Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised.
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  On Ideals of Regular Rings

  CHEN Huan Yin, LI Fu An

  Acta Mathematica Sinica. 2002, 18(2): 347-356. https://doi.org/10.1007/s101140100140

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  In this paper, we investigate ideals of regular rings and give several characterizations for an ideal to satisfy the comparability. In addition it is shown that, if I is a minimal two-sided ideal of a regular ring R, then I satisfies the comparability if and only if I is separative. Furthermore, we prove that, for ideals with stable range one, Roth's problem has an affirmative solution. These extend the corresponding results on unit-regularity and one-sided unit-regularity.
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  Rigidity of Proper Self-Mapping on Some Kinds of Generalized Hartogs Triangle

  CHEN Zhi Hua, XU De Kang

  Acta Mathematica Sinica. 2002, 18(2): 357-362. https://doi.org/10.1007/s101140200156

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  It is proved that every proper holomorphic self-mapping of some kinds of Generalized Hartogs Triangles is an automorphism, and its explicit expression is given.
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  On a Fixed Point Theorem of Ky Fan

  DE BLASI Francesco S., GEORGIEV Pando Gr.

  Acta Mathematica Sinica. 2002, 18(2): 363-374. https://doi.org/10.1007/s101140200165

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  We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function f:D→E, in several directions: (1) for noncompact sets D, when f(D) precompact; (2) for compact D and upper semicontinuous multifunction f; and more generally, (3) for noncompact D and upper semicontinuous multifunction f with f(D) Hausdorff precompact.In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions whose values are convex closed bounded, thus not necessarily compact.
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  Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

  COHN William S., LU Guo Zhen

  Acta Mathematica Sinica. 2002, 18(2): 375-390. https://doi.org/10.1007/s101140200159

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  We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in[18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups.Let G be any group of Heisenberg type whose Lie algebra is generated by m left invariant vector fields and with a q-dimensional center.Let Q = m+ 2q, Q
  = Q/Q−1 and 
  
  Then
  
  with A_Q as the sharp constant, where _G denotes the subelliptic gradient on G.This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group[18].
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  Continuity of Higher Order Commutators on Certain Hardy Spaces

  DING Yong, LU Shan Zhen, ZHANG Pu

  Acta Mathematica Sinica. 2002, 18(2): 391-404. https://doi.org/10.1007/s101140200160

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  In this paper, the autors study the continuity properties of higher order commutators generated by the homogeneous fractional integral and BMO functions on certain Hardy spaces, weak Hardy spaces and Herz-type Hardy spaces.
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  Asymptotic Properties of a Modified Semi-Parametric MLE in Linear Regression with Right-Censored Data

  YU Qi Qing, WONG George Y. C.

  Acta Mathematica Sinica. 2002, 18(2): 405-416. https://doi.org/10.1007/s101140200169

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  Consider a linear regression model, Y=β' X+ε where Y may be right censored and the cdf F_o of ε is unknown. We show that a modified semi-parametric MLE, denoted by , is strongly consistent under certain regularity conditions. Moreover, if F_o is discontinuous, then P(≠β i.o.)=0, which means that P( =β if the sample size is large)=1. The latter property has not been reported for the existing estimators. By contrast, most estimators, such as the Buckley-James estimator and M-estimators , satisfy that P( ≠β i.o.)=1.