15 June 1995, Volume 11 Issue 2

    

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  Semicontinuity of Multiple Integrals on W^k,p (Ω,R^m)(k≥1)

  Zhang Chengdian, Li Kaitai

  Acta Mathematica Sinica. 1995, 11(2): 113-127. https://doi.org/10.1007/BF02274054

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  This paper shows that W^1,p-quasiconvexity is a necessary and sufficient condition of swlsc (sw^*lsc) for multiple integrals with a Caratheodory function as the variational function,bounded by
  
  whereg(x,u) is also a Caratheodory function,or
  
  The result for W^k,p-quasiconvexity is also shown.
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  Bifurcation and Chaotic Dynamics of Homoclinic Systems in R³

  Sun Jianhua

  Acta Mathematica Sinica. 1995, 11(2): 128-136. https://doi.org/10.1007/BF02274055

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  We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems.We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits,and illustrate our results with two examples.
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  The Solubility of Certain Diophantine Inequalities

  Li Hongze

  Acta Mathematica Sinica. 1995, 11(2): 137-145. https://doi.org/10.1007/BF02274056

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  Let λ₁,λ₂,...,λ_s bes non-zero real numbers not all of the same sign and not all in rational ratio,andk be a natural number; letD(k) be the leasts for which the inequality < (maxn_j)^-δ has infinitely many solutions.In this paper,we give some new estimations of D(k) for k≥6.
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  The Existence of a Weak Solution of Inhomogeneous Quasilinear Elliptic Equation with Critical Growth Conditions

  Li Gongbao, Zhou Huansong

  Acta Mathematica Sinica. 1995, 11(2): 146-155. https://doi.org/10.1007/BF02274057

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  In this paper,we get the existence of a weak solution of the following inhomogeneous quasilinear elliptic equation with critical growth conditions:
  
  whrer
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  On the Exceptional Set for the Equation n=p+k²

  Wang Tianze

  Acta Mathematica Sinica. 1995, 11(2): 156-167. https://doi.org/10.1007/BF02274058

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  Denote by E(X) the number of integers n≤X which are not a sum of a prime and a square.Set θ=1-0.994428b,where b≤min(0.9365/c₃,0.02578/c₄),c₃,c₄ being given in Lemma 8.The result E(x)<<x^θ is proved in this paper.
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  Irregular but Nonfractal Drums and n-Dimensional Weyl Conjecture

  Chen Hua

  Acta Mathematica Sinica. 1995, 11(2): 168-178. https://doi.org/10.1007/BF02274059

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  In this paper,we study the asymptotics of the spectrum of the Dirichlet (or Neumann) Laplacian in a bounded open set Ω ⊂Rⁿ (n≥1) with irregular but nonfractal boundary ∂Ω.We give a partial resolution of the Weyl conjecture,i.e.for the counting function N_i(λ)(i=0:Dirichlet;i=1:bNeumann),we have got a precise estimate of the remainder term ψ_i(λ)=ψ(λ)-N_i(λ) for largeλ,whereψ(λ) is the Weyl term.This implies that for the irregular but nonfractal drum Ω,not only the volume |Ω|_n is spectral invariant but also the area of boundary |∂Ω|_n-1 might be spectral invariant as well.
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  On the Asymptotic Behavior of Nonlinear Schrödinger Equations with Magnetic Effect

  Guo Boling, Tan Shaobin

  Acta Mathematica Sinica. 1995, 11(2): 179-187. https://doi.org/10.1007/BF02274060

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  The purpose of this paper is to study the long time asymptotic behavior for a nonlinear Schrödinger equations with magnetic effect.Under certain conditions,we prove the existence and nonexistence of the non-trivial free asymptotic solutions.In addition,the decay estimates of the solutions are also obtained.
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  Random Weighting T-Statistics in Linear Regression Models

  Shi Jian, Zheng Zhongguo

  Acta Mathematica Sinica. 1995, 11(2): 188-199. https://doi.org/10.1007/BF02274061

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  In this paper,we have constructed a random weighting statistic to approximate the distribution of studentized least square estimator in a linear regression model with ideal accuracy o(n^-1/2).Thus,we have provided a more practical distribution approximating method.
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  Theta Function and Hilbert Modular Forms of Half Integral Weight

  Feng Xuning

  Acta Mathematica Sinica. 1995, 11(2): 200-210. https://doi.org/10.1007/BF02274062

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  In this paper a kind of theta function is constructed by means of spherical function.And we also obtain some Hilbert modular forms of half integral weight.
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  Combinatorial Invariants on Planar Graphs

  Liu Yanpei

  Acta Mathematica Sinica. 1995, 11(2): 211-220. https://doi.org/10.1007/BF02274063

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  This paper introduces three kinds of operators on planar graphs with binary weights on edges,for which combinatorial invariants on two kinds of equivalences are found.Further,it is shown that the Jones polynomial and the bracket polynomial which are proved to be new topological invariants on knots in topology become special cases.Moreover,these invariants are a kind of generalization of Tutte polynomial on graphs.
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  Finite Presentation of Alternating Groups

  Q. Mushtaq, F. Shaheen

  Acta Mathematica Sinica. 1995, 11(2): 221-224. https://doi.org/10.1007/BF02274064

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  Graham Higman posed the question: How small can the integers p and q be made,while maintaining the property that all but finitly many alternating and symmetric groups are factor groups of Δ(2,p,q)=<x,y:x²=y^p=3 (xy)^q=1>? He proved that for a sufficiently large n,the alternating group is a homomorphic image of the triangle group Δ(2,p,q) where p=3 and q=7.Later,his result was generalized by proving the result for p=3 and q≥7.Choosing p=4 and q≥17 in this paper we have answered the "Hiqman Question".