15 October 1998, Volume 14 Issue 4

    

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  Periodic Solutions for Second Order Systems at Resonance

  Tang Chunlei

  Acta Mathematica Sinica. 1998, 14(4): 433-440. https://doi.org/10.1007/BF02580399

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  The existence and multiplicity results are obtained for periodic solutions of second order systems at resonance with unbounded nonlinearity. The proofs rely on the minimax methods and an interesting integral inequality.
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  On Non-Decomposable Hermitian Forms over Gaussian Domain

  Yang Lihua, Sun Xingming

  Acta Mathematica Sinica. 1998, 14(4): 441-446. https://doi.org/10.1007/BF02580400

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  In this paper we study the kernels of a linear operator and its algebraic adjoint by studying their restriction on a subspace, a Banach space, such that the restriction is the difference of the identity and a compact operator under some conditions, and therefore some results on compact operator theory can be applied. As an example we study the M-scale subdivision operators.
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  On Non-Decomposable Hermitian Forms over Gaussian Domain

  Zhu Fuzu

  Acta Mathematica Sinica. 1998, 14(4): 447-456. https://doi.org/10.1007/BF02580401

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  Methods are presented for the construction of non-decomposable Hermitian forms over the ring of integers of an imaginary quadratic field.
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  Kneading Sequences for Unimodal Expanding Maps of the Interval

  Zeng Fanping

  Acta Mathematica Sinica. 1998, 14(4): 457-462. https://doi.org/10.1007/BF02580402

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  Let P and AC be two primary sequences with min {P, AC}≥RLR^∞,ρ(P) and ρ(AC) be the eigenvalues of P andAC, respectively. Let f∈C⁰ (I, I) be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved thatf has the kneading sequence K(f)≥(RC)^*m *P if λ≥(ρ(P))^1/2m, and K(f)>(RC)^*m*AC*E for any shift maximal sequenceE if λ>(ρ(AC))^1/2m. The value of (ρ(P))^1/2m or (ρ(AC))^1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one.
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  PBVP of Integrodifferential Equations with Carathéodory Functions

  Zhuang Wan, Chen Yubo

  Acta Mathematica Sinica. 1998, 14(4): 463-472. https://doi.org/10.1007/BF02580403

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  We study the periodic boundary value problems for nonlinear integro-differential equations of Volterra type with Carathéodory functions. For two situations relative to lower and upper solutions α and β:α≤β or β≤α, the existence of solutions and the monotone iterative method for establishing extreme solutions are considered.
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  The ACL Property of Homeomorphisms under Weak Conditions

  Fang Ainong

  Acta Mathematica Sinica. 1998, 14(4): 473-480. https://doi.org/10.1007/BF02580404

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  Suppose that D is a domain in Rⁿ(n ≥ 2) and y = f(x) : D → Rⁿ is a homeomorphism.We prove that if the modulus dilatation K(x, f) satisfies the condition A then f(x) is ACL.
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  A Problem about the Weak Hilbert Property

  Zeng Guangxing

  Acta Mathematica Sinica. 1998, 14(4): 481-486. https://doi.org/10.1007/BF02580405

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  In this paper, we prove the main result: Let both (K, S) and (K^*,S^*) be preordered fields, and let (K^*,S^*) be a finitely generated extension of (K, S). IfK^* is transcendental overK, then (K^*,S^*) has the weak Hilbert property. This result answers negatively an open problem posed by the author in reference [1]. Moreover, some results on the weak Hilbert property are established.
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  Separation Properties for MW-Fractals

  Li Wenxia

  Acta Mathematica Sinica. 1998, 14(4): 487-494. https://doi.org/10.1007/BF02580406

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  Corresponding to the irreducible 0-1 matrix (a_ij )_n×n , take similitude contraction mappings Ψ_ij for eacha_ij =1, ina_ij =1, in R^d with ratio 0<r_ij <1. There are unique nonempty compact sets F₁,…,F_n satisfying for each 1≤i≤n, F_i. We prove that open set condition holds if and only if F_{em is ans-set for some 1≤i≤n, where s is such that the spectral radius of matrix (rij ³)nxn is 1.}
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  Uniqueness Theorem of Solutions for Stochastic Differential Equation in the Plane

  Liang Zongxia

  Acta Mathematica Sinica. 1998, 14(4): 495-506. https://doi.org/10.1007/BF02580407

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  Let M={M_z, z ∈ R₊² be a continuous square integrable martingale and A={A_z, z ∈ R₊^{2dX_z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R₊2,
  X_z=Z_z, z∈∂R₊2,
  whereR₊2 =[0, +∞)×[0,+∞) and ∂R₊2 is its boundary,Z is a continuous stochastic process on ∂R₊2. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2].}
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  The Convergence of Krylov Subspace Methods for Large Unsymmetric Linear Systems

  Jia Zhongxiao

  Acta Mathematica Sinica. 1998, 14(4): 507-518. https://doi.org/10.1007/BF02580408

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  The convergence problem of many Krylov subspace methods,e.g., FOM, GCR, GMRES and QMR, for solving large unsymmetric (non-Hermitian) linear systems is considered in a unified way when the coefficient matrixA is defective and its spectrum lies in the open right (left) half plane. Related theoretical error bounds are established and some intrinsic relationships between the convergence speed and the spectrum ofA are exposed. It is shown that these methods are likely to converge slowly once one of the three cases occurs:A is defective, the distribution of its spectrum is not favorable, or the Jordan basis ofA is ill conditioned. In the proof, some properties on the higher order derivatives of Chebyshev polynomials in an ellipse in the complex plane are derived, one of which corrects a result that has been used extensively in the literature.
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  On the Construction of the Conjugate Hull of Brandt Semigroups

  Zhang Moucheng, Kan Haibing, Li Yonghua

  Acta Mathematica Sinica. 1998, 14(4): 519-526. https://doi.org/10.1007/BF02580409

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  In this paper the characterizations of conjugate hulls ψ (S). Φ(S),T(S) and θ(S) on a Brandt semigroupS are given. By using these results we can pro thatT(S) is self-conjugate in ψ(S) for a Brandt semigroupS.
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  The Lifespan for 3-D Spherically Symmetric Compressible Euler Equations

  Yin Huicheng, Qiu Qingjiu

  Acta Mathematica Sinica. 1998, 14(4): 527-534. https://doi.org/10.1007/BF02580410

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  In this paper, we study the lifespan of solutions for three dimensional compressible Euler equations with spherically symmetric initial data that is a small perturbation of amplitude ε from a constant state. From our result, the classical solutions have to blow up in finite time in spite of any small ε.
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  Some Results on the Multiplier Conjecture for the Case n=5n₁

  Wan Zhaoze

  Acta Mathematica Sinica. 1998, 14(4): 535-540. https://doi.org/10.1007/BF02580411

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  Using the method presented in [1], we obtain some new results which improve on the result of MacFarland's theorem (see [2]) in this case.
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  Crossed Products on the Flow of Groupoid with Quasi-Invariant Measures

  Fang Xiaochun

  Acta Mathematica Sinica. 1998, 14(4): 541-546. https://doi.org/10.1007/BF02580412

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  Let G be a second countable groupoid with Haar system {λ^u },A be an abelian group which left invariant acts on G. Then we have a C^*-dynamic system (C^* (G, A, β). In this paper we have studied the existence of quasi-invariant measure with certain properties; using these measures some important results about crossed products and groupoid C^*-algebras have been obtained.
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  Characterization of Periodic Multiresolution Analysis and an Application

  Li Dengfeng, Peng Silong

  Acta Mathematica Sinica. 1998, 14(4): 547-554. https://doi.org/10.1007/BF02580413

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  In this paper, we study the properties of periodic multiresolution analysis, and present a complete characterization of the scaling function sequence, which enables us to construct a new scaling function sequence from a given one. An application of the main results is given at the end.
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  Representation Type of Endomorphism Algebras of Exceptional Sequences of Type A_n

  Yao Hailou, Ping Yanru

  Acta Mathematica Sinica. 1998, 14(4): 555-562. https://doi.org/10.1007/BF02580414

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  Let F be an algebraically closed field, Δ be a quiver of typeA_n . In this paper we prove that the endomorphism algebras of exceptional sequences over Δ are sums of finitely many tilted algebras of type A_m wherem≤n by using perpendicular categories, and thus the endomorphism algebras of exceptional sequences of type A_n are representation-finite.
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  Unimodal Sequences of Gorenstein Ideals of Codimension 4

  Hyoung J. Ko, Yong Su Shin

  Acta Mathematica Sinica. 1998, 14(4): 563-568. https://doi.org/10.1007/BF02580415

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  We present Gorenstein ideals of condimension 4 which have unimodal Hilbert functions.
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  On Unique Range Sets for Meromorphic or Entire Functions

  Fang Mingliang, Guo Hui

  Acta Mathematica Sinica. 1998, 14(4): 569-576. https://doi.org/10.1007/BF02580416

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  This paper has studied the Gross uniqueness question ignoring multiplicity and relating multiple values, and extended and improved on some related results in [1-8].