15 June 2003, Volume 19 Issue 2

    

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  Quadratic Recursion Relations of Hodge Integrals via Localization

  Gang TIAN, Jian ZHOU

  Acta Mathematica Sinica. 2003, 19(2): 209-232. https://doi.org/10.1007/s10114-003-0260-8

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  We derive some quadratic recursion relations for some Hodge integrals by virtual localization and obtain many closed formulas. We apply our formulas to the local geometry of toric Fano surfaces in a Calabi-Yau threefold and compute some of the numbers n_β^g in Gopakumar-Vafa's formula for all g in this case.
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  Counting SL₂(F_2⁸) Representations of Torus Knot Groups

  Wei Ping LI (Weiping LI), Liang XU

  Acta Mathematica Sinica. 2003, 19(2): 233-244. https://doi.org/10.1007/s10114-003-0261-7

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  In this paper we count the number of SL ₂(F_2⁸) representations of torus knot groups up to a conjugacy. For the finite field F_2⁸ with character 2, the counting method is similar to that in our previous work[1]. Explicit formulae of the effective counting are given in this paper. Twisted Alexander polynomials related to those representations are discussed.
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  Seiberg-Witten-Floer Homology and Gluing Formulae

  Alan L. CAREY, Bai Ling WANG

  Acta Mathematica Sinica. 2003, 19(2): 245-296. https://doi.org/10.1007/s10114-003-0262-6

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  This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spin^c structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.
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  Calderón-Zygmund Operators on the Predual of a Morrey Space

  Yasuo KOMORI

  Acta Mathematica Sinica. 2003, 19(2): 297-302. https://doi.org/10.1007/s10114-002-0226-2

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  We consider the boundedness of Calderón-Zygmund operators from H^p,Ψ (Rⁿ)(the predual of a Morrey space) to h^p,Ψ (Rⁿ)(the local version of H^p,Ψ). We show that Calderón's commutator is bounded from H^p,Ψ to h^p,Ψ.
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  Schrödinger Flows on Compact Hermitian Symmetric Spaces and Related Problems

  Wei Yue DING, Hong Yu WANG, You De WANG

  Acta Mathematica Sinica. 2003, 19(2): 303-312. https://doi.org/10.1007/s10114-003-0263-5

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  In this note, we prove that the Schrödinger flow of maps from a closed Rieman surface into a compact irreducible Hermitian symmetric space admits a global weak solution. Also, we show the existence of weak solutions to the initial value problem of Heisenberg model wit Lie algebra values, which is closely related to the Schr?dinger flow on compact Hermitian symmetric spaces.
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  On Bounds for Spectra of Operator Pencils in a Hilbert Space

  Michael Iosif GIL

  Acta Mathematica Sinica. 2003, 19(2): 313-326. https://doi.org/10.1007/s10114-002-0225-3

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  A class of pencils (operator-valued functions of a complex argument) in a separable Hilbert space is considered. Bounds for the spectra are derived. Applications to differential operators, integral operators with delay and infinite matrix pencils are discussed.
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  Upper Bounds on Character Sums with Rational Function Entries

  Todd COCHRANE, Chun Lei LIU, Zhi Yong ZHENG

  Acta Mathematica Sinica. 2003, 19(2): 327-338. https://doi.org/10.1007/s10114-002-0230-6

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  We obtain formulae and estimates for character sums of the type S{χ,f,p^m) = Σ_x=1^pmχ(f(x)), where p^m is a prime power with m ≥ 2, χ is a multiplicative character (mod p^m), and f=f₁/f₂ is a rational function over Z. In particular, if p is odd, d=deg(f₁)+deg(f₂) and d* = max(deg(f₁), deg(f₂)) then we obtain |S(χ, f, p^m)|≤(d-1)}p^m(1-1/d*) for any non-constant f (mod p) and primitive character . For p = 2 an extra factor of 2√2 is needed.
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  Equivalence Bimodule between Spherical Noncommutative Tori

  Chun Gil PARK

  Acta Mathematica Sinica. 2003, 19(2): 339-348. https://doi.org/10.1007/s10114-002-0229-z

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  Let n_#em/em# ,m_j >1. In[5], the sperical noncommutative torus S_p^pd was defined by twisting ×Z^l in T^pd⊕C^*(Z^l) by a totally skew multiplier p on ×Z^l for T^pd a pd-homogeneous C*-algebra over . It is shown that S_p^pd is strongly Morita equivalent to .
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  Elliptic Equations with Degenerate Coercivity: Gradient Regularity

  Daniela GIACHETTI, Maria Michaela PORZIO

  Acta Mathematica Sinica. 2003, 19(2): 349-370. https://doi.org/10.1007/s10114-002-0235-1

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  In this paper, we prove higher integrability results for the gradient of the solutions of some elliptic equations with degenerate coercivity whose prototype is
  -div{a(x,u)Du) = f\ in D'(Ω),f∈L^r(Ω), r > 1,
  where for example, a(x,u)=(1+|u|)^-θ with θ ∈ (0,1). We study the same problem for minima of functionals closely related to the previous equation.
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  Morita Context and Morita-Like Equivalence for the xst-Rings

  Bai Yu OUYANG, Wen Ting TONG

  Acta Mathematica Sinica. 2003, 19(2): 371-380. https://doi.org/10.1007/s10114-002-0219-1

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  The notion of the xst-rings was introduced by García and Marín[5] in 1999. In this paper, we consider Morita context, Morita-like equivalence and the exchange property for the xst-rings. The results of the first Morita theorem are generalized to the xst-rings. So we obtain an important Morita-like equivalence of the xst-rings, from which, as an immediate consequence, we deduce the main result of Xu-Shum-Turner[4] and the standard Morita equivalence, A ～ M_n (A), for a unital ring A. Moreover, we describe the properties of those well-known intermediate matrix rings, and show that the exchange property of a unital ring A coincides with the one for any M_n(A) as well as any intermediate matrix ring sitting between FM_Γ(A) and FC_Γ(A), which is an extension of a well-known result obtained by Nicholson[7].
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  A Geometric Approach to the Calderón-Zygmund Estimates

  Li He WANG (Lihe WANG)

  Acta Mathematica Sinica. 2003, 19(2): 381-396. https://doi.org/10.1007/s10114-003-0264-4

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  In this note, we will show a new proof of the classical Calderón-Zygmund estimates established in[7] 1952. The estimates are among the fundamental estimates for elliptic equations. Parabolic equations are also considered.
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  L²(Rⁿ) Boundedness for a Class of Multilinear Singular Integral Operators

  Guo En HU

  Acta Mathematica Sinica. 2003, 19(2): 397-404. https://doi.org/10.1007/s10114-002-0231-5

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  The L²(Rⁿ) boundedness for the multilinear singular integral operators defined by
  T_Af(x) = ∫_RⁿΩ(x-y)|x-y|ⁿ⁺¹(A(x) -A(y)-∇A(y)(x-y)) f(y)dy
  is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO(Rⁿ). A sufficient condition based on the Fourier transform estimate and implying the L²(Rⁿ) boundedness for the multilinear operator T_A is given.
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  Convergence of Newton’s Method and Uniqueness of the Solution of Equations in Banach Spaces Ⅱ

  Xing Hua WANG, Chong LI

  Acta Mathematica Sinica. 2003, 19(2): 405-412. https://doi.org/10.1007/s10114-002-0238-y

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  Some results on convergence of Newton's method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L average.
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  The Proof of Inseparable Prime AW^*-Algebras Being Primitive C^*-Algebras

  Lun Chuan ZHANG

  Acta Mathematica Sinica. 2003, 19(2): 413-416. https://doi.org/10.1007/s10114-002-0192-8

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  The relation between the inseparable prime C^*-algebras and primitive C^*-algebras is studied, and we prove that prime AW*-algebras are all primitive C^*-algebras.