Dec 2006, Volume 1 Issue 4
    

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  • Riemann-Finsler geometry
    GUO Enli, MO Xiaohuan
    2006, 1(4): 485-498. https://doi.org/10.1007/s11464-006-0023-9
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    In this paper, a survey on Riemann-Finsler geometry is given. Non-trivial examples of Finsler metrics satisfying different curvature conditions are presented. Local and global results in Finsler geometry are analyzed.
  • Approximation of several dimensional functions by trigonometric polynomials
    CHENG Minde, CHEN Yonghe
    2006, 1(4): 499-516. https://doi.org/10.1007/s11464-006-0024-8
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  • Robust H∞ control for singular systems with time-varying uncertainties
    DONG Xinzhuang, ZHANG Qingling
    2006, 1(4): 517-525. https://doi.org/10.1007/s11464-006-0025-7
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    In this paper, we investigate the problem of robust H∞ control for singular systems with polytopic time-varying parameter uncertainties. By introducing the notion of generalized quadratic H∞ performance, the relationship between the existence of a robust H∞ dynamic state feedback controller and that of a robust H∞ static state feedback controller is given. By using matrix inequalities, the existence conditions of robust H∞ static state feedback and dynamic output feedback controllers are derived. Moreover, the design methods for such controllers are provided in terms of the solutions of matrix inequalities. An example is also presented to demonstrate the validity of the proposed methods.
  • On Hochschild cohomology rings of Fibonacci algebras
    FAN Jinmei, XU Yunge
    2006, 1(4): 526-537. https://doi.org/10.1007/s11464-006-0026-6
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    Let ∧ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of ∧ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of ∧ is proved to be trivial.
  • Global regular solutions for Landau-Lifshitz equation
    GUO Boling, HAN Yongqian
    2006, 1(4): 538-568. https://doi.org/10.1007/s11464-006-0027-5
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    In this note, we prove that there exists a unique global regular solution for multidimensional Landau-Lifshitz equation if the gradient of solutions can be bounded in space L2(0, T; L∞). Moreover, for the two-dimensional radial symmetric Landau-Lifshitz equation with Neumann boundary condition in the exterior domain, this hypothesis in space L2(0, T; L∞) can be cancelled.
  • On completely irreducible operators
    JIANG Zejian, SUN Shanli
    2006, 1(4): 569-581. https://doi.org/10.1007/s11464-006-0028-4
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    Let T be a bounded linear operator on Hilbert space H, M an invariant subspace of T. If there exists another invariant subspace N of T such that H = M + N and M ∩ N = 0, then M is said to be a completely reduced subspace of T. If T has a nontrivial completely reduced subspace, then T is said to be completely reducible; otherwise T is said to be completely irreducible. In the present paper we briefly sum up works on completely irreducible operators that have been done by the Functional Analysis Seminar of Jilin University in the past ten years and more. The paper contains four sections. In section 1 the background of completely irreducible operators is given in detail. Section 2 shows which operator in some well-known classes of operators, for example, weighted shifts, Toeplitz operators, etc., is completely irreducible. In section 3 it is proved that every bounded linear operator on the Hilbert space can be approximated by the finite direct sum of completely irreducible operators. It is clear that a completely irreducible operator is a rather suitable analogue of Jordan blocks in L(H), the set of all bounded linear operators on Hilbert space H. In section 4 several questions concerning completely irreducible operators are discussed and it is shown that some properties of completely irreducible operators are different from properties of unicellular operators.
  • Applications of Wigner’s theorem to positive maps preserving norm of operator products
    LI Jinxiu
    2006, 1(4): 582-588. https://doi.org/10.1007/s11464-006-0029-3
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    We apply Wigner s theorem to positive maps on standard operator algebras that preserve norm of operator products or sum of singular values of operator products. It follows that such preservers are of the form φ(A) = UAU* with U either a unitary or antiunitary operator.
  • Noncontinuous data boundary value problems for Schr╫dinger equation in Lipschitz domains
    TAO Xiangxing
    2006, 1(4): 589-603. https://doi.org/10.1007/s11464-006-0030-x
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    The noncontinuous data boundary value problems for Schrödinger equations in Lipschitz domains and its progress are pointed out in this paper. Particularly, the Lp boundary value problems with p > 1, and Hp boundary value problems with p < 1 have been studied. Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.
  • Orthogonal two-direction multiscaling functions
    XIE Changzhen, XIE Changzhen, YANG Shouzhi, YANG Shouzhi
    2006, 1(4): 604-611. https://doi.org/10.1007/s11464-006-0031-9
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    The concept of a two-direction multiscaling functions is introduced. We investigate the existence of solutions of the two-direction matrix refinable equation imagewhere r × r matrices {P+k } and {P-k } are called the positive-direction and negative-direction masks, respectively. Necessary and sufficient conditions that the above two-direction matrix refinable equation has a compactly supported distributional solution are established. The definition of orthogonal two-direction multiscaling function is presented, and the orthogonality criteria for two-direction multiscaling function is established. An algorithm for constructing a class of two-direction multiscaling functions is obtained. In addition, the relation of both orthogonal two-direction multiscaling function and orthogonal multiscaling function is discussed. Finally, construction examples are given.
  • On value distribution of f(k) - afn
    ZHANG Zhanliang
    2006, 1(4): 612-619. https://doi.org/10.1007/s11464-006-0032-8
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    The aim of this paper is to discuss the value distribution of the function f(k) - afn. Under the assumption that f(z) is a transcendental meromorphic function in the complex plane and a is a non-zero constant, it is proved that if n ≥ k + 3, then f(k) - afn has infinitely many zeros. The main result is obtained by using the Nevanlinna theory and the Clunie lemma of complex functions.
  • Projected Hessian algorithm with backtracking interior point technique for linear constrained optimization
    ZHU Detong
    2006, 1(4): 620-628. https://doi.org/10.1007/s11464-006-0033-7
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    In this paper, we propose a new trust-region-projected Hessian algorithm with nonmonotonic backtracking interior point technique for linear constrained optimization. By performing the QR decomposition of an affine scaling equality constraint matrix, the conducted subproblem in the algorithm is changed into the general trust-region subproblem defined by minimizing a quadratic function subject only to an ellipsoidal constraint. By using both the trust-region strategy and the line-search technique, each iteration switches to a backtracking interior point step generated by the trustregion subproblem. The global convergence and fast local convergence rates for the proposed algorithm are established under some reasonable assumptions. A nonmonotonic criterion is used to speed up the convergence in some ill-conditioned cases.
  • The Abel lemma and the q-Gosper algorithm
    Vincent Y. B. Chen, William Y. C. Chen, Nancy S. S. Gu
    2006, 1(4): 629-634. https://doi.org/10.1007/s11464-006-0034-6
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