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.

Let f(x, y) be a periodic function defined on the region D$0 \leqslant x \leqslant 2\pi ,0 \leqslant y \leqslant 2\pi $ with period 2π for each variable. If f(x, y) ∈ C^p (D), i.e., f(x, y) has continuous partial derivatives of order p on D, then we denote by ω_α,β(ρ) the modulus of continuity of the function $\frac{{\partial ^p f(x,y)}}{{\partial x^\alpha \partial y^\beta }}(\alpha ,\beta \geqslant 0,\alpha + \beta = p)$ and write $\omega _p (\rho ) = \mathop {\max }\limits_{\alpha ,\beta \geqslant 0,\alpha + \beta = p} \omega _{\alpha ,\beta } (\rho ).$ For p = 0, we write simply C(D) and ω(ρ) instead of C⁰(D) and ω₀(ρ).

Let T(x,y) be a trigonometrical polynomial written in the complex form $T(x,y) = \sum {C_{m,n} } e^{i(mx + ny)} .$ We consider R = max(m² + n²)^1/2 as the degree of T(x, y), and write T_R(x, y) for the trigonometrical polynomial of degree ⩾ R.

Our main purpose is to find the trigonometrical polynomial T_R(x, y) for a given f(x, y) of a certain class of functions such that $\mathop {\max }\limits_{x,y} \left| {f(x,y) - T_R (x,y)} \right|$ attains the same order of accuracy as the best approximation of f(x, y).

Let the Fourier series of f(x, y) ∈ C(D) be $f(x,y) \sim \sum\limits_{ - \infty }^\infty {C_{m,n} e^{i(mx + ny)} } ,$ and let $A_\nu (x,y) = \sum\limits_{m^2 + n^2 = \nu } {C_{m,n} e^{i(mx + ny)} } .$ Our results are as follows

Theorem 1 Let f(x, y) ∈ C^p(D (p = 0, 1) and $S_R^\delta (x,y;f) = \sum\limits_{\nu = R^2 } {\left( {1 - \frac{\nu }{{R^2 }}} \right)^\delta A_\nu (x,y)(\delta > {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2})} .$Then $S_R^\delta (x,y;f) - f(x,y) = O\left[ {\frac{1}{{R^p }}\omega _p \left( {\frac{1}{R}} \right)} \right](p = 0,1)$holds uniformly on D.

If we consider the circular mean of the Riesz sum S_R ^δ(x, y) ≡ S_R ^δ(x, y; f): $\mu _t \left[ {S_R^\delta (x,y)} \right] = \frac{1}{{2\pi }}\int_0^{2\pi } {S_R^\delta (x + t\cos \theta ,y + t\sin \theta )d\theta ,} $ then we have the following

Theorem 2 If f(x, y) ∈ C^p (D) and ω_p(ρ) = O(ρ^α (0 < α ⩾ 1; p = 0, 1), then $\mu _{\frac{{\lambda _0 }}{R}} [S_R^\delta (x,y)] - f(x,y) = O\left( {\frac{1}{{R^{p + \alpha } }}} \right)(p = 0,1;\delta \geqslant 0)$holds uniformly on D, where λ₀is a positive root of the Bessel function J₀(x)

It should be noted that either $S_R^\delta (x,y;f) - f(x,y) = o({1 \mathord{\left/ {\vphantom {1 {R^2 }}} \right. \kern-\nulldelimiterspace} {R^2 }})$ or $\mu _{\frac{{\lambda _0 }}{R}} [S_R^\delta (x,y)] - f(x,y) = o({1 \mathord{\left/ {\vphantom {1 {R^2 }}} \right. \kern-\nulldelimiterspace} {R^2 }})$ implies that f(x, y) ≡ const.

Now we consider the following trigonometrical polynomial $S_R^\delta (x,y;f) = \sum\limits_{\nu \leqslant R^2 } {\left( {1 - \frac{{\nu ^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{R^k }}} \right)^\delta A_\nu (x,y)(k \in \mathbb{Z}^ + )} .$ Then we have

Theorem 3 If f(x, y) ∈ C^p(D), then uniformly on D, $S_R^{(k)} (x,y;f) - f(x,y) = \left\{ \begin{gathered} o[\tfrac{1}{{R^p }}\omega _p (\tfrac{1}{R})],p = 0,1,...,k - 1forkeven, \hfill \\ o[\tfrac{1}{{R^p }}\omega _p (\tfrac{1}{R})lnR],p = k - 1forkodd. \hfill \\ \end{gathered} \right.$

Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem of Zygmund, which can be extended to the multiple case as follows

Theorem 3′ Let f(x₁, ..., x_n) ≡ f(P) ∈ C^pand let $S_R^{(k)} (P;f) = \sum\limits_{\nu \leqslant R^2 } {\left( {1 - \frac{{\nu ^{\tfrac{k}{2}} }}{{R^k }}} \right)^{\sigma _m } A_\nu (P)} ,$where $\sigma _m = \left[ {\frac{{m - 1}}{2}} \right] + 1$and $A_\nu (P) = \sum\limits_{n_1^2 + ... + n_m^2 = \nu } {C_{n_1 ,...,n_m } e^{i(n_1 x_1 + ... + n_m x_m )} } ,$$C_{n_1 , \cdots ,n_m } $$being the Fourier coefficients of f(P). Then $S_R^{(k)} (p;f) - f(P) = \left\{ \begin{gathered} o\left[ {\tfrac{1}{{R^p }}\omega _p \left( {\tfrac{1}{R}} \right)} \right](p \leqslant k - 2;p = k - 1forkeven), \hfill \\ o\left[ {\tfrac{1}{{R^p }}\omega _p \left( {\tfrac{1}{R}} \right)lnR} \right](p = k - 1forkodd) \hfill \\ \end{gathered} \right.$holds uniformly.

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.

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.

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.

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.

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) = U AU* with U either a unitary or antiunitary operator.

The noncontinuous data boundary value problems for Schrödinger equations in Lipschitz domains and its progress are pointed out in this paper. Particularly, the L^p boundary value problems with p > 1, and H^p boundary value problems with p < 1 have been studied. Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.

The concept of a two-direction multiscaling functions is introduced. We investigate the existence of solutions of the two-direction matrix refinable equation $\Phi (x) = \sum\limits_{k \in \mathbb{Z}} {P_k^ + \Phi (2x - k) + } \sum\limits_{k \in \mathbb{Z}} {P_k^ - \Phi (k - 2x)} ,$ where 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.

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.

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.