中国科学院数学与系统科学研究院期刊网

15 July 1979, Volume 22 Issue 4
    

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  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 389-403. https://doi.org/10.12386/A1979sxxb0037
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    Let be a vector space over a division ring F and R be the complete ring of linear transformations of In this paper we discribe a method to construct the ring R using complete rings of linear transformations of finite dimensional vector spaces over F. But here we omit discribing this method. In addition to the above results we give a geometrical meaning of an infinite matrix ring which contains matrices with both row-infinite and column-infinite. Now let {U_i}r be a basis of be the complete direct stun of {Fu_i}r. Suppose that is the soele of the complete ring R, then it is clear that RE_i, where RE_i are minimal left ideals of R and E_i~2=E_i, E_iE_j=0, i≠j, i, j ∈Γ. Consider the set {RE_i}r and write RE_i as the complete direct sum, then we have following definition.Definition:Let a ∈then aσTheorem: is the complete group of all linear transformations of onto Fu_i and the complete ring of linear transformations of is a subgroup
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 404-419. https://doi.org/10.12386/A1979sxxb0038
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    Consider the absolute stability of direct control system: x=Ax + bf(σ), σ=c'x, (·)=d/dt.(1) We point out an error of A.I. Lurie, I.G. Malkin and A. M. Letev and an error in "Non-linear differential equations of higher order" by R. Reissig, G. Sansone and R. Conti.(2) We give the following result: Under the assumption that A is stable, the necessary and suffieient conditions for to satisfy the condition of the LBK Theorem and hence guarantee the absolute stability is that there exists a matrix B>0, such that(i) u(x) = x'Bx-(c'A + 2b'P)x-c'b≥0 for c′x=0,(ii) c'B~(-1)(c'A + 2b'P)≤0, where P is determined by the Lipunov relation A'P + PA = -B.(3) Consider the systems in the canonical form We give a criterion for the absolute stability.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 420-437. https://doi.org/10.12386/A1979sxxb0039
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    Let be a measurable space and every single point set {x} belong to P(s, t, x, A) (0≤s≤t<∞, ) is said to be a Markow proeess,it(i) For fixed s, t, A, P(s, t,·, A) is a measmrable function of x;(ii) For fixed s, t, x, P(s, t, x,·)is a measure on ,and 0≤P(s,t,x,)≤1; (iii) (k-c) equation is satisfiedIn this paper the following main results are obtained:(1) P(+,t,x,A) is continuous on [o,t] uniformly for t≥0.(2) P(s,·,x,A) is continuous from right on [s, ∞] uniformly for(3) If P(s,t,x,A) satisfies:Moreover, under some additional conditions, we have where P(s,t,x,A) is a continuous kf and t on for fixed x and A.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 438-447. https://doi.org/10.12386/A1979sxxb0040
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    The results obtained so far on the stochastic control under the quadratic index are discussed in this paper and at the same time their defects are pointed out. To avoid these defects, the unique condition imposed on the matrix coefficients of the system is their integrability in some sense, and a suitable set of admissible controls is now selected. In this paper the existence and the uniqueness of solution of a Riccati equation are considered and the existent nonlinear filtering equation is rewritten into a form which we need. By means of these results for the linear stochastic systemwith the quadratic index the separation principle is proved and the optimal controlis given.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 448-458. https://doi.org/10.12386/A1979sxxb0041
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    In this paper we give a slight generalization of the imbedding theorem proved in [3]. And then we apply it to algebraic number fields, we obtaining a large sieve inequality. Finally we generalize to algebraic number fields the mean value theorem in [1] where with it we proved the Chen's theorem on Goldbaeh problem.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 459-470. https://doi.org/10.12386/A1979sxxb0042
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    Let E be a set in Banach space X. Let T be an average nonexpansive mapping of E into itself (i.e. ‖Tx- Ty‖≤ a‖x- y‖ + b(‖x - Tx‖ + ‖y - Ty‖) + e(‖x-Ty‖ + ‖y - Tx‖), where x, y ∈ E, a, b, e ≥ 0 and a + 2b + 2c ≤ 1). The present paper deals with existence problems of fixed point for this mapping. The main results are as follows.Theorem 1. Suppose E is a closed nonempty set. If b > 0 and c > 0, then for the mapping T, there exists a unique fixed point x in E, and for any x∈E,Theorem 2. Suppose E is a closed convex nonempty set, and the mapping T is also Lipschitzian in E for a constant A > 0.If b>0, then for any x_o∈E and 0<θ< min {2b/A(1-c), 1}, the successive sequence {x_n} determined by x_o ∈ E, x_(n+1)=θT~2x_n+(1-θ)Tx_n, (n = 0, 1,…) converges to a unique fixed point of the mapping T in E.Theorem 3. Suppose E is a quasi-weakly compact convex nonempty set (i.e. E is closed, and any bounded closed covex subset of the set E is weakly compact). If b > 0 and one of the following conditions is satisfied, then the mapping T has a unique fixed point in E:(1) The mapping T is continuous in .E.(2) 25 < 1.(3) The set E has close-to-normed structure.(4) In any subset K of E with T(K)K and δ(K) > 0, there exists a point x∈K and a weakly limit point y of the {T~nx} such that lim sup‖Ty-T~nx‖<δ(K).Theorem 4. Suppose E is a weakly compact convex nonempty set, and the mapping T satisfies the asymptotic normed condition (i.e. for any closed convex subset K of E with T(K)K and σ(K) >0, there exist x, y∈K such that lim sup‖y-T~nx‖<σ(K)). If b=0, then the mapping T has a fixed point in E.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 471-486. https://doi.org/10.12386/A1979sxxb0043
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    Denote by C the space of continuous functions defined on the l-dimensional Euclidean space R_l, and by C the subspace of C spanned by the funetions with limit zero at infinity. In this paper, we consider the differential expression (in general, unsymmetric and with unbounded coefficients) We have proved that G induces a closed operator A on a subspace C of C(C C C). A is shown to satisfy the Hille-yosida conditions, so it generates a contraction semigroup T_t on C.If the ergodic limit of Γ_tf(x) does not vanish (theorem 5), it can be proved that T_t possesses an invariant measure θ(dx) and is ergodie in the sense of L_p(θ(dx)). The semigroup T_t is also shown to have a Markov transition function p(t, x, Γ) which furnishes the basis for constructing a Markov process.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 487-494. https://doi.org/10.12386/A1979sxxb0044
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    Let k ≥ 3 and a_1,…, a_(k-1) be distinct non-vanishing integers. Let a be a positive integer such that (a, a_1,…, a_(k-1)) = 1. Further, let B = a + max(|a_1|, …,|a_(k-1)|) and E_i = e~(ai/a)( 1≤i≤k - 1).We haveTheorem. Suppose that y ≥B~(16k4)·B~(16k4) Then we have y‖yE_1‖…‖yE_(k-1)‖>y~(-12k2(k+1)(log B·(log log y)-1))1/2(1)This gives a slight modification of a theorem due to Mahler, whose original result was obtained by replacing the right-hand side of (1) with y~(-12k3(k-1) (log B·(log log y)-1)1/2
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(4): 495-501. https://doi.org/10.12386/A1979sxxb0045
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    First, we formulate the definition of local convexity. Let π and π be a surface and its boundary respectively. For a point P ∈ int π=π\π, if there is a small neighborhood about P in the topology of π such that it can be considered as part of a surface of some convex body, then we say π is locally convex at point P. If every point of int π is locally convex, we say that π has the property of local convexity. Evidently, if π processes the property of local convexity only, it may not be a convex surface, We are going to discuss the conditions under which π will be a convex surface. In this paper we stipulate that π is a n-connected surface, it means that πeorresponds topologically to a n-conneeted domain Ω in E_2 plane, where the boundary of π consists of n-closed Jordan curves. The main result of this paper is the followingTheorem 1. Let π be a n-connected surface belonging to C~o, λ=π (consisting of finite closed curves). If1° π is locally convex;2° For every point x∈γ, there is a plane S_x, x ∈S_x S_x is an entirely supportingplant for π, i.e. the points of π lie on S_x or in one half of the space divided by S_x.Then π is a convex surface. Conversely,if π is a convex surface then 1°and 2° arevalid.