中国科学院数学与系统科学研究院期刊网

15 November 1979, Volume 22 Issue 6
    

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  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 643-652. https://doi.org/10.12386/A1979sxxb0060
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    Let be a measurable space, q(t,x,A) (t≥0, x∈, A∈) is said to be a q-function, if(ii) For fixed t≥0, x∈, q(t,x,·)is a denumerable additive set funetion on, for fixed t≥0, A∈q(t,A) is a measurable function of x.The Markov process p(s, t,x,A) is said to be a q-process, ifIn this paper, the existence and uniqueness of q-processes is investigated when q-function q(t,x,A) is a continuous function of t.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 653-666. https://doi.org/10.12386/A1979sxxb0061
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    In this paper, we prove the following results.Theorem 1. The perfect space λ is a nuclear relative to the normal topology if and only if for any positive member W of the Kothe's dual spade λ there exist W ∈λ such that W≥W (i. e. W_k~(2)≥W_k~(1)for k=1,2,…andNote. This theorem was obtained independently by the author and Pietsch fourteen years ago.Theorem 3. The perfect space λ is symmetric and nuclear if and only if λ=ω or λ=φ.Theorem 4. For the perfect space λ the following propositions arc equvilent:(1) λ is a (F)-space relative to;(2) in λ there exists an enumerable sequence {U~(n_o)}, which absorbs any member U of λ, i.e. for any U∈λ there exist α_o>0 and U~(n_o) such that |u_k|≤α_o|u_k~(n_o)|(k=1,2,…);(3) λ is a Gestufen space, generated by the enumerable, non-negative and increasing Stufen system {B~((n))}.Theorem 5. Let λ be a Gestufen space generated by the enumerable,non-negative and increasing Stufen system {B~(n)}. Then λ is J-nuclear if and only if for any abstract function X(t)={x_1(t), x_2(t),…,x_k(t),…} on [0, 1] to λ, the conditions and imply
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 667-674. https://doi.org/10.12386/A1979sxxb0062
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    Let R be a partially ordered linear space (See [3]) and operator T be defined on R. We shall find conditions in order that the equation Tv+r=v has a solution for a given r∈R. We formulate the results as follows:Theorem 1.If u_o,ω_o∈R(u_o≤ω_o) are two given elements and Tu_o≥u_o, Tω_o≤ω_o, furthermore if there exists a constant N<1 such that for u, ω(u≤ω) in [u_o, ω_o] Tω-Tu≥N(ω-u),then and the equation Tv=v has a solution in [u_n, ω_n] .Theorem 2. If u_o,ω_o∈R(u_o≤ω_o) are two giren elements and Tu_o≤u_o,Tω_o≥ω_o, furthermore if there exists a constant M>1 such that for u, ω(u≤ω) in [u_o,ω_o] Tω-Tu≤M(ω-u), then u_o≤u_1≤u_2≤…≤u_n≤…≤ω_n≤…≤ω_2≤ω_1≤ω_o and the equation Tv=v has a solution in [u_n, ω_n].Let operator T_1 be increasing and operator T_2 be decreasing. Define T=T_1+T_2. We haveTheorem 3. If u_o, ω_o ∈R(u_o≤ω_o) are two given elements and T_1u_o+T_2ω_o+γ≥ u_o, T_1ω_o + T_2u_o + r≤ω_o, furthermore if T is additive,then u_o≤u_1≤u_2≤…≤u_n≤…≤ω_n≤…≤ω_2≤ω_1≤ω_o and the equation Tv+γ=v has a solution in [u_n, ω_n].
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 675-692. https://doi.org/10.12386/A1979sxxb0063
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    Let be a bounded domain in the 2n-dimensional space of n complex variables z = (z_1,…,z_n), n≥2, and its boundary Ω be a (2n-1)-dimensional smooth orientable manifold of class C~2. K(ζ, ξ)denotes the Bochner-Martinelli kernel, where ζξ∈Ω; dS denotes the volume element of Ω; |ζ- ξ|——the Euclidean distance between ζ and ξ.In this paper, we have proved:Theorem (Transformation formula). If φ(ξ, η)is a continuous complex-valued function defined on Ω, which satisfies Holder condition with index α(0<α<1) with respect to ξ and η, and if ζ∈Ω, then we have where the integral on the left and the inner integral of the first term on the right take on principle values, and the inner integral on the right is uniformly O(|ζ-η|~(-(2n-1-α/2)))dS_η with respect to ζ and η on Ω, when |ζ-η|→0; therefore the outer integral is an ordinary integral.With this formula, we have given the regularization theorem of the singular integral equations with Bochner-Martinelli kernel on Ω.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 693-712. https://doi.org/10.12386/A1979sxxb0064
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    In the present paper, we consider the maximum signal to noise ratio filtering proplems. The signal is assumed to be of the form and the nonstationary additive noise is where Ω_s(λ) and Ω_R(λ) are functions of bounded variations on [-π,π], and Z_ξ(λ) is a process with orthogonal increments.For every τ(-∞<τ<+∞), the maximum SNR is defined as where L_τ~2(dF_ξ)={e~(iλt)|t ≤τ}.The existence and uniqueness theorem of maximum SNR filteration is proved, the optimal Φ(λ) is given in terms of the functions Ω_s(λ), Ω_R(λ) and the spectral function F_ξ(λ).The corresponding problems in 2-dim case are also considered.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 713-718. https://doi.org/10.12386/A1979sxxb0065
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    The partial differential equations of non-principal type are far from being wellunderstood. In 1974, F. Treves studied the following Cauchy problem where λ represents a real parameter. By. using the concatenation method, a diserete phenomenon is disclosed. For almost all values of λ, there is uniqueness. However, non-uniqueness occurs only for λ=1, 3, 5,… In this paper, by using energy integral method, we deal with the general second order PDE's of non-principal type in two variables and prove that non-uniqueness is a rare event.
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 719-732. https://doi.org/10.12386/A1979sxxb0066
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    Courant R. and Friedrichs K. O. solved the Riemann Problem for the onedimensional adiabatic flow of polytropic gas, which represents the case with convexity. Wendroff B. and Tai-Ping Liu and this paper all study the case without convexity, but the following contents are not concerned in [4] and [5]:a. We clarify the relative position of two shoeh wave curves in phase space, where one curve has its begining point on another. This is the base of all our demonstration. (§2).b. We proof that there is no possibility of appearance of any limit point while the wave curves are extended (§3).c. We discover that the wave curve attains the maximum of entropy in a certain sense (§4).d. We give the formula of solution and an example of systems without convexity (§5-6).
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 733-742. https://doi.org/10.12386/A1979sxxb0067
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    Let u(z) be an algebroid function with v branches, and r be any positive integer. Remove those zeros whose orders of multiplicity exceed r from the zeros of u(z)-α, and count the rest only once. We obtain the following theorems.Theorem 1. Let u(z) be an algebroid function, and let thenTheorem 2. Let u(z) and v(z) be two algebroid functions, and let E_γ(α,u) and E_γ) (α,v) be the sets of the zeros of u(z) - a and v(z)-a respectively. If. E_(γi)(α_i,u) for i=1,2,…,q, and the following condition is satisfied, then u(z)≡v(z).Theorem 3. Let u(z) be an algebroid function in |z|<1, if then
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 743-750. https://doi.org/10.12386/A1979sxxb0068
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    The purpose of the present paper is to establish the following integral representation of analytie functions on the bounded convex domain D={z|Φ(z)<0} in C~n:where k~(k) denotes the following differential operater: where α=(α_1,α_2,…,α_n) is an arbitrary inner fixed point in the domain D.This integral representation denotes that the value of the analytic functions in the bounded convex domain may be defined by the boundary value of differential expression g (z).From the integral representation of bounded convex domain we can obtained two classes of integral representations of n-multiple circular domain: and which are different from the type,where and H~(k) denotes the following differential operater:
  • Acta Mathematica Sinica, Chinese Series. 1979, 22(6): 751-758. https://doi.org/10.12386/A1979sxxb0069
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    In this paper we study the relative position and number of limit cycles of the quadratic differential system. dx/dt=-y+lx~2+mxy+ny~2,dy/dt=x(1+ax+by)(1)for which O(0,0) is a fine focus. We prove that limit cycles of system (1) Can only be distributed around one of the two focus, provided that system (1) apart from these focus, has a third singuler point.On the other hand system (1) has only two singular points: a fine focus O(0,0) and a rough focus N(0, 1), then (1) may have limit cycles around the two singular points simultaneously. It is important to note that if we choose suitably the Coefficents in system (1), and add a sufficiently small term x to the right hand side of the first equation, there may exist three limit cycles around O(0, 0) and one limit cycle around N(0, 1). Then system (1) has at least four limit cycles.