15 March 1979, Volume 22 Issue 2

    

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  OPTIMUM SELECTION WITE TIME DELAY

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 129-139. https://doi.org/10.12386/A1979sxxb0010

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  This paper is concerned with the optimum selection problem which involves the sequential tests with time delays. Such tests can each be put at an unit time, but the result for each test can be obtained only after τ units of time (τ≥1). The purpose of this paper is to find the maximum length of the feasible test interval for time delay τ= 3 and to give a closer upper bound of this maximum length for each τ≥3.
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  ON THE SATURATION OF MULTIDIMENSIONAL SPLINE APPROXIMATION

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 140-145. https://doi.org/10.12386/A1979sxxb0011

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  In this paper we consider the problem of saturation by two-dimensional spline approximation associated with {△~((k))}, where {△~((k))} is a sequence of partition of Ω=[a, b] × [c, d].The main results we have obtained are:Theorem 1. Let f(x, y)∈C~((m,n)) (Ω), S_(△(k)) be the first Type of degree (m-1) × (n - 1) of blended Spline, R_(△(k)) ≤ β < ∞, (k = 1, 2,…) and ‖△_x~((k))‖)→0, ‖△_y~((k))‖→0(k→∞).If max holds for some(p,l),(0≤p≤m,0≤l≤n),thenTheorem 2. Let f(x,y)be Comtinuous On and S_(△(k) be the Second Type of degree(m-1)×(n-1)of spline andif max then
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  THE SPECTRAL REPRESENTATION

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 146-155. https://doi.org/10.12386/A1979sxxb0012

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  We give the following results.Theorem 1 Let {A_1|l∈A} be a commutative set of self-adjoint operators in Hilbert space with a sequence of cyclic vectors, where the index set A has arbitrary cardinal. Let all real numbers), S be the σ-Bool algebra generated by rectangular parallelepipeds of X. Then there existthe unique (in equivalent sense) bounded measure ν on S, measurable field of Hilbert spaces λ→(λ) on the Borel space (X, S), and a unitary isomorphism U from onto such thatU A_l U~(-1)=T_l, l ∈, where T_l is the operator multiplying λ_l in (λ= (λ_l.)_(l′∈)∈X).Theorem 2 we keep the notions of theoreml. Let T be a closed dense linear operator in such that TB BT for every bounded linear operator B satisfying BE~((l)) = E_l~((l)) B then there exists a unique (p.p.ν) S-measurable function X such that where E(·) is the spectral measure generated by {E_l~((l)0 |l∈A, t real}, and T_l is the operator multiplying f(λ) in.These results are the generalization of works [1], [2] and others.
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  ON THE CONVERGENCE CRITERION OF GALERKIN METHOD

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 156-169. https://doi.org/10.12386/A1979sxxb0013

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  Let H be a separable Hilbert space and A .D(f) → R(f) be a linear closed operator with D(f) =R(f) = H. We consider an independent sequence φ_j ∈ D(A), j = 1, 2,… Sequence Aφ_j, j =1, 2,…, is total in H. Let H_n be the subspace spanned by φ_1…,φ_n, and E_m :H →H_n be the projection mapping,and put A_n = E_nAE_n.Suppose the element f ∈ H and the functional equation Ax = f(1)are given. To replace the solution of Eq. (1), we consider the algebraic equation A_nX = E_nf. (2)The solution of Eq. (2) is cal]ed the Galerkin approximate solution. In this paper, the author gives the necessary and sufficient conditions and sufficient conditons for the Galerkin approximate solution to converge to the exact solution. The main results are as follows :1. The Galerkin approximate solutions x_n ∈H_n are convergent for n → ∞, iff there exists a constant ν > 0 such that where ν is independent of n. If the condition (3) is satisfied, and A~(-1) exists, then for f ∈H, Eq. (1) has the unique solution to which the Galerkin approximate solution converges. Moreover, for error estimate we have p. If there exists a positive constant ν > 0 such that then the Galerkin approximate solution covenges to the exact solution.
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  AUTOMORPHISMS AND HOMOMORPHISMS OF THE DE BRUIJN-GOOD GRAPH

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 170-177. https://doi.org/10.12386/A1979sxxb0014

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  The de Bruijn-Good graph of degree n G_n is a directed graph, with {(a_1, a_2,…, a_n)|a_i=0 or 1} as its vertex set and with {(a_1, a_2,…, a_n) → (a_2,a_3,…, a_(n+1))|a_i = 0 or 1} as its arc set, where (a_1,a_2,…, a_n) → (a_2, a_3,…, a_(n+1)) denotes an are starting at (a_1, a_2,…, a_n) and ending with (a_2, a_3,…, a_(n+1)). In this paper the following results are proved:1. G_n has only two graph automorptiisms, i.e. the identity automorphisms I and the dual automorphism D.2. There are only six two-to-one homomorphisms from G_n onto G_(n-1), denoted by where D is the dual automorphism of G_(n-1) and
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  ON A THEOREM OF BEURLING AND AHLFORS

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 178-184. https://doi.org/10.12386/A1979sxxb0015

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  A continuous strictly increasing function μ mapping the real line onto itself is called ρ-quasisymmetric, 1 ≤ ρ < ∞, if 1/ρ ≤ μ(x + t) - μ (x)/μ(x)-μ(x-t)≤ ρ(1) for all x and all t ≠ 0. Beurling and Ahlfors first proved that any given ρ-quasisymmetrie funetion μ has an extension to a K-quasiconformal homeomorphismfrom the upper half-plane onto itself with K≤ρ~2.(2) Reed then improved the inequality (2) as follows: K < 8ρ.(3)In the present paper we give a detailed exposition of the computation for the inequality (2) (for [2], such an exposition may be concerned with by Reed because in [3] Lehto and Virtanen obtained K ≤ 8ρ(ρ + 1)~2 only) and prove the following result :
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  ON THE BOUND FOR DOMINANCE RADIO OF NONNEGATIVE MATRICES

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 185-194. https://doi.org/10.12386/A1979sxxb0016

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  Let A be an n×n complex matrix, and let The radio d is called dominance ratio of matrix A. In 1964, A. M. Ostrowski [3] obtained a bound for dominance radio of positive matrices. In 1974, Ostrowski [4] also obtained a bound for dominance radio of irreducible nonnegative matrices.In this paper we have obtained a bound better than that of Ostrowski [4]. Our result is further extended to some other types of matrices such as irreducible nonnegative matrices, namely, sub-nonnegative matrices defined in our paper [1]. In addition, we also give a simple methodto determine primitive character γ(A) (in the sense that for A ≥ 0, γ(A) = min{K|A~k > 0}, where k is a positive integer). Thus better bounds for dominance radio may be obtained.
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  THE EXISTENCE OF NONMITOTIC a-RECURSIVELY ENUMERABLE SETS

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 195-203. https://doi.org/10.12386/A1979sxxb0017

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  By the use of the a-finite injury priority method of shore, we lift the theorem on the existence of nonmitotic, recursively enumerable sets of the ordinary recursion theory to the a-reeursion theory.Theorem: There exists an a-regular nonmitotie a-recursively enumerable set.
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  A THEORY OF RINGS THAT ARE ISOMORPHIC TO THE COMPLETE RINGS OF LINEAR TRANSFORMATIONS(Ⅰ)

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 204-218. https://doi.org/10.12386/A1979sxxb0018

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  An abstract ring which is isomorphic to the ring of all linear transformations of a vector space has been studied by wolfson. In this paper we follow his work and obtain a main theorem. Before formulating we first introduce the following.Definition: Let M be a (F, R)-module, i.e. M is a left F, right R-module, and let M' be a (K, R)-module, where F, K, R are rings. We say that M and M' are (ψ, I)isomorphic if there exists a mapping S satisfying the following conditions:(i) S is an isomorphism of (M, +) onto (M', +).(ii) There exists an isomorphism ψ of F onto K such that for all x ∈M, r ∈R we hav.e (fx)S = f~ψ(xS), (fxr)S = f~ψ(xS)r.Theorem: Let R be an abstract ring. Then R is isomorphic to the ring of linear transformations of a left vector space A over a division ring F if and only if the following conditions are satisfied:(i) R has a socle with and every non-zero ideal of R contains.(ii) Suppose that S_1⊕S_2,S_1 are minimal left ideals Of R, then(iii) R has an identity.Moreover, if R satisfies the above three conditions then every minimal right ideal A'= eR of R can be expressed as a left vector space over the division ring K and R is also the complete ring of K-linear transformations of A'. Furthermore there exists a (ψ, I)-isomorphism of A as (F, R)-module onto A' as (K, R)-module, and the rank of A is equal to that of Thus the left vector space of R except semi-linear transformation is uniquely determined by the soele.
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  SOME PROBLEMS ABOUT THE APPROXIMATE SOLUTION FOR OPERATOR EQUATIONS

  Acta Mathematica Sinica, Chinese Series. 1979, 22(2): 219-230. https://doi.org/10.12386/A1979sxxb0019

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  In §1 we study the acceleration of nonlinear projection method. For an isolated solution u of eq. we show that the piecewise linear finite ele ment method has a unique solution u~h in a neighborhood of u and ‖u~h- u‖1 =O(h). Subsequently, u~h is used to obtain a better approximation u:‖u-u‖1=O(h~2), where u is the solution of Poisson eq.In §2 we consider the non-negative integal operator A(see (19)) and two matrices A_n and A_n (see (20) (21)). The monotonity of spectral radius p(A_n) ≤ρ(A_(n+1))≤ ρ(A) ≤ρ(A_(n+1)) ≤ρ(A_n) has been obtained. An application of this principle to the continuous energy transport eq. (25) gives the same inequality, but in which ρ(A_n) and ρ(A_n) represent the spectralradius of two systems of multi-group eq. (26) and (27).