15 January 1960, Volume 10 Issue 1

    

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  TOPOLOGICAL STRUCTURE OF INTEGRAL CURVES OF THE DIFFERENTIAL EQUATION y′= a_0x~n+a_1x~(n-1)y+…+a_ny~n/b_ox~n+b_1x~(n-1)y+…+b_ny~n

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 1-21. https://doi.org/10.12386/A1960sxxb0001

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  Let n_1, n_2,…, n_m be m positive integers, g the g.c.d, of n_1,…,n_m. δ the positive factor of g (including 1, g). P_1, P_2,…,p_λ the positive prime factors of g. 1 < P_1 <
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  ON THE MODULO 2 IMBEDDING CLASS OF TRIANGULABLE COMPACT MANIFOLDS

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 22-32. https://doi.org/10.12386/A1960sxxb0002

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  For the realization problem of complexes or more general spaces in euclidean spaces, Whitney & Thorn have obtained the following results:Theorem. (Whitney) The necessary conditions for an n-dimensional compact differentiable manifold M~n to be differentiably realizable in R~N are W~k(M~n) = 0, k ≥ N -n. (1)Theorem. (Thom) The necessary conditions for a locally contractible compact Hausdorff space X with a countable basis to be topologically realizable in R~N are S_m~kH~r(X; I_2) = 0,2k+r≥N. (2)Besides, in the case of Hausdorff space, Wu Wen-tsun has introduced a system of cohomology invariants——imbedding classes Φ~k(X), and proved the followingTheorem. If a Hausdorff space X is topologically realizable in R~N, then Φ~k(x) = 0. He has also proved:Theorem. If M~n is an n-dimensional triangulable compact differentiable manifold and Theorem. If M~n is an n-dimensional triangulable compact differentiable manifold and (nod.2)],theTheorem. If M~n is an n-dimensional triangulable compact differentiable manifold,and W~k(M~n) = 0, k ≥N-n. ThenHence, it is natural to us to ask:Ⅰ. Is there any n-dimensional compact differentiable manifold which satisfies (2) but not (1)?Ⅱ. Is there any n-dimensional compact differentiable manifold M~n which satisfies (1) but ρ_2Φ~N(M~n) ≠0?In this paper I proveTheorem. Let M~n be an n-dimensional (n > 0) triangulable compact manifold. then ρ_2Φ~N(M~n) = 0 when and only When S_m~kH~r(M~n;I_2) = 0, 2k + r ≥ N.Hence we obtainTheorem. Let M~n began n-dimensional (n > 0) triangulable compact differentiable manifold. Then the following conditions are equivalent:(i) ρ2Φ~N(M~n) = 0;(ii) W~k(M~n) = 0, k≥N- n;(iii) S~kmH~r(M~n; I_2) = 0, 2k+r≥N.According to this theorem, we know that no manifold exists as in I or II discribed.I obtain the relations between imbedding classes and duality for manifold. According to this, and the method given by Wu [4, Theorem 4], I give a proof of the above theorem.From this theorem, I obtain the followingTheorem. Let M~n be an n-dimensional triangulable compact manifold. Then(i) Φ~(2n-1)(M~n) = 0, when n≠2~α(α≥0);(ii) Φ~(2n-1)(M~n) = 0, when n = 2~a(a>0), and M~n orientable.
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  SUMMABILITY OF THE SERIES OF ORTHOGONAL POLYNOMIALS

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 33-40. https://doi.org/10.12386/A1960sxxb0003

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  Let {p_n(x)}be a system of polynomials normalized and orthogonal in (a, b) with the weight τ(x), p_n(x) being of the degree n. The chief object of this paper is to discuss the summability of the series Letdng p_n(x) = a_o + a_1x + … + a_nx~n and supposing f(x) ∈ L_p(a,b), p > 1, we writeFirstly we prove the following theorem:1) If the integra converges then(1)converges p.p.2) If the integra dt converges,the(1)is summable(C,a) p. p. for any α > 0.The proposition (1) improves a theorem of HaTaHCOH. The proof of 2) is based on Menchoff's (C, a)-summability theorem.Secondly, using a theorem of Orlicz and writing l_1(t) =|log t|, l_ν+1(t)= |log l_ν(t)|, λ_i(t, p) = l_1(t)…l_i-1(t)l_i~p(t), we prove3) If the series converges for a number p greater than unity, then (1) unconditionally converges p. p.From this theorem, we derive4) Let i and k be two natural numbers, p > 1,k = 1, 2. If the integral converges then (1) unconditionally converges p. p.This result 4) improves author's previous theorem, and is equivalent to the following proposition:5) If the series converges, then (1) unconditionally converges P.P. The foregoing theorems hold also good for the trigonometric series where F(θ)= can establish with σ'(t) > 0 (t > 0). However, in the place of (1) we then (2) converges p. p., where f(θ)≡f(θ+2π).
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  МЕТОД СУММИРОВАНИЯ В ПРОСТРАНСТВЕ БАНАХА

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 41-54. https://doi.org/10.12386/A1960sxxb0004

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  AUTOMORPHISMS OF THE PROJECTIVE UNIMODULAR GROUP

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 55-65. https://doi.org/10.12386/A1960sxxb0005

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  Let be the group of all n ×n integral matrices of determinant ±1 and let be the subgroup of consisting of all matrices with determinant +1. Let be thefactor-group of by its centrum, and let be the factor-group of by its centrum. In [1] L. K. Hun and I. Reiner determined the group of automorphisms of and (m ≥ 1). In [2] Wan Cheh-Hsian determined the group, of automorphisms of for odd n. In the present paper the author have proved the following result.Theorem. Let n be even, n ≥ 6. Then the mappings X→AXA~(-1),A∈(1)and X→AX~('-1)A~(-1), A ∈ (2)from onto itself are autormorphisms of and conversely, the automorphisms of are either of the form (1) or of the form (2).
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  RECURSIVE ALGORITHMS THEORY OF RECURSIVE ALGORITHMS Ⅰ

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 66-88. https://doi.org/10.12386/A1960sxxb0006

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  An abstract of this paper in English has been published in Science Record 4 (1960) 91-98.
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  KERNEL FUNCTIONS THEORY OF RECURSIVE ALGORITHMS Ⅱ

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 89-97. https://doi.org/10.12386/A1960sxxb0007

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  An abstract in English of this paper has been published in Science Record 4 (1960), 99-101, where it is put together with the abstract of another paper: Normal Forms of Recursive Functions.
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  NORMAL FORMS OF RECURSIVE FUNCTIONS THEORY OF RECURSIVE ALGORITHMS Ⅲ

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 98-103. https://doi.org/10.12386/A1960sxxb0008

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  An abstract in English of this paper has been published in Science Record 4 (1960), 99-101, where it is put together with the abstract of another paper: Kernel Functions.
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  ON THE EQUIVALENCE PROBLEM OF DIFFERENTIAL EQUATIONS AND DIFFERENCE-DIFFERENTIAL EQUATIONS IN THE THEORY OF STABILITY OF THE FIRST CRITICAL CASE

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 104-124. https://doi.org/10.12386/A1960sxxb0009

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  The problem is to investigate the equivalence of stability between the system of differential equations and the system of difference-differential equations where the psσs, qsos, pss and qss are given constants, and δ(t)'s may be non-negative real constants or non-negative real continuous of t. In order to this problem, we consider first the case of Ps= 0, qs=0 (s=1,2,…,n), i.e. to study the eqivalence problem of stability between the system of differential equations and the system of difference-differential equations satisfying the following conditions:(1) X(x_1,…,x_n,x), Y(X_1,…,x_n, x), X_s(x_1, x_2,…, x_n,x) and Y_s(x, x_1, …, x_n) are analytic functions of the variables x_1, …, x_n, x in the neighbourhood of the origin Of coodinates, and the orders of the terms of their expansions are not less than two;(2) All the roots of the characteristic equation D(X)≡|P_(sσ)+q_(sσ)-δ_(sσ)X|=0 (s,σ= 1, 2,…,n) satisfy the conditions(4) m_s≥m.In the article the second type of method mentioned in [1] is used.§1. The equivalence of stabilityTheorem 1. If m is an odd positive integer and g + l < 0, then there exists a positive constant △ = △ (X, Y, p_(sσ), q_(sσ), X_s, Y_s) > 0, such that the trival solution of (2)' is asymptotically stable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ ≤ △.§2. The equivalence of instabilityTheorem 2. If m is an odd positive integer and g + l > 0, then there exists a positive constant △= △(X, Y, p_(sσ), q_(sσ), X_s, Y_s) > 0, such that the trivial solution of (2)' is unstable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ≤△.Theorem 3. If m is an even positive integer and g + l ≠0, then there exists a positive constant △= △(X, Y, p_(sσ), q_(sσ), X, Y) > 0, such that the trivial solution of (2)'is unstable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ ≤ △.§3. The equivalence of stability in the singular caseTheorem 4. The trivial solution of (1)' is stable, then there exists a positive constant △ = △(X, Y, p_(sσ), q_(sσ), X_s, Y_s) > 0, such that the trivial solution of (2)' is stable, provided that the δ(t)'s satisfy the inequality 0 ≤ δ ≤ △.§4. The general caseWe now return to the case between (1) and (2). We can solve this problem on the basis of the equivalence of (1)' and (2)'. Using non-linear transformation x_s = ξ_s + u_s(t) (s = 1, 2, …,n), the system (1) will remain in the same form as the syslem (1)': where u_s(x)'s (s = 1, 2, …, n) satisfy the following system of equations:where x_s = u_s(x) (s = 1,…, n) are analytic functions of x for sufficienfly small |x|.
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  UNCONDITIONAL STABILITY OF SYSTEMS WITH TIME-LAGS

  Acta Mathematica Sinica, Chinese Series. 1960, 10(1): 125-142. https://doi.org/10.12386/A1960sxxb0010

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  The problem of unconditional stability of systems with time-lags was proposed by H. S. Tsie.Definition.Given a system of constant coefficients.with constant time-lagsIf for any τ≥ 0, the trivial solution is asymptotically stable, then the system (1) is called unconditional stable.DenoteTheorem. The necessary and sufficient conditions that (1) is unconditional stable are that (i) The real parts of roots λ of △(λ; 0) ≡|α_(si) + b_(si) - δ_(si)λ| = 0 are all positive; (ii) For any real ω and real non zero y F(y ,ω) ≠ 0. Based upon this theorem we calculate the case n = 1, 2.Case n = 1.The necessary and sufficient conditions that the system dx(t)/dt=ax(t) + bx(t- τ) is unconditional stable, are that a + b< 0 and b-a≥0.The case n = 2.Denote (i) If A_1+A_2+A_3≤0 or A_4 +A_5≤0, then (3) is not unconditional stable. Hence in the following we assume that A_1 + A_2 + A_3>0, A_4 + A_5> 0. (ii) Denote H(x) ≡ [A_1 + A_2x + A_3(2x~2 - 1)][A_4 + A_5x]~2 + + A_5[A_2 + 2A_3x][A_4 + A_5x][1 - x~2] - [A_2 + 2A_3x]~2[1 - x~2]. If H(x) ≠ 0 for |x| ≤ 1, then (3) is unconditional stable. Hence in the following we assume that H(x) = 0 possesses real roots in |x| ≤ 1. (iii) If in |x| ≤ 1, there exists at least one real root x_o of H(x)= 0, such that [1 - x_o~2] [A_2 + 2A_3x_o] [A_4 + A_5x_o] ≠ 0,then (3) is not unconditional stable.Hence in the following we assume that all real roots of H(x) = 0 in |x| ≤ 1 satisfy [1 - x~2] [A_2 + 2A_3x] [A_4 + A_5x] = 0. (iv) If in |x| ≤ 1 all real roots of H(x) = 0 satisfy A_4 + A_5x ≠ 0, then (3) is unconditional stable.Hence in the following we assume that in |x| ≤ 1 some root x_o of H(x) = 0, such that A_4 + A_5 x_o= 0. (v) If x_o=-1, then the necessary and sufficient condition that (3) is unconditional stable is that A_1- A_2+ A_3≤0.If x_o≠ -1, then the necessary and sufficient condition that (3) is unconditional stable is that A_5~2-A_4~2+4(A_1- A_3) < 0.In conclusion we obtained that the problem of unconditional stability is reduced to that of roots of algebraic equations.