中国科学院数学与系统科学研究院期刊网

15 July 1954, Volume 4 Issue 3
    

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  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 263-278. https://doi.org/10.12386/A1954sxxb0016
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    Let the integrals exist in the sense of Denjoy. If the function f(x) is not summable in the interval (0,2π), then the relations a_n=o(1), b_n =o(1)(1) are in general untrue. On the other hand, from (1) we are unable to conclude that the trigonometrical series corresponds to some function f(x) which is integrable in (0,2π) in the sense of Lebesgue. Indeed Titchmarsh has, appealing only to elementary analysis, constructed an odd function f(t) possessing the following properties. exists for n = 1, 2, 3,…. γ) ∑|b_n|~(2+ε)converges for every ε>0.Titchmarsh's function f(t) is however neither of elementary character nor of absolutely continuous in the interval (ε,π), where 0<ε<π. In the present paper, Ⅰ establish the following result:Corresponding to a number δ<1/2, there exists an elementary function f(x) such that (i) f(x) is absolutely continuous in every interval (ε, 2π), where 0<ε<2π.In fact, take a number α>2δ/1-2δ,the function f(x)=1/x cos 1/x~αsatisfies all these conditions; in particular, the condition (iv) is a reduced form of the following relations: These relations are contained in the following inequality: where α>0,A>0, b>0, n>0, and the constant C (α, A) depends only upon α and A.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 279-294. https://doi.org/10.12386/A1954sxxb0017
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    Der Verfasser macht einen Versuch die bekannte, ebene Potentialtheorie elementaren Teils in die auf einer beliebigen geschlossenen, konformen Riemannschen Flache einzuordnen. Die Rolle, die fruher das Logarithmus spielt, spielt nun das Abelsche Potential dritter Gattung. Auf Grund der hierin eingefuhrten Riemannschen Metrik verallgemeinert man den ersten Nevanlinnaschen Hauptsatz u.s.w. Dabei bleiben einige Fragen ungelost.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 295-299. https://doi.org/10.12386/A1954sxxb0018
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    In dieser Note, beweist der Verfasser die Existenz einer eindeutigen, analytischen Funktion auf einem beliebigen Teilgebiet einer geschlossenen, konformen Riemannschen Flache. Die Beweismethode ist von der Koebeschen verschieden.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 301-304. https://doi.org/10.12386/A1954sxxb0019
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    Ein Satz von Myberg veranlasst die vorliegende Note. Wir gewinnen:Die Randstiickmenge einer nichtfortsetzbaren Riemannschen Flache ist vom absoluten harmonischen Nullmass.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 305-316. https://doi.org/10.12386/A1954sxxb0020
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    It is the purpose of this paper to investigate the asymptotic behavior of a class of integrals of the following form in which f(u)=f(u_1,…, u_n)>0 is assumed to attain an effective absolute maxima at a certain boundary point ξ= (ξ_1,…,ξ_n) of D,D being a simply connected n dimensional closed domain in Euclidean n space.In our investigation, three main types of boundary points ξ's have been distinguished, namely,Type Ⅰ. Let D be an n-dimensional closed domain. We shall call ξ = (ξ_1,…,ξ_n,) a boundary point of type Ⅰ, if it is an ordinary point of the boundary surface S of D and if S has a continuously turning tangent plane near ξ.Type Ⅱ.We shall call ξ = (ξ_1,…,ξ_n) a boundary point of type Ⅱ, if the following conditions are satisfied: (i)ξis a point belonging to the intersection of two (n-1) surfaces S_1 and S_2, where S_1 and S_2 constitutes a part of the boundary surface of D, (ii) ξ is an ordinary point for both S_1 and S_2, where S_1, S_2 have continuously lurning tangent planes near ξ, (iii) the intersection angle θ between tangent planes of S_1 and S_2 at ξ, as measured from the inside of D, is greater than 0 and less than πType Ⅲ. If in the above definition concerning a boundary point of type Ⅱ, the intersection angle θ defined by condition (iii) is equal to zero (i.e. S_1, S_2 have the same tangent plane at ξ), then ξ is called a boundary point of type Ⅲ.In two earlier prapers [1] and [2], the author has studied the case where the function f(u) attains an absolute maxima at a boundary point of type Ⅰ. We are therefore concerned ourselves with the cases for boundary points of types Ⅱ and Ⅲ in the present investigation. Clearly, without loss of generality we may assume a typical boundary point ξ to be origin of the (u)-system, viz.ξ=O=(0,…,0). Moreover, for our need, we assume that arc cos (…) only takes a value between 0 and π. Then our results may be stated as follows:Theorem 1. Let φ(u) and f(u) >0 be defined on D such that1°φ(u) [f(u)]~N is absolutely integrable over D for all N≥0,2°partial derivatives f_i′(u),f_(i,k)″(u) all exist and are continuous. 3°f(u) attains an absolute maxima at a boundary point ξ=(0,…, 0) of type Ⅱ,4°where u→ξ) denotes that u approaches ξ from the inside of D,5°φ(u) is continuous at ξ=(0,…,0) with φ(ξ)≠0. Then for N→∞ we have the asymptotic formula where A =[-ψ_(ik)″(ξ)]is the Hessian matrix of-ψ(u)=-log f(u),α and βare respectively the normal vectors orthogonal to S_1 and to S_2 at ξ such that their intersection angle is equal to that between tangent planes of S_1 and S_2 at ξ, as measured from the inside of D.Clearly, in the particular case n=2, the (n-1) surfaces S_1, S_2 of the above theorem reduce to two boundary curves C_1, C_2 of the plane region D.Theorem 2. Let f(u_1, u_2)>0, φ(u_1, u_2) and D satisfy all the conditions of the above theorem (with n=2) except condition 3° which is replaced by3 f(u_1, u_2) has an absolute maxima at a boundary point ξ=(0, 0) of type Ⅲ. Then for N→∞ we have the formulas according as the positive directions of two principal normals of C_1, C_2 at (0, 0) are just opposite or the same, where R_1, R_2 are respectively the radii of curvatures of C_1, C_2 at (0, 0) andα=-(f(0,0)~(-1){f_(11)"(0,0)cos~2θ+f_12″(0,0)sin2θ+f_(22)″(0,0)sin~2θ},θ being the angle made by the tangent of C_1 (or C_2) at (0, 0) with the u_1-axis.Proof of the first result consists of applying some known results of [1], using some simple matrix algebra and employing the n-dimensional spherical coordinates for the transformation and asymptotic estimation of an n-fold integral. The second result is proved by a method of so-called circle arc-wise integration in which the Frenet-Serret formula is used for an estimation of some arc length.The author has not yet found an explicit asymptotic formula for J(N) with regard to the general case of type Ⅲ.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 317-322. https://doi.org/10.12386/A1954sxxb0021
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    The following theorem is proved:Every bounded non-continuable domain with constant curvature can be mapped pseudoconformally into a unit sphere.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 323-346. https://doi.org/10.12386/A1954sxxb0022
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    We determine completely the Pontrjagin squares in the Grassmannian manifold. As a consequence, we prove that the Pontrjagin classes, reduced mod 4, and hence also mod 12, when combined with the preceding one in this series of papers, of a closed differentiable manifold are topological invariants of that manifold.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 347-357. https://doi.org/10.12386/A1954sxxb0023
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  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 359-364. https://doi.org/10.12386/A1954sxxb0024
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    This kind of Markofl process can be defined by its transition probabilities P_(ij)(t) satisfying conditions (1), (2), (3) and (4). In this note we have proved that without further assumptions, all the transition probabilities P_(ij)(t) possess right hand derivatives for t≥0, furthermore each of these derivatives is bounded from above.
  • Acta Mathematica Sinica, Chinese Series. 1954, 4(3): 365-379. https://doi.org/10.12386/A1954sxxb0025
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    Let X be an arcwise connected topological space and α, β, γ be elements of the homotopy groups π_ρ(X, x_o), π_q(X, x_o), π_r(X,x_o) respectively, where r ≥q≥p≥2. By [β, γ] we mean the Whitehead product and [α,[β, γ]] the repeated Whitehead product. The use of the method which the author has developed in [1] leads to the following consequence: (-1)~(r(p+1)) [α, [β, γ] ] + (-1)~(p(q+1)) [β, [γ,α]]+(-1)~(q(r+1)) [γ, [α,β] ] = 0,namely, the Jacobi identity.