中国科学院数学与系统科学研究院期刊网

15 October 1956, Volume 6 Issue 4
    

  • Select all
    |
    论文
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 515-541. https://doi.org/10.12386/A1956sxxb0041
    Abstract ( )   Knowledge map   Save
    Let d_k(n) be the number of expressions of n as k factors, and let D_k(x) R_k(x) = (a_(k,o) + a_(k.1) In x + …+ a_(k,k-1 In~(k-1) x) x (x > 0) be the residue of ξ~k(s) -x~s/s at s = 1. Let △_k(X) = D_k(x) - R_k(x). Define the average order of △_k(x)byLet σ_k be the lower bound of values of σ for which holds for every positive ε.It was proved by Titchmarsh in [1] that "β_k=k-1/2k if and only if σ_k≤k+1"/2k.The following theorem are proved in this paper.Theorem. If σ_k≤k+1/2k,then the relation holds for every positive ε.This theorem is better than the sufficient part of Titchmarsh's result mentioned above.The proof of the theorem is based on generalized identity given by the author in [2].
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 542-547. https://doi.org/10.12386/A1956sxxb0042
    Abstract ( )   Knowledge map   Save
    Only commutative rings with unity and no divisor of zero are considered. Such a ring is called a primal ring if every ideal of it is primal in the sense of L. Fuchs. The author showed that if R is a primal ring, then (1) The prime ideals of R are simply ordered; (2) R has at most one prime principal ideal (in addition to R itself); (3) the set of all non-units of R is a prime ideal P of R, and in case R has a prime principal ideal, this ideal is P; (4) the ideal P is the adjoint prime ideal of every principal ideal. The following theorems have also been proved.Theorem 1. R is a valuation ring, if and only if R is primal and every non-principal ideal cannot be generated by a finite number of elements.Theorem 2. R is a subring of a valuation ring Q of the quotient field K of R such that the set of all non-units of R coincides with the maximum prime ideal of Q if and only if R is primal such that whenever a, b ∈ R, a + b and b + a, the quotients (a) b~(-1) and (b) a~(-1) always equal to the maximum prime ideal of R.Theorem 3. An integral domain R is primal, if and only if the set of all prime ideals of R is simply ordered by inclusion.Theorem 4. A unique factorization domain R is primal, if and only if it is a valuation ring of its quotient field with respect to some discrete archimedean valuation.
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 548-564. https://doi.org/10.12386/A1956sxxb0043
    Abstract ( )   Knowledge map   Save
    According to L. Kalmar, the elementary functions are the functions obtainable from the initial functions x+y, |x-y|, x.y, [x/y] by means of the operations stitution. If only the narrowed summation and substitution are available, the functions obtained are called restricted elementary functions. We may show that the latter form a proper subclass of the class of elementary functions (this remains true even if the operation ai is also available).The main result of the present paper is to improve the well known Kleene's theorem as follows:Every gefieral reeursive function may be expressed in the form A(ey[B(x_1,x_2,…,x_r, y)=0]), where A and B are restricted elementary functions.In the additional note we get a still stronger result. Let us call basical functions those functions which are obtainable from the initial functions x+1, x⊕y(=sg x+sg y), x-y, x·sg y,by means of the operations ∑ and substitution alone. Evidently the basical functions form a proper subclass of the class of the restricted elementary functions. We show that the functions A and B mentioned above may be required to be basieal ones.
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 565-582. https://doi.org/10.12386/A1956sxxb0044
    Abstract ( )   Knowledge map   Save
    Assuming the truth of grand Riemann hypothesis, that is, assuming the real parts of all zeros of all Dirichlet L-functions L(s,X are ≤1/2. We have the following three theorems:Theorem 1. Every large even integer is a sum of a prime and a product of at most 4 primes.Theorem 2. There are infinitely many primes p, such that (p+2) is a product of at most 4 primes.Theorem 3. Let Z_2(N) denote the number of twin primes ≤ N. Then Z_2(N) ≤ (8+ε)(1-1/(P-1)~2)N/log~2N+O(N/log~3N), where ε is any given positive number and the constant implied by the symbol O depends on ε only.
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 583-597. https://doi.org/10.12386/A1956sxxb0045
    Abstract ( )   Knowledge map   Save
    Let G be a domain in the z-plane and a_1, a_2,…, a_n be n distinct points of G. Let G_1, G_2,…, G_n be a system of sub-domains of G, such that a_i ∈ G_i (i=1, 2,…, n) and that G_i G_i=0(i≠j). We denote the mapping radius of G_i with respect to a_i by R(a_i, G_i), Let x_1, x_2,…,x_n be a set of positive numbers and write Our problem is to investigate the properties of extremal domains G_1, G_2,…, G_n for the quantity I_G(a_1,…,a_n; x_1,…x_n). In the case of G being a circle or the whole plane, the problem was solved by Kolbina and others, for n =2. The aim of present note is to suggest a method for attacking the problem in general.The expression P(z)dz~2 is said to be a positive quadratic differential in the domain G, if it is positive or zero on the boundary of G, the function P(z) is to be assumed meromorphic on the closure G of G. For simplicity, we assume that the boundary of G consists only of analytic curves. Our main theorem isTheorem 1. Let g_1, g_2,…,g_n be a sy stem of non-overlapplng domains in G such that a_ν∈ g_ν, ν=1, 2,…, n. The necessary and sufficient condition for the truth of the equation is that the boundary curves of g_1, g_2, …, g_n, are all the trajectories with end points being the zeros of a positive quadratic differential P(z)dz~2 in G. Where P(z)dz~2 satisfies the following conditions:(1) P(z) is regular in G with the exception of the double poles a_1, a_2, …, a_n, whereat we have the Laurent expension(2) P(z) has at least one zero in each closed boundary curve of G, unless n=1 and G is simply connected,(3) the equation holds good for any pair of zeros z', z" of P(z). Under these conditions, one branch of the function is regular and schlicht in g_ν, and maps g_ν onto unit circle, and transforms the point a_ν into ζ=0. For the case that G is the circle |z| < R, we have the following Theorem 2. Let a_1, a_2,…, a_n be n distinct points in the circle |z| < R. Let b_1, b_2,…, b_(2(n-1)) be 2(n-1) points in the closed circle |z| ≤ R. Let the function satisfy the following conditions(1)there is at least one b_ν_o such that |b_ν_o| = R,Then the domains g_1 g_2,…, g_n having the boundary curves form a system of extremal domains for the quantity I_G(a_1, a_2,…, a_n; x_1, x_2,…, x_n), and the function maps g_ν onto |ζ|<1.Similar results can be obtained for the case G being a plane.We conclude this note by proving the following.Theorem 3. The necessary and suffident condition that the system g_1, g_2, …, g_n become extremal domains for the quantity I_G(a_1,…,a_n; x_1,…,x_n) is that the boundaries of these domains are identical with the level curves H(z)=0 of the function H(z), where H(z) is harmonic in G, two-valued, and satisfies the following conditions:(1) the modulus |H(z)| is single-valued and vanishes on the boundary of G,(2) it possesses only a finite number of branch points of order two in the closure G of G, and the pair of equations H(a)= 0 and lim |grad H(z)| = 0 hold at any branch point a,(3) On each dosed boundary curve of G, there exists at least one branch point of H(z),(4) H(z) is harmonic on the closure G of G with the exception of branch points and logrithmic poles a_1, a_2 ,…, a_n , with the residues ν=1, 2, …, n respectively.Under these conditions we haveAs a corollary of this theorem we have the inequality: where H(z, ζ) denotes the regular part of the Green function G(z,ζ) of G at z=ζ, i.e. H(z, ζ) = log |z-ζ1+ G(z, ζ).
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 598-618. https://doi.org/10.12386/A1956sxxb0046
    Abstract ( )   Knowledge map   Save
    Let G be a doubly-connected domain in the z-plane. If, G can be represented conformally on a circular ring 1 <|ζ|0 and z=Φ_α(ζ) maps the ring 1<|ζ|R onto the unit circle|z|<1 with the slit 0≤z≤α.Theorem 4. Let D be a n-ply-connected domain in the ζ-plane, with the boundary continua γ_1, γ_2,…, γ_n.Let(ζ, z_o), z_o ∈D, be a function regular and univalent in D suchthat; (i) (z_o, z_o) =0, (ii) |(ζ, z_o)| = 1 for ζ∈γ_ν, (iii) arg (ζ, z_o) = const, forζ∈γ_μ,μ≠ν. Then, for any function regular and univalent in D,f(z_o)=0, |f(ζ)|≥1 onγ_ν, the inequalityholds true.
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 619-630. https://doi.org/10.12386/A1956sxxb0047
    Abstract ( )   Knowledge map   Save
    T. Y. Thomas has shown that in general the mean curvature of a surface yields an algebraic determination of its second fundamental form b_(ii)dx~idx~i and has given the explicit ex pression of b_(ii) in terms of the metric tensor g_(ii), the mean curvature H and their derivatives.In this paper we consider the same problem for the surface V_2 in 3 dimensional Riemannian space V_3.At first, we consider the isometric correspondence which preserves the mean curvature between pairs of surfaces in V_3. By solving a system of exterior differential equations we find that the degree of freedom of such pairs of surfaces is 4 functions of single argmefit. Consequently, in general V_2 is determined by the mean curvature and the metric tensor.In order to get the expression of b_(ii), we choose such a coordinate by which the equation of V_2 is y~3=0 and the normal of the surface is ξ~α(0,0,1) then the equations of Peterson-Gauss Codazzi becomes R_(1212)=(b_(11)b_(22)-b_(12)~2+R_(1212)b_(ij,k)-b_(ik,i)=R_(iεik) (i, i, k = 1, 2)From the condition of inttgrability of b_(ij,k) and using the similar but much more complicated calculation as T. Y. Thomas we get A 2 b_(12) + B (b_(11)-b_(22)) = C, A (b_(22)-b_(11)) + B (2 b_(12)) = D. where A,B,C,D stand for the expression in (17), (18), (19), (20). Here also we notice that when B=C=D=0 it corresponds to the work of T. Y. Thomas of V_2 E_3.We discuss the problem at any arbitrary paint P and choose the system of coordinate to be orthogonal at that point.Let △=4b_(12)~2+(b_(11)-b_(22))~2=4(H~2-K). If △ = 0, we get the expression of b_(ii) in the following form b_(ii)= H g_(ii). Under the supposition of △≠0, we get a b_(12)~2 + β (b_(11)-b_(22)) = γ where α,β, γ, stand for the expression in (24).When α≠0, b_(ii) are expressed by (26),When α=0, there are three cases(i) βγ≠0(ii) γ = 0 β≠ 0(iii) β= 0, then γ= 0.In the case (i) and (ii), we can also express b_(ii) by the mean curvature and the coefficients of first fundamental form and their derivatives. In the case (iii), we have H~2-K=const. generally. All the surfaces which admit continuous deformation which preserves mean curvature are included in this case.We also obtain the equations which must be satisfied by the mean curvature.
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 631-637. https://doi.org/10.12386/A1956sxxb0048
    Abstract ( )   Knowledge map   Save
    Let S~(ni) (i = 1,…, r) be spheres of which the topological product S~(n1) ×…× S~(nr) has a unique top dimensional cell, e~(n1+…+nr). By Y we mean the topological space obtained from S~(n1)×…× S~(nr) by removing e~(n1+…+nr). Let x_i~o, e~o and x be reference points of S~(ni), Y and X respectively. If f: Y, e~o→X, x is a continuous map and g: E~(n1+…+nr-1), E~(n1+…+nr-1)→Y, e~o represents the homotopy boundary of the element of ∏_(n1+…+nr)(S~(n1)×…× S~(nr), Y) determined by the characteristic map of e~(n1+…+nr), then fg: E~(n1+…+nr-1), E~(n1+…+nr-1)→X, x gives us an element of ∏_(n1+…+nr-1) (X, x) called the (r-1) secondary product of the elements α_i(i = 1,…, r) represented by the maps f|S~(ni): S~(ni),x_i~o→X, x. All the continuous maps of (r-1)-dimensional spheres S~(r-1) into X, carrying the reference point of S~(r-1) to x, constitute a topological space Ω_(r-1). Write E~(n1+…+nr-1)= E~(n1+…+nr-r).E~(r-1). The map fg determines a map b being the map S~(r-1)→x, such that the map satisfieswhere x ∈ E~(nx+…+nr-r) and τ ∈ E~(r-1). Evidently the map (fg) determines a homology class ξ in H_(n1+…+nr-r) (Ω_(r-1)).On the other hand we take the product of spheres, S~(n1-1)×…×S~(nr-1), and define a map Φ: S~(n1-1)×…×S~(nr-1)→Ω_(r-1) as follows: If x_i ∈ S~(ni-1), which is considered as the equator of S~(ni), then there is a unique plane, π, passing through x_i, x_i~o and perpendicular to the equator. The intersection of π and S~(n)i is a circle, namely, S_(xi)~1. The product may be considered as the image of a continuous map in natural way. A map of the sphere E~r = (E_1~1×…×E_r~1) into X is defined by if t_i and where then is continuous and We define The map Φ has a corresponding homology class (Ω_(r-1)). The purpose of this note is to prove
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 638-650. https://doi.org/10.12386/A1956sxxb0049
    Abstract ( )   Knowledge map   Save
    On se propose de resoudre, par la methode de col, une equation non-lineaire F(x) = 0. (1) consideree dans un espace d'Hilbert. On suppose a ce propos que l'operateur F(x) est du type potentiel au sens de M. M. Vainberg et que la derivee F'(x) de F(x) au sens de M. Frechet est un operateur lineaire borne symetrique positif defini. Dans ce cas, la methode de col conduit au procede x_(n+1) = x_n- ε_n F(x_n), ou le nombre ε_n doit etre convenablement choisi. On demontre que ce procede converge vers la solution exacte de (1), et que la convergence sera au moins aussi vite que celle d'une progression geometrique, comme dans les cas lineaires. En changeant de proche en proche la metrique dans l'espace considere, on peut obtenir le procede de Newton comme un cas particulier de la methode de col, mais sous les conditions enumerees ci-dessus qui sont assez restrictives. Quelques indications sont donnees sur l'application de la methode proposee ici pour la resolutiondes problemes aux limites de certains types d'equations differentielles ordinaireset d'equations aux derivees partielles, et aussi pour la resolution de certains types d'equations integrales non lineaires. Cette methode a ete consideree par divers auteurs dans le cas particulier d'espace a dimensions finies.
  • Acta Mathematica Sinica, Chinese Series. 1956, 6(4): 651-664. https://doi.org/10.12386/A1956sxxb0050
    Abstract ( )   Knowledge map   Save
    Let f(z) be a regular function in the unit circle |z|<1 such that f(0)= 0, f′(0)= 1,f(-z) = - f(z) and that The signs of equality holds only for the functions, f(z)respectively, where "a". is a certain number in the inzerval [0, 1].In a like manner we also determine the upper bound as well as the lower bound of |f(z)|.