15 October 1955, Volume 5 Issue 4

    

• Select all
  |

论文
• Select
  REPRESENTATION OF ORDERED ABELIAN GROUPS AND ORDERED RINGS OF FINITE DEGREE

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 425-432. https://doi.org/10.12386/A1955sxxb0032

  Abstract ( )   Knowledge map   Save

  The present paper originates from an attempt to illustrate the usefulness of various ordered rings of n-dimensional rela vectors in representing abstract ordered rings. (The former rings were briefly discussed for the case n=2 in a previous noteIn an ordered abelian group G (with the group operation denoted by +), let a～b denote: "there exist positive integers h, k such that h|a|≥|b| and k|b|≥|a|."Let a<
• Select
  A PROBLEM ON THE k-ADIC REPRESENTATION OF POSITIVE INTEGERS

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 433-438. https://doi.org/10.12386/A1955sxxb0033

  Abstract ( )   Knowledge map   Save

  Let k≥1 be a fixed integer, then any positive integer x can be uniquely represented by the following form x = a_1 k~(n1) + a_2 k~(n2) + … + a_1 k~(n1), where n_1> n_2 > … > n_t ≥ 0 are integers, and a_1, …, a_t are also positive integers not greater than k-1. Define a(x)Theorem 1. For any k≥2, we have Moreover, the result is the best possible.Let m be a fixed integer, then the equation a(y)=m has infinite many solutions. Let B_m(x) be the number of solutions not greater than x, we haveTheorem 2.
• Select
  ON THE MODULI OF SOME CLASSES OF ANALYTIC FUNCTIONS

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 439-454. https://doi.org/10.12386/A1955sxxb0034

  Abstract ( )   Knowledge map   Save

  In this note we consider two classes C and L of regular functions on the unit circle. Each function f(ζ) of the class C satisfies the condition f(ζ_1)f(ζ_2) ≠1, |ζ_1|<1, |ζ_2|<1. And if f(ζ) ∈ L, then the condition f(ζ_1) f(ζ_2)≠1 with |ζ_1|<1,|ζ_2|<1 holds. We prove with the extremal functionsFor the class L, we prove the equality
• Select
  SUR L’UNICIT DU PROBLME DE TRICOMI DE L’QUATION DE CHAPLYGIN

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 455-461. https://doi.org/10.12386/A1955sxxb0035

  Abstract ( )   Knowledge map   Save

  Dans cette note, nous avons demontre: Si yo (yo < 0) est le zero reel (d'ordre impair) le plus grand de F=2(K/K~′)~′+ 1, a toute droite ι~'parellele a l'axe des y, on peut toujours trouver un nombre positif ε (qui ne depend que de la distance de A a ι), de sorte que ι'unicite du Probleme de Tricomi: Γ_1 etant un arc caracteristique dans le domaine hyperbolique partant de A(x_o, 0) jusqu'a C(( X_o+ X_o)/2 , yo-ε et Γ_2 est l'autre caracteristique de C qui rencontrera l'axe des x au point B(x_o, 0), σ est unecourbe de Jordan dans le domaine elliptique a droite de 1, soit assuree.Nous avons aussi etudie le cas ou F a un nombre fini de zeros reels (ou plus general encore le cas ou le plus grand des zeros de F n'en est pas le point limite). Nous sommes arrives de demontrer: dans ce cas, K a necessairement un point singulier entre les deux premiers zeros de F.
• Select
  INEQUALITIES INVOLVING DETERMINANTS

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 463-470. https://doi.org/10.12386/A1955sxxb0036

  Abstract ( )   Knowledge map   Save

  In the study of the theory of functions of several complex variables, we discovered the following pure algebraic inequality:If I-ZZ~′>0 and I-WW~′>0, then d(I-zz~′) d(I-WW~′)≤| d(I-ZW~′)|~2.(1) We use capital Latin letters to denote n-rowed matrices with complex elements, and use Z~' to denote the transposed and conjugate complex matrix of Z. If H is Hermitian, we use H>0 to denote that H is positive definite and H≥0 to denote that H is positive semi-definite. We use also d(Z) to denote the determinant of Z. Since (I-ZW~') (I-WW~')~(-1) (I-ZW~')~' - (I-ZZ~') = (Z-W) (I-W~' W)~(-1) (Z-W)~', we deduce that |d(I-ZW~') |~2≥d(I-ZZ~') d(I-WW~')- |d(Z-W)|~2, and consequently, we have (1). The inequality is also generalized to the following more general form:Let X_1, …, X_m be m n-rowed matrix. Let ρ be a positive number. If I-X_i X_i~'>0 for 1 ≤ i ≤ m, then the Hermitian matrix is positive semi-definite.The proof of this result is different from that of (1), it requires some lemmas related to the representation theory of linear group. It seems to be interesting to find a pure algebraic proof of it.
• Select
  ON THE ISOMORPHIC TRANSFORMATIONS OF MINIMAL HYPERSURFACES IN A FINSLER SPACE

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 471-488. https://doi.org/10.12386/A1955sxxb0037

  Abstract ( )   Knowledge map   Save

  In a recent paper W. Barthel has investigated the geometry in a Finsler space with new Euclidean connections. The coefficients of the affine connection F_(ikh) there established are different from those utilized by E. Cartan and they obey a new postulate, namely,In such a Finsler space so called "gefaserte" Finsler space we can define a minimal hypersurface by setting the extremal mean curvature of the hypersurface, M, to zero:Consider the infinitesimal deformation x~i = x~i + ξ~i(x) δt, which carries on the point (x) into the point (x) infinitely near (x), δt being an infinitesimal. If a minimal hypersurface S is deformed to a nearby minimal hypersurface S, then we shall call ξ~i(x) the extremal deviation of the isomorphic transformation of the minimal hypersurface S. Just as in a regular Cartan space based on the notion of area we propose to solve the following problem:How depends the deviation of a minimal hypersurface in a Finsler-Barthd space upon the curvature of the space, when the minimal hypersurface is subjected to an isomorphic transformation?The present paper deals with the calculation of the extremal mean curvature M, which corresponds to the formula of the author concerning the extremal deviation in a geometry based on the notion of area. The result runs: δM/δt=E-MG, whence the equation of deviation of a minimal hypersurface turns out as E=O.In rewriting these quantities, especially E, in terms of fundamental tensors, curvature tensors and allied affinors of the space and the minimal hypersurface we arrive at an invariant, analogous to the invariant U of Barthel in the normal form of the second variation of an (n-1)-ple surface-integral.It should be noted that depends upon both the curvature-tensor R_(ρoσo) and the affinor G~(ρσ)≡g~(ρσ) + A~σ A~σ + x_k~ρ A~k ‖~σ.Here we can also generalize the result to the general case where the extermal deviation ξ~i contains not only the position of the point in space, but also the directioncoefficients of the hypersurface. The investigation of such transformations will lead to the normal deviation of a minimal hypersurface.
• Select
  PSEUDOKONVERGENZ UND DIE PERFEKTHEIT BEWERTETER KRPER

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 489-495. https://doi.org/10.12386/A1955sxxb0038

  Abstract ( )   Knowledge map   Save

  Die Beziehungen zwischen dem Perfektheitsbegriff Krulls und den von Ostrowski entwickelten Pseudokonvergenten Folgen sind hier betrachtet. Als ein endgiiltiges Resultat beweist der Verfasser:,,Damit ein bewerteter Korper K perfekt ist, ist notwendig und hinreichend, dass jede pseudokonvergente Folge mit primer Breite in K mindestens einen Pseudolimes besitzt".Als eine unmittelbare Folgerung davon erhalten wir die MacLanesche Charakterization fur die Maximalitat diskret bewerteter Korper.
• Select
  LA METHODE DE COL DANS LE PROBLEME DE RELAXATION

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 497-504. https://doi.org/10.12386/A1955sxxb0039

  Abstract ( )   Knowledge map   Save

  On considere ici pour la resolution des equations lineaires dans un espace d'Hilbert une methode, qui est, comme on pourrait dire, une combinaison de la methode de col et la mehode de relaxation proposee par R. V. Southwell. En effet, si l'on choisit la direction de la plus vite descente non pas dans l'espace entier en consideration, mais dans un sousespace a dimensions finies convenablement choisi, on peut faire disparaitre toutes les composantes du vecteur residu qui correspondent a ce sousespace au lieu d'une seule composante, comme dans le cas de la methode de relaxation. La choix du sousespace doit garantir l'aneantissement des composantes dominantes du vecteur residu; ou plus precisement, pour resoudre l'equation lineaire Ax=b on commence par un vecteur x qui jouera le role d'une approximation initiale a la solution exacte, et dans chaque pas du procede on choisit le sousespace H_n tel que la norme ‖P_n c_n‖de la projection du vecteur residu c_n sur H_n satisfait a la condition suivante :ou k est un nombre fixe compris entre 0 et 1. On demontre pour le procede propose ici la convergence dont la vitesse est aussi grande que celle d'une progression geometrique, et on donne quelques indications sur la comparaison du procede considere avec les methodes diverses usuellement employees. En particulier, on donne une demonstration du procede de M. F. H. Chao considere dans un espace d'Hilbert.
• Select
  ON THE REALIZATION OF COMPLEXES IN EUCLIDEAN SPACES I

  Acta Mathematica Sinica, Chinese Series. 1955, 5(4): 505-552. https://doi.org/10.12386/A1955sxxb0040

  Abstract ( )   Knowledge map   Save

  It was early known that any n-dimensional abstract complex may be realized in an (2n + 1) -dimensional euclidean space R~(2n+1). From this theorem, whose proof is quite simple, it follows that the (2n + 1)-dimensional euclidean space contains in reality all imaginable n-dimensional complexes. However, the complete recognization of all n-dimensional complexes in an euclidean space of given dimension m where m<2n+1 is a problem much more difficult which cannot, it seems, be solved completely in the near future. Among the miscellaneous results so far obtained along this line the most remarkable one is no doubt that of Van Kampen and Flores, who first proved the existence of n-dimensional complexes which, even under further subdivisions, cannot be realized in an R~(2n).The invariant by means of which Van Kampen was able to conclude the non-realizability of a (finite simplicial) n-dimensional complex in an R~(2n) may be described as follows. Denote the k-dimensional simplexes of the given n-dimensional complex K by S_i~k. Any two simplexes of K with no vertices in common will be said to be disjoint. Let A be the set of all unordered index pairs (i,j), corresponding to pairs of disjoint n-dimensional simplexes S_i~n and S_i~n. Construct a vector space on the ring of integers with dimension equal to the number of elements in A. Any vector of may then be represented by a system of integers (a_(ij)) where (i, i)∈ A. To each pair of disjoint simplexes S_a~(n-1) and S_l~n in K a certain vector V_(la)= (a_(ij)) of may be determined in the following manner. If both i, j≠lor one of them, say j=l, but S_a~(n-1) is not a face of S_i~n, then we put a_(ij)= 0. Otherwise we put a_(il) = ±1 (with sign conveniently chosen). Two vectors P, P of will then be said to be equivalent if P-P is a certain linear combination with integral coefficients of vectors of form V_(la) above defined. The vectors of are thus distributed in such equivalence classes.Take now an arbitrary simplicial subdivision K_1 of K and try to realize K_1 in R~(2n) as much as possible. We will obtain then some "almost true" realization such that parts ′S_i~k and ′S-i~l, corresponding to disjoint S_i~k and S_i~l of K will be disjoint in R~(2n) when k + l < 2n, while they intersect only in isolated points when k=l=n. With respect to an orientation arbitrarily chosen of R~(2n), 'S_i~n and 'S_i~n determine then a definite intersection number ±a_(ij) (with sign conveniently chosen). These numbers determine in turn a vector P = (a_(ij)) of Van Kampen's work shows that, whatever be the subdivision K_1 of K and the "almost true" realization of K_1 in R~(2n), the corresponding vectors P belong always to one and the same equivalence class in It follows that this equivalence class is an invariant of the complex K. It is evident that the belongness of the zero vector to this invariant equivalence class is a necessary condition for the existence of "true" realization of K in R~(2n). It is this invariant which has enabled Van Kampen to assure the existence of n-dimensional complexes non-realizable in R~(2n). On the other side, Van Kampen failed to ascertain whether the above necessary condition is also sufficient and the problem of characterizing n-dimensional complexes in R~(2n) remains unsettled up to the present. Moreover the method of Van Kampen-Flores cannot be seen to be readily generalizable to the realizability in R~m, m being arbitrary. We remark also that whether Van Kampen's invariant is a "topological" invariant of the space of K, or even whether it is a combinatorial invariant of K_1 cannot be decided from his works.At the time of Van Kampen and Flores the cohomology theory has not yet been created. To get a deeper insight of their results we will reformulate them in the modern terminology of cohomology. Their statements will then become clear and natural as follows. From the given simplicial complex (of any dimension) let us construct a subcomplex K of K×K, consisting of all cells σ×τ such that σ, τ are disjoint in K. Identify each pair σ×τ and τ×σ of K to t