中国科学院数学与系统科学研究院期刊网

15 April 1957, Volume 7 Issue 2
    

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  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 167-182. https://doi.org/10.12386/A1957sxxb0012
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    In the present paper,we consider the following classes of functions:S_p(ρ):the class of functions f(z)=■regular and schlichtin the unit circle |z|<1,and being such thatR(zf’(z)/f(z))≥ρ,0≤ρ<1,|z|<1.For simplicity,we write S_1(ρ)=S(ρ),S(0)=S,S(1/2)=S.K(ρ):the sub-class of S,whereof each function f(z)be such thatzf'(z)∈S(ρ),so that(?)|z|<1.Every function f(z)of S(ρ)(0≤ρ<1)is star-like with respect tothe origin f(0)=0,in particular,any function w=f(z)of K(ρ)maps|z|<1 onto a convex domain in the w-plane.Our main results are as follows:Theorem A.Corresponding to α function f(z)of S(ρ),0≤ρ<1,there exists an increasing function a(θ)with 1/2π∫_0~2πda(θ)=1 satisfying.■Conversely,if the increasing function a(θ)satisfies 1/2π∫_0~2πda(θ)=1,thenthis formula of representation implies f(z)∈S(ρ). Cor.1.A necessary and sufficient condition for f(z)∈(ρ) is thatf(z)can be written as■with some increasing function α(θ)satisfying 1/2π∫_0~2πdα(θ)=1Cor 2.Supposing 0≤ρ_1≤ρ_2<1,if f_1(z)S(ρ_1),then there existsf_2(z)∈S(ρ_2)satisfying(?)In particular~([3]),f(z)∈S implies 2(?).(?)Theorem B.If f(z)=(?),0≤ρ<1,then(?)Theorem C.If f(z)∈S(ρ),o≤ρ0.The signs of equality can hold whenand only when f(z)=z(1-ηz~p)~((-2/p)(1-ρ)),|η|=1.In particular,if f(z)∈S,then■and if f(z)∈S,then■Theorem F.Let p be an odd integer.■ 0≤ρ
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 183-199. https://doi.org/10.12386/A1957sxxb0013
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    We define in § l the local products,which relates the homology of aclosed subset X_0 of a compact Hausdorff space X and that of the neigh-bourhoods of X_0 in X.Duality theorems of Alexander,Lefschetz and Poin-carétype in manifolds and manifolds with boundary are expressed in termsof these local products.In § 3,we study a class of manifold-like spaces.The use of local homology or cohomology groups was implicit in[3,pp.233—263;8].Products given in the“local”way seem natural and useful.Let X be a compact Hausdorff space and X_0,E closed subsets of X·Let E_0=X_0∩E and let G be a coefficient group.Local homology groupsH_i(X_0|X,E_0|E;G)are defined as limits of direct system of relative homo-logy groups of the pairs(X,(X—W)U E)where W are neighbourhoodsof X-0 in X.Local cohomology groups H~i(X_0|X,E_0|E;G)are defined aslimits of inverse systems of relative cohomology groups of such pairs.Localcup and cap products are established in a natural way asH~i(X_0,G_1)(?)H~j(X_0|X,E_0|E;G_2)→H~(i+J)(X_0|X,E_0|E;G_0),H~i(X_0,G_1)⌒H~(i+j)(X_0|X,E_0|E;G_2)→H_j(X_0|X,E_0|E;G_0),H~i(X_0|X,E_0|E;G_1)⌒H_(i+j)(X_0|X,E_0|E;G_2)→H_j(X_0,G_0),where G_1 and G_2 are given coefficient groups paired to a coefficient gorupG_0.We summarize here some of the results.For instance,(i)if X is anorientable n-dimensional manifold with regular boundary E(n≥1),and if(?)(X)is an integral fundamental homology class of X,then local cap pro-ducts H~i(X_0|X,E_0|E;G)⌒(?)(X)etc.yield isomorphisms⌒(?)(X):H~i(X_0|X,E_0|E;G)≈H_(n-i)(X_0,G)etc.When X_0=X,this reduces to the ordinary duality of Poincaré-Lefschetztype.(ii)If X is of finite dimension and satisfies suitable local homology properties,then,with coefficients in a field(?),the following three condi-tions(a),(b),(c)are equivalent:(a)There is an element (?)(X)of H_n(X,(?))such that⌒(?)(X):H~i(X_0,(?))≈H_(n-i)(X_0|X,(?))for every closed subset X_0 of X and every integer i.(b)There is an element(?)(X)of H_n(X,(?))such that⌒(?)(X):H~i(X_0|X,(?))≈H_(n-i)(X_0,(?))for every closed subset X_0 of X and every integer i.(c)For every closed subset X_0 of X and every integer i,(?)is a dual pairing.(iii)If X is a non-empty connected compactum,which is lc~1 over(?),and if there is(?)(X)∈H_2(X,(?))such that(?)for every closed subset X_0 of X and for 0≤i≤2,then X is a closed sur-face.This paper was completed in the United states of America.Abstractappears also in a few lines in the notes“periodic transformations andfixed point theoremsⅠ,Ⅱ”,Science Record,Vol.1(1957),No.1.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 200-228. https://doi.org/10.12386/A1957sxxb0014
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    The inequality(?)(1)is usually called Schwarz's inequality,although it was stated first byBuniakowsky~[1].In this paper,the author generalises(1)as follows|det(f~i,g_j)|~2≤det(f_i,f_j)det(g_i,g_j)(2)(i,j=1,2,...,n)where we use the notationdet u_(ij)(i,j=1,2,...,n)to denote the determinant of the n-th order of which the element in thei-th row and the j-th column is u_(ij)and,{f_i}and{g_i}(i=1,2,...,n)denote two sets of elements of an arbitrary Hilbert space,and(f_i,g_j)isthe inner product of f_i and g_j etc.Two special cases are that:i)for any two sets of L~2 functions{f_i(x)}and{g_i(x)}(i=1,2,...,n)defined in the interval α≤x≤b we have:(?)(i,j=1,2,...,n) (When n=1,(3)becomes(1)).ii)For any 2n sets of complex numbers{a_(ih)},{b_(hj)},(h=1,2,3,…;i,j=1,2,…,n),satisfying the conditions(?)we have(?)(4)(i,j=1,2,…,n)which is a generalization of the Cauchy's inequality(?)(5)By the aid of these inequalities,some new properties of integral equa-tions and Hilbert spaces are obtained.The main results are given in thefollowing theorems:Theorem 1.Let(?)be an abstract Hilbert space,and let K be a boundedpositive transformation on the space(?),then(?)(6)(i,j=1,2,…,n)for all elements f_i,g_j∈(?).Theorem 2.Let A(x,y)and B(x,y)be two arbitrary L~2 kernels,sothat(?)then for any positive integer n,we have:(?)(7)(?)(8)(i,i=1,2,…,n),where(?)(?)(?)(?)Applying(8)we obtain the inequality(?)(9) where{λ_h[A]}(h=1,2,3,…)denotes the complete set of singular values(i.e.E.Schmidt characteristic values)of A(x,y)satisfying 0<λ_i[A]≤≤λ_2[A]≤….More accurately,we shall prove that(?)(h_11 we have:(?)(17)Further,if a function of several variables f(x_1,x_2,
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 229-234. https://doi.org/10.12386/A1957sxxb0015
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    The main purpose of this paper is to prove the following formula(?)(1)and the following two inequalities(?)(2)(?)(3)where m, n_1,n_2,…,n_m denote arbitrary finite positive integers n_0=1,(?)a_(ij)denotes the determinant of the n-th order of which the element in the i-throw and the j-th column is a_(ij) and(?)In(3)and(4),α_(s_1s_2)…s_m,and β_(t_1t_2)…t_m denote elements of an arbitraryHilbert space(complete or non-comptete,separable or non-separable),(α,β)denotes the inner product of the two elements α andβof this Hilbertspace.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 235-241. https://doi.org/10.12386/A1957sxxb0016
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    The Steenrod powers(p a prime and I_p the field of mod p integers)St_(p)~k=St~k:H~r(K,I_p)→H~(r+k)(K,I_p)in a complex K were discovered by Steenrod~[1]from the consideration of theproduct complex(?)under the cyclic transformation t(x_1,…,x_p)=(x_p,x_1,…,X_(p-l)),X_i ∈|K|.On the other hand,by Smith's theory,[2,3]withrespect to K~p under t,we may introduce in a natural manner,a system ofhomomorphisms.(?)The question of the relations between Smith operetions Sm_k and Steenrodpowers St~k arises naturally.The author discovered~[4]that these two systemsof oPerations are actually equivalent,the one being determined by the other.This furnishes a more natural and simple definition of Steenrod powers andmakes it directly connected with the theory of Smith.The previous proofof the author depends on the intrinsic axiomatic theory of Steenrod powersof Thom.The present paper aims at givfng a direct proof independent ofthe work of Thorn.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 242-267. https://doi.org/10.12386/A1957sxxb0017
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    The theory of Mellin transforms is extended to the distributions on thehalf-line(0,∞).The distributions on(0,∞)(∈P')are linear continuous functionalsover the space P of infinitely differentiable functions with compact supportsin(0,∞).An exponential substitution establishes the isomorphism betweenP and K(viz.P'and K').Let Q be the space of all integral functions ψ(S)=ψ,(a+it)satisfying1.|ψ(σ+it)|≤Ae~(B|σ|),2.ψ(σ+it)∈S[t] for each fixed a;and Q' be itsdual.If f(s)is analytic and of finite order in the stripe a<(?)s
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 271-276. https://doi.org/10.12386/A1957sxxb0019
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    Es sei f(x)∈C[α,b]und es bezeichne A_x die Menge aller Funktionen{f(x)}mit(?)Sei B ein einfach zusammenh(?)ngendes schlichtes Gebiet,dessen Beran-dung die rektifizierbare Jordankurve Fist,und seiξ=ψ(z)die Funktionwelche den Kreis|z|<1 auf das Gebiet B abbildet.Wir nehmen an da(?)die Randkurve Γin jedem Punkt eine Tangent hat,und bezeichnen mit(?)(s)den Winkel,den die positive Richtung der Tangente mit der x-Ache bildet.s ist dabei die Bogenl(?)nge von Γ,von einem festen Punkte ab gerechnet.Satz I.Ist(?)~(n)(s)∈Λ_s,(n=0,1,2,…),So ist die Funktionψ~(n+1)(z)in dem abgeschlossenen Kreisgebiet|z|≤1 stetig und(?)wo A,B,C'Konstanter sind.Es sei f(x)∈Z~p[α,b],p≥l undes bezeichneΛ_x(p)die Menge allerFunktionen{f(x)}mit(?)(?)Dann gilt der Satz:SatzⅡ.Wenn (?)(n-1)(s),(n=1,2,…)totatstetig ist und(?)~(n)(s)derKlasse Λ_s(p)(p>1)angeh(?)rt,dann ist die Funktionψ~(n)(z)in dem abges- chlossenen Kreisgebiet|z|≤1 stetig,und totalstetig auf der Begrenzung|z|=1;ψ~(n+1)(z) geh(?)rt der Klasse H_p an,seine Grenzwerte sin f.ü.(?)und(?)
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 277-284. https://doi.org/10.12386/A1957sxxb0020
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    We say,for simplicity,a closed space curve with finite angular pointsa curvilinear polygon.The purpose of this paper is to establish the fol-lowingTheorem:For the integral of the frst curvature of any curvilinearpolygon C in m-dimensional Euclidean space S_m holds the followinginequality:(?)where θ_1,θ_2,…,θ_n are the interior angles of C.The equality holds onlyfor a plane convex curvilinear polygon.Corollary 1.For the integral curvature of any curvilinear polygon C inordinary space S_3 holds the following inequality: (?)Corollary 2.The integral of the first curvature of any closed spacecurve in space S_m is not less than 2π.The corresponding theorem of corollary 2 for a closed curve in ordinaryspace S_3 is due to W.Fenchel.Corollary 3.The integral of the first curvature of any closed curvewith an angular point in space S_m is not less than π.Corollary 4.The integral of the first curvature of any closed curvewith one k-multiple point in space S_m is not less than kπ.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 285-294. https://doi.org/10.12386/A1957sxxb0021
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    In a previous paper of mine and Ku Chao-hao(1952),we have consi-dered certain affinely connected spaces with given areal metric.Letx~i=x~i(u~α)(i=1,…,N;α=1,…,K)be the equations of a differentiable K-dimensional variety V_k in an N-di-mensional space S_N,and let the'area'of a certain portion R of the varietygiven by a K-ple integral(?)where(?)is an abbreviation for du~1,du~2,…,du~k and the func-tion F satisfies certain conditions of invariance.The connection coefficients Γ_(jk)~i there introduced are functions of(x~i)aswell as the K-ple supporting element(p_α~i),and are supposed to satisfy aset of conditions which suffice to insure that(?)(*)These Γ's are related to the metric function F by the equations of connec-tion(?)(**)where we have placed(?)In Riemannian spaces these conditions(*)and(**)are satisfied by theChristoffel symbols of the second species(?)and the metric function(?)of a K-dimensional differentiable variety VK in the space S_N,where gλ_udenotes the induced metric tensor of V_k,so that the general formula for thesecond variation of the'area'gives immediately the one due to E.T.Daviesas its special case.It is natural to inquire whether or not our theory contains the corres-ponding theories for Finsler and Cartan spaces.In the present paper,we demonstrate that the equations of connectionstill hold good in the geometries of Finsler space and a regular Cartanspace as a necessary consequence of the generalized Ricci Lemmas in thesespaces. On the contrary,the conditions(*)are by no means valid in Finsler orCartan spaces.For the purpose of finding more extensive conditions inorder to include both Finsler and Cartan geometries,we have to investigateEulerian vector E_i in each of these spaces.In the former,it is readily shown that(*)should be replaced by thefollowing ones:(?)where Γ_(jh)~(*k)denotes the connection coefficient of Cartan as well as that ofBerwald and therefore that E_i is equal to the covariant curvature vector ofthe curve in consideration.Denoting the integrand of the second variation of the are under theinfinitesimal transformation(?)by F"and assuming,in particular,thatξ~i is independent of t,we obtain(?)(F2)where R_(jikh) denotes the curvature tensor of the space.In a regular Cartan space we have to put K=N-l and obtain that(?)(C_1)These relations suggest us to consider a further generalization of af-finely connected spaces with areal metric in the following manner:(Ⅰ)The coefficients of affine connections,Γ_(jh)~(*k),are functions of position(x~i)as well as K-ple areal element(p_a~i).(Ⅱ)The metric function F(x,p)is related to these Γ's by the conditionthat the metric of any K-ple areal element should be invariant with respectto the parallel transport of the connection when the element itself is takenfor the supporting element.This naturally leads to the equations of con-nection.(Ⅲ)The Eulerian vector E_i is given by(?)(E)where we have placed(?)There is no difficulty in showing that(E)is equivalent to(?)(E')which implies(C_1).Thus we have extended the spaces to such ones which may be seen ascontaining Riemannian,Finslerian and Cartannian geometries.The formulafor the second variation of the'area'as established in the previous paperremains valid.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 295-308. https://doi.org/10.12386/A1957sxxb0022
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    By an A_n~3-system we mean a(μ,△,γ)-system consisting of cohomology-groups H~r(3≤n≤r≤n+4),H~s(2)(n≤s≤n+3)and homomorphisms(?)(?)(?)(?)(?)only,where △~(-1)(0)=μH~r and △ has a right inverse △*.This system playsan important part[3]in the homotopy type of A_n~3-polyhedra(n≥3).Herewe investigate the complete system of invariants of proper isomorphismclass of A_n~3-systems and reach the following:Theorem.The complete set of invariants of proper isomorphism classof A_n~3-systems are Betli numbers,torsions,block invariants,relative blockinvariants characteristic polynomials and characteristic coefficients.Characteristic polynomials and characteristic coefficients are homotopyinvariants of polyhedra.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 309-312. https://doi.org/10.12386/A1957sxxb0023
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    Let the function(?)be regular andschlicht in the domain l<|ξ|<∞.All such functions form a class Σ.Let Γ_F be the boundary of the image of|ξ|>1 represented by w=F(ξ).Let α_1,α_2,…,α_n be n(n>l)distinct points on Γ_F,P_n(F)the maximumof the quantity(?)as the points vary.The maximum problem of the mean diameter P_n(F)hasbeen considered by Bieberbach,Schiffer and Golusin[1][2].The aim ofthe present note is to discuss the minimum Q_n of P_n(F).Fixing a point α_1=a of Γ_F,the maximum of(l)will be denoted by P_n(F,a),when a_2,……,a_n,vary on Γ_F.The following result is attained.Theorem.There exists a function F_1 of Σ such that Q_n=P_n(F_1).Forthis extremal function F_1,P_n(F_1,a)is a constant for any point a on Γ_F_1.By means of this proposition,we can establish the followingCorollary.If Q_2=P_2(F_1).Then the boundary of the image W_F_1 of|ξ|>1 mapped by F_1 is a closed convex curve.Moreover,there exist pointsexterior to the image W_F_1.
  • Acta Mathematica Sinica, Chinese Series. 1957, 7(2): 313-326. https://doi.org/10.12386/A1957sxxb0024
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    Let S be the class of functions f(Z)=Z+a_2Z~2+… regular and schlichtin the unit circle |Z|l.The chief object of this work is to establish the following two theorems.Theorem 1.If f(ξ)∈Σ,thenThe sign of equality holds only whenIn the case a_1=0,the equality(?)holds when and only when(?)Theorem 2.If F(ξ)∈Σ,then(?)where(?)and x_0=0,92402…is the positive root of the equation 5x~3+27x~2-27=0,Equality holds only for(?)The proofs of theorem 1 and theorem 2 are based on a fundamentallemma concerning the extremum property of the functional H(x_1,…,X_N;y_1,…,y_N)withx_n+iy_n=a_n,n=1,…,Nwe prove also the following two theorems.Theorem 3.If(?)then(?)the estimates are sharp.Theorem 4.If(?)then(?)2n+lthe estimates are sharp.These propositions are reduced to the theorems of G.M.Golusin[3]whenM→∞.