15 October 1957, Volume 7 Issue 4

    

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  ON THE SCHWARZ LEMMA IN EINSTEIN SPACES OF SEVERAL COMPLEX VARIABLES

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 471-476. https://doi.org/10.12386/A1957sxxb0033

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  The main theorem of the present note isTheorem 1.Let D be a simple domain of n complex variables(z)=(z~1,…,z~n)with Bergman metric(?)where(?)Let D be an Einstein space with constant Ricci curvature-1.Let w~((j))= f~((j))(z~1,…,z~n),j= 1,2,…,n be a pseudo-conformal mappingwhich maps D onto a domain D′in w-space.If we can define a Hermi-tian metric(?) in D′ such that the Ricci curvature of D′ defined by this metric is notgreater than -1 and h
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  COMPLETENESS OF SETS OF FUNCTIONS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 477-491. https://doi.org/10.12386/A1957sxxb0034

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  If the real function g(x)is non-decreasing and of bounded variation on[-π,π].The set {B_n(x)}_1~∞(?)∠~p(-π,π;dg),p>1,is complete ∠~p(-π,π;dg), p>1,if(?)and f(x)∈∠~q(-π,π;dg),1/p+1/q=1,imly that f(x)=0 almost everywhere on [-π,π].In this paper,we suppose the following conditions are usually satisfied:g(x-0)=g(x);(?) If 0<λ_1<λ_2…and let λ(r)be the number of λ_n0,then(?),θ ∈ E,is complete∠~p(-π,π;dg),p>1,where E(?)[-π,π]is some set of points havingpositive Lebesgue measure.7)If G(z)∈(M_1)is analytic in |z|<1 and continuous on |z|≤1,let us denot the modulus of continuity of G(e~(ix))by ω(t),then either(?);(?)θ ∈ E, is complete ∠~p(-π,π;dg),p>1.If(?),01,where ψ_E(t)=mes E_t,and E_t is the closed set of pointswith distance≤t from E.8)Let G(z)∈(M_1)be an analytic function defined in unit circle.If|α_n|,n=0,1,2,…,lim α_n=α,|a|<1; |α_n-α|≤1-|α|; ∑|α_n--α_(n+1)|<∞,then(?),p>1.9)Let(?)and μ(r) be the number ofβ_n≤r.For an entire function G(z)∈(M_1),if(?)then(?)is complete ∠~p(-π,π;dg),p>1.10)If{α_n}possesses a limite point α,|α|≠1,∞,then(?)complete ∠~p(-π,π;dg),p>1.If{α_n}possesses the properties describedin 8),then(?)is complete ∠~p(-π,π;dg)p>1.In the second part,for the class ∠~p(0,l),we have considered theproblem of the completeness of a set Of functions of the following forms:{G(α_ne~(ix))};(?);{G(α_nx)};(?);(?);(?)Naturally,some similar results are obtained.The completeness of above sets of functions with respect to the classesH_2(D)or E_2(D)is also discussed.
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  КАРАКТЕРИСТИКА АЛΓЕБРАИЧЕСКОЙ ПОВЕРХНОСТИ И МЕТОД ОПРЕДЕЛЕНИЯ МНОЖИТЕЛЯ ЦЕЛОΓО ВЫРАЖЕНИЯ

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 492-512. https://doi.org/10.12386/A1957sxxb0035

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  ON THE BLOCH CONSTANT FOR MULTIPLY CONNECTED REGIONS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 513-519. https://doi.org/10.12386/A1957sxxb0036

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  Let D be a bounded multiply connected region in z-plane.The curvatureof Bergman metric of this region takes the minimum value τ_α at z=α.Let w=f(z)be an analytic function in D with(?),which mapsD onto D_f in w-plane.The object of the present note is to prove that D_f must contain a circlewith radius(?)where T_b is the value of the Bergman metric T of theregion D at z=b.Further,if the function f(z)is simple in D,then D_f mustcontain a circle with radius(?)
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  SOME PROPERTIES CONCERNING COEFFICIENTS OF DOUBLE FOURIER SERIES

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 520-532. https://doi.org/10.12386/A1957sxxb0037

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  Let f(x,y)be an integrable function,periodic with period 2π in eachvariable.The object of this paper is to discuss some properties concerningcoefficients of double Fourier series of the class of functions of boundedvariation in the sense of Hardy-Krause(c.f.Clarkson,J.A.and Adams,C.R.,Trans.of the Amer.Math.Soc.,35(1933),824—854),or of the classof absolutely continuous functions in the sense of Gergen(c.f.Gergen,J.J.,Trans.of the Amer.Math.Soc.,35(1933),27—63).We shall say func-tions of bounded variations or absolutely continuous functions for short.The following results have been established:Theorem 1.The function(?) (1)defined in the fundamental region[0,2π;0,2π]is continuous and of bound-ed variation,but it is not absolutely continuous.Theorem 2.The Fourier coefficients of a function which is continuousand of bounded variation should satisfy(?) (2)where(?).The fact that the condition(2)is not sufficient for a function ofbounded variation to be also continuous has been shown by an example.On the other hand,we have the following theorem:Theorem 3.If the Fourier coefficients of a function f(x,y)of boundedvariation satisfy(2),thenf(x+h,y+k)-f(x-h,y+k)-f(x+h,y-k)++f(x-h,y-k)→ (3)as h→+0,k→+0.
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  ON THE AUTOMORPHISM OF LINEAR GROUPS OVER A NON-COMMUTATIVE PRINCIPAL IDEAL DOMAIN OF CHARACTERISTIC≠2

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 533-573. https://doi.org/10.12386/A1957sxxb0038

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  Let R be a non-commutative principal ideal domain.Let GL_n(R)bethe group formed by all n×n invertible matrices over R,and let SL_n(R)bethe subgroup of GL_n(R)generated by all elements of the formΤ_(ij)(λ)=Ι+Ε_(ij),i≠j, i,j=1,2,…,n,where Ιdenotes the n×n identity matrix,Ε_(ij)denotes the matrix with 0elsewhere except a 1 situated in the(i,j)position,λ∈R and λ≠0.Then thefollowing theorems are proved.Theorem 1.Let R be a non-commutative Euclidean ring of characteris-tic≠2 and n≥3.Then every automorphism of SL_n(R)is either of theformX→AX°A~(-1)or of the formX→A(X~τ)′~(-1)A~(-1),where А∈GL_n(R),σ is an automorphism of R and τ is an anti-automor-phism of R.Theorem 2.Let R be a non-commutative principal ideal domain of cha-racteristic≠2 and n≥3.Then every automorphism of GL_n(R)is eitherof the form(?)or of the form(?)where χ is a homomorphism of GL_n(R)into the center of the multiplicativesemi-group of R with the property that χ(ΥI) =Υ~(-1)for ΥΙ ∈ the center ofGL_n(R)implies Υ=1 and A,σ,τ has the same meaning as in Theorem 1.Theorem 3.Let R be a commutative principal ideal domain of charac-teristic≠2 and n≥3.Then every automorphism of the group UL_n(R)formed by all n×n matrices with determinant 1 over R is either of theform(?)or of the form(?)where A ∈ GL_n(R),σ is an automorphism of R and χ is a homomorphismOf UL_n(R)into the center of the multiplicative semi-group of R with theproperties:(i)if χ(ΥI)=Υ~(-1),then Υ=1 for all Υ ∈ R such that Υ~n=1,(ii)(χ(X))~n=1 for all X ∈ UL_n(R).
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  AN IMPROVEMENT OF ВИНОΓРАДОВ'S MEAN VALUE THEOREM AND SOME APPLICATIONS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 574-589. https://doi.org/10.12386/A1957sxxb0039

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  As it was pointed out by one of the authors that the Виноградов's me-an value theorem is a fundamental tool for the estimation of exponentialsums.The latest form of the Виноградов's mean value theorem was givenin Hua[2].In this paper,we shall give a slightly better improvement of the meanvalue theorem in the following form:Theorem 1.Letf(x)=α_κx~κ+…+α_1x,and(?)If(?)then(?)where A,B are absolutely constants,and(?)Theorem 1 has many applications,here we give two of them.NamelyTheorem 2.When x→∞,(?)where π(x)denotes the number of primes not exceeding x,and(?)AndTheorem 3.When t→∞,(?)
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  PROPERTIES OF THE SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH SINGULAR COEFFICIENTS IN THE NEIGHBORHOOD OF SINGULAR LINE(SINGULAR SURFACE)

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 590-630. https://doi.org/10.12386/A1957sxxb0040

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  In the first part of this paper we consider the partial differential equa-tion as a generalized Euler-Poisson equation:(?) (1.1)where β,β′are constants, and a(x,y),b(x,y),c(x,y),d(x,y)are all regularfunctions in Hadamard's sense.Therefore x=y is the singular line of thecoefficients.The behaviors of the solutions of(1.1)in the neighborhood ofthe singular line x=y are described by introducing the concepts of“index”and the“regular part”:Let ρ be a constant and υ(x,y)be a regularfunction(υ(x,x)≠0)such thatu(x,y)=(x-y)~ρυ(x,y)is a solution of(1.1),then the constant ρ is said to be the“index”andρ(x,y)the“regular part”of the solution.It is shown that all the possibleindexes must satisfy the indicial equation(?)and if F(ρ+1)≠0,then the normal derivative of the regular part on thesingular line x=y is determined completely by the value itself,i.e.(?)The regular part υ(x,y)satisfies the equation of a particular form of(1.1),in which γ=0,and therefore it is sufficient to study the equation of theform(?) (?) (3.2)We define the singular Cauchy prob em as follows:to find a functionυ(x,y)continuous together with its first derivatives and twice differentiablein the region ACBD(cf.figure 1 p.518),and satisfying the equation(3.2)in the region ACBD,except the singular line AB,on which it takes anygiven regular funtion u_0(2x)as its initial value.We give the existence proof of such singular Cauchy problem in thegeneral case(β+β′≠0),and it follow that,the solution of the equation(1.1)may,in general,be expressed as.(?)where ρ_1 and ρ_2 are different roots of the indicial equation;or(?)where ρ_1 is the double root of indicial equation.The second part of this paper,deals with the singular equation in spa-ce,especially the equation of the following form:(?) (15.5)where A_σ is any linear operator which (?)epends only on the variables σ==(σ_1,…,σ_n),such that,the Cauchy problem for the associated regular equation(?) (15.6)and the initial data(?)has a unique soluion υ(x,σ_,…,σ_n).The solution of singular Cauchy pro-blem for equation(15.5),with initial data(?)can be expressed by υ(x,σ_1,…,σ_n)in the form(?)where K(τ,t)is a kernel well defined by the operator(?)For example,the kerne for Euler-Poisson-Darboux opera-tor(?)is(?). The same method can be applied to solve the Cauchy problem for thegeneralized Chapligin equation(?)(where K(t)is an increasing function,and K(0)=0),with initial data(?)The solution is given explicitly by(17.12).(p.550).
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  CHAINS OF ADMISSIBLE SUBGROUPS OF GROUPS WITH OPERATORS

  Acta Mathematica Sinica, Chinese Series. 1957, 7(4): 631-640. https://doi.org/10.12386/A1957sxxb0041

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  A group G with a left operator domain Ω is called an Ω-group.A groupG with both a left operator domain Ω and a right operator domain ∑ iscalled an(Ω,∑)-group.In this note some theorems in a recent paper of the author are gene-ralized as follows:1°If N is an Ω-normal subgroup of an Ω-group G,then the ascending(or descending)chain condition is satisfied by the Ω-subgroups of G whenand only when the ascending(or descending)chain condition is satisfiedby the Ω-subgroups of the Ω-group G/N as well as the Ω-subgroups of theΩ-group N.2°The ascending(or descending)chain condition is satisfied by theΩ-subgroups of an Ω-group G if and only if there exist two Ω-normal sub-groups M and N of G such that(a)The ascending(or descending)chain condition is satisfied by the Ω-subgroups of the Ω-group G/M as well as the Ω-subgroups of the Ω-group G/N;(b)M∩N={e},where e is the identity of G.3°The ascending(or descending)chain condition is satisfied by theΩ-subgroups of an Ω-group G if and only if there exist Ω-normal subgroupsN_1,N_2,…,N_n,(n≥1)of G such that(a)The ascending(or descending)chain condition is satisfied by theΩ-subgroups of the Ω-groups G/N_i,i=1,2,…,n;(b)The ascending(or descending)chain condition is satisfied by the Ω-subgroups of the Ω-group(?).As a special case of 3°we get a more general form result of Theorem6(Hsieh,1956)in the following:4°The ascending(or descending)chain condition is satisfied by theleft ideals of a non-associative ring Ω if and only if there exist two-sidedideals N_1,N_2,…,N_n, of Ω such that(a)Theascending(or descending)chain condition is satisfied by theleft ideals of the residue class rings Ω/N_i,i= 1,2,…,n;(b)The ascending(or descending)chain condition is satisfied by theleft ideals of Ω contained in(?).If the words “Ω-subgroups”are replaced by the words “Ω-normal sub-groups”in assertions 1°, 2°and 3°,then some new theorems are obtained.5°If G_i are respectively Ω_i-groups with ascending(or descending)chain condition on Ω_i-subgroups i=1,2,…,n,then the direct product G ofG_i is an Ω-group with ascending(or descending)chain condition on Ω-sub-groups,where Ω is defined as the set of all elements ωω=(ω_1,ω_2,…,ω_n)ω_i∑Ω_i,i=1,2,…,n.such that if α=(α_1,α_2,…,α_n)∈G,then ωα=(ω_1α_1,ω_2α_2,…,ω_nα_n).6°If G is an additive Ω-group with ascending(or descending)chaincondition on Ω-subgroups,then G_(n×n)is an Ω_(n×n)-group with ascending(ordescending)chain condition on Ω_(n×n)-subgroups,where G_(n×n)and Ω_(n×n)aredefined as the sets of elements a and ω respectively:(?)(where α_(ij)∈G,ω_(ij)∈Ω,i,j=1,2,…,n)such that(?)By the above result 6°Theorem 2.2E(Artin-Nesbitt-Thrall,1946)isgeneralized as follows:If Ω is a non-associative ring with maximum(orminimum)condition on left ideals,then the total matrix ring Ω_n has maxi-mum(or minimum)condition on left ideale.If we replace the chain conditions by another kind of chain conditionsas defined in“The theory of rings”(Jacobson,1943),then all the corres-ponding theorems remain true.If the words“Ω”are replaced by the words“Ω,∑”in all the aboveassertions concerning Ω-groups,then some new theorems are obtained,