15 April 1958, Volume 8 Issue 2

    

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  FUNDAMENTAL FINITE MODAL SYSTEMS

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 153-180. https://doi.org/10.12386/A1958sxxb0011

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  A finite sequence of non-negative integers is called a modal sequence ifany of its three consecutive terms of the form“aOb”is identified as a singleterm“a+b”,for example,13045 is identified as 175.For any two modal sequences α=a_1 a_2…a_h and β=b_1 b_2…b_k,if we canget the same sequence by replacing some(or none)even terms α_(2i)and some(or none)odd terms b_(2j+1)by smaller integers,then we say that a precedes β,denoted by α(?)β.It is easy to see that the relation(?)is an order relation.If α(?)βand β(?)α,then we say that α is equivalent to β,and denoted byEvidently we have infinitely many non-equivalent modal sequences.If weassume some new order relations,it may happen that in the result systemsonly a finite number of non-equivalent modal sequences can exist.Such sys-tems will be called finite modal systems.In any finite modal system thereare some properties which may be characterized by non-negative numbers(call-ed parameters).When the values of parameters are given,we may deducesome order relations in the system.Such order relations are called essentialrelations,which hold necessarily,in some sense,in every finite modal systems.For example,the relations n=n+1 and α=α~h+α~k are essential relations,whereα~h denoted αα…α(h in number).The number n is called the order of the sys-tem.The number pair(h,k)is called the type of the sequence α.Amongthe various types of all the sequences of the system,the strongest ones arecalled the types of the system.The present paper is to construct finite modal systems when the parame- ters alone are given,in other words,to construct the finite modal systems fromthe essential relations alone.Such systems constructed are called fundamentalfinite modal systems.The main results are as follows:We may construct the systems of order 0 and of order 1 readily.Theycontain 2 and 14 modal sequences respectively.When the type(h,2k)of the sequence 1 and the type(r_i,1)of the se-quence 21~(2i-1)(i=1,2,…,m,where m is the integral part of h/2)are given,we may construct the system of order two.The total number of the modalsequences in it may be computed out.We may construct the systems of the type(1,2),and the systems of order3 with type(2,2).In such construction,however,the sequences which havethe type(1,2)or(2,2)cannot be given arbitrarily.By means of zu-sequences(characterized by z(?)u,z~3=z~5,and zuz=zuz~3==z~3uz)we may construct infinitely many fundamental finite modal systems oforder three.Finally,we prove that(under a very broad assumption)if wecan construct all the fundamental finite modal systems of order three withtype(2,2),then we can construct all the fundamental finite modal systems.
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  EXTENSIONS OF SOME THEOREMS OF ROBINSON AND GOLUSIN

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 181-189. https://doi.org/10.12386/A1958sxxb0012

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  1.Let the p-symmetric function(?)schlicht in the unit circle |z|<1.The totality of all such functions forms aclass which shall be denoted by S_p.For brevity we denote S_1 by S.Let S_pbe the subclass of S_p consisting of all functions W=f(z)of S_p which map|z|<1 onto regions star-shaped with respect to W=0.Supposing f(z)∈S_p,there exists an x=x(f,z),01,and obtain thefollowing precise estimations.(a)Suppose f(z)∈S_p,|z|<1,then(?)where x is determind by(1)or(2).(b)If:0
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  APPROXIMATION TO FUNCTIONS ON THE POSITIVE REAL AXIS BY GENERALIZED POLYNOMIALS

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 190-199. https://doi.org/10.12386/A1958sxxb0013

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  We generalize in this paper some results of S.Mandelbrojt and S.Ag-mon.Theorem 1.Let F(x)be a positive function(x≥0)and log F(x)be aconvex function of logx(x>0).Let{υ_}n(n=1,2,…)be a sequence ofcomplex numbers such that:A)0<|ν_1|<|ν_2|<…|ν_n|<…;B)(?),where K is a positive integer;C)(?)D){ν_n}lie in a half-strip in the s=σ+it plane:|t|<α,σ>0,whereα is a positive contant;E)(?)Suppose that(?)(1)where 00,we can find a polynomial of the formP(x)=a_0+a_1x~(ν_1)+a_2x~(ν_3)+…+a_nx~(ν_n),(2)such that for x≥0,(?)β)If f(x)is a function continuous on[0,∞),and if (?)then given any ε>0,we can find a polynomial P(x)of the form(2)suchthat for x≥0,|f(x)-P(x)|<εF(x).γ)If f(x)∈L~p[0,∞)(1≤p<∞),then given any ε>0,we can finda polynomial P(x)of the form(2)such that(?)The results of S.Mandelbrojt and S.Agmon on uniqueness of the solu-tion of the moment problem can be generalized as follows:Theorem 2.Let{ν_n}(n=1,2,…)be a strictly increasing sequence ofpositive numbers verifying the conditions B),C)and E)mentioned above.Let{m_n}(n=0,1,2,…)be a sequence of positive numbers.PutB(σ)=sup(ν_nσ-log m_n)(ν_0=0).If(?)(3)where 0
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  HOMOTOPY GROUPS AND CUP PRODUCT OF COHOMOLOGY GROUPS

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 200-209. https://doi.org/10.12386/A1958sxxb0014

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  For any arcwised connected space X,the m-th Betti number of X is denotedby ρ_m(X).If R is the field of rational numbers, we define P_m(X)to be thevector subspace of H~m(X,R)generated by all those elements of the form a·bwhere“·”denotes the cup product,a∈H~i(X,R),b∈H~j(X,R),i+j=m and0
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  ON THE ESTIMATION OF REGRESSION COEFFICIENTS OF A RANDOM FIELD WITH HOMOGENEOUS RESIDUAL

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 210-221. https://doi.org/10.12386/A1958sxxb0015

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  U.Grenander has investigated the problem of estimating the regression coefficients of a time series with stationary residual.Recently M.Rosenblatthas generalized to the case of vector valued time series with stationary resi-dual.In this paper we investigate the problem of estimating the regressioncoefficients of a random field defined on the positive lattice points in R_k.Byour method we can extend the asymptotic efficiency formula to more generaltype of spectral functions of the residual field.For simplicity we consideronly the case that there is only one regression coefficient.Let the randomfield{x_(n_1),…,n_k}n_j=0,1,2,…j=1,2,…,k with expectation cψ_(n_1)…n_k beexpressed asx_(n_1)…n_k=cψ_(n_1)…n_k+y_(n_1)…n_k, (1)where{ψ_(n_1)…n_k}is a known sequence,they are called the regression variablesand c an unknown constant,it is called the regression coefficient.The resi-dual field{y_(n_1)…n_k}is assumed to be a homogeneous random field with spectralfunction F(λ_1…λ_k).Hence the correlation function of the random field canbe written as(?)(2)We consider two kinds of estimates of c;one is the least square estimateC_Ls,the other is the best linear unbiassed estimate C_(BLS).Let(?)We first assume that as N_j→∞(j=1,…,k)(denoted by N→∞),Φ_N→∞.Some conditions of this type is required if we are to be able to estimate cconsistently.We also require that for any h=(h-1,…,h_k),(?)(3)If one of the ν's is a negative integer,we set ψ_(ν1)…νk=0,Let the limit(?)(4)exist for all(ν_1…ν_k).Then the condition(3)is equivalent to(?)for any(h_1…h_k),and the sequence{ψ_(ν1)…νk}is called homogeneous,thenR_(ν1)…νk has the following spectral representation:(?)(5)where the function ψ(λ)=ψ(λ_1…λ_k)is a distribution law on the k-dimensionalinterval[-1/2,1/2;…;-1/2,1/2),ψ(λ)is called the spectral function of thehomogeneous sequence{ψ_(ν1…νk)}.In this paper are have proved the following:Theorem 3.Let{ψ_(n_1)…n_k}be a homogeneous sequence with sepectral function ψ(λ),then at every continuity interval I of ψ,we have(?)(6)Using this theorem and under furthor assumptions on the spectral functionF(λ)of the homogeneous residual we can deduce the following theorems,whichare generalizations of Grenander's~([1])results about the asymptotic efficiency ofthe least square estimate C_(LS)of c.Theorem 6.Let the regression variables ψ_(ν1)…νk of the random field x_(ν_1)…νkbe homogeneous,whose spectral function be ψ(λ)=ψ(λ_1…λ_k).Let the spec-tral function F(λ)of the homogeneous residual random field y_(ν1)…νk)can bedecomposed into the following form(?)(7)where f(λ)=f(λ_1…λ_k)is a positive continuous function in[-1/2,1/2;…;-1/2,1/2],H is a singular measure satisfying the following condition:(?)(8)Then the asymptotic efficiency of the least square estimate C_(LS) is given by thefollowing expression:(?)Corollary 6.1.Under the conditions of Theorem 6,in order that theleast square estimate shall be asymptotically efficient whatever be the functionf(λ)as described in Theorem 6,it is necessary and sufficient that the wholemass of ψ must be concentrated at a single point.From our method,it seems impossible to extend Theorem 6 and its corol-lary to a class of residual with spectral function considerably general than thosewe have treated here.However we can prove the following theorem in whichthe conditions imposed on the spectral function of the residual is extended tosuch a level of generality as in U.Grenander's work on the equidistributedestimate of the mean value of a process[6].Theorem 7.Let the regression variables{ψ_ν1…νk}be homogeneous withspectral function ψ(λ)=ψ(λ_1…λ_k).Suppose that the mass of ψ is concen-trated at a point P_0.Further let the spectral function F(λ)of the residualy_(ν1…νk) can be decomposed into the following form(?)where H is a singular measure satisfying the condition(8),and f(λ)=f(λ_1…λk)is a positive bounded function continuous at a neighborhood of P_0,then theleast square estimate is asymptotically efficient.
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  ON THE ISOMETRIC CORRESPONDENCE OF RIEMANNIAN SPACES OF CATEGORY n—2

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 222-230. https://doi.org/10.12386/A1958sxxb0016

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  A regular Riemannian space is said to be of category p,if it admits pand only p functionally independent invariants.Such a space we shall denoteby R(n,p).The aim of the present paper is to establish the necessary andsufficient conditions so that two spaces of category n-2 may be isometricallyequivalent.But the equivalence problem is solved here only in the restrictedsense that,when two spaces of invariants I_1,I_2,…,I_(n-2);(?)_1,…,(?)_(n-2)respectivelywith the same structure are isometric,we have necessarilyI_1=(?)_1,I_2=(?)_2,…,I_(n-2)=(?)_(n-2).The main procedures consist of the following.According to a result of A.Komar,every R(n,p)admits a group of mot-ions of order n-p with n-p linearly independent Killing vectors.When then Ricci characteristic roots of the Riemannian space are distinct,this group isalso maximal.We can choose the local coordinates (x~i)in such a manner,thatx~(n-p+1),…,x~n will be the p invariants in question,and that at the same timethe line element of the space R(n,p)is reduced to the form(?)where λ,μ,ν=1,…,n-p;α,β,γ=n-p+1,…,n(Theorem 1).We shallrefer to this form as the standard form of the metric,and the coordinates(x~i)employed as the proper coordinates associated to the space R(n,p).In particular,for p=n-2 the local group of motions G_2 will be given by(Ⅰ)(?)or(Ⅱ)(?)according as it is abelian or nonabelian.The standard form of the line ele-ment is reducible tods~2=g_(ij)(x~α)dx~idx~j (1)in case(Ⅰ),and tods~2=a(dx)~2+2(ax+b)dxdy+(ax~2+2bx+c)dy~2++2h_αdxdx~α+2(h_αx+k_α)dydx~α+g_(αβ)dx~αdx~β;(2)in case(Ⅱ),where a,b,c,hα,kα,g_(αβ)are functions of x~3,…,x~n only(Theorem 2).Based upon these preliminary considerations,the author then establishedthe following theorems.Theorem 3.A Riemannian space R(n,n-2)admits α 3-dimensional localLie group as the complete group of motions,when and only when by properchoice of coordinates x,y,x~3,…,x~n its line element is reducible to the stan- dard formds~2=g~(11)(x~α)(dx~2+dy~2)+g_(αβ)(x~r)dx~αdy~β.(3)Theorem 4.Let ν_n and (?)_n be two Riemannian spaces of category n-2,eachadmiring α 3-dimensional local Lie group as the complete group of motions.After their line elements have been reduced to the standard forms:ds~2=g_(11)(x~α)(dx~2+dy~2)+g_(αβ)(x~r)dx~αdx~β,(?)υ_n and (?)_n are isometric,when and only when(?)on putting x~α=(?)~α.Theorem 5.Let υ_n and (?)_n be two Riemannian spaces of category n-2,each admiring an abelian group G_2 as the complete group of motions.Aftertheir line elements have been reduced to the standard forms:(?)υ_n and (?)_n are isometric,if and only if(i)g~(αβ)=(?)~(αβ),(ii)(?)_0/g_0=Const,whereg_0=det |g_(λμ)|,(?)_0=det |(?)_(λμ)|(λ,μ=1.2).(iii)there exist four constants d_(λμ),to reduce n-5n+9 equations(n-5n+8 equations being linear and onequadratic in d_(λμ))into identities.Theorem 6.Let υ_n and (?)_n be two Riemannian spaces of category n-2,eachadmiring an nonabelian group G_2 as the complete group of motions.Aftertheir line elements have been reduced to the standard forms(2),υ_n and(?)_nare isometric if and only if there exists a constant γ,so that by the transfor-mation(?)the metric of the one space will be reduced to that of the other.
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  COMPUTATIONS OF SECONDARY PRODUCTS

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 231-238. https://doi.org/10.12386/A1958sxxb0017

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  According to[1,2]the definition of secondary product is as follows:Let E~N=E~(r_1)×…×E~(r_n)(N=∑r_i).Let ψ_i:E~(r_i),(?)→S~(r_i),pi be a map ofdegree 1.Define a mapf:E~N→S~(r_1)×…×S~(r_n),such thatf(x_1×…×x_n)=ψ_1(x_1)×…×ψ_n(x_n),then(?)where(?)is the space obtained from S~(r_1)×…×S~(r_n)by removing its top dim-entional cell.If there is a map~(1))φ:(?),q_0→X,x_0,then φg:(?)~N,p_0→X,x_0 determines anelement ∑ of(?).Identify(?)with E~(r_1)×…×E~(r_(i-1))××yi×E~(r_(i+1)×E~(r_n), yi being a point of(?).Then φg induces a map,φi,of(?)into X.On the other hand(?).determines an elements α_i of π_(r_i)(X)(i=1,…,n).The element ∑ is denotedby[φ_1,…,φ_n]and called a secondary product~(2))of α_1,…,α_n.As an example of the existence of non-trivial secondary product,S.C.Chang proves that the secondary product[φ_1,…,φ_n]is a free generator ofπ_(N-1)((?)).The purpose of this note is to determine the algebraic structureand geometrical representations of the group π_r((?)),if r<2r_1+2r_2+r_3++…+r_n-3. In the space S~lU S~mU S~n,we have the Whitehead products([6])[S~l,S~m],[S~m S~n],[S~n S~l]and repeated Whitehead products[S~l,[S~m S~n]],[S~m[S~n S~l]],[S~n,[S~l,S~m]].Between these repeated products,the Jacobi identity(-1)~(l_(n+1))[S~l,[S~m,S~n]]+(-1)~(m(l+1))[S~m,[S~n,S~l]]++(-1)~(n(m+1))[S~n,[S~l,S~m]]=0holds [3,5].Let(?),where(?)is attachedto(?)by Whitehead product(?).Then it is natural to askwhether there exists any relation between the secondary products(?)(?)in Y?The answer is negative.All the results are stated as follows:Theorem 1.(?),then(?)Denote the Whitehead product of S~r and[S~r,[φ_1,…,φ_n]]by[(S~r)~2,[φ_1,…,φ_n,]]and then define(?)Theorem 2.The elements of the group π_r,((?))mentioned in theorem 1 aregeometrically realized by the following mappings:(i)i_kπ_r(S~(r_k),(ii)[φ_1,…,φ_n,](?)π_r(S~(Σr_i-1)),(iii)[Sr_k,[φ_1,…,φ_n]](?)π_r(S~(r_k+Σr_i-2),and(iv)[(S~(r_1)~((p-1)),[φ_1,…φ,φ_n]](?)π_r(?),k=1,…,n,p=3,4,….Theorem 3.The relation(?)holds in Y if,and only if,p_1=p_2=P_3=P_4=0.
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  ON DEMOULIN TRANSFORMS OF PROJECTIVE MINIMAL SURFACES(Ⅳ)

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 239-242. https://doi.org/10.12386/A1958sxxb0018

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  As a suppliment to the result given in a former paper(Ⅲ)we shall heredemonstrate the following theorems:Theorem 1.If the corresponding points of the Godeaux sequences of aprojective minimal surface S and one of its Demoulin transforms(?)be ar-ranged in three rows,(?)then in the space S_5 the join of any two consecutive points of the second rowmust intersect the join of the two consecutive points standing in the same co-lumns of the first or third row,and these points of intersection(?)and(?)are Laplace sequences of the two rectilinear congruences W,each of which hasS and one of the two second D transforms for its focal surfaces.Theorem 2.Suppose that four Demoulin transforms of a projective mi-nimal surface are distinct;they may be divided into two pairs such that in thearrange of Godeaux sequences of each pair(?)the corresponding joins of any two pairs of consecutive points in each rowintersect each other and these points of intersection are Laplace sequencesstated in Theorem 1.Theorem 3.In the arrange(?)the corresponding joins of any two pairs of consecutive points in each row in=tersect each other and these points of intersection are points of the Laplacesequence recently considered by L.Godeaux(1953).
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  AN ANALYTIC INVARIANT AND ITS CHARACTERISTIC PROPERTIES

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 243-252. https://doi.org/10.12386/A1958sxxb0019

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  In this paper the author gives a detailed proof of the results announcedpreviously (Sc.Rec.,Vol.1,No.2).Besides,we add the theorem(§4)relatingto the invariants L((?))and U((?))of a reducible classical domain(?):If(?),where each(?)is a classical domain,then(?)and(?)
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  ON WARING'S PROBLEM FOR n-th POWERS

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 253-257. https://doi.org/10.12386/A1958sxxb0020

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  The aim of the note is to prove thatG(n)≤n(3 log n+5.2),which is an unsubstantial improvement of a result due to Vinogradow.
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  РАСЛРЕДЕЛЕНИЯ ИНДЕКСОВ ОСОБЫХ ТОЧЕК И ПОЛОЖЕНИЯ ПРЕ ДЕЛЬНЫХ ЦИКЛОВ СИСТЕМЫ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ■

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 258-268. https://doi.org/10.12386/A1958sxxb0021

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  A CHARACTERIZATION OF A PROJECTIVE FLAT SPACE

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 269-271. https://doi.org/10.12386/A1958sxxb0022

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  The aim of the present paper is to give a characterization of a projectiveflat space.It has an intimate connection with Cartan's theorem on the axiomof plane and Beltrami's on the geodesic correspondence of a Riemannian spaceand a fiat space.The main result is as follows:Theorem:If an m-dimensional affine connected space without torsioncontains m+2 families of totally geodesic hypersurfaces,such that there existsa mapping of V_m into affine space A_m which brings the m+2 families ofhypersurfaces onto m+2 families of planes in general position,then the spacemust be projective flat and conversely.Especially,in the case of a Riemannian space,we obtain a characterizationof a space of constant curvature.It is also shown that the number m+2 cannot be replaced by a smaller one.Even when the images of m+1 families among m+2 families are hyperplanesand the remaining one is not,the space is not always projectively flat.
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  О ВЛОЖЕНИИ МНОΓООБРАЗИЙ ФИНСЛЕРА В ПРОСТРАНСТВА МИНКОВСКОΓО

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 272-275. https://doi.org/10.12386/A1958sxxb0023

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  ON DEMOULIN TRANSFORMS OF PROJECTIVE MINIMAL SURFACES(Ⅴ)

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 276-280. https://doi.org/10.12386/A1958sxxb0024

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  The present paper is a sequel to a previous one in which we have consi-dered the two rectilinear congruences W associated with a projective minimalsurface S and one of its D-transforms(?).In the space S_5 there are corres-ponding Laplace sequences (?)(W)and(?)If we construct the second image in S_5 of the osculating linear complex of(W),that is,the pole P of the hyperplane(?)with respect tothe Klein hyperquadric Q,then the point P must belong to a Laplace sequence(?)(P)where every point is the transform of the preceding along the sense u.In asimilar way we obtain another Laplace sequence(?)where every point is the transform of the preceding along the same sense.It is shown that(P)and(P)can be obtained from the Godeaux sequences(?)(L)and(?)of S and(?)as intersections of joins,namely,(?)(n=1,2,…).The points(?)and(?)in 85 are the images of two sides of Demoulinquadrilateral of S which intersect each other at the corresponding point of(?).We denote by(?)and(?)the second focal sheets of the congruences(W)and(W),so that they are the second D-transforms of S.The Laplace sequence(W)corresponding to the pair(S,(?))is the second transform along the senseν of the Laplace sequence(W)corresponding to the pair((?),(?))and theLaplace sequence(?)corresponding to(S,(?))is similarly the second transformalong the sense u of the Laplace sequence(W)corresponding to((?)).Several remarkable relations between the sequences of Godeaux quadrics(?)and(?)of the surfaces S,(?)and(?)are obtained.For example,Φ_n and(?)touch at four points such that Φ_n and(?)alsotouch at two of them,and Φ_n and(?)at the remaining two points.
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  HARMONIZABLE STOCHASTIC PROCESS AND LINEAR TRANSLATABLE STOCHASTIC FUNCTIONAL EQUATIONS

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 281-289. https://doi.org/10.12386/A1958sxxb0025

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  Let x(t)be a complex-valued stochastic process,the process x(t)is said tobe a regular stochastic process,if the variance(?)exists and(?)for all t (-∞
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  ЗАДАЧА РИМАНА-ΓИЛЬБЕРТА ДЛЯ МНОΓОСВЯЗНОЙ ОБЛАСТИ

  Дун Γуан-чан

  Acta Mathematica Sinica, Chinese Series. 1958, 8(2): 297-304. https://doi.org/10.12386/A1958sxxb0027

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  ~~