中国科学院数学与系统科学研究院期刊网

15 July 1959, Volume 9 Issue 3
    

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  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 227-242. https://doi.org/10.12386/A1959sxxb0023
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    An English abstract of this paper was published in Science Record,11(1958),355—357.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 243-263. https://doi.org/10.12386/A1959sxxb0024
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    An English abstract of this paper was published in Science Record,11(1958),358—363.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 264-270. https://doi.org/10.12386/A1959sxxb0025
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    Letψ_k(x)denote the polynomial of the title,and let g(ψ_kdenote the Least r withthe property that every positive integer is representable as a sum of at most r values ofψ_k(x),arising from non-negative integral value of x.Necaker,V.1 proved thatfor K≥12.In the present paper we improves the right-hand side to K(5 log K+12),and the left-hand side to K logK—K.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 271-280. https://doi.org/10.12386/A1959sxxb0026
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    This paper extends S.N.Mergelyan's converse theorems of Tchebycheff approximation in the complex domain to that of the corresponding theorem of one real variable given by De la Vall(?)e Poussin,and gives some corollaries.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 281-291. https://doi.org/10.12386/A1959sxxb0027
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    H.Milloux and Professor K.L.Hiong had given two different inequalities whichdo not contain N(r,f)and which extend the fundamental inequality of Nevanlinna bydifferent methods.Professor K.L.Hiong has pointed out that we may extend the aboveresults to a general case.We obtain the following two theorems:Theorem I.Let f(x) and ψ_v(x)(v=1,2,3)be meromorphic functions,such thatfor r→∞T(r,(?)= o[T(r,f~(k))],T(r,ψ_2)=o[T(r,f~(k)]andT(r,(?))=o[T(r,f~(k)],and the ψ_v are different from one another.If (?)(O)≠∞;f(O)≠O,∞,(?)(O)andf~(K)(O)≠ψ_1~((k))(O),ψ_2(O),then the inequality(?)is satisfied for|x|=γ< p except,in case when ψ_v is of infinite order,a sequence ofthe intervals that their total lenagth is finite.The remainder S_k satisfies the conditions ofNevanlinna.Theorem Ⅱ.Let f(x) and ψ_v (x)(v=1,2,3)be meromorphic functions,such.that for γ→∞T(r,(?))=o[T(γ,f)],(v=1,2,3)and (?),(?)are distinct from(?)and do not reduce to zero.If(?)(O)≠∞,(?)(O)——(?)(O)=0,f(O)≠O,∞;F(O)≠0,1 and F′(O)≠0(?)then the inequality(?)is satisfied for |x|=γ
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 292-294. https://doi.org/10.12386/A1959sxxb0028
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    In dieser Note wollen wir den folgenden Satz beweisen,der von Goodman konjiziertwurde.Die Beweisfhrung dieses Satzes selbst enthalt einen einfacheren und elementa-rischen Beweis des Bieberbach-Eilenbergschen Satzes.Satz.Sei G die lineare Transformationsgruppe,die der H-Kondition[1,p.84]und die Transformation(?)enthalt.(?)f(0)=0genfigtregular und in bezug auf G beinahe beschrinkt[1,p,83]Dann gilt(?)wobei das Gleichheitszeichen nut fǔr die Funktionen f(Z)=ηz |η|=1 gilt.Wir zeigen noeh,dass die Shtze 3—5 von Goodman[1,§5]auch fǔt die regulairenFunktionen gelten.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 295-305. https://doi.org/10.12386/A1959sxxb0029
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    Let R_(11),(n) be the domain(?)and let (?)(n)be the points Z of the closure (?)(n) such that(?)is of rank r.Obviously(?)(n)=(?)(n),and (?)(n)=(?)(n)is the characteristic manifold of(?)(n)(c.f.Hua[3]).Each (?)(n)is invariant under the group of motion of (?)(n).Let U~[2] denote an 1/2n(n+1)×1/2n(n×1)matrix defined by an n×n unitarymatrix U=(Uαβ)1≤αβ≤n.such that the elements of U~[2]are(?)with P_aa=(?)All the matrices U~[2] forma group U~[2].Its subset of elements(?)where T~(s)is an.s×s real orthogonal matrix and U~(n—s)an(n—s)×(n—s)unitary matrix,is asubgroup (?)of(?).Denote (?)the quotient splice (?).We prove thatThere is an one-to-one real analytic transformation carrying(?)(n)onto the topologicalproduct (?)(r)×(?).Let (?)and (?)real-valued conti-(?)nuous function defined on (?)(n).If a point Z of (?)(n)approaches Q,then we have thefollowing formula:(?) where V((?)(n))and S denote the total volume and the volume element of(?)(n)respectively.Applying this formula and the theorem above,we solve the“Dirichlet problem”of thepartial differential equation(?)associated to (?)(n).
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 306-314. https://doi.org/10.12386/A1959sxxb0030
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    Before to establish the boundary value problem of the partial differential equation ofsecond order(?)which is of elliptic type in the domain (?)(n):(?)αβ=1,…n,and is degenerated on the bourdary (?)(n)of R_m(n),we first considerthe geometrical structure of the botundary.(?)Let(?)(n)be the points Z of(n)(the closure of (?))such that(?)of rank n—2r(r=0.1,…,(n/2))Then(?)where(?)is the characteristic manifold of (?)(n)(c.f.Hua andLook~[1]The points of(?)(n)are of the type K=U′F(n)U,U running over all unitarymatrices of order n and F(2v)=(?)For a matrix U(uαβ)of order n,we define a matrix U~{2}of order 1/2n(n—1)such the element of U~{2} at the(αβ)row and the(λμ)colume is given by(?)(c.f.Hua~[3],p,93 and Look~[1],p.625).when U runs over all unitary matrices,the matrices U~{2} form a group U~{2} whichhas a subgroup,(?)generating by the elements(?)where V~(2r)are unitary matrices,of order 2r satisfying the relation V′F(2r)V=F(2r)and(?)(n)can be mapped by anU(n—2r)are arbitrary unitary matrices of order n—2r.(?)(n)can be mapped by an one-to-one real analytic transformation onto the topological product (?)(n—2r)×m~(2r)where m~(2r) denotes the quotient space(?)When n is even, for h real-valued func.tion cp(K) continuous on (?)m(n),we have,asin the two previous parts of this paper,(?)where V(?)and K are the total volume and the volume element of(?)respectively.When n is odd,the closure of Rm(n)can be imbedded into the closure of R_m(n+1)such that (?)(n)appears as a submanifold of(?)(n+1).Then we can apply the resultsfor even n and obtain the corresponding formulae for odd n.After the above consideration,it is clear how to suggest and then solve the boundaryvalue problem of the equation(1)in (?)_m(n).Denote (?)the class of real-valued functions(?)u(Z), each ou(Z).each of Which in (?)_m(n)is continuousand on(?)is harmonic with respect to the coordinates of R_m(n—2r),i.e.,u(Z)satisfies the dif-equation(1)corresponding to R_m(n—2r)(r≡O,1,…,[n/2]—1).Then,ferential equation(1)corresponding to (?)(n—2).Then,for any real-valued functionψ(K)continuous on(?)tll(n),there is a unique function u(Z)of(?)which takes the given boundary values ψ(K)on(?)(n).Moreover,this functioncan be represented explicitly by the“Poisson integral” of ψ(K):where a=(n-1)/2 for even n and a=n/2 for odd n,and V(?))is the total volumeof (?)(n).
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 315-329. https://doi.org/10.12386/A1959sxxb0031
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    In this paper,we have the following theorem.Theorem.Every large odd integer can be expressed asN=p_1+p_2+p_3.1.Pi=1/3N+O(?),where s>0.c=15/92.2.P1≤N.P2≤N.P3≤N(?)3.P1≤(?),P2≤(?),N—(?)
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(3): 333-363. https://doi.org/10.12386/A1959sxxb0033
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    Ⅰ.THE PROBLEM AND THE METHODSThe equivalence problem was proposed and solved systematically in[1]for the casen=1.This article treats the case for general n.The problem is to investigate the equivalence problem of stability between the systemof differential equations(?)and the system of difference-differential equations(?)where att'where(?)and bil's are given constants,and Tii(t)'s may be non-negative real constantsor non-negative real continuous functions of i.The equivalence problem between nonlinear systems will be considered correspondingly.In order to solve this problem,theorem of Hayes[2]was used in[1]while cor-responding complete theorems for general n do not seem to appear in literature.Forsolving our problem we use two types of methods.The first-type method is used to treat the case when Tij(t)'s are non-negative realconstants.In this case the relation between the roots of the characteristic equationsis investigated,and relations between(1)and(2)are deduced.The second-type method is to rewrite(2)in the following form In case tij(t)'s are small,(?)(t) may be regarded as small disturbances,hence properties of(2)will be deduced from(1).Each method has its own merits and shortcomings.Ⅱ.THE EQUIVALENCE OF STABILITYLemma 1.If all the roots of(3)possess negative real parts,then there exist twopositive numbers △=△(aij,bij)>(aij,bil)>0 and T=T(aij,bij)>0 such that all the roots of(4)satisfy the inequality(?)provided that(?)On the basis of Lemma 1 we haveTheorem 1.If the trivial solution of(1)is asymptotically stable,then there existsa positive constant △=△(aij,bij)>0 such that the trivial solution of(2)is alsoasymptotically stable,provided that Tij(t)'s are constants Tij and satisfy the inequality(?)Using the second-type method,this theorem will be improved as follows:Theorem 2.The hypothesis that Tij(t)'s are constants can be omitted,and theconclusion of Theorem 1 still holds,provided that(?)Theorems 1 and 2 are natural generalizations of Theorem 1 of [Ⅰ].Ⅲ.THE EQUIVALENCE OF INSTABILITYLemma 2.If(3)possesses at least one root with positive real part,then thereexists a positive constant △=△ such that(4)possesses at least one rootwith positive real part,provided that(?)On the basis of Lemma 2 we haveTheorem 3.If(3)possesses at least one root with positive real part,hence thetiivial solution of(1)is unstable,then there exists a real positive constant △=△ A(aij,bij)>0 such that the trivial solution of(2)is also unstable,provided that the Tij(t)'sare constants and satisfy the inequality(?)Using the second,type method, this theorem will be improved as follows:Theorem 4.The hypothesis that Tij(t)'s,are constants can be omitted,and theconclusion of theorem 3 still holds,provided that(?)Theorems 3 and 4 are only partial generalizations of Theorem 2 in [1].In orderto extend the range of τ,we have the following extensions: Lemma 3.Let D(0)and(-1)~n-be of different signs,i.e.,equation(3)has oddnumber of roots with positive real parts,and λ=0 is not a root;then for any systemof realnumbers τ_(ij)≥0 (i,j=1,2,…,n),equation(4)possesses at least one root withpositive real part.Theorem 5.Let D(O)and(—1)~n be of different signs.The trivial solution of(1)is unstable,then the trivial solution of(2)is also unstable.In case that equation(3)has even number of roots with positive real parts,andλ=0 is not a root,counter example is given to show that for certain Tij≥0,(4)hasonly roots with negative real parts.Theorem 6.Given a differential-difference equation(?)where the constants a and b satisfy the conditions:(i)a+b>0,(ii)T(t)is a non-negative bounded real continuous function of t(here the uppebound of i(t)can be any finite positive number and has no connection with a and b)then the trivial solution of(5)is unstable.IV.NONLINEAR CASETheorem 7.Let the trivial solution of(1)be asymptotically stable.Given a non-linear system.(?)where(?)and there exists an (?)>O,such that(?)then there exists a positive number △=△(aij,bji,F_2~(i)>0,such that the trivial solution,of(6)is also asymptotically stable,provided thatTheorem 8.Given an