15 January 1952, Volume 2 Issue 1

    

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  ON THE AUTOMORPHISMS AND ISOMORPHISMS OF LINEAR GROUPS

  LOO-KENG HUA,HEH-HSTAN&nb

  Acta Mathematica Sinica, Chinese Series. 1952, 2(1): 25-32. https://doi.org/10.12386/A1952sxxb0002

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  A THEOREM ON THE DENSITY OF SETS OF GAUSSIAN INTEGERS

  L.CHEO

  Acta Mathematica Sinica, Chinese Series. 1952, 2(1): 33-44. https://doi.org/10.12386/A1952sxxb0003

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  This is a continuation of the author's“On the density of sets of Gaus-sian integers”in which the author extended the concept of Schnirelman'sdensity of sets of positive integers to that of Gaussian integers.Let A beαset of Gaussian integers α+α'i,with α≧0,α'≧0 but not both 0.Thedensity of A is defined to be thetaken over all Gaussian integers x+yi,x≧0,y≧0 but not both 0,whereA(x+yi)denotes the number of elements α+α'i in A with α≦x,andα'≦y.Let A,B be two sets of Gaussian integers.The sum A+B isdefined to be the set censisting of all elements in A,all elements in B andall sums(α+b)+(α'+b')i with α+α'i being in A,and b+b'i being inB.Let the densities of A,B and A+B be α,βand γrespectively.Theauthor proved the following theorems:Theorem I.If α+β≧1,then γ=1.Theorem Ⅱ.If B contains all numbers ji,j=1,2,...,then γ≧α_0+β-α_0β,where α_0=g.l.b.(?).Theorem Ⅲ.γmay be less than α+β,even if α+β<1.In this paper,the author proved another theorem.Theoren Ⅳ.Letα'=g.l.b.(?)in A,and in B for all positive integral j,then γ≧α'+β'.
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  AN ASYMPTOTIC INVESTIGATION OF A TYPE OF EXPONENTIAL INTEGRALS

  L.C.HSU(1),J.R.RAVETZ(2)

  Acta Mathematica Sinica, Chinese Series. 1952, 2(1): 45-49. https://doi.org/10.12386/A1952sxxb0004

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  The key result contained in this paper is the following:Lemma.Let f(x)εL(0,l),(00 the asymptotic formulaholds uniformly as |s|→∞ in any angular region |arg s|≦π/2-ε,(ε>0).This is an improvement over Hartman-Wintner's result concerningthe asymptotic integral of the type mentioned above(see[1]and[2]).Itis noteworthy that a much stronger condition involved in Hartman'ssecond theorem of[1],namely,|f(x)-f(+0)|·|logx|~(lδ)→0(x→+0),has been now released by our lemma.Moreover,our treatment of thecase δ>0 is a considerable simplification of the original method.Finally,as a consequence of our lemma we prove a more generaltheorem in which the interval of integration is replaced by a linear meas-urable set E whose left-end point x=0 is a point of density belongingto the Lebesgue set of f(x).
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  ON p-GROUPS WITH CYCLIC COMMUTATOR SUBGROUP BELONGING TO THE CENTER

  Acta Mathematica Sinica, Chinese Series. 1952, 2(1): 50-64. https://doi.org/10.12386/A1952sxxb0005

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  Let (?) be a p-group of order p~n,its commutator group D(?) beingcyclic and belonging to the center(?),of(?) Suppose that the orderof D(?)is p~m,and D(?)=(t) where t is a generating element ofD(?).It is proved that the factor group(?)is an Abelian group oftype{p~(m1),p~(m1),p~(m2),p~(m2),...,P~(mr),p~(mr)} where m=m_1≧m_2≧...≧m_r.The order of(?) is p~(2r),r=m_1+m_2+m_r.(?) is the product of its sub-groups (?)When i≠j,elements of (?) commute with elements of (?) and (?)The group (?) (i=1,2,...,r) is defined as(?)the commutator of (?) and(?)Any element X of (?) can be uniquely represented asSuch elements (?) are called basis of (?)The transformation of two bases can be represented by a matrixof (?) where (?) is the set of all 2n-rowed square matrices with integralelements.It is proved that the necessary and sufficient condition for the matrix T to represent such a transformation is that TPT~0≡P(modp~m),where P=diag{p~(m-m_1),p~(m-m_1),p~(m-m_2),p~(m-m_2),...ρ~(m-m_r),ρ~(m-m_r)},and θ be atransformationθ:T←→T~0,(T and T~0 in (?))of(?)specially defined such that(A+B)~0=A~0+B~0,(AB)~0=B~0 A~0,(A~0)~0=A for every A and B of(?) The totality of such matrices Tform a group.
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  ON TARRY'S PROBLEM

  Acta Mathematica Sinica, Chinese Series. 1952, 2(1): 65-132. https://doi.org/10.12386/A1952sxxb0006

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  Let k≧2 and let t_0 be defined by the following table:Let r_t(P)be the number of the system of Diophantine eqnationswith the restriction that1≦xi yi≦pOne of the purposes of the paper is to prove that,for t>t_0,we haveWherewe use the abbreviation e(x)for(?) In 1939,the author[1]prevent that (*)holds forwhich is asymptotically equal to 1.5k~3 logk for large k.(It was publishedin 1947.Particular attentions are also paid to the convergence problem of thereal density(?)and the ρ-adic density(?)which is a factor of our singularseries.To handle the convergence problem of(?)the author uses Youngand Hausdorff's theorem on multiple Fourier integrals.The exponent ofconvergence so obtained is sharper than that obtained previously from theestimation of exponential integrals(Vinogradow[2]).For the ρ-adic density and the convergence of(?)the author introducesa new method by means of which he obtains the best possible value of theconvergence exponent.