中国科学院数学与系统科学研究院期刊网

15 April 1953, Volume 3 Issue 2
    

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  • Acta Mathematica Sinica, Chinese Series. 1953, 3(2): 87-100. https://doi.org/10.12386/A1953sxxb0007
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    Es wird in der vorliegenden Arbeit beabsichtigt,die Methode derAhlfors'schen Theorie der (?)berlagerungsfl(?)chen in ihrer Anwendung auf dieWertverteilungstheorie der meromorphen Funktionen weiter auszubauen umdadurch einig Resultate,die bisher nut im“Punktfall”bekannt sind,auch auf“Gebietsfall”zu verallgemeinern.Der Ⅱ.Abschnitt befasst,sich mit der Defektrelation in hyperbolischenFalle,wobei eine der Nevanlinnaschen ganz entsprechende Form erzielt wird.Der Ⅲ.Abschnitt befasst sich mit der Millouxschen“Cercles de remplissage”entsprechenden Erscheinungen bei meromorphen Funktionen nullter,Ordnungim parabolischen Falle,w(?)hrend der Ⅳ.Abschnitt mit der BorelschenRichtung bei Funktionen endlicher Ordnung.
  • CHANG HSIAO-LI
    Acta Mathematica Sinica, Chinese Series. 1953, 3(2): 123-132. https://doi.org/10.12386/A1953sxxb0009
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  • Acta Mathematica Sinica, Chinese Series. 1953, 3(2): 133-141. https://doi.org/10.12386/A1953sxxb0010
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    These notes are devoted to the solutions of two simple problems con-cerning the permutability of congruence relations on lattices and on quasigroups,which,were announced by G.Birkhoff as unsolved in his“Lattice Theory”(revised edition,1948.We shall refer to it simply as LT_2.)In §1,we prove at first:Theorem 1.1.Any two congruence relations on α relatively complementedlattice are permutable.In LT_2,this fact was a conjecture(see p.86,ex.3).As an applicationof it,we prove:Theorem 1.2.A necessary(and sufficient)condition that the congruence.relations on α complemented modular lattice L should form α Boolean algebrais that all neutral ideals of L be principal.This theorem is relevant to Problem 72 of LT_2(see p.153).In §2 is given a complete solution to Problem 31 of LT_2(see p.86).~(1))For the finite case of this problem we prove:Theorem 2.1.Any two congruence relations on α finite quasigroup arepermutable.As to the infinite case,we first construct a class of quasigroups and loopsin number,where c is the cardinal number of the continuum of N_0 elementswith non-permutable congruence relations.Next we prove the further result:Theorem.2.2.For any cardinal number α≧N_0,there exist(many)quasigroups and loops of α elements with non-permutable congruence relations.(The proof of Theorem 2.2 is based on commonly accepted formulas ofset theory.)1)The referee points out that a solution of this problem has already been reported according toa 1952 article of G.Birkhoff.But as the details of the reported solution are unavailable in China,the author cannot determine whether the present proof is superfluous.
  • Acta Mathematica Sinica, Chinese Series. 1953, 3(2): 142-147. https://doi.org/10.12386/A1953sxxb0011
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    It is known that the Laplace,Fourier and Mellin transforms can all beinverted under the principal assumption(Jordan's condition)that the deter-mining function in question is of bounded variation in a neighborhood of thepoint of inversion.In this note,inversion theorems for the Fourier and Mellintransforms have been established,under more general conditions in Which theterm 'of bounded variation'is used in the sense of Saks.A typical result,as stated in a less general form is the following(cf.Corollary ofTheorem 2 of [1]):Theorem.~(l)) If t_(γ-l) f(t)∈L(0,∞)and if there is α linear measurable setE contained in some interval I(?)(0,∞)such that f(t)is of bounded variation(in the sense of Saks)on both E and CE(with regard to I),then for anyinterior point t=x of increasing density of E,the Mellin transformF(s)=(?)f(t)t~(s-1)dt (s=γ=iw)can always be inverted as~(2))1/2[f(x+0)+f(x-0)]=1/2πi (?)F(s)x~(-s)ds.The proof depends partly upon an analogous theorem for the Fouriertransform,and partly upon the following lemma:If t_0 is an interior pointof density of a measurable set E and if the average measure function Δ(t_0,t)of E is of bounded variation near t_0,then,correspondingly,the set G formedby all the points ξ=log t(t∈E)contains ξ_0=logt_0 as its point of densityand the average measure function (?)(ξ_0,ξ)of G is of bounded variation near ξ_0.1)In fact,the theorem is true for any density poin t=x of E such that the average measurefunction Δ(x,l)of E is of bounded variation near t=x.2)Here f(x+0)denotes the limit of f(x)as x→0+in the set E.The meaning off(x-0)is similar.
  • Acta Mathematica Sinica, Chinese Series. 1953, 3(2): 148-153. https://doi.org/10.12386/A1953sxxb0012
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    In the first section of this note,we deduce,with the aid of theapproximation theorem for an integrable function,the following result from atheorem of Wilkins concerning Maréchal's integration scheme:If G(x)∈L(a,b)and if F(x,X,Y)is continuons on α≦x≦b,|X|≦1,|Y|≦1,then(1)(?)F(x,cosλx,sinλx)G(x)dx=1/2π(?)F(x,cosφ,sinφ)G(x)dx dφ.The Riemann-Lebesgue theorem is a special case with F(x,cosφ,sinφ)=cosφor sinφ.In particular,if(?)where S(z) is analytic inside,and on the unit circle |z|=1,except at afinite number,of singularities in |z|<1,then an application of the residuetheorem gives (2)(?)F(cosλx,sinλx)G(x)dx=i∑R_k·(?)G(x)dx,where ΣR_k is the sum of residues of S(z)inside |z|=1.In the second section,we consider a type of integration which approxi-mates a double intehral over a region D,where D is bounded by continuouscurves y=f_1(x),y=f_2(x)and lines x=α,x=b.Under general conditions itis shown that(3)(?)F(x,g_λ(x))h_λ(x)dx=∫∫_D F(x,y)dx dy,where g_λ(x)=cos~2(λx)f_1(x)+sin~2(λx)f_2(x),h_λ(x)=(f_1(x)-f_2(x))|sin(2λx)|,f_1(x)≥f_2(x),(a≤x≤b).
  • Acta Mathematica Sinica, Chinese Series. 1953, 3(2): 154-165. https://doi.org/10.12386/A1953sxxb0013
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    Given a metric space E,let us denote by T(E)its“complete envelope”(vollst(?)ndige H(?)lle,as called by Hausdorff~[4]).By a further investigationon the relation between E and Γ(E),we obtain the following main results:1)Let E_i be a finite or infinite sequence of metric spaces with the metricd_i on each term E_i.Then for properly chosen metrics on the product spacesΠmultiply from i E_i and Πmultiply from i Γ(E_i),Γ(Πmultiply from i E_i)andΠmultiply from i Γ(E_i)are isometric,in symbols:2)A metric space which is“convex”in Menger's sense~[6] will bereferred to,for obvious reasons,as quasi-convex.We mean here by a convexspace a metric space E which is such that for each pair of points of E thereis a mid-point.With this definition,it is easily shown that the convexity ofE implies the connectedness of Γ(E).Given a convex space E,it remains aninteresting open question whether or not its complete envelope Γ(E)is convex.To this question,we can give an affirmative answer in case that the convexspace E satisfies the axiom that every bounded subset is totally bounded.Hence for such spaces E,Γ(E)are arcwise connected by Menger's ExistenceTheorem of Geodesics~[6].3)By means of completion we obtain the following criterion for localcompleteness of D.Montgomery~[7]:E is locally complete if and only if E isopen in Γ(E).