15 July 1953, Volume 3 Issue 3

    

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  ASYMPTOTIC NETS AND HARMONIC CONGRUENCES

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 167-176. https://doi.org/10.12386/A1953sxxb0014

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  As a generalization of the results given by Su(1946),Hsiung(1951)andP.O.Bell(1952)we establish the following main theorem:Let P_1 and P_2 be two points on the asymptotic tangents of α non-ruled non-degenerate surface(P)at α generic point P,and let the congruence(P_1 P_2)be harmonic to the asymptotic net of the surface(P).If PP_3 is the linedrawn from P to intersect the remaining tangents(besides the common tangentP_1 P_2)of the focal nets(P_1)and(P_2)at P_1 and P_2 respectively,then PP_3and P_1P_2 must be conjugate lines with respect to the Wilczynski quadric of thesurface(P)at P.The tangent planes of the focal nets(P_1)and(P_2)at P_1 and P_2 cannotcoincide with each other except when they are tangent to the Wilczynskiquadric of the surface(P)at P,when they are one and the same fixed plane in space which touch all the Wilczynski quadrics of the surface(P).In the case of a conjugate net N_x on a surface(x)(not necessarily inordinary space)the congruence M(?) considered by Bell is harmonic to the netN_x,where M and(?)are two points on the tangents of N_x such that the tan-gent plane of the surface(?)at M(?)passes through(?)(M).In fact,let the Laplace equation of N_x bey=μx-(?),(?)=νx-(?)and let the points M and(?)bey=μx-(?),(?)=νx-(?);then the conditions quoted above give immediately that(?)=(?)=-μν+αμ+bν+c,so that we can putμ=(?),ν=(?),where θ is to satisfy the Laplace equation of N_x.Thus the coordinates ofM aud M are of the formy=θ(?)-(?)x,(?)=θ(?)-(?)x.Hence the nets N_M and(?)are Levy transforms of the net N_x.
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  EXPLICIT FORMULA FOR THE STABLE LAW OF DISTRIBUTION

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 177-185. https://doi.org/10.12386/A1953sxxb0015

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  The stable law of distribution is one whose characteristic function isf(t)=e~(-(c_0-ic_1 sgn t)|t|~α) (0<α<2,c_0>0,|c_1|≤c_0|tan(?)|).(1)Without assuming that(1)represents a characteristic function,but preservingthe conditions α>0(excluding the easy case α=1)and c_0>0,we derive in thispaper the series expansions and asymptotic expansions of the following functions,corresponding to the“probability density function”and the“distribution function”:p(x)=(?)e(-itx)f(t)dt,F(x)=(?)p(y)dy.The summability of p(x),which no longer follows automatically as in theprobability-theoretic case,is proved in passing.Letc=(?),ω=tan(?) (|ω|<π/2)).The results about p(x)are(0 denoting any quantity with |0|<1):(i)0<α<1.p(x)=1/πsum from n=1 to ∞(?)sin n((?)+ωsgn x),(x≠0)(2)p(x)=(?)sum from n=0 to N(?)sin(n+1)(?)+ +(?)|x|~(N=1).(ii)α>1.p(x)=(?)sum from n=0 to ∞(?)sin(n+1)(?),p(x)=(?)sum from n=1 to N(?)sin n((?)+ωsgn x)++(?).(5)By an indirect method we obtain the formulaF(0)=(?).(6)Formulae (2),(3),(4),(5),(6)and F(∞)=1 are sufficient for the derivationof the series expansion and asymptotic expansion of F(x).The method used is elementary,including a contour integration and somelimit passages that can be readily dealt with.
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  ON THE HOMOTOPY GROUPS OF S~ρUS~q Ⅰ.

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 186-189. https://doi.org/10.12386/A1953sxxb0016

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  Let S~p∪S~q be a topological space consisting of two spheres,S~p and S~q,touching at a point x~*.In this note the author contributes the following formula:π_r(S~p∪S~q)=π_r(S~p)+π_r(S~q)+π_(r+1)(S~(p+q))+π_(r+2)(S~(2p+q))+sum from i=2 to ∞π_(r+i)(S~(p+iq)),p>q≥2,r≤2(p+q)-4,where the sign+denotes direct summation and sum from i=2 to ∞ π_(r+i)(S~(p+iq))is the directsum of the non trivial groups concerned.
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  ON THE HOMOTOPY GROUPS OF S~pUS~q,Ⅱ.

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 190-199. https://doi.org/10.12386/A1953sxxb0017

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  Let E_1~r×E_2~r×E_3~r be the product of three euclidean cells and let f_i∈a_i∈π_(ri)(X),i=1,2,3.If the Whitehead products [α_1,α_2],α_2,α_3] and [α_1,α_3] aretrivial,there exist extensions φ_1:E_2~r×E_3~r→X,φ_2:E_l~rxE_3~r→X,and φ_3:E_1~r×E_2~r→X of the Whitehead products [f_2,f_3],[f_1,f_3] and [f_1,f_2] respectively.Define a mapF:(E_1~r×E_2~r×E_3~r)→Xsuch thatF(x×y×z)=φ_1(y×z),if x∈(?)_1~r,y∈E_2~r,z∈E_3~r,=φ_2(x×z),if x∈E_1~r,y∈(?)_2,z∈E_3~r,=φ_3(x×y),if x∈E_1~r,y∈(?)_2,z∈(?)_3~r.The map F contributes an element in π_(r1+r2+r3-1)(X),namely[φ_1,φ_2,φ_3]. Let Y be the topological space obtained from S_1~r×S_2~r×S_3~r by removingthe top dimensional cell.Let ψ_i(i=1,2,3)denote the characteristic maps ofthe r_2+r_3,r_1+r_3 and r_1+r_2 dimensional cells of Y respectively.Then it isproved thatπ_(r_1+r_2+r_3-1)(Y)=π_(r_1+r_2+r_3-1)(S_1~r)+π_(r_1+r_2+r_3-1)(S_2~r)+π_(r_1+r_2+r_3-1)(S_3~r)+[Ψ_1,Ψ_2,Ψ_3],where the sign+denotes direct summation and [ψ_1,ψ_2,ψ_3] is a free groupgenerated by one generator only.In the space X there may be another extension φ' of [f_2,f_3](=0).Thenit is proved that[φ_1,φ_2,φ_3]=[φ_1~',φ_2,φ_3]+∈[f_1,d(φ_1,φ_1~')],∈being+1 or-1 determined by the orientation concerned.The complexes {S~p×S~q} and S~p×S~q are distinguished by their p+q dimen-sional cells e~(p+q) and e_1~(p+q) only.The injectionj:S~p×S~q,S~p∪S~q→{S~p×s~q}∪e_1~(p+q),{S~p×S~q}induces the homomorphismε_(r+1):π_(r+1)(S~p×S~q,S~p∪S~q)→π_(r+1)({S~p×S~q}∪e_1~(p+q),{S~p×S~q}).Here ε_(2(p+q)-2)~(-1)(0) is proved to be at most a 2-group,if p+q is even.Parti-cularly,ε_(2(p+q)-2)is an isomorphism onto if p+q=4 or δ.In virtue of thearguments in[1]together with the new technique about[φ_1,φ_2,φ_3],the authorcalculatedπ_5(S~2∪S~2)and π_(13)(S~p∪S~(8-p)).They areπ_(4r-3)(S~r∪S~r)=π_(4r-3)(S~r)+π_(4r-3)(S~r)+π_(4r-2)(S~(2r))+π_(4r-1)(S~(3p))+π_(4r-1)(S~3p)+π_(4r)(S~(4p))+π_(4r)(S~(4p))+π+(4r)(S~(4p)),if r=2 or 4,andπ_(13)(S~p∪S~(8-p))=π_(13)(S~p)+π_(13)(S~(8-p))+π_(14)(S~8)+π_(15)(S~(8-p))+π_(16)(S~16)+sum from i=0 to ∞π_(16+i)(S~(16-p+i(8-p))),p=5 or 6.
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  A RELATION BETWEEN CHARACTERISTIC VALUES AND SINGULAR VALUES OF LINEAR INTEGRAL EQUATIONS

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 200-207. https://doi.org/10.12386/A1953sxxb0018

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  Let K(x,y)be a L~2 kernel so that (?)there are two kinds of eigen values connected with K(x,y)the characteristicvalues and the singular values.The characteristic values of K(x,y)are thecomplex numbers μ such that(?)for at least one real non-null L~2 function (?)(x).The singular values of K(x,y)are the non-negative real numbers λ such that λ~2 is one of the characteristicvalues of(?)Arranging the μ's as well as the λ's in the usual way so that |μ_1|≦|μ_2|≦……and λ_1≦k_2≦……,we have the inequalityλ_1≦|μ_1|,(1)and its immediate consequenceλ_1λ_2……λ_n≦|μ_1μ_2……μ_n|(n=1,2,3,……).(2)(1) and (2) were proved by the author~[1] in 1949.Later,in a paper~[10] ofH.Weyl,by means of these inequalities he proved a theorem of the author~[2].In this note,I shall present my original proof as follows.As a further deduction,I shall also prove the following theorem which givesa new property of determinants.Theorem:For any L~2 function K(x,y)there exists a decomposition ofthe determinant det|K(s_i,t_j)|of the n-th order into at least 2n factors inthe sense that(?)(S,T),where(?)(S,T)denotes(?)(?)(S,T)denotes(?)(S,T)etc. and each(?)(S,T)is a L~2function so that (?)If further(?)(ρ=0,1,2,……,p)are continuous,then there is a decomposition of the determinant det |K(s_i,t_j)|of n-th order into at least 2np factorsdet|K(s_i,t_j)|=(?)(S,T),where each(?)(S,T)is a L~2 function.
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  THE PRINCIPLE OF AREA IN THE THEORY OF UNIVALENT FUNCTIONS.

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 208-212. https://doi.org/10.12386/A1953sxxb0019

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  Let F(?) be regular in the domain G:1<|ξ|<∞,Q_m(t)be a polynomial of degree m.Write(?)Wolibner~[1] provedthat if the inequality (?)holds for any Q_m(t),then F(ξ)is univalent in G.On the otherhand,Golousin~[2] proves by the Principle of area that Wolib-ner's condition is aslo necessay for the univalency of F(ξ).In this paper,we prove that if F(ξ)satisfies Wolibner's condition,thenthe area of the image in the Z plane of the domain G by(?)is at most equal to(?)whereξ_μ,γ_μare arbitrary constants,and|ξ_μ|>1.From this theorem,we con-clude that if F(ξ)satisfies Wolibner's condition then(?)and(?)where λ_m(m=1,2,……)are arbitrary constants and d_(mn) are the coefficientsin the expression(?)And hence F(ξ) must be univalent in G.This gives another proof of Wolibner'stheorem.The two known results(2)and(3)have been proved by Golousin~[3,6],Schiffer~[5] and the present author~[4] by the method of variation as well as themethod of L(?)wner's parametric representation.It should be noticed that,wehave established(2)and(3)only using the principle of area.
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  ON OVALS OF n-TYPE

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 213-217. https://doi.org/10.12386/A1953sxxb0020

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  Ler ρ(?)be the radius of curvature of an oval C at a generic point andlet(?)be the Fourier coefficients;then C is said of n-type if α_1=α_2=……=α_n=0,b_1=b_2=……=b_n=0,but at least one of α_(n+1) and b_(n+1) is different from zero.We prove the following two theorems:Theorem 1.An oval of 2(2m+1)-type has at least 2m+3 squarescircumscribed to it(m=0,1,2…).Theorem 2.On an oval of n-type there are at least (n+1) or (n+2)pairs of opposite points such that the sum of the radii of curvature at eachpair is an extremum according as n is odd or even.These furnish a generalization of the theorems given by P.Ganapthi forSu ovals(m=0).
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  A NOTE ON DIOPHANTINE INEQUALITY WITH PRIME UNKNOWN

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 218-224. https://doi.org/10.12386/A1953sxxb0021

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  The object of this note is to prove the following theorem.Let n≧2,f(x)=α_(n+1)1~(x~(n+1))+……+α_1x be a polynomial with real coefficients,x be a posi-tive constant,α_r is one of α_n,……,α_2.(?)then there is a positive constant C such that as q>C,the system of inequa-lities:(?)has a prime solution in p and an integral solution in v for any real number A,where(?)The theorem has its origin in a paper of Vinogradow.There he establish-ed the theorem by using Fourier series method and without adding restrictionon the unknown.
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  THE FUNCTION k(t)IN GOLUSIN AND L(?)WNER'S DIFFERENTIAL EQUATION

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 225-230. https://doi.org/10.12386/A1953sxxb0022

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  Let s be the class of function f(z)=z+c_2 z~2+……,each of which is re-gular and schlicht in the unit circle |z|<1.It is well-kown that~[1,2]any functionof s can be approximated by the functions of a certain subclass s' of s.Cor-responding to any function f(z) of s',there is a continuous function k(t),|k(t)|=1,with possible exception of a finite number of discontinuities for t≧0,suchthat the solution f(z,t)of the equation(?)with the initial condition (?)satisfies the relation(?)Conversely,given a piece-wise continuous function k(t),|k(t)|=1, t≥0,thesolution w=f(z,t),(?) of the corresponding differential equation(1)isregular and schlicht in |z|<1 and bounded by 1 in |z|<1.Moreover,the limit(2)exists~[4,5].In the present note,we give all such functions of s',which correspond tothe functions k(t) having the type k(t)=e(~iθ)p(t),where p(t) is a real function,θis a real constant,and |p(t)|=1.We obtain the followingTheorem.If the function k(t) in the equation(1)is of the form:k(t)=e~(iθ)p(t),p(t)(?)0,|k(t)|=|p(t)|=1,then the solutioh f(z,t),with f(z,0)=z,satisfies the relation(?)where c is a real constant such that |c|≤2.The converse holds true.
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  DISTORSION THEOREMS AND COEFFICIENTS OF SCHLICHT FUNCTIONS

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 231-250. https://doi.org/10.12386/A1953sxxb0023

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  Let S be the class of functions f(z)=z+c_2z~2+……each of which isregular and schlicht in the unit circle |z|<1.The functions f(z)of S with thefixed coefficient c_2 form a subclass of S.We denote this subclass by S(c_2).Evidently,any function of S must belong to a certain S(c_2).It is interest to discuss the distorsion theorem for the functions of S(c_2).In 1920,Gronwall~[1] estimated |f(z)|and|f'(z)|.In 1951,Lebedjef andMielin~[2] point out that the result of Gronwall for the upper bound of |f(z)|when |c_2|<1 is not true,and prove that(?)and that(?)where f(z)∈S(c_2),r=[z].On the other hand,Golusin~[3] established the in-equality(?)In this paper,we improve Lebedjef and Mielin's results(1)and(2)andaccordingly we estimate the coefficients of functions in S(c_2).We prove thefollowing theorems.Theorem 1.If f(z)=z+c_2 Z~2+……S,then(?) where(?)Corollary.If f(z)=z+c_2z~2+……∈S,then(?)where(?)Theorem 2.If f(z)=z+c_2z~2+……S,then(?)where |Z|=γ,0<γ<1,|c_2|≤2.Corollary.If f(z)=z+c_2z~2+……S,then(?)where |z|=r,0<γ<1,|c_2≤2.It should be observed that our inequalities improve Lebedjef and Mielin'sresults(1)and(2)in various ways.Theorem 3.let f(Z)=z+c_2z~2+……S.(i)If(?)then(?) (?)whereCorollary.Let f(z)=z+c_2z~2+……∈S.(i)If(?)then(?)(?)0<γ<1,(?)where |z|=γ,0<γ<1.Theorem 5.If (?)then,for n>2,(?)Corollary. Let(?)then(?) and,in particular,if c_2=0,then(?)It follows from this corollary that in case(?)(?)Theorem(?)(?)The proofs of these theorems depend upon the following lemmae.Lemma 1.Let f(z)=z+c_2z~2+……∈S,then(?)where γ=|z|.Corollary.Let f(z)=z+c_2z~2+……∈S(c_2),then(?)where γ=|z|.Lemma 2.Let f(z)=z+c_2 z~2+c_3z~3+……∈S,then(?)where(?)Corollary.Let f(z)=z+c_2z~2+c_3z~3+……∈S,If c_2~2=c_3,then(?)where(?)Lemma 3.If f(z)∈S,|z|=γ<1,then(?)This result improves the well-known inequality(?)Corollary. If f(z)∈S,then (?)implies(?)where |z|=γ.
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  SOME THEOREMS ON SYMMETRIC SCHLICHT FUNCTIONS

  Acta Mathematica Sinica, Chinese Series. 1953, 3(3): 251-260. https://doi.org/10.12386/A1953sxxb0024

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  Let the k-symmetric funtions(?)be regular andschlicht in the unit circle |z|<1,and the k-symmetric functions F_k(?)(?)be regular and schlicht in the region 1<|ξ|<∞.In this note,we obtain the following theorems:(?)In particular if a_3~((2))=0,then(?)In particular if a_4~((3))=0,then(?)(?)which is equivalent to(?)(?) (?)(?)whereξ_1,ξ-2 are any two points in the region 1<|ξ|<∞,and w_1,w_2,……,w_(2k)are the roots of x~(2k)-1=0.The theorems 1,3 and 2,improve Takahashi's results~[4](?)andLevin's results~[3] |α_n~((2))|<3.39 respectively.The theorem 4 is an extension of Basilewitsch's~[8] inequality(?)