中国科学院数学与系统科学研究院期刊网

15 April 1959, Volume 9 Issue 2
    

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  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 87-100. https://doi.org/10.12386/A1959sxxb0010
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    In this paper,we give the details of the proofs of the following three theorems(Cf.Science Record,Academia Sinica,New Ser.Vol.I,No.3,1957,pp.1—5).Theorem 1.Let F(x)be an irreducible integral valued polynomial of degree k withoutany fixed prime divisor.Let(?)where w is the least integer satisfying(?)Then there are infinitely many integers x such that F(x)is a product of at most n primes.Theorem 2.Let k be a positive integer.Let n be an integer satisfying(1)and(2).Then for sufficiently large x,there is always an integer between x and x+x~(1/k)which has at most n prime factors.Theorem 3.For sufficiently large x
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 101-113. https://doi.org/10.12386/A1959sxxb0011
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    Given a polynomial with real Coefficients,it is well-known that Routh's method can be usedto determine the number of complex roots of the given polynomial in the right half-plane.It isobvious that this method can be used to determine the real part of a pair of conjugate imaginarytools.In this paper,the author proves that:(1),the imaginary part of the pair of conjugate imaginary roots with the largest real partcan also be found by Routh's method.(2)When we have a pair oLd'roots;real or complex,in the neighborhood of the origin,Routh's method gives in geaneral better results than roots determined by the tail quadratic formof the given polynomial.(3)When the imaginary part of the roots is not small,linear interpolation can be usedto determine a better approximation.Finally,using Routh's method,a method for determiningthe number of imaginary roots is devised.Numerous illustrative examples are given.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 114-120. https://doi.org/10.12386/A1959sxxb0012
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    The main purpose of:the present paper is to investigate some theorems as following:Let f(z)be an analytic function,regular in a region D,which is an interior regionbounded by a certain rectifiable Jordan curve Γ,composed of two curves Γ′and Γ″,takinga b as its two end points.Let f(z)be cintinuous on Γ′(including the end points).Funthermore,we assume there e.xists a Sequence of simple rectifiable curves {λ_n},whichconverges to curve Γ″,belonging to D except its end poiuts z′_n and z″_n belonging to Γ″, and such that(?)then f(z)≡0 in D.Assume the same assumptions of D,г,г′,г″and {λ_n},and assum{f_n(z)} besequence of analytic functions,regular in D,and each f_n(z)be continuous and boundedon Γ′uniformly,i.e.(?)Furthermore suppose,that there exists a certain sequence of positive.numbers {ε_n}→0,as n→+∞,and that for any fixed ε_k and λ_k,there exists a positive integer number N,for m,n≥N,we havethen {f_n(z)} is uniformly convergent in any closed region within D.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 121-142. https://doi.org/10.12386/A1959sxxb0013
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    We make use of Eukasjewicz symbols with the following addition:□a(it is necessarythat a) F_(αβ)(=□C_(αβ)and G_(αβ)(=KF_(αβ)F_(βα)).We start with the weakest(with quite few exceptions) basical modal system A,axiomatized as follows:1.Whenever a is an axiom of the traditional two-valued system,then □a is anaxiom of the system A.2.F□αα3.F_(αβ)→β.4.F_(αβ),□α→□β5.a,β→K_(αβ).6.G_(αβ)→G□α□β. We strengthen thesystem A by addition of the respective axioms or rules:B:F_(αβ)→C□α□β.C:CF_(αβ)□α□β D:FFαβC□α□β.E:F_(αβ)→F□α□βF:C + E.G: D + E.H:CF_(αβ)F□α□βI:D + H.J:FF_(αβ)□α□β.The relation of them may be schematized as follows(from weak to strong):(?)When the system X is added the axiom/rule CG_(αβ)G□α□β, FG_(αβ)G□α□βα→FβK_α(equivalent to α→□α),α→ FF_(αββ),C_αFF_(αββ),F_αFF_(αββ),then the resulting systems willdenoted X_1,X_2,X,X_a,X_b,X_c respectively.By combining some of these additions,we would have,apparently,240 systems.Yetmany of them are identacal.In fact,we have(1)H=H_1,I=I_1,J=J_1=J_2.(2)B=E,C=D=F=G,H=I=J, X_1= X_2.(3)Bi=C_1,B_2=C_2,E_1=H_1(= H)=F_1,G_1=I_1(=I).(4)E_2= F_2=G_2=H_2=I_2=J_2.(5)X_b=X_c,X_a+□F_(aa)=X(Hence X_a=X)(6)B_b+□F_(aa)=E_c,both are identical With two-valued system Hence we have atmost 65 distinct systems (two-valued system excluded).We show that G=S2,J=S3,and the asserted propositions in J are the same asthat in S4,yet the rules of procedure of them are different.We would meet paradoxes of implication even in the weakest system A,e.g.,wehave □α→Fβα and □N_α→F_(αβ).Hence the writer of the present paper would propose-two new implication systems instead.In these two new systems,the concepts F_(αβ)and □αare both primitive,neither of them can be defined by the other.The first implication system may be axiomatized as follows:1.Let the axioms of two-valued system be given in the form C_(αβ),then F_(αβ)are theaxloms of this system.2.FF_(αβ)C_(αβ).3.F_(αβ),α→β.4.FC_(αβ)C_(γδ),F_(αβ).5.α,β→K_(αβ).6.GC_(αβ)C(γδ)→GF_(αβ)F(γδ).7.FK□α□βK_(αβ).8.FF_(αβ)F□α□β.9.F□□αα.10.FF_(αβ)□αβ.11.F□Kαβ□α.The second implication system reads as follows:The rute of detachment and the rule of conjunctin.1.FF_pF_(pq)F(pq).2.F_pFF_(pqq).3.,FF_(pq)FF_(qr)F_(pr),.4.F_(pp).5.FF_pN_qF_qN_p.6.FNN_(pp).7.FK_(pq)K_(qp).8.FK_pK_(qr)KK_(pqr).9.FK_(ppp).10.F_pK_pp.11.FF_()pqFK_(pr),K_(qr),.12—16(the same as 7—11,with“K” replaced by“A”)17.FK_p,A_(qr)AK_(pqr).18.FF_(pq)AN_(pq).19.FK_pNK_(pq)N_q.20.AN_pA_(pq).21.FKN_pN_qNA_(pq).22.FNA_(pq)KN_p,N_q.23.FK□α□βK_(αβ).24.F□K_(αβ)□α.25.FF_(αβ)F□α□β.26.F□aa.27.FF_(αβ)□_(αβ).
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 156-169. https://doi.org/10.12386/A1959sxxb0016
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    (?)[2].(?)(E2)(?)(?)(i=1,2,3).(?)(?)(?)(p)(?)a,b,c,d,x_o,(?)(?) (?)
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 170-173. https://doi.org/10.12386/A1959sxxb0017
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    Let A=(?)be an arbitrary Square matrix,w denotes the maximum absolute valueof characteristic roots of the matrix A,and write(?)It was proved by Farnell and Gautschi that(?)In the present paper we proved that either(?)or(?)where(?)(?)
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 174-180. https://doi.org/10.12386/A1959sxxb0018
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    This paper is a continuation of my paper~[16].Let A=(?)be a square matrix of order n and(?)its characteristic roots.We define(?) (?)(?)where A is the conjugate of the transpose of A and K may assume any integral value.I.Schur~[17] proved the sharper resultso thatThis is sometimes better than the following resultst~[16]:(?)We prove the following four theorems:Theorem 9.(?)Theorem 10.(?)(i)A~k is a normal matrix if and only if,ρ(A,k)=1.(ii)The characteristic roots are all real or all pure imaginary,if and only ifp(A,k)=—1.(iii)A~k is a Hermitan matrix or skew-Hermitan matrix,if and only if p(A:k)=c,where c is an arbitrary real number in [—1,1].Theorem 11.Let F_k be a closed curve:(?)Each characteristic root λ of A lies in the initerior or on the boundary of T_k.If{T_k}→Tthan least one of the characteristtic roots λ of A lies on the boundary of T.Theorem 12.If p_1(α_1,β_1), P_2(α_2,β_2),…, P_A(α_n,β_n)are vertex of a polygon 6,we have(?)
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 181-190. https://doi.org/10.12386/A1959sxxb0019
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    In this paprr we consider the topological structure of the distribution of the integralcurves with one singular point on the torus.First of all,we define three kinds of sectors.We consider the neighborhood of thesingular point by separating it into sectors.Using the property of the torus and the factthat the sum of the index of the singular point is equal to zero,we obtain three classes offoundamental figures(see Tables Ⅰ,Ⅱand Ⅲ,on page 186—187).We define five classes of operations.We proved that from the three classes of founda-mental figures and the five classes of operations we can obtain all possible topologicalstructure with one singular point on the torus.Finally,we proved the differentdability of the figure and considered also rile case of afinite number of singular points on the torus.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 191-198. https://doi.org/10.12386/A1959sxxb0020
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    Ia Riemannian space,E.Cartan had introduced the notions of symmetric transformation ofpoints and symmetric transpose of vectors reference to a fixed point O along the geodesics passingthrough O.Furthermore,he had calculated the difference between the parallel and symmetrictranspose of a vector,by means of which,a special class of Riemannian spaces so called sym-metric Riemannian space was defined in which the difference is an infinitesimal of order three,andmany geometric properties was inserted.In this note,we give a direct extension of these resultsin Finsler spaces,and establish the following theorem:THEOREM In order that the difference in question is an infinitesimal of order twoin a rigion D,it is necessary and sufficient that the covariant derivatives of torsion tensor vanishesin the rigion D.In this theorem, the family of geodesics in the definition of symmetric transpose cannot bereplaced by any other family of curves,though we can give a more general definition of symmetrictranspose of veclors independ on family of geodesics.We call the space in theorem the subsymmetric Finsler space.By a simple calculation,thegeometric definition of another special Finsler space——symmetric Finsler space with its tensorcharacter was given.
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 199-212. https://doi.org/10.12386/A1959sxxb0021
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    (?) (?)
  • Acta Mathematica Sinica, Chinese Series. 1959, 9(2): 213-226. https://doi.org/10.12386/A1959sxxb0022
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    Method of constrnction of concrete examples for existence of limit cycles for the system(dx)/(dt)=X_2(x,y),(dy)/(dt)=Y_2(x,y),where X_2(x,y) and Y_2(x,y)are polynomials of x and y of second degree is given.Basedon this method a concrete example is given as follows:(?)where(?)and it is shown that this system possessesthree limit cycles around the origin(0, 0),hence also on the whole xy plane.Furthermore,it is shown that by rotatng the vector field an angle a=tan~(-1)λ,the resulting systempossesses two and only two limit cycles on the whole xy plane.