中国科学院数学与系统科学研究院期刊网

15 May 1952, Volume 2 Issue 3
    

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  • Acta Mathematica Sinica, Chinese Series. 1952, 2(3): 133-138. https://doi.org/10.12386/A1952sxxb0007
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    An analytic proof for the equivalence of two affine spaces and twometric spaces in terms of their curvature tensors was given in a previouspaper[1]by making use of a known theorem about differential equationsof mixed system.[2]In the present note we shall give a proof for thesame purpose without any application of the theorem just mentioned,instead of which some identities[3][4]in curvature tensor and its covariantderivatives are used.
  • Acta Mathematica Sinica, Chinese Series. 1952, 2(3): 139-143. https://doi.org/10.12386/A1952sxxb0008
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    Having succeeded in solving Taiski's“plank problem”,T.Bang re-marked at the end of his note[1],that the following deeper problem was stillunsolved:Whether is the sum of the relative widths of the strips alwaysgreater than or equal to 1,when a convex body in n-dimensional Euclideanspace,is entirely covered by these strips,where by a strip of width h wemean the part of space lying between two parallel hyperplanes whose dis-tance is h,while by the relative width of a strip we mean the ratio of itswidth with the width in the same direction of the covered convex body?We give here an affirmative answer to this problem.
  • Acta Mathematica Sinica, Chinese Series. 1952, 2(3): 144-156. https://doi.org/10.12386/A1952sxxb0009
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    Es wird in dieser Arbeit die folgenden S(?)tze bewiesen.Satz 1.Wenn zwischen zweier Fl(?)chen eine Abbildung besteht,welche die beiden Scharen der isotropen Linien der ersten Fl(?)che auf zweiScharen von komplexen geod(?)tischen Linien der zweiten Fl(?)che abbildet,und zugleich die beiden Scharen der isotropen Linien der zweiten Fl(?)cheauf zwei Scharen ven komplexen geod(?)tischen Linien der ersten Fl(?)cheabbildet,so ist die Abbildung selbst geod(?)tisch.Die Fundamentalformen der beiden Fl(?)chen lassen sich auf die klas-sische Dinische Formel bringen.Satz 2.Angenommen zwei Riemannsche R(?)ume hagen ein gemein-sames Orthogonalbezugssystem und es besteht eine Abbildung der beidenR(?)ume aufeinander,welche ein gewisses System von 2~(n-1)isotropenKurvenkongruenzen des ersten Raumes auf 2~(n-1) komplexe geod(?)tischeNormalenkongruenzen des zweiten Raumes abbildet,und zugleich eingewisses System von 2~(n-1)isotropen Kurvenkongruenzen des zweitenRaumes auf 2~(n-1)komplexe geod(?)tische Normalenkongruenzen des erstenRaumes abbildet,so ist die Abbildung selbst geod(?)itiscb.Die Funda-mentaltensoren der beiden R(?)ume lassen sich auf die klassiche Formel vonLevi-civita bringen.
  • Acta Mathematica Sinica, Chinese Series. 1952, 2(3): 157-166. https://doi.org/10.12386/A1952sxxb0010
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    The projective differential geometry of certain pairs of plane curvesat ordmary points has been investigaled by Hsiung[1],[2].The purposeof the present paper is to use the same method to study the projetive dif-ferential geometry of certain pairs of plane curves for the following cases:(1)Two curves in a plane intersecting at an inflexion point((?)2.3.4).(2)Two curyes in a plane having a common tangent at two infle,ion points((?)5).In the present paper we give a detailed discussion about the curves incase(1),while the theory aboter the curves in case(2)is easily obtainedfrom this.For the previous case we obtain a projective invariant whichis determined by the neighborhood of the fifth order of the curves at thepoint in question,and by using Bompiani's osculants[3]of the curves atthis point we give the projective invariant a simple geometric charac-terization.Moreover,by suitably choosing a covariant point for the,unitpoint at the,coordinate system,we obtain the canonical power series ex-pansions of the curves at the considered point.According to the invariantobtained vanishes or not,we have four different types.The interpretationof the absolute invariants in the expansions of each type is given interms of certain cross ratios alone.
  • Acta Mathematica Sinica, Chinese Series. 1952, 2(3): 167-202. https://doi.org/10.12386/A1952sxxb0011
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    If■is a group of order g=pg',(p a prime number,(g',p)=1),rather complete results can be ebtaiued describing the behaviour of thecharacters with regard to the prime number p.In this paper,groups oforder g=p~2g',((p,g')=1)are inverstigated.The situation here is farmore complicated,since not only blocks of defects 0 and 1 but also blocksof defect 2 appear about which not much is known.The group(?)contains a group μof older p~2.We have here twocares:μmay be abelian of type(p,p)or(?)may be cyclic.We shall re-strict our work to the first case.Actually the method apply to the secondcase too and the situation there is much simpler than in the first case.We shall consider the nomalizers and centralizers of(?)and of the sub-groups of order p of(?).Those groups may be considered as of a less com-plicated nature than(?),since we can describe them easily by means ofgroups whose orders contain the given prime p only to the first power.The aim of our inveratigation is to show that the behaviour of thecharacters of (?) depends strongly on the structure of these subgroups.Anumber of connections between the characters of (?) and the characters ofthe snbgroups mentioned are obtained.We begin with a discussion of the centralizer (?) and the norma-lizer (?).The group (?) is the direct product (?)×(?) of (?) and a group(?) of an order prime to p.The faotor group is either abelian or a groupof well known type.The blocks of defect 2 of (?) are in one-to-one cor- respondence to the classes of irreducible characters (?) of (?) which areassociated in (?).Every block of defect 1 has a defect group (?) of orderp which is determined uniquely,if coujugate groups are considered as notessontially different.The centralizer (?) is again a direct product(?)×(?) and the blocks with the defect group (?) are in one-to-one corre-spondence to the classes of characters of defect zero of (?) associated in thenormalizer (?)of (?).The values of the characters of (?) are described largely by the de-composition numbers d_(μ(?))~i.The d_(μ(?))~i are elements of a cyclotomic field.Inthe later sections,the nature of the occuring irrationallities is studiedmore closely.In particular,we are again interested in the question towhat extent the nature of the irrationalities depends on the structure ofthe groups (?),(?),(?)and(?).I an indebted to Professor Richard Brauer for his help.