Showing papers in "Osaka Journal of Mathematics in 1974"

The affine structures on the real two-torus. I

Abstract: In these two papers we intend to study the space of all affine structures on the real 2-dimensional torus T, a problem suggested by C. Ehresmann in 1936, or more specifically by S. S. Chern in one of his lectures and attacked by N. H. Kuiper [6] among others. An affine structure on a mainfold is a maximal atlas whose coordinate transformations belong to the affine transformation group A(ri) on the affine space. Our main purpose is to describe the set {Γ} of all affine structures on T module the group Diίf \\T2]e\\ here Diff [T ],, is the group of all diίfeomorphisms of T which induce the identity on the fundamental group π^T). The space {r}/Diff[T2]e, equipped with an appropriate topology, is regarded as an affine version of the Teichmϋller space. In the usual case the holonomy group H of an affine structure on a mainfold is defined as a subgroup of the affine transformation group A(n) up to the conjugate class. In this work, however, we construct a modified holomony group if* for an affine structure so that in the case of 2-dimensional affine torus the group H* is a subgroup ofA(2)e, the universal covering group of the identity component of A(2). We do this in such a way that the modified holonomy group H* is mapped onto the usual holonomy group by the projection mapping. With this modification of the holonomy group the first main result in the paper could be summarized as follows (Theorem 3.3 and 4.15): the affine structures on T are completely determined by their modified holonomy groups #*. Carrying out the determination of holonomy groups Jΐ*, we describe the space {Γh} IΌiff[T 2]e of the homogeneous affine structures on T . As Y. Matsushima [7] discusses for complex tori in a somewhat different way, we show the following (Theorem 3.10): the space {ΓA}/DiίF[Γ 2]β is an affine algebraic or, more precisely a 4-dimensional quadratic cone in R without singularities variety, except at the vertex, the vertex itself corresponding to the natural affine

On the spectral distribution of a disordered system and the range of a random walk

Abstract: ( 1 ) (H\"u)(a) = (Hu)(a)-q(a, ω)u(ά) , The operator —H, considered as a linear transform over C0(Z ), is a nonnegative definite symmetric operator on L(Z) and has a unique self-adjoint extension —ΐf. Express — H as — H=\\ xdEl by the associated spectral J[o,«0 family {£\"\", — oo <ΛJ< 00} and put pω(^)=(£'\"/0, 70), where ( , ) is the L -inner product and /β(β )δββ> > tfeZ. Denote by < > the expectation with respect to the probability measure P and set

Korteweg-de Vries equation: construction of solutions in terms of scattering data

Abstract: Gardner, Greene, Kruskal and Miura (GGKM) [4] have discovered that the solution of the KdV equation can be described in terms of the scattering data of one-dimensional Scrodinger operator. They also proposed that the formalism of inverse scattering theory gives a certain explicit realization of solutions. If the reflection coefficient identically vanishes, the fundamental integral equation of the inverse problem reduces to linear algebraic equations and the potential is expressed in closed form in terms of exponentials. GGKM have also recognized that solutions of the KdV equation associated with reflectionless potentials play an important role in the study of asymptotic behavior of general solutions. See [13] for detailed discussion of reflectionless solutions of the KdV equatino. An analogue of the GGKM theory for (0.1) has been initiated by Zakharov and Shabat [19]. They found that solutions of (0.1) are de-

On symmetric structure of a finite set

Abstract: A symmetric structure of a finite set A is defined to be a mapping S of A into the group of permutations on A (the image of an element a in A by S is denoted by Sa or by S[a] and the image of an element b in A by a permutation Sa is denoted by bSa) such that (i) aSa=a, (ii) Sl=I (the identity permutation) and (iii) S[bSa]=SaSbSa for a and b in A. A set with a symmteric structure is called a symmetric set (with a given symmetric structure). Every group G has a symmetric structure S defined by bSa=ab~ a for a and b in G, and when we regard a group as a symmetric set we always take this symmetric structure. Generally a symmetric set has a more complicate structure than a group and to develop a structure theory of a symmetric set seems to be an open problem. In this note, we first investigate the following two conditions.

On the hypoellipticity and the global analytic-hypoellipticity of pseudo-differential operators

Abstract: In the recent paper [13] Kumano-go and Taniguchi have studied by using oscillatory integrals when pseudo-differential operators in R are Fredholm type and examined whether or not the operators Lk(x, Dx, Dy)=Dx-\-ix kDy in Mizohata [15] and L+(x, Dx, Dy)=Dx±ixDl in Kannai [6] are hypoelliptic by a unified method. In the present paper we shall give the detailed description for results obtained in [13] and study the hypoellipticity for the operator of the form L= X] a^r^x^J^Ό^Ό'ζ with semi-homogeneity in (x9 y> Dχy Dy) | | + | ' ' | ^ l by deriving the similar inequality to that of Grushin [4] for the elliptic case. Then we can treat the semi-elliptic case as well as the elliptic case. We shall also give a theorem on the global analytic-hypoellipticity of a non-elliptic operator, and applying it give a necessary and sufficient condition for the operator L(x> Dχy Dy) to be hypoelliptic, when the coefficients of L are independent of Ψ* (see Theorem 3.1). In Section 1 we shall describe pseudo-differential operators of class 5 ^ p δ which is defined by using a basic weight function \ = χ ( x , ξ) varying in x and ξ (cf. [13] and also [1]). In Section 2 we shall study the global analytic-hypoellipticity of a non-elliptic pseudo-differential operator and give an example which indicates that the condition (2.3) is necessary in general. In Section 3 we shall consider the local hypoellipticity for the operator L and give some examples. The author wishes to thank Prof. H. Kumano-go for suggesting this problem and his helpful advice.

On strongly invariant coefficient rings

Abstract: Let A and B be rings with an identity. After P. Eakin and W. Heinzer we shall say that A and B are stably equivalent if there is an integer n such that polynomial rings A[Xίy •••, Xn] and B[Yly • ••, Yn] are isomorphic [3]. A number of recent investigations have been published concerning the study of this equivalence. One of the interesting question is the one, called the cancellation problem for rings, which asks when the stably equivalence implies the isomorphism. A ring with this property will be called an invariant ring. This paper contains some contribution to this problem. In §1 we shall take up the notions of strongly invriant rings which are defined by several authors in their own way and make it clear the relationship among them. In §2 we shall consider the following problem: Let A be a strongly invariant ring. What conditions on A guarantees the invariance of the polynomial ring A[X] ? Several conditions will be given. In the last section we shall give some examples of strongly invariant rings which have not such a nice property. As a result we can give examples of non invariant rings which are two dimensional affine domains over a field of positive characteristic.

On a characterization of a class of the regular graphs of diameter 2

Abstract: S.S. Schrikhande [2], [3], W.S. Conner [4] and AJ. Hoffman [5] determined all of the graphs with intersection matrices B(2t—2, t—2y 2) for t^2 and B (2ί-4, t-2, 4) for t^4. The lattice graphs of dimension 2 with intersection matrices B(2t—2> t—2,2) for £̂ >3 and the triangle graphs with intersection matrices B(2t—4, t—2, 4) for t^>6 have the remarkable property that for any three vertices which are not joined to each other, no vertex is joined to all of these three vertices. The purpose of this paper is to prove the following.

On the Cauchy problem for parabolic pseudo-differential equations

Abstract: In the recent paper [8] S. Kaplan has obtained an analogue of Garding's inequality for parabolic differential operators and applied it to a Hubert space treatment of the Cauchy problem. D. Ellis [3] has extended those results to higher order parabolic differential operators (see also [4]). On the other hand in [13] the author has studied a Hubert space treatment of the Cauchy problem for parabolic pseudo-differential equations and generalized the results of S. Kaplan [8]. In the present paper we shall study the Cauchy problem for higher order parabolic pseudo-differential equations of the form

On multiply transitive permutation groups iii

Abstract: This Theorem 1 is a kind of (but not full even in the case of generalization of the result (Theorem 1) in [1]. In the case of p^ 11, the Main Theorem in [2] (i.e., the determination of 2/>-ply transitive permutation groups whose stabilizer of 2p points is of order prime to p) is also completed alternatively by combining this Theorem 1 with the result of Miyamoto [6]. The brief outline of the proof of Theorem 1 is as follows. The proof will be done by the way of contradiction. First we will show that the symmetric group Sp is not involved in the group GL(p—3, p), if />> 11 (Theorem A). This is proved by the similar argument as in [1, §1], by exploiting the (ordinary, modular and projective) representation theories of the symmetric groups. Next, we will restrict the structure of the Sylow^ subgroup Po of G12...p. That is, if I Po I >p p+1 then we have | Z(P0) \\ >p, and moreover we can lead a contradiction by using the well known theorem of Burnside on fusion of elements in the center of a sylow p subgroup and by using a consequence (Theorem B) of Theorem A. If | P 0 | , then we can show (also by using Theorem A) that we have only one possibility for the structure of Po, namely, P o is isomorphic to the extraspecial/) group of order p and of exponent/). Finally, we exclude this remaining case and complete the proof of Theorem 1. This is done by considering the fusion of p elements in Po. The proof of this final step was provided by T. Yoshida. The author thanks Mr. Tomoyuki Yoshida for providing the argument of

Modules over dedekind prime rings iii

Abstract: This paper is mainly concerned with the investigation of modules over Dedekind prime rings. Throughout this paper R will denote a Dedekind prime ring and P will denote a nonzero prime ideal of R. For an exact sequence (E): 0-+L—>M->N^0 of right /^-modules, we shall define the concepts of P "-purity (ntίω), P°°-purity and T°°-purity as follows: (i) (E) is P-pure if and only if MP Π L=LP for every natural number

The complex bordism of cyclic groups

Abstract: Introduction. In their book, Differentίable Periodic Maps [2], P.E. Conner and E.E. Floyd initiated the study of cobordism groups of periodic maps and succeeded in determining the additive structure of the cobordism groups of free orientation-preserving ^-actions on manifolds for odd primes p and of free i n actions preserving a stably almost-complex structure for arbitrary primes by calculating MSO*{BZP) and MU*(BZP) respectively. Kamata [5] obtained the same results for MU*(BZP) using slightly different methods. We extend these results to a determination of MU*(BG) where G is an arbitrary cyclic group. The main result is Proposition 16: