Kodai Mathematical Journal

On the Schwarzian differential equation {w, z}=R(z, w)

Katsuya Ishizaki

1997 Volume 20 Issue 2 Pages 67-78
Published: 1997
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2996/kmj/1138043745

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It is showed in this note that if the Schwarzian differential equation (*) {w, z}=R(z, w)=P(z, w)/Q(z, w), where P(z, w) and Q(z, w) are polynomials in w with meromorphic coefficients, possesses an admissible solution w(z), then w(z) satisfies a first order equation of the form (**) (w')²+B(z, w)w'+A(z, w)=0, where B(z, w) and A(z, w), are polynomials in w having small coefficients with respect to w(z), or by a suitable Möbius transformation (*) reduces into {w, z}=P(z, w)/(w+b(z))² or {w, z}=c(z). Furthermore, we study the equation (**).

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Note on BP-theory for extensions of cyclic groups by elementary abelian p-groups

Nobuaki Yagita

1997 Volume 20 Issue 2 Pages 79-84
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043746

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On the first eigenvalue of minimal submanifolds

Qing Chen

1997 Volume 20 Issue 2 Pages 85-91
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043747

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Equivariant category of the free part of a G-manifold and of the sphere of spherical harmonics

Waclaw Marzantowicz

1997 Volume 20 Issue 2 Pages 92-106
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043748

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In this work we study the G-category of a G-manifold M by taking in consideration the fixed point set of a maximal torus of a compact Lie group G. The used method let us compute the G-category of sphere of every real irreducible, odd indexed representation V_l of the group G=SO(3). An application to a nonlinear Dirichlet problem, one of several possible, is given. Simplifying a proof of estimate of the G-category of the free part of a sphere we also show that the complement of saturation of fixed point set of a maximal torus is an open invariant subset of larger G-category than the free part of action and give particular computation for the spherical harmonics.

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Bordism of oriented manifolds with local T² action

Piotr Mikrut

1997 Volume 20 Issue 2 Pages 107-119
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043749

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The notion of pure T-structure of rank 2 with singularities is one of the notions introduced by Cheeger and Gromov in [CG1]. It is a generalization of both: a manifold with effective action of a torus T² and a manifold being the total space of a T² bundle with Aff(T²) as a structure group. Using well known properties of the group SL₂(Z)≅Z_6*z₂Z₄ and a smooth substitute of a classifying map we show that a compact orientable manifold with local T² action with suitable assumptions on orbit types is equivariantly cobordant with CP² bundle over a manifold, where T² acts in a standard way on fibers. The result is an important step towards calculating bordism group of manifolds with mixed singular T-structures. In dimensions 4, 5 and 6 we calculate explicit generators.

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Asymptotic properties of solutions of a class of impulsive differential equations of second order with a retarded argument

D. D. Bainov, M. B. Dimitrova, A. B. Dishliev

1997 Volume 20 Issue 2 Pages 120-126
Published: 1997
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2996/kmj/1138043750

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Some asymptotic properties are studied for the solutions of a class of impulsive differential equations of second order with retarded argument and fixed moments of impulse effect. Sufficient conditions are found for oscillation of all bounded solutions.

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Meromorphic functions that share two sets

Hong-Xun Yi, Lian-Zhong Yang

1997 Volume 20 Issue 2 Pages 127-134
Published: 1997
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2996/kmj/1138043751

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This paper studies the problem of uniqueness of meromorphic functions that share two sets and obtains some unicity theorems which improve some theorems given by H. X. Yi, G. D. Song and N. Li and other authors.

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Submanifolds whose quadric representations satisfy Δ˜{x}=B˜{x}+C

Jitan Lu

1997 Volume 20 Issue 2 Pages 135-142
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043752

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Let x : Mⁿ→E^m be an isometric immersion of an n-dimensional Riemannian manifold into the m-dimensional Euclidean space. Then the map ˜{x}=xx^t (where t denotes transpose) is called the quadric representation of Mⁿ. In this paper, we give some results on submanifolds in the Euclidean space E^m which satisfy Δ˜{x}=B˜{x}+C, where B and C are two constant matrices.

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On the frequency of zeros of a fundamental solution set of complex linear differential equations

Shupei Wang

1997 Volume 20 Issue 2 Pages 143-155
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043753

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We give a complete characterization of those equations of the form (1.1) which possess a fundamental solution set having few zeros. Some other results are also obtained.

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An n-dimensional flat torus in S²ⁿ⁻¹ whose extrinsic diameter is equal to π

Yoshihisa Kitagawa

1997 Volume 20 Issue 2 Pages 156-160
Published: 1997
Released on J-STAGE: January 21, 2009

DOIhttps://doi.org/10.2996/kmj/1138043754

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