Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents

We describe special asymptotic structures of solutions of the semilinear heat equation

     

An obstruction to the positivity of relative Yamabe invariants

For a supergroup , we describe an obstruction to the existence of positive scalar curvature metrics with minimal boundary condition on a compact n-dimensional -manifold W with nonempty boundary M, , in terms of the bordism class [M] in the Stolz obstruction group associated to [St2]. In par ticular, when W is a 5-dimensional spin manifold and the -invariant of a connected component of M is nonzero, we prove that W does not admit a positive scalar curvature metric with minimal boundary condition. Received: 4 July 2001; in final form: 5 February 2002 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by the Grants-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science, No. 11640070.     

On the Metrical Theory of Continued Fraction Mixing Fibred Systems and Its Application to Jacobi-Perron Algorithm

 We study the metrical theory of fibred systems, in particular, in the case of continued fraction mixing systems. We get the limit distribution of the largest value of a continued fraction mixing stationary stochastic process with infinite expectation and some related results. These are analogous to J. Galambos, W. Philipp, and H. G. Diamond–J. D. Vaaler theorems for the regular continued fractions. As an application, we see that these theorems hold for Jacobi-Perron algorithm. Received September 30, 2001; in revised form January 8, 2002     

Semilinear elliptic problems and concentration compactness on non-compact Riemannian manifolds

Existence of solution for semilinear problem with the Laplace-Beltrami operator on non-compact Riemannian manifolds with rich symmetries is proved by concentration compactness based on actions of the manifold's isometry group.     

Linear optical properties of frequency doubling polymers

Measurements of linear optical properties of methylmethacrylate copolymers with non-linear optical 4-alkoxy-4-alkylsulfone stilbene side-chains are presented. From there two methods for phase matching are discussed; one aiming to equalize the refractive indices of the fundamental and the second harmnic wave, the other to suppress the destructive character of the interference. The matter of photo stability of frequency doubling polymers is also discussed.     

Presentations of the first homotopy groups of the unitary groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n 2 we supply n homotopic maps that each represent the (n - 1)!-th power of a suitable generator of _2n SU(n) _n!. The product of these n commuting maps is the constant map to the identity matrix.     

On an Archimedean analogue of Tate's conjecture

We consider an Archimedean analogue of Tate's conjecture, and verify the conjecture in the examples of isospectral Riemann surfaces constructed by Vignéras and Sunada. We prove a simple lemma in group theory which lies at the heart of T. Sunada's theorem about isospectral manifolds.     

On the uniqueness of minimizing harmonic maps to a closed hemisphere

We study the uniqueness of minimizing harmonic maps to a closed hemisphere. We are able to describe the boundary data for which there are more than one minimizer, and to describe in these cases the corresponding set of minimizers. This is a limiting case for a former result of W.Jäger and H.Kaul.This article was processed by the author using the Springer-Verlag T_EX PJourlg macro package 1991.     

New elliptic potentials

We construct all tangential covers as zeroes of a particular set of polynomials. The resulting pointed curves give rise, through the Krichever dictionary, to elliptic KP solitons. We then give a criterion to detect the so-called hyperelliptic-tangential covers, which give rise to elliptic KdV solitons. Finally we construct new examples of the latter covers and write down the corresponding (doubly periodic finite-gap) source potentials.Dedicated to the memory of J.-L. Verdier     

Holomorphic maps of Kähler manifolds with constrained energy density

We consider holomorphic and antiholomorphic maps of Kähler manifoldsM andN withM compact. In view of bounds on the Ricci curvature ofM and the holomorphic bisectional curvature ofN, the energy density of the map is constrained to satisfy certain inequalities. One inequality implies that the map is constant. Another specifies the image ofM as a totally geodesic real surface of constant Gaussian curvature inN.