We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an ‘elliptic end’ (respectively a ‘hyperbolic end’) if the monodromy matrix at the end is diagonalizable with eigenvalues in the unit circle (respectively in the reals). Although the existence of the surfaces is numerical, the types of ends are mathematically determined.

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Let {σ_t}_{t∈(-∞,∞)} be a one-parameter family of hyperbolic Riemannian metrics on an open annulus which is continuouswith respect to the Gromov-Hausdorff topology. We consider a system E_t of ordinary differential equations with singular points which depends on the Riemannian metric σ_t. If t ≠ 0, all of the singular points of E_t are regular. If t = 0, E₀ has an irregular singular point. In this paper, we investigate the behavior of the singular points of E_t . We show that a regular singular point of E_t , together with another regular singular point of E_t , becomes the irregular singular point of E₀ as t (›0) tends to zero and that the irregular singular point of E₀ becomes a non-singular point of E_t as t decreases from zero.

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The Howe duality correspondence for a unitary dual pair over R is usually formulated for the determinant cover of the dual pair. For applications to automorphic theory, one has to use the Weil representation of the unitary dual pair (not of the coverings) instead, which is constructed by the doubling argument of Harris et al (Theta dichotomy for unitary groups. J. Amer. Math. Soc. 9(4) (1996), 941-1004). In this paper, we calculate explicit formulae for the Fock model of this latter representation, and determine the Howe correspondence between K-types under this. This yields a description of the Howe duality correspondence between unitary groups in terms of certain ε-factors.

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In this paper, we consider the space of entire functions for a given proximate order and a semi-norm, and show that a continuous endomorphism of this space is represented naturally as a partial differential operator of infinite order with entire function coefficients. Then, in the case of higher order, we study the relationship between the continuous endomorphisms and the linear partial differential operators of infinite order by means of the growth of the symbol of the operators.

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In this paper, first, we re-establish the notion of non-local pseudo-differential operators introduced by the author, more intrinsically by means of the Fourier-Sato transformation of the sheaf of holomorphic forms. By using this, we propose the construction of the composition of non-local pseudo-differential operators functorially.

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In this paper we study terminating and ill-defined Gauss hypergeometric functions. For their hypergeometric equations, the set of 24 Kummer′s solutions degenerates. We describe those solutions and relations between them.

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In a preceding paper we have introduced the t-singular homology with integral coefficient of orbifolds, which captures the orbifold structures. In this paper we introduce the s-singular homology of orbifolds, and study the s-singular homology of orbifolds and the t-singular homology with rational coefficient of orbifolds. The main theorems are the following: (i) the s-singular homology group of an orbifold is isomorphic to the usual singular homology group of its underlying space;(ii) the t-singular homology groupwith rational coefficient of an orbifold is isomorphic to the usual singular homology group with rational coefficient of its underlying space.

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We solve the extra zero problem in cylindrical domains in Cⁿ. We apply it to study domains of holomorphy in Cⁿ which are not rationally convex. This result generalizes Oka′s results.

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We give a probabilistic expression of the α-determinant by using the Wishart distribution on the cone of positive definite matrices. As a corollary, we give a partial answer to a positivity problem for the α-determinant which is equivalent to the existence problem of certain multivariate probability distributions. We also give a concrete example to support our conjecture for the positivity.

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The approach taking advantage of Jacobi fields to represent explicitly stochastic oscillatory integrals with quadratic phase function, introduced by Ikeda, Kusuoka, and Manabe, is completed in the general scheme and is testified in several examples.

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Kaneko and Yoshida introduced the kappa function κ by the property J(κ(z)) = λ(z), where J and λ are the elliptic modular functions. This paper constructs several variants of the kappa function using other pairs of modular functions, and gives their explicit Fourier expansions.

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We develop a method to construct diffusions on singular sets. Our method can be applied to various fractal sets including Sierpinski carpets, the Sierpinski gasket and the Menger sponge.

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Let (X, Y ) be a smooth non-projective Moishezon compactification of C³ with b₂(X) = 1. Then Y is a non-normal irreducible divisor on X with Kx = -rY (r = 1, 2). In this paper, we mainly study the case where Y is not nef, that is, there is a curve C such that (Y · C)_X < 0. First, we investigate the structure of the boundary divisor Y (Theorem 1) under the mild assumption that b₃(X) = 0. Next we define the invariant δ(X) and compute them for the examples with non-nef boundaries (Theorems 2 and 3).

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