In this paper we will construct a certain action of Aut$(\mathbb{C})$ on holomorphic modular forms (of any weights) with respect to an arbitrary unitary group, which is compatible with Hecke operators. We can write this action explicitly and simply. The image of the action in general is a holomorphic modular form for another unitary group.

We consider the Euler type integral associated to the configuration space of points on an elliptic curve, which is an analogue of the hypergeometric function associated to the configuration space of points on a projective line. We calculate the twisted homology group with coefficients in the local system associated to a power function $g^{\alpha}$ of an elliptic function $g$,and the related intersection form. Applying these calculations, we describe the connection matrices representing the linear isomorphisms induced from analytic continuations of the functions defined by integrations of $g^{\alpha}$ over twisted cycles.

We consider the spatial isosceles three-body problem where two masses are equal and the other may be different. We discuss free-fall orbits in the planar case and show that there exists a countable family of orbits converging to triple collisions forward and backward in time. In Devaney's coordinates, the orbits correspond to topologically transverse heteroclinic orbits between equilibria on the collision manifold in the blown-up equations (Theorem 1.1). By the "window theory" of Moeckel [SIAM J. Math. Anal., 15 (1984), 857-876], we see that there exists an orbit shadowing a prescribed path on the graph consisting of these new heteroclinic orbits and already known, heteroclinic and periodic orbits. In particular, choosing appropriate infinite paths, we obtain a variety of new oscillatory orbits.

We extend the Mehler-Heine type formula of Jacobi polynomials to a class of multiple orthogonal polynomials of type II. The Mehler-Heine type formulas show standard orthogonal polynomials or multiple orthogonal polynomials near the endpoints of the interval of orthogonality.

Let $A$ be a commutative noetherian ring. In this paper, we interpret localizing subcategories of the derived category of $A$ by using subsets of Spec $A$ and subcategories of the category of $A$-modules. We unify theorems of Gabriel, Neeman and Krause.

We give some adequate extension, in the framework of a general Lévy process, of our previous construction of processes with one-dimensional martingale marginals, done originally in the set-up of Brownian motion. The Lévy process framework allows us to streamline our previous arguments, as well as to reach a larger class of such processes, even in the Brownian case. We give some illustrations of our construction when the Lévy process is either a Gamma process, or a Poisson process. We also work in the fractional Brownian and stable frameworks.

We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings.

Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.