We first construct compatible actions of the product of the unit interval and the unit circle as a monoid on a semi-stable degeneration of pairs and on the associated log topological spaces. Then we show that the log topological family is locally trivial in piecewise smooth category over the base, i.e., the associated log topological family recovers the vanishing cycles of the original semi-stable degeneration in the most naive sense. Using this result together with the log Riemann-Hilbert correspondence, we introduce two types of integral structure of the variation of mixed Hodge structure associated to a semi-stable degeneration of pairs.

For an extension of number fields, we define the group of relative units, and determine its rank when the extension is a Galois extension. For this purpose we need to determine all the finite groups of which every abelian subgroup is cyclic.

We show the existence of weak solutions of the Navier-Stokes equations with test functions in the weak-$L^n$ space. As an application, we give a new criterion on uniqueness and regularity of weak solutions which covers the previous results.

The objective of this paper is to investigate the $p$-th moment asymptotic stability decay rates for certain finite-dimensional Itô stochastic differential equations. Motivated by some practical examples, the point of our analysis is a special consideration of general decay speeds, which contain as a special case the usual exponential or polynomial type one, to meet various situations. Sufficient conditions for stochastic differential equations (with variable delays or not) are obtained to ensure their asymptotic properties. Several examples are studied to illustrate our theory.

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric local complete intersection singularities. Our strikingly simple proof makes use of Nakajima's classification theorem and of some techniques from toric and discrete geometry.

Proved are transference results that show connections between: a) multipliers for the Fourier-Bessel series and multipliers for the Hankel transform; b) maximal operators defined by Fourier-Bessel multipliers and maximal operators given by Hankel transform multipliers; c) Fourier-Bessel transplantation and Hankel transform transplantation. In some way the connections described in a) and b) can be seen as multi-dimensional extensions of the classical results of Igari, and Kenig and Tomas for the one dimensional Fourier transform. We prove our results for the non-modified Hankel transform in the power weight setting, and this allows to translate them also to the context of the modified Hankel transform. Together with Gilbert's transplantation theorem, our transference shows that harmonic analysis results for the Hankel transform of arbitrary order are consequences of corresponding results for the cosine expansions.

We prove that a half dimensional, totally real and totally geodesic submanifold of a compact Riemannian 3-symmetric space is expressed as an orbit of a Lie subgroup of the isometry group of the ambient manifold. Moreover, we associate such submanifolds with graded Lie algebras of the second kind.

We prove that the rational homology of decorated Torelli groups and Torelli spaces are infinite dimensional when the genus of the reference surface is at least seven, thereby extended one of the main results of "Homological infiniteness of Torelli groups," Topology, v. 40 (2001), pp. 213-221.

Let $M$ be an oriented three-dimensional manifold of constant sectional curvature $-1$ and with positive injectivity radius, and $T^1M$ its tangent sphere bundle endowed with the canonical (Sasaki) metric. We describe explicitly the periodic geodesics of $T^1M$ in terms of the periodic geodesics of $M$: For a generic periodic geodesic $(h,v)$ in $T^1M$, $h$ is a periodic helix in $M$, whose axis is a periodic geodesic in $M$; the closing condition on $(h,v)$ is given in terms of the horospherical radius of $h$ and the complex length (length and holonomy) of its axis. As a corollary, we obtain that if two compact oriented hyperbolic three-manifolds have the same complex length spectrum (lengths and holonomies of periodic geodesics, with multiplicities), then their tangent sphere bundles are length isospectral, even if the manifolds are not isometric.

It is shown that every component of the Fatou set of an entire function with certain gaps is bounded.