Volume 23, Issue 2

In this paper we study the normal bundle of the embedding of subvarieties of dimension n – 1 in the Grassmann variety of lines in . Making use of some results on the geometry of the focal loci of congruences ([Arrondo, Bertolini and Turrini, Asian J. of Math. 5: 535–560, 2001] and [Arrondo, Bertolini and Turrini, Asian J. of Math. 9: 449–472, 2005]), we give some criteria to decide whether the normal bundle of a congruence is ample or not. Finally we apply these criteria to the line congruences of small degree in .

In this paper we study the (equivariant) topological types of a class of 3-dimensional closed manifolds (i.e., 3-dimensional small covers), each of which admits a locally standard (ℤ 2 ) 3 -action such that its orbit space is a simple convex 3-polytope. We introduce six equivariant operations on 3-dimensional small covers. These six operations are interesting because of their combinatorial natures. Then we show that each 3-dimensional small cover can be obtained from ℝ P 3 and S 1 × ℝ P 2 with certain (ℤ 2 ) 3 -actions under these six operations. As an application, we classify all 3-dimensional small covers up to (ℤ 2 ) 3 -equivariant unoriented cobordism.

We consider the null controllability problem with a finite number of constraints on the state for a nonlinear heat equation involving gradient terms in a bounded domain of . The control is distributed along a bounded subset of the domain and the nonlinearity is assumed to be of class and globally Lipschitz. Interpreting each constraint in terms of the notion of adjoint state, we transform the linearized problem into an equivalent null controllability problem with constraint on the control. Using a Carleman inequality adapted to the constraint, we prove first the null controllability of the linearized problem. Then, by a fixed-point method, we show that the same result holds when the nonlinearity is of class and globally Lipschitz.

Let be a Hölder-continuous linear cocycle with a discrete-time, μ -measure-preserving driving flow ƒ: X × ℤ → X on a compact metric space X . We show that the Lyapunov characteristic spectrum of (, μ ) can be approached arbitrarily by that of periodic points. Consequently, if all periodic points have only nonzero Lyapunov exponents and such exponents are uniformly bounded away from zero, then (, μ ) also has only non-zero Lyapunov exponents. In our arguments, an exponential closing property of the driving flow is a basic condition. And we prove that every C 1 -class diffeomorphism of a closed manifold obeys this closing property on its any hyperbolic invariant subsets.

We develop a general theory of reflection systems and, more specifically, partial root systems which provide a unifying framework for finite root systems, Kac–Moody root systems, extended affine root systems and various generalizations thereof. Nilpotent and prenilpotent subsets are studied in this setting, based on commutator sets and the descending central series. We show that our notion of a prenilpotent pair coincides, for Kac–Moody root systems, with the one defined by Tits in terms of positive systems and the Weyl group.

Let V be a Euclidean Jordan algebra and let Ω be the associated symmetric cone, a self-dual homogeneous open convex cone, which is a symmetric space of noncompact type under G (Ω) (the linear automorphism group)-invariant Riemannian metric. We show that the radius of the largest ball centered at a ∈ Ω inscribed in Ω coincides with its minimum eigenvalue and then provide a proof of the problem of finding a point x ∈ Ω to maximize the product of the radii of the largest balls centered at a, b ∈ Ω and inscribed in Ω of the tangent space T x (Ω) and its dual space T x –1 (Ω), respectively. We obtain an explicit formula for the maxima; it is precisely the minimal eigenvalue of P ( a 1/2 ) b where P denotes the quadratic representation of V . This provides an affirmative answer to a question of Todd on the maxima.

Spectral asymptotics of operators of the form are investigated. In the case of self-similar measures μ and ν it turns out that the eigenvalue counting function N ( x ) under both Dirichlet and Neumann conditions behaves like x γ as x → ∞, where the spectral exponent γ is given in terms of the scaling numbers of the measures. More precisely, it holds that In the present paper, we give a refinement of this spectral result, i.e. we give a sufficient condition under which the term N ( x ) x – γ converges. We show, using renewal theory, that the behaviour of N ( x ) x – γ depends essentially on whether the set of logarithms of the scaling numbers of the measures is arithmetic.