International Journal of Mathematics

We explore a relationship between topological properties of the orbits of 2-cycles in the symplectomorphism group Symp(M) and the existence of rational curves in M. Under the absence of rational curves hypothesis, we show that the evaluation map vanishies on π₂ and obtain a Gottlieb-type vanishing theorem for toroidal cycles in Symp(M).

The moduli space of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from to the space of even 4Θ divisors on the Jacobian variety of the curve.

In this paper, we investigate the existence of compact random attractors and their fractal dimension for the random dynamical systems determined by damped stochastic wave equations of Sine–Gordon type with linear multiplicative noise.

In this paper, we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as κ₁ = mκ₂ + n, where m and n are real numbers and κ₁ and κ₂ denote the principal curvatures at each point of the surface. We investigate the existence of such surfaces parametrized by a uniparametric family of circles. We prove that the only surfaces that exist are surfaces of revolution and the classical examples of minimal surfaces discovered by Riemann. The latter situation only occurs in the case (m, n) = (-1, 0).

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

Higher-rank versions of Wold decomposition are shown to hold for doubly commuting isometric representations of product systems of C*-correspondences over , generalizing the classical result for a doubly commuting pair of isometries due to Słociński. Certain decompositions are also obtained for the general, not necessarily doubly commuting, case and several corollaries and examples are provided. Possibilities of extending isometric representations to fully coisometric ones are discussed.

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.