Band 12, Heft 5

The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.

Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].

We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a $$\mathfrak{g}\mathfrak{l}_n$$ partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.

We consider a centered Gaussian random field X = {X t : t ∈ T} with values in a Banach space $$\mathbb{B}$$ defined on a parametric set T equal to ℝm or ℤm. It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = conv{X t : t ∈ T n}, where (T n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n)n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W n), where f is some function, is also studied.

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.

For all convolution algebras L 1[0, 1); L loc1 and A(ω) = ∩n L 1(ωn), the derivations are of the form D μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D μ to a natural dual space is weak-star continuous.

Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.

Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $$\mathbb{D}$$, we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $$\mathbb{D}$$ and on the boundary $$\partial \mathbb{D}$$ respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C n that satisfy 0 ≤ C n ≤ Σj=1n ‖x j‖ − ‖Σj=1n x j‖, x 1,...,x n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.