Notes on paratopological groups

In this paper, we prove that if a remainder of a non-locally compact paratopological group G   has a _Gδ$G_{\delta }$-diagonal and every compact subset of G is first countable, then G   has a _Gδ$G_{\delta }$-diagonal of infinite rank. This improves a result of Chuan Liu and Shou Lin [Chuan Liu, Shou Lin, Generalized metric spaces with algebraic structure, Topology Appl. 157 (2010) 1966–1974]. We also construct an open continuous homomorphism f from a non-metrizable paratopological group G onto a metrizable topological group H such that the kernel of f is metrizable. This result gives a negative answer to an open problem posed in [A.V. Arhangel?skii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Scientific, 2008].     

Direct Electrochemistry and Bioelectrocatalysis of Microperoxidase‐11 Immobilized on Chitosan‐Graphene Nanocomposite

A bioelectrochemical platform has been constructed for the direct electron transfer and biosensing purposes of microperoxidase‐11 (MP‐11) immobilized on the chitosan dispersed multilayer graphene nanocomposite. The immobilized MP‐11 at the modified gold electrode displays a well‐defined and quasireversible redox peaks, with a formal potential of ?0.38 V/SCE in a buffer solution (pH 7.0). MP‐11 absorbed on the electrode surface exhibits high electrocatalytic activity toward the reduction of both oxygen and hydrogen peroxide and also shows good analytical performance for the amperometric detection of H₂O₂ with a linear range from 2.5 to 135 μM. These results indicate the graphene modified electrode might be used as a third generation biosensor for H₂O₂ detection.     

Corrigendum to “Note on zeros of the derivative of the Selberg zeta-function”

We correct constants in asymptotic formulas of our paper “Note on zeros of the derivative of the Selberg zeta-function”, Arch. Math. 91 (2008), 238–246. The online version of the original article can be found under doi:.     

On the solutions of a boundary value problem arising in free convection with prescribed heat flux

For given , c < 0, we are concerned with the solution f _b of the differential equation f ′′′ + ff ′′ + g(f ′) = 0 satisfying the initial conditions f(0) = a, f ′ (0) = b, f ′′ (0) = c, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists b _* > 0 such that f _b exists on [0, + ∞) and is such that as t → + ∞, if and only if b ≥ b _*. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.      

Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums.      

A lower bound for the error term in Weyl’s law for certain Heisenberg manifolds

This article provides an Omega-result for the remainder term in Weyl’s law for the spectral counting function of certain rational (2ℓ + 1)-dimensional Heisenberg manifolds. Received: 22 September 2008     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

Spectral theory for algebraic combinations of Toeplitz and composition operators

We determine the essential spectra of algebraic combinations of Toeplitz operators with continuous symbol and composition operators induced by a class of linear-fractional non-automorphisms of the unit disk. The operators in question act on the Hardy space H² on the unit disk. Our method is to realize the C^*-algebra that they generate as an extension of the compact operators by a concrete C^*-algebra whose invertible elements are easily characterized.