Surfaces in $${\mathbb{S}^2\times\mathbb{R}}$$ with a canonical principal direction

We show a way to choose nice coordinates on a surface in and use this to study minimal surfaces. We show that only open parts of cylinders over a geodesic in are both minimal and flat. We also show that the condition that the projection of the direction tangent to onto the tangent space of the surface is a principal direction, is equivalent to the condition that the surface is normally flat in . We present classification theorems under the extra assumption of minimality or flatness. J. Fastenakels is a research assistant of the Research Foundation—Flanders (FWO). J. Van der Veken is a postdoctoral researcher supported by the Research Foundation—Flanders (FWO). This work was partially supported by project G.0432.07 of the Research Foundation—Flanders (FWO).     

Surfaces with high contact number and their characterization

The notion of contact number of a Euclidean submanifold was introduced in an earlier article (Proc. Edinb. Math. Soc. 47:69–100, 2004) as the highest order of contact of geodesics and normal sections on the submanifold. It was proved in (Proc. Edinb. Math. Soc. 47:69–100, 2004) that the contact number relates closely with the notions of isotropic submanifolds and holomorphic curves. One important problem concerning contact number is to construct Euclidean submanifolds with high contact number. The purpose of this article is thus to construct Euclidean surfaces with high contact number and to provide simple geometric characterization of such surfaces. Mathematics Subject Classification (2000) Primary 53C40, 53A10 Secondary 53B25, 53C42     

Convolution with measures on flat curves in low dimensions

We prove Lp→Lq convolution estimates for the affine arclength measure on certain flat curves in Rd when d∈{2,3,4}. For d=2,3, we also establish certain related Lorentz space estimates.     

Exact number of solutions of stationary reaction-diffusion equations

We study both existence and the exact number of positive solutions of the problem

     

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ? T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.     

Flat lightlike hypersurfaces in Lorentz–Minkowski 4-space

The lightlike hypersurfaces in Lorentz–Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.     

Flat Riemannian Manifolds as Torus Bundles

Vasquez showed that for any finite group G there exists a number n(G) such that for every flat Riemannian manifold M with holonomy group G there exists a fiber bundle T → M → N, where T is a flat torus and N is a flat manifold of dimension less than or equal to n(G). We show that n(H) ≤ n(G) if H Δ leftG or G = N ? H and use this result to describe groups with the Vasquez number equal to 2 or 3.     

Exact borel subalgebras of quasi-hereditary algebras. II

In this article, we consider the class of flat G-modules in the category of discrete modules over a profinite group G. We will appeal to a recent result of Enochs to prove that we have flat covers in this situation.     

The volume measure for flat connections as limit of the Yang–Mills measure

We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang–Mills measure defined in terms of the heat-kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space.     

Low-E probe for (19)F-(1)H NMR of dilute biological solids

Sample heating induced by radio frequency (RF) irradiation presents a significant challenge to solid state NMR experiments in proteins and other biological systems, causing the sample to dehydrate which may result in distorted spectra and a damaged sample. In this work we describe a large volume, low-E (19)F-(1)H solid state NMR probe, which we developed for the 2D (19)F CPMG studies of dilute membrane proteins in a static and electrically lossy environment at 600MHz field. In (19)FCPMG and related multi-pulse (19)F-(1)H experiments the sample is heated by the conservative electric fields E produced in the sample coil at both (19)F and (1)H frequencies. Instead of using a traditional sample solenoid, our low-E (19)F-(1)H probe utilizes two orthogonal loop-gap resonators in order to minimize the conservative electric fields responsible for sample heating. Absence of the wavelength effects in loop-gap resonators results in homogeneous RF fields and enables the study of large sample volumes, an important feature for the dilute protein preparations. The orthogonal resonators also provide intrinsic isolation between the (19)F and (1)H channels, which is another major challenge for the (19)F-(1)H circuits where Larmor frequencies are only 6% apart. We detail steps to reduce (19)F background signals from the probe, which included careful choice of capacitor lubricants and manufacture of custom non-fluorinated coaxial cables. Application of the probe for two-dimensional (19)F CPMG spectroscopy in oriented lipid membranes is demonstrated with Flufenamic acid (FFA), a non-steroidal anti-inflammatory drug.