A Nonlinear Force-Free Magnetic Field Approximation Suitable for Fast Forward-Fitting to Coronal Loops. I. Theory

We derive an analytical approximation of nonlinear force-free magnetic field solutions (NLFFF) that can efficiently be used for fast forward-fitting to solar magnetic data, constrained either by observed line-of-sight magnetograms and stereoscopically triangulated coronal loops, or by 3D vector-magnetograph data. The derived NLFFF solutions provide the magnetic field components $B_x({\bf x})$, $B_y({\bf x})$, $B_z({\bf x})$, the force-free parameter $\alpha({\bf x})$, the electric current density ${\bf j}({\bf x})$, and are accurate to second-order (of the nonlinear force-free $\alpha$-parameter). The explicit expressions of a force-free field can easily be applied to modeling or forward-fitting of many coronal phenomena.Comment: Solar Physics (in press), 26 pages, 11 figure

Bisector and zero-macrospin co-rotational systems for shell elements

A principal issue in any co-rotational approach for large displacement analysis of plates and shells is associated with the specific choice of the local reference system in relation to the current deformed element configuration. Previous approaches utilised local co-rotational systems, which are invariant to nodal ordering, a characteristic that is deemed desirable on several fronts; however, the associated definitions of the local reference system suffered from a range of shortcomings, including undue complexity, dependence on the local element formulation and possibly an asymmetric tangent stiffness matrix. In this paper, new definitions of the local co-rotational system are proposed for quadrilateral and triangular shell elements, which achieve the invariance characteristic to the nodal ordering in a relatively simple manner and address the aforementioned shortcomings. The proposed definitions utilise only the nodal coordinates in the deformed configuration, where two alternative definitions, namely, bisector and zero-macrospin definitions, are presented for each of quadrilateral and triangular finite elements. In each case, the co-rotational transformations linking the local and global element entities are presented, highlighting the simplicity of the proposed approach. Several numerical examples are finally presented to demonstrate the effectiveness and relative accuracy of the alternative definitions proposed for the local co-rotational system

On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

We study the Laplacian in deformed thin (bounded or unbounded) tubes in ?$\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on $h(s)$ is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator

1,1′-Fc(4-C6H4CO2Et)2and its unusual salt derivative withZ′ = 5,catena-[Na+]2[1,1′-Fc(4-C6H4CO2−)2]·0.6H2O [1,1′-Fc = (η5-(C5H4)2Fe]

The neutral diethyl 4,4'-(ferrocene-1,1'-diyl)dibenzoate, Fe[[eta]5-(C5H4)(4-C6H4CO2Et)]2 (I), yields (II) (following base hydrolysis) as the unusual complex salt poly[disodium bis[diethyl 4,4'-(ferrocene-1,1'-diyl)dibenzoate] 0.6-hydrate] or [Na+]2[Fe{[eta]5-(C5H4)-4-C6H4CO_2^-}2]·0.6H2O with Z' = 5. Compound (I) crystallizes in the triclinic system, space group P\bar 1, with two molecules having similar geometry in the asymmetric unit (Z' = 2). The salt complex (II) crystallizes in the orthorhombic system, space group Pbca, with the asymmetric unit comprising poly[decasodium pentakis[diethyl 4,4'-(ferrocene-1,1'-diyl)dibenzoate] trihydrate] or [Na+]10[Fe{[eta]5-(C5H4)-4-C6H4CO_2^-}2]5·3H2O. The five independent 1,1'-Fc[(4-C6H4CO2)-]2 dianions stack in an offset ladder (stepped) arrangement with the ten benzoates mutually oriented cisoid towards and bonded to a central layer comprising the ten Na+ ions and three water molecules [1,1'-Fc = [eta]5-(C5H4)2Fe]. The five dianions differ in the cisoid orientations of their pendant benzoate groups, with four having their -C6H4- groups mutually oriented at interplanar angles from 0.6 (3) to 3.2 (3)° (as [pi]...[pi] stacked C6 rings) and interacting principally with Na+ ions. The fifth dianion is distorted and opens up to an unprecedented -C6H4- interplanar angle of 18.6 (3)° through bending of the two 4-C6H4CO2 groups and with several ionic interactions involving the three water molecules (arranged as one-dimensional zigzag chains in the lattice). Overall packing comprises two-dimensional layers of Na+ cations coordinated mainly by the carboxylate O atoms, and one-dimensional water chains. The non-polar Fc(C6H4)2 groups are arranged perpendicular to the layers and mutually interlock through a series of efficient C-H...[pi] stacking contacts in a herringbone fashion to produce an overall segregation of polar and non-polar entities

Finite element analyses of lipped chanel beams with web openings in shear

Cold-formed steel members are increasingly used as primary structural elements in buildings due to the availability of thin and high strength steels and advanced cold-forming technologies. Cold-formed lipped channel beams (LCB) are commonly used as flexural members such as floor joists and bearers. Shear behaviour of LCBs with web openings is more complicated and their shear capacities are considerably reduced by the presence of web openings. However, limited research has been undertaken on the shear behaviour and strength of LCBs with web openings. Hence a numerical study was undertaken to investigate the shear behaviour and strength of LCBs with web openings. Finite element models of simply supported LCBs with aspect ratios of 1.0 and 1.5 were considered under a mid-span load. They were then validated by comparing their results with test results and used in a detailed parametric study. Experimental and numerical results showed that the current design rules in cold-formed steel structures design codes are very conservative for the shear design of LCBs with web openings. Improved design equations were therefore proposed for the shear strength of LCBs with web openings. This paper presents the details of this numerical study of LCBs with web openings, and the results

Definable equivalence relations and zeta functions of groups

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math. So