Optimal Meshes for Finite Elements of Arbitrary Order

Given a function f defined on a bounded domain Ω⊂ℝ² and a number N>0, we study the properties of the triangulation T_N\mathcal{T}_{N} that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the norm X=L ^p for 1≤p≤∞, and we consider Lagrange finite elements of arbitrary polynomial degree m−1. We establish sharp asymptotic error estimates as N→+∞ when the optimal anisotropic triangulation is used, recovering the results on piecewise linear interpolation (Babenko et al. in East J. Approx. 12(1), 71–101, 2006; Babenko, submitted; Chen et al. in Math. Comput. 76, 179–204, 2007) and improving the results on higher degree interpolation (Cao in SIAM J. Numer. Anal. 45(6), 2368–2391, 2007, SIAM J. Sci. Comput. 29, 756–781, 2007, Math. Comput. 77, 265–286, 2008). These estimates involve invariant polynomials applied to the m-th order derivatives of f. In addition, our analysis also provides practical strategies for designing meshes such that the interpolation error satisfies the optimal estimate up to a fixed multiplicative constant. We partially extend our results to higher dimensions for finite elements on simplicial partitions of a domain Ω⊂ℝ ^d .     

Synthesis and Diels-Alder Reactions of a New Kind of Chiral Dienophiles: Cyclic Vinyl-p-tolylsulfilimines

A new kind of chiral dienophiles, cyclic vinyl-p-tolylsulfilimines (2a and 2b), were obtained from the corresponding (Z)-sulfinylacrylonitriles with HBF(4) and methanol. The asymmetric Diels-Alder reaction of optically pure 2a with cyclopentadiene under mild thermal or catalyzed conditions afforded only the endo-4a adduct with complete endo and pi-facial selectivities. The ability of the sulfilimine moiety to enhance the dienophilic reactivity of the double bond is similar to that of the sulfinyl group.     

Neutron Diffraction Investigation of Liquid Alkali Metal—Gallium Alloys. Giant Cluster Formation?

Neutron diffraction experiments were performed on the liquid alloys NaGa, NaGa₃, KGa₃ and CsGa₃. The structure factors of KGa₃ and CsGa₃ display prepeaks at small wavenumbers (0.64 and 0.61 Å^‐1, respectively). This may indicate the existence of extremely large aggregates of atoms in the liquid, which can not be associated with any of the crystal structures known for the solid counterparts of these alloys.     

Appearance of long-range magnetic order in a system of small ferromagnetic particles

Using x-ray and magnetic methods as well as small-angle neutron scattering, it is shown for the example of the decomposing alloy Cu₆₄Mn₉Al₂₇ that the appearance of long-range ferromagnetic order in a system of small superparamagnetic particles dissolved in a nonmagnetic matrix is due to cooperative ordering of their magnetic moments. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 959–962 (25 June 1996)     

Pressure and field induced magnetic order in the spin liquid Tb2Ti2O7 as studied by single crystal neutron diffraction

We have studied the spin liquid Tb2Ti2O7 by single crystal neutron diffraction under high pressure up to 2.8 GPa, together with uniaxial stress, down to 0.1 K, in zero and high magnetic fields up to 7 T. In zero magnetic field, a long-range ordered antiferromagnetic structure is induced by pressure. The Néel temperature and ordered magnetic moment can be tuned by the anisotropic pressure component. Under magnetic field, the antiferromagnetic structure transforms into a canted ferromagnetic one at 0.6 T. Spin canting persists even at 7 T. The magnetic phase diagram under pressure shows a strong increase of the Néel temperature with the field.     

Geminal bis-ureas as gelators for organic solvents: gelation properties and structural studies in solution and in the gel state

Several geminal bis-urea compounds were synthesised by means of an acid-catalysed condensation of various benzaldehydes with different monoalkylureas. Many of these compounds form thermoreversible gels with a number of organic solvents at very low concentrations (<3mM) and which are stable to temperatures higher than 100 degrees C. Electron microscopy revealed a three-dimensional (3D) network of intertwined fibres, which are several tens of micrometers long and have a width ranging from approximately 30 to 300 nm. The possible aggregate forms and aggregate symmetries were evaluated by means of molecular mechanics calculations. 1H NMR, 2D NMR, 13C NMR and 13C-CP/MAS NMR techniques were used to obtain information about the aggregation and possible aggregate symmetry of geminal bis-ureas in solution, in the gel state, and in the solid state.     

Long-range sigma-pi interactions in tetrahydro-4H-thiopyran end-capped oligo(cyclohexylidenes). Photo-electron spectroscopy, ab initio SCF MO calculations, and natural bond orbital analyses

Long-range sigma-pi interactions in tetrahydro4H-thiopyran end-capped oligo(cyclohexylidenes) were identified by He(I) photoelectron spectroscopy (PES) and ab initio RHF/6-31G* calculations. The vertical ionization energies Ivj of the highest occupied molecular orbitals (MO's) were assigned using Koopmans' theorem (Iv,j = -epsilonj) and by correlation with the ionizations of related reference compounds. The experimental (PES) and theoretical (RHF/6-31G*) results are in good agreement. For tercyclohexylidene derivatives which contain two nonconjugated pi-bonds splittings deltaIv,j of the pi-bands in the range from approximately 0.5 to 0.7 eV (delta-epsilonj approximately 0.6 to 0.9 eV). For the bi- and tercyclohexylidene compounds containing two sulfur atoms at their alpha- and omega-end positions the pi-type sulfur lone pair bands [Lppi(S)] split significantly by deltaIvj approximately 0.3 to 0.4 eV (delta-epsilonj approximately 0.3 to 0.4 eV), i.e. sigma-pi interactions over distances of ca. 8 and 12 A, respectively, occur. The magnitude of the interactions and the observed splittings are independent of the anti and syn conformations of the oligo(cyclohexylidene) hydrocarbon skeletons. RHF/6-31G* Natural Bond Orbital analyses reveal that the Hax-C-C-Hax precanonical MO's (PCMO's) centered on the cyclohexyl-type rings are paramount for the relay of the through-bond sigma-pi interactions; no through-space sigma-pi interactions were identified.