Symmetry Properties of the Elastic Energy of a Woven Fabric with Bending and Twisting Resistance

A woven fabric can be described as a surface made of two families of fibers: in this work we study how the geometry of the weave pattern affects the symmetry properties of the elastic energy of the surface. Four basic symmetry classes of weave patterns are possible, depending on the angle between the fibers and their material properties. The properties of the pattern determine the material symmetry group of the network, under which the elastic energy is invariant. We derive representations for the energy of a woven fabric that are invariant under the symmetry group of the network, and discuss the relation of these invariants with the curvature and twist of the fibers.      

Effects of potassium on the supramolecular structure and electronic properties of eumelanin thin films

The role of potassium in the formation of synthetic eumelanin aggregates is investigated by atomic force microscopy and soft-X-ray spectroscopy. Control over the thin film granularity is achieved by using K salts, in both drop casting and electrodeposition of eumelanin thin films. Further control over orientation is made possible by a suitable choice of the substrate: evidence of self-assembly is found for thin films deposited on gold. Finally, it is shown that the potassium content affects not only the samples morphology, but also the low-lying states in the valence band, where a transfer of spectral weight across the HOMO-LUMO gap is observed, disclosing possible applications of this multifunctional biomacromolecule.     

The crystal and molecular structure of the tetraethylammonium salt of tetrahydridopentadecacarbonyltetrarhenate(2−) in two crystallographic forms

The structure of the title compound, (NEt₄)₂[H₄Re₄(CO)₁₅], is reported in two crystallographic modifications, I and II. Both forms axe monoclinic and the cell constants are as follows: I, a 11.355(2), b 21.204(4), c 17.416(3) Å, β 94.15(2)°, space group P2₁/c; II, a 21.831(4), b 17.584(3), c 11.446(2) Å, β 96.02(2)°, space group P2₁,/n. Two sets of 3042 (I) and 2870 (II) independent diffraction intensities, collected by counter methods, were used for the solution and refinement of the two structures. The final conventional R factors have values 5.5% (I) and 6.3% (II), respectively. The crystal packings are compared, showing different conformations of the (NEt₄)⁺ cations. The anions contain a tetrametal cluster formed by an isosceles triangle plus an apically bound metal atom; the carbonyl groups are all terminally bonded to the rhenium atoms. Some differences, present both in the metal atom clusters and in the carbonyl dispositions, are discussed and compared with a third, previously reported, crystallographic modification of the same compound.     

Synthesis, Chemical Characterization, and Molecular Structure of Au8{Fe(CO)4}4(dppe)2 and Au6Cu2{Fe(CO)4}4(dppe)2

The new Au₈{Fe(CO)₄}₄(P^P)₂ and Au₆Cu₂{Fe(CO)₄}₄(P^P)₂ (P^P=dppm, dppe) neutral cluster compounds were isolated in good yields by condensation of the [Au₃{Fe(CO)₄}₂(P^P)]^- anions with Au(SEt₂)Cl and CuCl, respectively, and have been characterized by IR, NMR and microanalyses. The molecular structures of Au₈{Fe(CO)₄}₄(dppe)₂ and Au₆Cu₂{Fe(CO)₄}₄(dppe)₂ have been determined by X-ray diffraction studies. Both molecules adopt a stereogeometry of the heavy atoms consisting of a triangulated and corrugated ribbon twisted around the elongation direction. Contrary to the expectations the latter displays the two copper atoms in the sites of highest connectivity. This implies that site exchange between copper and gold occurs during the synthesis.     

Propagation of Singularities¶for Solutions of Nonlinear First Order¶Partial Differential Equations

Few results are available in the mathematical literature for studying the structure of the singular set of a weak solution u of F(x,u,Du)=0. This paper provides new techniques to analyse such a set when u is semiconcave and F is a nonlinear convex function with respect to p. The main objective achieved here is a classification of the singularities of u that propagate along Lipschitz arcs. Such a propagation phenomenon is also described by means of a generalized characteristics inclusion.     

Analytic stratifications and the cut locus of a class of distance functions

We study the analytic singularities of viscosity solutions of equations of eikonal type and obtain that the analytic singular support of these functions has an analytic stratification. The singular support can be identified with the cut locus of the distance to the boundary of an open set, when the interior is equipped with a degenerate Riemannian metric. We apply the result to elliptic equations as well as to model operators of Grušin type.     

Boundary conditions and non-equilibrium thermodynamics

The theory of non-equilibrium thermodynamics is applied to a system of two immiscible fluids and their interface. A singular energy density at the interface, which is related to the phenomenon of surface tension, is taken into account. Furthermore the momentum and the heat currents are allowed to be singular at the interface. Using the conservation laws and the Gibbs' relation for the surface, an expression for the singular entropy production density at the interface is obtained. The linear phenomenological laws between fluxes and thermodynamic forces occurring in this singular entropy production density are given. Some of these linear laws are boundary conditions for the solution of the differential equations governing the evolution of the state variables in the bulk.