Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

Isolation of a Novel Polyketide from Neodidymelliopsis sp.

Fungi have become an invaluable source of bioactive natural products, with more than 5 million species of fungi spanning the globe. Fractionation of crude extract of Neodidymelliopsis sp., led to the isolation of a novel polyketide, (2Z)-cillifuranone (1) and five previously reported natural products, (2E)-cillifuranone (2), taiwapyrone (3), xylariolide D (4), pachybasin (5), and N-(5-hydroxypentyl)acetamide (6). It was discovered that (2Z)-cillifuranone (1) was particularly sensitive to ambient temperature and light resulting in isomerisation to (2E)-cillifuranone (2). Structure elucidation of all the natural products were conducted by NMR spectroscopic techniques. The antimicrobial activity of 2, 3, and 5 were evaluated against a variety of bacterial and fungal pathogens. A sodium [1-¹³C] acetate labelling study was conducted on Neodidymelliopsis sp. and confirmed that pachybasin is biosynthesised through the acetate polyketide pathway.     

LiCoO(2) Concaved Cuboctahedrons from Symmetry-Controlled Topological Reactions

Morphology control of functional materials is generally performed by controlling the growth rates on selected orientations or faces. Here, we control particle morphology by "crystal templating": by choosing appropriate precursor crystals and reaction conditions, we demonstrate that a material with rhombohedral symmetry-namely the layered, positive electrode material, LiCoO(2)-can grow to form a quadruple-twinned crystal with overall cubic symmetry. The twinned crystals show an unusual, concaved-cuboctahedron morphology, with uniform particle sizes of 0.5-2 μm. On the basis of a range of synthetic and analytical experiments, including solid-state NMR, X-ray powder diffraction analysis and HRTEM, we propose that these twinned crystals form via selective dissolution and an ion-exchange reaction accompanied by oxidation of a parent crystal of CoO, a material with cubic symmetry. This template crystal serves to nucleate the growth of four LiCoO(2) twin crystals and to convert a highly anisotropic, layered material into a pseudo-3-dimensional, isotropic material.     

Comparative study of the electronic structure of the ordered and disordered Cu3Au (100) and Cu3Au (110) surfaces

The influence of structural changes on the electronic structure has been investigated by a comparison of the angle-integrated valence and core level photoelectron spectra of ordered and disordered Cu₃Au(100) and (110) surfaces. The total width of the Au 5d and Cu 3d bands does not change with the ordering state or surface orientation. The spectra for the (100) surface are compared with selfconsistent calculations and good agreement is found, for the ordered state, however with a 0.6 eV correction of the calculated Fermi level position. We observe three Au 5d derived bands at 5.1, 6.0 and 6.9 eV, in contrast to previous experimental findings. Our results indicate the existence of shortrange order above the critical temperature.     

New cesium titanate layer structures

A phase study of the Cs₂OTiO₂ system in the composition range 75–100 mole% TiO₂ and the temperature range 850–1200°C revealed the existence of two new cesium titanates, with compositions Cs₂Ti₅O₁₁ and Cs₂Ti₆O₁₃. The former compound undergoes a reversible hydration reaction below 200°C to form Cs₂Ti₅O₁₁ · (1 + x)H₂O, 0.5 < x < 1. The structures of the three phases have been determined. They are based on corrugated layers of edge-shared octahedra, with cesium ions (and H₂O) packing between the layers. In Cs₂Ti₆O₁₃, the layers are continuous in two dimensions, whereas in Cs₂Ti₅O₁₁ and Cs₂Ti₅O₁₁ · (1 + x)H₂O, the layers are periodically stepped to give 5-octahedra wide, corner-linked ribbons.     

Poly(diiododiacetylene): preparation, isolation, and full characterization of a very simple poly(diacetylene)

Poly(diiodiacetylene), or PIDA, is a conjugated polymer containing the poly(diacetylene) (PDA) backbone but with only iodine atom substituents. The monomer diiodobutadiyne (1) can be aligned in the solid state with bis(nitrile) oxalamide hosts by hydrogen bonds between oxalamide groups and weak Lewis acid-base interactions (halogen bonds) between nitriles and iodoalkynes. The resulting cocrystals start out pale blue but turn shiny and copper-colored as the polymerization progresses. The development of a crystallization methodology that greatly improves the yield of PIDA to about 50% now allows the full characterization of the polymer by X-ray diffraction, solid-state (13)C MAS NMR, Raman, and electron absorption spectroscopy. Comparison of a series of hosts reveals an odd-even effect in the topochemical polymerization, based on the alkyl chain length of the host. In the cocrystals formed with bis(pentanenitrile) oxalamide (4) and bis(heptanenitrile) oxalamide (6), the host/guest ratio is 1:2 and the monomer polymerizes spontaneously at room temperature, while in the case of bis(butanenitrile) oxalamide (3) and bis(hexanenitrile) oxalamide (5), where the host and guest form cocrystals in a 1:1 ratio, the polymerization is disfavored and does not go to completion. The topochemical polymerization can also be observed in water suspensions of micrometer-sized 6.1 cocrystals; the size distribution of these microcrystals, and the resulting polymer chains, can be controlled by sonication. Completely polymerized PIDA cocrystals show a highly resolved vibronic progression in their UV/vis absorption spectra. Extensive rinsing of the crystals in organic solvents such as methanol, THF, and chloroform separates the polymer from the soluble host. Once isolated, PIDA forms blue suspensions in a variety of solvents. The UV/vis absorption spectra of these suspensions match the cocrystal spectrum, without the vibronic resolution. However, they also include a new longer-wavelength absorption peak, associated with aggregation of the polymer chains.     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).