Multiplicity theorems for superlinear elliptic problems

In this paper we study second order elliptic equations driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti–Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. Finally we indicate how in the semilinear case, Morse theory can be used to produce six nontrivial solutions. This paper was completed while the first author was visiting the University of Aveiro as an Invited Scientist. The hospitality and financial support of the host institution are gratefully acknowledged. The second and third authors acknowledge the partial financial support of the Portuguese Foundation for Science and Technology (FCT) under the research project POCI/MAT/55524/2004.     

Direct time-resolved detection of singlet oxygen in zeolite-based photocatalysts

Singlet oxygen has been characterized spectroscopically as a product of the exposure of suspensions of zeolites containing oxidation catalysts. Spectroscopic and lifetime studies show that a part of the singlet oxygen formed reacts within the zeolite porous structure, while a significant fraction escapes and becomes available for reaction in the bulk media. The liquid phase plays a key role in determining intra- and extracavity dynamics.     

Heterogeneous gold catalysts for efficient access to functionalized lactones

A novel class of heterogeneous gold catalysts supported on zeolite beta-NH4+ was prepared by the deposition-precipitation method. This new class of catalyst showed interesting catalytic activities for the intramolecular cycloisomerization of gamma-acetylenic carboxylic acids leading to functionalized gamma-alkylidene gamma-butyrolactones. Analysis of the supported gold species with in situ X-ray photoelectron spectroscopy (in situ XPS) suggests that cationic Au (possibly AuIII) can play an important role in such reactions. The high discrepancy in catalyst stability in favor of the Au supported on the zeolite system over bulk Au2O3 is explained by 1) the size of the particles and 2) the reversibility of the redox deactivating process (AuIII-->AuI) in the presence of oxygen for the supported system. The efficiency of this system allowed reaction under mild heterogeneous conditions. The potential for catalyst recycling was also highlighted.     

Highly Conjugated Bis(benzo[b]phosphole)-P-oxides: Synthesis and Electrochemical,Optical, and Computational Studies

The first examples of a π-conjugated benzo[b]phosphole P-oxide in which two phosphole P-oxide units are connected by a carbon-carbon double bond are described. The molecules are synthesized as E isomers with respect to the carbon-carbon double bond and exist as stable cis and trans isomers (chiral and meso one respectively) relatively to the two stereogenic P atoms. The optical and electrochemical properties of both isomers have been investigated by experiment and computations.     

Rotational fluctuations of water confined to layered oxide materials: nonmonotonous temperature dependence of relaxation times

The rotational molecular dynamics of water confined to layered oxide materials with brucite structure was studied by dielectric spectroscopy in the frequency range from 10(-2) to 10(7) Hz and in a broad temperature interval. The layered double hydroxide samples show one relaxation process, which was assigned to fluctuations of water molecules forming a layer, strongly adsorbed to the oxide surface. The temperature dependence of the relaxation rates has an unusual saddlelike shape characterized by a maximum. The model of Ryabov et al. (J. Phys. Chem. B 2001, 105, 1845) recently applied to describe the dynamics of water molecules in porous glasses is employed also for the layered materials. This model assumes two competing effects: rotational fluctuations of water molecules that take place simultaneously with defect formation, allowing the creation of free volume necessary for reorientation. The activation energy of rotational fluctuations, the energy of defect formation, a pre-exponential factor, and the defect concentration are obtained as main parameters from a fit of this model to the data. The values of these parameters were compared with those found for water confined to nanoporous molecular sieves, porous glasses, or bulk ice. Several correlations were discussed in detail, such as the lower the value of the energy of defect formation, the higher the number of defects. The pre-exponential factor increases with increasing activation energy, as an expression of the compensation law, and indicates the cooperative nature of the motional process. The involvement of the surface OH groups and of the oxygen atoms of the interlayer anions in the formation of hydrogen bonds was further discussed. For the birnessite sample, the relaxation processes are probably overlaid by a dominating conductivity contribution, which is analyzed in its frequency and temperature dependence. It is found that the conductivity of birnessite obeys the characteristics of semiconducting disordered materials. Especially the Barton/Nakajima/Namikawa relationship is fulfilled. Analyzing the temperature dependence of the direct current (dc) conductivity sigma0 in detail gives some hint that sigma0(T) has also an unusual saddlelike form.     

Self-assembled lamellar structures with functionalized single wall carbon nanotubes

Highly ordered self-assembled multi-layer structures with denatured collagen wrapped single wall carbon nanotubes and surfactant systems were obtained through bioinspired methodology.     

Dinuclear amine-thiophenolate complexes coligated by ferrocenemonocarboxylate and 1,1'-ferrocenedicarboxylate anions: preparation, characterization and structures of trinuclear [LMII 2(O2CC5H4FeCp)]+ and Pentanuclear [(LMII 2)2(O2CC5H4)2Fe]2+ complexes (M=Co, Ni, Zn)

A series of novel tri- and pentanuclear complexes composed of dinuclear LM(2) units (M=Co, Ni, Zn; L=24-membered macrocyclic hexaazadithiophenolate ligand) and ferrocenecarboxylate ([CpFeC(5)H(4)CO(2)](-)) or 1,1'-ferrocenedicarboxylate ([Fe(C(5)H(4)CO(2))(2)](2-)) groups is reported. The complexes [LM(II) (2)(O(2)CC(5)H(4)FeCp)](+) (M=Co (6), Ni (7), Zn (8)) and [(LM(II) (2))(2)(O(2)CC(5)H(4))(2)Fe](2+) (M=Co (9), Ni (10)) have been prepared by substitution reactions from labile [LM(II) (2)L'](+) precursors (L'=Cl, OAc) and the respective ferrocenecarboxylate anions in methanol. Mixed-valent [(LCo(II)Co(III))(2)(O(2)CC(5)H(4))(2)Fe](4+) (11) was prepared by oxidation of 9 with bromine. Complexes 7[BPh(4)], 8[BPh(4)], 9[BPh(4)](2), 10[BPh(4)](2), and 11[ClO(4)](4) have been characterized by X-ray crystallography; showing that the ferrocenyl carboxylates act as bidentate (7, 8) or bis-bidentate (9-11) bridging ligands towards one or two bioctahedral LM(2) subunits, respectively. The structures are retained in solution as indicated by NMR spectroscopic studies on the diamagnetic Zn(2)Fe complex 8[ClO(4)]. Electrochemical studies reveal significant anodic potential shifts for the oxidation potential of the ferrocenyl moieties upon complexation and the magnitude of the potential shift appears to correlate with the charge of the LM(2) subunits. This is qualitatively explained in terms of destabilizing electrostatic (Coulomb) interactions between the M(2+) ions of the LM(2) unit and the proximate ferrocenium fragment. An analysis of the temperature-dependent magnetic susceptibility data for 10[BPh(4)](2) shows the presence of weak ferromagnetic magnetic exchange interactions between the Ni(II) ions in the LNi(2) units. The exchange coupling across the ferrocenedicarboxylate bridge is negligible.     

High-Field Electron Paramagnetic Resonance Reveals a Stable Glassy Fraction up to Melting in Semicrystalline Poly(dimethylsiloxane)

The reorientation of the guest 4-methoxy-TEMPO (spin probe) in the disordered fraction of semicrystalline poly(dimethylsiloxane) (PDMS) is investigated by high-field electron paramagnetic resonance (HF-EPR) at 190 and 285 GHz. Accurate numerical simulations of the HF-EPR lineshapes evidence that the reorientation times of the spin probes are distributed between the melting temperature \(T_{\rm m}\) and \(T_{\rm m}\)—30 K. The distribution exhibits, in addition to a broad component, a narrow component with low mobility up to the PDMS melting point. It is shown that the temperature dependence of the reorientation time of the spin probes with low mobility is the same of the spin probes in glassy PDMS. The result suggests that the low-mobility fraction is localized in the so-called rigid amorphous fraction.     

A Class of Integrable Flows on the Space of Symmetric Matrices

For a given skew symmetric real n × n matrix N, the bracket [X, Y] _N = XNY − YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n × n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system , or equivalently in Lax form, the equation on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure. If N is invertible, several remarkable identifications can be implemented. First, (Sym(n, N), [·, ·]) is Lie algebra isomorphic with the symplectic Lie algebra associated to the symplectic form on given by N ⁻¹. In this case, the system is the reduction of the geodesic flow of the left invariant Frobenius metric on the underlying symplectic group Sp(n, N ⁻¹). Second, the trace of the product of matrices defines a non-invariant non-degenerate inner product on Sym(n, N) which identifies it with its dual. Therefore Sym(n, N) carries a natural Lie-Poisson structure as well as a compatible “frozen bracket” structure. The Poisson diffeomorphism from Sym(n, N) to maps our system to a Mischenko-Fomenko system, thereby providing another proof of its integrability if N is invertible with distinct eigenvalues. Third, there is a second ad-invariant inner product on Sym(n, N); using it to identify Sym(n, N) with itself and composing it with the dual of the Lie algebra isomorphism with , our system becomes a Mischenko- Fomenko system directly on Sym(n, N). If N is invertible and has distinct eigenvalues, it is shown that this geodesic flow on Sym(n, N) is linearized on the Prym subvariety of the Jacobian of the spectral curve associated to a Lax pair formulation with parameter of the system. If, on the other hand, N has nullity one and distinct eigenvalues, in spite of the fact that the system is completely integrable, it is shown that the flow does not linearize on the Jacobian of the spectral curve. Research partially supported by NSF grants CMS-0408542 and DMS-0604307. Research partially supported by the Swiss SCOPES grant IB7320-110721/1, 2005-2008, and MEdC Contract 2-CEx 06-11-22/25.07.2006. Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932. Research partially supported by the Swiss NSF and the Swiss SCOPES grant IB7320-110721/1.