Periodic solutions to some problems of n-body type

We prove the existence of at least one T-periodic solution to a dynamical system of the type $$ - m_i \ddot u_i = \sum\limits_{j = 1,j \ne i}^n {\triangledown V_{ij} (u_i - u_j ,{\text{ }}t)}$$ (1) where the potentials V _ij are T-periodic in t and singular at the origin, u _i ε R ^k i=1, ..., n, and k≧3. We also provide estimates on the H ¹ norm of this solution. The proofs are based on a variant of the Ljusternik-Schnirelman method. The results here generalize to the n-body problem some results obtained by Bahri & Rabinowitz on the 3-body problem in [6].     

Synthesis of a Benzannulated Pyrrolizidine by a Copper‐Catalyzed Intramolecular α‐Arylation Reaction

A synthetic route to the pyrrolo[1,2‐a]indole ring system (benzannulated pyrrolizidine) involving a base‐induced intramolecular aza‐Michael reaction as the key C? N bond‐forming penultimate step, followed by a Cu‐catalyzed intramolecular α‐arylation reaction, to provide the tricyclic framework over six steps is described.     

Coordination Chemistry‐Assembled Multicomponent Systems Built from a Gable‐Like Bis‐Porphyrin: Synthesis and Photophysical Properties

Multiporphyrinic assemblies were quantitatively formed, in one step, from a gable‐like zinc(II) bis‐porphyrin ZnP₂ and free‐base porphyrins bearing pyridyl groups. The different fragments are held together by axial 4′‐N(pyridyl)–Zn interactions. Formation of a macrocycle ZnP₂?(4′‐cisDPyP) and a bis‐macrocycle (ZnP₂)₂?(TPyP) is discussed. The macrocycle and the bis‐macrocycle were crystallized and studied by X‐ray diffraction, which confirmed the excellent complementarity between the various components. Spectrophotometric and spectrofluorimetric titrations and studies reveal high association constants for both multiporphyrinic assemblies due to the almost perfect geometrical match between the interacting units. As expected, energy transfer from the zinc porphyrin component to the free‐base porphyrin quenches the fluorescence of the zinc porphyrin components in both compounds. But while in ZnP₂?(4′‐cis DPyP) sensitization of the emission of the free‐base porphyrin was observed, in (ZnP₂)₂?(TPyP) excitation of the peripheral Zn porphyrin units does not lead to quantitative sensitization of the luminescence of the free‐base porphyrin acceptor. An unusual HOMO–HOMO electron transfer reaction from ZnP₂ to the excited TPyP unit was detected and studied.     

Hydrophilic interaction liquid chromatography/tandem mass spectrometry for the analysis of diallyldimethylammonium chloride in water

We report a selective, sensitive and fast liquid chromatography/tandem mass spectrometry (LC/MS/MS) method for the determination of diallyldimethylammonium chloride (DADMAC) in water. Hydrophilic interaction liquid chromatography (HILIC) was used to avoid ion-pairing reagents, which are generally employed to retain cationic compounds. The complementary information obtained in a triple quadrupole mass spectrometer and in an ion trap Orbitrap has been used to study the fragmentation of the DADMAC cation [M](+) and for the correct assignment of the products ions. The HILIC/MS/MS method developed, using electrospray ionization in positive ion mode and selected reaction monitoring (SRM) acquisition mode, led to a reliable determination and confirmation of the DADMAC cation in water samples down to 50 ng L(-1). The low detection limit achieved, in combination with the absence of matrix effects, allowed the direct analysis of samples without any pretreatment, preconcentration or clean-up step. DADMAC was determined in samples collected in a drinking water treatment plant (DWTP) in Barcelona (Spain) and it was found in the influent at the μg L(-1) level.     

Electrospray ionization ion-trap multiple-stage mass spectrometry of Quillaja saponins

Fifteen identified C-18 fatty acyl-containing saponin structures from Quillaja saponaria Molina have been investigated by electrospray ionization ion-trap multiple-stage mass spectrometry (ESI-IT-MS(n)) in positive ion mode. Their MS(1)-MS(3) spectra were analyzed and ions corresponding to useful fragments, important for the structural identification of Quillaja saponins, were recognized. A few key fragments could describe the structural variations in the C-3 and the C-28 oligosaccharides of the Quillaja saponins. A flowchart involving a stepwise procedure based on key fragments from the MS(1)-MS(3) spectra of these saponins, together with key fragments from these saponins and 13 previously investigated saponins, was constructed for the identification of structural elements in Quillaja saponins. Peak intensity ratios in MS(3) spectra were found to be correlated to structural features of the investigated saponins and is therefore of value for the identification of regioisomers.     

Poly(ester urethane) guides for peripheral nerve regeneration

A biocompatible and elastomeric PU was synthesized from low-molecular-weight PCL as macrodiol, CMD as chain extender and HDI as chain linker for applications in the field of peripheral nerve repair. PU cast films supported in vitro attachment and proliferation of NOBEC. The in vitro adhesion and proliferation of S5Y5 neuroblastoma cells on the inner surface of uncoated, gelatin- and PL-coated PU guides were compared. Due to their superior in vitro performance, PL-coated PU guides were tested in vivo for the repair of 1.8 cm-long defects in rat sciatic nerves. The progressive regeneration was confirmed by EMG and histological analysis showing the presence of regenerating fibers in the distal stumps.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

Finding all pure strategy Nash equilibria in a planar location game

In this paper, we deal with a planar location-price game where firms first select their locations and then set delivered prices in order to maximize their profits. If firms set the equilibrium prices in the second stage, the game is reduced to a location game for which pure strategy Nash equilibria are studied assuming that the marginal delivered cost is proportional to the distance between the customer and the facility from which it is served. We present characterizations of local and global Nash equilibria. Then an algorithm is shown in order to find all possible Nash equilibrium pairs of locations. The minimization of the social cost leads to a Nash equilibrium. An example shows that there may exist multiple Nash equilibria which are not minimizers of the social cost.     

Radical scavenging activity of sterically hindered catecholate and o-amidophenolate complexes of LSbPh3 type

The radical scavenging effect of the substituted catecholates (1-3, 6) and o-amidophenolates (4, 5) of triphenylantimony(V) in reactions with DPPH^• radical and in a process of oleic acid peroxidation was studied in details. Complexes 1-6 show the high activity in radical scavenging reactions with DPPH^• radical leading to disappearance of radical species. Complexes were demonstrated to be high-efficient inhibitors of chain-radical process of the peroxidation of oleic acid as well as the effective destructors of the formed hydroperoxides. It was found that the effectiveness of complexes studied in the inhibition of the peroxidation of oleic acid depends on the first oxidation potential of complex.