Novel solvable many-body problems

Novel classes of dynamical systems are introduced, including many-body problems characterized by nonlinear equations of motion of Newtonian type (“acceleration equals forces”) which determine the motion of points in the complex plane. These models are solvable, namely their con?guration at any time can be obtained from the initial data by algebraic operations, amounting to the determination of the zeros of a known time-dependent polynomial in the independent variable z. Some of these models are multiply periodic, isochronous or asymptotically isochronous; others display scattering phenomena.     

A Solvable N-body Problem of Goldfish Type Featuring N2 Arbitrary Coupling Constants

A N-body problem “of goldfish type” is introduced, the Newtonian (“acceleration equal force”) equations of motion of which describe the motion of N pointlike unit-mass particles moving in the complex z-plane. The model—for arbitrary N—is solvable, namely its configuration (positions and velocities of the N “particles”) at any later time t can be obtained from its configuration at the initial time by algebraic operations. It features specific nonlinear velocity-dependent many-body forces depending on N² arbitrary (complex) coupling constants. Sufficient conditions on these constants are identified which cause the model to be isochronous—so that all its solutions are then periodic with a fixed period independent of the initial data. A variant with twice as many arbitrary coupling constants, or even more, is also identified.     

A Semi Rigid Novel Hydroxamate AMPED-Based Ligand for 89Zr PET Imaging

In this work, we designed, developed, characterized, and investigated a new chelator and its bifunctional derivative for ⁸⁹Zr labeling and PET-imaging. In a preliminary study, we synthesized two hexadentate chelators named AAZTHAS and AAZTHAG, based on the seven-membered heterocycle AMPED (6-amino-6-methylperhydro-1,4-diazepine) with the aim to increase the rigidity of the ⁸⁹Zr complex by using N-methyl-N-(hydroxy)succinamide or N-methyl-N-(hydroxy)glutaramide pendant arms attached to the cyclic structure. N-methylhydroxamate groups are the donor groups chosen to efficiently coordinate ⁸⁹Zr. After in vitro stability tests, we selected the chelator with longer arms, AAZTHAG, as the best complexing agent for ⁸⁹Zr presenting a stability of 86.4 ± 5.5% in human serum (HS) for at least 72 h. Small animal PET/CT static scans acquired at different time points (up to 24 h) and ex vivo organ distribution studies were then carried out in healthy nude mice (n = 3) to investigate the stability and biodistribution in vivo of this new ⁸⁹Zr-based complex. High stability in vivo, with low accumulation of free ⁸⁹Zr in bones and kidneys, was measured. Furthermore, an activated ester functionalized version of AAZTHAG was synthesized to allow the conjugation with biomolecules such as antibodies. The bifunctional chelator was then conjugated to the human anti-HER2 monoclonal antibody Trastuzumab (Tz) as a proof of principle test of conjugation to biologically active molecules. The final ⁸⁹Zr labeled compound was characterized via radio-HPLC and SDS-PAGE followed by autoradiography, and its stability in different solutions was assessed for at least 4 days.     

Weakly <Emphasis Type="Italic">φ</Emphasis>-pairs and common fixed points in cone metric spaces

In this paper we introduce a weak contractive condition, called weakly φ-pair, for two mappings in the framework of cone metric spaces and we prove a theorem which ensures existence and uniqueness of common fixed points for such mappings. Also we obtain a result on points of coincidence. These results extend and generalize well-known comparable results in the literature. The authors are supported by Università degli Studi di Palermo, R. S. ex 60%.     

Dynamics of Bianchi Type I Solutions of the Einstein Equations with Anisotropic Matter

We analyze the global dynamics of Bianchi type I solutions of the Einstein equations with anisotropic matter. The matter model is not specified explicitly but only through a set of mild and physically motivated assumptions; thereby our analysis covers matter models as different from each other as, e.g., collisionless matter, elastic matter and magnetic fields. The main result we prove is the existence of an ‘anisotropy classification’ for the asymptotic behaviour of Bianchi type I cosmologies. The type of asymptotic behaviour of generic solutions is determined by one single parameter that describes certain properties of the anisotropic matter model under extreme conditions. The anisotropy classification comprises the following types. The convergent type A₊: Each solution converges to a Kasner solution as the singularity is approached and each Kasner solution is a possible past asymptotic state. The convergent types B₊ and C₊: Each solution converges to a Kasner solution as the singularity is approached; however, the set of Kasner solutions that are possible past asymptotic states is restricted. The oscillatory type D₊: Each solution oscillates between different Kasner solutions as the singularity is approached. Furthermore, we investigate non-generic asymptotic behaviour and the future asymptotic behaviour of solutions. Submitted: October 28, 2008.; Accepted: January 26, 2009.     

Synthesis and characterization of chromium(I) bis(η-toluene) derivatives containing sterically demanding anions

The reaction of Cr(η⁶-CH₃C₆H₅)₂ with 1-benzoyl-6-hydroxy-6-phenyl fulvene, dbcpH, and with pentakis(methoxycarbonyl)cyclopentadiene, pcmcpH, proceeds with evolution of dihydrogen and the formation of the ionic derivatives [Cr(η⁶-CH₃C₆H₅)₂][X] ([X]⁻ = 1,2-dibenzoylcyclopentadienyl, [dbcp]⁻, pentakis(methoxycarbonyl)cyclopentadienyl, [pcmcp]⁻), which have been characterized by IR and EPR spectroscopies, X-ray diffraction and electrochemical techniques. The sterically demanding anions do not affect the structural and electronic properties of the cations in solution but strongly influence crystal packing. In fact, a rare cis-eclipsed conformation of the toluene rings is found for [Cr(η⁶-CH₃C₆H₅)₂][dbcp] · THF, whereas two independent complexes are observed in the unit cell of [Cr(η⁶-CH₃C₆H₅)₂][pcmcp], one with toluene rings in a cis-eclipsed conformation and the other in a staggered conformation (projections of methyl groups form an angle of 151°).     

Multiple scattering theory for non-local and multichannel potentials

Methodological advances in multiple scattering theory (MST) in both wave and Green's function versions are reported for the calculation of electronic ground and excited state properties of condensed matter systems with an emphasis on core-level photoemission and absorption spectra. Full-potential MST is reviewed and extended to non-local potentials. Multichannel MST is reformulated in terms of the multichannel density matrix whereby strong electron correlation of atomic multiplet type can be accounted for in both ground and excited states.     

COMMON FIXED POINT THEOREMS FOR MULTI-VALUED MAPS

We establish some results on coincidence and common fixed points for a twopair of multi-valued and single-valued maps in complete metric spaces.Presented theorems generalize recent results of Gordji et...     

Meir-Keeler Type Contractions for Tripled Fixed Points

In 2011, Berinde and Borcut [6] introduced the notion of tripled fixed point in partially ordered metric spaces. In our paper, we give some new tripled fixed point theorems by using a generalization of Meir-Keeler contraction.