Smoothing estimates for the Schrödinger equation with unbounded potentials

We prove a local in time smoothing estimate for a magnetic Schrödinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two derivatives. The proof is based on the multiplier method and no pseudodifferential techniques are required.     

Investigation of the Homologous Series of 4-Acetyl-4′-n-Alkanoyloxyazobenzenes by X-ray Diffraction

This paper reports an investigation by X-ray diffraction of the homologous series of liquid crystalline 4-acetyl-4′-n-alkanoyloxyazobenzenes. By using two independent parameters, namely the temperature and the paraffinic chain length, it was possible to obtain some structural information on two solid phases for comparison with a previously reported model for the S_A phase of the same series. In particular, the molecular layer thickness and, by using previously reported dilatometric data, the aromatic sublayer thickness, the methylene group thickness, the molecular area and the aromatic stem inclination angle were obtained for the different phases.     

Controlled Hopf bifurcation of a storage-ring free-electron laser

Local bifurcation control is a topic of fundamental importance in the field of nonlinear dynamical systems. We discuss an original example within the context of storage-ring free-electron laser physics by presenting a new model that enables analytical insight into the system dynamics. The transition between the stable and the unstable regimes, depending on the temporal overlapping between the light stored in the optical cavity and the electrons circulating into the ring, is found to be a Hopf bifurcation. A feedback procedure is implemented and shown to provide an effective stabilization of the unstable steady state.     

Phase transitions and structural studies of new compounds belonging to homologous series

Summary Since few years, we have been studying the synthesis and the physical properties of homologous series with the following general structure: CH₃-(CH₂) _m -CO-C₆H₄-N=N-C₆H₄-O-CO-(CH₂) _n -CH₃, withn ranging from zero to sixteen. In this work we report the results of the diffractometric study of the liquid crystalline series 4-butanoyl-4′-n-alkanoyloxy azobenzenes (m=2) and 4-pentanoyl-4′-n-alkanoyloxy azobenzenes (m=3): in particular all the ordered smectic phases of them=3 series have been identified and some structural parameters are discussed. Considering the smecticA phase, we have determined the layer thickness as a function ofn for both them=2 andm=3 series: the results have been compared with those of the homologous series withm=0 andm=1. In particular an appreciable decrease of the thickness increment due to a CH₂ group and a corresponding increase of the thickness of aromatic sublayer were observed: these results are interpreted by means of a proposed model.

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Riassunto Da alcuni anni si stanno studiando la sintesi e le proprietà fisiche di alcune serie omologhe aventi la seguente struttura generale: CH₃-(CH₂) _m -CO-C₆H₄-N=N-C₆H₄-O-CO-(CH₂) _n -CH₃, conn variabile da zero a sedici. In questo lavoro vengono descritti i risultati di uno studio diffrattometrico della serie liquido cristallina dei 4-butanoil-4′-n-alkanoilossi azobenzeni (m=2) e dei 4-pentanoil-4′-n-alkanoilossi azobenzeni (m=3): in particolare le fasi smettiche ordinate della seriem=3 sono state identificate e ne vengono discussi i parametri strutturali. Per quanto riguarda la fase smetticaA, abbiamo misurato lo spessore degli strati in funzione din sia nella seriem=2 chem=3: i risultati ottenuti sono stati confrontati con quelli delle serie omologhe conm=0 em=1. In particolare sono state osservate un'apprezzabile diminuzione dell'incremento dello spessore dovuto al gruppo CH₂ ed un corrispondente aumento dello spessore del substrato aromatico: il modello che viene proposto permette di interpretare tali fenomeni.

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Резюме В течение нескольких лет мы исследовали синтез и физические свойства гомологических рядов со следующей структурой: CH₃-(CH₂) _m -CO-C₆H₄-N=N-C₆H₄-O-CO-(CH₂) _n -CH₃, гдеn находится в области от 0 до 16. В этой работе мы приводим результаты дифрактометрических исследований жидких кристаллических соединений сm=2 сm=3, в частности, индентифицируются все упорядоченные смектические фазы дляm=3 и обсуждаются некоторые структурные параметры. Рассчитывая смектическуюА фазу, мы определяем зависимость толщины слоя от для рядов сm=2 и сm=3. Полученные результаты сравниваются с результатами для гомологических рядов сm=0 и сm=1. В частности, наблюдается заметное уменьшение приращения толщины, обусловленное группой CH₂, и соответствующее увеличение толщины ароматического подслоя. Полученные результаты интерпретиружтся с помощью предложенной модели.

     

An unresolved conjecture on nonmaximal planar graphical sequences

In this work it is shown that the Euler nonmaximal sequences, 55555555555553 755555555555555555 (more briefly 5¹³3¹ 7¹5¹⁷) are not planar graphical, partly proving an unresolved conjecture by Schmeichel and Hakimi.     

Liquid Crystalline Phases in 4-Acetyl-4′-n-Alkanoyloxy-Azobenzenes

Abstract

The preparation and melting behaviour of the homologous series of 4-acetyl-4′-n-alkanoylox-azobenzenes, CH₃-CO-C₆H₄-N?N-C₆H₄-O-CO-(CH₂) _n -CH₃, (with n ranging from zero to 16) is reported. The melting behaviour has been investigated by means of thermal microscopy and differential scanning calorimetry. The experimental results indicate for almost all the compounds the presence of solid state polymorphism and a type A smectic; for one compound a type B smectic phase is also present. Some homologs present, in addition, a nematic phase.