Noncommutative classical and quantum mechanics for quadratic Lagrangians (Hamiltonians)

Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ _eff which depends on additional noncommutativity.     

Remarks on “Quasi-contraction on a cone metric space”

Recently, D. Ili? and V. Rako?evi? [D. Ili?, V. Rako?evi?, Quasi-contraction on a cone metric space, Appl. Math. Lett. (2008) doi:10.1016/j.aml.2008.08.011] proved a fixed point theorem for quasi-contractive mappings in cone metric spaces when the underlying cone is normal. The aim of this paper is to prove this and some related results without using the normality condition.     

Quantitative study of fluorescence excitation and emission spectra of bean leaves

A quantitative and comprehensive knowledge of leaf fluorescence is required for the interpretation of fluorescence signals at the canopy level and also for the modelling of leaf and canopy fluorescence. In this work we present full range fluorescence excitation and emission spectra of intact leaves, expressed in units of apparent spectral fluorescence yield, from both the adaxial and the abaxial sides of the leaves, and for both front-side and back-side geometries. Emission spectra were measured for incident radiations in the blue and the green spectral range. The red/far-red fluorescence ratio depended on the measurement geometry and on the excitation wavelength. Excitation spectra were measured for emissions at 687 and 760 nm. When the abaxial side was illuminated, the measured spectra always had a larger intensity compared to adaxial side that is explained by the higher scattering of the spongy tissues. At 760 nm, the spectra had the same shape for front-side and back-side geometry, indicating that scattering predominated. At 687 nm, the shape of the spectra was very different for front-side and back-side geometry due to re-absorption of red fluorescence within the leaf. The comparison of excitation spectra measured from the adaxial or the abaxial side revealed differences in carotenoid absorption.     

Brachistochrone with limited reaction of constraint in an arbitrary force field

The paper considers brachistochronic motion of a particle along a curve y=y(x) in an arbitrary force field in the vertical plane of Cartesian coordinate system. The curve is treated as a bilateral or unilateral constraint that can be smooth or rough. The projection of the reaction force of the curve onto the normal to the curve is confined to the fixed limits. A control variable u is given as the second derivative of the function y(x) relative to the horizontal coordinate?x of the particle, i.e., u=d ² y/dx ². Applying Pontryagin??s maximum principle and singular optimal control theory, the problem is reduced to numerical solving of the corresponding two-point boundary value problem. The procedure based on the shooting method is used to solve the boundary value problem. Two examples with friction forces of the viscous friction and Coulomb friction type have been solved.     

Hysteretic behavior of rectangular tube (box) sections based on Preisach model

In the present paper, the Preisach model of hysteresis is applied to model cyclic behavior of elasto-plastic material. Rate of loading and viscous effects will not be considered. The problem of axial loading of rectangular cross-section and cyclic bending of rectangular tube (box) will be studied in details. Hysteretic stress–strain loop for prescribed history of stress change is plotted for material modeled by series connection of three unite element. Also moment–curvature hysteretic loop is obtained for a prescribed curvature change of rectangular tube (box). All obtained results clearly show advantages of the Preisach model for describing cyclic behavior of elasto-plastic material.     

Theoretical analysis of the experimental UV-Vis absorption spectra of some phenolic Schiff bases

The time-dependent density functional theory (TDDFT) was applied in conjunction with the natural bond orbital analysis to examine the UV-Vis properties of 10 phenolic Schiff bases. The calculations were performed with different functionals, but main discussion refers to results obtained at the B3LYP/6-311+G(d,p) level of theory. The approach based on the natural localised molecular orbital clusters indicates similar behaviour for majority of examined compounds. The HOMO (“highest occupied molecular orbital”) cluster is delocalised over the ring which is electron richer, the HOMO-1 cluster is spread over the other ring, whereas the LUMO (“lowest unoccupied molecular orbital”) cluster is situated on the imino group. The two bands at long wavelengths correspond to the HOMO → LUMO and HOMO-1 → LUMO transitions, i.e. from both A and B rings to the imino group. The next band originates from a transition from the imino group to the imino group. The band at the smallest wavelengths originates from a transition from the A ring to the A ring, or from the B ring to the B ring. Our findings are in very good agreement with the existing literature data.     

Channeling of fast ions through the bent carbon nanotubes:The extended two-fluid hydrodynamic model

We investigate the interactions of charged particles with straight and bent single-walled carbon nanotubes(SWNTs)under channeling conditions in the presence of dynamic polarization of the valence electrons in carbon. This polarization is described by a cylindrical, two-fluid hydrodynamic model with the parameters taken from the recent modelling of several independent experiments on electron energy loss spectroscopy of carbon nano-structures. We use the hydrodynamic model to calculate the image potential for protons moving through four types of SWNTs at a speed of 3 atomic units. The image potential is then combined with the Doyle–Turner atomic potential to obtain the total potential in the bent carbon nanotubes.Using that potential, we also compute the spatial and angular distributions of protons channeled through the bent carbon nanotubes, and compare the results with the distributions obtained without taking into account the image potential.     

Desmotropy, polymorphism, and solid-state proton transfer: four solid forms of an aromatic o-hydroxy Schiff base

The Schiff base derived from salicylaldehyde and 2-amino-3-hydroxypyridine affords a diversity of solid forms, two polymorphic pairs of the enol-imino (D1?a and D1?b) and keto-amino (D2?a and D2?b) desmotropes. The isolated phases, identified by IR spectroscopy, X-ray crystallography, and (13)C cross-polarization/magnetic angle spinning (CP/MAS) NMR spectroscopy, display essentially planar molecular conformations characterized by strong intramolecular hydrogen bonds of the O-H???N (D1) or N-H???O (D2) type. A change in the position of the proton within this O???H???N system is accompanied by substantially different molecular conformations and, subsequently, by divergent supramolecular architectures. The appearance and interconversion conditions for each of the four phases have been established on the basis of a number of solution and solvent-free experiments, and evaluated against the results of computational studies. Solid phases readily convert into the most stable form (D1?a) upon exposure to methanol vapor, heating, or by mechanical treatment, and these transformations are accompanied by a change in the color of the sample. The course of thermally induced transformations has been monitored in detail by means of temperature-resolved powder X-ray diffraction and infrared spectroscopy. Upon dissolution, all forms equilibrate immediately, as confirmed by NMR and UV/Vis spectroscopy in several solvents, with the equilibrium shifted far towards the enol tautomer. This study reveals the significance of peripheral groups in the stabilization of metastable tautomers in the solid state.     

Desmotropy,Polymorphism, and Solid‐State Proton Transfer: Four Solid Forms of an Aromatic o‐Hydroxy Schiff Base

The Schiff base derived from salicylaldehyde and 2‐amino‐3‐hydroxypyridine affords a diversity of solid forms, two polymorphic pairs of the enol‐imino ( D1 a and D1 b ) and keto‐amino ( D2 a and D2 b ) desmotropes. The isolated phases, identified by IR spectroscopy, X‐ray crystallography, and ¹³C cross‐polarization/magnetic angle spinning (CP/MAS) NMR spectroscopy, display essentially planar molecular conformations characterized by strong intramolecular hydrogen bonds of the O? H???N ( D1 ) or N? H???O ( D2 ) type. A change in the position of the proton within this O???H???N system is accompanied by substantially different molecular conformations and, subsequently, by divergent supramolecular architectures. The appearance and interconversion conditions for each of the four phases have been established on the basis of a number of solution and solvent‐free experiments, and evaluated against the results of computational studies. Solid phases readily convert into the most stable form ( D1 a ) upon exposure to methanol vapor, heating, or by mechanical treatment, and these transformations are accompanied by a change in the color of the sample. The course of thermally induced transformations has been monitored in detail by means of temperature‐resolved powder X‐ray diffraction and infrared spectroscopy. Upon dissolution, all forms equilibrate immediately, as confirmed by NMR and UV/Vis spectroscopy in several solvents, with the equilibrium shifted far towards the enol tautomer. This study reveals the significance of peripheral groups in the stabilization of metastable tautomers in the solid state.     

Some graphs whose second largest eigenvalue does not exceed 2

We determine all trees whose second largest eigenvalue does not exceed $\sqrt{2}$. Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to $\sqrt{2}$. Some additional results are also given.