On the Recursive Sequence

The boundedness, global attractivity, oscillatory and asymptotic periodicity of the nonnegative solutions of the difference equation?in the title is investigated, where all the coefficients are nonnegative real numbers. The paper is motivated by an open problem proposed by Ladas [Open problems and conjectures, J. Differ. Equations?Appl., 5 (1999), 211–215].     

Synthesis of tin oxides SnO2–x in the entire composition range (x = 0 to 1) by ion‐beam sputter‐deposition

Ion‐beam sputter‐deposition (IBSD) was used to reactively deposit tin oxide crystalline films at oxygen fluxes of 3–15 sccm and at substrate temperatures of 100–600 °C. Analysing the samples by X‐ray diffraction and Raman spectro‐ scopy yields a map of the crystalline structures in dependence on the growth parameters. In addition to SnO₂, pure SnO films of high quality and an intermediate phase such as Sn₂O₃ or Sn₃O₄ can be reproducibly obtained. Thus, IBSD is, to our knowledge, the only thin‐film deposition technique verified yet to reliably produce samples in the entire composition range of tin oxides. (© 2015 WILEY‐VCH Verlag GmbH &Co. KGaA, Weinheim)     

Polyazapodands Derived from Biphenyl. Study of their Behaviour as Conformationally Regulated Fluorescent Sensors

Eight new polyazapodands containing a 4,4′-substituted biphenyl moiety have been synthesised. Four (7, 8, 9 and 11) are functionalised on positions 4 and 4′ with a nitro group and four (1, 2, 3 and 10) with a dimethylamino substituent. Comparison of the emission behaviour of 1, 2, 3 with that of the reference compounds 10 and tetramthylbenzidine, clearly suggests that a modification in the dihedral angle between the biphenyl rings is an important factor in determining the fluorescent response of the molecule. The fluorescence is pH dependent, due to the formation of intramolecular hydrogen bonds between protonated aliphatic nitrogens and a carbonyl oxygen, which influences the aforementioned dihedral angle. A crystal structure resolved by X-ray diffraction of 7·2HCl has been determined, and confirms the dependence of the angle and the rigidity on the hydrogen bonding. The complexation properties of these ligands have been studied with Zn²⁺, Cd²⁺, Ni²⁺, Cu²⁺ and Pb²⁺, which show that the number of amino groups within the pendants has a strong influence on the nature of the complexation and the fluorescent response of each ligand.     

A substructure-based SAR model for odor perception in humans relevant to health risk assessments

The ability of humans to perceive odors is a very complex phenomenon involving the selective binding of molecules to approximately 1000 olfactory receptors. Accordingly, the derivation of a substructure-based SAR model can be expected to be problematic. Yet, based upon published data on odor thresholds of volatile organic chemicals, we were able to derive such an SAR model. An examination of the structural determinants and related modulators indicates that lipophilicity is a major contributor to olfactory perception. The availability of a substructure-based SAR model permits an examination of the relationship between the presence in the environment of odorous chemicals and public health risks.     

Quasistatic Thermoviscoelastic Frictional Contact Problem with Damped Response

We analyze a problem which describes the frictional contact between a thermoviscoelastic body and a rigid foundation. The process is assumed to be quasistatic and the contact is modeled by a general normal damped response condition with friction law and heat exchange. Then we present a variational formulation of the problem, which is set in an abstract form as a system of evolution equations for the displacements and temperature. We establish the existence and uniqueness of the weak solution, using general results on evolution equations with monotone operators and fixed point arguments. Finally, we study the continuous dependence of the solution with respect to the initial data and contact conditions.     

A quadratic convergence scheme for determining families of periodic solutions

Global and local boundedness results for the solutions of a certain class of A₂-degenerate parabolic equations are proved.     

Further observations on the anharmonic motion equation

Peters [1] discussed some theoretical aspects of the ‘ringing’ sound associated with barbershop quartet singing. These combination tones were first investigated theoretically by Helmholtz [2] who showed that the transient displacement of the ear membrane is governed by the anharmonic motion equation. Rayleigh [3] modelled the situation as an initial value problem and this is now referred to as the Rayleigh problem. Following [1] there has been a renewed interest in the problem (see [4‐8]). The intention of the present paper is to:

• illustrate the qualitative behaviour of the solutions of the Rayleigh problem by a phase plane analysis;

• describe how the method of multiple scales provides a framework within which previous analyses can be seen;

• describe an improvement to the method of multiple scales, which may also prove useful in other applications.

     

Asymptotic behavior of the potential and existence of a periodic solution for a second order differential equation

We establish the existence of a 2π-periodic solution for a second order semilinear equation in terms of the asymptotic behavior of the potential of the nonlinearity. Our condition includes the case in which the nonlinearity is asymptotically linear with slopes at infinity of the jumping nonlinearity between the positive axes and the first Fucik spectrum curve (see [S. Fucik (1976 Fucik, S. 1976. Boundary value problems with jumping nonlinearities. Casopis Pest. Mat., 101: 69–87.  [Google Scholar]). Boundary value problems with jumping nonlinearities. Casopis Pest. Mat., 101, 69–87.]). Our results extend those of [L. Fernandez and F. Zanolin (1988 Fernandez, L. and Zanolin, F. 1988. Periodic solutions of a second order differential equation with one-sided growth restriction on the restoning term. Arch. Math., 51: 151–163.  [Google Scholar]). Periodic solutions of a second order differential equation with one-sided growth restriction on the restoning term. Arch. Math., 51, 151–163.].