Dynamics of non-autonomous difference equation

In this paper we study the asymptotic behavior and periodicity of the equation \(x_{n+1}=p_{n}+\frac{x_{n}}{x_{n-1}}\), where \(x_{0}\ge 0, x_{-1}>0\) and \(p_{n}\) is a positive bounded sequence.     

Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics

The current study examines whether the engagement of mathematics teachers in modelling activities and subsequent changes in their conceptions about these activities affect their beliefs about mathematics. The sample comprised 52 mathematics teachers working in small groups in four modelling activities. The data were collected from teachers' Reports about features of each activity, interviews and questionnaires on teachers' beliefs about mathematics. The findings indicated changes in teachers' conceptions about the modelling activities. Most teachers referred to the first activity as a mathematical problem but emphasized only the mathematical notions or the mathematical operations in the modelling process; changes in their conceptions were gradual. Most of the teachers referred to the fourth activity as a mathematical problem and emphasized features of the whole modelling process. The results of the interviews indicated that changes in the teachers' conceptions can be attributed to structure of the activities, group discussions, solution paths and elicited models. These changes about modelling activities were reflected in teachers' beliefs about mathematics. The quantitative findings indicated that the teachers developed more constructive beliefs about mathematics after engagement in the modelling activities and that the difference was significant, however there was no significant difference regarding changes in their traditional beliefs.     

Multiplasmon replicas of the emission lines of bound excitons in ZnSe crystals

This paper reports on the results of investigations into the cathodoluminescence spectra of ZnSe crystals grown from a vapor phase and annealed in a bismuth melt [ZnSe(Bi) crystals], as well as in a bismuth melt containing aluminum [ZnSe(Bi,Al) crystals], at a temperature of 1200 K for 120 h. The longitudinal optical (LO) phonon replicas of the emission lines of free excitons are dominant in the cathodoluminescence spectra of the samples subjected to annealing in the bismuth melt with subsequent quenching. The emission spectra of all the samples studied, including the initial crystals, contain series of I ₁ ^s,d -nLO lines of bound excitons. It is revealed that plasmon replicas are characteristic not only of the I ₁ ^s -nLO lines. A weak exciton-plasmon interaction manifests itself in the form of single-plasmon Stokes satellites of the I ₁ ^d -nLO spectral lines. A theory describing the multiphoton optical transitions of bound excitons with the participation of mixed plasmonphonon vibrational modes is proposed. This theory makes it possible to calculate the shape function of an emission spectrum without recourse to model concepts. The results of the theoretical calculations are in agreement with the experimental data.     

Radio- and thermoluminescence of single-doped and codoped lithium-sodium sulfate

Radioluminescence at room temperature and thermoluminescence (TL) measurements of single-doped and codoped LiNaSO₄ above room temperature are reported here. The codoped samples were studied to investigate the possibility of enhancing the TL sensitivity of LiNaSO₄:Eu. This objective was not satisfied and the codopants (Ce, Sm, Ho and Er) decrease the TL sensitivity and slightly shift the dosimetric peak to lower temperatures. Samples doped with Mg, K, Bi and Tl were used with the hope that they may alter the trapping centers stability and introduce new peaks in the temperature range 430–500 K to observe any TL discontinuity or wavelength shift in their spectra as observed in CL measurements. This objective was fulfilled with Tl and Bi, where there is a discontinuity and/or wavelength shift at about 460 K. Such intensity and/or wavelength variations are ascribed to microstructural phase changes within the LiNaSO₄ crystals that may result from twin boundaries behaving like Na₂SO₄.     

Structural,M?ssbauer spectroscopy,magnetic properties,and thermal measurements of Y(3-x)Dy_x Fe_5O_(12)

Yttrium iron garnet powder samples((3-x)Dy_x Fe_5O_(12)), where part of yttrium ions are substituted by dysprosium ions with different concentrations are prepared by the solid state reaction method. The properties of the prepared samples are examined by different methods such as x-ray diffraction(XRD), Mssbauer spectroscopy, macroscopic magnetization measurements, and thermal measurements. The XRD measurements show that all the samples reveal the presence of a single garnet phase with a BCC structure. Room temperature Mssbauer spectra indicate that iron ions occupy three magnetic sites, i.e., two octahedral sites and one tetrahedral site. The saturation magnetization and the initial magnetic susceptibility decrease with the increase of Dy~(3+) substitution. The Curie temperature obtained from the thermal measurements seems to be independent of Dy~(3+) substitution.     

Identification problems for degenerate parabolic equations

This paper deals with multivalued identification problems for parabolic equations. The problem consists of recovering a source term from the knowledge of an additional observation of the solution by exploiting some accessible measurements. Semigroup approach and perturbation theory for linear operators are used to treat the solvability in the strong sense of the problem. As an important application we derive the corresponding existence, uniqueness, and continuous dependence results for different degenerate identification problems. Applications to identification problems for the Stokes system, Poisson-heat equation, and Maxwell system are given to illustrate the theory.