A method for solving two-point boundary-value problems of difference equations

The paper proposes an iterative solution method for discrete-time, nonlinear, two-point boundary-value problems (TPBVP) of the form: $$\begin{gathered} x(k) - x(k - 1) = f(k, x(k - 1), p(k)), \hfill \\ p(k) - p(k - 1) = g(k, x(k - 1), p(k)), \hfill \\ \end{gathered} $$ subject to $$h(x(0), p(0)) = 0,e(x(N), p(N)) = 0.$$ It is a counterpart of a method recently proposed by the authors for similar continuous-time TPBVPs with ordinary differential equations. The method, based on invariant imbedding and a generalized Riccati transformation, reduces the TPBVP to a pair of approximate initial-value problems with ordinary difference equations. Numerical tests are run on two examples originating in optimal control problems.     

Determination of strange quark suppression in hadronic vacuum

We study the forward-backward rapidity correlation slope in \(\bar p\) p interactions at high energies. We show that the measurement of this slope in different rapidity intervals, yields direct information on multiple-scattering. It turns out that the particles produced in inelastic rescattering are concentrated neary=0. This is a key feature of the multi-chain dual parton model, and leads to a simple explanation of the main features of multi-hadron production at very high energies, such as the energy dependence of the height and shape of the rapidity distribution, approximate K.N.O. scaling etc. We also show that the large values of the forward-backward correlation slope measured at ISR and SPS, are obtained in the model using the standard idea of short-range order in a single inelastic collision.     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

Analysis of topological cross sections in pp interactions at 22.4 GeVc

The multiplicity distribution for 11558 events in $\text{p}\text{p}$ interactions at 22.4 $\text{GeV}\text{c}$ was analyzed. The average charged multiplicity 〈n_ch〉 for inelastic events was found to be 4.69 ±0.05 and $\text{〈n}{}_{\mathrm{ch}}\text{〉}\text{D=2.04 ± 0.05}$. The early KNO scaling of normalized topological cross sections has been tested.     

Planar chalcogenide quarter wave stack filter for near-infrared

A quarter wave stack dielectric filter with normal incidence pass band in near-infrared range was prepared from alternating high index contrast chalcogenide films. The prepared filter consists of a low index Ge–S spacer layer surrounded by two 4.0 pairs Sb–Se/Ge–S reflectors. Films were deposited using flash and thermal evaporation techniques. After deposition, the filter was annealed at 165 °C for 1 h. Optical reflectivity measurements of the annealed filter revealed a ～63% normal incidence passband near 1540 nm. An ～80% passband was recorded after illumination of the filter by the light with s-polarization at angles 35°, 45° and 55°, while its position shifted to 1451, 1476 and 1505 nm, respectively. A ～75% passband appeared near 1436, 1467 and 1498 nm in response to illumination of the filter by the light with p-polarization at the same angles. The angular dependence of the reflectivity of dielectric multilayer can be exploited for filtering of incident light.     

In vitro degradation of porous poly(dl-lactide-co-glycolide) (PLGA)/bioactive glass composite foams with a polar structure

In vitro degradation of porous 50/50, 70/30 and 90/10 PLGA (poly(dl-lactide-co-glycolide)) foams and PLGA/bioactive glass (20 wt%) composite foams was studied up to 16 weeks in TRIS (pH 7.4; 37 °C). Polar PLGA/bioactive glass composite films were prepared by applying the bioactive glass (S53P4) on one side of the composite. Porous foams were made by solvent casting and pressure quenching with CO₂. The fabricated foams had an initial pore size of 50-500 μm and thickness of 2-2.5 mm. In vitro degradation of the prepared foams was evaluated after 1, 2, 4, 6, 8, 12 and 16 weeks. Weight loss, water uptake, molecular mass and the amount of dissolved bioactive glass were measured after each time period. Changes in pore morphology were analysed with SEM. The present in vitro results will be evaluated and compared with the results from ongoing animal studies where comparable implants are used for bone defect treatment under non-load-bearing conditions.     

Interval length continuation method for solving two-point boundary-value problems

We consider the solution of the following two-point boundary-value problem: $$\begin{gathered} \dot x(t) = f(t,x(t),p(t)), \dot p(t) = g(t,x(t),p(t)), t \in [0,T], \hfill \\ h(x(0),p(0)) = 0, p(T) = q. \hfill \\ \end{gathered} $$ We propose a combination technique consisting of the interval length continuation method and the back-and-forth shooting method. Certain alternative ways of employing continuation are discussed, and some of them are well suited for the problem under consideration. As a test for the method, a numerical example of a problem originating in optimal control is given.     

Fixed point of polynomial type nonlinear operators

A concept of generalized polynomial operators is considered, and a fixed-point problem with these operators is posed. The existence of a fixed point, with a minimum norm property, is stated and a power series representation is obtained. Problems of this kind appear, for instance, in some two-point boundary-value problems in optimal control.     

Spectroscopic properties of Ni and rare-earth codoped Ge-Ga-Sb-S glass

The infrared emission properties of rare-earth (i.e., Dy³⁺, Pr³⁺ or Sm³⁺) and nickel-codoped Ge-Ga-Sb-S glasses were investigated by photoluminescence measurements. It was found that the emission intensities of the rare-earth doped glasses were significantly weakened even with very few amount of nickel (i.e., 500 at. ppm). On the other hand, the nickel-doped glasses show larger indices of refraction, which property will be desirable in nonlinear optics. Our results, however, suggest that to obtain rare-earth doped chalcogenide glasses of better infrared emission property it is necessary to keep the nickel dopant as low as possible.     

Multilayer planar structures prepared from chalcogenide thin films of As–Se and Ge–Se systems and polymer thin films using thermal evaporation and spin-coating techniques

We report the preparation of multilayers based on polyamide–imide polymer and As–Se or Ge–Se chalcogenide thin films. Chalcogenide films of As–Se and Ge–Se systems were deposited using a thermal evaporation method periodically alternating with spin-coated Polyamide–imide films. Fifteen layers of PAI + As–Se system and nineteen layers of PAI + Ge–Se system were coated. Optical properties of prepared multilayers have been established using UV–vis–NIR and Ellipsometric spectroscopy. Both, PAI + As–Se and PAI + Ge–Se multilayer systems, exhibited the high-reflection bands centered around 830 nm and 1350 nm, respectively. The shift of the band position of PAI + Ge–Se multilayers to lower energies was caused by higher thickness of Ge–Se films. The bandwidth of reflection band of 8 PAI + 7 As–Se multilayer was ～90 nm while bandwidth of PAI + Ge–Se system decreased to ～70 nm because Ge–Se films have 0.1 lower refractive index against As–Se films. Design of 1D-photonic crystals based on alternating chalcogenide and polymer films is a new opportunity for application of chalcogenide thin films as optical materials for near-infrared region.