Abstract:
For a family of Poisson algebras, parametrized by by an integer number r, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the r-th dispersionless KP, r-th dispersionless Dym and r-th dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential r-th dispersionless Dym equation, for which a complete integral is presented and therefore an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the r-th dispesionless modified KP equation.

Abstract:
In this paper the Galilean, scaling and translational self--similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non--scaling limit analysis of the one--matrix model are shown to correspond to the Galilean self--similarity condition for this hierarchy. We describe, in terms of the initial data for the zero--curvature 1--form of the AKNS hierarchy, the moduli space of these self--similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal--Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura--Hirota rational solutions of the NLS equation. An explicit 1--parameter family of Galilean self--similar solutions of the AKNS equation and the associated solution to the NLS equation is determined.

Abstract:
Purpose: of this paper is to present the results of research programme on improve polymer materials properties by irradiation. This method can be helpful in improving some of mechanical properties and thermal stability of polymers.Design/methodology/approach: Radiation processing involves the use of natural or man-made sources of high energy radiation on an industrial scale. The principle of the radiation processing is the ability of the high energy radiation to produce reactive cations, anions, and free radicals in materials. The industrial application of the radiation processing of plastic and composites includes polymerization, cross-linking and grafting. Radiation processing involves mainly the use of either electron beams from electron accelerators or gamma radiation from Cobalt-60 sources. The big advantage of radiation processing is, that does not make the product radioactive. In this research programme, the properties of natural (not irradiated) and irradiated polypropylene (PP), both unfilled and filled with 25% of glass fibres, were compared. Flexural strength, tensile strength, impact strength, thermal stability and complex modulus E* were researched. The injection moulding machine DEMAG – EGROTECH 50 – 200 was used for sample preparation. Irradiation was carried out in the company BGS Beta Gamma Service GmbH Co, KG, Saal am Donau, Germany with the electron rays, electron energy 10 MeV, doses of 15 and 33 kGy.Findings: The most important results are the enormous improvement of the thermal stability and some mechanical properties of irradiated PP.Practical implications: From the practical point of view the most important is the enormous improvement of the thermal stability of irradiated PP. The majority of industrial applications of radiation processing are cross-linking of wire and cable insulations, tube, heat shrink cables, components of tires, composites, moulded products for automotive and electrical industry etc.Originality/value: It is necessary to use engineering polymers or even high performance polymers in some application. In many cases it would be possible to use standard or engineering polymers and to improve their properties, e.g. by irradiation.

Abstract:
Purpose: The description of a new method of rubber parts wear testing especially wear of tyre treads is the main aim of this paper. Understanding of wear procedure could help to improve the quality of tyres and other rubber parts working in heavy terrain conditions.Design/methodology/approach: For that purpose testing equipment was designed and constructed. New method of testing of wear resistance based on gravimetric determination of mass loss of testing part during the test period was prepared and well – proven. Behaviour of testing samples during the test was monitored using high speed video camera.Findings: Because of complexity of this problem it would be very useful to continue this research and to describe in details the wear procedure using the new testing methods. Monitoring of wear progress by high speed video – camera may be one of the significant methods.Practical implications: The main benefit for praxis could be seen in new testing method which makes comparing different rubber compounds possible from the point of view of their wear (Chip - Chunk) resistance.Originality/value: Completely new in this paper is also monitoring of wear process using high speed video - camera.

Abstract:
In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots.

Abstract:
Darboux transformation is reconsidered for the supersymmetric KdV system. By iterating the Darboux transformation, a supersymmetric extension of the Crum transformation is obtained for the Manin-Radul SKdV equation, in doing so one gets Wronskian superdeterminant representations for the solutions. Particular examples provide us explicit supersymmetric extensions, super solitons, of the standard soliton of the KdV equation. The KdV soliton appears as the body of the super soliton.

Abstract:
We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the $n$-th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy we get the line soliton, the lump solution and the Johnson-Thompson lump, and the corresponding determinant formulae for the non-linear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases.

Abstract:
The vectorial fundamental transformation for the Darboux equations is reduced to the symmetric case. This is combined with the orthogonal reduction of Lame type to obtain those vectorial Ribaucour transformations which preserve the Egoroff reduction. We also show that a permutability property holds for all these transformations. Finally, as an example, we apply these transformations to the Cartesian background.

Abstract:
Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogues: the multidimensional quadrilateral lattices, i.e. lattices x: Z^N -> R^M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analogue of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogues of the Laplace, Combescure, Levy, radial and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between "transformations" and "discretizations" is also investigated for quadrilateral lattices. We finally interpret these results within the D-bar formalism.

Abstract:
We make a rigorous exploration of the profiles of a few diagonal and
off-diagonal components of linear (α_{xx}, α_{yy}, α_{xy} andα_{yx}), first nonlinear (β_{xxx}, β_{yyy}, β_{xyy} andβ_{yxx}), and second nonlinear (γ_{xxxx}, γ_{yyyy}, γ_{xxyy}andγ_{yyxx})
polarizabilities of quantum dots under the influence of external pulsed field.
Simultaneous presence of additive white noise has also been considered. The
quantum dot contains dopant described by a Gaussian potential. The numbers of
pulse and the dopant location have been found to fabricate the said profiles
jointly. The βcomponents display greater complexity in their
profiles in comparison with the αandγcounterparts. The presence of noise
prominently enhances the influence of dopant coordinate on the polarizability
profiles, particularly for αand γcomponents. However,
for βcomponents, the said influence becomes quite
evident both in the presence and absence of additive noise. The study reveals
some means of achieving stable, enhanced, and often maximized output of
noise-driven linear and nonlinear polarizabilities.