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The article proposes a method of complex comparison of equipment for electrolyte-plasma polishing of surfaces. The Harrington function is used to evaluate the equipment. The values of the function are limited by the numerical interval 0.2 - 0.8. The parameters of the equipment are selected as the evaluation parameters: maximum polishing area per cycle, productivity, power, area occupied by the installation, cost. As a result of a comprehensive assessment, the best type of equipment for electrolyte-plasma polishing, which provides a resource-saving and energy-saving mode of operation was selected. A computer program has been developed with the help of which a generalized evaluation parameter has been calculated and a comparative analysis of the equipment has been performed.

When feeding broiler ducklings of the Agidel 343 breed, the most optimal raw protein level in the pre-start (0-10 days old), start (11-21 days old) mixed fodder of the first phase of fattening can be considered 22 and 21% and the second phase 21% (22-49 days old). It was found that at the end of fattening with increasing the protein level in mixed fodder of the second phase from 17 to 19 and 21%: increase in live weight of young ducklings by 4.5-8.8%; decrease in feed expenses per 1 kg of live weight gain of ducklings by 4.4-9.5%; increase in preservation of young animals by 5.3%; stimulation of protein and carbohydrate exchange in ducklings body by increasing total protein content in blood serum by 3.1-9.2% and glucose by 5.5-8.5%; improvement in nitrogen exchange by increasing total nitrogen absorbed in body by 6.8-18.7%; increasing slaughter yield from 68 to 68.5-69.8%, increasing muscle yield from 46.7 to 47.5-49.1% and edible body parts from 55.7 to 56.2-58.0%; increasing European fattening efficiency from 233 to 277 and 300 units and increasing economic efficiency by increasing the profitability of duck meat production from 29.2 to 39.1-41.4%.

The accident rate in road transport remains very high and it is necessary to consider all the factors that affect this process to reduce it. In this regard, special attention should be paid to the process of long-term braking, the negative manifestations of which require the development of additional braking systems (retarding retarders) for vehicles operating in mountainous areas, primarily in the sphere of passenger transport. Transmission retarder brakes, which provide sufficient braking performance, have a number of disadvantages that hinder their use. The existing motor brakes-retarders show insufficient retardation and Kuban State Agrarian University conducted researches to improve their effectiveness. The compressor brake mode was experimentally studied after the theoretical analysis, in which an increased pressure was created in the inlet collector, and at the end of the compression stroke, the air from the cylinder was released through a special valve back into the system, which created the braking effect. The experiments have confirmed the possibility of a significant increase in the braking torque of the engine in the compressor brake mode, when both flaps are closed - the exhaust flap after the exhaust collector and the inlet collector before the carburetor, and compressed air is supplied to the inlet collector at different pressures. Then the braking torque increases in comparison with the engine braking more than 3 times.

This paper presents a comprehensive program for modelling airframe structures made of composite materials using the finite element method. The purpose of this study is to apply the comprehensive approach to the creation of models in accordance with the main directions of numerical modelling. The “computational and experimental research pyramid” and its relationship with the complex modelling program are presented. Methods for calculating aggregates and structures using micromechanics, static, dynamic and resource strength methods are described, as well as the influence of the manufacturing cycle on the final performance of a product made of a composite material. This integrated approach allows one to optimize all the stages in the design process of the aircraft. Using this approach, one can replace part of the certification tests with modelling according to the “computational and experimental research pyramid”, which will reduce the cost of certification tests.

We have studied the ground state phase diagram of a spin ladder with alternating rung exchange $J^{n}_{\perp} = J_{\perp}[1 + (-1)^{n} \delta ]$ in a magnetic filed, in the limit where the rung coupling is dominant. In this limit the model is mapped onto an $XXZ$ Heisenberg chain in a uniform and staggered longitudinal magnetic fields, where the amplitude of the staggered field is $\sim \delta$. We have shown that the magnetization curve of the system exhibits a plateau at magnetization equal to the half of the saturation value. The width of a plateau scales as $\delta^{\nu}$, where $\nu =4/5$ in the case of ladder with isotropic antiferromagnetic legs and $\nu =2$ in the case of ladder with isotropic ferromagnetic legs. We have calculated four critical fields ($H^{\pm}_{c1}$ and $H^{\pm}_{c2}$) corresponding to transitions between different magnetic phases of the system. We have shown that these transitions belong to the universality class of the commensurate-incommensurate transition.

We give a graphical prescription for obtaining and characterizing all separable coordinates for which the Schr?dinger equation admits separable solutions for one of the superintegrable potentials Here xn+1 is a distinguished Cartesian variable. The algebra of second-order symmetries of the resulting Schr?dinger equation is given and, for the first potential, the closure relations of the corresponding quadratic algebra. These potentials are particularly interesting because they occur in all dimensions n ? 1, the separation of variables problem is highly nontrivial for them, and many other potentials are limiting cases.

The intensities of the noise in a bicrystal high-TC (HTS) Josephson junction have been precision-measured at 1–2 GHz frequency band at bias voltages up to 50 mV at T = 4.2 K. At large bias voltages, V > 30 mV, the dependence of current noise density was found exactly coinciding with the Schottky shot noise asymptote 2eI. At relatively low voltages, V < 4 mV, a noticeable noise rise has been registered. The broadening of Josephson oscillation linewidths ΔfJ over the values ΔfRSJ predicted by the RSJ model has been experimentally studied at different frequencies in the mm and submm wave range up to voltages V = 2 mV in connection with low-voltage noise rise. Both the features observed, the linewidth broadening and the excess noise over the noise level of thermal fluctuations, are discussed in terms of multiple Andreev reflection, giving rise to a nonequilibrium shot noise—the case which may take place in the d-wave superconducting junctions. Experimental results on noise performance are also compared with the qualitatively similar dependences of the current noise, known for the s-superconducting ballistic point-like or diffusive-type SNS junctions, where the excess low-voltage noise is manifested due to multiple Andreev reflections. Increasing the operating temperature, the thermal (equilibrium) fluctuations were found to predominate, resulting in a decrease of ratio ΔfJ/ΔfRSJ. The characteristics of the ac Josephson effect in HTS junctions measured at submm wave frequencies at temperatures close to the transition temperature TC are also discussed.

A finite two-dimensional oscillator is built as the direct product of two finite one-dimensional oscillators, using the dynamical Lie algebra su(2)x⊕su(2)y. The position space in this model is a square grid of points. While the ordinary `continuous' two-dimensional quantum oscillator has a symmetry algebra u(2), the symmetry algebra of the finite model is only u(1)x⊕u(1)y, because it lacks rotations in the position (and momentum) plane. We show how to `import' an SO(2) group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid. We thus propose a finite analogue for fractional U(2) Fourier transforms.

A finite two-dimensional radial oscillator of (N + 1)2 points is proposed, with the dynamical Lie algebra so(4) = su(2)x⊕su(2)y examined in part I of this work, but reduced by a subalgebra chain so(4)⊃so(3)⊃so(2). As before, there are a finite number of energies and angular momenta; the Casimir spectrum of the new chain provides the integer radii 0≤ρ≤N, and the 2ρ + 1 discrete angles on each circle ρ are obtained from the finite Fourier transform of angular momenta. The wavefunctions of the finite radial oscillator are so(3) Clebsch-Gordan coefficients. We define here the Hankel-Hahn transforms (with dual Hahn polynomials) as finite-N unitary approximations to Hankel integral transforms (with Bessel functions), obtained in the contraction limit N→∞.

Lie algebra contractions from o(n + 1) to e(n) are used to obtain asymptotic limits of interbase expansions between bases corresponding to different subgroup chains for the group O(n + 1). The contractions lead to interbase expansions for different subgroup chains of the Euclidean group E(n). They provide asymptotic formulae for quantities such as Wigner rotation matrices, Clebsch-Gordan coefficients and Racah coefficients.

The possibility that Schrodinger's equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability. Examples of this type of system are well known. In this paper we demonstrate how to establish a complete list of such potentials using essentially algebraic means. Our approach is to classify all nondegenerate potentials that admit a pair of second-order constants of motion. Here `nondegenerate' means that the potentials depend on four independent parameters. This is carried out for two-dimensional complex Euclidean space, though the method generalizes to other spaces and dimensions. We show that all these superintegrable systems correspond to quadratic algebras, and we work out the detailed structure relations and their quantum analogues.

The graded contractions of pseudo-Euclidean Lie algebra are studied. The non-equivalent gradings of of type and a are extended to the entire Lie algebra , using the action of on the Abelian ideal (the translations.). The graded contractions embed into a large family of six-dimensional Lie algebras. The family includes solvable, nilpotent and nonsolvable Lie algebras, both decomposable and indecomposable ones. The distinction between graded and Inonu-Wigner contractions is analysed. The physically most interesting Lie algebras obtained by the contractions are the inhomogeneous Galilei and pseudo-Galilei.

We investigate the consequences of contraction of the Lie algebras of the orthogonal groups to the Lie algebras of the Euclidean groups in terms of separation of variables for Laplace-Beltrami eigenvalue equations, and the solutions of these equations that arise through separation of variables techniques, on the N-sphere and in N-dimensional Euclidean space. General ellipsoidal and paraboloidal coordinates are included, not just the subgroup-type coordinates that have been the concern of most investigations of contractions as applied to special functions. We pay special attention to the case N = 2 where we show in detail, for example, how Lame polynomials contract to periodic Mathieu functions. Our point of view emphasizes the characterization of separable polynomial eigenfunctions in terms of the zeros of these eigenfunctions. We also consider all possible separable coordinate systems on the complex two-sphere (which includes real hyperboloids as special cases) and their contraction to flat space coordinates.

We study the free quantum motion on the three-dimensional sphere in ellipso-cylindrical coordinates, where we distinguish between prolate elliptic and oblate elliptic coordinates. The oblate and prolate elliptic wavefunctions are constructed in terms of the hyper-spherical and cylindrical wavefunctions. A perturbation theory for the elliptic wavefunctions in powers of the elliptic parameter is established. Finally, the interbasis expansions are used to set up path integral identities for the two ellipso-cylindrical coordinate systems on .

We extend recent work by Tremblay, Turbiner, and Winternitz which analyzes an infinite family of solvable and integrable quantum systems in the plane, indexed by the positive parameter k. Key components of their analysis were to demonstrate that there are closed orbits in the corresponding classical system if k is rational, and for a number of examples there are generating quantum symmetries that are higher order differential operators than two. Indeed they conjectured that for a general class of potentials of this type, quantum constants of higher order should exist. We give credence to this conjecture by showing that for an even more general class of potentials in classical mechanics, there are higher order constants of the motion as polynomials in the momenta. Thus these systems are all superintegrable.

Expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined. Rcpresentations of the elliptic basis as a superposition of polar and parabolic states are obtained. Certain interesting limiting cases are investigated.

It is shown that non-bijective quadratic transformation generated by the Kelly matrix changes the problem of a five-dimensional hydrogen atom into the problem of an eight-dimensional isotropic oscillator.

The Fock method is applied to the problem of a one-dimensional hydrogen atom. Integral Fock equations are obtained in discrete and continuous spectra; the case of zero energy is studied and wavefunctions and normalisation constants are calculated in the momentum representation.

A hydrogen atom is analysed in spheroidal coordinates. Limiting transformations R to 0 and R to infinity , R being a dimensional parameter characterising the spheroidal coordinates, are considered in trinomial recurrence relations and spheroidal wavefunctions. Expansions are found for normalised spheroidal wavefunctions of the hydrogen atom over spherical and parabolic bases, and relevant limiting transformations are studied.

It is shown that, unlike the model based only on the potential singularity U(x)=- mod x mod -1, the group of hidden symmetry explains not only the double degeneration of the energy spectrum but also the explicit form of the spectrum, wavefunctions and an extra constant of motion analogous to the Runge-Lenz vector.

A closed expression is obtained for the n-dimensional Wigner oscillator function, and a diagrammatic technique is developed for finding the explicit dependence of the function on Euler angles at any dimension n.

Results are reported for an experimental investigation of the generation of the difference frequency of partially mode-locked neodymium laser radiation in a nonlinear lithium niobate (LiNbO/sub 3/) crystal placed inside a rectangular waveguide. Difference-frequency radiation was detected at a wavelength of about 8 mm and also in the wavelength range from 2 to 4.5 mm the measurements indicate that the radiation was maximum in the 70-GHz range and that the spectral density of the radiation was independent of crystal length. Focusing with a cylindrical lens is found to increase the conversion efficiency of the laser radiation approximately twofold in comparison with the case of a homogeneous distribution of the laser beam across the waveguide. The experimental results are compared with theoretical calculations of difference-frequency generation in a rectangular waveguide.