Reduction Operators of Linear Second-Order Parabolic Equations

The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The ``no-go'' result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.Comment: 31 pages, minor misprints are correcte

Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem

The quasi-geostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Where possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.Comment: Extended version, 24 pages, 1 figur

Business in metaverse, or how we will communicate in the next 5 years

Even if the idea of Metaverse sounds unreal, all of us still have a number of questions about this whole system. But at the same time, it is obvious, that this will be a new state-of-the-art way to communicate, to travel, to run business, we will might need some time to get used to it. Augmented reality is really about closing the gap between the Internet and the world around us by overlaying digital content and information onto our physical world, worlds will ultimately come together as the Metaverse where users can freely and instantaneously hop between worlds, including our own

ЕФЕКТИВНІСТЬ ЕКСПЛУАТАЦІЇ СПЕЦІАЛЬНОЇ ТЕХНІКИ ДЛЯ ТРАНСПОРТУВАННЯ НЕБЕЗПЕЧНИХ ВІДХОДІВ У СКЛАДІ ПОБУТОВИХ

One of the key principles of state environmental safety is the efficient waste management. A significant factor in the impact on the environment is the presence of hazardous waste in the household materials. There is a tendency to accumulate hazardous waste in containers together with household ones in Ukraine's settlements. Hazardous waste in the composition of household waste is collected separately from other types of household waste; it should be also separated at the stage of collection or sorting and passed by consumers and operators of household waste disposal services to specialized enterprises having licenses for operations in the field of hazardous waste management. This situation is primarily due to the lack of specialized containers for hazardous waste and specialized equipment. Therefore, the purpose of the research is to define technical and operational indicators of specialized vehicles for hazardous substances transportation and to choose the most optimal model. The authors used mathematical and statistical methods in the course of the study. We present the results of investigation of efficiency of operation of the specialized equipment for transportation of hazardous waste in household waste composition. The issue of the efficiency of the operation of specialized equipment for the transportation of hazardous waste in the household waste composition is extremely relevant; hence it is being researched by many scientists. However, the study of the efficiency of modern vehicles operation requires in-depth research and calculations. Consequently, the best indicators of the overall operational and technical characteristics of the vehicles for the transportation of hazardous stuff are as follows: according to the specific weight of specialized vehicle equipment: KRAZ K16.2–0.4 tons; for specific engine power: OT-20 (ISUZU NQR 71P) – 30.7 kW; by the coefficient of compactness: OT-20 (ZIL-433360) – 3.04. The highest annual average productivity for OT-20 (ZIL-433360) is 96780.2 t km/year. By the load capacity (16 tons) and the volume of the container for hazardous waste in the solid household waste (30 m3), a specialized KRAZ K16.2 has the best technical characteristics.Небезпечні відходи у складі побутових відходів потрібно збирати окремо від інших видів побутових відходів, а також відокремлювати на етапі збирання чи сортування і споживачі та виконавці послуг із вивезення побутових відходів мають їх передавати спеціалізованим підприємствам, що одержали ліцензії на здійснення операцій у сфері поводження з небезпечними відходами. Наведено результати досліджень ефективності експлуатації спеціальної техніки для транспортування небезпечних відходів у складі побутових. Питання ефективності експлуатації спеціальної техніки для транспортування небезпечних відходів у складі побутових є надзвичайно актуальними, тому їх досліджують багато науковців. Проте вивчення ефективності експлуатації сучасних транспортних засобів потребують глибших досліджень та розрахунків. Встановлено, що найкращі показники загальної експлуатаційно-технічної характеристики транспортних засобів для перевезення небезпечних вантажів належать: за питомою масою спеціального обладнання спецавтомобіля: КрАЗ К16.2 ― 0,4 т; за питомою потужністю двигуна: ОТ-20(ISUZU NQR 71P) – 30,7 кВт; за коефіцієнтом компактності: ОТ-20 (ЗІЛ-433360) – 3,04. Найвищою є середньорічна продуктивність для ОТ-20 (ЗІЛ-433360) і становить 96780,2 т км/рік. За вантажопідйомністю (16 т) та об'ємом контейнера для небезпечних відходів у складі твердих побутових (30 м3) найкращі технічні характеристики належать спецавтомобілю КрАЗ К16.2

Social and economic trends for leaving home among young people

In this article we will look into the main trends for starting an independent life among young people in different countries and study the social and economic reasons for the changes

Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.Comment: 25 page

New results on group classification of nonlinear diffusion-convection equations

Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient $(1+1)$-dimensional nonlinear diffusion-convection equations of the general form $f(x)u_t=(D(u)u_x)_x+K(u)u_x.$ We obtain new interesting cases of such equations with the density $f$ localized in space, which have large invariance algebra. Exact solutions of these equations are constructed. We also consider the problem of investigation of the possible local trasformations for an arbitrary pair of equations from the class under consideration, i.e. of describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page

Computation of Invariants of Lie Algebras by Means of Moving Frames

A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature. Bases of invariants of the real solvable Lie algebras up to dimension five, the real six-dimensional nilpotent Lie algebras and the real six-dimensional solvable Lie algebras with four-dimensional nilradicals are newly calculated and listed in tables.Comment: 17 pages, extended versio

Mineral waters, metabolism, neuro-endocrine-immune complex, s. 252.

Dedicated to the 75th anniversary of Volodymyr Illich Kozyavkin with gratitude for the support of the Truskavetsian Scientific School of BalneologyThe monograph systematizes these writers and highlights the results of their own priority experimental and clinical-physiological studies of the impact of drinking mineral waters of Ukraine on neuroendocrine regulation, metabolism and immunity of healthy rats and patients in the process of rehabilitation of chronic pyelonephritis and cholecystitis in remission. In line with the concepts of functional-metabolic continuum and neuroendocrine-immune complex using the methods of factor, discriminant and canonical correlation analysis, it is demonstrated that mineral waters have both similar and specific physiologically favorable modulating effects on the parameters of the studied body systems. For specialists in medical rehabilitation, endocrinologists, immunologists, biochemists, pathophysiologists