Group classification of heat conductivity equations with a nonlinear source

We suggest a systematic procedure for classifying partial differential equations invariant with respect to low dimensional Lie algebras. This procedure is a proper synthesis of the infinitesimal Lie's method, technique of equivalence transformations and theory of classification of abstract low dimensional Lie algebras. As an application, we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms. We have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry. It is shown that there are three, seven, twenty eight and twelve inequivalent classes of partial differential equations of the considered type that are invariant under the one-, two-, three- and four-dimensional Lie algebras, correspondingly. Furthermore, we prove that any partial differential equation belonging to the class under study and admitting symmetry group of the dimension higher than four is locally equivalent to a linear equation. This classification is compared to existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach. Furthermore, a number of new invariant equations are constructed which have rich symmetry properties and, therefore, may be used for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page

Berry phases for 3D Hartree type equations with a quadratic potential and a uniform magnetic field

A countable set of asymptotic space -- localized solutions is constructed by the complex germ method in the adiabatic approximation for 3D Hartree type equations with a quadratic potential. The asymptotic parameter is 1/T, where $T\gg1$ is the adiabatic evolution time. A generalization of the Berry phase of the linear Schr\"odinger equation is formulated for the Hartree type equation. For the solutions constructed, the Berry phases are found in explicit form.Comment: 15 pages, no figure

THE LEFT VENTRICLE DIASTOLIC FUNCTION IN PATIENTS WITH HYPERTENSION UNDER THE USE OF DIFFERENT DRUG GROUPS

The objective: To assess the prevalence of diastolic dysfunction in patients with hypertension and preserved left ventricular ejection fraction under pharmacological correction (monotherapy) with angiotensin converting enzyme inhibitors, angiotensin II receptor blockers and β-blockers. Materials and methods: 82 patients (58 women and 24 men) with stage 2 hypertension were examined. The diastolic function was assessed via echocardiography in accordance with the European Association of Cardiovascular Imaging guidelines (2017). Echocardiography was performed before the onset of the treatment and 6 months after its onset. The treatment onset was considered to start after a 2-week period of elimination of previously used pharmacological substance and 2 weeks of assessing tolerability, dose and regimen adjustment. Results: For all selected drugs, target values of blood pressure were achieved, and no adverse effects were identified. The average values of the left atrial volume index before and after the treatment course did not show significant differences. In the majority of the examined patients, this parameter did not exceed the threshold value of 34 ml/m2 . Values exceeding the specified threshold were observed in Group 1 in 4 patients, in Group 2 in 3 patients and in Group 3 in 8 patients. According to the Tissue Doppler echocardiography results on the velocity of myocardial motion at the early diastolic filling, which was measured at the level of the lateral segments of mitral valve and the interventricular septum, positive, but unreliable changes were observed in the Groups of bisoprolol and valsartan, and no changes — in the Group of perindopril. According to the traditional criteria, diastolic dysfunction was observed in 80 % of patients, while according to the criteria of the European Association of Cardiovascular Imaging (2017) — in 21 % of patients. Conclusion: The same efficacy of all three drugs is observed in terms of achieving target blood pressure values. The most pronounced effect on the morphometric parameters of the left atrium and intracardiac hemodynamics is shown in the Groups of bisoprolol and valsartan

Investigation of the blood cooling process cattle

Perfection of machinery and technology for the processing of secondary raw materials of the meat industry is a very urgent task. One of the most important methods in the processing of blood of cattle is the effect of low temperatures, used both for preserving blood by freezing, and for its concentration by the method of freezing moisture. In this regard, the article examines the process of cooling the blood of cattle, which allows the most complete assessment of processes and phenomena occurring in the blood under the influence of low temperature. The features of studying the process of temperature reduction based on the main provisions of thermal analysis are considered. As a result of the experiment, the dependences reflecting the nature of the temperature change in the blood sample of cattle and the reference element have been obtained, allowing one to accurately record the features of the phase phenomena observed in this biotechnological environment. It is shown that the crystallization of moisture in the blood, which is a complex multicomponent solution, is due to the presence of eutectic compounds that arise when the temperature is lowered and water is released in the form of pure ice. Guided by the obtained temperature dependences, we obtained informative data on the necessary energy potential for the realization of cooling and freezing processes. The features of the mechanism of ice freezing in the blood under investigation from the point of view of the molecular-kinetic theory of molecular motion are explained. The carried out researches have allowed to define activation energy of processes of crystallization of a moisture in investigated blood of large horned livestock