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

ДОПЛЕРОМЕТРИЧНЕ ДОСЛІДЖЕННЯ МАТКОВО-ПЛАЦЕНТАРНО-ПЛОДОВОГО КРОВОПЛИНУ У ВАГІТНИХ З ПЕРВИННОЮ АРТЕРІАЛЬНОЮ ГІПОТОНІЄЮ

The aim of the study is to research the uterine-placental-fetal hemodynamic in pregnant women with arterial hypotension. Materials and methods. Doppler study of blood flow in the uterine-placenta-fetal system is presented in 43 pregnant women with arterial hypotension and 32 pregnant women with normal arterial tension during pregnancy weeks – 20–24, 28–32, 37–41. For the assessment of haemodynamics in the mother-placenta-fetus we used a method doplerometria to study the blood flow using the device “SA-8000 EX” (Medison, South Korea) in obstetric programs with convex transduser (3.5 MHz). Doplerometric study included the observation and the assessment of blood flow in uterine arteries, artery of the umbilical cord and middle cerebral artery of the fetus. Results and discussion. The presence of irregularities in the uterine-placental-fetal system in the both examinated groups was feund In the first group indexes of vessel's resistance in the right and level uterine arterias were much more higher than in the second group. The most of changer were presented in the right uterine arteria. Hemodynamic irregularities in pregnant women with arterial hypotension were presented by increasing of the intervillage space, transversal size of regional sine, varicose expansion of parametral veins. The phenomen of venous stagnation were accompanied by the high indexes of vascular resistance in uterine arterias and more often at a dynamic supervision brought a delay of height of fetus. Study of hemodynamic indexer in the artery of umbilical cord revealed the tendency of increase of all indexes during the pregnancy in the first group and their reduction – in the second ( control) group. And it was presented increasing of all indexes in the third trimester of pregnancy in pregnant women with arterial hypotension comparing with the control group. The study of fetal hemodynamics educed the decline of the blood flow resistance in the middle cerebral artery of fetus with progress of pregnancy and centralization of blood circulation. Strengthening of cerebral circulation of the fetus is a compensatory reaction to chronic tissue hypoxia. Conclusions. Irregularities of the utero-placental blood flow that is developing in pregnant women with primary arterial hypotension, starting from the second trimester of pregnancy lead to fetal distress.Цель исследования – изучение гемодинамических процессов в системе мать–плацента–плод у беременных с первичной артериальной гипотонией. Материалы и методы. Проведено допплерометрическое исследование маточно-плацентарного и плодово-плацентарного кровотока у 43 беременных с первичной артериальной гипотонией и 32 здоровых беременных с нормальным артериальным давлением в сроки гестации – 20–24 нед., 28–32 нед., 37–41 нед. Для оценки состояния гемодинамики в системе мать–плацента–плод проводилось допплерометрическое исследование кровотока с помощью прибора «SA-8000 EX» (Medison, Южная Корея) в акушерских программах конвексным датчиком (3,5 МГц). Допплерометрическое исследование включало оценку кровотока в маточных артериях, артериях пуповины и средней мозговой артерии плода.Результаты исследования и их обсуждение. Выявлено закономерности становления гемодинамических процессов в системе мать–плацента–плод в обеих группах наблюдения. В группе беременных с первичной артериальной гипотонией индексы сосудистого сопротивления в правой и левой маточных артериях были достоверно выше, чем в контрольной группе, с преобладанием изменений в правой маточной артерии. Гемодинамические проявления при первичной артериальной гипотонии со стороны маточно-плацентарного контура выглядели при ультразвуковом исследовании как расширение интервиллезного пространства, расширение поперечного размера краевого синуса, варикозное расширение параметральных вен. Явления венозного застоя в маточно-плацентарном контуре сопровождались более высокими показателями индексов сосудистого сопротивления в маточных артериях и чаще при динамическом наблюдении приводили к развитию задержки роста плода. Изучение гемодинамических показателей в артерии пуповины выявило тенденцию к росту всех индексов в течение беременности в I группе и уменьшение – во II группе; при этом у беременных с первичной артериальной гипотонией в III триместре наблюдалось достоверное увеличение показателей по сравнению с контрольной группой. Изучение плодовой гемодинамики выявило снижение резистентности кровотока в средней мозговой артерии плода с прогрессированием беременности и развитием централизации кровообращения. Усиление мозгового кровообращения плода является компенсаторной реакцией на хроническую тканевую гипоксию. Выводы. Нарушение маточно-плацентарного кровотока, которые развиваются у беременных с первичной артериальной гипотонией, начиная со II триместра беременности, в дальнейшем приводят к фетальному дистрессу.Мета дослідження – вивчення гемодинамічних процесів у системі мати – плацента – плід у вагітних з первинною артеріальною гіпотонією.Матеріали та методи. Проведено доплерометричне дослідження матково-плацентарного та плодово-плацентарного кровоплину у 43 вагітних із первинною артеріальною гіпотонією та 32 здорових вагітних з нормальним артеріальним тиском у терміни гестації – 20–24 тиж., 28–32 тиж., 37–41 тиж. Для оцінки стану гемодинаміки в системі мати – плацента – плід проводилось доплерометричне дослідження кровоплину з допомогою приладу «SA-8000 EX» (Medison, Південна Корея) в акушерських програмах конвексним датчиком (3,5 МГц). Доплерометричне дослідження включало оцінку кровотоку в маткових артеріях, артеріях пуповини та середній мозковій артерії плода.Результати дослідження та їх обговорення. Виявлено закономірності становлення гемодинамічних процесів у системі мати – плацента – плід у двох групах спостереження. У групі вагітних з первинною артеріальною гіпотонією індекси судинного опору в правій і лівій маткових артеріях були достовірно вищими, ніж у контрольній групі, з переважанням змін у правій матковій артерії. Гемодинамічні прояви при первинній артеріальній гіпотонії з боку матково-плацентарного контуру виглядали при ультразвуковому дослідженні як розширення інтервільозного простору, збільшення поперечного розміру крайового синуса, варикозне розширення параметральних вен. Явища венозного застою у матково-плацентарному контурі супроводжувались вищими показниками індексів судинного опору у маткових артеріях і частіше при динамічному спостереженні призводили до розвитку затримки росту плода. Вивчення гемодинамічних показників в артерії пуповини виявило тенденцію до зростання всіх індексів упродовж вагітності у І групі та зменшення – у ІІ групі; при цьому у вагітних з первинною артеріальною гіпотонією в ІІІ триместрі спостерігалось достовірне збільшення показників порівняно з контрольною групою. Вивчення плодової гемодинаміки виявило зниження резистентності кровоплину в середній мозковій артерії плода із прогресуванням вагітності та розвитком централізації кровообігу. Посилення мозкового кровообігу плода є компенсаторною реакцією на хронічну тканинну гіпоксію.Висновки. Порушення матково-плацентарного кровоплину, які розвиваються у вагітних з первинною артеріальною гіпотонією, починаючи з ІІ триместру вагітності, в подальшому призводять до фетального дистресу.

Interpersonal differences between of the entropies of EEG, HRV, immunocytogram and leukocytogram

Background. Previously, we have shown that in humans entropy of the normalized parameters of the HRV and SPD of loci of EEG significantly correlate with the entropy of Leukocytogram and Immunocytogram as well as parameters of immunity, which testifies to their modulating regulatory effects. This article is a direct extension of the previous one. Its purpose is the distribution of the observed contingent into groups that are homogeneous in terms of entropy of the HRV, EEG as well as Leukocytogram and Immunocytogram. Material and methods. In basal conditions in 37 men and 14 women with chronic pyelonephritis and cholecystitis in remission as well as without clinical diagnose but with dysfunction of neuro-endocrine-immune complex and metabolism, we recorded twice, before and after balneotherapy at the spa Truskavets’, EEG (“NeuroCom Standard”) and HRV (“Cardiolab+VSR”). In blood we determined relative content of components (RCC) of Immunocytogram (ICG) (T helper, T cytolytic, B and NK lymphocytes) and Leukocytogram (LCG) (Eosinophils, Stub and Segmentonucleary Neutrophils, Lymphocytes and Monocytes). Than we calculated for each locus of EEG and HRV as well as for ICG and LCG the Entropy (h) of normalized spectral power density (SPD) or RCC using Shannon’s formula. Results. The method of cluster analysis is revealed that in members of the major cluster (60%), the entropy of EEG, HRV, ICG and LCG varies within the normal range (-0,5σ ÷ +0,5σ). The members of the next largest cluster (23%) are characterized by a moderately increased entropy of the SPD of EEG in conjunction with the normal entropy of the ICG and the moderately reduced entropy of HRV and LCG. The members of the third cluster (9%) noted a significantly lower entropy (negentropy) of the SPD in loci F3, F4, T3 and C4; in addition, there is a moderate decrease in the entropy of the LCG. Instead, members of the last cluster (8) noted the negentropy of SPD in paired loci Fp1 and Fp2, T5 and T6, T3 and T4, F7 and F8 as well as O1 and O2; in addition, there is a moderate decrease in entropy of the ICG. The entropy of other EEG locus as well as of HRV and LCG is within the normal range. Conclusion. The enropy of HRV and SPD of loci of EEG as well as ICG and LCG is characterized by a large variation, which corresponds to the known wide variance of parameters of the ICG and LCG, which are subordinate to the regulatory influences of the central and autonomic nervous system

Features of the immunotropic effects of partial components of the balneotherapeutic complex of SPA Truskavets’

Background. The arsenal of therapeutic factors of the spa Truskavets’ is not limited to bioactive water Naftussya, but also includes ozokerite applications and mineral baths. The purpose of this study is to quantify the partial immunotropic effects of these balneofactors and also the organic substances and microbiota of Naftussya water. Material and methods. The object of observation were 41 men and 10 women aged 24-70 years old, who came to the spa Truskavets’ for the treatment of chronic pyelonephritis combined with cholecystitis in remission. Immune status evaluated on a set of I and II levels recommended by the WHO. The survey was conducted twice, before and after 7-10-days balneotherapy. 23 patients drinked bioactive water Naftussya (BAWN) by 3,2 mL/kg for 1 hour before meals three times a day, while 7 others volunteers drinked Ozokerite extract by 2 mL dissolved in sweet water (2,5-2,6 mL/kg) one time daily. 8 patients in the third group received BAWN and baths with mineral water (Cl--SO42--Na+-Mg2+ containing salt concentration 25 g/L) and for the other 13 patients in the balneotherapeutic complex included additionally application of Ozokerite on the lumbar region. Results. The most pronounced as stimulating (blood level of total, active and cytolytic T-lymphocytes, Circulating Immune Complexes, IgA and Microbial Count for Staph. aureus) and suppressor (blood level of helper T-lymphocytes and Neutrophils, theirs Killing Index and Bactericydity vs both Staph. aureus and E. coli) action are the organic substances of Ozokerite that contact the surface of the skin, whereas their contact with the mucous of the digestive tract causes less pronounced immunotropic effect. Naftussya water has a stronger effect than the water solution of Ozokerite, apparently due to the additional effects of microbes and organic matter produced by them. In contrast, the bath factors affect the immune parameters of the opposite influences. Microbiota has the most pronounced enhancing effects on the Phagocytose Index of Neutrophils vs. Staph. aureus and blood level of Natural Killers as well as Entropy of Leukocytogram. Instead, organic substances of Ozokerite have the same tangible but opposite effects on these parameters

Singular reduction operators in two dimensions

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+1)-dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in-depth prior study of the corresponding singular vector fields.Comment: 22 pages, minor correction

Relationships between the entropies of EEG, HRV, immunocytogram and leukocytogram

Background. In mathematics the entropy is a measure of uncertainty of a random function; in the theory of information entropy is a measure of uncertainty in a situation, any experience (test) that can have different consequences. The entropy is also a measure of disorder, the degree of chaos present in the system. CE Shannon linked the mathematical dependence of the concept of information and entropy, which characterizes the degree of ordering of the system. This estimate of the amount of information coincides with the estimation of the quantitative measure of elimination of uncertainty of entropy, the degree of organization of the system. It is well known about functional interactions between central and autonomic nervous and immune systems. In the context of this concept, we have prioritized research on the interconnections between the entropies of these systems. Material and methods. In basal conditions in 37 men and 14 women with chronic pyelonephritis and cholecystitis in remission as well as without clinical diagnose but with dysfunction of neuro-endocrine-immune complex and metabolism, we recorded twice, before and after balneotherapy at the spa Truskavets’, EEG (“NeuroCom Standard”) and HRV (“Cardiolab+VSR”). In blood we determined relative content of components (RCC) of Immunocytogram (ICG) (T helper, T cytolytic, B and NK lymphocytes) and Leukocytogram (LCG) (Eosinophils, Stub and Segmentonucleary Neutrophils, Lymphocytes and Monocytes). Than we calculated for each locus of EEG and HRV as well as for ICG and LCG the Entropy (h) of normalized spectral power density (SPD) or RCC using Shannon’s formula: Results. There was a complete absence of correlation between hHRV and hLCG (r=-0,03) as well as hICG (r=-0,03), whereas the relationship between hLCG and hICG is significant (r=-0,40; p<0,001). Accepting the entropy of EEG HRV as a factor, using the correlation analysis with step-by-step exclusion, we obtain the equations for dependent variables. Canonical correlation between hHRV&EEG, on the one hand, and hLCG&Immunity, on the other hand, is strong: R=0,814; R2=0,663; χ2(240)=296; p=0,008. Conclusion. The enropy of HRV and EEG significantly correlate with the entropy and parameters of immunity, which testifies to their modulating regulatory effects

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

Features of the neurotropic effects of partial components of the balneotherapeutic complex of SPA Truskavets’

Background. In the previous article we outlined the results of comparative evaluation of immunotropic effects of partial components of the balneotherapeutic complex of spa Truskavets’. In the previous article we outlined the results of comparative evaluation of neurotropic effects of partial components of the complex. Material and methods. The object of observation were the same 41 men and 10 women aged 24-70 years old, who came to the spa Truskavets’ for the treatment of chronic pyelonephritis combined with cholecystitis in remission. We recorded simultaneously parameters of HRV and EEG before and after course of balneotherapy. 23 patients drank bioactive water Naftussya (BAWN); 7 others volunteers drank Ozokerite extract; 8 patients in the third group received BAWN and baths with mineral water and for the other 13 patients in the balneotherapeutic complex included additionally application of Ozokerite on the lumbar region Results. It has been shown that the use of the water solution of Ozokerite dramatically increases the vagus tone and reduces the reciprocally the sympathetic tone, which is accompanied by an increase in the Amplitude of both β- and δ-rhythms as well as in the Frequency of the latter, on the one hand, and a decrease in the Index and the Deviation of θ-rhythm as well as the Amplitude of α-rhythm, on the other hand. At the same time, there is a left-sided shift in the Laterality of β- and α-rhythms. Instead, the Microbiota of Naftussya water, together with the transformed by microbes of organic substances that are related to Ozokerite, have the same pronounced but opposite effect on the listed parameters of HRV and EEG. As a result, BAWN, which contains both neurogenic antipodes in its composition, has a very moderate neurotropic effect on the listed parameters of HRV and EEG. Mineral baths activate other vagus tone markers and suppress the sympathetic marker, coupled with a decrease in the Entropy of HRV bands and β-rhythm Frequency, as well as a significant left-sided shift in Laterality both θ- and δ-hythms. The consistent use of BAWN somewhat weakens the listed effects of Baths due to the slight opposite effect of its organic substances, but not Microbes. Instead, the application of organic substances to the skin causes a much more pronounced opposing neurotropic action. Organic substances of Ozokerite, applied to the skin, causes a increase in Bayevskiy's ARS Index as well as vagal tone and a decline in Bayevskiy's Stress Index coupled with a decrease in Indexes both β- and α-rhythms. A similar, but much weaker effect makes contact with the skin of mineral water, while, as native and transformed by microbes organic matter on the side of the mucous of the digestive tract has the opposite effects on the listed parameters of HRV and EEG. Conclusion. The hypothesis that as native and transformed by microbes organic matter related to Ozokerite on the side of the gut activate chemoreceptors of vagus terminals and/or TL-Receptors of Macrophages of GALT. Activated Macrophages release cytokines, which too activate vagus afferents. When applying Ozokerite or taking Baths, organic substances and mineral salts activate the skin nerve terminals and/or TL-Receptors of Langerhans cells (as variety of Macrophages of SALT) which also release cytokines

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

Interindividual differences in parameters of the EEG and HRV in the humans with various levels of the entropy of EEG, HRV, immunocytogram and leukocytogram

Background. Previously, we have shown that the entropy of the normalized parameters of the HRV and spectral power density (SPD) of loci of EEG significantly correlate with the entropy and parameters of immunity, which testifies to their modulating regulatory effects. Individual analysis revealed that the entropy of HRV and EEG as well as Immunocytogram and Leukocytogram is characterized by considerable variability. The method of cluster analysis was the distribution of the observed contingent into four groups that are homogeneous in terms of entropy. The purpose of this study is to identify the spectral parameters and indices of HRV, the amplitude-frequency and spectral parameters of the rhythms of EEG as well as the indices of asymmetry and lateralization of rhythms, which together are four clusters of entropy significantly different from each other. Material and methods. In basal conditions in 37 men and 14 women with chronic pyelonephritis and cholecystitis in remission as well as without clinical diagnose but with dysfunction of neuro-endocrine-immune complex and metabolism, we recorded twice, before and after balneotherapy at the spa Truskavets’, EEG (“NeuroCom Standard”) and HRV (“Cardiolab+VSR”). Than we calculated for each locus of EEG and HRV the Entropy of normalized SPD using Shannon’s formula. Results. As a result of screening relationships between the normalized EEG entropy levels on the one hand, and the EEG and HRV parameters and indices on the other, three pairs of quasi-mirror patterns were detected. According to the results of the discriminant analysis 37 parameters were identified as characteristic of the entropy clusters, 11 of which relate to delta rhythm, 8 to theta rhythm, 8 to beta rhythm and 4 to alpha rhythm of EEG, 6 more represent HRV. Conclusion. The entropy clusters of SPD of EEG that we have discovered previously are quantitatively and qualitatively distinct from each other by at least 37 amplitude-frequency and spectral parameters of EEG as well as of HRV