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.

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

Peculiarities of spectral parameters of EEG, HRV and routine parameters of immunity in patients with various levels of the entropy of EEG, HRV, immunocytogram and leukocytogram

Background. We have previously shown that the entropy levels of spectral parameters of EEG and HRV as well as of Immunocytogram and Leukocytogram are well dispersed. The method of cluster analysis was the distribution of the observed contingent into four groups that are homogeneous in terms of entropy. Have been identified 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. The purpose of this study is to identify the recognition parameters of immunity together with spectral parameters of EEG and HRV. 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 parameters of Immune status on a set of I and II levels recommended by the WHO. Than we calculated for each locus of EEG and HRV as well as for Immunocytogram and Leukocytogram the Entropy (h) using Shannon’s formula. Results. The cluster analysis method created four groups, homogeneous in entropy parameters. Discriminant analysis revealed 25 SPD of EEG parameters, 5 HRV parameters, and 9 immunity parameters, by the totality of which four entropy clusters clearly different from each other. Conclusion. Quantitatively and qualitatively different levels of entropy of the spectral parameters of EEG and HRV as well as the Immunocytogram and Leukocytogram are accompanied by characteristic constellations of the parameters of EEG, HRV and Immunity

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

Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities

We perform the complete group classification in the class of nonlinear Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+|\psi|^\gamma\psi+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$ $\gamma$ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equations have non-trivial Lie symmetries using a combination of algebraic and compatibility methods. The proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics.Comment: 10 page

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

Modulating Neuro-Endocrine Effects Of Bioactive Water Naftussya From Layers Truskavets’ And Pomyarky At Humans With Dysfunction Of Neuro-Endocrine-Immune Complex

Background. Previously we have been carry out comparative investigation long-term effects of Bioactive Water Naftussya from layers Truskavets’, Pomyarky and Skhidnyts’a on some metabolic and biophysical parameters at men with dysfunction of neuro-endocrine-immune complex. The aim of this study is the investigation long-term effects on neuro-endocrine regulation at similar patients. Materials and methods. The object of observation were 20 volunteers: ten women and ten men aged 33-76 years without clinical diagnose but with dysfunction of neuro-endocrine-immune complex and metabolism. We evaluated the trait and reactive anxiety, recorded EEG, HRV and blood pressure, determined levels of Cortisol, Aldosterone, Testosterone, Triiodothyronine and Calcitonin. After examination volunteers within 7 days used bioactive water Naftussya (250 mL three times a day) from Truskavets’ or Pomyarky layer, then repeated the tests listed. Results. Weekly drinking of Bioactive Water Naftussya normalizes initially elevated levels of testosterone in women and reduces initially normal its level in men, in both sexes normalizes initially elevated levels of triiodothyronine and activity of the autonomic regulatory systems as the parasympathetic, so sympathetic, combined with a slight increase in the normal levels of aldosterone and trend to increase (under low) reactive anxiety as well as post occlusive reactivity of vascular tone. Among the 164 registered parameters quantitative EEG revealed noticeable changes only 8. This is, above all, normalizing increase initially reduced spectral power density (SPD) α-rhythm in loci T6 by 93%, P4 by 33%, F7 by 29%, increase by 10% in the normal range modal frequency of β-rhythm and further enhance SPD θ-rhythm in locus T5 by 42%. On the other hand, showed a reduction by 28% initially normal PSD of β-rhythm in locus in O2 and further reduction its by 20% in locus P4. Moreover, there was left-sided lateralization initially symmetric α-rhythm. No significant differences between the effects of Naftussya both deposits were found. Conclusion. Bioactive water Naftussya both Truskavets’ and Pomyarky layers causes favorable normalizing effects on abnormalities neuro-endocrine parameters, which is a manifestation of its adaptogenic properties

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

Relationships between changes in parameters of the manual function and electroencephalogram, heart rate variability as well as gas discharge visualization in children with spastic cerebral palsy caused by the Kozyavkin method

Introduction. It was reported earlier that in children with spastic forms of cerebral palsy (SFCP) after two-week course of rehabilitation by the Kozyavkin method the neural component of muscle tone (NCMT) was reduced in 79,3% cases, while in 13,8% cases changes were not detected and in 2 children even increased. It was assumed that such a variety of changes in NCMT is due to differently directed changes in the background activity of the nerve centers. Objectives. To analyse the relationships between changes (Ch) in NCMT as well as manual functional tests on the one hand, and parameters of EEG, HRV and Gas Discharge Visualization (GDV) on the other one. Material and Methods. The object of observations were 14 children (6 girls and 8 boys) aged 8÷15 years with SFCP. The state of motor development at GMFCS was on II÷IV level. The functional status of the hand with MACS was on II÷III level. The estimation of hand function was carried out by Dynamometry (D), Box and Block Test (B&B) and Nine Hole Peg Test (NHP). NCMT was also registered by the device ‘NeuroFlexor’ (Aggero MedTech AB, Sweden), HRV and EEG were tested simultaneosly by the hardware-software complex ‘Cardiolab+VSR’ and ‘NeuroCom Standard’ respectively (KhAI Medica, Kharkiv, Ukraine) as well as GDV by ‘GDV Chamber’ (‘Biotechprogress’, St-Pb, RF). Results. After two-week course of rehabilitation NCMT was reduced in 9 children from 19,8 ± 3,4 to 12,3 ± 2,8 Newtons (Ch: -7,5 ± 2,0 N), in 3 children NCMT was 8,2 ± 3,3 before and 7,9 ± 3,5 after rehabilitation (Ch: -0,3 ± 0,3 N) while in one girl NCMT increased from 15,1 to 17,9 N and in one boy from 6,1 to 19,4 N. Manual functional tests also changed ambiguously. The Ch in NCMT are correlated with Ch in parameters HRV&EEG (R2=0,786) as well as GDV (R2=0,556). The Ch in functional tests for the left hand are correlated with Ch in parameters HRV&EEG to the stronger extent (R2=0,931) and for the right hand the correlation is maximal (R2=0,997). As to GDV parameters, connections are weaker but they are also stronger for the right hand. Conclusion. In children with spastic forms of cerebral palsy caused by the Kozyavkin method, changes in manual functional tests and the neural component of the muscle tone are determined by changes in parameters of EEG and HRV as well as GDV which is a completely suitable non-invasive method for assessing the effectiveness of rehabilitation