Explicit integration of some integrable systems of classical mechanics

The main objective of the thesis is the analytical and geometrical study of several integrable finite-dimentional dynamical systems of classical mechanics, which are closely related, namely: - the classical generalization of the Euler top: the Zhukovski-Volterra (ZV) system describing the free motion of a gyrostat, i.e., a rigid body carrying a symmetric rotator whose axis is fixed in the body; - the Steklov-Lyapunov integrable case of the Kirchhoff equations describing the motion of a rigid body in an ideal incompressible liquid; - a nontrivial integrable generalization of the Steklov-Lyapunov system found by V.Rubanovskii: it describes the motion of a gyrostat in an ideal fluid in presence of a non-zero circulation. In our study we obtained explicit solution of the Zhukovski-Volterra ([2] and the Steklov-Lyapunov systems in terms of sigma- or theta-functions, and performed a bifurcation analysis of these systems, as well as of the Rubanovskii generalization. One should note that the solution of the ZV system was first given by V. Volterra, who, however, presented only its structure, but not the explicit formulas. The thesis gives a new alternative solution of this system by using an algebraic parametrization of the angular momentum. This allowed us to find poles and zeros of angular momentum in an algebraic way. The parametrization was also used to find an explicit solution for the Euler precession angle, and, as a consequence, to solve the Poisson equations describing the motion of the gyrostat in space. Similarly, by giving a geometric interpretation of the separating variables, and using the Weierstrass root functions, we reconstructed the thetafunctional solution of the Steklov-Lyapunov systems, which was first given by F. Kötter in 1899 without a derivation ([3]). In the study of bifurcations and singularities of the ZV system we used its bi-Hamiltonian structure ([1]. According the new method, the solution is critical, if there exist a parameter of corresponding family of Poisson brackets, for wich the rang of the brackets with this parameter drops. Applying new technics, based on the property of the system of being bi-Hamiltonian, we construct the bifurcation diagram of the ZV system. We also find the equilibrium points of the system, check the non-degeneracy condition for such points in the sense of the singularity theory of Hamiltonian systems, determine the types of equilibria points, and verify whether they are stable or not. We also describe the topological type of common levels of the first integrals of the ZV system. Similar problems have been discussed in many papers, but the goal of our work is to study the system and demonstrate the above techniques. It is a remarkable fact that using the bi-Hamiltonian property makes it possible to answer all the above questions practically without any difficult computations. The same method is applied to construct the bifurcation diagram for the Steklov-Lyapunov system, describe the zones of real motion, and analyze stability of critical periodic solutions. Then the bifurcation analysis is extended to the Rubanovskii generalizaton. Here the main difficulty is that the number of different types of the bifurcation diagram is quite high, so we only describe general properties of the bifurcation curves, do stability analysis for closed trajectories, and equilibria.El objetivo principal de la tesis es el estudio analítico y geométrico de varios sistemas integrables dinámicos y finito-dimensionales de la mecánica clásica que están estrechamente vinculados, a saber: -La generalización clásica de Euler top: el sistema Zhukovski-Volterra (ZV) que describe el movimiento libre de un giróstato, es decir, un cuerpo rígido que lleva un rotor simétrico cuyo eje es fijo al cuerpo. - El caso del sistema integrable de Steklov-Lyapunov de las ecuaciones de Kirchhoff que describen el movimiento de un cuerpo rígido en un líquido incompresible ideal; - Una generalización no trivial del sistema integrable de Steklov-Lyapunov encontrado por V. Rubanovskii que describe el movimiento de un giróstato en un fluido ideal en presencia de una circulación distinta de cero. En nuestro estudio hemos obtenido una solución explícita de los sistemas de Zhukovski-Volterra [2] y de Steklov-Lyapunov en términos de funciones sigma- o theta y hemos realizado un análisis de la bifurcación de estos sistemas, así como de la generalización de Rubanovskii. Hay que señalar que la solución del sistema de ZV fue dado por primera vez por V. Volterra, que, sin embargo, presenta sólo su estructura, pero no las fórmulas explícitas. La tesis ofrece una nueva solución alternativa de este sistema mediante el uso de una parametrización algebraica del momento angular. Esto nos ha permitido encontrar polos y ceros del momento angular en forma algebraica. La parametrización también se utilizó para encontrar una solución explícita para el ángulo de precesión de Euler, y, en consecuencia, para resolver las ecuaciones de Poisson que describen el movimiento de un giróstato en el espacio. Del mismo modo, al dar una interpretación geométrica de las variables de separación, y utilizando las funciones de las raíces Weierstrass, hemos reconstruido la solución thetafunctional de los sistemas de Steklov-Lyapunov, que fue dado por primera vez por F. Kotter en 1899 sin una derivación ([3]). En el estudio de las bifurcaciones y las singularidades del sistema ZV hemos utilizado su estructura bi-Hamiltoniana ([1]). Según el nuevo método, la solución es crítica, si existe un parámetro de la familia correspondiente del paréntesis de Poisson, para que el rango de las paréntesis con este parámetro se disminuye. Aplicando las nuevas técnicas, basadas en la propiedad del sistema de ser bi-Hamiltoniana, construimos el diagrama de bifurcación del sistema ZV. También hemos encontrado los puntos de equilibrio del sistema, verificando la condición de no-degeneración de estos puntos, en el sentido de la teoría de singularidad de los sistemas hamiltonianos, determinando los tipos de puntos de equilibrio, y comprobando si son estables o no. También hemos descrito el tipo topológico de los niveles comunes de los primeros integrales del sistema de ZV. Problemas similares se han discutido en muchas obras, pero el objetivo de nuestro trabajo es estudiar el sistema y demostrar las técnicas anteriormente mencionadas. Es un hecho notable que el uso de la propiedad bi-Hamilton permite responder a todas las preguntas anteriores, prácticamente sin ningún cálculo difícil. El mismo método se aplica para construir el diagrama de bifurcación para el sistema de Steklov-Lyapunov, describir las zonas de movimiento real, y analizar la estabilidad de soluciones periódicas críticas. A continuación, el análisis de bifurcación se extiende a la generalización Rubanovskii. Aquí la principal dificultad consiste en que el número de diferentes tipos del diagrama de bifurcación es bastante alto, por lo que sólo se describen las propiedades generales de las curvas de bifurcación, y el análisis de estabilidad se hace para trayectorias cerradas, y equilibrios

LYME DISEASE. CLINICAL AND EPIDEMIOLOGICAL STUDY FOR THE PERIOD 1989-1991 IN VARNA REGION

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ACTIVATION OF PHAGOCYTOSIS IN VITRO WITH FRACTIONS OF FASEOLOSAXIN

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Evaluation of some parameters of copper metabolism and leipzig scoring system in the diagnosis of wilson disease

PURPOSE: Wilson disease (WD) is an autosomal recessively inherited disorder of copper accumulation and toxicity. Its recognition is easy in the presence of typical clinical presentations. Unexplained liver test abnormalities are a diagnostic challenge and require more examinations. The objective of this study is to assess the diagnostic value of ceruloplasmin, 24-hour urine copper excretion and Leipzig scoring system in WD.MATERIAL AND METHODS: Sixty-five patients with WD (22 females and 43 males) and control group of 17 patients with other chronic liver diseases (CLD) were analyzed. The values of the parameters of copper metabolism and Leipzig scoring system were evaluated.RESULTS: Average ceruloplasmin level was under 0,2 g/L and 24-hour urinary copper concentration was increased. D-penicillamine challenge test showed a mean value of 17,4 µmol/24 hours of urinary copper excretion in WD patients versus 5,46 µmol/24 hours in CLD ones. According to the Leipzig scoring diagnostic criteria, 58 WD patients (89,23% of the cases) presented with a score ≥4 (maximal value of 12). Score 3 was found out in seven patients, however, the exclusion of other etiology and the clinical course of the disease confirmed the diagnosis. The control subjects presented with a score ≤3 as it was ≤2 in 76,5% of the cases.CONCLUSION: Our results confirm the diagnostic value for WD of the Leipzig scoring system combined with clinical symptoms, laboratory parameters of copper metabolism, genetic testing and liver biopsy in clinical practice

CHANGES IN THE ACTIVITY OF SOME SERUM ENZYMES IN PATIENTS WITH EXTRAHEPATIC CHOLESTASIS

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A Personalized Diet Plan for Inflammatory Bowel Disease. Current Evidence and Future Insights

The prevalence of inflammatory bowel disease (IBD) has been rising year by year. Crohn's disease (CD) and ulcerative colitis (UC) are increasingly diagnosed in both pediatric and adult populations. Since the early 21st century, IBD has become a global health problem. Despite the advances in medical treatment, clinical remission and mucosal healing are still not achieved in many patients. The cause is likely rooted in the fact that the immune system and microbiome of genetically susceptible, with the environment also playing a role, fail to provide an appropriate response, which subsequently leads to chronic inflammation of the gut. Recently increasing evidence has emerged that one of the major environmental factors that play a role in the onset and course of the disease is the diet. This leads to the conclusion that it can be used as part of the strategy to reduce clinical symptoms and intestinal inflammation.The aim of this article is to analyze the available information in the field of personalized dietary patterns with an accent on their level, focus, and scope. The ever-changing evidence for modifiable dietary approaches will give us the opportunity to walk the path from generic advice on a healthy diet to truly personalized dietary plans tailored to meet the needs of each individual patient with IBD

Non-alcoholic fatty liver disease in lean individuals or another rare differential diagnosis - a clinical case

Въведение: Неалкохолната Мастна Чернодробна Болест (НАМЧБ) е значим обществен здравен проблем, засягаща  близо един милиард хора по цял свят. НАМЧБ е най-честата причина за чернодробно увреждане в световен мащаб и представлява чернодробна изява на метаболитния синдром. НАМЧБ е основно свързана със затлъстяването и свързаните му коморбидности, но също така би могло да се наблюдава и при индивиди с нормален или граничен BMI. Това е т.н. НАМЧБ „при слаби“ и представлява субфенотипна изява на Неалкохолната мастна чернодробна болест. Слабите индивиди с НАМЧБ показват по-леки признаци на Метаболитния синдром, сравнено с тези със затлъстяване, но са с по-висока честота на метаболитни промени, като дислипидемия, артериална хипертония, инсулинова резистентност и диабет сравнено със здрави контроли. Поставянето на диагнозата на НАМЧБ изисква изключване на вторични причини за чернодробна стеатоза, както и изключване на дневна алкохолна консумация ≥ 30 гр. абсолютен алкохол за мъже и ≥ 20 гр. абсолютен алкохол за жени.Клиничен случай: Представяме клиничен случай на мъж на 29 години, постъпил в клиниката с оглед диагностично уточняване във връзка с ехографски данни за изразена чернодробна стеатоза, съпроводена с лекостепенна хепатоспленомегалия. Липсват данни за известен Захарен диабет, ХБ или затлъстяване. Пациента е с наднормено тегло, но без затлъстяване: ръст 187 см, телесно тегло 108 кг, BMI – 30, коремна обиколка 111 см. От лабораторните изследвания нормална ПКК, CRP, нормални АСАТ, АЛАТ, АФ  и билирубин, леко повишен ГГТ, нормална синтетична чернодробна функция. Изследването на липидната обмяна показва повишени нива на LDL холестерол и ниско ниво на HDL холестерол. Проведен е лабораторен хепатологичен скрининг, при който са изключени – хронични вирусни хепатити, Болест на Уилсън и хемохроматоза, автоимунна чернодробна болест. Неинвазивните cкоровете за доказване наличие на стеатоза показаха високи стойности. NAFLD fibrosis score: -3,35 и FIB-4 index: 0,52, отхвърлят наличието на значима  фиброза. При ехографското изследване данни за изразена чернодробна стеатоза – 3 ст, горно-граничен размер на черния дроб и лекостепенна спленомегалия. Фиброскан еластографията показва нормална плътност на черния дроб – 4,8 кРа, при САР – 304 db/m, което доказва наличие на високостепенна стеатоза. Имайки предвид физикалните данни, категоричните резултати от неинвазивните скорове за стеатоза,  ехографското изследване, данните от фиброскан еластографията и след изключване на алкохолна консумация и на други по-чести причини за стеатоза се прие диагноза Неалкохолна мастна чернодробна болест при слаби.  Направи се и изследване на суха капка кръв  за лизозомални ензими, при което се изключи Болест на Гоше. Установи се ниско ниво на кисела сфингомиелиназа, което е индикация за Niemann Pick A/B болест, която представлява рядка наследствена автозомно-рецесивна болест засягаща липидния метаболизъм.Заключение: Клиничният случай представя пациент без затлъстяване, с категорични лабораторни, ехографски и фиброскан данни за Неалкохолна мастна чернодробна болест. В диференциално диагностичен план установихме данни за рядка наследствена болест – Болестта на Niemann Pick, протичаща с нарушение в обмяната на мазнините. Клиничният случай доказва широкия спектър от диференциални диагнози при слаби пациенти с НАМЧБ.abstractIntroduction:    Non-alcoholic    fatty    liver    disease    (NAFLD)  is  a  significant  health  problem  in  society  and it affects around a billion people on a global scale. NAFLD is the most common reason for liver damage worldwide and is considered the hepatic manifestation of the metabolic syndrome.. Nonalcoholic fatty liver disease (NAFLD) is generally associated with obesity and the related comorbidities but it can also develop in subjects with a normal body mass index (BMI). This sub-phenotype of NAFLD is so-called “lean” NAFLD. Lean subjects with NAFLD have milder features of the metabolic syndrome when compared with obese patients. Nonetheless they have a higher prevalence of metabolic alterations (e.g.,dyslipidemia, arterial hypertension, insulin resistance, and diabetes) compared with healthy controls. The diagnosis of NAFLD requires the exclusion of both secondary causes and of a daily alcohol consumption ≥ 30 g for men and ≥ 20 g for women.Clinical case: We present the clinical case of a 29-year-old man, who was admitted to the hospital because of diagnostic evaluation of ultrasound findings of severe liver steatosis accompanied with mild hepatosplenomegaly. No evidence of known Diabetes mellitus, Hypertensive disease or Obesity. The patient is overweight but without obesity: height – 187 sm., weight – 108 kg, BMI - 30, waist circumference – 111 sm. Laboratory examination revealed: normal blood count, normal level of CRP, AST, ALT, AF, total bilirubin, mild elevation of GGT and normal synthetic liver function tests. Lipid profile showed elevated level for LDL cholesterol and low HDL cholesterol. Underlying chronic viral hepatitis, autoimmune liver diseases, Wilson's disease and Hemohromatosis were ruled out. Calculated non-invasive scores for steatosis were high. NAFLD fibrosis score: -3,35 and FIB-4 index: 0,52, ruled out presence of significant fibrosis. Abdominal ultrasound examination showed liver steatosis grade 3, upper limit of normal size, mild splenomegaly. Liver transient elastography (TE) with fibroscan revealed normal liver stiffness - 4,8 kPa and CAP – 304 db/m, that proved presence of severe steatosis. Based on physical examination data, definite results of non-invasive scores for steatosis, ultrasound examination and fibroscan elastography, and after excluding of alcohol consumption and other common causes for steatosis we established the diagnosis „lean“ Non-alcoholic fatty liver disease. We performed an examination for Lysosomal Enzymes from dried blood, that rule out classical Gaucher disease. Low activity of acid sphingomyelinase was found, which is indicative of Niemann Pick disease A/B – rare inherited autosomal recessive condition involving lipid metabolism.Conclusion: Clinical case represents a patient without obesity, with definite laboratory, ultrasound and fibroscan data for Non-alcoholic fatty liver disease. As a differential diagnosis we found a rare inherited condition - Niemann Pick disease, characterized with impaired lipid metabolism. Our clinical case proves the broad spectrum of differential diagnosis of patients with NAFLD. Introduction:  Non-alcoholic  fatty  liver  disease  (NAFLD) is a socially significant health problem and it affects around a billion people on a global scale. NAFLD is the most common reason for liver damage worldwide and is considered the hepatic manifestation of the metabolic syndrome. Non-alcoholic fatty liver disease is generally associated with obesity and the related comorbidities, but it can also develop in subjects with a normal body mass index (BMI). This sub-phenotype of NAFLD is called lean NAFLD. Lean subjects with NAFLD have milder features of the metabolic syndrome when compared with obese patients. Nonetheless, they have a higher prevalence of metabolic alterations (e.g., dyslipidemia, arterial hypertension, insulin resistance, and diabetes) compared with healthy controls. The diagnosis of NAFLD requires the exclusion of both secondary causes and of a daily alcohol consumption ≥ 30 g for men and ≥ 20 g for women.Clinical Case: We present the clinical case of a 29-year-old man, who was admitted to the hospital because of diagnostic evaluation of ultrasound findings of severe liver steatosis accompanied with mild hepatosplenomegaly. There was no evidence of diabetes mellitus, hypertensive disease or obesity. The patient was overweight but without obesity: height – 187 cm., weight – 108 kg, BMI – 30, waist circumference – 111 cm. Laboratory examination revealed: normal blood count, normal level of CRP, AST, ALT, AF, total bilirubin, mild elevation of GGT and normal synthetic liver function tests. The lipid profile showed elevated level for LDL cholesterol and low HDL cholesterol. Underlying chronic viral hepatitis, autoimmune liver diseases, Wilson's disease and hemochromatosis were ruled out. The calculated non-invasive scores for steatosis were high. The NAFLD fibrosis score: -3.35 and FIB-4 index: 0.52, ruled out presence of significant fibrosis. Abdominal ultrasound examination showed liver steatosis grade 3, upper limit of normal size, mild splenomegaly. Liver transient elastography (TE) with fibroscan revealed normal liver stiffness – 4.8 kPa and CAP – 304 db/m that proved the presence of severe steatosis. Based on physical examination data, definite results of non-invasive scores for steatosis, ultrasound examination and fibroscan elastography, and after excluding alcohol consumption and other common causes for steatosis, we established the diagnosis lean non-alcoholic fatty liver disease. We performed an examination for lysosomal enzymes from dried blood that ruled out classic Gaucher disease. Low activity of acid sphingomyelinase was found, which was indicative of Niemann-Pick disease A/B – rare inherited autosomal recessive condition involving lipid metabolism.Conclusion: The clinical case presents a patient without obesity, with definite laboratory, ultrasound and fibroscan data for non-alcoholic fatty liver disease. As a differential diagnosis we found a rare inherited condition – Niemann-Pick disease, characterized by impaired lipid metabolism. Our clinical case proves the broad spectrum of differential diagnoses of patients with NAFLD