Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every non-trivial measurable set $S$ of real numbers, there is a point at which both the lower densities of $S$ and of the complement of $S$ are at least $\delta$. The problem of determining the supremum of possible values of this $\delta$ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.Comment: 45 page

Kernels and best approximations related to the system of ultraspherical polynomials

22 pages, no figures.-- MSC2000 codes: 41A10; 41A17; 41A25; 42C10; 33C45.MR#: MR2129477 (2005k:41087)Zbl#: Zbl 1074.41003We study the uniformly bounded orthonormal system \Cal U_\lambda of functions u_n (\lambda )}(x)=\varphi _n (\lambda )}(\cos x) (\sin x) lambda , \quad x \in [0,\pi], where \{\varphi _n (\lambda )} \}_{n=0} infty ($\lambda \geq 0$) is the normalized system of ultraspherical polynomials. We investigate some approximation properties of the system \Cal U_\lambda and we show that these properties are similar to one's of the trigonometric system. First, we obtain estimates of $L$-norms of the kernels of the system \Cal U_\lambda . These estimates enable us to prove Nikol'skij-type inequalities for \Cal U_\lambda -polynomials. Next, we prove directly that \Cal U_\lambda is a basis in each $L _w$, $1<p<\infty$, where $w$ is an arbitrary $A_p$-weight function. Finally, we apply these results to get sharp inequalities for the best \Cal U_\lambda -approximations in $L$ in terms of the best \Cal U_\lambda -approximations in $L$ ($1\leq p<q<\infty$). For the trigonometric system such inequalities have been already known.This research was partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grants BFM 2000–0206–C04–01 and C04–03.Publicad

Адаптивна парадигма моделювання економіки

Викладено ідеологію і принципи адаптивного підходу до проблем математичного моделювання економіки. Термінологія і основні поняття адаптації витлумачено і реалізовано на прикладі створення адаптивної математичної моделі (ММ) економічного стану, яка на поточний момент є універсальною за своїми можливостями і призначенням. Вона охвачує різноманітні ефекти і процеси стану економіки.Ideology and principles of the adaptive going is expounded near the problems of mathematical design of economy. Terminology and basic concepts of adaptation it is interpreted and realized on the example of creation of adaptive mathematical model (MM) of economic position, which on a present moment is universal after the possibilities and setting. It engulfs various effects and processes of the state of economy

DISEASE-ASSOCIATED HLA-DR POLYMORPHISM, CLINICAL AND IMMUNOLOGICAL CHARACTERISTICS OF MULTIPLE SCLEROSIS PATIENTS IN THE NORTHEASTERN REGION OF UKRAINE

The development of multiple sclerosis is the result of complex interactions between environmental factors, genetic factors that determine individual disease susceptibility, and immunological and physiological characteristics of the patient. multiple sclerosis treatment requires obtaining information about the main pathological processes, therefore, the evaluation of biochemical, immunological, and genetic markers of such processes, and not just clinical indicators, can become the basis of improved approaches for diagnosis and monitoring of the disease. Polymorphism of the disease-associated genes is considered as one of the key factors in multiple sclerosis pathogenesis, capable of influencing the risk of development, clinical manifestations and nature of the course of the disease, treatment response. The HLA genes polymorphism was found to be the most important and playing an essential role in the development of autoimmune diseases. There are data on the population characteristics of these types of polymorphisms, which require conducting relevant research in Ukraine and its regions to identify relevant genetic markers. The aim of the study was to provide a comparative clinical and immunological characteristic of multiple sclerosis patients in the northeastern region of Ukraine, depending on the presence of the disease-associated HLA-DR polymorphism. Materials and methods. 39 patients with multiple sclerosis, inhabitants of Kharkiv and Kharkiv region, were examined, of which 7 men and 32 women of medium age of 33.7 ± 7.7 and 42.1 ± 11.9 years, respectively; control group was consisted of 27 practically healthy persons of both sexes with an average age of 30.1 ± 8.2 years. Clinical characteristics of multiple sclerosis patients included the determining the form and the actual type of the disease, its duration, and disability assessment based on the EDSS scale. The rate of the disease progression was determined as the ratio of the EDSS score to the duration of the disease. Biological material was blood and buccal epithelium samples from multiple sclerosis patients and practically healthy people. In addition to assessing the clinical status of the patients, design of the study also included evaluation of the systemic immunity indicators, levels of nonspecific and antinuclear antibodies in the serum, and analysis of HLA polymorphism, in particular, detection of the HLA-DR15 haplotype for its specific marker SNP rs9271366. Isolation of high molecular DNA was performed on a magnetically sensitive sorbent using the NeoPrep100 DNA Magnet kit (NeoGene, Ukraine). SNP polymorphism rs9271366 A/G was typed by the allele-specific amplification method with subsequent electrophoretic detection of the results. Primer selection was performed using the method by Liu Jing et al. (2012) adding a mismatch in the third position at the 3’ends. The primers MS92АF 5`-CACGTAATATAA-ATGGTTGCAAAGGA-3`, MS92GF 5`-CACGTAATATAAATGGTTGCA-AAGGG-3` and MS92R5` AACCCTGATGTAACAGA(C/T)CTCTA-3` (Eurofins Genomics), as well as Taq-mut polymerase (Liteh, Russian Federation), were used in the study. The amplification was performed on the multichannel amplifier Tercyc (Russian Federation). Amplification mode: denaturation at 96°C for 3 minutes and 35 cycles that included denaturation at 95°C for 30 seconds, annealing at 58°C for 30 seconds, and synthesis at 72°C for 30 seconds. The length of the amplicon was 233 bp. Electrophoresis was performed in 2% agarose gel in TAE buffer. Electrophoregram analysis was carried out on transilluminator UVT 1 (Biocom, Russian Federation). Serum IgM, IgG, IgA levels were evaluated by ELISA using a test system by NPL Granum (Ukraine) and the immunoassay analyzer Stat-Fax 303 (USA). The determination of peripheral blood lymphocytes subpopulations was carried out using the test systems by NPL Granum, Ukraine in accordance with the manufacturer's instructions. Circulating immune complexes in serum were evaluated by selective precipitation of antigen complexes with 3.5% PEG-6000 solution (AppliChem GmbH, Germany). The content of lymphocytotoxic autoantibodies was determined by a lymphocytotoxic macrotest. The complementary activity of the blood serum was evaluated by means of 50% of sheep erythrocytes hemolysis in the presence of homologous antibodies. The statistical treatment was performed using STATISTICA 11.0 (StatSoft, Inc). To determine the reliability of the differences between the indexes in the studied samples, the non-parametric Mann-Whitney's criterion was used, while the Student's T-criterion was used for the normal distribution. Results. At the stage of clinical examination of multiple sclerosis patients, 25 patients (64.1%) were diagnosed with RRMS, 4 patients (10.3%) with PPMS, and another 10 persons (25,6%) with SPMS. The familial form of the disease was established in 11 cases (28.2%), in 28 cases (71.8%) the form of the disease was classified as sporadic. The average disease duration of the examined persons was 10.16 ± 9.57 years. The disability score assessed according to the EDSS scale was 3.54 ± 1.67 points; the rate of progression of the disease was 0.93 ± 1.26 and 0.83 ± 0.85 points, respectively. Analysis of the HLA polymorphism indicated that the disease-associated (minor) allele G SNP rs9271366 was present in 43.6% of the subjects examined (haplotype AG, G+ group), and another 56.4% of the patients were homozygous by the major allele A (haplotype AA, G- group). The form of the disease, the EDSS score, and the rate of the disease progression in the examined patients did not have association with the allele G SNP rs9271366. Patients with familial and sporadic forms of multiple sclerosis did not have significant differences in immune status and clinical indicators. The average EDSS score was 3.45 ± 1.95 points in G+ patients and 3.65 ± 1.26 points in G- patients and the rate of disease progression was 0.83 ± 0.85 points and 0.93 ± 1.26 points, respectively. The study of cellular immunity indicators revealed a decrease in relative number of CD4+ and CD8+ lymphocytes in multiple sclerosis patients compared to that in control group, and it was more pronounced in subjects with the presence of the SNP rs9271366 minor allele. The relative content of CD4 + cells was 28.00 ± 3.45% in heterozygous patients, 30.94 ± 4.42% in patients homozygous for allele A, in the control group 42.3 ± 1.8% (p<0.05), and the CD8+ cells content was 18,55 ± 4.08%, 20.65 ± 3.52% and 29.4 ± 2.2%, respectively (p<0.05). At the same time, the reduction in the relative number of CD4+ lymphocytes was observed in 90.9% of G+ patients, and in 64.7% cases in G- patients (p<0.05). Decreased CD8+ lymphocyte content was also more common in patients with G+ (77.3% of cases) than in G- (41.2% of cases), p<0.05. Acute inflammatory process was determined in 38.5% of patients with multiple sclerosis, as evidenced by an elevated IgM level in 20.5% of cases, as well as an increase in complement activity in 62.5% of subjects. The levels of lymphocytotoxic autoantibodies in patients with multiple sclerosis were significantly higher compared to the control group (18.33 ± 3.67% of subjects and 5.82 ± 3.12% of subjects, respectively, p<0.05), and the levels of circulating immune complexes were elevated in 10.3% of patients. The study of the levels of antinuclear antibodies in multiple sclerosis patients revealed a diagnostically significant levels of antibodies to native DNA found in 92.3% of the cases, including all G + patients and 83.3% of G- patients. In this case, 16.7% of patients homozygous for allele A and 14.3% heterozygous patients were also positive for the presence of antibodies to denatured DNA. Similar results were also obtained concerning the presence of antibodies to formalinized DNA - 16.7% positive results in G- patients and 28.6% in G+ patients (p<0.05). At the same time, elevated levels of antibodies to native, denatured, and formalinized DNA were determined predominantly in patients with SPMS and PPMS (15.4% of the number of examined patients). The elevated levels of antinuclear antibodies may indicate an unfavorable course of the disease or its transition to the active phase. Conclusion. In the surveyed multiple sclerosis patients from the northeastern region of Ukraine, the prevalence of SNP rs9271366 and its association with a decrease in the relative number of CD4+ and CD8+ cells against the background of elevated levels of antinuclear and lymphocytotoxic antibodies were detected

Excitation of elastic oscillations in solids by a pulsed proton beam

At present, the beams of charged particles are used not only in fundamental researches but they have found a wide application in solving the plasma electronics problems, modification of material surfaces properties and in a number of other applications

Low carbon global economy: Scenarios of sustainable development, power consumption and greenhouse gas emission control

This paper is devoted to problems of a gradual transfer to renewable energy sources and decarbonization of power systems. It presents the analysis of the structure and the trend in global power consumption, greenhouse gas emissions, and target values of decreased emissions by the largest greenhouse gas emitters. A trend in renewable energy sources in total global power consumption is analyzed. The authors study a potential for carbon capture and storage as an instrument of CO2 emission neutralization in power and industrial processes. The paper contains indicators to evaluate efficiency of power resource utilization and greenhouse gas emission control. © 2019 IOP Publishing Ltd. All rights reserved

НЕЙРОННЫЕ СЕТИ КОНЕЧНОГО КОЛЬЦА НА ОСНОВЕ РЕДУКЦИОННОЙ СХЕМЫ ПОЗИЦИОННО-МОДУЛЯРНОГО КОДОВОГО ПРЕОБРАЗОВАНИЯ

The article studies the problem of creating a neural network of modular computing structures for highperformance expressions in the field of information security. The main attention is paid to the reduction technology of position-modular transformation of scalable integers, which serves as the basis for constructing the so-called neural networks of the finite ring (NNFR). To increase the speed of convergence of the reduction scheme used to reduce the number of elements of the generated sequence of residues, an effective tabular method is proposed. The developed approach makes it possible to reduce the number of iterations of the reduction process to a theoretical minimum. This is achieved through flexible adaptive mechanism check botheration deductions to a special range, allowing a tabular decomposition of its elements into pairs of residues in modules of the modular number system. On the basis of a modified reduction method there was synthesized a fast algorithm and a parallel structure of the NNFR with feedback, which ensures the implementation of the reduction scheme in a time order (S(⌈log2b⌉+1) +2)tsum, were S – the number of iterations, b – the bit width of the input number, – the duration of the addition operation of two deductions. Рассматривается проблема создания нейросетевых модулярных вычислительных структур для высокопроизводительных выражений в области защиты информации. Главное внимание уделяется редукционной технологии позиционно-модулярного преобразования масштабируемых целых чисел, которая служит основой для построения так называемых нейронных сетей конечного кольца (НСКК). Для повышения скорости сходимости используемой редукционной схемы понижения разрядности элементов формируемой последовательности вычетов предложен эффективный табличный метод. Разработанный подход позволяет свести к теоретическому минимуму количество итераций редукционного процесса. Это достигается за счет применения гибкого адаптивного механизма проверки принадлежности поитерационных вычетов к специальному диапазону, допускающему табличную декомпозицию его элементов на пары остатков по модулям модулярной системы счисления. На базе модифицированного редукционного метода синтезированы быстрый алгоритм и параллельная структура НСКК с обратной связью, обеспечивающая реализацию редукционной схемы за время (S(⌈log2b⌉+1) +2)tсл , где S – число итераций, b – разрядность входного числа, – длительность операции сложения двух вычетов