Measure Theory Bogachev, Vladimir I
2007, 2006, 2007-02-01
eBook
Measure theory is a classical area of mathematics born more than two thousand years ago. Nowadays it continues intensive development and has fruitful connections with most other fields of mathematics ...as well as important applications in physics. This book gives an exposition of the foundations of modern measure theory and offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course (the material of this level corresponds to a variety of special courses), and, finally, more specialized topics partly covered by more than 850 exercises. Volume 1 (Chapters 1-5) is devoted to the classical theory of measure and integral. Whereas the first volume presents the ideas that go back mainly to Lebesgue, the second volume (Chapters 6-10) is to a large extent the result of the later development up to the recent years. The central subjects of Volume 2 are: transformations of measures, onditional measures, and weak convergence of measures. These three topics are closely interwoven and form the heart of modern measure theory. The organization of the book does not require systematic reading from beginning to end, in particular, almost all sections in the supplements are independent of each other and are directly linked only to specific sections of the main part. The target readership includes graduate students interested in deeper knowledge of measure theory, instructors of courses in measure and integration theory, and researchers in all fields of mathematics. The book may serve as a source for many advanced courses or as a reference.
Abstract
The process of multimodal freight transportation on a railway loop is investigated, for which a multicriteria optimization model with time indicators is created, which is a nonlinear integer ...programming problem. The developed approach is based on egalitarian and utilitarian principles in the welfare theory and allows (along with Pareto optimal transportation plans) to find other plans that can be considered rational from the point of view of maintaining the balance of interests of the transportation process participants. A mathematical experiment is an effective heuristic tool in research. The algorithm for solving the optimization problem is implemented in the environment of a computer algebra system and brought to numerical results.
We study the existence of solutions to the Kantorovich optimal transportation problem with a nonlinear cost functional generated by a cost function depending on the transport plan. We also consider ...the case of a cost function depending on the conditional measures of the transport plan. Broad sufficient conditions are obtained for the existence of optimal plans for Radon marginal distributions on completely regular spaces and a lower semicontinuous cost function.
This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal ...transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse applications of these problems in non-linear analysis, probability theory, and differential geometry are discussed.
The creation of a caesium vapour laser with closed-cycle circulation of the laser-active medium is first reported. The power of the laser radiation amounted to {approx}1 kW with the 'light-to-light' ...conversion efficiency of {approx}48 %. Quasi-two-dimensional computational model of the laser operation that provides adequate description of experimental results is considered. Calculated and experimental dependences of the laser radiation power on the temperature of the cuvette walls, laser medium pressure and pump power are presented.
A multi-agent approach has been developed to find the optimal distribution of freight traffic on the considered railway loop for multimodal transportation carried out within the framework of an ...oligopolistic market. At the first stage of research, the method of economic-geographical delimitation of the «influence areas» of loading stations, developed earlier by the authors, is applied. The obtained territorial picture of the rational distribution of traffic flows is used further when introducing restrictions in a multi-criteria optimization problem, which is solved as an integer programming problem in the environment of the system of analytical calculations. As a result, on the basis of an assessment of the transport and technological infrastructure of the loop and the cost of transport services, a mathematical model is built that allows to optimize the process of freight transportation.
An aqueous colloidal solution of dextran-coated magnetite nanoparticles was studied by nonlinear second-harmonic magnetic response (M2), transmission electron microscopy (TEM), dynamic light ...scattering (DLS) and electron magnetic resonance (EMR). Nanoparticles were found to aggregate. A set of magnetic parameters of the aggregates, such as the mean magnetic moment, the magnetization damping constant, the longitudinal relaxation time, the field and energy of magnetic anisotropy, and others were evaluated from M2 measurements with the data processing formalism based on the Gilbert–Landau–Lifshitz (GLL) equation for the stochastic dynamics of superparamagnetic (SP) particles. Combined with TEM and DLS, the M2 technique additionally enabled the differentiation between magnetic and nonmagnetic components of the colloid. To achieve full numerical consistency between the parameters obtained from the M2 and TEM data, magnetic correlations of nanoparticles inside the aggregates were taken into account and their correlation radius was evaluated. The observed crossover in the magnetic field dependence of the EMR signal occurring due to the break of the dipole–dipole (d–d) coupling between nanoparticles in the aggregates was described using the M2 and TEM data.
MRI-Guided Therapy Bogachev, Yu. V.; Nikitina, A. V.; Frolov, V. V. ...
Technical physics,
09/2020, Letnik:
65, Številka:
9
Journal Article
Recenzirano
Magnetic resonance (MR) theranostics is a new trend in modern medicine that combines rich diagnostic potentialities of the magnetic resonance method with the therapeutic action of physical fields of ...different origins on various organs. We consider peculiarities and clinical applications of MR theranostics methods and equipment, including magnetic resonance imaging (MRI), MRI-guided high-intensity focused ultrasound surgery (tumor ablation), radiation (X-ray) therapy, proton therapy, and rf therapy.
Olfactory dysfunction is an early marker of COVID-19 infection. However, individuals may develop chronic olfactory impairment for more than six months in 1-10 % of cases. The study's objective is to ...evaluate the efficacy and safety of intranasal immunotherapy using bioactive substances produced by M2 macrophages for the treatment of people with long-term post-COVID-19 hyposmia. Seven individuals with long-term persistent hyposmia (7 to 24 months), associated with PCR-confirmed coronavirus infection were evaluated for olfactory function at baseline, one, and six to twelve months after therapy. The intranasal inhalation of M2 macrophage conditioned medum (one time per day for 28-30 days) was well tolerated. Furthermore, olfactometry demonstrated that the patients restored their capacity to perceive (Kruskal-Wallis H test 14.123, p = 0.0009) and recognize odours (H = 11.674, p = 0.0029). In addition, the subjective evaluation of smell significantly improved (H = 11.935, p = 0.0026). At the 6- to 12-month follow-up, the majority of patients (5/7) reported extremely high levels of satisfaction with the outcomes, and the remaining two patients also felt generally positive about the therapy's success. Overall, our study showed that the use of intranasal inhalations as a method of delivering bioactive factors and the conditioned medium of M2 macrophages as a therapeutic agent are both safe, well tolerated and, according to preliminary data, clinically effective in the treatment of patients with long-term post-COVID-19 hyposmia.
We have used data from the space telescope SOHO/EIT and the spectrometer VEIS on the Wind spacecraft to compare the solar wind (SW) speed near Earth's orbit with changes in the area of polar coronal ...holes (CHs) on the Sun during the 1996 solar activity minimum. We have found that in March 1996 the SW speed correlated with the southern CH area by a factor of 0.64. In September and October 1996, a correlation was revealed between the SW speed and the area of the northern CH (the coefficients are 0.64 and 0.85 respectively). We believe that this confirms the assumption that the solar wind from polar CHs can penetrate into the ecliptic plane at solar minimum. The SW speed was 460–500 km/s, which is lower than that from equatorial CHs (600–700 km/s).
PDF
