Elements of mathematics in problems. Through olympiads and circles to profession

This is a collection of teaching materials used in several Russian universities, schools, and mathematical circles. Most problems are chosen in such a way that in the course of the solution and discussion a reader learns important mathematical ideas and theories. The materials can be used by pupils and students for self-study, and by teachers. This is an abridged pre-copyedit version of the published book submitted with the permission of the publisher. Each included individual material is self-contained and ready-for-use. Solutions to problems are not included intentionally. This collection consolidates updates of several arXiv submissions, e.g., arXiv:1305.2598.Comment: Edited by A. Skopenkov, M. Skopenkov, A. Zaslavsky. In Russian. Optimized for printing on A5 paper. Moscow Center for Continuous Mathematical Education, 2018, 592pp (in Russian

Thermodynamic interpolation for the simulation of two-phase flow of non-ideal mixtures

This paper describes the development and application of a technique for the rapid interpolation of thermodynamic properties of mixtures for the purposes of simulating two-phase flow. The technique is based on adaptive inverse interpolation and can be applied to any Equation of State and multicomponent mixture. Following analysis of its accuracy, the method is coupled with a two-phase flow model, based on the homogeneous equilibrium mixture assumption, and applied to the simulation of flows of carbon dioxide (CO2) rich mixtures. This coupled flow model is used to simulate the experimental decompression of binary and quinternary mixtures. It is found that the predictions are in good agreement with the experimental data and that the interpolation approach provides a flexible, robust means of obtaining thermodynamic properties for use in flow models

Comparative Analysis of the Subventricular Zone in Rat, Ferret and Macaque: Evidence for an Outer Subventricular Zone in Rodents

The mammalian cerebral cortex arises from precursor cells that reside in a proliferative region surrounding the lateral ventricles of the developing brain. Recent work has shown that precursor cells in the subventricular zone (SVZ) provide a major contribution to prenatal cortical neurogenesis, and that the SVZ is significantly thicker in gyrencephalic mammals such as primates than it is in lissencephalic mammals including rodents. Identifying characteristics that are shared by or that distinguish cortical precursor cells across mammalian species will shed light on factors that regulate cortical neurogenesis and may point toward mechanisms that underlie the evolutionary expansion of the neocortex in gyrencephalic mammals. We immunostained sections of the developing cerebral cortex from lissencephalic rats, and from gyrencephalic ferrets and macaques to compare the distribution of precursor cell types in each species. We also performed time-lapse imaging of precursor cells in the developing rat neocortex. We show that the distribution of Pax6+ and Tbr2+ precursor cells is similar in lissencephalic rat and gyrencephalic ferret, and different in the gyrencephalic cortex of macaque. We show that mitotic Pax6+ translocating radial glial cells (tRG) are present in the cerebral cortex of each species during and after neurogenesis, demonstrating that the function of Pax6+ tRG cells is not restricted to neurogenesis. Furthermore, we show that Olig2 expression distinguishes two distinct subtypes of Pax6+ tRG cells. Finally we present a novel method for discriminating the inner and outer SVZ across mammalian species and show that the key cytoarchitectural features and cell types that define the outer SVZ in developing primates are present in the developing rat neocortex. Our data demonstrate that the developing rat cerebral cortex possesses an outer subventricular zone during late stages of cortical neurogenesis and that the developing rodent cortex shares important features with that of primates

Measuring 129Xe transfer across the blood‐brain barrier using MR spectroscopy

Purpose This study develops a tracer kinetic model of xenon uptake in the human brain to determine the transfer rate of inhaled hyperpolarized 129Xe from cerebral blood to gray matter that accounts for the effects of cerebral physiology, perfusion and magnetization dynamics. The 129Xe transfer rate is expressed using a tracer transfer coefficient, which estimates the quantity of hyperpolarized 129Xe dissolved in cerebral blood under exchange with depolarized 129Xe dissolved in gray matter under equilibrium of concentration. Theory and Methods Time‐resolved MR spectra of hyperpolarized 129Xe dissolved in the human brain were acquired from three healthy volunteers. Acquired spectra were numerically fitted with five Lorentzian peaks in accordance with known 129Xe brain spectral peaks. The signal dynamics of spectral peaks for gray matter and red blood cells were quantified, and correction for the 129Xe T1 dependence upon blood oxygenation was applied. 129Xe transfer dynamics determined from the ratio of the peaks for gray matter and red blood cells was numerically fitted with the developed tracer kinetic model. Results For all the acquired NMR spectra, the developed tracer kinetic model fitted the data with tracer transfer coefficients between 0.1 and 0.14. Conclusion In this study, a tracer kinetic model was developed and validated that estimates the transfer rate of HP 129Xe from cerebral blood to gray matter in the human brain

HARDENING OF THE ELECTRODESIEGED IRON CHEMICAL HEAT TREATMENT

Summary. Currently in the repair and manufacture at the stage of recovery of steel parts, widely used special coatings formed by electrolytic effects on ferrous ions. This technique offers high performance, ease of implementation, low cost of technological equipment and materials used, as well as easy automation of the process. However, this method has several disadvantages: low fatigue strength of reconditioned parts, insufficiently strong grip of the iron coating to the substrate, particularly in alloy steels, insufficient wear resistance. For the purpose of increasing durability and wear resistance of parts, restored through electrochemical action, it is proposed to use chemical-heat treatment, consisting in the application of carbonitriding. Investigated the efficacy of different modes of carbonitriding in the highly carburizing paste-and their influence on the structure and properties of iron plating. It is established that the nitrocarburizing both low and high temperatures repeatedly (6-7.5 times) increases the microhardness of the coatings. The highest hardness is obtained by low-temperature carbonitriding with direct quenching in water. Conducting the carbonitriding process at low temperatures (650 °C), significantly increases the hardness of the iron coatings, increasing the limit of its fluidity, a and also greatly increases its endurance limit. Nitrocarburized fatigue strength of samples with iron precipitation on the surface, as shown by our studies, not only higher strength of the same samples without carbonitriding (more than 2 times), but higher than the fatigue strength of the base metal without coatings. Raising the temperature of the carbonitriding did not increase the hardness of electrolytic iron. Developed a rational technology of hardening of steel parts, re-chain iron fortification. Selected optimum conditions for carbonitriding hardening restored iron fortification, with the purpose of increasing durability of machine parts Optimal process temperature re-benching of the process of carbonitriding in which is possible to obtain maximum fatigue strength and wear resistance from the restored detail, is the process of carbonitriding at 650°C, followed by quenching and tempering at 150 °C

Development of two-stage grain grinder

The most important task in the development of the diet of farm animals feeding is a selection of the most balanced in its composition and most nutritious feeds, which are safe and meet all the necessary requirements at the same time. To evaluate the productive value of feeds and their effectiveness the rate of food productive action η was proposed. This ratio reflects the productive part of the total value of the exchange energy of the daily feed ration and is an essential criterion of the feed quality indicators. In the feed rations of animals the most expensive, but energy-rich feed is a mixed fodder, a mixture of grinded seeds of agricultural crops and protein, mineral and vitamin additives. In the diet for its nutritional value, this feed product is for cattle – 50, pigs – 60… 100 and birds – 100%. The basic operation in the production of mixed fodder is seeds grinding, i.e. their destruction under the influence of external forces, exceeding the forces of molecular adhesion of the grains particles. To grind the grain different ways are used: chopping, grinding, impact «in flight», crushing, etc. In the production of mixed fodder on the existing production equipment, there is the problem of getting the grain mixed fodder the necessary degree of grinding and uniform in its particle size distribution at the same time. When receiving too coarse grinding there is a problem of difficult digestibility of mixed fodder by farm animals. Moreover grinding process is accompanied by a high energy consumption. Grain grinder, the principle of which is based on the implementation of two ways of grinding grain: splitting and impact «in flight» is proposed. The proposed constructive solutions allow to obtain a high-performance technical means for crushing seeds of crops, as well as reduce energy costs that arise during the course of the process of obtaining of mixed fodder. The methodology justification of degree of grain grinding by the steps of the proposed grinder is shown in the article