### Book

## Basics of modern mathematical statistics: exercises and solutions

- Presents numerous exercises with solutions to help the reader better understand different aspects of modern statistics
- Applications with R and Matlab code show how to practically use the methods
- Includes numerous explanations and tips on how to apply modern statistical methods

The complexity of today’s statistical data calls for modern mathematical tools. Many fields of science make use of mathematical statistics and require continuous updating on statistical technologies. Practice makes perfect, since mastering the tools makes them applicable. Our book of exercises and solutions offers a wide range of applications and numerical solutions based on R. In modern mathematical statistics, the purpose is to provide statistics students with a number of basic exercises and also an understanding of how the theory can be applied to real-world problems. The application aspect is also quite important, as most previous exercise books are mostly on theoretical derivations. Also we add some problems from topics often encountered in recent research papers. The book was written for statistics students with one or two years of coursework in mathematical statistics and probability, professors who hold courses in mathematical statistics, and researchers in other fields who would like to do some exercises on math statistics.

In theory, a poverty line can be defined as the cost of a common (inter-personally comparable) utility level across a population. But how can one know if this holds in practice? For groups sharing common consumption needs but facing different prices, the theory of revealed preference can be used to derive testable implications of utility consistency knowing only the "poverty bundles" and their prices. Heterogeneity in needs calls for extra information. We argue that subjective welfare data offer a credible means of testing utility consistency across different needs groups. A case study of Russia's official poverty lines shows how revealed preference tests can be used in conjunction with qualitative information on needs heterogeneity. The results lead us to question the utility consistency of Russia's official poverty lines.

This article analyzes the issues of crime statistics, it`s showing particular use in criminal law and criminology, disclosed reserves replenishment of criminal law, criminology and criminology resource - a resource of criminal law, argues the need for a substantial update as one and the other sciences, formulated conclusions on enhancing their effectiveness in the context of the stabilization of the country's political, economic and social situation.

The paper continues research into words denoting everyday life objects in the Russian language. This research is conducted for developing a new encyclopedic thesaurus of Russian everyday life terminology. Working on this project brings up linguistic material which leads to discovering new trends and phenomena not covered by the existing dictionaries. We discuss derivation models which gain polularity: clipped forms (komp < komp’juter ‘computer’, nout < noutbuk ‘notebook computer’, vel < velosiped ‘bicycle’, mot<motocikl ‘motorbike’), competing masculine and feminine con- tracted nouns derived from adjectival noun phrases (mobil’nik (m.) / mo- bilka (f.) < mobil’nyj telefon (m.) ‘mobile phone’, zarjadnik (m.) / zarjadka (f.) < zarjadnoe ustrojstvo (n.) ‘AC charger’), hybrid compounds (plat’e- sviter ‘sweater dress’, jubka-brjuki ‘skirt pants’, shapkosharf ‘scarf hat’, vilkolozhka ‘spork, foon’). These words vary in spelling and syntactic behav- iour. We describe a newly formed series of words denoted multifunctional objects: mfushkaZ< MFU < mnogofunkcional’noe ustrojstvo ‘MFD, multi- function device’, mul’titul ‘multitool’, centr ‘unit, set’. Explaining the need to compose frequency lists of word meanings rather than just words, we of- fer a technique for gathering such lists and provide a sample produced from our own data. We also analyze existing dictionaries and perform various experiments to study the changes in word meanings and their comparative importance for speakers. We believe that, apart from the practical usage for our lexicographic project, our results might prove interesting for research in the evolution of the Russian lexical system.

The purpose of this paper is the presentation of the ideas and concepts that

form the basis of the concept of mathematical model control some processes

occurring in the Russian market of cereals. The estimated model must have a

stochastic nature, i.e. constitute some random process. Indeed, in a free market

there are objectively random factors that cannot be described by deterministic.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.