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## Normal Forms, Inner Products, and Maslov Indices of General Multimode Squeezings

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension.

This paper is aimed at applying and analyzing international active ageing indices in Russia, including the Active Ageing Index (AAI), developed by European Centre Vienna, and Global AgeWatch Index by HelpAge International, to provide the base for cross-national comparison and development of a comprehensive national policy on active ageing. Our research was motivated by the following questions (1) to what extent can the international approaches to measure active ageing be applied to the Russian context and data? (2) to what extent a country’s position in the ranking is sensitive to the index methodology and data used? (3) whether and under what conditions Russia can improve its positions in the active ageing indices? To answer these questions, we estimated the AAI for Russia based on eight data sources and recalculated some of the AgeWatch Index results based on reliable data. The methodology of both indices and the quality and adequacy of the data used are discussed in detail in the paper. The results show that ranking of Russia according to these indices varies considerably from the 65th place out of 96 countries by the Global AgeWatch Index to the 18th place among 29 countries (28 EU countries plus Russia) by the AAI. Nevertheless, both indices draw rather similar pictures of active ageing potential in Russia. We provide some recommendations on how the indicators can be modified to capture some peculiarities of the ageing context in Russia and other countries with similar demographic, economic and social context.

This book is based on a lecture course given by the author at the Educational Center of the Steklov Mathematical Institute in 2011. It is designed for a one-semester course for undergraduate students familiar with basic differential geometry and complex and functional analysis.

The universal Teichmüller space T is the quotient of the space of quasisymmetric homeomorphisms of the unit circle modulo Möbius transformations. The first part of the book is devoted to the study of geometric and analytic properties of T. It is an infinite-dimensional Kähler manifold which contains all classical Teichmüller spaces of compact Riemann surfaces as complex submanifolds, which explains the name "universal Teichmüller space". Apart from classical Teichmüller spaces, T contains the space S of diffeomorphisms of the circle modulo Möbius transformations. The latter space plays an important role in the quantization of the theory of smooth strings.

The quantization of T is presented in the second part of the book. In contrast with the case of diffeomorphism space S, which can be quantized in frames of the conventional Dirac scheme, the quantization of T requires an absolutely different approach based on the noncommutative geometry methods.

The book concludes with a list of 24 problems and exercises which can used to prepare for examinations.

In this paper, we consider several compression techniques for the language modeling problem based on recurrent neural networks (RNNs). It is known that conventional RNNs, e.g., LSTM-based networks in language modeling, are characterized with either high space complexity or substantial inference time. This problem is especially crucial for mobile applications, in which the constant interaction with the remote server is inappropriate. By using the Penn Treebank (PTB) dataset we compare pruning, quantization, low-rank factorization, tensor train decomposition for LSTM networks in terms of model size and suitability for fast inference.

In the last years native RDF stores made enormous progress in closing the performance gap compared to RDBMS. This albeit smaller gap, however, still prevents adoption of RDF stores in scenarios with high requirements on responsiveness. We try to bridge the gap and present a native RDF store “OntoQuad” and its fundamental design principles. Basing on previous researches, we develop a vector database schema for quadruples, its realization on index data structures, and ways to efficiently implement the joining of two and more data sets simultaneously. We also offer approaches to optimizing the SPARQL query execution plan which is based on its heuristic transformations. The query performance efficiency is checked and proved on BSBM tests. The study results can be taken into consideration during the development of RDF DBMS’s suitable for storing large volumes of Semantic Web data, as well as for the creation of large-scale repositories of semantic data.

Methodology of automated forming of weak-formalized manufacturing documents for support the production processes of an enterprise on the basis of the cybernetic methods is considered. A model of elaboration of automaed documents preparation technology with use of automated lexicological synthesis is presented. The technology being proposed allows substantially reduce the man-hours at making of full-text documents.

We define and study the Hochschild (co)homology of the second kind (known also as the Borel-Moore Hochschild homology and the compactly supported Hochschild cohomology) for curved DG categories. An isomorphism between the Hochschild (co)homology of the second kind of a CDG-category B and the same of the DG category C of right CDG-modules over B, projective and finitely generated as graded B-modules, is constructed. Sufficient conditions for an isomorphism of the two kinds of Hochschild (co)homology of a DG-category are formulated in terms of the two kinds of derived categories of DG-modules over it. In particular, a kind of “resolution of the diagonal” condition for the diagonal CDG-bimodule B over a CDG-category B guarantees an isomorphism of the two kinds of Hochschild (co)homology of the corresponding DG-category C. Several classes of examples are discussed. In particular, we show that the two kinds of Hochschild (co)homology are isomorphic for the DG-category of matrix factorizations of a regular function on a smooth affine variety over a perfect field provided that the function has no other critical values but zero.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.