The materials of The International Scientific – Practical Conference is presented below. The Conference reflects the modern state of innovation in education, science, industry and social-economic sphere, from the standpoint of introducing new information technologies. It is interesting for a wide range of researchers, teachers, graduate students and professionals in the field of innovation and information technologies.

This book constitutes the proceedings of the 8th International Conference on Analysis of Images, Social Networks and Texts, AIST 2019, held in Kazan, Russia, in July 2019.

The 24 full papers and 10 short papers were carefully reviewed and selected from 134 submissions (of which 21 papers were rejected without being reviewed). The papers are organized in topical sections on general topics of data analysis; natural language processing; social network analysis; analysis of images and video; optimization problems on graphs and network structures; analysis of dynamic behaviour through event data.

Provides an overview of the developments and advances in the field of network clustering and blockmodeling over the last 10 years

This book offers an integrated treatment of network clustering and blockmodeling, covering all of the newest approaches and methods that have been developed over the last decade. Presented in a comprehensive manner, it offers the foundations for understanding network structures and processes, and features a wide variety of new techniques addressing issues that occur during the partitioning of networks across multiple disciplines such as community detection, blockmodeling of valued networks, role assignment, and stochastic blockmodeling.

Written by a team of international experts in the field, Advances in Network Clustering and Blockmodeling offers a plethora of diverse perspectives covering topics such as: bibliometric analyses of the network clustering literature; clustering approaches to networks; label propagation for clustering; and treating missing network data before partitioning. It also examines the partitioning of signed networks, multimode networks, and linked networks. A chapter on structured networks and coarsegrained descriptions is presented, along with another on scientific coauthorship networks. The book finishes with a section covering conclusions and directions for future work. In addition, the editors provide numerous tables, figures, case studies, examples, datasets, and more.

Offers a clear and insightful look at the state of the art in network clustering and blockmodeling Provides an excellent mix of mathematical rigor and practical application in a comprehensive manner Presents a suite of new methods, procedures, algorithms for partitioning networks, as well as new techniques for visualizing matrix arrays Features numerous examples throughout, enabling readers to gain a better understanding of research methods and to conduct their own research effectively Written by leading contributors in the field of spatial networks analysis

Advances in Network Clustering and Blockmodeling is an ideal book for graduate and undergraduate students taking courses on network analysis or working with networks using real data. It will also benefit researchers and practitioners interested in network analysis.

This book deals with mathematical modeling, namely, it describes the mathematical model of heat transfer in a silicon cathode of small (nano) dimensions with the possibility of partial melting taken into account.  This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type free boundary problems. The approach used is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the mathematical model including its parallel implementation. The results of numerical simulation concludes the book. The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.​

This book constitutes the refereed proceedings of the 11th International Conference on Intelligent Data Processing, IDP 2016, held in Barcelona, Spain, in October 2016.  

 

The 11 revised full papers were carefully reviewed and selected from 52 submissions. The papers of this volume are organized in topical sections on machine learning theory with applications; intelligent data processing in life and social sciences; morphological and technological approaches to image analysis.

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided.

We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contractible space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications.

After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc.

The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.

In the last 30 years a new pattern of interaction between mathematics and physics emerged, in which the latter catalyzed the creation of new mathematical theories. Most notable examples of this kind of interaction can be found in the theory of moduli spaces. In algebraic geometry the theory of moduli spaces goes back at least to Riemann, but they were first rigorously constructed by Mumford only in the 1960s. The theory has experienced an extraordinary development in recent decades, finding an increasing number of connections with other fields of mathematics and physics. In particular, moduli spaces of different objects (sheaves, instantons, curves, stable maps, etc.) have been used to construct invariants (such as Donaldson, Seiberg-Witten, Gromov-Witten, Donaldson-Thomas invariants) that solve longstanding, difficult enumerative problems. These invariants are related to the partition functions and expectation values of quantum field and string theories. In recent years, developments in both fields have led to an unprecedented cross-fertilization between geometry and physics. These striking interactions between geometry and physics were the theme of the CIME School Geometric Representation Theory and Gauge Theory. The School took place at the Grand Hotel San Michele, Cetraro, Italy, in June, Monday 25 to Friday 29, 2018. The present volume is a collection of notes of the lectures delivered at the school. It consists of three articles from Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively.

We propose a novel machine-learning-based approach to detect bid leakage in first-price sealed-bid auctions. We extract and analyze the data on more than 1.4 million Russian procurement auctions between 2014 and 2018. As bid leakage in each particular auction is tacit, the direct classification is impossible. Instead, we reduce the problem of bid leakage detection to Positive-Unlabeled Classification. The key idea is to regard the losing participants as fair and the winners as possibly corrupted. This allows us to estimate the prior probability of bid leakage in the sample, as well as the posterior probability of bid leakage for each specific auction. We find that at least 16% of auctions are exposed to bid leakage. Bid leakage is more likely in auctions with a higher reserve price, lower number of bidders and lower price fall, and where the winning bid is received in the last hour before the deadline.

Workshop concentrates on an interdisciplinary approach to modelling human behavior incorporating data mining and expert knowledge from behavioral sciences. Data analysis results extracted from clean data of laboratory experiments will be compared with noisy industrial datasets from the web e.g. Insights from behavioral sciences will help data scientists. Behavior scientists will see new inspirations to research from industrial data science. Market leaders in Big Data, as Microsoft, Facebook, and Google, have already realized the importance of experimental economics know-how for their business. 

In Experimental Economics, although financial rewards restrict subjects preferences in experiments, exclusive application of analytical game theory is not enough to explain the collected data. It calls for the development and evaluation of more sophisticated models. The more data is used for evaluation, the more statistical significance can be achieved. Since large amounts of behavioral data are required to scan for regularities, along with automated agents needed to simulate and intervene in human interactions, Machine Learning is the tool of choice for research in Experimental Economics. This workshop is aimed at bringing together researchers from both Data Analysis and Economics in order to achieve mutually beneficial results.

This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.

The Third Workshop on Computer Modelling in Decision Making (CMDM 2018) was held in Saratov State University (Saratov, Russia) within the VII International Youth Research and Practice Conference ‘Mathematical and Computer Modelling in Economics, Insurance and Risk Management’. The workshop 's main topic is computer and mathematical modeling in decision making in finance, insurance, banking, economic forecasting, investment and financial analysis. Researchers, postgraduate students, academics as well as financial, bank, insurance and government workers participated in the Workshop.

Modal logics, both propositional and predicate, have been used in computer science since the late 1970s. One of the most important properties of modal logics of relevance to their applications in computer science is the complexity of their satisfiability problem. The complexity of satisfiability for modal logics is rather high: it ranges from NP-complete to undecidable for propositional logics and is undecidable for predicate logics. This has, for a long time, motivated research in drawing the borderline between tractable and intractable fragments of propositional modal logics as well as between decidable and undecidable fragments of predicate modal logics. In the present thesis, we investigate some very natural restrictions on the languages of propositional and predicate modal logics and show that placing those restrictions does not decrease complexity of satisfiability. For propositional languages, we consider restricting the number of propositional variables allowed in the construction of formulas, while for predicate languages, we consider restricting the number of individual variables as well as the number and arity of predicate letters allowed in the construction of formulas. We develop original techniques, which build on and develop the techniques known from the literature, for proving that satisfiability for a finite-variable fragment of a propositional modal logic is as computationally hard as satisfiability for the logic in the full language and adapt those techniques to predicate modal logics and prove undecidability of fragments of such logics in the language with a finite number of unary predicate letters as well as restrictions on the number of individual variables. The thesis is based on four articles published or accepted for publication. They concern propositional dynamic logics, propositional branchingand alternating-time temporal logics, propositional logics of symmetric rela tions, and first-order predicate modal and intuitionistic logics. In all cases, we identify the “minimal,” with regard to the criteria mentioned above, fragments whose satisfiability is as computationally hard as satisfiability for the entire logic.

This volume collects the referred papers based on plenary, invited, and oral talks, as well on the posters presented at the Third International Conference on Computer Simulations in Physics and beyond (CSP2018), which took place September 24-27, 2018 in Moscow. The Conference continues the tradition started by an inaugural conference in 2015. It took place on the campus of A.N. Tikhonov Moscow Institute of Electronics and Mathematics in Strogino, was jointly organized by the National Research University Higher School of Economics, the Landau Institute for Theoretical Physics and Science Center in Chernogolovka.

The Conference is a multidisciplinary meeting, with a focus on computational physics and related subjects. Indeed, methods of computational physics prove useful in a broad spectrum of research in multiple branches of natural sciences, and this volume provides a sample.

We hope that this volume will interest readers, and we are already looking forward to the next conference in the series.

Moscow, Russia

November, 2018

CSP2018 Conference Chair and Volume Editor

Lev Shchur

This book covers the classical theory of Markov chains on general state-spaces as well as many recent developments. The theoretical results are illustrated by simple examples, many of which are taken from Markov Chain Monte Carlo methods. The book is self-contained, while all the results are carefully and concisely proven. Bibliographical notes are added at the end of each chapter to provide an overview of the literature.

In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with the cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a uid under gravity in such a domain. These so- lutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (in quadratures). The behavior of the so- lutions at far distances is studied by use of the saddle point technique. The corresponding eigenoscil- lations of a uid are then interpreted as eigenfunctions of the continuous spectrum.

The monograph provides a detailed presentation of the theory of weak convergence of measures. 

This work discusses a possibility to assess the probability of company default using system dynamic model. This approach is based on Monte Carlo Simulation with various inputs for a system dynamic model. The results are compared with the estimations of rating agencies.

Detection of positive selection signatures in populations around the world is helping to uncover recent human evolutionary history as well as the genetic basis of diseases. Most human evolutionary genomic studies have been performed in European, African and Asian populations. However, populations with Native American ancestry have been largely underrepresented. Here, we used a genome-wide local ancestry enrichment approach complemented with neutral simulations to identify post-admixture adaptations underwent by admixed Chileans through gene flow from Europeans into local Native Americans. The top significant hits (P = 2.4 x 10−7) are variants in a region on chromosome 12 comprising multiple regulatory elements. This region includes rs12821256, which regulates the expression of KITLG, a well-known gene involved in lighter hair and skin pigmentation in Europeans. Another variant from that region is associated with the long noncoding RNA RP11-13A1.1, which has been specifically involved in the innate immune response against infectious pathogens (Riege, et al. 2017). Our results suggest that these genes were relevant for adaption in Chileans following the Columbian exchange.

This paper is a review of results on multiple flag varieties, i.e., varieties of the form G/P1×· · ·×G/Pr. We provide a classification of multiple flag varieties of complexity 0 and 1 and results on the combinatorics and geometry of B-orbits and their closures in double cominuscule flag varieties. We also discuss questions of finiteness for the number of G-orbits and existence of an open G-orbits on a multiple flag variety.

There are only 10 Euclidean forms, that is  flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of $n$-fold coverings over the orientable Euclidean manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$, and calculate the numbers of non-equivalent coverings of each type. The manifolds $\mathcal{G}_{3}$ and $\mathcal{G}_{5}$ are uniquely determined among other forms by their homology groups $H_1(\mathcal{G}_{3})=\ZZ_3\times \ZZ$ and $H_1(\mathcal{G}_{5})= \ZZ$.

We classify subgroups in the fundamental groups $\pi_1(\mathcal{G}_{3})$ and $\pi_1(\mathcal{G}_{5})$ up to isomorphism.  Given index $n$,  we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating functions for the above sequences.

There are only 10 Euclidean forms, that is  flat closed three dimensional manifolds: six are orientable and four are non-orientable. The aim of this paper is to describe all types of $n$-fold coverings over orientable Euclidean manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$, and calculate the numbers of non-equivalent coverings of each type. We classify subgroups in the fundamental groups $\pi_1(\mathcal{G}_{2})$ and $\pi_1(\mathcal{G}_{4})$ up to isomorphism and calculate the numbers of conjugated classes of each type of subgroups for index $n$. The manifolds $\mathcal{G}_{2}$ and $\mathcal{G}_{4}$ are uniquely determined among the others orientable forms by their homology groups $H_1(\mathcal{G}_{2})=\ZZ_2\times \ZZ_2 \times \ZZ$ and $H_1(\mathcal{G}_{4})=\ZZ_2 \times \ZZ$.  

This paper investigates whether a root lattice can be similar to a lattice O of integer elements of a number field K endowed with the inner product (x,y):=Trace_{K/\mathbb Q}(x\cdot \theta(y)), where \theta is an involution of the field K. For each of the following three properties (1), (2), (3), a classification of all the pairs K, \theta with this property is obtained: (1) O is a root lattice; (2) O is similar to an even root lattice; (3) O is similar to the lattice \mathbb Z^{[K:\mathbb Q]}. The necessary conditions for similarity of O to a root lattice of other types are also obtained. It is proved that O cannot be similar to a positive definite even unimodular lattice of rank \leqslant 48, in particular, to the Leech lattice.

The development of advanced electrochemical devices for energy conversion and storage requires fine tuning of electrode reactions, which can be accomplished by altering the electrode/solution interface structure. Particularly, in case of an alkali-salt electrolyte the electric double layer (EDL) composition can be managed by introducing organic cations (e.g. room temperature ionic liquid cations) that may possess polar fragments. To explore this approach, we develop a theoretical model predicting the efficient replacement of simple (alkali) cations with dipolar (organic) ones within the EDL. For the typical values of the molecular dipole moment ($2-4~D$) the effect manifests itself at the surface charge densities higher than 30 $\mu C/cm^2$. We show that the predicted behavior of the system is in qualitative agreement with the molecular dynamics simulation results.

We consider difference and differential Stackelberg game theoretic models with several followers of opinion control in marketing networks. It is assumed that in the stage of analysis of the network its opinion leaders have already been found and are the only objects of control. The leading player determines the marketing budgets of the followers by resource allocation. In the basic version of the models both the leader and the followers maximize the summary opinions of the network agents. In the second version the leader has a target value of the summary opinion. In all four models we have found the Stackelberg equilibrium and the respective payoffs of the players analytically. It is shown that the hierarchical control system is ideally compatible in all cases.

The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called ‘Lambek’s restriction’, i.e. the antecedent of any provable sequent should be non-empty. In this paper, we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek’s restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek’s restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek’s restriction and cut elimination and another system with Lambek’s restriction and substitution. For both calculi, we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek’s restriction is folklore and known to be undecidable.

The article discusses construction of traveling wave type solutions for the Frenkel-Kontorova model on the propagation of longitudinal waves. For the first time, based on the existence and uniqueness theorem of traveling wave type solutions, as well as the approximation theorem, a complete family of traveling wave type solutions is constructed in the form of subfamilies of bounded solutions (horizontals) and unbounded solutions (verticals).

Subword complexes were defined by A.Knutson and E.Miller in 2004 for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S.Assaf and D.Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres.

We study smooth Fano weighted complete intersections with respect to the new invariant -- the variance var(X) = coindex(X) - codim(X).

Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, in this paper we obtain a description of Λˆ+G(G/H) via the set of simple spherical roots of~G/H together with certain combinatorial data explicitly computed from the pair (P,H). As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing Λˆ+G(G/H) in the case where H is strongly solvable.

Let P=A_1...A_n be a generic polygon in three-dimensional space and let v_1,v_2,...,v_n be vectors A_1A_2,A_2A_3,...,A_nA_1, respectively. P will be called regular, if there exist vectors u_1,...,u_n such that cross products  [u_1,u_2],[u_2,u_3],...,[u_n,u_1] are equal to vectors v_2,v_3,...,v_1, respectively. In this case the polygon P', defined be vectors u_2-u_1,u_3-u_2,...,u_1-u_n will be called the derived polygon or the derivative of the polygon P. In this work we formulate conditions for regularity and discuss geometric properties of derived polygons for n=4,5,6.

We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers  of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdot\theta(y))$, where $\theta$ is an involution of $K$. We classify all  pairs $K$, $\theta$ such that $\mathscr O$ is similar to either an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also classify all  pairs $K$, $\theta$ such that $\mathscr O$ is a root lattice. In addition to this, we show that $\mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $\leqslant 48$, in particular, $\mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathscr O$.

Multilevel modeling (MLM, also known as hierarchical linear modeling, HLM) is a methodological frameworkwidely usedin the social sciences to analyze data with a hierarchical structure,where lower units of aggregation are ‘nested’ in higher units, including longitudinal data.In economics, however, MLM is usedvery rarely.Instead, economists use separate econometric techniques including cluster-robust standard errors and fixedeffects models. In this paper, we reviewthe methodological literature and contrast the econometrictechniques typically used in economics with the analysis ofhierarchical data using MLM. Ourreview suggests that economic techniques are generally lessconvenient, flexible, and efficientcompared to MLM. Theimportant limitation of MLM, however, isits inability to deal with the omitted variable problem at the lowest level of data, while standard economic techniques may be complemented by quasi-experimental methods mitigatingthis problem.It is unlikely, though, that this limitation can explain and justify the rare use of MLM in economics.Overall, we conclude that MLM has been unreasonably ignored in economics,and we encourage economists to apply this frameworkbyproviding ‘when and how’ guidelines.

We consider NP-hard multi{machine scheduling problems with the criterion of minimizing the maximum penalty, e.g. maximum lateness. For such problems, we introduce a metric which delivers an upper bound on the absolute error of the objective function value. Taking the given in- stance of some problem and using the introduced metric, we determine the nearest instance for which a polynomial or pseudo-polynomial algorithm is known. For this determined instance, we construct a schedule which is then applied to the original instance. It is shown how this approach is applied to di_erent scheduling problems.

Main subject of the paper is a (strong) Morse function on a compact manifold with boundary. We construct a cellular structure and discuss its algebraic properties in this paper.

Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous shift of space variables. Dynamics of singularities of the KdV and Sinh--Gordon equations, as well as rational cases of the Calogero--Moser and Ruijsenaars--Schneider models are shown to provide examples of such induced dynamics. Some other examples are given to demonstrates highly nontrivial collisions of particles and Liouville integrability of induced dynamical systems.

We consider quadrangles of perimeter 2 in the plane with marked directed edge. To such quadrangle  Q two-dimensional plane  P with orthonormal base is corresponded. Orthoganal plane defines  a plane quadrangle R of perimeter 2 and with marked directed edge. This quadrangle is definded uniquely ( up to rotation and symmetry ). Quadrangles Q and R will be called dual to each ather. The following properties of duality are prooved: a)duality preserves qonvexity, non convexity and self-intersection;b) duality preserves the length of diagonals; c) the sum of length of corresponding edges in Q and R is 1.l

In this paper, we consider single machine scheduling problems with given release dates and the objective to minimize the maximum penalty which is NP-hard in the strong sense. For this problem, we introduce a dual and an inverse problem and show that both these problems can be solved in polynomial time. Since the dual problem gives a lower bound on the optimal objective function value of the original problem, we incorporate the optimal function value of a sub-problem of the dual problem into a branch and bound algorithm for the original single machine scheduling problem. We present some initial computational results for instances with up to 20 jobs.

Presburger Arithmetic PrA is the true theory of natural numbers with addition. We consider linear orderings interpretable in Presburger Arithmetic and establish various necessary and sucient conditions for interpretability depending on dimension n of interpretation. We note this problem is relevant to the interpretations of Presburger Arithmetic in itself, as well as the characterization of automatic orderings. For n = 2 we obtain the complete criterion of interpretability.

We consider quadrangles of perimeter 2 in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^bot$ defines a plane quadrangle $Q^\circ$ of perimeter 2 and with marked directed edge. This quadrangle is defined uniquely (up to rotation and symmetry). Quadrangles $Q$ and $Q^\circ$ will be called dual to each other. The following properties of duality are proved: a) duality preserves convexity, non convexity and self-intersection; b) duality preserves the length of diagonals; c) the sum of lengths of corresponding edges in $Q$ and $Q^\circ$ is 1.

Buchstaber and Terzic introduced a notion of universal space of parameters F for a manifold M^2n, which has an effective action of compact torus T^k , k≤n with some additional properties. with special properties. This space is needed to construction of factor M^2n/T^k. Buchstaber and Terzic constructed the universal space of parameters for G_5,2. In this work we construct universal space of parameters for complex Grassmann manifold G_q+1,2. Our construction is based on the construction of moduli space of stable curves of genus zero with q+1 marked points due to Salamon, McDuff and Hofer.

Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott-Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.