This is an advanced guide to optimal stopping and control, focusing on advanced Monte Carlo simulation and its application to finance. Written for quantitative finance practitioners and researchers in academia, the book looks at the classical simulation based algorithms before introducing some of the new, cutting edge approaches under development.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z) diagram by the second sheet.

Logic deals with the fundamental notions of truth and falsity. Modal logic arose from the philosophical study of “modes of truth” with the two most common modes being “necessarily true” and “possibly true”. Research in modal logic now spans the spectrum from philosophy, computer science and mathematics using techniques from relational structures, universal algebra, topology, and proof theory. These proceedings record the papers presented at the 2016 conference on Advances in Modal Logic, a biennial conference series with an aim to report on important new developments in pure and applied modal logic. As indicated above, there are new developments in using modal logic to reason about obligations, about programs, about time, about combinations of modal logics and even about negation itself.

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart.

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.

The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces.

The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.

Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework.

This book explores the theory and application of locally nilpotent derivations, which is a subject of growing interest and importance not only among those in commutative algebra and algebraic geometry, but also in fields such as Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the entire theory is based. These are used to establish classical results, such as Rentschler s Theorem for the plane, right up to the most recent results, such as Makar-Limanov s Theorem for locally nilpotent derivations of polynomial rings. Topics of special interest include: progress in the dimension three case, finiteness questions (Hilbert s 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. The reader will also find a wealth of pertinent examples and open problems and an up-to-date resource for research.

This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations.

The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.

More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.

A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.

This is the second part of a 2-year course of abstract algebra for students beginning a professional study of higher mathematics.1 This textbook is based on courses given at the Independent University of Moscow and at the Faculty of Mathematics at the National Research University Higher School of Economics. In particular, it contains a large number of exercises that were discussed in class, some of which are provided with commentary and hints, as well as problems for independent solution that were assigned as homework.Working out the exercises is of crucial importance in understanding the subject matter of this book.

This book is the first volume of an intensive “Russian-style” two-year graduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them.

The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses.

This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.

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.

The main research subject of this book is the phenomenon of the "positive deviation" in Sabaic epigraphy, i.e. the use of the plural in the places where one would expect the singular or dual. The quantitative analysis of this phenomenon undertaken in this book leads me to the supposition that its main causes are social and not purely linguistic, though the linguistic trend towards the supplanting of the dual by the plural observed in Middle Sabaic epigraphy can partly (but only partly) explain the positive deviation from the dual. Hence, the study of this phenomenon leads me to the following suppositions with respect to the social history of ancient Yemen: (1.) Clan organization seems to have played an important role in the social life of Middle Sabaean society (= the Middle Sabaean cultural-political area = the Northern part of the area of the Middle Sabaic epigraphy, the 1st century BC - the 4th century AD): **(1.a.)** All the main types of immovable property (fields, vineyards, houses, irrigation structures, wells &c) were considered as a rule almost without exceptions to be the property of clan groups, but not of the individuals. **(1.b.)** Clan groups (not individuals) were considered to be chiefs of the tribes.

**1.c.** Clan groups were often considered to be both objects of the client dependence, and the patrons of the clients ('dm).

**1.d.** Tribes were often considered to consist of clan groups (not of individuals).

**2.** In the Ancient Sabaean cultural-political area (the 1st millennium BC) the role of the clan organization was remarkably less important.

**2.a.** It is impossible to say that almost all kinds of immovable property were considered here to be in the possession of clans. In the majority of the cases individual (not clan) possessions are mentioned in the Ancient Sabaean inscriptions. Though private ownership might not have become completely universal in the Ancient Period, it is quite evident that the process of the formation and proliferation of this form of ownership went quite far in this Period.

**2.b.** In the Ancient Period the individual forms of cliental dependence seem to have played a much more important role than the clan ones. In the majority of the cases individual persons (not clients) were considered to be both "patrons" and "clients".

**2.c.** Individual persons (not clans) were usually considered to be leaders of tribes and communities in the Ancient Period.

**2.d.** Tribes were always considered to consist of individuals (not clans) in this period.

**3.** One may suppose that the process of the formation of the state and civilization in the Lowlands went far enough in the Ancient Period to cause a considerable decline of the clan organization and the ejecting of it to the periphery (both in the spatial and social senses of this word) of the social system.

**4.** Hence, it is possible to suppose that with the transition from the Ancient to Middle Period the clan organisation in the "North" significantly consolidated, its social importance considerably grew.

**5.** The "archaization" of the social life in the Southern (Himyarite-Radmanite) part of the area of the Middle Sabaic epigraphy (most of which was a part of the Qatabanian cultural-political area in the Ancient Period) was less strong than in the Northern ("Sabaean") part. The Ancient "individualized" tradition survived in the South to some extent, and the positions of the clan organization were not so solid here as they were in the North.

**6.**The above-mentioned social changes fit quite well in the general picture of the Pre-Islamic Yemeni history.

**6.a.** Several factors described in Chapter 4 caused a significant decline of the Sabaean state and civilization by the end of the 1st millennium BC. The weakening state organization seems to have become incapable of providing guarantees of life and property to individuals, and it was the clan organization that took on these functions to a considerable extent. As a result we can see by the Middle Period the consolidation of the clan organization which acted as a partial substitute for the weak state. This process can be also considered as quite an adequate social adaptation to the new situation which appeared in the Sabaean cultural-political area by the end of the 1st millennium BC with the relative decline of the Sabaean Lowlands (caused by the above-mentioned factors) and the rise of the importance of the "Sabaean" Highlands. Indeed, the Middle "Sabaean" political system, which was much less like a regular state than the Ancient one which included strong clan and tribal structures as its integral elements, turned out to be a really effective form of socio-political organization for a complex society in the Northern Highlands. Most political entities which appeared in this region from that time till the present have showen evident similarities to the Middle "Sabaean" socio-political organization.

**6.b.** The Middle Sabaean political system may be also characterized as consisting of a weak state in its centre and strong chiefdoms on its periphery. However, there is no doubt that this was a real system, i.e. it had some integrative properties which could not be reduced to the characteristics of its elements. It should be also taken into consideration that the state and chiefdoms were not the only elements of this political system. It included as well e.g. a sub-system of temple centres and the civil community of M_rib, as well as some true tribes (not chiefdoms) in the area of the Sabaean Lowlands, primarily the tribes of the Amirite confederation. With the transition from the Ancient to Middle Period the Sabaean political system was essentially transformed, becoming as a whole very different from the "state", but remaining, however, on basically the same level of political complexity. Without losing any political complexity and sophistication, the Middle "Sabaeans" managed to solve in quite different ways the problems which in complex societies are normally solved by states, such as the mobilization of resources for the functioning of the governing sub-system, the territorial organization of a vast space and the provision of guarantees of life and property. The Middle "Sabaean" experience seems to demonstrate that a large, complex, highly developed (in comparison with for example an average chiefdom) and integrated territorial entity need not necessarily be organized politically as a state. This appears to show that for the "early state" the transition to the "mature state" or complete "degeneration" into "tribes" and "chiefdoms" were not the only ways of possible evolution. One of the possible alternatives was its transformation into a "political system of the Middle Sabaean type". The real processes of political evolution seem to have been actually much less "unilinear" than is sometimes supposed. A significant transformation appears to have occurred in the area in the Early Islamic Period, and by the late Middle Ages the political system of the former "Sabaean" region seems to have consisted mainly of a stronger state in its centre and true tribes (not chiefdoms) on its periphery, whereas regular state structures persisted in the Southern (former Himyarite) cultural-political area.

**6.c. **The decline of the Ancient Qatabanian state took place significantly later than that of the Ancient Sabaean one. As a result the social continuity between the Ancient and the Middle Period in the Qatabanian cultural-political area was stronger, and the social transformation in the "South" turned out to be less dramatic. As a result in the Middle Period the state organization in the "South" appears considerably stronger than in the "North"; whereas the clan organization seems to have been much weaker. Quite an impressive feature of Yemeni history is that we find a more or less similar picture in 20th century Yemen: very strong clan-tribal structures and very weak state ones in the Yemeni Uplands to the north of Naq_l Yili (in the "Sabaean Highlands") and relatively weak clan-tribal structures and relatively strong state ones to the south of it, in the "Himyarite Highlands". Thus the above described picture appears as almost invariable in Yemeni history since the first centuries AD. This fact leads one to the supposition that there must be some fundamental basis for such a stable difference between the "North" and the "South". Its main objective factor is evident: the significant difference in the geographical conditions. It is really remarkable to find that the Highland territories of the two Middle Period cultural-political areas are practically identical with two main ecological zones of the Yemeni uplands.

**7.** The clan organization was not universal, even in the Middle Sabaean cultural-political area. The dense network of the clan relations was considerably weaker near the king and, perhaps, the most important temple centres, as they stood outside the clan organization and above it. In spatial dimensions, the zone of the weaker clan relations could be localized in the area of Marib and, perhaps, Nashq, Nashshan and San'a'.

The volume is to contain the proceedings of the 13th conference AGCT as well as the proceedings of the conference Geocrypt. The conferences focus on various aspects of arithmetic and algebraic geometry, number theory, coding theory and cryptography. The main topics discussed at conferences include the theory of curves over finite fields, theory of abelian varieties both over global and finite fields, theory of zeta-functions and L-functions, asymptotic problems in number theory and algebraic geometry, algorithmic aspects of the theory of curves and abelian varieties, the theory of error-correcting coding and particularly that of algebro-geometric codes, cryptographic issues related to algebraic curves and abelian varieties.

The text introduces all main subjects of "naive" (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.

In recent decades, the regions of Russia have taken different paths of regime transition. Despite the consolidation of an autocratic regime at national level and the centralization steered by Vladimir Putin’s government, the variation across sub-national regimes persists. Using an innovative theoretical framework, this book explores both causes and consequences of democratization in the regions of Russia. It is the first study in the field to systematically integrate structural and agency approaches in order to account for economic, social, historical and international causes of democratization and to trace its consequences. By focusing on the challenging and under-studied topic of sub-national regimes, the book provides a unique perspective on regime transition and the new theoretical framework contributes to a better understanding of democratization world-wide. The book will be of key interest to scholars and students of democratization, sub-national regimes, East European politics, comparative politics, post-communism, and international relations.

Concept discovery is a Knowledge Discovery in Databases (KDD) research field that uses human-centered techniques such as Formal Concept Analysis (FCA), Biclustering, Triclustering, Conceptual Graphs etc. for gaining insight into the underlying conceptual structure of the data. Traditional machine learning techniques are mainly focusing on structured data whereas most data available resides in unstructured, often textual, form. Compared to traditional data mining techniques, human-centered instruments actively engage the domain expert in the discovery process. This volume contains the contributions to CDUD 2011, the International Workshop on Concept Discovery in Unstructured Data (CDUD) held in Moscow. The main goal of this workshop was to provide a forum for researchers and developers of data mining instruments working on issues with analyzing unstructured data. We are proud that we could welcome 13 valuable contributions to this volume. The majority of the accepted papers described innovative research on data discovery in unstructured texts. Authors worked on issues such as transforming unstructured into structured information by amongst others extracting keywords and opinion words from texts with Natural Language Processing methods. Multiple authors who participated in the workshop used methods from the conceptual structures field including Formal Concept Analysis and Conceptual Graphs. Applications include but are not limited to text mining police reports, sociological definitions, movie reviews, etc.