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## Exact dynamics for a mutator gene model

We investigate the dynamics of a molecular evolution model related to the mutator gene phenomenon. Here mutation in one gene drastically changes the properties of the whole genome. We investigated the Crow-Kimura version of the model, which can be mapped into a Hamilton-Jacobi equation. For the symmetric fitness landscape, we calculated the dynamics of the maximum of the total population distribution. We found two phases in the dynamics: a simple one when the maximum of distribution moves along a characteristics, and more involved one when the maximum jumps to another characteristic at some turnout point T.

A macroevolutionary approach and the new field of Big History seek ** to develop an inclusive view of the Cosmos, Earth, life and humanity** by erasing boundaries between disciplines. Big History is a versatile study that brings together constantly updated information from Astronomy, Physics, Geology, Biology, Chemistry, Anthropology, Psychology and other scientific disciplines, and then merges it with the contemplative realms of Philosophy and the Humanities. Big History evolved from the academic need to transcend the straight-jacket of university disciplines in the early 20th century, beginning with subjects like Biochemistry and Astrophysics (Christian and McNeill 2008; Rodrigue and Stasko 2011).

The history of the mankind is closely connected with the universal processes and to a large extent determined by natural laws. History is made up of the process of interchanges of dominance between Countryside (e.g. ancient east societies, early medieval European states), with slow development, and Town (city-states of Antiquity with its protobourgeoisie and electiveness of governmental structures, modern Europe and America), with much more rapid one.

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'.

New data on the relative elasticity of soft materials for linings for dentures and obturators with rigid inserts was obtained. The problems of improving the quality of dental prosthetic rehabilitation of oncological patients with acquired defects of the upper jaw were discussed.

The author argues on expediency and mutual conditionality of evolutionary changes in the nature and in society. In the article three major factors of the evolution are allocated, namely: the accident, the factor of coincidence of circumstances and the factor of acceleration of social evolution.

Optimization, simulation and control are very powerful tools in engineering and mathematics, and play an increasingly important role. Because of their various real-world applications in industries such as finance, economics, and telecommunications, research in these fields is accelerating at a rapid pace, and there have been major algorithmic and theoretical developments in these fields in the last decade.

This volume brings together the latest developments in these areas of research and presents applications of these results to a wide range of real-world problems.

- Collection of selected contributions giving a state-of-the-art account of recent developments in the field - Covers a broad range of topics in optimization and optimal control, including unique applications - Written by an international group of experts in their respective disciplines - Broad audience of researchers, practitioners, and advanced graduate students in applied mathematics and engineeringWe construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S|/ + U(x, t) = 0, where > 1 and the potential U(x, t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential de“ned on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O(log T)2−2/. We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton…Jacobi or Burgers equations are discussed.

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.

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.