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## Compact non-contraction semigrous of affine operators

We analyze compact multiplicative semigroups of affine operators acting in a finite-dimensional space. The main result states that every such semigroup is either contracting, that is, contains elements of arbitrarily small operator norm, or all its operators share a common invariant affine subspace on which this semigroup is contracting. The proof uses functional difference equations with contraction of the argument. We look at applications to self-affine partitions of convex sets, the investigation of finite affine semigroups and the proof of a criterion of primitivity for nonnegative matrix families.

This paper examines partition as a solution to ethnic civil wars and modifies the ethnic security dilemma, suggesting that strong state institutions are more important than demographically separating ethnic groups to achieve an enduring peace. The paper starts with a puzzle: if ethnic separation is required for peace, how do some partitions that leave minorities behind maintain peace? The paper compares post-partition Georgia–Abkhazia, which experienced violence renewal within five years of the partition, with post-partition Moldova–Transnistria, which maintained peace. Both countries had ‘stay-behind’ ethnic minorities. The paper also disaggregates and compares the territories within post-partition Abkhazia, which contain ethnic Georgians: Lower Gali experienced violence while neighboring Upper Gali did not. The paper argues that state institutions create an incentive for ethnic minorities to collaborate with the state, regardless of minority preferences, and this helps maintain peace. However, preferences become important where institutions are weak and members of the ethnic minority have the opportunity to defect; this increases the likelihood of violence. The results build on the ethnic security dilemma by specifying micro-mechanisms and challenging the theory's reliance on intransigent ethnic identities in explaining the causes of post-partition violence.

In this study we use the crossing tree of a signal for the purpose of analysis of H-sssi processes. The crossing tree performs an ad-hoc decomposition of a signal adapted to its dynamics, and represents a natural tool for the analysis of its local fluctuations. We present here a new multifractal formalism and a novel approach for estimating the spectrum of singularities of H-sssi processes using the crossing-tree. The performance of the crossing-tree based method is demonstrated in a numerical study. Its performance is also compared with state-of-the-art techniques based on wavelets, including wavelet-leaders.

A series of bilinear identities on the Schur symmetric functions is obtained with the use of Plucker relations.

We present a new class of multifractal process on R, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change. In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step n, we can generate step n+1 in O(log n) operations. Detailed pseudo-code for this algorithm is provided.R

This paper the experimental research of dynamic characteristics self-similar scales of measurement financial time lines and quality check statistical, econometrics and intellectual methods of their analysis and forecasting is described. Research was carried out on 25 various financial time lines, including on lines of the prices of actions of the Russian and foreign companies, the prices for gold, oil, indexes of the MMVB, S&P, exchange rates, etc. The Analysis of these lines has confirmed presence of the common laws in change of structure of lines depending on scale.

In this paper, we consider the model of server traffic when the traffic is separated into several streams. The amount of transferred data differs for different streams. Based on real traffic measurements we proposed the server traffic model where traffic of every stream is described by the same independent processes, but each process has its own time scale. We show that for traffic analysis as well as for developing of the most effective methods of control of this traffic, it is necessary to correctly identify the time scale for each stream, as well as the time scale of traffic fluctuations those have a significant effect to QoS.

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.