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## Limit theorems for the alloy-type random energy model

In this paper, we consider limit laws for the model, which is a generalisation of the random energy model (REM) to the case when the energy levels have the mixture distribution. More precisely, the distribution of the energy levels is assumed to be a mixture of two normal distributions, one of which is standard normal, while the second has the mean \(\sqrt{n}a\) with some \(a\in \R,\) and the variance \(\sigma \ne 1\). The phase space \((a,\sigma) \subset \R \times \R_+\) is divided onto several domains, where after appropriate normalisation, the partition function converges in law to the stable distribution. These domains are separated by the critical surfaces, corresponding to transitions from the normal distribution to \(\alpha-\)stable with \(\alpha \in (1,2)\), after to 1-stable, and finally to \(\alpha-\)stable with \(\alpha \in (0,1).\) The corresponding phase diagram is the central result of this paper.

This chapter discusses in some detail the possibility of the Singularity being a product of biased human perception described by the Weber–Fechner law. It is shown that though the Weber–Fechner effect can produce series with a hyperbolic shape, the hyperbolic acceleration pattern with the twenty-first century Singularity detected in Panov and Modis–Kurzweil series is explained first of all by the actual hyperbolic acceleration of the global megaevolution.

Digital technologies provide new possibilities for studying cultural heritage. Thus, literature research involving large text corpora allows to set and solve theoretical problems which previously had no prospects for their decision. For example, it has become possible to model the literary system for some defi-nite literary period (i.e., for the Silver Age of Russian literature) and to classify prose writers according to their stylistic features. And more than that, it allows to solve more general theoretical problems. The given research was conducted on Russian literary texts of the early 20th century. The sample included 100 short stories by 100 different writers. The measurements were carried out for 5 syntactic variables. For each of these distributions, the most popular statistics were calculated. Basing on these data, we consider empirical verification of Lyapunov's central limit theorem (CLT). The article validates the effectiveness of CLT theorem and the conditions for its implementation. Besides the normal (Gaussian) function we used another analytical model — the Hausstein func-tion. It turned out that both theoretical distributions for each of five variables do not contradict the experimental data. However, the alternative analytical model (Hausstein function) has shown even better agreement with the experimental data. The obtained results may be used in computational linguistic studies and for research of Russian literary heritage.

The article studies the important (for resurrection of Russian tradition of economic analysis) problem of relationship between the ideas of N.D. Kondratieff and E.E. Slutsky during the period 1910-1930s. The problem is considered in the frames of statistical method used by both scientists. The attention is focused on two features of method's application, namely the correlation theory and interpretations of the large numbers law. Instead of widely-known summative approach to the intellectual heritage of the scientists, genetic approach is used. Historical context related with A.A. Tchouprow and his adepts - N.S. Chetverikov and О.N. Anderson - is added. The article continues the series of research submitted in the previously published collections: Kondratieff 's Suzdal Letters, 1932-1938 (2004), Kondratieff 's Conjuncture Institute: Selected works (2010) and Slutsky's Selected economic and statistical works (2010).

After the beginning of the Arab Spring in 2011, explosive global growth was observed for the majority of indicators of sociopolitical destabilization in all parts of the World System. In order to identify the structure of this destabilization wave, we apply a series of statistical techniques such as trend analysis and t-tests to study the degrees of intensification of various instability indicators (as recorded by the Cross-National Time Series database). We reveal explosive global growth in anti-government demonstrations, riots, general strikes, terrorist attacks/guerrilla warfare and purges, as well as in the global integral index of sociopolitical destabilization. On the other hand, no statistically significant growth has been detected for assassinations and major government crises, whereas for such an important indicator of global sociopolitical destabilization, as the global number of coups and coup attempts, we find a statistically significant decrease.

An improved version of the functional limit theorem is proved establishing weak convergence of random walks generated by compound doubly stochastic Poisson processes (compound Cox processes) to Lévy processes in the Skorokhod space under more realistic moment conditions. As corollaries, theorems are proved on convergence of random walks with jumps having finite variances to Lévy processes with variance-mean mixed normal distributions, in particular, to stable Lévy processes.

In this paper we consider the product of two independent random matrices X^(1) and X^(2). Assume that X_{jk}^{(q)},1\le j,k \le n,q=1,2,, are i.i.d. random variables with \EX_{jk}^{q}=0, VarX_{jk}^{(q)}=1/ Denote by s_1(W),…,s_n(W) the singular values of W:=n^{-1}X^(1)X^(2). We prove the central limit theorem for linear statistics of the squared singular values s_1^2(W),…,s_n^2(W) showing that the limiting variance depends on \kappa_4:=\E(X_{11}^{(1)})^4−3.

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.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

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.