For every algebraically closed field k of characteristic different from 2, we prove the following: (1) Finite-dimensional (not necessarily associative) k-algebras of general type of a fixed dimension, considered up to isomorphism, are parametrized by the values of a tuple of algebraically independent (over k) rational functions of the structure constants. (2) There exists an “algebraic normal form” to which the set of structure constants of every such algebra can be uniquely transformed by means of passing to its new basis—namely, there are two finite systems of nonconstant polynomials on the space of structure constants, {f_i}_i∈I and {b_j}_j∈J , such that the ideal generated by the set {f_i}_i∈I is prime and, for every tuple c of structure constants satisfying the property b_j(c) = 0 for all j ∈ J, there exists a unique new basis of this algebra in which the tuple c' of its structure constants satisfies the property f_i(c') = 0 for all i ∈ I.

The main result of this note shows that Palm distributions of the determinantal point process governed by the Bessel kernel with parameter s are equivalent to the determinantal point process governed by the Bessel kernel with parameter s+2. The Radon-Nikodym derivative is explicitly computed as a multiplicative functional on the space of configurations.

We discuss applications of generating functions for colored graphs to asymptotic expansions of matrix integrals. The described technique provides an asymptotic expansion of the Kontsevich integral. We prove that this expansion is a refinement of the Kontsevich expansion, which is the sum over the set of classes of isomorphic ribbon graphs. This yields a proof of Kontsevich’s results that is independent of the Feynman graph technique.

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in sense of Ehresmann and Reeb.

In our previous papers (2002, 2017), we derived conditions for the existence of a strong Nash equilibrium in multistage nonzero-sum games under additional constraints on the possible deviations of coalitions from their agreed-upon strategies. These constraints allowed only one-time simultaneous deviations of all the players in a coalition. However, it is clear that in real-world problems the deviations of different members of a coalition may occur at different times (at different stages of the game), which makes the punishment strategy approach proposed by the authors earlier inapplicable in the general case. The fundamental difficulty is that in the general case the players who must punish the deviating coalition know neither the members of this coalition nor the times when each player performs the deviation. In this paper, we propose a new punishment strategy, which does not require the full information about the deviating coalition but uses only the fact of deviation of at least one player of the coalition. Of course, this punishment strategy can be realized only under some additional constraints on stage games. Under these additional constraints, it was proved that the punishment of the deviating coalition can be effectively realized. As a result, the existence of a strong Nash equilibrium in the game was established.

A new strongly time-consistent (dynamically stable) optimality principle is proposed in a cooperative differential game. This is done by constructing a special subset of the core of the game. It is proposed to consider this subset as a new optimality principle. The construction is based on the introduction of a function V^ that dominates the values of the classical characteristic function in coalitions. Suppose that V (S, x¯ (τ), T −τ) is the value of the classical characteristic function computed in the subgame with initial conditions x¯ (τ), T −τ on the cooperative trajectory. Define V^(S;X0,T−t0)=maxt0≤τ≤TV(S;x∗(τ),T−τ)V(N;X∗(τ),T−τ)V(N;x0,T−t0) Using this function, we construct an analog of the classical core. It is proved that the constructed core is a subset of the classical core; thus, we can consider it as a new optimality principle. It is also proved that the newly constructed optimality principle is strongly time-consistent.

The classical parametric and semiparametric Bernstein-von Mises (BvM) results are reconsidered in a nonclassical setup allowing finite samples and model misspecification. In the parametric case and in the case of a finite-dimensional nuisance parameter, we establish an upper bound on the error of Gaussian approximation of the posterior distribution of the target parameter; the bound depends explicitly on the dimension of the full and target parameters and on the sample size. This helps to identify the so-called *critical dimension p* *n* of the full parameter for which the BvM result is applicable. In the important special i.i.d. case, we show that the condition “*p^*3 /*n* is small” is sufficient for the BvM result to be valid under general assumptions on the model. We also provide an example of a model with the phase transition effect: the statement of the BvM theorem fails when the dimension *p* approaches *n^*{1/3}.

We prove the following: (1) the existence, for every integer *n* ≥ 4, of a noncompact
smooth *n*-dimensional topological manifold whose diffeomorphism group contains an isomorphic
copy of every finitely presented group; (2) a finiteness theorem for finite simple subgroups of
diffeomorphism groups of compact smooth topological manifolds.

We study the question of realisability of iterated higher Whitehead products with a given form of nested brackets by simplicial complexes, using the notion of the moment–angle complex $\mathcal{Z_K}$. Namely, we say that a simplicial complex $\mathcal{K}$ realises an iterated higher Whitehead product *w* if *w*is a nontrivial element of $\pi_*(\mathcal{Z_K})$. The combinatorial approach to the question of realisability uses the operation of substitution of simplicial complexes: for any iterated higher Whitehead product $w$ we describe a simplicial complex $\partial\Delta_w$ that realises $w$. Furthermore, for a particular form of brackets inside w, we prove that $\partial\Delta_w$ is the smallest complex that realises $w$. We also give a combinatorial criterion for the nontriviality of the product $w$. In the proof of nontriviality we use the Hurewicz image of $w$ in the cellular chains of $\mathcal{Z_K}$ and the description of the cohomology product of $\mathcal{Z_K}$. The second approach is algebraic: we use the coalgebraic versions of the Koszul and Taylor complexes for the face coalgebra of $\mathcal{K}$ to describe the canonical cycles corresponding to iterated higher Whitehead products $w$. This gives another criterion for realisability of $w$.

We study interpolation properties of provability logics. We prove the Lyndon interpolation for GL and the uniform interpolation for GLP.

This review is devoted to the domains of holomorphy invariant under holomorphic actions of real Lie groups. We have collected here the results on this subject obtained during the last twenty years, which have passed since the publication of the first review of the authors on this topic. This first review was mainly devoted to the case of compact transformation groups, while the first two sections of the present review deal mostly with noncompact groups. In Section 3 we discuss the problem of rigidity of automorphism groups of domains of holomorphy invariant under compact transformation groups.

We consider approximations of an arbitrarymap *F*: *X* → *Y* between Banach spaces *X* and *Y* by an affine operator *A*: *X* → *Y* in the Lipschitz metric: the difference *F* — *A* has to be Lipschitz continuous with a small constant *ɛ* > 0. In the case *Y* = ℝ we show that if *F* can be affinely *ɛ*-approximated on any straight line in *X*, then it can be globally 2*ɛ*-approximated by an affine operator on *X*. The constant 2*ɛ* is sharp. Generalizations of this result to arbitrary dual Banach spaces *Y* are proved, and optimality of the conditions is shown in examples. As a corollary we obtain a solution to the problem stated by Zs. Páles in 2008. The relation of our results to the Ulam-Hyers-Rassias stability of the Cauchy type equations is discussed.

In the introductory part of this survey, we briefly discuss the problems of nonequilibrium statistical physics that arise in the study of energy transport in solids as well as the results available at the moment. In the main part of the survey, we explain, compare, and generalize results obtained in our previous works. We study the dynamics and energy transport in Hamiltonian systems of particles where each particle is weakly perturbed by the interaction with its own stochastic Langevin thermostat. Such systems can be regarded as models of solids that interact weakly with a continuum.

For every pair (G, V ) where G is a connected simple linear algebraic group and V is a simple algebraic G-module with a free algebra of invariants, the number of irreducible components of the nullcone of unstable vectors in V is found.

Conditional Kolmogorov complexity of a word $a$ given a word $b$ is the minimum length of a program that prints $a$ given $b$ as an input. We generalize this notion to quadruples of strings $a,b,c,d$: their joint conditional complexity $\K((a\to c)\land(b\to d))$ is defined as the minimum length of a program that given $a$ outputs $c$ and given $b$ outputs $d$. In this paper, we prove that the joint conditional complexity cannot be expressed in terms of usual conditional (and unconditional) Kolmogorov complexity. This result provides a negative answer to the following question, asked by A.Shen on a session of Kolmogorov seminar at Moscow State University in 1994: Is there a problem of information processing whose complexity is not expressible in terms of conditional (and unconditional) Kolmogorov complexity? We show that a similar result holds for classical Shannon entropy. We provide two proofs of both results, an effective one and a ``quasi-effective'' one. Finally we present a quasi-effective proof of a strong version of the following statement: there are two strings whose mutual information cannot be extracted. Previously, only a non-effective proof of that statement was known. The results concerning Kolmogorov complexity appeared, in a preliminary form, in the Proceedings of the 16th Annual IEEE Conference on Computational Complexity in 2001. [A. Muchnik and N. Vereshchagin. ``Logical operations and Kolmogorov complexity. II''. Proc. of 16th Annual IEEE Conference on Computational Complexity, Chicago, June 2001, pp. 256--265.]