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Working paper

The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebras, Siegel modular forms and affine Lie algebras.

Added: Mar 3, 2015

Working paper

We list all finite abelian groups which act effectively on smooth cubic fourfolds.

Added: Nov 19, 2013

Working paper

We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.

Added: Apr 12, 2012

Working paper

We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.

Added: Oct 16, 2015

Working paper

We present a new combinatorial formula for Hall-Littlewood functions associated with the affine root system of type A~n−1, i.e. corresponding to the affine Lie algebra slˆn. Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation.Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion's theorem and then apply it to our polyhedron to prove the formula.

Added: Aug 7, 2015

Working paper

We find a criterion for an effective divisor D on a smooth surface to be left-orthogonal or strongly left-orthogonal (i.e. for the pair of line bundles (O;O(D)) to be exceptional or strong exceptional).

Added: Oct 15, 2016

Working paper

Let M be a G2-manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.

Added: Nov 2, 2012

Working paper

For slow–fast quantum systems, we compute first corrections to the quantum action and to the effective slow Hamiltonian.

Added: Apr 9, 2014

Working paper

We consider tilings of the plane by two prototiles which are right triangles. They are called the small and the large tiles. The small tile is similar to the large tile with some similarity coefficient ψ. The large tile can be cut into two pieces so that one piece is a small tile and the other one is similar to the small tile with the same similarity coefficient ψ. Using this cut we define in a standard way the substitution scheme, in which the large tile is replaced by a large and a small tile and the small tile is replaced by a large tile. To every substitution of this kind, there corresponds a family of the so-called substitution tilings of the plane in the sense of [C. Goodman-Strauss, Matching Rules and Substitution Tilings, Annals of Mathematics 147 (1998) 181-223]. All tilings in this family are non-periodic. It was shown in the paper [N. Vereshchagin. Aperiodic Tilings by Right Triangles. In: Proc. of DCFS 2014, LNCS vol. 8614 (2014) 29--41] that this family of substitution tilings is not an SFT. This means that looking at a given tiling trough a bounded window, we cannot determine whether that tiling belongs to the family or not, however large the size of the window is. In the present paper, we prove that this family of substitution tilings is sofic. This means that we can color the prototiles ina finite number of colors and define some local rules for colored prototiles so that the following holds. For any tiling from the family, we can color its tiles so that the resulting tiling (by colored tiles) satisfies local rules. And conversely, for any tiling of the plane satisfying the local rules, by removing colors we obtain a tiling from the family. Besides, the considered substitution can be generalized to colored tiles so that the family of substitution tilings for the resulting substitution coincides with the family of tilings satisfying our local rules.

Added: Jul 24, 2020

Working paper

We show that affine cones over smooth cubic surfaces do not admit non-trivial
$\mathbb{G}_a$ -actions.

Added: Dec 27, 2013

Working paper

A non-degenerate two-dimentional linear operator Ф transforms the unit circle into ellipse.We define the coefficient of deformation k(Ф), as the relation of the lendth of the smaller ellipses axis to its bigger one. In this work we compute the mean value of k(Ф).Analogously, we define the deformation coefficient k(Ф) in three-dimensional case and give an estimation of its mean value.

Added: Jun 28, 2017

Working paper

The notion of a (metric) modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by the author [V.V. Chistyakov, Metric modulars and their application, Dokl. Math. 73 (1) (2006) 32–35, and Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (1) (2010) 1–14]. In this paper we establish a fixed point theorem for contractive maps in modular spaces. It is related to contracting rather “generalized average velocities” than metric distances, and the successive approximations of fixed points converge to the fixed points in a weaker sense as compared to the metric convergence.

Added: Feb 6, 2013

Working paper

The gamma kernels are a family of projection kernels K(z,z′)=K(z,z′)(x,y) on a doubly infinite 1-dimensional lattice. They are expressed through Euler's gamma function and depend on two continuous parameters z,z′. The gamma kernels initially arose from a model of random partitions via a limit transition. On the other hand, these kernels are closely related to unitarizable representations of the Lie algebra 𝔰𝔲(1,1). Every gamma kernel K(z,z′) serves as a correlation kernel for a determinantal measure M(z,z′), which lives on the space of infinite point configurations on the lattice.
We examine chains of kernels of the form
…,K(z−1,z′−1),K(z,z′),K(z+1,z′+1),…,
and establish the following hierarchical relations inside any such chain:
Given (z,z′), the kernel K(z,z′) is a one-dimensional perturbation of (a twisting of) the kernel K(z+1,z′+1), and the one-point Palm distributions for the measure M(z,z′) are absolutely continuous with respect to M(z+1,z′+1).
We also explicitly compute the corresponding Radon-Nikodým derivatives and show that they are given by certain normalized multiplicative functionals.

Added: May 25, 2019

Working paper

We prove a version of the Aleksandrov-Fenchel inequality for mixed volumes of coconvex bodies. This version is motivated by an inequality from commutative algebra relating intersection multiplicities of ideals.

Added: Oct 6, 2013

Working paper

In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with a spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of Theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer such natural topological questions as whether a given singular solution is stable or not. In the present paper, the problem of stability for equilibrium points is considered, and it is shown that this problem can also be approached by means of algebraic geometry.

Added: Nov 19, 2013

Working paper

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) equals to that of a family which is open in X.

Added: Jul 24, 2017

Working paper

It is proved that any strictly exceptional collection generating the derived category of coherent sheaves on a smooth projective variety X with \rk K_0(X) = \dim X + 1 constists of locally free sheaves up to a common shift.

Added: Feb 23, 2014

Working paper

Durand B., Shen A.,

arxiv.org. math. Cornell University, 2012. No. 2896.
We describe all Ammann tilings of a plane, a half-plane and a quadrant. In our description, every tiling is associated to an infinite sequence of two letters. We provide simple criteria of (a) whether a tiling associated with a sequence tiles the entire plane, a half-plane or a quadrant and (b) whether tiling associated with two sequences are congruent. It is well known that all Ammann tilings are aperiodic; we show how one can use this fact to construct an aperiodic 2-dimensional sub-shift of finite type

Added: Dec 11, 2013

Working paper

Let K be a field and G be a group of its automorphisms. If K is algebraic over the subfield KG fixed by G then, according to Hilbert's Theorem 90, any smooth (i.e. with open stabilizers) K-semilinear representation of the group G is isomorphic to a direct sum of copies of K. If K is not algebraic over KG then there exist non-semisimple smooth semilinear representations of G over K, so Hilbert's Theorem 90 does not hold. Let now G be the group of all permutations of an infinite set S acting naturally on the field k(S) freely generated over a subfield k by the set S. The goal of this note is to present three examples of G-invariant subfields K\subseteq k(S) such that the smooth K-semilinear representations of G of {\sl finite length} admit an explicit description, close to Hilbert's Theorem 90. Namely, (i) if K=k(S) then any smooth K-semilinear representation of G of finite length is isomorphic to a direct sum of copies of K, (ii) if K\subset k(S) is the subfield of rational homogeneous functions of degree 0 then any smooth K-semilinear representation of G of finite length splits into a direct sum of one-dimensional K-semilinear representations of G, (iii) if K\subset k(S) is the subfield generated over k by x-y for all x,y\in S then there is a unique isomorphism class of indecomposable smooth K-semilinear representations of G of each given finite length.

Added: Nov 16, 2015

Working paper

The paper puts forth an axiomatic description of the complexity of an object of sociological investigation. The proposed axioms allow us to determine complexity within the framework of mathematical sociology such as the variational principle, which is formed relative to the state of the object of sociological investigation. On the basis of this principle we can conclude the equation of state, which coincides with the stationary forward Kolmogorov equation. Based on the results of an empirical study of Russian doctorate holders, a scientific capital value was determined for each respondent by calculating an empirical distribution function for each respondent’s active properties. The goal of this study is to develop a phenomenological theory of Scientific Capital in the form of a hierarchy of variational principles. On the micro level, the principle of the maximum enables us to examine the numerous meanings of Scientific Capital based on the measurement of the actual (i.e., factually realized) distribution of active properties of a scientific field’s agent among the multitude of possible distributions. On the macro level, the principle of the minimum energy functional describes the distribution of Scientific Capital among the agents of a scientific field.

Added: Jun 13, 2013

Working paper

This is the first part of a two parts paper dedicated to global bifurcations in the plane. In this part we construct an open set of three parameter families whose topological classification has a numerical invariant that may take an arbitrary positive value. In the second part we construct an open set of six parameter families whose topological classification has a functional invariant. Any germ of a monotonically increasing function may be realized as this invariant. Here "families" are "families of vector fields in the two-sphere".

Added: Jun 24, 2015