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Working paper

Let X be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold X′, deformation equivalent to X, which is not an analytification of any projective variety, if and only if H^0(X, Ω^2)≠0. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat Kähler manifolds with vanishing first Betti number.

Added: Aug 19, 2020

Working paper

Honoré I.,

math. arxive. Cornell University, 2016. No. 1605.08525.
We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure ν of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f -- ν(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr{\"o}benius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.

Added: Oct 21, 2016

Working paper

Abstract. We propose the new construction of complex surfaces with h^1,0 = h^2,0 = 0 from smoothings of normal crossing surfaces with non- collapsible dual complexes and carry it out for the simplest case of the duncehat complex, obtaining the surface with h^1,1 = 9 (presumably Barlow surface).

Added: Nov 1, 2019

Working paper

Chamorro D.,

math. arxive. Cornell University, 2016. No. 1610.05537.
Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{\"o}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful.

Added: Oct 21, 2016

Working paper

We present a sequent calculus for the Grzegorczyk modal logic Grz allowing cyclic and other non-well-founded proofs and obtain the cut-elimination theorem for it by constructing a continuous cut-elimination mapping acting on these proofs. As an application, we establish the Lyndon interpolation property for the logic Grz proof-theoretically.

Added: Apr 4, 2018

Working paper

In the previous papers we present a construction of the set U_SBS in the direct product B_S×PΓ(M, L) of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle L→M over a given compact simply connected symplectic manifold M. Canonical projections p:U_SBS→PΓ(M, L) and q:U_SBS→ B_S are studied in the present text: first, we show that the differential dp at a given point is an isomorphism, which implies that a natural complex structure can be defined on U_SBS; second, the projection q:U_SBS→ B_S splits as the combination U_SBS→ TB_S→ B_S such that the fibers of the first map are complex subsets in U_SBS. This implies that an appropriate section of the first map should define a complex structure on TB_S; therefore it can be seen as a complexification of the moduli space B_S. The construction can be exploited in the Lagrangian approach to Geometric Quantization.

Added: Oct 15, 2018

Working paper

The algebra of big zeta values we introduce in this paper is an intermediate object between multiple zeta values and periods of the multiple zeta motive. It consists of number series generalizing multiple zeta values, the simplest examples, which are not multiple zeta series, are Tornheim sums. We show that convergent big zeta values are periods of the moduli space of stable curves of genus zero on one hand and multiple zeta values on the other hand. It gives an alternative way to prove that any such period may be expressed as a rational linear combination of multiple zeta values and a simple algorithm for finding such an expression.

Added: Nov 11, 2020

Working paper

In a very recent paper Sidorenko stated the following problem: Let Gk be a graph whose vertices are functions f : Zk → Zk. A pair of vertices {f, g} forms an edge in Gk if f − g is a bijection. Lemma 2 restates the fact that Gk has no triangles when k is even. For odd k, the problem of counting triangles in Gk has been solved asymptotically in [1]. Let p(k) be the smallest prime factor of k. The p(k) functions f0, f1, . . . , fp(k)−1, where fi(j) := i· j mod k, form a complete subgraph in Gk. It is very tempting to conjecture that p(k) is indeed the size of the largest clique in Gk. We know that this is true for even k and for prime k. Computer search confirms that this is also true for k = 9. It turns out that there is a counterexample for k = 15.

Added: Oct 21, 2019

Working paper

In [3] L. Zapponi studied the arithmetic of plane bipartite trees with prime number of edges. He obtained a lower bound on the degree of tree's definition field. Here we obtain a similar lower bound in the following case. There exists a prime p such, that: a) the number of edges is divisible by p, but not by p2; b) for any proper subset of white (black) vertices the sum of their degrees is not divisible by this p.

Added: Nov 10, 2017

Working paper

We show that every reductive subgroup of the automorphism group of a quasi-smooth well formed weighted complete intersection is a restriction of a subgroup in the automorphism group in the ambient weighted projective space. Also, we provide examples demonstrating that an automorphism group of a quasi-smooth well formed Fano weighted complete intersection may be infinite and even non-reductive.

Added: Aug 19, 2020

Working paper

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The alternative technique is applicable to other types of PDEs including parabolic and time-dependent Schr\"{o}dinger ones. The schemes are implicit and three-point in each spatial direction and time and include a scheme with a splitting operator for $n\geq 2$. For $n=1$ and the mesh on characteristics, the 4th order scheme becomes explicit and close to an exact four-point scheme. We present a conditional stability theorem covering the cases of stability in strong and weak energy norms with respect to both initial functions and free term in the equation. Its corollary ensures the 4th order error bound in the case of smooth solutions to the IBVP. The main schemes are generalized for non-uniform rectangular meshes. We also give results of numerical experiments showing the sensitive dependence of the error orders in three norms on the weak smoothness order of the initial functions and free term and essential advantages over the 2nd approximation order schemes in the non-smooth case as well.

Added: Dec 1, 2020

Working paper

We study contraction of points on ℙ1(ℚ¯) with certain control on local ramification indices, with application to the unramified curve correspondences problem initiated by Bogomolov and Tschinkel.

Added: Jul 14, 2016

Working paper

Added: Oct 9, 2017

Working paper

In this paper we present a family of values of the parameters of the third Painlevé equation such that Puiseux series formally satisfying this equation -- considered as series of z^{2/3} -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them in sectors with the vertices at infinity.

Added: Feb 21, 2017

Working paper

In work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with 2n vertices is C2n ∙ Cn+1, where Ck is Catalan number. In this work a marked directed edge is not demanded, i.e. we enumerate tree-rooted planar cubic maps. Formulas are more complex, of course, but not significantly.

Added: Mar 14, 2017

Working paper

We study family of dynamical systems on 2-torus modeling over-damped Josephson junction in superconductivity. It depends on three parameters (B,A;ω): B (abscissa), A(ordinate), ω (a fixed frequency).We study the rotation numberρ(B,A;ω) as a function of (B,A) withfixedω. Aphase-lock areais the level set Lr:={ρ=r}, if it has an on-empty interior. This holds for r∈Z (a result by V.M.Buchstaber, O.V.Karpov and S.I.Tertychnyi). It is known that each phase-lock area is an infinite garland of domains going to infinity in the vertical direction and separated by points called constrictions (expect for the separation points with A= 0). We show that all the constrictions in Lr lie in its axis {B=ωr}, confirming an experimental fact (conjecture) observed numerically by S.I.Tertychnyi, V.A.Kleptsyn, D.A.Filimonov, I.V.Schurov. We prove that each constriction is positive: the phase-lock area germ contains the vertical line germ (confirming another conjecture). To do this, we study family of linear systems on the Riemann sphere equivalently describing the model: the Josephson type systems.We study their Jimbo isomonodromic deformations described by solutions of Painleve 3 equations. Using results of this study and a Riemann–Hilbert approach, we show that each constriction can be analytically deformed to constrictions with the same l:=Bω and arbitrarily small ω. Then non-existence of ”ghost” constrictions (nonpositive or with ρ not equal to l) with a given l for small ω is proved by slow-fast methods.

Added: Nov 26, 2020

Working paper

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as subalgebras of this algebra.

Added: Dec 3, 2018

Working paper

Added: Dec 3, 2018

Working paper

We prove that a quasi-smooth Fano threefold hypersurface is birationally rigid if and only if it has Fano index one.

Added: Aug 19, 2020

Working paper

We construct counterexamples to lifting properties of Hamiltonian and contact isotopies

Added: Dec 7, 2016

Working paper

We show that the monodromy group acting on $H^1(\cdot,\mathbb Z)$ of a smooth
hyperplane section of a del Pezzo surface over $\mathbb C$ is the entire
group $\mathrm{SL}_2(\mathbb Z)$. For smooth surfaces with $b_1=0$ and hyperplane section
of genus $g>2$, there exist examples in which a similar assertion is
false. Actually, if hyperplane sections of a smooth surface are
hyperelliptic curves of genus $g\ge3$, then the monodromy group
acting on the integer $H^1$ on hyperplane sections is a proper
subgroup of $\mathrm{Sp}_{2g}(\mathbb Z)$.

Added: Jun 14, 2017