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Alexander Ivanov

Publications and source records attributed to Alexander Ivanov.

36 records · Page 2Linked to original sources
arXiv2016

Steiner Ratio and Steiner-Gromov Ratio of Gromov-Hausdorff Space

In the present paper we investigate the metric space $\cal M$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that for any finite subset $M$ from a sufficiently small neighborhood of a generic finite metric space, providing $M$ consists of finite metric spaces with the same number of points, each Steiner minimal tree in $\cal M$ connecting $M$ is a minimal filling for $M$. As a consequence, we prove that the both Steiner ratio and Gromov-Steiner ratio of $\cal M$ are equal to $1/2$.

math.MG↗
arXiv2016

Local Structure of Gromov-Hausdorff Space, and Isometric Embeddings of Finite Metric Spaces into this Space

We investigate the geometry of the family $\cal M$ of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that sufficiently small neighborhoods of generic finite spaces in the subspace of all finite metric spaces with the same number of points are isometric to some neighborhoods in the space ${\mathbb R}^N_{\infty}$, i.e., in the space ${\mathbb R}^N$ with the norm $\|(x_1,\ldots,x_N)\|=\max_i|x_i|$. As a corollary, we get that each finite metric space can be isometrically embedded into $\cal M$ in such a way that its image belongs to a subspace consisting of all finite metric spaces with the same number $k$ of points. If the initial space has $n$ points, then one can take $k$ as the least possible integer with $n\le k(k-1)/2$.

math.MG↗
arXiv2016

Gromov--Hausdorff Distance, Irreducible Correspondences, Steiner Problem, and Minimal Fillings

We introduce irreducible correspondences that enables us to calculate the Gromov--Hausdorff distances effectively. By means of these correspondences, we show that the set of all metric spaces each consisting of no more than $3$ points is isometric to a polyhedral cone in the space $R^3$ endowed with the maximum norm. We prove that for any $3$-point metric space such that all the triangle inequalities are strict in it, there exists a neighborhood such that the Steiner minimal trees (in Gromov-Hausdorff space) with boundaries from this neighborhood are minimal fillings, i.e., it is impossible to decrease the lengths of these trees by isometrically embedding their boundaries into any other ambient metric space. On the other hand, we construct an example of $3$-point boundary whose points are $3$-point metric spaces such that its Steiner minimal tree in the Gromov-Hausdorff space is not a minimal filling. The latter proves that the Steiner subratio of the Gromov-Hausdorff space is less than 1. The irreducible correspondences enabled us to create a quick algorithm for calculating the Gromov-Hausdorff distance between finite metric spaces. We carried out a numerical experiment and obtained more precise upper estimate on the Steiner subratio: we have shown that it is less than $0.857$.

math.MG↗
arXiv2016

Realizations of Gromov-Hausdorff Distance

It is shown that for any two compact metric spaces there exists an "optimal" correspondence which the Gromov-Hausdorff distance is attained at. Each such correspondence generates isometric embeddings of these spaces into a compact metric space such that the Gromov-Hausdorff distance between the initial spaces is equal to the Hausdorff distance between their images. Also, the optimal correspondences could be used for constructing the shortest curves in the Gromov-Hausdorff space in exactly the same way as it was done by Alexander Ivanov, Nadezhda Nikolaeva, and Alexey Tuzhilin in arXiv:1504.03830, where it is proved that the Gromov-Hausdorff space is geodesic. Notice that all proofs in the present paper are elementary and use no more than the idea of compactness.

math.MG↗
arXiv2015

Stable sets of primes in number fields

We define a new class of sets -- stable sets -- of primes in number fields. For example, Chebotarev sets $P_{M/K}(σ)$, with $M/K$ Galois and $σ\in \Gal(M/K)$, are very often stable. These sets have positive (but arbitrary small) Dirichlet density and generalize sets with density 1 in the sense that arithmetic theorems like certain Hasse principles, the Grunwald-Wang theorem, the Riemann's existence theorem, etc. hold for them. Geometrically this allows to give examples of infinite sets $S$ with arbitrary small positive density such that $\Spec \mathcal{O}_{K,S}$ is algebraic $K(π,1)$ (for all $p$ simultaneous).

math.NT↗
arXiv2015

On a mechanical lens

In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems.

physics.class-ph↗
arXiv2015

Reconstructing decomposition subgroups in arithmetic fundamental groups using regulators

Our main goal in the present article is to explain how one can reconstruct the decomposition subgroups and norms of points on an arithmetic curve inside its fundamental group if the following data are given: the fundamental group, a part of the cyclotomic character and the family of the regulators of the fields corresponding to the generic points of all étale covers of the given curve. The approach is inspired by that of Tamagawa for curves over finite fields but uses Tsfasman-Vlăduţ theorem instead of Lefschetz trace formula. To the authors' knowledge, this is a new technique in the anabelian geometry of arithmetic curves. It is conditional and depends on still unknown properties of arithmetic fundamental groups. We also give a new approach via Iwasawa theory to the local correspondence at the boundary.

math.NT↗
arXiv2015

Analytic Deformations of Minimal Networks

A behavior of extreme networks under deformations of their boundary sets is investigated. It is shown that analyticity of a deformation of boundary set guarantees preservation of the networks types for minimal spanning trees, minimal fillings and so-called stable shortest trees in the Euclidean space.

math.DG↗
arXiv2015

Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for $GL_2$

In the present article we define coverings of affine Deligne-Lusztig varieties attached to a connected reductive group over a local field of characteristic $p > 0$. In the case of $\GL_2$, the unramified part of the local Langlands correspondence is realized in the $\ell$-adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell-Henniart. All proofs are purely local.

math.AG↗
arXiv2014

Densities of primes and realization of local extensions

In this paper we introduce new densities on the set of primes of a number field. If $K/K_0$ is a Galois extension of number fields, we associate to any element $x \in {\rm Gal}_{K/K_0}$ a density $δ_{K/K_0,x}$ on primes of $K$. In particular, the density associated to $x = 1$ is the usual Dirichlet density on $K$. After establishing some properties of these densities, we use them to show that the maximal solvable extension of a number field unramified outside an almost Chebotarev set realize the maximal local extension at each prime lying outside this set.

math.NT↗
arXiv2013

On a generalization of the Neukirch-Uchida theorem

In this paper we generalize a part of Neukirch-Uchida theorem for number fields from the birational case to the case of curves $\Spec \caO_{K,S}$ with $S$ a stable set of primes of a number field $K$. In particular, such sets can have arbitrarily small (positive) Dirichlet density. The proof consists of two parts: first one establishes a local correspondence at the boundary $S$, which works as in the original proof of Neukirch. But then, in contrast to Neukirchs proof, a direct conclusion via Chebotarev density theorem is not possible, since stable sets are in general too small, and one has to use further arguments.

math.NT↗
arXiv2013

On some anabelian properties of arithmetic curves

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicate to describe the position of decomposition groups of points at the boundary of the scheme $\Spec \caO_{K,S}$, where $K$ is a number field and $S$ a set of primes of $K$, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally with the fundamental group $π_1(\Spec \caO_{K,S})$: the location of decomposition groups of boundary points inside it, the $p$-part of the cyclotomic character, the number of points on the boundary of all finite etale covers, etc. Under certain finiteness hypothesis on Tate-Shafarevich groups with divisible coefficients, one can reconstruct all this quantities from the fundamental group alone.

math.NT↗
arXiv2012

On a finite type property of certain affine Deligne-Lusztig varieties

We give a new proof of the fact that affine Deligne-Lusztig varieties for an algebraic group of adjoint type, associated with superbasic elements, are of finite type. The proof uses a property of the associated Hecke algebra, which we reprove, and an estimate of the length of certain elements in the affine Weyl group.

math.AG↗
arXiv2009

The cohomology of the affine Deligne-Lusztig varieties in the affine flag manifold of $GL_2$

This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of GL_2. At first we determine all such varieties up to isomorphy. After this we investigate the representations of the sigma-stabilizer of an element b of the group on the etale cohomology of the affine Deligne-Lusztig variety X_w(b). We describe such representations as inductions from compact subgroups and in terms of noncuspidal representations.

math.AG↗
arXiv2006

Measurement of Optical Response of a Detuned Resonant Sideband Extraction Interferometer

We report on the optical response of a suspended-mass detuned resonant sideband extraction (RSE) interferometer with power recycling. The purpose of the detuned RSE configuration is to manipulate and optimize the optical response of the interferometer to differential displacements (induced by gravitational waves) as a function of frequency, independently of other parameters of the interferometer. The design of our interferometer results in an optical gain with two peaks: an RSE optical resonance at around 4 kHz and a radiation pressure induced optical spring at around 41 Hz. We have developed a reliable procedure for acquiring lock and establishing the desired optical configuration. In this configuration, we have measured the optical response to differential displacement and found good agreement with predictions at both resonances and all other relevant frequencies. These results build confidence in both the theory and practical implementation of the more complex optical configuration being planned for Advanced LIGO.

gr-qc↗