Related papers: Mityagin's Extension Problem. Progress Report
An algebraic extension K of the rationals has the Bogomolov property if the absolute logarithmic height of non-torsion points of K* is bounded away from 0. We define a relative extension L/K to be Bogomolov if this holds for points of L\K.…
Let $\mathfrak{L}$ be a Leibniz algebra, $E$ a vector space and $\pi : E \to \mathfrak{L}$ an epimorphism of vector spaces with $ \mathfrak{g} = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Leibniz…
We consider the problem of constructing a weakly-continuous mapping extending continuous mapping defined on a dense set of a topological space to the entire space. Theorem on necessary and sufficient conditions for the existence of such an…
We prove that for $d\ge 2,\, k\ge 2$, if the Hausdorff dimension of a compact set $E\subset \mathbb{R}^d$ is greater than $\frac{d^2}{2d-1}$, then, for any given $r > 0$, there exist $(x^1, \dots, x^{k+1})\in E^{k+1}$, $(y^1, \dots,…
Let $\mathfrak{g}$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g}$ as a subspace. The paper is devoted to the \emph{extending structures problem} which asks for the classification of all Lie algebra structures on $E$ such…
The extension problem asks whether positive semi-definite functions on a symmetric unital subset of a discrete group can be extended to positive semi-definite functions on the whole group. It has been known at least since the work of Rudin…
In this paper, we prove the existence of a bounded linear extension operator $T: L^{2,p}(E) \rightarrow L^{2,p}(\mathbb{R}^2)$ when $1<p<2$, where $E \subset \mathbb{R}^2$ is a certain discrete set with fractal structure. Our proof makes…
Based on a result of Makeev, in 2012 Blagojevi\'c and Karasev proposed the following problem: given any positive integers $m$ and $1\leq \ell\leq k$, find the minimum dimension $d=\Delta(m;\ell/k)$ such that for any $m$ mass distributions…
We prove non-extendability results for Lipschitz maps with target space being jet spaces equipped with a left-invariant Riemannian distance, as well as jet spaces equipped with a left-invariant sub-Riemannian Carnot-Caratheodory distance.…
We obtain a minimal supersymmetric extension of the Snyder algebra and study its representations. The construction differs from the general approach given in Hatsuda and Siegel ({\tt hep-th/0311002}), and does not utilize super-de Sitter…
We give a wedge removability theorem for metrically thin sets of two codimensional Hausdorff null measure. This removability theorem combined with the wedge removability theorem of Merker for closed subsets of two codimensional manifolds,…
In this work, firstly in the Hilbert space of vector-functions L^2 (H,(-\infty,a)\bup(b,+\infty)),a<b all selfadjoint extensions of the minimal operator generated by linear singular symmetric differential expression l(\cdot)=i d/dt+A with a…
We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both…
Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed…
The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory Series B 1985), is an indispensable tool for converting the combinatorial condition of graph expansion to an algebraic condition on the eigenvalues of the graph…
We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive…
A Banach space is said to be Grothendieck if weak and weak$^*$ convergent sequences in the dual space coincide. This notion has been quantificated by H. Bendov\'{a}. She has proved that $\ell_\infty$ has the quantitative Grothendieck…
It is known that if a compact set $E$ in $\mathbb{R}^d$ has Hausdorff dimension greater than $(d+1)/2$, then its $n$-chain distance set $$\Delta^n(E) = \left\{\left(\left|x^1-x^2\right|,\cdots, \left|x^{n}- x^{n+1}\right|\right)\in…
We revisit the non-perturbative renormalization of a class of simple polaron models with resting fermions. The considered dispersion relations and form factors are allowed to be highly singular, such that infinite self-energies and wave…
We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set…