Related papers: On the containment problem
This is a brief introduction to control theory in finite-dimensional spaces. The material is partly based on my lectures for the Master 1 program in Math\'ematiques et applications at Sorbonne University, delivered over the past few years.…
Compared with constraint satisfaction problems, counting problems have received less attention. In this paper, we survey research works on the problems of counting the number of solutions to constraints. The constraints may take various…
This is a pedagogical introduction to the main concepts of the sterile neutrino - a hypothetical particle, coined to resolve some anomalies in neutrino data and retain consistency with observed widths of the $W$ and $Z$ bosons. We briefly…
Several complementary approaches to investigate knotted solutions of Maxwell's equations in vacuum are now available in literature. However, only partial results towards a unified description of them have been achieved. This is potentially…
We point out that the horizon problem encountered in standard text-books or review papers on cosmology is, in general, derived for world models based on Robertson-Walker line element where homogeneity and isotropy of the universe -- \`a la…
We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and…
The purpose of this note is to attach a name to a natural class of combinatorial problems and to point out that this class includes many important special cases. We also show that a simple problem of placing nonoverlapping labels on a…
Various theoretical obstacles are associated with a homogeneous and isotropic distribution of ``charge'' which is subject to a repulsive, long-range force. We show how these can be overcome, for all practical purposes, by the simple device…
An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…
Advantages of inhomogeneous cosmological models that are exact solutions of Einstein's equations over linearised perturbations of homogeneous models are presented. Examples of effects that can be described in the inhomogeneous ones are…
We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving…
We continue the study of the rich family of norm-closed, automorphism invariant ideals of a continuous nest algebra. First we present a unified framework which captures all stable ideals as the kernels of limits of diagonal compressions. We…
In this paper, we focus on the associated primes of powers of monomial ideals and asymptotic behavior properties such as normally torsion-freeness, normality, the strong persistence property, and the persistence property. In particular, we…
The universe we observe is homogeneous on super-horizon scales, leading to the ``cosmic homogeneity problem''. Inflation alleviates this problem but cannot solve it within the realm of conservative extrapolations of classical physics. A…
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms…
We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the…
Generic inhomogeneous steady states in an asymmetric exclusion process on a ring with a pair of point bottlenecks are studied. We show that, due to an underlying universal feature, measurements of coarse-grained steady-state densities in…
By an easy application of Skoda's theorem on ideal generation, a non-local version of the Briancon-Skoda theorem is obtained. In particular, the symbolic powers $I^{(p)}$ of a zero dimensional radical ideal $I$ generated by $r$ holomorphic…
We discuss several aspects of cosmic censorship hypothesis. There is evidence both in favor and against the hypothesis. On one hand one can prove that cosmic censorship holds in several special cases and on the other hand there is a number…
One point compactification is studied in the light of ideal of subsets of $\mathbb{N}$. $\mathcal{I}$-proper map is introduced and showed that a continuous map can be extended continuously to the one point $\mathcal{I}$-compactification if…