Related papers: A note on a spectral constant associated with an a…

We establish the best (minimum) constant for Ulam stability of first-order linear $h$-difference equations with a periodic coefficient. First, we show Ulam stability and find the Ulam stability constant for a first-order linear equation…

Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi and Stein, Aharoni and Berger…

We prove that $K_G<\frac{\pi}{2\log(1+\sqrt{2})}$, where $K_G$ is the Grothendieck constant.

Consider an algebraic number field, $K$, and its ring of integers, $\mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ we can find a prime ideal, $\mathfrak{p}$, in $\mathcal{O}_K$ with norm $N(\mathfrak{p})$ in the…

In this article, we provide the spectral analysis of a Dirac-type operator on $\mathbb{Z}^2$ by describing the behavior of the spectral shift function associated with a sign-definite trace-class perturbation by a multiplication operator. We…

Given a positive real number $x$, we consider the smallest base $q_s(x)\in(1,2)$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that \[ x=\sum_{i=1}^\infty\frac{d_i}{(q_s(x))^i}. \] In this paper we give complete…

A transversal set of a graph $G$ is a set of vertices incident to all edges of $G$. The transversal number of $G$, denoted by $\tau(G)$, is the minimum cardinality of a transversal set of $G$. A simple graph $G$ with no isolated vertex is…

Let $r$, $m$ and $k\geq 2$ be positive integers such that $r\mid k$ and let $v \in \left[ 0,\lfloor \frac{k-1}{2r} \rfloor \right]$ be any integer. For any integer $\ell \in [1, k]$ and $\epsilon \in \{0,1\}$, we let…

The two-dimensional signed small ball inequality states that for all possible choices of signs, $$ \left\| \sum_{|R| = 2^{-n}}{ \varepsilon_R h_R} \right\|_{L^{\infty}} \gtrsim n,$$ where the summation runs over all dyadic rectangles in the…

We consider discrete one-dimensional Schr\"odinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to $\log$-Hausdorff measures. We apply this result to operators with Sturmian…

Suppose $\mathcal{A}$ is a compact normal operator on a Hilbert space $H$ with certain lacunarity condition on the spectrum (which means, in particular, that its eigenvalues go to zero exponentially fast), and let $\mathcal{L}$ be its rank…

This note proves that there exists positive constants $c_1$ and $c_2$ such that for all finite $A \subset \mathbb R$ with $|A+A| \leq |A|^{1+c_1}$ we have $|AAA| \gg |A|^{2+c_2}$.

This paper studies the \(k^{th}-\)order slant Toeplitz and slant little Hankel operators on the weighted Bergman space \(\mathcal{A}_\alpha^2(\mathbb{D})\). These operators are constructed using a slant shift operator \(W_k\) composed with…

In this paper, we prove the relation $\frac{r_{A}(T) + r_{A}(T^{\diamond}) + |r_{A}(T^{\diamond}) - r_{A}(T)|}{2} = \sup \{ |\lambda|: \lambda \in \sigma_{A}(T)\}$, where $A$ is a positive semidefinite operator (not necessarily to have a…

Let $r$ be an integer with $r\ge 2$ and $G$ be a connected $r$-uniform hypergraph with $m$ edges. By refining the broken cycle theorem for hypergraphs, we show that if $k>\frac{m-1}{\ln(1+\sqrt{2})}\approx 1.135 (m-1)$ then the $k$-list…

The Erd\"os-Hajnal conjecture states that for every graph $H$, there exists a constant $\delta(H) > 0$ such that every graph $G$ with no induced subgraph isomorphic to $H$ has either a clique or a stable set of size at least…

Given a real $n \times m$ matrix $B$, its operator norm can be defined as $$|B|=\max_{|v|=1}|Bv|.$$ We consider a matrix "small" if it has non-negative integer entries and its operator norm is less than $2$. These matrices correspond to…

For each $r\ge 4$, we show that any graph $G$ with minimum degree at least $(1-1/100r)|G|$ has a fractional $K_r$-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional…

Each bounded operator T on an infinite dimensional Hilbert space H is a sum of three operators that are similar to positive operators; two such operators are sufficient if T is not a compact perturbation of a scalar. The spectra of L\"uders…

In this paper, we introduce the notion of $k^{th}$-order slant little Hankel operator on the Bergman space with essentially bounded harmonic symbols on the unit disc and obtain its algebraic and spectral properties. We have also discussed…