Related papers: Closed numerical ranges

Several upper and lower bounds for the numerical radius of $2 \times 2$ operator matrices are developed which refine and generalize the earlier related bounds. In particular, we show that if $B,C$ are bounded linear operators on a complex…

We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$ is an upper bound on the state complexity of the shuffle…

In this paper, we study the linear mapping which sends the sequence $x=(x_n)_{n \in \mathbb{N}}$ to $y=(y_n)_{n \in \mathbb{N}}$ where $y_n = \sum_{k=1}^\infty f(n/k)x_k$ for $f: \mathbb{Q}^+ \to \mathbb{C}$. This operator is the…

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved…

We show that the de Bruijn-Erd\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq…

For any truncated path algebra $\Lambda$ of a quiver, we classify, by way of representation-theoretic invariants, the irreducible components of the parametrizing varieties $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ of the $\Lambda$-modules with…

We study the number of degree $n$ number fields with discriminant bounded by $X$. In this article, we improve an upper bound due to Schmidt on the number of such fields that was previously the best known upper bound for $6 \leq n \leq 94$.

The sequence of middle divisors is shown to be unbounded. For a given number $n$, $a_{n,0}$ is the number of divisors of $n$ in between $\sqrt{n/2}$ and $\sqrt{2n}$. We explicitly construct a sequence of numbers $n(i)$ and a list of…

In this paper, we investigate the multilinear boundedness properties of the higher ($n$-th) order Calder\'on commutator for dimensions larger than two. We establish all multilinear endpoint estimates for the target space…

It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 \pi n_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k…

Let $\{\lambda_n\}_n \in \ell^\infty(\mathbb{N})$. In 1960, R. Schatten \cite{SCHATTEN} studied operators of the form $\sum_{n=1}^{\infty}\lambda_n (x_n\otimes \bar{y_n})$, where $\{x_n\}_n$, $\{y_n\}_n$ are orthonormal sequences in a…

Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade' approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) =…

For any increasing function $f: {\Bbb N} \rightarrow {\Bbb N}_{\ge 2}$ which takes only finitely many distinct values, a connected finite dimensional algebra $\Lambda$ is constructed, with the property that $\text{fin.dim}_n\, \Lambda =…

We prove prime exponential sums have no better than square root cancellation on average on short intervals, in the sense that $$\frac{1}{x} \sum_{-y< n\le x} \left|\sum_{\substack{n< m \le n+y\\ 1\le m \le x}} \Lambda(m) \mathrm{e}(\alpha…

We are interested in the asymptotics of the number of standard Young tableaux $f^{\lambda/\mu}$ of a given skew shape $\lambda/\mu$. We mainly restrict ourselves to the case where both diagrams are balanced, but investigate all growth…

In this paper, we apply our minimax theory ([4], [5], [6]) with the one developed by A. Moameni in [2] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical…

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…

We establish improved sample-complexity bounds for sample-based Lindbladian simulation based on the Wave Matrix Lindbladization (WML) algorithm. For a jump operator $L$ with dimension $d$, we derive an explicit non-asymptotic sample…

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= \lambda_n x^n + \nu_n x^{n-2}$ for $\lambda_n$ and $\nu_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = \lambda_n…

Let $A$ be an isotropic, sub-gaussian $m \times n$ matrix. We prove that the process $Z_x := \|Ax\|_2 - \sqrt m \|x\|_2$ has sub-gaussian increments. Using this, we show that for any bounded set $T \subseteq \mathbb{R}^n$, the deviation of…