Related papers: Closed numerical ranges

Two sets $A, B \subseteq \{0, 1\}^n$ form a Uniquely Decodable Code Pair (UDCP) if every pair $a \in A$, $b \in B$ yields a distinct sum $a+b$, where the addition is over $\mathbb{Z}^n$. We show that every UDCP $A, B$, with $|A| =…

Given a rational number $x$ and a bound $\varepsilon$, we exhibit $m,n$ such that $|x-12 s(m,n)|<\varepsilon$. Here $s(m,n)$ is the classical Dedekind sum and the parameters $m$ and $n$ are completely explicit in terms of $x$ and…

We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower…

We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is attained for n = 1,5,9. Furthermore, we prove that certain pairs of mutually…

We develop a new simple approach to prove upper bounds for generalizations of the Heilbronn's triangle problem in higher dimensions. Among other things, we show the following: for fixed $d \ge 1$, any subset of $[0, 1]^d$ of size $n$…

We consider Riemannian metrics compatible with the natural symplectic structure on T^2 x M, where T^2 is a symplectic 2-Torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its…

We show that for large enough $n$, the number of non-isomorphic pseudoline arrangements of order $n$ is greater than $2^{c\cdot n^2}$ for some constant $c > 0.2604$, improving the previous best bound of $c>0.2083$ by Dumitrescu and Mandal…

We study the problem of obtaining asymptotic formulas for the sums $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$ and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$, where $\Lambda$ is the von Mangoldt function, $d_k$ is the $k^{\operatorname{th}}$…

We present a new approach to the problem of enumerating permutations of length n that avoid a fixed consecutive pattern of length m. We use this idea to give explicit upper and lower bounds on the number of permutations avoiding a pattern…

We study the asymptotic behavior of a bounded solution of an inhomogeneous delay linear difference equation in a Banach space by using the spectrum of bounded sequences. We get a significant extension of excellent results in [1]. A new…

We study approximability of regular constraint satisfaction problems, i.e., CSPs where each variable in an instance has the same number of occurrences. In particular, we show that for any CSP $\Lambda$, existence of an $\alpha$…

Let $\beta_1,...,\beta_n$ be distinct points in the open unit disc in the complex plane, none of which is the origin, and let $H^1$ be the Hardy space. Define a closed convex set in $\mathbb{C}^{n}$ by $\Lambda = \{…

Let $A$ be a commutative and unital $\mathbb{R}$-algebra, and $M$ be an Archimedean quadratic module of $A$. We define a submultiplicative seminorm $\|\cdot\|_M$ on $A$, associated with $M$. We show that the closure of $M$ with respect to…

We investigate the boundedness of solutions of the first order linear difference equation of the form $x_{n+1} = Ax_{n} + y_{n}, \; n \geq 1$ where $A$ is a square matrix with complex entries, sequence $\{y_{n}\}_{n\geq 1}$ and initial…

For an $n\times n$ complex matrix $C$, the $C$-numerical range of a bounded linear operator $T$ acting on a Hilbert space of dimension at least $n$ is the set of complex numbers ${\rm tr}(CX^*TX)$, where $X$ is a partial isometry satisfying…

We show that the equation $\lambda_1 n_1^2 + ... + \lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\log N)^{-c_s}$, provided that $s \geq 7$ and the coefficients $\lambda_i$ sum to zero and satisfy…

Consider the $2n$-by-$2n$ matrix $M=(m_{i,j})_{i,j=1}^{2n}$ with $m_{i,j} = 1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1| \leq 2n$ and $m_{i,j} = 0$ for all other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n \geq…

The simplex method for linear programming is known to be highly efficient in practice, and understanding its performance from a theoretical perspective is an active research topic. The framework of smoothed analysis, first introduced by…

Let $d \geq 1$ and $s \leq 2^d$ be nonnegative integers. For a subset $A$ of vertices of the hypercube $Q_n$ and $n\geq d$, let $\lambda(n,d,s,A)$ denote the fraction of subcubes $Q_d$ of $Q_n$ that contain exactly $s$ vertices of $A$. Let…

For a positive integer $n$ and a real number $\xi$, let $\lambda_n (\xi)$ denote the supremum of the real numbers $\lambda$ for which there are arbitrarily large positive integers $q$ such that $|| q \xi ||, || q \xi^2 ||, \ldots , ||q…