Related papers: Simultaneous packing and covering in sequence spac…
The ball number of a link $L$, denoted by $ball(L)$, is the minimum number of solid balls (not necessarily of the same size) needed to realize a necklace representing $L$. In this paper, we show that $ball(L)\leq 5 cr(L)$ where $cr(L)$…
In the classic maximum coverage problem, we are given subsets $T_1, \dots, T_m$ of a universe $[n]$ along with an integer $k$ and the objective is to find a subset $S \subseteq [m]$ of size $k$ that maximizes $C(S) := |\cup_{i \in S} T_i|$.…
Finding nonoverlapping balls with given centers in any metric space, maximizing the sum of radii of the balls, can be expressed as a linear program. Its dual linear program expresses the problem of finding a minimum-weight set of cycles…
We obtain exact for order estimates of best uniform approximations and uniform approximations by Fourier sums of classes of convolutions the periodic functions belong to unit balls of spaces $L_{p}, \ {1\leq p<\infty}$, with generating…
In this paper, we provide an efficient method for computing the Taylor coefficients of $1-p_n f$, where $p_n$ denotes the optimal polynomial approximant of degree $n$ to $1/f$ in a Hilbert space $H^2_\omega$ of analytic functions over the…
We study some sequences of functions of one real variable and conjecture that they converge uniformly to functions with certain positivity and growth properties. Our conjectures imply a conjecture of Cohn and Elkies, which in turn implies…
$E$-optimal experimental designs for a second-order response surface model with $k\geq1$ predictors are investigated. If the design space is the $k$-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1 (1977a) 121-132]…
We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is…
This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify…
We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$…
In this paper we study the following problems: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects…
We give a short linear--algebraic proof of the inequality \[ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2, \] valid for every \(x\in\mathbb{R}^p\). This inequality relates three fundamental norms on finite-dimensional spaces…
The following theorem is the main result of this note. Theorem 1. Let $(E, \|\cdot\|_E) $ be a rearrangement invariant Banach function space on the interval $[0, 1]$. If $E$ is isometric to $\L_p [0, 1]$ for some $1\le p<\infty$, then $E$…
This paper determines the optimal upper bound for the simultaneous packing and covering constants of the two-dimensional centrally symmetric convex domains. It solved a problem opening for more than thirty years.
We study the extreme $L_p$ discrepancy of infinite sequences in the $d$-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star $L_p$ discrepancy, which uses…
We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent…
Motivated by the construction of confidence intervals in statistics, we study optimal configurations of $2^d-1$ lines in real projective space $RP^{d-1}$. For small $d$, we determine line sets that numerically minimize a wide variety of…
We study the problem of Reinforcement Learning (RL) with linear function approximation, i.e. assuming the optimal action-value function is linear in a known $d$-dimensional feature mapping. Unfortunately, however, based on only this…
We present a short, direct proof of the uniform convexity of L^p spaces for 1<p<\infty.
The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff's inflations…