arXiv subjects

Michael J. Mossinghoff

Publications and source records attributed to Michael J. Mossinghoff.

24 records · Page 2Linked to original sources

arXiv2014

Most Reinhardt polygons are sporadic

A \textit{Reinhardt polygon} is a convex $n$-gon that, for $n$ not a power of $2$, is optimal in three different geometric optimization problems, for example, it has maximal perimeter relative to its diameter. Some such polygons exhibit a particular periodic structure; others are termed \textit{sporadic}. Prior work has described the periodic case completely, and has shown that sporadic Reinhardt polygons occur for all $n$ of the form $n=pqr$ with $p$ and $q$ distinct odd primes and $r\geq2$. We show that (dihedral equivalence classes of) sporadic Reinhardt polygons outnumber the periodic ones for almost all $n$, and find that this first occurs at $n=105$. We also determine a formula for the number of sporadic Reinhardt polygons when $n=2pq$ with $p$ and $q$ distinct odd primes.

math.MG↗

arXiv2014

Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function

We prove that the Riemann zeta-function $ζ(σ+ it)$ has no zeros in the region $σ\geq 1 - 1/(5.573412 \log|t|)$ for $|t|\geq 2$. This represents the largest known zero-free region within the critical strip for $3.06\cdot10^{10} < |t|<\exp(10151.5)$. Our improvements result from determining some favorable trigonometric polynomials having particular properties, and from analyzing the error term in the method of Kadiri. We also improve an upper bound in a question of Landau regarding nonnegative trigonometric polynomials.

math.NT↗

arXiv2013

Wieferich pairs and Barker sequences, II

We show that if a Barker sequence of length $n>13$ exists, then either $n=3979201339721749133016171583224100$, or $n > 4\cdot10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly 2000. We also obtain 18 additional integers $n<10^{50}$ that cannot be ruled out as the length of a Barker sequence, and find more than 237000 additional candidates $n<10^{100}$. These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on $n$, to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.

math.CO↗

arXiv2012

Sporadic Reinhardt polygons

Let $n$ be a positive integer, not a power of two. A \textit{Reinhardt polygon} is a convex $n$-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all $n$, there are many Reinhardt polygons with $n$ sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of $n$, additional Reinhardt polygons exist that do not possess this structured form. We call these polygons \textit{sporadic}. We completely characterize the integers $n$ for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when $n=pqr$ with $p$ and $q$ distinct odd primes and $r\geq2$. We also prove that a positive proportion of the Reinhardt polygons with $n$ sides are sporadic for almost all integers $n$, and we investigate the precise number of sporadic Reinhardt polygons that are produced for several values of $n$ by a construction that we introduce.

math.MG↗

arXiv2008

Slices, slabs, and sections of the unit hypercube

Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Polya's Ph.D. thesis, and we discuss several applications of these formulas.

math.MG↗

arXiv2005

Generalizations of Goncalves' inequality

If $F$ is a polynomial with complex coefficients, leading term $a_N$, and roots $α_1$, ..., $α_N$, then Gonçalves' inequality states that $\|F\|_2^2$ is bounded below by $\abs{a_N}^2 (\prod_{n=1}^N \max\{1, \abs{α_n}^2\} + \prod_{n=1}^N \min\{1, \abs{α_n}^2\})$. We establish generalizations of this inequality for other $L_p$ norms, and derive additional lower bounds on the $L_p$ norms of a polynomial in terms of its coefficients.

math.CA↗