arXiv subjects

Michael J. Miller

Publications and source records attributed to Michael J. Miller.

4 recordsLinked to original sources

arXiv2025

Seeking a quadratic refinement of Sendov's conjecture

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $β$ is one of those roots, then within one unit of $β$ lies a root of the polynomial's derivative. If we define $r(β)$ to be the greatest possible distance between $β$ and the closest root of the derivative, then Sendov's conjecture claims that $r(β) \le 1$. In this paper, we conjecture that there is a constant $c>0$ so that $r(β) \le 1-cβ(1-β)$ for all $β\in [0,1]$. We find such constants for complex polynomials of degree $2$ and $3$, for real polynomials of degree $4$, for all polynomials whose roots lie on a line, for all polynomials with exactly one distinct critical point, and when $β$ is sufficiently close to $1$. In addition, we show that experimental data suggests that $c\approx0.233$.

math.CV↗

arXiv2021

Overview of Cherenkov Telescope on-board EUSO-SPB2 for the Detection of Very-High-Energy Neutrinos

We present the status of the development of a Cherenkov telescope to be flown on a long-duration balloon flight, the Extreme Universe Space Observatory Super Pressure Balloon 2 (EUSO-SPB2). EUSO-SPB2 is an approved NASA balloon mission that is planned to fly in 2023 and is a precursor of the Probe of Extreme Multi-Messenger Astrophysics (POEMMA), a candidate for an Astrophysics probe-class mission. The purpose of the Cherenkov telescope on-board EUSOSPB2 is to classify known and unknown sources of backgrounds for future space-based neutrino detectors. Furthermore, we will use the Earth-skimming technique to search for Very-High-Energy (VHE) tau neutrinos below the limb (E > 10 PeV) and observe air showers from cosmic rays above the limb. The 0.785 m^2 Cherenkov telescope is equipped with a 512-pixel SiPM camera covering a 12.8° x 6.4° (Horizontal x Vertical) field of view. The camera signals are digitized with a 100 MS/s readout system. In this paper, we discuss the status of the telescope development, the camera integration, and simulation studies of the camera response.

astro-ph.IM↗

arXiv2010

On minimal Rolle's domains for complex polynomials

Define a subset of the complex plane to be a Rolle's domain if it contains (at least) one critical point of every complex polynomial P such that P(-1)=P(1). Define a Rolle's domain to be minimal if no proper subset is a Rolle's domain. In this paper, we investigate minimal Rolle's domains.

math.CV↗

arXiv2007

Unexpected local extrema for the Sendov conjecture

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P) to be the maximum of the distances from each of the roots of a polynomial P to that root's nearest critical point. In this notation, Sendov's conjecture asserts that d(P)<=1 for every P in S(n). Define P in S(n) to be locally extremal if d(P)>=d(Q) for all nearby Q in S(n), and note that maximizing d(P) over all locally extremal polynomials P would settle the Sendov conjecture. Prior to now, the only polynomials known to be locally extremal were of the form P(z)=c(z^n+a) with |a|=1. In this paper, we determine sufficient conditions for real polynomials of a different form to be locally extremal, and we use these conditions to find locally extremal polynomials of this form of degrees 8, 9, 12, 13, 14, 15, 19, 20, and 26.

math.CV↗