Splitting Behavior of SnS_n-Polynomials

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z$ evaluated at the primes p in S. We study properties of these measures. We deduce that there exist degree n extensions of the rationals with Galois closure having Galois group $S_n$ with a given finite set of primes S having given Artin symbols, with some restrictions on allowed Artin symbols for p<n. We compare the distributions of these measures with distributions formulated by Bhargava for splitting probabilities for a fixed prime $p$ in such degree $n$ extensions ordered by size of discriminant, conditioned to be relatively prime to $p$.Comment: 33 pages, v2 34 pages, introduction revise

Non-conservation of dimension in divergence-free solutions of passive and active scalar systems

For any $h\in(1,2]$, we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in $\mathbb{R}^2$ that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension $h$. These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory. We then construct active scalar systems in $\mathbb{R}^2$ and $\mathbb{R}^3$ with measure-valued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.Comment: 32 pages, 3 figures. This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of Record of this article is published in Arch Rational Mech Anal, and is available online at https://doi.org/10.1007/s00205-021-01708-

More than Dollars for Scholars: The Impact of the Dell Scholars Program on College Access, Persistence and Degree Attainment

Although college enrollment rates have increased substantially over the last several decades, socioeconomic inequalities in college completion have actually widened over time. A critical question, therefore, is how to support low-income and first-generation students to succeed in college after they matriculate. We investigate the impact of the Dell Scholars Program which provides a combination of generous financial support and individualized advising to scholarship recipients before and throughout their postsecondary enrollment. The program's design is motivated by a theory of action that, in order to meaningfully increase the share of lower-income students who earn a college degree, it is necessary both to address financial constraints students face and to provide ongoing support for the academic, cultural and other challenges that students experience during their college careers. We isolate the unique impact of the program on college completion by capitalizing on an arbitrary cutoff in the program's algorithmic selection process. Using a regression discontinuity design, we find that although being named a Dell Scholar has no impact on initial college enrollment or early college persistence, scholars at the margin of eligibility are significantly more likely to earn a bachelor's degree on-time or six years after high school graduation. These impacts are sizeable and represent a nearly 25 percent or greater increase in both four- and six-year bachelor's attainment. The program is resource intensive. Yet, back-of-theenvelope calculations indicate that the Dell Scholars Program has a positive rate of return

Cardiac Memory-induced T-wave Inversions

Introduction: Cardiac memory refers to T-wave inversions that result when normal ventricular activation resumes following a period of abnormal ventricular activation.Case Report: We present a case of a 29-year-old man with a pacemaker who presented with new, deep symmetric T-wave inversions caused by cardiac memory.Discussion: Abnormal ventricular activation is most commonly induced by ventricular pacing but can also occur in the setting of transient left bundle branch blocks, ventricular tachycardia, and intermittent ventricular pre-excitation.Conclusion: Recognition of this phenomenon may help to reduce unnecessary admissions, cardiac testing, and cardiac catheterizations