The Error Term in the Sato-Tate Conjecture

Let $f(z)=\sum_{n=1}^\infty a(n)e^{2\pi i nz}\in S_k^{new}(\Gamma_0(N))$ be a newform of even weight $k\geq2$ that does not have complex multiplication. Then $a(n)\in\mathbb{R}$ for all $n$, so for any prime $p$, there exists $\theta_p\in[0,\pi]$ such that $a(p)=2p^{(k-1)/2}\cos(\theta_p)$. Let $\pi(x)=\#\{p\leq x\}$. For a given subinterval $I\subset[0,\pi]$, the now-proven Sato-Tate Conjecture tells us that as $x\to\infty$, $$\#\{p\leq x:\theta_p\in I\}\sim \mu_{ST}(I)\pi(x),\quad \mu_{ST}(I)=\int_{I} \frac{2}{\pi}\sin^2(\theta)~d\theta.$$ Let $\epsilon>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\to\infty$, $$\#\{p\leq x:\theta_p\in I\}=\mu_{ST}(I)\pi(x)+O\left(\frac{x}{(\log x)^{9/8-\epsilon}}\right),$$ where the implied constant is effectively computable and depends only on $k,N,$ and $\epsilon$.Comment: 9 page

The Message is the Medium: Electronically Helping Writing Tutors Help Electronically

The history of online writing centers is a history of doubt. I experienced those reservations in 2009, when, in addition to traditional face-to-face peer tutoring, I launched my own online peer tutoring program and began training undergraduates to respond to student submissions. Online writing centers were already common, but the decision the begin tutoring online was not all mine—the university administration was encouraging faculty to create online and web-assisted courses, and it expected its academic support keep up with the pace of technology, distance learning, and even fears that a future pandemic could hinder face to face learning. After consulting with tutors and instructional technology staff, I decided on asynchronous peer tutoring: students would fill out an intake form and questionnaire about their assignment and writing process, and then they would upload what they had written; tutors would then respond via email within 24 hours, even on weekends. This system allowed us to help as many students as quickly as possible, particularly non-traditional, commuting, and working students unable to meet face to face.University Writing Cente

Non-governmental organizations and multi-sited marine conservation science: A case study

Non-governmental organizations (NGOs) are now major players in the realm of environmental conservation. While many environmental NGOs started as national organizations focused around single-species protection, governmental advocacy, and preservation of wilderness, the largest now produce applied conservation science and work with national and international stakeholders to develop conservation solutions that work in tandem with local aspirations. Marine managed areas (MMAs) are increasingly being used as a tool to manage anthropogenic stressors on marine resources and protect marine biodiversity. However, the science of MMA is far from complete. Conservation International (CI) is concluding a 5 year, $12.5 million dollar Marine Management Area Science (MMAS) initiative. There are 45 scientific projects recently completed, with four main “nodes” of research and conservation work: Panama, Fiji, Brazil, and Belize. Research projects have included MMA ecological monitoring, socioeconomic monitoring, cultural roles monitoring, economic valuation studies, and others. MMAS has the goals of conducting marine management area research, building local capacity, and using the results of the research to promote marine conservation policy outcomes at project sites. How science is translated into policy action is a major area of interest for science and technology scholars (Cash and Clark 2001; Haas 2004; Jasanoff et al. 2002). For science to move policy there must be work across “boundaries” (Jasanoff 1987). Boundaries are defined as the “socially constructed and negotiated borders between science and policy, between disciplines, across nations, and across multiple levels” (Cash et al. 2001). Working across the science-policy boundary requires boundary organizations (Guston 1999) with accountability to both sides of the boundary, among other attributes. (Guston 1999; Clark et al. 2002). This paper provides a unique case study illustrating how there are clear advantages to collaborative science. Through the MMAS initiative, CI built accountability into both sides of the science-policy boundary primarily through having scientific projects fed through strong in-country partners and being folded into the work of ongoing conservation processes. This collaborative, boundary-spanning approach led to many advantages, including cost sharing, increased local responsiveness and input, better local capacity building, and laying a foundation for future conservation outcomes. As such, MMAS can provide strong lessons for other organizations planning to get involved in multi-site conservation science. (PDF contains 3 pages

On discretely entropy conservative and entropy stable discontinuous Galerkin methods

High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEs. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matrices. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions

Separated at Birth: Jet Maximization, Axis Minimization, and Stable Cone Finding

Jet finding is a type of optimization problem, where hadrons from a high-energy collision event are grouped into jets based on a clustering criterion. As three interesting examples, one can form a jet cluster that (1) optimizes the overall jet four-vector, (2) optimizes the jet axis, or (3) aligns the jet axis with the jet four-vector. In this paper, we show that these three approaches to jet finding, despite being philosophically quite different, can be regarded as descendants of a mother optimization problem. For the special case of finding a single cone jet of fixed opening angle, the three approaches are genuinely identical when defined appropriately, and the result is a stable cone jet with the largest value of a quantity J. This relationship is only approximate for cone jets in the rapidity-azimuth plane, as used at the Large Hadron Collider, though the differences are mild for small radius jets.Comment: 7 pages, 2 tables; v2: references added; v3: small clarifications and table 2 added to match journal versio