Jensen polynomials for the Riemann zeta function and other sequences

In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at its point of symmetry. This hyperbolicity has been proved for degrees $d\leq 3$. We obtain an asymptotic formula for the central derivatives $\zeta^{(2n)}(1/2)$ that is accurate to all orders, which allows us to prove the hyperbolicity of a density $1$ subset of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all $d\leq 8$. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the GUE random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.Comment: 11 page

Proof of the Umbral Moonshine Conjecture

The Umbral Moonshine Conjectures assert that there are infinite-dimensional graded modules, for prescribed finite groups, whose McKay-Thompson series are certain distinguished mock modular forms. Gannon has proved this for the special case involving the largest sporadic simple Mathieu group. Here we establish the existence of the umbral moonshine modules in the remaining 22 cases.Comment: 56 pages, to appear in Research in the Mathematical Science

On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements

In the context of operator valued W*-free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as $vx$ where $x$ is self-adjoint and $v$ is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that $v$ and $x$ are *-free from each other. In particular, we prove, when B=C^2, that if a $B$-valued circular element has a free bipolar decomposition with $v$ unitary, then it has one where $v$ normalizes $B$

What Works and What Doesn’t When Teaching Large Classes?

What we want to do is talk in this 20 Minute Mentor, share some ideas about what you can do actually during the semester. Once it’s begun, you’re over the shock of having been assigned a large classroom, if it was a surprise, and you’re actually into the semester. Kenneth L. Alford, Ph.D., is an Associate Professor of Church History and Doctrine at Brigham Young University. After serving almost 30 years on active duty in the U.S. Army, he retired as a Colonel in 2008. While on active duty, Ken served in numerous personnel, automation, acquisition, and education assignments, including eight years teaching computer science and information technology at the United States Military Academy at West Point, New York and four years as Professor and Department Chair of the Strategic Leadership Department at the National Defense University in Washington, DC. Tyler J. Griffin, Ph.D., is an assistant professor at Brigham Young University. With degrees in Electrical Engineering and Instructional Technology, combined with 18 years of professional teaching experience, Tyler has three major focal points in his work: (1) Best practices for teaching & learning (2) Best uses of technology to increase the scope and scale of learning, and (3) best practices for teacher development/inservice.https://knowledge.e.southern.edu/onlineseminars/1031/thumbnail.jp

Using the Medical Research Council framework for the development and evaluation of complex interventions in a theory-based infant feeding intervention to prevent childhood obesity:The baby milk intervention and trial

Introduction. We describe our experience of using the Medical Research Council framework on complex interventions to guide the development and evaluation of an intervention to prevent obesity by modifying infant feeding behaviours. Methods. We reviewed the epidemiological evidence on early life risk factors for obesity and interventions to prevent obesity in this age group. The review suggested prevention of excess weight gain in bottle-fed babies and appropriate weaning as intervention targets; hence we undertook systematic reviews to further our understanding of these behaviours. We chose theory and behaviour change techniques that demonstrated evidence of effectiveness in altering dietary behaviours. We subsequently developed intervention materials and evaluation tools and conducted qualitative studies with mothers (intervention recipients) and healthcare professionals (intervention deliverers) to refine them. We developed a questionnaire to assess maternal attitudes and feeding practices to understand the mechanism of any intervention effects. Conclusions. In addition to informing development of our specific intervention and evaluation materials, use of the Medical Research Council framework has helped to build a generalisable evidence base for early life nutritional interventions. However, the process is resource intensive and prolonged, and this should be taken into account by public health research funders. This trial is registered with ISRTCN: 20814693 Baby Milk Trial

Heights of points on elliptic curves over Q\mathbb Q

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $$\Psi_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$\widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D).$$Comment: The revision includes a new remark on the proportion of curves for which Proposition 1.3 applie