Naming the largest number: Exploring the boundary between mathematics and the philosophy of mathematics

What is the largest number accessible to the human imagination? The question is neither entirely mathematical nor entirely philosophical. Mathematical formulations of the problem fall into two classes: those that fail to fully capture the spirit of the problem, and those that turn it back into a philosophical problem

Extrinsic Diophantine approximation on manifolds and fractals

Fix $d\in\mathbb N$, and let $S\subseteq\mathbb R^d$ be either a real-analytic manifold or the limit set of an iterated function system (for example, $S$ could be the Cantor set or the von Koch snowflake). An $extrinsic$ Diophantine approximation to a point $\mathbf x\in S$ is a rational point $\mathbf p/q$ close to $\mathbf x$ which lies $outside$ of $S$. These approximations correspond to a question asked by K. Mahler ('84) regarding the Cantor set. Our main result is an extrinsic analogue of Dirichlet's theorem. Specifically, we prove that if $S$ does not contain a line segment, then for every $\mathbf x\in S\setminus\mathbb Q^d$, there exists $C > 0$ such that infinitely many vectors $\mathbf p/q\in \mathbb Q^d\setminus S$ satisfy $\|\mathbf x - \mathbf p/q\| < C/q^{(d + 1)/d}$. As this formula agrees with Dirichlet's theorem in $\mathbb R^d$ up to a multiplicative constant, one concludes that the set of rational approximants to points in $S$ which lie outside of $S$ is large. Furthermore, we deduce extrinsic analogues of the Jarn\'ik--Schmidt and Khinchin theorems from known results

The Importance of Screening for, and managing, Gestational Diabetes Mellitus in Malta

The detection and management of gestational diabetes mellitus (GDM) has been a source of controversy for many years. Evidence has now accumulated that dietary and insulin therapy are effective and reduce the risk of macrosomia and Caesarean section. Studies are underway to assess the impact of screening and of the different diagnostic criteria for GDM. However, studies to date have reported only an impact on obstetric, neonatal and fetal outcomes. It is now possible to prevent or at least delay the onset of maternal Type 2 diabetes, and interventions targeting women with a history of GDM are likely to have a substantive impact on the current diabetes epidemic. An even greater impact may result from preventing excessive intra-uterine exposure to hyperglycaemia, increasingly implicated as a cause of obesity and diabetes in the offspring of women with past GDM. Developing and implementing approaches to preventing long term risks to mother and baby across populations will take many years and possibly decades. In the meantime, all women should be screened for GDM so that the need for long term follow up, and, where possible, intervention for mother and baby can be identified. Such action requires knowledge of the diagnosis not only by the health care team but also the woman herself.peer-reviewe

Unconventional height functions in simultaneous Diophantine approximation

Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and $H(\mathbf r)$. Here $H(\mathbf r)$ is the so-called "standard height" of the rational point $\mathbf r$. In this paper the authors ask: What changes if we replace the standard height function by a different one? As it turns out, this change leads to dramatic differences from the classical theory and requires the development of new methods. We discuss three examples of nonstandard height functions, computing their exponents of irrationality as well as giving more precise results. A list of open questions is also given

A high-fidelity N-body ephemeris generator for satellites in Earth orbit

A program is currently used for mission planning called the Analytic Satellite Ephemeris Program (ASEP), which produces projected data for orbits that remain fairly close to Earth. Lunar and solar perturbations are taken into account in another program called GRAVE. This project is a revision of GRAVE which incorporates more flexible means of input for initial data, provides additional kinds of output information, and makes use of structured programming techniques to make the program more understandable and reliable. The computer program ORBIT was tested against tracking data for the first 313 days of operation of the CRRES satellite. A sample graph is given comparing the semi-major axis calculated by the program with the values supplied by NORAD. When calculated for points at which CRRES passes through the ascending node, the argument of perigee, the right ascension of the ascending node, and the mean anomaly all stay within about a degree of the corresponding values from NORAD; the inclination of the orbital plane is much closer. The program value of the eccentricity is in error by no more than 0.0002

Carving Out the Space of 4D CFTs

We introduce a new numerical algorithm based on semidefinite programming to efficiently compute bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. Using our algorithm, we dramatically improve previous bounds on a number of CFT quantities, particularly for theories with global symmetries. In the case of SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal technicolor. In N=1 superconformal theories, we place strong bounds on dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive anomalous dimensions in this region. We also place novel upper and lower bounds on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we find examples of lower bounds on central charges and flavor current two-point functions that scale with the size of global symmetry representations. In the case of N=1 theories with an SU(N) flavor symmetry, our bounds on current two-point functions lie within an O(1) factor of the values realized in supersymmetric QCD in the conformal window.Comment: 60 pages, 22 figure