Effective Bounds for the Andrews spt-function

In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function $\mathrm{spt}(n)$. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function $p(n)$ and $\mathrm{spt}(n)$. Further, we strengthen one of the conjectures, and prove that for every $\epsilon>0$ there is an effectively computable constant $N(\epsilon) > 0$ such that for all $n\geq N(\epsilon)$, we have \begin{equation*} \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}{\pi}+\epsilon\right) \sqrt{n}\,p(n). \end{equation*} Due to the conditional convergence of the Rademacher-type formula for $\mathrm{spt}(n)$, we must employ methods which are completely different from those used by Lehmer to give effective error bounds for $p(n)$. Instead, our approach relies on the fact that $p(n)$ and $\mathrm{spt}(n)$ can be expressed as traces of singular moduli.Comment: Changed the title. Added more details and simplified some arguments in Section

Crustal deformation in northwestern Arabia from GPS measurements in Syria: Slow slip rate along the northern Dead Sea Fault

New Global Positioning System (GPS) measurements in NW Syria provide the first direct observations of near-field deformation associated with the northern Dead Sea fault system (DSFS) and demonstrate that the kinematics of the northern section of this transform plate boundary between the Arabian and Sinai plates deviate significantly from plate model predictions. Velocity estimates based on GPS survey campaigns in 2000, 2007 and 2008, demonstrate left-lateral shear along the northern DSFS with 1σ uncertainties less than 0.7 mm yr-1. These velocities are consistent with an elastic dislocation model with a slip rate of 1.8-3.3 mm yr-1 and a locking depth of 5-16 km. This geodetically determined slip rate is about half of that reported farther south along the central section (Lebanese restraining bend) and the southern section (Jordan Valley and Wadi Araba) of the transform and consequently requires some deformation to occur away from the transform along other geological structures. The factor of two difference in slip rates along the transform is also consistent with differing estimates of total fault slip that have occurred since the mid Miocene: 20-25 km along the northern DSFS (in NW Syria) versus about 45 km along the southern DSFS segment. Some of the strain deficit may be accommodated by north-south shortening within the southwestern segment of the Palmyride fold belt of central Syria. Additionally, a distinct change in velocity occurs within the Sinai plate itself. These new GPS measurements, when viewed alongside the palaeoseismic record and the modest level of present-day seismicity, suggest that the reported estimates of recurrence time of large earthquakes (M > 7) along the northern section of the DSFS may be underestimated owing to temporal clustering of such large historical earthquakes. Hence, a revised estimate of the earthquake hazard may be needed for NW Syria

Effects of hospital facilities on patient outcomes after cancer surgery: an international, prospective, observational study

Background Early death after cancer surgery is higher in low-income and middle-income countries (LMICs) compared with in high-income countries, yet the impact of facility characteristics on early postoperative outcomes is unknown. The aim of this study was to examine the association between hospital infrastructure, resource availability, and processes on early outcomes after cancer surgery worldwide.Methods A multimethods analysis was performed as part of the GlobalSurg 3 study-a multicentre, international, prospective cohort study of patients who had surgery for breast, colorectal, or gastric cancer. The primary outcomes were 30-day mortality and 30-day major complication rates. Potentially beneficial hospital facilities were identified by variable selection to select those associated with 30-day mortality. Adjusted outcomes were determined using generalised estimating equations to account for patient characteristics and country-income group, with population stratification by hospital.Findings Between April 1, 2018, and April 23, 2019, facility-level data were collected for 9685 patients across 238 hospitals in 66 countries (91 hospitals in 20 high-income countries; 57 hospitals in 19 upper-middle-income countries; and 90 hospitals in 27 low-income to lower-middle-income countries). The availability of five hospital facilities was inversely associated with mortality: ultrasound, CT scanner, critical care unit, opioid analgesia, and oncologist. After adjustment for case-mix and country income group, hospitals with three or fewer of these facilities (62 hospitals, 1294 patients) had higher mortality compared with those with four or five (adjusted odds ratio [OR] 3.85 [95% CI 2.58-5.75]; p&lt;0.0001), with excess mortality predominantly explained by a limited capacity to rescue following the development of major complications (63.0% vs 82.7%; OR 0.35 [0.23-0.53]; p&lt;0.0001). Across LMICs, improvements in hospital facilities would prevent one to three deaths for every 100 patients undergoing surgery for cancer.Interpretation Hospitals with higher levels of infrastructure and resources have better outcomes after cancer surgery, independent of country income. Without urgent strengthening of hospital infrastructure and resources, the reductions in cancer-associated mortality associated with improved access will not be realised

Some applications of classical modular forms to number theory

In this thesis we use classical modular forms to study several problems in number theory. In chapter 2 we use non-holomorphic Eisenstein series for the Hilbert modular group to obtain a formula for the relative class number of certain abelian extensions of CM number fields. In chapter 3 we compute the scattering determinant for the Hilbert modular group, and explain how this can be used to prove that the subspace of cuspidal, square integrable eigenfunctions for the Laplacian on products of rank one symmetric spaces is infinite dimensional. In chapter 4 we use zeta functions of quadratic forms over number fields to sharpen a certain constant appearing in C. L. Siegel’s lower bound for the residue of the Dedekind zeta function at s = 1.Mathematic