When is the Bloch-Okounkov q-bracket modular?

We obtain a condition describing when the quasimodular forms given by the Bloch-Okounkov theorem as $q$-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator {\Delta}. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.Comment: 12 pages; corrected typo

Quantitative Results on Diophantine Equations in Many Variables

We consider a system of integer polynomials of the same degree with non-singular local zeros and in many variables. Generalising the work of Birch (1962) we find quantitative asymptotics (in terms of the maximum of the absolute value of the coefficients of these polynomials) for the number of integer zeros of this system within a growing box. Using a quantitative version of the Nullstellensatz, we obtain a quantitative strong approximation result, i.e. an upper bound on the smallest integer zero provided the system of polynomials is non-singular.Comment: Accepted for publication in Acta Arithmetica. Added a few pages so that familiarity with Birch's work is no longer assumed; 24 page

Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions

Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramification profiles. This yields generalisations of several results known for Hurwitz numbers. When the target surface is of genus one, we show that the generating series of these interpolated Hurwitz numbers are quasimodular forms. In the case that all ramification is simple, we refine this result by writing this series as a sum of quasimodular forms corresonding to tropical covers weighted by Gromov-Witten invariants. Moreover, we derive a quantum curve for monotone and Grothendieck dessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary but fixed ramification profile. Thus, we obtain spectral curves via the semiclassical limit as input data for the CEO topological recursion. Astonishingly, we find that the CEO topological recursion for the genus $1$ spectral curve of the strictly monotone Hurwitz numbers compute the monotone Hurwitz numbers in genus $0$. Thus, we give a new proof that monotone Hurwitz numbers satisfy CEO topological recursion. This points to an unknown relation between those enumerants. Finally, specializing to target surface $\mathbb{P}^1$, we find recursions for monotone and Grothendieck dessins d'enfants double Hurwitz numbers, which enables the computation of the respective Hurwitz numbers for any genera with one arbitrary but fixed ramification profile.Comment: 41 page

When is the Bloch–Okounkov q-bracket modular?

We obtain a condition describing when the quasimodular forms given by the Bloch–Okounkov theorem as q-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator Δ. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials

Hedgehogs In Lehmer's Problem

Motivated by a famous question of Lehmer about the Mahler measure we study and solve its analytic analogue.Comment: 3! page

When is the Bloch–Okounkov q-bracket modular?

We obtain a condition describing when the quasimodular forms given by the Bloch–Okounkov theorem as q-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator Δ. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials

Routinely measuring symptom burden and health-related quality of life in dialysis patients: first results from the Dutch registry of patient-reported outcome measures

Background. The use of patient-reported outcome measures (PROMs) is becoming increasingly important in healthcare. However, incorporation of PROMs into routine nephrological care is challenging. This study describes the first experience with PROMs in Dutch routine dialysis care. Methods. A pilot study was conducted in dialysis patients in 16 centres. Patients were invited to complete PROMs at baseline and 3 and 6 months. PROMs consisted of the 12-item short-form and Dialysis Symptom Index to assess health-related quality of life (HRQoL) and symptom burden. Response rates, HRQoL and symptom burden scores were analysed. Qualitative research methods were used to gain insight into patients' views on using PROMs in clinical practice. Results. In total, 512 patients (36%) completed 908 PROMs (24%) across three time points. Response rates varied from 6 to 70% among centres. Mean scores for physical and mental HRQoL were 35.6 [standard deviation (SD) 10.2] and 47.7 (SD 10.6), respectively. Patients experienced on average 10.8 (SD 6.1) symptoms with a symptom burden score of 30.7 (SD 22.0). Only 1-3% of the variation in PROM scores can be explained by differences between centres. Patients perceived discussing their HRQoL and symptom scores as insightful and valuable. Individual feedback on results was considered crucial. Conclusions. The first results show low average response rates with high variability among centres. Dialysis patients experienced a high symptom burden and poor HRQoL. Using PROMs at the individual patient level is suitable and may improve patient-professional communication and shared decision making. Further research is needed to investigate how the collection and the use of PROMs can be successfully integrated into routine care to improve healthcare quality and outcomes