Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the class $C^2([0,1])$ and study the minimal number n(\e) of function evaluations or queries that are necessary to compute an \e-approximation of the smallest eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic) worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$ matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in $n$ is an open issue.Comment: 33 page

Initial Conditions, European Colonialism and Africa's Growth

We investigate the role of initial conditions at colonial independence on economic growth in Africa in the post-independence period using Bayesian Model Averaging (BMA). A key innovation in our estimation methodology is that we incorporate parameter heterogeneity in model averaging as well as try to mitigate the endogeneity problem present in growth regressions. In order to ensure that differences in the growth determinants between Africa and the world are not driven by experiences of an alternative group of countries, we also control for the presence of OECD countries and former European colonies in the global sample. We find that the impact of different initial conditions on growth in Africa is strikingly different from the world. We argue that these initial conditions reflect the state of development at the close of the colonial era and are therefore inherently related with the legacy of colonialism.

Average Case Tractability of Non-homogeneous Tensor Product Problems

We study d-variate approximation problems in the average case setting with respect to a zero-mean Gaussian measure. Our interest is focused on measures having a structure of non-homogeneous linear tensor product, where covariance kernel is a product of univariate kernels. We consider the normalized average error of algorithms that use finitely many evaluations of arbitrary linear functionals. The information complexity is defined as the minimal number n(h,d) of such evaluations for error in the d-variate case to be at most h. The growth of n(h,d) as a function of h^{-1} and d depends on the eigenvalues of the covariance operator and determines whether a problem is tractable or not. Four types of tractability are studied and for each of them we find the necessary and sufficient conditions in terms of the eigenvalues of univariate kernels. We illustrate our results by considering approximation problems related to the product of Korobov kernels characterized by a weights g_k and smoothnesses r_k. We assume that weights are non-increasing and smoothness parameters are non-decreasing. Furthermore they may be related, for instance g_k=g(r_k) for some non-increasing function g. In particular, we show that approximation problem is strongly polynomially tractable, i.e., n(h,d)\le C h^{-p} for all d and 0<h<1, where C and p are independent of h and d, iff liminf |ln g_k|/ln k >1. For other types of tractability we also show necessary and sufficient conditions in terms of the sequences g_k and r_k

On the dc Magnetization, Spontaneous Vortex State and Specific Heat in the superconducting state of the weakly ferromagnetic superconductor RuSr2_{2}GdCu2_{2}O8_{8}

Magnetic-field changes $<$ 0.2 Oe over the scan length in magnetometers that necessitate sample movement are enough to create artifacts in the dc magnetization measurements of the weakly ferromagnetic superconductor RuSr$_{2}$GdCu$_{2}$O$_{8}$ (Ru1212) below the superconducting transition temperature $T_{c} \approx$ 30 K. The observed features depend on the specific magnetic-field profile in the sample chamber and this explains the variety of reported behaviors for this compound below $T_{c}$. An experimental procedure that combines improvement of the magnetic-field homogeneity with very small scan lengths and leads to artifact-free measurements similar to those on a stationary sample has been developed. This procedure was used to measure the mass magnetization of Ru1212 as a function of the applied magnetic field H (-20 Oe $\le$ H $\le$ 20 Oe) at $T < T_{c}$ and discuss, in conjunction with resistance and ac susceptibility measurements, the possibility of a spontaneous vortex state (SVS) for this compound. Although the existence of a SVS can not be excluded, an alternative interpretation of the results based on the granular nature of the investigated sample is also possible. Specific-heat measurements of Sr$_{2}$GdRuO$_{6}$ (Sr2116), the precursor for the preparation of Ru1212 and thus a possible impurity phase, show that it is unlikely that Sr2116 is responsible for the specific-heat features observed for Ru1212 at $T_{c}$.Comment: 17 pages, 6 figure

Bilinear structure and Schlesinger transforms of the qq-PIII_{\rm III} and qq-PVI_{\rm VI} equations

We show that the recently derived ($q$-) discrete form of the Painlev\'e VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.Comment: 10 pages, Plain Te

Individual differences in infant fixation duration relate to attention and behavioral control in childhood

Individual differences in fixation duration are considered a reliable measure of attentional control in adults. However, the degree to which individual differences in fixation duration in infancy (0–12 months) relate to temperament and behavior in childhood is largely unknown. In the present study, data were examined from 120 infants (mean age = 7.69 months, SD = 1.90) who previously participated in an eye-tracking study. At follow-up, parents completed age-appropriate questionnaires about their child’s temperament and behavior (mean age of children = 41.59 months, SD = 9.83). Mean fixation duration in infancy was positively associated with effortful control (β = 0.20, R2 = .02, p = .04) and negatively with surgency (β = −0.37, R2 = .07, p = .003) and hyperactivity-inattention (β = −0.35, R2 = .06, p = .005) in childhood. These findings suggest that individual differences in mean fixation duration in infancy are linked to attentional and behavioral control in childhood

Supporting material for co-researchers

This pack has been designed to be used alongside the Peer Research Training Resource (https://doi.org/10.25561/94819) and includes: • Skills, experience, and training reviews for Advisory Group Members and Peer Researchers • Zoom Interviews: Guide for Peer Researchers • Useful COVID-19 resources for people living with HIV The pack is suitable for academics and public involvement practitioners who are involving people with lived experience as co-researchers in research. The material presented here was developed for a participatory research study on COVID-19 experiences among people living with HIV where interviews were conducted online

First-year Accounting students’ confidence levels during Covid-19: Evidence over a period of two years at a South African university

The Coronavirus Disease (Covid-19) outbreak has caused serious and continuous concerns globally within the higher education sector. One of these is the confidence levels of students to perform specific academic-related tasks. This study explored first-year accounting students’ confidence levels during Covid-19 over a two-year period at a South African university to determine if any associations exist between students’ gender, age, choice of study, language, and method of study funding, compared to confidence level, whilst applying a quantitative research design. Mixed results were obtained. Areas for further research identified include collaboration with other universities, and to expand the scope of the population, to determine the status of students’ confidence levels for different accountancy-related modules during and post-Covid-19. The study will inform and sensitise lecturers, students and academic institutions regarding the confidence levels of students to perform specific academic-related tasks within accounting education and training under Covid-19 conditions