The Grading of Earth Roads

Exact date of bulletin unknown.PDF pages: 1

Evaluation of The Careers & Enterprise Company’s teacher encounters programme.

Teachers and the wider education workforce play a key role in supporting young people’s career decision making. However, the pace of change in the economy and industries can make it challenging for teachers to stay up to date with the skills employers need from a future workforce and the variety of opportunities and career pathways available to their students. The Teacher Encounters Programme is designed to help teachers increase their confidence and knowledge of the range of opportunities in today’s world of work. It provides an opportunity for teachers to engage directly with employers to see and learn about the different career pathways relevant to their subject and how that subject is applied practically in the workplace of today. This report is an analysis of the impact of the year-long national pilot programme which delivered over 1000 encounters between teachers and employers. The programme was operated through The Careers & Enterprise Company’s Careers Hubs in partnership with combined authorities, local authorities and LEPs. The Careers Hubs bring together schools, colleges, employers, apprenticeship providers, combined and local authorities and LEPs to increase the ability of education providers to improve how they prepare young people for their next steps and their career

Sums of two squares and a power

We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for $k\not\equiv 0 \bmod 4$ a congruence condition is imposed on $z$. These examples are of interest as there is no congruence obstruction itself for the representation of these $n$. This way we provide a new family of counterexamples to the Hasse principle or strong approximation.Comment: 6 pages, to appear in the memorial volume "From Arithmetic to Zeta-Functions - Number Theory in Memory of Wolfgang Schwarz

Character sums for primitive root densities

It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a primitive root modulo $q$ can be written as an infinite product $\prod_p \delta_p$ of local factors $\delta_p$ reflecting the degree of the splitting field of $X^p-r$ at the primes $p$, multiplied by a somewhat complicated factor that corrects for the `entanglement' of these splitting fields. We show how the correction factors arising in Artin's original primitive root problem and some of its generalizations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors. The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL$_2$-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.Comment: 23 pages. This version is to appear in the Mathematical Proceedings of the Cambridge Philosophical Societ

International Centre for Guidance Studies (iCeGS) Annual Review 2022

This publication offers a brief insight into the wide range of activities the iCeGS team are involved with over the year. It explores our contribution to policy, research and practice within the career development sector both in the UK and wider afield. iCeGS Annual Review also gives the team an opportunity to reflect on our many achievements over the last year. This year, like other years, we feel particularly proud of several activities

Child Maltreatment, Non-Suicidal Self-Injury, and the Mediating Role of Self-Criticism

We examined the relation between child maltreatment and non-suicidal self-injury (NSSI). Participants were 86 adolescents who completed measures of child maltreatment, self-criticism, perceived criticism, depression, and NSSI. Analyses revealed significant, small-to-medium associations between specific forms of child maltreatment (physical neglect, emotional abuse, and sexual abuse) and the presence of a recent history of NSSI. Emotional and sexual abuse had the strongest relations with NSSI, and the data supported a theoretical model in which self-criticism mediates the relation between emotional abuse and engagement in NSSI. Specificity for the mediating role of self-criticism was demonstrated by ruling out alternative mediation models. Taken together, these results indicate that several different forms of childhood maltreatment are associated with NSSI and illuminate one mechanism through which maltreatment may be associated with NSSI. Future research is needed to test the temporal relation between maltreatment and NSSI and should aim to identify additional pathways to engagement in NSSI.Psycholog

Sums and differences of four k-th powers

We prove an upper bound for the number of representations of a positive integer $N$ as the sum of four $k$-th powers of integers of size at most $B$, using a new version of the Determinant method developed by Heath-Brown, along with recent results by Salberger on the density of integral points on affine surfaces. More generally we consider representations by any integral diagonal form. The upper bound has the form $O_{N}(B^{c/\sqrt{k}})$, whereas earlier versions of the Determinant method would produce an exponent for $B$ of order $k^{-1/3}$ in this case. Furthermore, we prove that the number of representations of a positive integer $N$ as a sum of four $k$-th powers of non-negative integers is at most $O_{\epsilon}(N^{1/k+2/k^{3/2}+\epsilon})$ for $k \geq 3$, improving upon bounds by Wisdom.Comment: 18 pages. Mistake corrected in the statement of Theorem 1.2. To appear in Monatsh. Mat