Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

Exceptional sets for Diophantine inequalities

We apply Freeman's variant of the Davenport-Heilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a positive number c

Policy shift or program drift? Implementing housing first in Australia

This Essay critically analysed how the Housing First approach can be successfully applied to the system of supported housing in Australia

Exiting unsheltered homelessness and sustaining housing: a human agency perspective

An emerging body of evidence has demonstrated the extent to which social programs and housing initiatives have successfully helped people exit chronic and unsheltered homelessness. Contemporary research shows that people with both health and social problems are able to exit homelessness and sustain housing over several years. Although the existing evidence is robust and often based on studies employing rigorous experimental designs, clients of programs are presented as passive service recipients whose exits from homelessness are attributed to outside intervention. Drawing on a multisite Australian study with people who had exited chronic and unsheltered homelessness, this article adopts a theoretical framework of human agency to demonstrate how people exiting homelessness play active roles in shaping the outcomes they achieve. Extending the existing evidence base, we show how they explain their outcomes in terms of imagined future trajectories and an evaluation of their options to achieve change

A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM

We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1-o(1) that takes O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW PRAM parallel model of computation. We believe this to be the first randomized sublinear time algorithm on the EREW PRAM for this problem

Soluble oligomerization provides a beneficial fitness effect on destabilizing mutations

Mutations create the genetic diversity on which selective pressures can act, yet also create structural instability in proteins. How, then, is it possible for organisms to ameliorate mutation-induced perturbations of protein stability while maintaining biological fitness and gaining a selective advantage? Here we used a new technique of site-specific chromosomal mutagenesis to introduce a selected set of mostly destabilizing mutations into folA - an essential chromosomal gene of E. coli encoding dihydrofolate reductase (DHFR) - to determine how changes in protein stability, activity and abundance affect fitness. In total, 27 E.coli strains carrying mutant DHFR were created. We found no significant correlation between protein stability and its catalytic activity nor between catalytic activity and fitness in a limited range of variation of catalytic activity observed in mutants. The stability of these mutants is strongly correlated with their intracellular abundance; suggesting that protein homeostatic machinery plays an active role in maintaining intracellular concentrations of proteins. Fitness also shows a significant correlation with intracellular abundance of soluble DHFR in cells growing at 30oC. At 42oC, on the other hand, the picture was mixed, yet remarkable: a few strains carrying mutant DHFR proteins aggregated rendering them nonviable, but, intriguingly, the majority exhibited fitness higher than wild type. We found that mutational destabilization of DHFR proteins in E. coli is counterbalanced at 42oC by their soluble oligomerization, thereby restoring structural stability and protecting against aggregation