The induced diffusion of tracers in a bacterial suspension is studied theoretically and experimentally at low bacterial concentrations. Considering the swimmer-tracer hydrodynamic interactions at low-Reynolds number and using a kinetic theory approach, it is shown that the induced diffusion coefficient is proportional to the swimmer concentration, their mean velocity and a coefficient $\beta$, as observed experimentally. The coefficient $\beta$ scales as the tracer-swimmer cross section times the mean square displacement produced by single scatterings. The displacements depend on the swimmer propulsion forces. Considering simple swimmer models (acting on the fluid as two monopoles or as a force dipole) it is shown that $\beta$ increases for decreasing swimming efficiencies. Close to solid surfaces the swimming efficiency degrades and, consequently, the induced diffusion increase. Experiments on W wild-type {\em Escherichia coli} in a Hele-Shaw cell under buoyant conditions are performed to measure the induced diffusion on tracers near surfaces. The modification of the suspension pH vary the swimmers' velocity in a wide range allowing to extract the $\beta$ coefficient with precision. It is found that the solid surfaces modify the induced diffusion: decreasing the confinement height of the cell, $\beta$ increases by a factor 4. The theoretical model reproduces this increase although there are quantitative differences, probably attributed to the simplicity of the swimmer models.

G. L. Miño J. Dunstan A. Rousselet E. Clement R. Soto

The AC Josephson effect predicted in 1962 and observed experimentally in 1963 as quantised voltage steps (the Shapiro steps) from photon assisted tunnelling of Cooper pairs is among the most fundamental phenomena of quantum mechanics and is vital for metrological quantum voltage standards. The physically dual effect, the AC coherent quantum phase slip (CQPS), photon assisted tunnelling of magnetic fluxes through a superconducting nanowire, is envisaged to reveal itself as quantised current steps. The basic physical significance of the AC CQPS is also complemented by practical importance in future current standards; a missing element for closing the Quantum Metrology Triangle. In 2012, the CQPS was demonstrated as superposition of magnetic flux quanta in superconducting nanowires. However the direct sharp current steps in superconductors; the only unavailable basic effect of superconductivity to date, was unattainable due to lack of appropriate materials and challenges in circuit engineering. Here we report the direct observation of the dual Shapiro steps in a superconducting nanowire. The sharp steps are clear up to 26 GHz frequency with current values 8.3 nA and limited by the present setup bandwidth. The current steps have been theoretically predicted in small Josephson junctions (JJs) 30 years ago. However, broadening unavoidable in JJs prevents their direct experimental observation. We solve this problem by placing a thin NbN nanowire in an inductive environment.

R. S. Shaikhaidarov K. H. Kim J. W. Dunstan I. V. Antonov S. Linzen M. Ziegler D. S. Golubev V. N. Antonov E. Il'ichev O. V. Astafiev

For a real number $t$, let $s_t$ be the multiplicative arithmetic function defined by $\displaystyle{s_t(p^{\alpha})=\sum_{j=0}^{\alpha}(-p^t)^j}$ for all primes $p$ and positive integers $\alpha$. We show that the range of a function $s_{-r}$ is dense in the interval $(0,1]$ whenever $r\in(0,1]$. We then find a constant $\eta_A\approx1.9011618$ and show that if $r>1$, then the range of the function $s_{-r}$ is a dense subset of the interval $\displaystyle{\left(\frac{1}{\zeta(r)},1\right]}$ if and only if $r\leq \eta_A$. We end with an open problem.

Colin Defant

For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{\alpha-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $\alpha$. The function $S_1$ is simply Euler's totient function $\phi$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient numbers generalize to results concerning Schemmel nontotient numbers. We also invoke Zsigmondy's Theorem in order to generalize a result due to Mendelsohn.

Colin Defant

Quantum state diffusion shows how stochastic interaction with the environment may cause localisation of the wave-function, and thereby demonstrates that quantum mechanics need not invoke a separate axiom of measurement to explain the emergence of the classical world. It has not been clear whether quantum state diffusion requires some new physics. We set up an explicit numerical calculation of the evolution of the wave-function of a two-state system under interaction using only the physics explicitly contained in quantum mechanics without an axiom of measurement. The wave-function does indeed localise, as proposed by quantum state diffusion, on eigenstates of the perturbation. The mechanism appears to be the superposition of histories evolving under different Hamiltonians.

D. J. Dunstan

Herein an alternative model to reptation to describe concentrated polymer dynamics is developed. The model assumes that the chains act as blobs that are able to diffuse past each other in a compressed state. Allowing that the local viscosity experienced by a blob is less than the macroscopic viscosity allows the viscosity-molecular weight and diffusion coefficient-molecular weight relationships to be determined.

Dave E. Dunstan

The range of the divisor function $\sigma_{-1}$ is dense in the interval $[1,\infty)$. However, the range of the function $\sigma_{-2}$ is not dense in the interval $\displaystyle{\left[1,\frac{\pi^2}{6}\right)}$. We begin by generalizing the divisor functions to a class of functions $\sigma_{t}$ for all real $t$. We then define a constant $\eta\approx 1.8877909$ and show that if $r\in(1,\infty)$, then the range of the function $\sigma_{-r}$ is dense in the interval $[1,\zeta(r))$ if and only if $r\leq\eta$. We end with an open problem.

Colin Defant

In 1974, Gugenheim and May showed that the cohomology $\text{Ext}_A(R,R)$ of a connected augmented algebra over a field $R$ is generated by elements with $s = 1$ under matric Massey products. In particular, this applies to the $E_2$ page of the $H\mathbb{F}_p$-based Adams spectral sequence. By studying a novel sequence of deformations of a presentably symmetric monoidal stable $\infty$-category $C$, we show that for a variety of spectral sequences coming from filtered spectra, the set of elements on the $E_2$ page surviving to the $E_k$ page is generated under matric Massey products by elements with degree $s < k.$ This work is the author's PhD thesis, completed under the supervision of Peter May.

Colin Aitken

Over the last few years, the power law distribution has been used as the data generating mechanism in many disparate fields. However, at times the techniques used to fit the power law distribution have been inappropriate. This paper describes the poweRlaw R package, which makes fitting power laws and other heavy-tailed distributions straightforward. This package contains R functions for fitting, comparing and visualising heavy tailed distributions. Overall, it provides a principled approach to power law fitting.

Colin S Gillespie

The coexistence of superconductivity and ferromagnetism has brought about the phenomena of ferromagnetic superconductors. The theory needed to understand the compatibility of such antagonistic phenomena cannot be built until the pairing symmetry of such superconductors is correctly identified. The proper and unambiguous identification of the pairing symmetry of such superconductors is the subject of this paper. This work shows that crossed Andreev reflection can be a very effective tool in order to identify the pairing symmetry of these superconductors.

Colin Benjamin