A few representation formulas for solutions of fractional Laplace equations

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We first establish a representation formula for the partial derivatives of the solutions of the homogeneous Dirichlet problem. Along the way, we obtain a Pohozaev-type identity for the fractional Green function and of the fractional Robin function. The latter extends to the fractional setting a formula obtained by Br\'ezis and Peletier, see \cite{Bresiz}, in the classical case of the Laplacian. As an application we consider the particle system extending the classical point vortex system to the case of a fractional Laplacian. We observe that, for a single particle in a bounded domain, the properties of the fractional Robin function are crucial for the study of the steady states. We also extend the classical Hadamard variational formula to the fractional Green function as well as to the shape derivative of weak solution to the homogeneous Dirichlet problem. Finally we turn to the in homogeneous Dirichlet problem and extend a formula by J.L. Lions, see \cite{Lions}, regarding the kernel of the reproducing kernel Hilbert space of harmonic functions to the case of $s$-harmonic functions. We observe that, despite the order of the operator is not $2$, this formula looks like the Hadamard variational formula, answering in a negative way to an open question raised in \cite{ELPL}

User Manual for MOLSCAT, BOUND and FIELD, Version 2020.0: programs for quantum scattering properties and bound states of interacting pairs of atoms and molecules

MOLSCAT is a general-purpose package for performing non-reactive quantum scattering calculations for atomic and molecular collisions using coupled-channel methods. Simple atom-molecule and molecule-molecule collision types are coded internally and additional ones may be handled with plug-in routines. Plug-in routines may include external magnetic, electric or photon fields (and combinations of them). Simple interaction potentials are coded internally and more complicated ones may be handled with plug-in routines. BOUND is a general-purpose package for performing calculations of bound-state energies in weakly bound atomic and molecular systems using coupled-channel methods. It solves the same sets of coupled equations as \MOLSCAT, and can use the same plug-in routines if desired, but with different boundary conditions. FIELD is a development of BOUND that locates external fields at which a bound state exists with a specified energy. One important use is to locate the positions of magnetically tunable Feshbach resonance positions in ultracold collisions. Versions of these programs before version 2019.0 were released separately. However, there is a significant degree of overlap between their internal structures and usage specifications. This manual therefore describes all three, with careful identification of parts that are specific to one or two of the programs.Comment: 206 pages. Program source code available from https://github.com/molscat/molscat This is the full program documentation for the programs described in the journal papers Comp. Phys. Commun. 241, 1-8 (2019) (arXiv:1811.09111) and Comp. Phys. Commun. 241, 9-16 (2019) (arXiv:1811.09584). There is significant text overlap between some parts of the documentation and the (much shorter) journal paper

Antibunched photons emitted by a dc-biased Josephson junction

We show experimentally that a dc biased Josephson junction in series with a high-enough-impedance microwave resonator emits antibunched photons. Our resonator is made of a simple microfabricated spiral coil that resonates at 4.4 GHz and reaches a 1.97kΩ characteristic impedance. The second order correlation function of the power leaking out of the resonator drops down to 0.3 at zero delay, which demonstrates the antibunching of the photons emitted by the circuit at a rate of 6×10^7 photons per second. Results are found in quantitative agreement with our theoretical predictions. This simple scheme could offer an efficient and bright single-photon source in the microwave domain

Making molecules by mergoassociation: Two atoms in adjacent nonspherical optical traps

Mergoassociation of two ultracold atoms to form a weakly bound molecule can occur when two optical traps that each contain a single atom are merged. Molecule formation occurs at an avoided crossing between a molecular state and the lowest motional state of the atom pair. We develop the theory of mergoassociation for pairs of nonidentical nonspherical traps. We develop a coupled-channel approach for the relative motion of the two atoms and present results for pairs of cylindrically symmetrical traps as a function of their anisotropy. We focus on the strength of the avoided crossing responsible for mergoassociation. We also develop an approximate method that gives insight into the dependence of the crossing strength on aspect ratio

Lumped element kinetic inductance detectors maturity for space-borne instruments in the range between 80 and 180 GHz

This work intends to give the state-of-the-art of our knowledge of the performance of LEKIDs at millimetre wavelengths (from 80 to 180~GHz). We evaluate their optical sensitivity under typical background conditions and their interaction with ionising particles. Two LEKID arrays, originally designed for ground-based applications and composed of a few hundred pixels each, operate at a central frequency of 100, and 150~GHz ($\Delta \nu / \nu$ about 0.3). Their sensitivities have been characterised in the laboratory using a dedicated closed-circle 100~mK dilution cryostat and a sky simulator, allowing for the reproduction of realistic, space-like observation conditions. The impact of cosmic rays has been evaluated by exposing the LEKID arrays to alpha particles ($^{241}$Am) and X sources ($^{109}$Cd) with a readout sampling frequency similar to the ones used for Planck HFI (about 200~Hz), and also with a high resolution sampling level (up to 2~MHz) in order to better characterise and interpret the observed glitches. In parallel, we have developed an analytical model to rescale the results to what would be observed by such a LEKID array at the second Lagrangian point.Comment: 7 pages, 2 tables, 13 figure

Niobium Silicon alloys for Kinetic Inductance Detectors

We are studying the properties of Niobium Silicon amorphous alloys as a candidate material for the fabrication of highly sensitive Kinetic Inductance Detectors (KID), optimized for very low optical loads. As in the case of other composite materials, the NbSi properties can be changed by varying the relative amounts of its components. Using a NbSi film with T_c around 1 K we have been able to obtain the first NbSi resonators, observe an optical response and acquire a spectrum in the band 50 to 300 GHz. The data taken show that this material has very high kinetic inductance and normal state surface resistivity. These properties are ideal for the development of KID. More measurements are planned to further characterize the NbSi alloy and fully investigate its potential.Comment: Accepted for publication on Journal of Low Temperature Physics. Proceedings of the LTD15 conference (Caltech 2013

Social attention biases in juvenile wild vervet monkeys: implications for socialisation and social learning processes.

The concept of directed social learning predicts that social learning opportunities for an individual will depend on social dynamics, context and demonstrator identity. However, few empirical studies have examined social attention biases in animal groups. Sex-based and kinship-based biases in social learning and social attention towards females have been shown in a despotic and female philopatric primate: the vervet monkey (Chlorocebus pygerythrus). The present study examined social attention during the juvenile period. Social attention was recorded through 5-min focal observations during periods of natural foraging. Kin emerged as the most important focus of social attention in juveniles, intensified by biased spatial proximity towards matrilineal related members. The highest-ranking conspecifics were more frequently observed by juveniles than low-ranking ones. Additionally, younger and orphaned juveniles showed higher levels of social attention overall, compared to other age categories. No effect of the juvenile's hierarchical rank was detected, suggesting that the variation in social attention recorded reflects different biases and stages of social learning and socialisation, rather than social anxiety. Juvenile females tended to exhibit a dominance-based bias more strongly than did males. This might be explained by a greater emphasis on attaining social knowledge during juvenile socialisation in the philopatric sex. Moreover, despite a preferred association between juveniles, social attention was more often directed to adults, suggesting that adults may still be more often chosen as a target of attention independent of their dominance rank

Making molecules by mergoassociation: two atoms in adjacent nonspherical optical traps

Mergoassociation of two ultracold atoms to form a weakly bound molecule can occur when two optical traps that each contain a single atom are merged. Molecule formation occurs at an avoided crossing between a molecular state and the lowest motional state of the atom pair. We develop the theory of mergoassociation for pairs of nonidentical nonspherical traps. We develop a coupled-channel approach for the relative motion of the two atoms and present results for pairs of cylindrically symmetrical traps as a function of their anisotropy. We focus on the strength of the avoided crossing responsible for mergoassociation. We also develop an approximate method that gives insight into the dependence of the crossing strength on aspect ratio

When, why and how tumour clonal diversity predicts survival

The utility of intratumour heterogeneity as a prognostic biomarker is the subject of ongoing clinical investigation. However, the relationship between this marker and its clinical impact is mediated by an evolutionary process that is not well understood. Here, we employ a spatial computational model of tumour evolution to assess when, why and how intratumour heterogeneity can be used to forecast tumour growth rate and progression‐free survival. We identify three conditions that can lead to a positive correlation between clonal diversity and subsequent growth rate: diversity is measured early in tumour development; selective sweeps are rare; and/or tumours vary in the rate at which they acquire driver mutations. Opposite conditions typically lead to negative correlation. In cohorts of tumours with diverse evolutionary parameters, we find that clonal diversity is a reliable predictor of both growth rate and progression‐free survival. We thus offer explanations—grounded in evolutionary theory—for empirical findings in various cancers, including survival analyses reported in the recent TRACERx Renal study of clear‐cell renal cell carcinoma. Our work informs the search for new prognostic biomarkers and contributes to the development of predictive oncology