MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms

We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of MacMahon about their general form by relating them to quasi-modular forms. These functions arise as solutions to a curve-counting problem on Abelian surfaces.Comment: 6 Page

Game theory of mind

This paper introduces a model of ‘theory of mind’, namely, how we represent the intentions and goals of others to optimise our mutual interactions. We draw on ideas from optimum control and game theory to provide a ‘game theory of mind’. First, we consider the representations of goals in terms of value functions that are prescribed by utility or rewards. Critically, the joint value functions and ensuing behaviour are optimised recursively, under the assumption that I represent your value function, your representation of mine, your representation of my representation of yours, and so on ad infinitum. However, if we assume that the degree of recursion is bounded, then players need to estimate the opponent's degree of recursion (i.e., sophistication) to respond optimally. This induces a problem of inferring the opponent's sophistication, given behavioural exchanges. We show it is possible to deduce whether players make inferences about each other and quantify their sophistication on the basis of choices in sequential games. This rests on comparing generative models of choices with, and without, inference. Model comparison is demonstrated using simulated and real data from a ‘stag-hunt’. Finally, we note that exactly the same sophisticated behaviour can be achieved by optimising the utility function itself (through prosocial utility), producing unsophisticated but apparently altruistic agents. This may be relevant ethologically in hierarchal game theory and coevolution

Fluctuations of a holographic quantum Hall fluid

We analyze the neutral spectrum of the holographic quantum Hall fluid described by the D2-D8' model. As expected for a quantum Hall state, we find the system to be stable and gapped and that, at least over much of the parameter space, the lowest excitation mode is a magneto-roton. In addition, we find magneto-rotons in higher modes as well. We show that these magneto-rotons are direct consequences of level crossings between vector and scalar modes.Comment: 20 pages, 8 figures; v.2 figures improved, 2 figures added, and text clarified particularly in Sec. 5, to appear in JHE

Incremental Costs and Cost Effectiveness of Intensive Treatment in Individuals with Type 2 Diabetes Detected by Screening in the ADDITION-UK Trial: An Update with Empirical Trial-Based Cost Data.

BACKGROUND: There is uncertainty about the cost effectiveness of early intensive treatment versus routine care in individuals with type 2 diabetes detected by screening. OBJECTIVES: To derive a trial-informed estimate of the incremental costs of intensive treatment as delivered in the Anglo-Danish-Dutch Study of Intensive Treatment in People with Screen-Detected Diabetes in Primary Care-Europe (ADDITION) trial and to revisit the long-term cost-effectiveness analysis from the perspective of the UK National Health Service. METHODS: We analyzed the electronic primary care records of a subsample of the ADDITION-Cambridge trial cohort (n = 173). Unit costs of used primary care services were taken from the published literature. Incremental annual costs of intensive treatment versus routine care in years 1 to 5 after diagnosis were calculated using multilevel generalized linear models. We revisited the long-term cost-utility analyses for the ADDITION-UK trial cohort and reported results for ADDITION-Cambridge using the UK Prospective Diabetes Study Outcomes Model and the trial-informed cost estimates according to a previously developed evaluation framework. RESULTS: Incremental annual costs of intensive treatment over years 1 to 5 averaged £29.10 (standard error = £33.00) for consultations with general practitioners and nurses and £54.60 (standard error = £28.50) for metabolic and cardioprotective medication. For ADDITION-UK, over the 10-, 20-, and 30-year time horizon, adjusted incremental quality-adjusted life-years (QALYs) were 0.014, 0.043, and 0.048, and adjusted incremental costs were £1,021, £1,217, and £1,311, resulting in incremental cost-effectiveness ratios of £71,232/QALY, £28,444/QALY, and £27,549/QALY, respectively. Respective incremental cost-effectiveness ratios for ADDITION-Cambridge were slightly higher. CONCLUSIONS: The incremental costs of intensive treatment as delivered in the ADDITION-Cambridge trial were lower than expected. Given UK willingness-to-pay thresholds in patients with screen-detected diabetes, intensive treatment is of borderline cost effectiveness over a time horizon of 20 years and more.ADDITION-Cambridge was supported by the Wellcome Trust (grant reference No G061895) the Medical Research Council (grant reference no: G0001164), National Health Service R&D support funding (including the Primary Care Research and Diabetes Research Networks), and the National Institute for Health Research. We received an unrestricted grant from University of Aarhus, Denmark, to support the ADDITION-Cambridge trial. Bio-Rad provided equipment to undertake capillary glucose screening by HbA1c in general practice. SG is a National Institute for Health Research (NIHR) Senior Investigator. The Primary Care Unit is supported by NIHR Research funds. SJG received support from the Department of Health NIHR Programme Grant funding scheme (RP-PG-0606-1259)

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.Comment: 46 pages, 11 figure

Occasional errors can benefit coordination

The chances solving a problem that involves coordination between people are increased by introducing robotic players that sometimes make mistakes. This finding has implications for real-world coordination problems

Single-atom imaging of fermions in a quantum-gas microscope

Single-atom-resolved detection in optical lattices using quantum-gas microscopes has enabled a new generation of experiments in the field of quantum simulation. Fluorescence imaging of individual atoms has so far been achieved for bosonic species with optical molasses cooling, whereas detection of fermionic alkaline atoms in optical lattices by this method has proven more challenging. Here we demonstrate single-site- and single-atom-resolved fluorescence imaging of fermionic potassium-40 atoms in a quantum-gas microscope setup using electromagnetically-induced-transparency cooling. We detected on average 1000 fluorescence photons from a single atom within 1.5s, while keeping it close to the vibrational ground state of the optical lattice. Our results will enable the study of strongly correlated fermionic quantum systems in optical lattices with resolution at the single-atom level, and give access to observables such as the local entropy distribution and individual defects in fermionic Mott insulators or anti-ferromagnetically ordered phases.Comment: 7 pages, 5 figures; Nature Physics, published online 13 July 201

Grand Challenges: Improving HIV Treatment Outcomes by Integrating Interventions for Co-Morbid Mental Illness.

In the fourth article of a five-part series providing a global perspective on integrating mental health, Sylvia Kaaya and colleagues discuss the importance of integrating mental health interventions into HIV prevention and treatment platforms. Please see later in the article for the Editors' Summary

Сутність та класифікація ризиків інвестиційної діяльності

Наводиться визначення поняттю "ризики інвестиційної діяльності" за рахунок поєднання його сутнісних характеристик, виконано узагальнення класифікації цих ризиків, запропоновано введення нової класифікаційної групи – "корпоративні ризики", які пов'язані з можливістю втрати контролю над підприємством інвестором-акціонером

Qubit portrait of the photon-number tomogram and separability of two-mode light states

In view of the photon-number tomograms of two-mode light states, using the qubit-portrait method for studying the probability distributions with infinite outputs, the separability and entanglement detection of the states are studied. Examples of entangled Gaussian state and Schr\"{o}dinger cat state are discussed.Comment: 20 pages, 6 figures, TeX file, to appear in Journal of Russian Laser Researc