Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.Comment: 24 pages, 16 figure

Power computation for the triboelectric nanogenerator

We consider, from a mathematical perspective, the power generated by a contact-mode triboelectric nanogenerator, an energy harvesting device that has been well studied recently. We encapsulate the behaviour of the device in a differential equation, which although linear and of first order, has periodic coefficients, leading to some interesting mathematical problems. In studying these, we derive approximate forms for the mean power generated and the current waveforms, and describe a procedure for computing the Fourier coefficients for the current, enabling us to show how the power is distributed over the harmonics. Comparisons with accurate numerics validate our analysis

A comparative study of selected saxophone quartet repertoire from the Mule, Rascher and Apollo Saxophone Quartets between 1928 and 1995

This research will investigate the nature of saxophone quartet repertoire that has resulted from the careers of the Mule, Rascher and Apollo saxophone quartets, who have played important roles in creating a base of concert repertoire for the ensemble. This research aims to identify trends in the styles of repertoire produced, along with the techniques that are required of the performers. This research will discuss nine works that have been commissioned by, or dedicated to, three professional saxophone quartets; the Marcel Mule Quartet, the Rascher Saxophone Quartet, and The Apollo Saxophone Quartet. The quartets have been chosen as they have all played an important role in creating new repertoire for the saxophone quartet. With the Marcel Mule Quartet existing from 1928-1967, the Rascher quartet existing from 1969 to the present day and the Apollo Saxophone Quartet existing from 1985 to the present day, their careers will provide an insight into the development of the saxophone quartet since 1928

Illustrating field emission theory by using Lauritsen plots of transmission probability and barrier strength

This technical note relates to the theory of cold field electron emission (CFE). It starts by suggesting that, to emphasize common properties in relation to CFE theory, the term 'Lauritsen plot' could be used to describe all graphical plots made with the reciprocal of barrier field (or the reciprocal of a quantity proportional to barrier field) on the horizontal axis. It then argues that Lauritsen plots related to barrier strength (G) and transmission probability (D) could play a useful role in discussion of CFE theory. Such plots would supplement conventional Fowler-Nordheim (FN) plots. All these plots would be regarded as particular types of Lauritsen plot. The Lauritsen plots of -G and lnD can be used to illustrate how basic aspects of FN tunnelling theory are influenced by the mathematical form of the tunnelling barrier. These, in turn, influence local emission current density and emission current. Illustrative applications used in this note relate to the well-known exact triangular and Schottky-Nordheim barriers, and to the Coulomb barrier (i.e., the electrostatic component of the electron potential energy barrier outside a model spherical emitter). For the Coulomb barrier, a good analytical series approximation has been found for the barrier-form correction factor; this can be used to predict the existence (and to some extent the properties) of related curvature in FN plots.Comment: Based on a poster presented at the 25th International Vacuum Nanoelectronics Conference, Jeju, S. Korea, July 2012. Version 3 incorporates small changes made at proof stag

Bulletin No. 15: Integration in DPMP: An organising principle and an expanded set of tools

DPMP involves various levels of integration: • across law enforcement, treatment, harm reduction and prevention, as both approaches and sectors; • across global, national, state and local scales; • between research, policy and other practice; and • across an array of disciplinary and epistemological research approaches. Although the need for integration to better deal with complex problems, like illicit drug use, is now widely recognised and discussed, formalised processes for achieving this have been slow to develop. A unique aspect of DPMP is its close link with the new cross-cutting specialisation of Integration and Implementation Sciences. The specialisation draws from a range of disciplines, such as political science, systems thinking, complexity science, participatory methods, management, and information science, and aims to help make better-founded decisions on complex social problems by applying integrative methods to: • tackling problems systemically; • deepening understanding of problems based on all the relevant disciplines and interests; • applying knowledge management strategies to cope with both information overload and diverse epistemologies; and • applying understanding of how action occurs, in other words how policy is made, how business operates, how activism succeeds; as well as how action can be influenced by evidence. Thus the new specialisation provides theory, methods and skills to facilitate comprehensive examination of issues and problems, as well as effective mobilisation to action. Integration and Implementation Sciences also seeks to incorporate the effective use of research-based knowledge to help bring about change. Specifically, Integration and Implementation Sciences is developing theory and methods for: 1. Comprehensive scoping of problems and issues. 2. Application of appropriate integrative concepts and methods. 3. Involvement of the strengths of different research epistemologies. 4. Attention to emergent properties, i.e. to identifying and understanding properties that disappear when a system is studied in disaggregated segments. 5. Understanding of policy, product development and action and how these can be influenced by research. 6. Application of knowledge management concepts and tools. 7. Development and application of expanded ways of taking uncertainty into account. 8. Managing the inevitability of less than perfect outcomes. 9. Application of concepts and methods from change management and innovation, including developing new roles such as boundary spanners and knowledge brokers to apply research to changed practice. 10. Development and application of collaborative processes. DPMP has built on the insights gained as Integration and Implementation Sciences have developed. This has allowed some of the best leading edge ideas and methods developed in other areas to be introduced to the illicit drugs field

Effect of Zr additions on thermal stability of Al-Cu precipitates in as-cast and cold worked samples

While Zr is frequently added to Al alloys to control grain size with the formation of large (\u3e1 μm) primary precipitates, little research has been conducted on the effect of nanoscale Al3Zr precipitates on Al alloys. By comparing the precipitation and corresponding strength evolution between Al-Cu-Zr alloys with different Zr concentrations, the effects of Zr on Al-Cu precipitation with and without primary Al3Zr precipitates can be observed. In the absence of these large precipitates, all Al3Zr phases can be formed, through high temperature aging treatments, as a dispersion of nanoprecipaites inside the Al grains. In this study, Al-Cu-Zr ternary alloys were produced and heat treated to determine whether an increase in the coarsening resistance of Al-Cu precipitate phases would be observed with a distribution of the more thermally stable Al3Zr nanoprecipitates. Generally, properly aged Al-Cu alloys will coarsen when encountering elevated temperatures higher than ~473 K (~200 °C). Diluted Al-Zr alloys (\u3c0.07 at % Zr) resist coarsening behavior until the significantly higher temperatures of ~673 K (~400 °C), but are comparatively limited in strength because of a limited solubility of Zr in the Al matrix. Hardness testing and transmission electron microscope (TEM) results are discussed, in which it is found that even very small additions of Zr, when properly accounted for during heat treating, produce a finer microstructure and higher strength than in similar Al-Cu binary alloys. No significant change in the thermal stability of strengthening was observed, indicating that the finer precipitate microstructure is resultant from a higher nucleation density, as opposed to a decrease in coarsening behavior

Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model which has been recently introduced in a series of papers mainly by Efroimsky and Makarov. We present numerical and analytical results concerning the nature of the librations of Mercury’s spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time, consisting of a slow oscillation, with an amplitude of order of arcminutes, superimposed on the 88-day libration. This contrasts with recent astronomical observations and hence suggests that the 3:2 resonance in which Mercury has been trapped might have been originally described by a large-amplitude quasi-periodic libration which, only at a later stage, with the formation of a molten core, evolved into the small-amplitude libration which is observed nowadays

Considerations on bubble fragmentation models

n this paper we describe the restrictions that the probability density function (p.d.f.) of the size of particles resulting from the rupture of a drop or bubble must satisfy. Using conservation of volume, we show that when a particle of diameter, D0, breaks into exactly two fragments of sizes D and D2 = (D30−D3)1/3 respectively, the resulting p.d.f., f(D; D0), must satisfy a symmetry relation given by D22 f(D; D0) = D2 f(D2; D0), which does not depend on the nature of the underlying fragmentation process. In general, for an arbitrary number of resulting particles, m(D0), we determine that the daughter p.d.f. should satisfy the conservation of volume condition given by m(D0) ∫0D0 (D/D0)3 f(D; D0) dD = 1. A detailed analysis of some contemporary fragmentation models shows that they may not exhibit the required conservation of volume condition if they are not adequately formulated. Furthermore, we also analyse several models proposed in the literature for the breakup frequency of drops or bubbles based on different principles, g(ϵ, D0). Although, most of the models are formulated in terms of the particle size D0 and the dissipation rate of turbulent kinetic energy, ϵ, and apparently provide different results, we show here that they are nearly identical when expressed in dimensionless form in terms of the Weber number, g*(Wet) = g(ϵ, D0) D2/30 ϵ−1/3, with Wet ~ ρ ϵ2/3 D05/3/σ, where ρ is the density of the continuous phase and σ the surface tension

Spectral evidence for a powerful compact jet from XTE J1118+480

We present observations of the X-ray transient XTE J1118+480 during its Low/Hard X-ray state outburst in 2000, at radio and sub-millimetre wavelengths with the VLA, Ryle Telescope, MERLIN and JCMT. The high-resolution MERLIN observations reveal all the radio emission (at 5 GHz) to come from a compact core with physical dimensions smaller than 65*d(kpc) AU. The combined radio data reveal a persistent and inverted radio spectrum, with spectral index \~+0.5. The source is also detected at 350 GHz, on an extrapolation of the radio spectrum. Flat or inverted radio spectra are now known to be typical of the Low/Hard X-ray state, and are believed to arise in synchrotron emission from a partially self-absorbed jet. Comparison of the radio and sub-millimetre data with reported near-infrared observations suggest that the synchrotron emission from the jet extends to the near-infrared, or possibly even optical regimes. In this case the ratio of jet power to total X-ray luminosity is likely to be P_J/L_X >> 0.01, depending on the radiative efficiency and relativistic Doppler factor of the jet. Based on these arguments we conclude that during the period of our observations XTE J1118+480 was producing a powerful outflow which extracted a large fraction of the total accretion power.Comment: Accepted for publication as a Letter in MNRA