Harmonic forms on ALF gravitational instantons

We study the space of square-integrable harmonic forms over ALF gravitational instantons of type $A _{ K -1 }$ and of type $D _K$. We first calculate its dimension making use of a result by Hausel, Hunsicker and Mazzeo which relates the Hodge cohomology of a gravitational instanton $M$ to the singular cohomology of a particular compactification $X _M$ of $M$. We then exhibit an explicit basis, exact for $A _{ K -1 }$ and approximate for $D _K$, and interpret geometrically the relations between $M$, $X _M$ and their cohomologies

Frederick’s “Greatness”

This essay attempts to identify the various qualities that made Frederick II of Prussia’s just appellation ‘the Great’. Frederick employed a completely new type of rule, which was not only unique in the eighteenth century but also prefigured modern governance in many respects. Frederick personified the "raison d’etat" and came to exemplify the rational use of state power for the creation of a completely new standard of judicious kingship. As a visionary ruler of his day, Frederick foreshadowed modern principles of the state. To highlight Frederick’s innovations, the essay not only shows Frederick’s brilliant leadership in the scene of eighteenth-century Europe, but it also refers to rarely quoted contemporary sources; by doing so, the essay contrasts the prodigious divide between the crumbling culture of the "Ancien régime" and that of Frederick’s Prussia—the former still feudal and the latter possessing a vision that rulers are the ‘first servants of the state’

A Reconsideration of Werner Sombart’s "Luxury and Capitalism"

Werner Sombart’s classic text "Luxury and Capitalism" is revisited in the light of recent economic historians’ works that have analyzed luxury’s role in the development of capitalism. Most of these works, as well as Sombart’s book itself, are focused on the eighteenth century, since it was then that the proliferation—and availability—of luxury manifested itself for the first time most conspicuously. By employing secondary texts by economic historians and primary sources from the debates on luxury in the eighteenth century—some of which overlooked by a number of historians—the essay attempts a renewed outlook on a text that was controversial when written over a century ago and remains a prominent argument in the eternal discussion and dispute on the cause(s) of the rise of capitalism

Harmonic Forms and Spinors on the Taub-bolt Space

This paper studies the space of $L ^2$ harmonic forms and $L ^2$ harmonic spinors on Taub-bolt, a Ricci-flat Riemannian 4-manifold of ALF type. We prove that the space of harmonic square-integrable 2-forms on Taub-bolt is 2-dimensional and construct a basis. We explicitly find a 2-parameter family of $L ^2$ zero modes of the Dirac operator twisted by an arbitrary $L ^2$ harmonic connection. We also show that the number of zero modes found is equal to the index of the Dirac operator. We compare our results with those known in the case of Taub-NUT and Euclidean Schwarzschild as these manifolds present interesting similarities with Taub-bolt. In doing so, we slightly generalise known results on harmonic spinors on Euclidean Schwarzschild.Comment: Updated to match the published versio

Harmonic Spinors on a Family of Einstein Manifolds

The purpose of this paper is to study harmonic spinors defined on a 1-parameter family of Einstein manifolds which includes Taub-NUT, Eguchi-Hanson and $P^2(C)$ with the Fubini-Study metric as particular cases. We discuss the existence of and explicitly solve for spinors harmonic with respect to the Dirac operator twisted by a geometrically preferred connection. The metrics examined are defined, for generic values of the parameter, on a non-compact manifold with the topology of $C^2$ and extend to $P^2(C)$ as edge-cone metrics. As a consequence, the subtle boundary conditions of the Atiyah-Patodi-Singer index theorem need to be carefully considered in order to show agreement between the index of the twisted Dirac operator and the result obtained by counting the explicit solutions.Comment: Updated to match the published versio

Propagation of numerical noise in particle-in-cell tracking

Particle-in-cell (PIC) is the most used algorithm to perform self-consistent tracking of intense charged particle beams. It is based on depositing macro-particles on a grid, and subsequently solving on it the Poisson equation. It is well known that PIC algorithms occupy intrinsic limitations as they introduce numerical noise. Although not significant for short-term tracking, this becomes important in simulations for circular machines over millions of turns as it may induce artificial diffusion of the beam. In this work, we present a modeling of numerical noise induced by PIC algorithms, and discuss its influence on particle dynamics. The combined effect of particle tracking and noise created by PIC algorithms leads to correlated or decorrelated numerical noise. For decorrelated numerical noise we derive a scaling law for the simulation parameters, allowing an estimate of artificial emittance growth. Lastly, the effect of correlated numerical noise is discussed, and a mitigation strategy is proposed.Comment: 14 pages, 12 figure

Fast and accurate object detection in high resolution 4K and 8K video using GPUs

Machine learning has celebrated a lot of achievements on computer vision tasks such as object detection, but the traditionally used models work with relatively low resolution images. The resolution of recording devices is gradually increasing and there is a rising need for new methods of processing high resolution data. We propose an attention pipeline method which uses two staged evaluation of each image or video frame under rough and refined resolution to limit the total number of necessary evaluations. For both stages, we make use of the fast object detection model YOLO v2. We have implemented our model in code, which distributes the work across GPUs. We maintain high accuracy while reaching the average performance of 3-6 fps on 4K video and 2 fps on 8K video.Comment: 6 pages, 12 figures, Best Paper Finalist at IEEE High Performance Extreme Computing Conference (HPEC) 2018; copyright 2018 IEEE; (DOI will be filled when known

Fix-lines and stability domain in the vicinity of the coupled third order resonance

The single particle stability in a circular accelerator is of concern especially for operational regimes involving beam storage of hours. In the proximity to a resonance this stability domain shrinks, and the phase space fragments into a jungle of exotic objects like for instance "fix-lines". The concept of fix-points is easily understandable in a 2D phase space. It becomes quite challenging when the effect of resonances is considered in the 4D phase space, which leads then to the concept of fix-lines. In this paper we investigate the fix-lines in the proximity of a coupled third order resonance and find the relation of these objects with the stability of motion.Comment: 35 pages, 29 figure

Adiabatic dynamics of instantons on S4S ^4

We define and compute the $L^2$ metric on the framed moduli space of circle invariant 1-instantons on the 4-sphere. This moduli space is four dimensional and our metric is $SO(3) \times U(1)$ symmetric. We study the behaviour of generic geodesics and show that the metric is geodesically incomplete. Circle-invariant instantons on the 4-sphere can also be viewed as hyperbolic monopoles, and we interpret our results from this viewpoint. We relate our results to work by Habermann on unframed instantons on the 4-sphere and, in the limit where the radius of the 4-sphere tends to infinity, to results on instantons on Euclidean 4-space.Comment: 49 pages, 11 figures. Significant improvements in the discussion of framing in v