On small homotopies of loops

Two natural questions are answered in the negative: (1) If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic? (2) Can adding arcs to a space cause an essential curve to become nulhomotopic? The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and $\pi_1$-shape injective.Comment: 12 pages, 5 figure

Seeking critical nodes in digraphs

The Critical Node Detection Problem (CNDP) consists in finding the set of nodes, defined critical, whose removal maximally degrades the graph. In this work we focus on finding the set of critical nodes whose removal minimizes the pairwise connectivity of a direct graph (digraph). Such problem has been proved to be NP-hard, thus we need efficient heuristics to detect critical nodes in real-world applications. We aim at understanding which is the best heuristic we can apply to identify critical nodes in practice, i.e., taking into account time constrains and real-world networks. We present an in-depth analysis of several heuristics we ran on both real-world and on synthetic graphs. We define and evaluate two different strategies for each heuristic: standard and iterative. Our main findings show that an algorithm recently proposed to solve the CNDP and that can be used as heuristic for the general case provides the best results in real-world graphs, and it is also the fastest. However, there are few exceptions that are thoroughly analyzed and discussed. We show that among the heuristics we analyzed, few of them cannot be applied to very large graphs, when the iterative strategy is used, due to their time complexity. Finally, we suggest possible directions to further improve the heuristic providing the best results

Effect of toroidal field ripple on plasma rotation in JET

Dedicated experiments on TF ripple effects on the performance of tokamak plasmas have been carried out at JET. The TF ripple was found to have a profound effect on the plasma rotation. The central Mach number, M, defined as the ratio of the rotation velocity and the thermal velocity, was found to drop as a function of TF ripple amplitude (3) from an average value of M = 0.40-0.55 for operations at the standard JET ripple of 6 = 0.08% to M = 0.25-0.40 for 6 = 0.5% and M = 0.1-0.3 for delta = 1%. TF ripple effects should be considered when estimating the plasma rotation in ITER. With standard co-current injection of neutral beam injection (NBI), plasmas were found to rotate in the co-current direction. However, for higher TF ripple amplitudes (delta similar to 1%) an area of counter rotation developed at the edge of the plasma, while the core kept its co-rotation. The edge counter rotation was found to depend, besides on the TF ripple amplitude, on the edge temperature. The observed reduction of toroidal plasma rotation with increasing TF ripple could partly be explained by TF ripple induced losses of energetic ions, injected by NBI. However, the calculated torque due to these losses was insufficient to explain the observed counter rotation and its scaling with edge parameters. It is suggested that additional TF ripple induced losses of thermal ions contribute to this effect

Deuterium Balmer/Stark spectroscopy and impurity profiles: first results from mirror-link divertor spectroscopy system on the JET ITER-like wall

For the ITER-like wall, the JET mirror link divertor spectroscopy system was redesigned to fully cover the tungsten horizontal strike plate with faster time resolution and improved near-UV performance. Since the ITER-like wall project involves a change in JET from a carbon dominated machine to a beryllium and tungsten dominated machine with residual carbon, the aim of the system is to provide the recycling flux, equivalent, to the impinging deuterium ion flux, the impurity fluxes (C, Be, O) and tungsten sputtering fluxes and hence give information on the tungsten divertor source. In order to do this self-consistently, the system also needs to provide plasma characterization through the deuterium Balmer spectra measurements of electron density and temperature during high density. L-Mode results at the density limit from Stark broadening/line ratio analysis will be presented and compared to Langmuir probe profiles and 2D-tomography of low-n Balmer emission [1]. Comparison with other diagnostics will be vital for modelling attempts with the EDGE2D-EIRENE code[2] as the best possible data sets need to be provided to study detachment behaviour.Comment: 18 pages, 11 figure