The Engel elements in generalized FC-groups

We generalize to FC*, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002), 241-254], a result of Baer on Engel elements. More precisely, we prove that the sets of left Engel elements and bounded left Engel elements of an FC*-group G coincide with the Fitting subgroup; whereas the sets of right Engel elements and bounded right Engel elements of G are subgroups and the former coincides with the hypercentre. We also give an example of an FC*-group for which the set of right Engel elements contains properly the set of bounded right Engel elements.Comment: to appear in "Illinois Journal of Mathematics

Unsteady end-wall pressure measurements using near-field diy sensors on fouled fan rotor blade

The fouling is identifiable by the presence of dust on rotor and stator blades, and its main origin, in industrial turbomachinery, is the presence of a film of moist or lubricant driven to the trailing edge by the near-wall flow, or centrifuged toward the casing by impeller rotation. Solid particles pile up on them, leading to eccentricity and load unbalance. The formation of build-up results in performance reduction, and the chance of a deposit detachment while the impeller spun, may cause damages due to the impact on the machine parts. In industrial fans, the presence of fouling influences the characteristic curve and could anticipate stall when the flow rate is throttled. Rotating stall is an aerodynamic instability with a typical frequency about half the rotor frequency, acoustically identifiable from the changes in the emitted rotor noise, due to displacement from the stability. This work investigates rotating stall dynamics on an axial fan with fouled blades. The stall is identified with time-resolved pseudo-sound measurements in the end-wall region using DIY sensors. The signals have been analysed in frequency domain, and time domain using a phase space reconstruction technique. It is demonstrated a modification of the dynamic to stall and are identified diverse stall precursors

A nilpotency criterion for some verbal subgroups

The word $w=[x_{i_1},x_{i_2},\dots,x_{i_k}]$ is a simple commutator word if $k\geq 2, i_1\neq i_2$ and $i_j\in \{1,\dots,m\}$, for some $m>1$. For a finite group $G$, we prove that if $i_{1} \neq i_j$ for every $j\neq 1$, then the verbal subgroup corresponding to $w$ is nilpotent if and only if $|ab|=|a||b|$ for any $w$-values $a,b\in G$ of coprime orders. We also extend the result to a residually finite group $G$, provided that the set of all $w$-values in $G$ is finite

Groups generated by a finite Engel set

A subset $S$ of a group $G$ is called an Engel set if, for all $x,y\in S$, there is a non-negative integer $n=n(x,y)$ such that [x,\,_n y]=1. In this paper we are interested in finding conditions for a group generated by a finite Engel set to be nilpotent. In particular, we focus our investigation on groups generated by an Engel set of size two.Comment: to appear in Journal of Algebr

Analytical ray-tracing in planetary atmospheres

Ground-based astro-geodetic observations and atmospheric occultations, are two examples of observational techniques requiring a scrutiny analysis of atmospheric refraction. In both cases, the measured changes in observables are geometrically related to changes in the photon path and the light time of the received electromagnetic signal. In the context of geometrical optics, the change in the physical properties of the signal are related to the refractive profile of the crossed medium. Therefore, having a clear knowledge of how the refractivity governs the photon path and the light time evolution is of prime importance to clearly understand observational features. Analytical studies usually focused on spherically symmetric atmospheres and only few aimed at exploring the effect of the non-spherical symmetry on the observables. In this paper, we analytically perform the integration of the photon path and the light time of rays traveling across a planetary atmosphere. We do not restrict our attention to spherically symmetric atmospheres and introduce a comprehensive mathematical framework which allows to handle any kind of analytical studies in the context of geometrical optics. To highlight the capabilities of this new formalism, we carry out five realistic applications for which we derive analytical solutions. The accuracy of the method of integration is assessed by comparing our results to a numerical integration of the equations of geometrical optics in the presence of a quadrupolar moment $J_2$. This shows that the analytical solution leads to the determination of the light time and the refractive bending with relative errors at the level of one part in $10^8$ and one part in $10^5$, for typical values of the refractivity and the $J_2$ parameter at levels of $10^{-4}$ and $10^{-2}$, respectively