QFT on the non-commutative Moyal space and combinatorics

In this chapter we present the Phi? QFT model on the non-commutative Moyal space and the UV/IR mixing issue, which prevents it from being renormalizable. We then present the Grosse–Wulkenhaar Phi? QFT model on the non-commutative Moyal space, which changes the usual propagator of the Phi? model (based on the heat kernel formula) to a Mehler kernel based propagator. This Grosse–Wulkenhaar model is perturbatively renormalizable but it is not translation-invariant (translation-invariance being a usual property of high-energy physics models). We then show how the Mellin transform technique can be used to express the Feynman integrals of the Grosse-Wulkenhaar model. In the last part of the chapter, we present another Phi? QFT model on the non-commutative Moyal space, which is however both renormalizable and translation-invariant. We show the relation between the parametric representation of this model and the Bollobás–Riordan polynomial.

Cellular automata in operational probabilistic theories

The theory of cellular automata in operational probabilistic theories is developed. We start introducing the composition of infinitely many elementary systems, and then use this notion to define update rules for such infinite composite systems. The notion of causal influence is introduced, and its relation with the usual property of signalling is discussed. We then introduce homogeneity, namely the property of an update rule to evolve every system in the same way, and prove that systems evolving by a homogeneous rule always correspond to vertices of a Cayley graph. Next, we define the notion of locality for update rules. Cellular automata are then defined as homogeneous and local update rules. Finally, we prove a general version of the wrapping lemma, that connects CA on different Cayley graphs sharing some small-scale structure of neighbourhoods.

Lissajous trajectories in electromagnetically driven vortices

An experimental and theoretical study of laminar vortical flows driven by oscillating electromagnetic forces that act in orthogonal directions in a shallow electrolytic fluid layer is presented. Forces are generated by the interaction of the field of a dipolar permanent magnet and two imposed alternating electric currents perpendicular to each other with independent frequencies varying in the range of 10–30 mHz. Velocity fields of the time-dependent flow are obtained using particle image velocimetry, while particle tracking allows exploration of the Lagrangian trajectories and time maps. An approximate two-dimensional analytical solution is obtained for the laminar creeping regime so that Lagrangian trajectories are integrated explicitly. These trajectories resemble Lissajous figures with the usual property that, when the ratio of the frequencies of the imposed currents is rational, closed paths are found, while non-closed paths occur when this ratio is irrational. Deviations of this regime that account for slight increase of inertial effects are explored through a quasi-two-dimensional numerical simulation. In this case, non-closed paths are found even for rational frequency ratios. This case was observed in the experiment. Lagrangian trajectories calculated numerically show a qualitative agreement with experimental particle tracking. Furthermore, numerical time maps obtained for increasing inertial effects and rational frequency ratios reveal a chaotic behaviour. Some features of the Lagrangian trajectories are validated experimentally. In particular, topological properties of the calculated and observed time maps are in qualitative agreement. In a characteristic case, a partial time map calculated numerically is compared with the section acquired from the experimental tracking of one particle.

III. Experimental researches in electricity.—Twenty-first series

2343. The magnetic characters of iron, nickel and cobalt, are well known; and also the fact that at certain temperatures they lose their usual property and become, to ordinary test and observation, non-magnetic; then entering into the list of diamagnetic bodies and acting in like manner with them. Closer investigation, however, has shown me that they are still very different to other bodies, and that though inactive when hot, on common magnets or to common tests, they are not so absolutely, but retain a certain amount of magnetic power whatever their temperature; and also that this power is the same in character with that which they ordinarily possess. 2344. A piece of iron wire, about one inch long and 0·05 of an inch in diameter, being thoroughly cleaned, was suspended at the middle by a fine platinum wire connected with the suspending thread (2249.) so as to swing between the poles of the electro-magnet. The heat of a spirit-lamp was applied to it, and it soon acquired a temperature which rendered it quite insensible to the presence of a good ordinary magnet, however closely it was approached to the heated iron. The temperature of the iron was then raised considerably higher by adjustment of the flame, and the electro-magnet thrown into action. Immediately the hot iron became magnetic and pointed between the poles. The power was feeble, and in this respect the state of the iron was in striking contrast with that which it had when cold; but in character the force was precisely the same.