Unitary rotation and gyration of pixellated images on rectangular screens

In the two space dimensions of screens in optical sy stems, rotations, gyrations, and fractional Fourier transformations form the Fourier subgroup of the symplectic group of linear canonical transformations: U(2) F $\subset$ Sp(4,R). Here we study the action of this Fourier group on pixellated images within generic rectangular $N_x$ $\times$ $N_y$ screens; its elements here compose properly and act unitarily, i.e., without loss of information.Comment: 7 pages, 5 figure

Dynamics and symmetries of flow reversals in turbulent convection

Based on direct numerical simulations and symmetry arguments, we show that the large-scale Fourier modes are useful tools to describe the flow structures and dynamics of flow reversals in Rayleigh-B\'enard convection (RBC). We observe that during the reversals, the amplitude of one of the large-scale modes vanishes, while another mode rises sharply, very similar to the "cessation-led" reversals observed earlier in experiments and numerical simulations. We find anomalous fluctuations in the Nusselt number during the reversals. Using the structures of the RBC equations in the Fourier space, we deduce two symmetry transformations that leave the equations invariant. These symmetry transformations help us in identifying the reversing and non-reversing Fourier modes.Comment: 4 pages, 3 figure

Linear Canonical Transformations in Relativistic Quantum Physics

Linear Canonical Transformations (LCTs) are widely known in signal processing and optics as transformations generalizing some integral transforms such as Fourier and fractional Fourier transforms. In our previous works, LCTs have been identified, in the framework of quantum theory, to be the linear transformations which keep invariant the canonical commutation relations between coordinates and momenta operators. In this work, we extend this approach to establish that LCTs can be considered as the elements of a symmetry group for relativistic quantum physics. It is established that Lorentz transformations and multidimensional generalization of Fourier transforms can be considered as particular case of the multidimensional LCTs and some of the main symmetry groups currently considered in relativistic theories can be obtained from the contractions of LCT groups. It is also shown that a natural link can be established between the spinorial representation of LCTs and some properties of elementary fermions. Some possible applications of the obtained results are discussed. From a simplistic physical point of view, LCT group mixes spacetime with energy-momentum in a manner analogous to the action of the Lorentz group on space and time.Comment: 15 page

Fourier analysis on the affine group, quantization and noncompact Connes geometries

We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.Comment: 37 pages, Latex, uses TikZ package to draw 3 figures. Two new subsections, main results unchange