CELLULAR AUTOMATA ALGORITHMS FOR PSEUDORANDOM NUMBERS GENERATION

Work describes four permutation algorithms of square matrices based on cyclic rows and columns shifts. This choice of discrete transformation algorithms is justified by the convenience of the cellular automaton (CA) formulation. Output matrices can be considered as pseudo-random sequences of numbers. As a result of numerical calculation, empirical formulas are obtained for the permutation period and the function of the period of a single CA-cell on the order of the matrix n. As a parameter of CA dynamics, we analyze two "mixing metrics" on permutations of the matrix (compared to the initial matrix).

On $C^*$-algebras associated to actions of discrete subgroups of $\operatorname{SL}(2,\mathbb{R})$ on the punctured plane

Dynamical conditions that guarantee stability for discrete transformation group $C^*$-algebras are determined. The results are applied to the case of some discrete subgroups of $\operatorname{SL} (2,\mathbb{R} )$ acting on the punctured plane by means of matrix multiplication of vectors. In the case of cocompact subgroups, further properties of such crossed products are deduced from properties of the $C^*$-algebra associated to the horocycle flow on the corresponding compact homogeneous space of $\operatorname{SL} (2,\mathbb{R} )$.

Signature-causality reflection generated by Abelian gauge transformations

In this paper, we want to better understand the causality reflection that arises under a subset of Abelian local gauge transformations in geometrodynamics. We proved in previous papers that in Einstein–Maxwell spacetimes, there exist two local orthogonal planes of gauge symmetry at every spacetime point for non-null electromagnetic fields. Every vector in these planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. The vectors that span these local orthogonal planes are dependent on electromagnetic gauge. The local group of Abelian electromagnetic gauge transformations has been proved isomorphic to the local groups of tetrad transformations in these planes. We called LB1 the local group of tetrad transformations made up of SO(1, 1) plus two different kinds of discrete transformations. One of the discrete transformations is the full inversion two by two which is a Lorentz transformation. The other discrete transformation is given by a matrix with zeroes on the diagonal and ones off-diagonal two by two, a reflection. The group LB1 is realized on this plane, we call this plane one, and is spanned by the time-like and one space-like vectors. The other local orthogonal plane is plane two and the local group of tetrad transformations, we call this LB2, which is just SO(2). The local group of Abelian electromagnetic gauge transformations is isomorphic to both LB1 and LB2, independently. It has already been proved that a subset of local electromagnetic gauge transformations that leave the electromagnetic tensor invariant induces a change in sign in the norm of the tetrad vectors that span the local plane one. The reason is that one of the discrete transformations on the local plane one that belongs to the group LB1 is not a Lorentz transformation, it is a flip or reflection. It is precisely on this kind of discrete transformation that we have an interest since it has the effect of changing the signature and the causality. This effect has never been noticed before.

Digital signal processing in telecommunications based on parametric discrete Fourier transform

A generalization of the discrete Fourier transform in the form of a parametric discrete Fourier transform is proposed. The analytical and stochastic properties of the introduced discrete transformation are investigated. An example of the application of the parametric discrete Fourier transform in telecommunications is given - a generalization of the well-known Herzel algorithm

A dichotomy for groupoid -algebras

We study the finite versus infinite nature of C$^{\ast }$-algebras arising from étale groupoids. For an ample groupoid$G$, we relate infiniteness of the reduced C$^{\ast }$-algebra$\text{C}_{r}^{\ast }(G)$to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid$S(G)$which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C$^{\ast }$-algebra of$G$in the sense that if$G$is ample, minimal, topologically principal, and$S(G)$is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for$\text{C}_{r}^{\ast }(G)$. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph$\text{C}^{\ast }$-algebras as well.