THE PROFINITE COMPLETION OF A PROFINITE PROJECTIVE GROUP

We prove that the profinite completion of a profinite projective group is projective.

UNIVERSAL INVARIANT ECHOCOMPENSATION METHOD IN DUPLEX COMMUNICATION SYSTEMS

Предлагается универсальный метод эхокомпенсации в дуплексных системах связи, инвариантный относительно амплитудной характеристики паразитного эхотракта устройств разделения направлений передачи. Теоретическое обоснование метода базируется на отображении линейных и нелинейных измерений сигналов передатчика в эхотракте проективной группой преобразований и использовании инварианта этой группы для расчета копий сигналов передатчика на выходе эхотракта. It is proposed the universal method of echocompensation in duplex communication systems, which is invariant with respect to the amplitude characteristic of the parasitic echo path of transmission direction separation devices. The theoretical justification of the method is based on the mapping of linear and nonlinear measurements of the transmitter signals in the echo path by a projective group of transformations and using the invariant of this group to calculate copies of the transmitter signals at the output of the echo path.

Projective properties of Lorentzian surfaces

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non-flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite non-compact surface can be endowed with a metric having a non-isometric projective transformation of infinite order.

Applications of a Particular Four-Dimensional Projective Geometry to Galactic Dynamics

Relativistic localizing systems that extend relativistic positioning systems show that pseudo-Riemannian space-time geometry is somehow encompassed in a particular four-dimensional projective geometry. The resulting geometric structure is then that of a generalized Cartan space (also called Cartan connection space) with projective connection. The result is that locally non-linear actions of projective groups via homographies systematically induce the existence of a particular space-time foliation independent of any space-time dynamics or solutions of Einstein’s equations for example. In this article, we present the consequences of these projective group actions and this foliation. In particular, it is shown that the particular geometric structure due to this foliation is similar from a certain point of view to that of a black hole but not necessarily based on the existence of singularities. We also present a modified Newton’s laws invariant with respect to the homographic transformations induced by this projective geometry. Consequences on galactic dynamics are discussed and fits of galactic rotational velocity curves based on these modifications which are independent of any Modified Newtonian Dynamics (MOND) or dark matter theories are presented.