Hilbert Topology Research Articles

Generation of broadcasting for fractal adaptive Internet of things reconfiguration under the swarm intelligence paradigm

Recently, a wide range of small devices, such as Wi-Fi Internet of things development boards, which are a kind of the microcontroller units in a general purpose board, are interrelated throughout the planet. In addition, certain microcontroller units interact inside our homes when turning lights on or detecting movements, measuring various parameters, such as gas concentrations, [Formula: see text], humidity, and the temperature inside a room, or adjusting the intensity of the lights inside and outside of the house. Likewise, there is a great diversity of microcontroller units, ranging from smart cellular telephones or reduced general purpose devices, ESP8266 or RaspberryPi3 to any type of Internet of things devices. Therefore, the general way of connecting the microcontroller units to the Internet is through hub nodes, so that the information can be propagated and shared among them. The main purpose of this article is to yield an adaptive reconfiguration algorithm to link all the sensor nodes (microcontroller units) of a network based on the fractal topology, avoiding the use of hub nodes, in order for the microcontroller units to share all the parameters established in the Internet of things network only through two adjacent sensor nodes as long as any sensor node in the network knows all the parameters of the other ones, even if the sensor nodes are not adjacent. To achieve the above, in this work, an Internet of things network was built based on the Hilbert fractal for being a filling-space curve yielded from the L-systems paradigm, so this fractal Hilbert topology allows access to the entire Internet of things network in a dynamic way, and it is possible to reconfigure the network topology when a new sensor node is attached by applying artificial intelligence to intelligent and dynamic environments.

ASYMPTOTIC SYMMETRIES IN 3-DIM GENERAL RELATIVITY: THE B(2,1) CASE

The ordinary Bondi—Metzner—Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space—times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). Strongly continuous unitary irreducible representations (IRs) of B(2, 1), the analogue of B in three space—time dimensions, are analysed in the Hilbert topology. It is proved that all IRs of B(2, 1) are induced from IRs of compact ‘little groups’, which are the closed subgroups of SO(2). It is proved that all IRs of B(2, 1) are obtained by Wigner—Mackey’s inducing construction notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

On the representation theory of the Bondi–Metzner–Sachs group and its variants in three space–time dimensions

The original Bondi–Metzner–Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space–times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, McCarthy classified all relativistic B-invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of B. Here we introduce the analogue B(2, 1) of the BMS group B in 3 space–time dimensions. B(2, 1) itself admits thirty-four analogues both real in all signatures and in complex space–times. In order to find the IRS of both B(2, 1) and its analogues, we need to extend Wigner–Mackey’s theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2, 1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space–time dimensions, d≥3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2, 1). The main results of the representation theory are as follows: The IRS are induced from “little groups” which are compact. The finite “little groups” are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

Representations of the Bondi—Metzner—Sachs group in three space—time dimensions

The original Bondi–Metzner–Sachs group B is the common asymptotic symmetry group of all asymptotically at Lorentzian 4–dim space–times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B–invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of B. Here, we construct the IRS of B(2, 1), the analogue of B, in 3 space–time dimensions. The IRS are induced from ‘little groups’ which are compact. The finite ‘little groups’ are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

First results on the representation theory of the Ultrahyperbolic BMS group UHB(2, 2)

The Bondi–Metzner–Sachs (BMS) group B is the common asymptotic group of all asymptotically flat (lorentzian) space–times, and is the best candidate for the universal symmetry group of General Relativity (G.R.). B admits generalizations to real space–times of any signature, to complex space–times, and supersymmetric generalizations for any space— time dimension. With this motivation McCarthy constructed the strongly continuous unitary irreducible representations (IRs) of B some time ago, and he identified B(2,2) as the generalization of B appropriate to the to the ultrahyperbolic signature (+,+,−,−) and asymptotic flatness in null directions. We continue this programme by introducing a new group UHB(2, 2) in the group theoretical study of ultrahyperbolic G.R. which happens to be a proper subgroup of B(2, 2). We report on the first general results on the representation theory of UHB(2, 2). In particular the main general results are that the all little groups of UHB(2, 2) are compact and that the Wigner–Mackeys inducing construction is exhaustive despite the fact that UHB(2, 2) is not locally compact in the employed Hilbert topology.

Representations of the ultrahyperbolic BMS group UHB(2, 2). I. General Results

The ordinary Bondi—Metzner—Sachs (BMS) group B is the common asymptotic symmetry group of all radiating, asymptotically flat, Lorentzian space—times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). However, in studying quantum gravity, space—times with signatures other than the usual Lorentzian one, and complex space-times, are frequently considered. Generalisations of B appropriate to these other signatures have been defined earlier. In particular, the generalisation B(2, 2) appropriate to the ultrahyperbolic signature (+,+, —,—) has been described in detail, and the study of its irreducible unitary representations (IRs) of B(2, 2) has been initiated. We continue this programme by introducing a new group UHB(2, 2) in the group theoretical study of ultrahyperbolic G.R. which happens to be a proper subgroup of B(2, 2). In this paper we report on the first general results on the representation theory of UHB(2, 2). In particular the main general results are that the all little groups of UHB(2, 2) are compact and that the Wigner—Mackey's inducing construction is exhaustive despite the fact that UHB(2, 2) is not locally compact in the employed Hilbert topology.

On the Representation Theory of the Ultrahyperbolic BMS group UHB(2, 2). I. General Results

The Bondi-Metzner-Sachs (BMS) group B is the common asymptotic group of all asymptotically flat (lorentzian) space-times, and is the best candidate for the universal symmetry group of General Relativity (G.R.). B admits generalizations to real space-times of any signature, to complex space-times, and supersymmetric generalizations for any space- time dimension. With this motivation McCarthy constructed the strongly continuous unitary irreducible representations (IRs) of B some time ago, and he identified B(2,2) as the generalization of B appropriate to the to the 'ultrahyperbolic signature' (+,+,−,−) and asymptotic flatness in null directions. We continue this programme by introducing a new group UHB(2, 2) in the group theoretical study of ultrahyperbolic G.R. which happens to be a proper subgroup of B(2, 2). In this short paper we report on the first general results on the representation theory of UHB(2, 2). In particular the main general results are that the all little groups of UHB(2, 2) are compact and that the Wigner-Mackey's inducing construction is exhaustive despite the fact that UHB(2, 2) is not locally compact in the employed Hilbert topology. At the end of the paper we comment on the significance of these results.

Representations of the Bondi-Metzner-Sachs group with the Hilbert topology

All unitary irreducible representations of the Bondi-Metzner-Sachs group with the Hilbert topology are shown to be derivable by the inducing construction, which proves the exhaustivity of Mc Carthy's list of representations.

The Bondi—Metzner—Sachs group in the nuclear topology

The Bondi-Metzner-Sachs group is topologized as a nuclear Lie group, and it is shown th at irreducible representations arise from either (i) transitive SL{2,C) actions on supermomentum space, or (ii) cylinder measures in supermomentum space with respect to which the SL(2,C) action is strictly ergodic. The irreducibles arising from transitive actions are shown to be induced, and most of the theorems from a previous analysis (in which the group was given a Hilbert topology) are generalized so as to apply here. All non-discrete closed subgroups of 2, C) are found, and this analysis is used to construct all induced representations whose little groups are not both discrete and infinite. In the previous analysis, there were exactly two connected little groups, SU{2) and r (T double covers $0(2)). In the present analysis, exactly one additional connected little group A (which double covers E{2)) arises for faithful representations (that is, those for which the mass squared is defined); the associated mass squared value is zero. Exactly one further connected little group arises;