Technical Notes on Classical Electromagnetism, with exercises

The present technical notes offer a brief summary of the essential points of electromagnetism at the undergraduate physics level. Some problems are presented at the end of each section; those with solutions are marked with an asterisk.Comment: 34 page

An Approach to Loop Quantum Cosmology Through Integrable Discrete Heisenberg Spin Chains

The quantum evolution equation of Loop Quantum Cosmology (LQC) -- the quantum Hamiltonian constraint -- is a difference equation. We relate the LQC constraint equation in vacuum Bianchi I separable (locally rotationally symmetric) models with an integrable differential-difference nonlinear Schr\"odinger type equation, which in turn is known to be associated with integrable, discrete Heisenberg spin chain models in condensed matter physics. We illustrate the similarity between both systems with a simple constraint in the linear regime.Comment: 6 pages, accepted for publication in Foundations of Physics; minor changes in accordance with referee's suggestio

An Inhomogeneous Space-Time Patching Model Based on a Nonlocal and Nonlinear Schrodinger Equation

We consider an integrable, nonlocal and nonlinear, Schr\"odinger equation (NNSE) as a model for building space-time patchings in inhomogeneous loop quantum cosmology (LQC). We briefly review exact solutions of the NNSE, specially those obtained through "geometric equivalence" methods. Furthemore, we argue that the integrability of the NNSE could be linked to consistency conditions derived from LQC, under the assumption that the patchwork dynamics behaves as an integrable many-body system.Comment: 29 pages, 2 figures, accepted for publication in Foundations of Physic

A Note on Norton's Dome

"Norton's Dome" is an example of a Newtonian system that violates the Lipschitz condition at a single point, leading to non-unique solutions (indeterminism). Here we reformulate this problem into a "weak" form (in the sense of distributions). In our description the indeterminism manifests through the problematic interpretation of initial conditions, since distributions (as linear functionals on the space of test functions) do not have values at individual points.Comment: 11 pages, no figures, added a new reference, typos corrected; comments welcom

FC-groups with finitely many automorphism orbits

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called "automorphism orbits" of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. In this paper we prove that if $G$ is an FC-group with finitely many automorphism orbits, then the derived subgroup $G'$ is finite and $G$ admits a decomposition $G = Tor(G) \times D$, where $Tor(G)$ is the torsion subgroup of $G$ and $D$ is a divisible characteristic subgroup of $Z(G)$. We also show that if $G$ is an infinite FC-group with $\omega(G) \leqslant 8$, then either $G$ is soluble or $G \cong A_5 \times H$, where $H$ is an infinite abelian group with $\omega(H)=2$. Moreover, we describe the structure of the infinite non-soluble FC-groups with at most eleven automorphism orbits.Comment: Submitted to an internacional journa

On state-closed representations of restricted wreath product of groups of type G_{p,d}=C_{p}wrC^{d}

Let $G_{p,d}$ be the restricted wreath product $C_{p} wr C^{d}$ where $C_{p}$ is a cyclic group of order a prime $p$ and $C^{d}$ a free abelian group of finite rank $d$. We study the existence of faithful state-closed (fsc) representations of $G_{p,d}$ on the $1$-rooted $m$-ary tree for some finite $m$. The group $G_{2,1}$, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for $d \geq 2$ there are no fsc representations of $G_{p,d}$ on the $p$ -adic tree. We characterize all fsc representations of $G=G_{p,1}$ on the $p$-adic tree where the first level stabilizer of the image of $G$ contains its commutator subgroup. Furthermore, for $d \geq 2$, we construct uniformly fsc representations of $G_{p,d}$ on the $p^{2}$ -adic tree and exhibit concretely the representation of $G_{2,2}$ on the $4$-tree as a finite-state automaton group

Energy Ranking Preservation in a N-Body Cosmological Simulation

In this paper we present a study of the cosmic flow from the point of view of how clusterings at different dynamical regimes in an expanding universe evolve according to a `coarse-grained' partitioning of their ranked energy distribution. By analysing a Lambda-CDM cosmological simulation from the Virgo Project, we find that cosmic flows evolve in an orderly sense, when tracked from their coarse-grained energy cells, even when nonlinearities are already developed. We show that it is possible to characterize scaling laws for the Pairwise Velocity Distribution in terms of the energy cells, generally valid at the linear and nonlinear clustering regimes.Comment: 8 pages, 3 figures, accepted for publication in the MNRA

Dependence of microwave absorption properties on ferrite volume fraction in MnZn ferrite/rubber radar absorbing materials

We report the analysis of measurements of the complex magnetic permeability ($\mu_r$) and dielectric permittivity ($\epsilon_r$) spectra of a rubber radar absorbing material (RAM) with various MnZn ferrite volume fractions. The transmission/reflection measurements were carried out in a vector network analyzer. Optimum conditions for the maximum microwave absorption were determined by substituting the complex permeability and permittivity in the impedance matching equation. Both the MnZn ferrite content and the RAM thickness effects on the microwave absorption properties, in the frequency range of 2 to 18 GHz, were evaluated. The results show that the complex permeability and permittivity spectra of the RAM increase directly with the ferrite volume fraction. Reflection loss calculations by the impedance matching degree (reflection coefficient) show the dependence of this parameter on both thickness and composition of RAM.Comment: 9 pages, 6 figures; accepted for the Journal of Magnetism and Magnetic Material

A Self-Consistent Extrapolation Method for the Complex Permittivity and Permeability Based on Finite Frequency Data

We describe a method of extrapolation based on a "truncated" Kramers-Kronig relation for the complex permittivity ($\epsilon$) and permeability ($\mu$) parameters of a material, based on finite frequency data. Considering a few assumptions, such as the behavior of the loss tangent and the overall nature of corrections, the method is robust within a small relative error, if the assumed hypotheses hold at the extrapolated frequency range.Comment: 27 pages, 19 figures, accepted for publication in the Journal of Computational Interdisciplinary Sciences (JCIS); this version matches the accepted versio

The two-component virial theorem and the acceleration-discrepancy relation

We revisit the "two-component virial theorem" (2VT) in the light of recent theoretical and observational results related to the "dark matter"(DM) problem. This modification of the virial theorem offers a physically meaningful framework to investigate possible dynamical couplings between the baryonic and DM components of extragalactic systems. In particular, we examine the predictions of the 2VT with respect to the "acceleration-discrepancy relation" (ADR). Considering the combined data (composed of systems supported by rotation and by velocity dispersion), we find that: (i) the overall behavior of the 2VT is consistent with the ADR; and (ii) the 2VT predicts a nearly constant behavior in the lower acceleration regime, as suggested in recent data on dwarf spheroidals. We also briefly comment on possible differentiations between the 2VT and some modified gravity theories.Comment: 8 pages, 4 figures, appendices. Submitted to MNRAS. A substantially shortened and clarified version in order to address the referee's repor