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 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

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

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

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

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

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

Counts-in-Cells of subhaloes in the IllustrisTNG simulations: the role of baryonic physics

We present an analysis of the Counts-in-Cells (CiC) statistics of subhaloes in the publicly available IllustrisTNG cosmological simulations (TNG100-1, TNG100-3 and TNG300-3), considering their full and dark-only versions, in redshifts ranging from $z = 0$ to $z=5$, and different cell sizes. We evaluated two CiC models: the gravitational quasi-equilibrium distribution (GQED) and the negative binomial distribution (NBD), both presenting good fits, with small detectable differences in the presence of baryons. Scaling and time dependencies of the best-fit parameters showed similar trends compared with the literature. We derived a matter density-in-cells probability distribution function (PDF), associated with the GQED, which was compared to the PDF given in Uhlemann et al. (2016), for the IllustrisTNG 100-3-Dark run at $z=0$. Our results indicate that the simplest gravithermodynamical assumptions of the GQED model hold in the presence of baryonic dissipation. Interestingly, the smoothed (density-in-cells) version of the GQED is also adequate for describing the dark matter one-point statistics of subhaloes and converges, to subpercentage levels (for an interval of parameters), to the Uhlemann et al. PDF in the high density range.Comment: Revised version, 17 pages, 9 figures, with clarifications and new results concerning the PDFs of density-in-cells statistic

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

Non-linear (loop) quantum cosmology

Inhomogeneous quantum cosmology is modeled as a dynamical system of discrete patches, whose interacting many-body equations can be mapped to a non-linear minisuperspace equation by methods analogous to Bose-Einstein condensation. Complicated gravitational dynamics can therefore be described by more-manageable equations for finitely many degrees of freedom, for which powerful solution procedures are available, including effective equations. The specific form of non-linear and non-local equations suggests new questions for mathematical and computational investigations, and general properties of non-linear wave equations lead to several new options for physical effects and tests of the consistency of loop quantum gravity. In particular, our quantum cosmological methods show how sizeable quantum corrections in a low-curvature universe can arise from tiny local contributions adding up coherently in large regions.Comment: 20 page