Limit cycles for a class of eleventh Z12−\mathbb{Z}_{12}-equivariant systems without infinite critical points

We analyze the complex dynamics dynamics of a family of $\mathbb{Z}_{12}-$equivariant planar systems, by using their reduction to an Abel equation. We derive conditions in the parameter space that allow uniqueness and hyperbolicity of a limit cycle surrounding either $1,~13$ or $25$ equilibria.Comment: 12 pages, 1 figur

Application of K-integrals to radiative transfer in layered media

Simple yet accurate results for radiative transfer in layered media with discontinuous refractive index are obtained by the method of K-integrals, originally developed for neutron transport analysis. These are certain weighted integrals applied to the angular intensity distribution at the refracting boundaries. The radiative intensity is expressed as the sum of the asymptotic angular intensity distribution valid in the depth of the scattering medium and a transient term valid near the boundary. Integral boundary conditions are obtained from the vanishing of the K-integrals of the boundary transient, yielding simple linear equations for the intensity coefficients (two for a halfspace, four for a slab or an interface), enabling the angular emission intensity and the diffuse reflectance (albedo) and transmittance of the scattering layer to be calculated. The K-integral method is orders of magnitude more accurate than diffusion theory and can be applied to scattering media with a wide range of scattering albedoes. For example, near five figure accuracy is obtained for the diffuse reflectance of scattering layers of refractive index n = 1.5 with single scattering albedo in the range 0.3 to 1Comment: 38 pages, 8 tables, 14 figure

Oscillation patterns in tori of modified FHN neurons

We analyze the dynamics of a network of electrically coupled, modified FitzHugh-Nagumo (FHN) oscillators. The network building-block architecture is a bidimensional squared array shaped as a torus, with unidirectional nearest neighbor coupling in both directions. Linear approximation about the origin of a single torus, reveals that the array is able to oscillate via a Hopf bifurcation, controlled by the interneuronal coupling constants. Group theoretic analysis of the dynamics of one torus leads to discrete rotating waves moving diagonally in the squared array under the influence of the direct product group $\mathbb{Z}_N\times\mathbb{Z}_N\times\mathbb{Z}_2\times\mathbb{S}^1.$ Then, we studied the existence multifrequency patterns of oscillations, in networks formed by two coupled tori. We showed that when acting on the traveling waves, this group leaves them unchanged, while when it acts on the in-phase oscillations, they are shifted in time by $\phi.$ We therefore proved the possibility of a pattern of oscillations in which one torus produces traveling waves of constant phase shift, while the second torus shows synchronous in-phase oscillations, at $N-$ times the frequency shown by the traveling waves.Comment: 23, 5 figure

The Energetics of Particle Acceleration Using High Intensity Lasers

We point out that even the most intense laser beams available today can provide only a very small fraction of the beam energy required to reach the design luminosity for a future e+e- linear collider. This fact seems to have been overlooked in the extensive literature on laser acceleration of charged particles

The Cyclic Hopf H mod K Theorem

The $H~\mathrm{mod}~K$ theorem gives all possible periodic solutions in a $\Gamma-$equivariant dynamical system, based on the group-theoretical aspects. In addition, it classifies the spatio temporal symmetries that are possible. By the contrary, the equivariant Hopf theorem guarantees the existence of families of small-amplitude periodic solutions bifurcating from the origin for each $\mathbf{C}-$axial subgroup of $\Gamma\times\mathbb{S}^1.$ In this paper we identify which periodic solution types, whose existence is guaranteed by the $H~\mathrm{mod}~K$ theorem, are obtainable by Hopf bifurcation, when the group $\Gamma$ is finite cyclic.Comment: 6 pages in Mathematical Reports (2015

Use of Slow Light to test the Isotropy of Space

It is suggested that slow light could be used to test for relative motion with respect to an absolute reference frame at a sensitivity v ~ 10^{-3} m/s

Search for Higher Dimensions through their Gravitational Effects in High Energy Collisions

We consider the use of a microwave parametric converter for the direct detection of gravitational effects at the LHC. Because of the extra dimensions the strength of the gravitational interaction in the bulk grows at high energies. This leads to possibly detectable signals

Saturable absorption and 'slow light'

Quantitative evaluation of some recent 'slow light' experiments based on coherent population oscillations (CPO) shows that they can be more simply interpreted as saturable absorption phenomena. Therefore they do not provide an unambiguous demonstration of 'slow light'. Indeed a limiting condition on the spectral bandwidth is not generally satisfied, such that the requirements for burning a narrow spectral hole in the homogeneously broadened absorption line are not met. Some definitive tests of 'slow light' phenomena are suggested, derived from analysis of phase shift and pulse delay for a saturable absorberComment: 11 pages 4 figures (data analysis of bacteriorhodopsin 'slow light' experiment by Wu and Rao: Phys Rev Letters 95 253601 (2005) added

The refractive index of the vacuum and the dark sector

We discuss a recent result about the refractive index of the vacuum and compare it with existing limits. We consider a possible connection with the dark sector.Comment: 4 page

Path Integral Quantization of Volume

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $R$ be a compact set inside $\mathbf{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $V_R(e)$ as the volume of $R$, for a given choice of $e$. The Einstein-Hilbert action $S(e,\omega)$ is defined on $e$ and $\omega$. We will quantize the volume of $R$ by integrating $V_R(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $R$, and the exponential of the Einstein-Hilbert action, over the space of vierbein $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the volume operator can be computed by counting the number of half-twists in the projected hyperlink, which lie inside $R$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the volume operator gives the total kinetic energy, which comes from translational and angular momentum