Negative index of refraction, perfect lenses and transformation optics -- some words of caution

In this paper we show that a negative index of refraction is not a direct implication of transformation optics with orientation-reversing diffeomorphisms. Rather a negative index appears due to a specific choice of sign freedom. Furthermore, we point out that the transformation designed lens, which relies on the concept of spacetime folding, does not amplify evanescent modes, in contrast to the Pendry-Veselago lens. Instead, evanescent modes at the image point are produced by a duplicated source and thus no imaging of the near field (perfect lensing) takes place.Comment: 13 pages, 3 figures, LaTe

Light propagation in local and linear media: Fresnel-Kummer wave surfaces with 16 singular points

It is known that the Fresnel wave surfaces of transparent biaxial media have 4 singular points, located on two special directions. We show that, in more general media, the number of singularities can exceed 4. In fact, a highly symmetric linear material is proposed whose Fresnel surface exhibits 16 singular points. Because, for every linear material, the dispersion equation is quartic, we conclude that 16 is the maximum number of isolated singularities. The identity of Fresnel and Kummer surfaces, which holds true for media with a certain symmetry (zero skewon piece), provides an elegant interpretation of the results. We describe a metamaterial realization for our linear medium with 16 singular points. It is found that an appropriate combination of metal bars, split-ring resonators, and magnetized particles can generate the correct permittivity, permeability, and magnetoelectric moduli. Lastly, we discuss the arrangement of the singularities in terms of Kummer's (16,6)-configuration of points and planes. An investigation parallel to ours, but in linear elasticity, is suggested for future research.Comment: 7 pages, 3 figure

The Kummer tensor density in electrodynamics and in gravity

Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\cal T}^{ijkl} = - {\cal T}^{jikl} = - {\cal T}^{ijlk}$. Thus, ${\cal K}\sim {\cal T}^3$, see Eq.(46). (i) If $\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If $\cal T$ is identified with the {\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then ${\cal K}\sim R^3$ and, in the special case of general relativity, ${\cal K}$ reduces to the Kummer tensor of Zund (1969). This $\cal K$ is related to the {\it principal null directions} of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose $\cal K$ irreducibly under the 4-dimensional linear group $GL(4,R)$ and, subsequently, under the Lorentz group $SO(1,3)$.Comment: 54 pages, 6 figures, written in LaTex; improved version in accordance with the referee repor

Comment on: Reply to comment on `Perfect imaging without negative refraction'

Whether or not perfect imaging is obtained in the mirrored version of Maxwell's fisheye lens is debated in the comment/reply sequence [Blaikie-2010njp, Leonhardt-2010njp] discussing Leonhardt's original paper [Leonhardt-2009njp]. Here we show that causal solutions can be obtained without the need for an "active localized drain", contrary to the claims in [Leonhardt-2010njp].Comment: v2 (added MEEP ctl file), v3 (publisher statement

Infinitely wide limits for deep Stable neural networks: sub-linear, linear and super-linear activation functions

There is a growing literature on the study of large-width properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed parameters or weights, and Gaussian stochastic processes. Motivated by some empirical and theoretical studies showing the potential of replacing Gaussian distributions with Stable distributions, namely distributions with heavy tails, in this paper we investigate large-width properties of deep Stable NNs, i.e. deep NNs with Stable-distributed parameters. For sub-linear activation functions, a recent work has characterized the infinitely wide limit of a suitable rescaled deep Stable NN in terms of a Stable stochastic process, both under the assumption of a ``joint growth" and under the assumption of a ``sequential growth" of the width over the NN's layers. Here, assuming a ``sequential growth" of the width, we extend such a characterization to a general class of activation functions, which includes sub-linear, asymptotically linear and super-linear functions. As a novelty with respect to previous works, our results rely on the use of a generalized central limit theorem for heavy tails distributions, which allows for an interesting unified treatment of infinitely wide limits for deep Stable NNs. Our study shows that the scaling of Stable NNs and the stability of their infinitely wide limits may depend on the choice of the activation function, bringing out a critical difference with respect to the Gaussian setting.Comment: 20 pages, 2 figure