A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.

We describe a new method to reconstruct the permittivity distribution, of an object to image, from the remotely measured electromagnetic field. We propose to use the remote fields measured before and after injecting locally in the medium plasmonic nanoparticles. Such a technique is known in the framework of imaging using contrast agents where, in optical imaging, the nanoparticles play the role of these contrast agents. The plasmonic nanoparticles are known to enjoy resonant effects, as enhancing the applied incident field, while excited at certain particular frequencies called plasmonic resonances. These resonant frequencies encode the values of the unknown permittivity at the location of the injected nanoparticles. The imaging methods we propose mainly use this resonant effect. We show that the imaging functional build up from contrasting the fields before and after injecting the nanoparticles, measured at one single back-scattered direction, and in an explicit band of incident frequencies, reaches its maximum values, in terms of the incident frequency, precisely at the mentioned plasmonic resonances. Such a behaviour allows us to recover these plasmonic resonances from which we recover the point-wise values of the permittivity distribution. In this work, we describe the method and provide the mathematical justification of this resonant effect and its use for the optical inversion using plasmonic nanoparticles as contrast agents.

We consider equations involving the truncated Laplacians ${\cal P}_k^\pm$ and having lower-order terms with singular potentials posed in punctured balls. We study both the principal eigenvalue problem and the problem of classification of solutions, in dependence on their asymptotic behaviour near the origin, for equations having also superlinear absorption lower order terms. In the case of ${\cal P}_k^+$, owing to the mild degeneracy of the operator, we obtain results which are analogous to the results for the Laplacian in dimension $k$. On the other hand, for operator ${\cal P}_k^-$, we show that the strong degeneracy in ellipticity of the operator produces radically different results.