Topology of the space of conormal distributions

Given a closed manifold $M$ and a closed regular submanifold $L$, consider the corresponding locally convex space $I=I(M,L)$ of conormal distributions, with its natural topology, and the strong dual $I'=I'(M,L)=I(M,L;\Omega)'$ of the space of conormal densities. It is shown that $I$ is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and $I'$ is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace $K\subset I$ of conormal distributions supported in $L$ and for its strong dual $K'$. We construct a locally convex Hausdoff space $J$ and a continuous linear map $I\to J$ such that the sequence $0\to K\to I\to J\to 0$ as well as the transpose sequence $0\to J'\to I'\to K'\to 0$ are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that $I\cap I'=C^\infty(M)$ in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow $\phi=\{\phi^t\}$ on a compact foliated manifold $(M,F)$. It describes a Lefschetz distribution $L_{\text{\rm dis}}(\phi)$ defined by the induced action $\phi^*=\{\phi^{t\,*}\}$ on the reduced cohomologies $\bar H^\bullet I(F)$ and $\bar H^\bullet I'(F)$ of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $\phi$.Comment: 55 pages, index of notatio

A Trace Formula for Foliated Flows (working paper)

The talk, based on work in progress, will be about our progress to show a trace formula for foliated flows on foliated spaces, which has been conjectured by V. Guillemin, and later by C. Deninger with more generality. It describes certain Leftchetz distribution of the foliated flow, acting on some version of the leafwise cohomology, in terms of local data at the closed orbits and fixed points

Zeta invariants of Morse forms

Given a closed real 1-form $\eta$ on a closed Riemannian manifold $(M,g)$, let $d_z$, $\delta_z$ and $\Delta_z$ be the induced Witten's type perturbations of the de~Rham derivative and coderivative and the Laplacian on differential forms on $M$, parametrized by $z\in\mathbb C$, and let $\zeta(s,z)$ be the zeta function of $s\in\mathbb C$ given by $\zeta(s,z)=\operatorname{Tr}^s({\eta\wedge}\,\delta_z\Delta_z^{-s})$ when $\Re s\gg0$. For a class of Morse forms $\eta$, we prove that $\zeta(s,z)$ is smooth at $s=1$ for $|\Re z|\gg0$, and the zeta invariant $\zeta(1,z)$ converges to some $\mathbf z\in\mathbb R$ as $\Re z\to+\infty$, uniformly on $\Im z$. We describe $\mathbf z$ in terms of the instantons of an auxiliary Smale gradient-like vector field $X$ and the Mathai-Quillen current on $TM$ defined by $g$. Any real cohomology class has a representative $\eta$ satisfying the needed hypothesis. If $n$ is even, we can prescribe any real value for $\mathbf z$ by perturbing $g$, $\eta$ and $X$; moreover, we can also achieve the same limit as $\Re z\to-\infty$. This is used to define and describe certain tempered distributions induced by $g$ and $\eta$. These distributions appear in another publication as the contributions from the compact leaves preserved by the flow in a trace formula for simple foliated flows on closed foliated manifolds, which gives a solution to a problem proposed by C.~Deninger.Comment: 69 pages, 1 figur

Antenna measurement and diagnostics processing techniques using unmanned aerial vehicles

European Conference on Antennas and Propagation (EuCAP) (13th. 2019. Krakow, Poland

Unmanned aerial system for antenna measurement and diagnosis: evaluation and testing

This contribution analyses the performance of an unmanned aerial system for antenna measurement (UASAM) for different kinds of measurement scenarios. UASAM is conceived for antenna diagnostics and characterisation at the operational location of the antenna under test (AUT). The system measures the amplitude of the near field radiated by the AUT. Then, these measurements are post-processed using phase retrieval techniques and equivalent currents methods to obtain an electromagnetic model of the AUT. This model can be used for antenna diagnostics and for evaluating the far field pattern. Similar to antenna measurement systems in anechoic chamber, UASAM allows defining different acquisition grids depending on the type of AUT (planar, cylindrical, arc cylindrical), which also influences the flight time. In addition to this, the capability to measure circularly polarised antennas from amplitude-only measurements is presented, discussing the limitations found during the tests, and comparing the results with those from measurements at a spherical range in an anechoic chambe