"Partial" Fidelities

For pairs, omega, rho, of density operators on a finite dimensional Hilbert space of dimension d I call k-fidelity the d - k smallest eigenvalues of | omega^1/2 rho^1/2 |. k-fidelities are jointly concave in omega, rho. This follows by representing them as infima over linear functions. For k = 0 known properties of fidelity and transition probability are reproduced. Partial fidelities characterize equivalence classes which are partially ordered in a natural way.Comment: LATEX2e, 14 page

H\"older regularity for Maxwell's equations under minimal assumptions on the coefficients

We prove global H\"older regularity for the solutions to the time-harmonic anisotropic Maxwell's equations, under the assumptions of H\"older continuous coefficients. The regularity hypotheses on the coefficients are minimal. The same estimates hold also in the case of bianisotropic material parameters.Comment: 11 page

On multiple frequency power density measurements II. The full Maxwell's equations

We shall give conditions on the illuminations $\varphi_{i}$ such that the solutions to Maxwell's equations $$\left\{ \begin{array}{l} {\rm curl} E^{i}=i\omega\mu H^{i}\qquad\text{in }\Omega,\\ {\rm curl} H^{i}=-i(\omega\varepsilon+i\sigma)E^{i}\qquad\text{in }\Omega,\\ E^{i}\times\nu=\varphi_{i}\times\nu\qquad\text{on }\partial\Omega, \end{array}\right.$$ satisfy certain non-zero qualitative properties inside the domain $\Omega$, provided that a finite number of frequencies $\omega$ are chosen in a fixed range. The illuminations are explicitly constructed. This theory finds applications in several hybrid imaging problems, where unknown parameters have to be imaged from internal measurements. Some of these examples are discussed. This paper naturally extends a previous work of the author [Inverse Problems 29 (2013) 115007], where the Helmholtz equation was studied.Comment: 24 page

Absence of Critical Points of Solutions to the Helmholtz Equation in 3D

The focus of this paper is to show the absence of critical points for the solutions to the Helmholtz equation in a bounded domain $\Omega\subset\mathbb{R}^{3}$, given by $$\left\{ \begin{array}{l} -\rm{div}(a\,\nabla u_{\omega}^{g})-\omega qu_{\omega}^{g}=0\quad\text{in $\Omega$,}\\ u_{\omega}^{g}=g\quad\text{on $\partial\Omega$.} \end{array}\right.$$ We prove that for an admissible $g$ there exists a finite set of frequencies $K$ in a given interval and an open cover $\overline{\Omega}=\cup_{\omega\in K}\Omega_{\omega}$ such that $|\nabla u_{\omega}^{g}(x)|>0$ for every $\omega\in K$ and $x\in\Omega_{\omega}$. The set $K$ is explicitly constructed. If the spectrum of the above problem is simple, which is true for a generic domain $\Omega$, the admissibility condition on $g$ is a generic property.Comment: 14 page

On Multiple Frequency Power Density Measurements

We shall give a priori conditions on the illuminations $\phi_i$ such that the solutions to the Helmholtz equation $-div(a \nabla u^i)-k q u^i=0$ in \Omega, $u^i=\phi_i$ on $\partial\Omega$, and their gradients satisfy certain non-zero and linear independence properties inside the domain \Omega, provided that a finite number of frequencies k are chosen in a fixed range. These conditions are independent of the coefficients, in contrast to the illuminations classically constructed by means of complex geometric optics solutions. This theory finds applications in several hybrid problems, where unknown parameters have to be imaged from internal power density measurements. As an example, we discuss the microwave imaging by ultrasound deformation technique, for which we prove new reconstruction formulae.Comment: 26 pages, 4 figure

Enforcing local non-zero constraints in PDEs and applications to hybrid imaging problems

We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on the solutions of the PDE do not vanish locally inside the domain. Suitable boundary conditions are classically determined by using complex geometric optics solutions. This work focuses on an alternative approach to this issue based on the use of multiple frequencies. Simple boundary conditions and a finite number of frequencies are explicitly constructed independently of the coefficients of the PDE so that the corresponding solutions satisfy the required constraints. This theory finds applications in several hybrid imaging modalities: some examples are discussed.Comment: 24 pages, 2 figure

Animating archaeology: local theories and conceptually open-ended methodologies

Animists’ theories of matter must be given equivalence at the level of theory if we are to understand adequately the nature of ontological difference in the past. The current model is of a natural ontological continuum that connects all cultures, grounding our culturally relativist worldviews in a common world. Indigenous peoples’ worlds are thought of as fascinating but ultimately mistaken ways of knowing the world. We demonstrate how ontologically oriented theorists Eduardo Viveiros de Castro, Karen Barad and Tim Ingold in conjuncture with an anti-representationalist methodology can provide the necessary conditions for alternative ontologies to emerge in archaeology. Anthropo-zoomorphic ‘body-pots’ from first-millennium ad northwest Argentina anticipate the possibility that matter was conceptualized as chronically unstable, inherently undifferentiated, and ultimately practice dependent

Self-gravitating Newtonian models of fermions with anisotropy and cutoff energy in their distribution function

Systems of self-gravitating fermions constitute a topic of great interest in astrophysics, due to the wide field of applications. In this paper, we consider the gravitational equilibrium of spherically symmetric Newtonian models of collisionless semidegenerate fermions. We construct numerical solutions by taking into account the effects of the anisotropy in the distribution function and considering the prevalence of tangential velocity. In this way, our models generalize the solutions obtained for isotropic Fermi-Dirac statistics. We also extend the analysis to equilibrium configurations in the classical regime and in the fully degenerate limit, recovering, for different levels of anisotropy, hollow equilibrium configurations obtained in Maxwellian regime. Moreover, in the limit of full degeneracy, we find a direct expression relating the anisotropy with the mass of the particles composing the system.Comment: 32 pages, 10 figures, accepted for publication in Physical Review

Quantum walk of a Bose-Einstein condensate in the Brillouin zone

We propose a realistic scheme to implement discrete-time quantum walks in the Brillouin zone (i.e., in quasimomentum space) with a spinor Bose-Einstein condensate. Relying on a static optical lattice to suppress tunneling in real space, the condensate is displaced in quasimomentum space in discrete steps conditioned upon the internal state of the atoms, while short pulses periodically couple the internal states. We show that tunable twisted boundary conditions can be implemented in a fully natural way by exploiting the periodicity of the Brillouin zone. The proposed setup does not suffer from off-resonant scattering of photons and could allow a robust implementation of quantum walks with several tens of steps at least. In addition, onsite atom-atom interactions can be used to simulate interactions with infinitely long range in the Brillouin zone.Comment: 9 pages, 4 figures; in the new version, added a discussion about decoherence in the appendi