Extremal covariant POVM's

We consider the convex set of positive operator valued measures (POVM) which are covariant under a finite dimensional unitary projective representation of a group. We derive a general characterization for the extremal points, and provide bounds for the ranks of the corresponding POVM densities, also relating extremality to uniqueness and stability of optimized measurements. Examples of applications are given.Comment: 15 pages, no figure

Optimization of quasi-normal eigenvalues for Krein-Nudelman strings

The paper is devoted to optimization of resonances for Krein strings with total mass and statical moment constraints. The problem is to design for a given $\alpha \in \R$ a string that has a resonance on the line \alpha + \i \R with a minimal possible modulus of the imaginary part. We find optimal resonances and strings explicitly.Comment: 9 pages, these results were extracted in a slightly modified form from the earlier e-print arXiv:1103.4117 [math.SP] following an advise of a journal's refere

Rotating Electromagnetic Waves in Toroid-Shaped Regions

Electromagnetic waves, solving the full set of Maxwell equations in vacuum, are numerically computed. These waves occupy a fixed bounded region of the three dimensional space, topologically equivalent to a toroid. Thus, their fluid dynamics analogs are vortex rings. An analysis of the shape of the sections of the rings, depending on the angular speed of rotation and the major diameter, is carried out. Successively, spherical electromagnetic vortex rings of Hill's type are taken into consideration. For some interesting peculiar configurations, explicit numerical solutions are exhibited.Comment: 27 pages, 40 figure

The nonconforming virtual element method for eigenvalue problems

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L^2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice

Nonlinear eigenvalue problem for optimal resonances in optical cavities

The paper is devoted to optimization of resonances in a 1-D open optical cavity. The cavity's structure is represented by its dielectric permittivity function e(s). It is assumed that e(s) takes values in the range 1 <= e_1 <= e(s) <= e_2. The problem is to design, for a given (real) frequency, a cavity having a resonance with the minimal possible decay rate. Restricting ourselves to resonances of a given frequency, we define cavities and resonant modes with locally extremal decay rate, and then study their properties. We show that such locally extremal cavities are 1-D photonic crystals consisting of alternating layers of two materials with extreme allowed dielectric permittivities e_1 and e_2. To find thicknesses of these layers, a nonlinear eigenvalue problem for locally extremal resonant modes is derived. It occurs that coordinates of interface planes between the layers can be expressed via arg-function of corresponding modes. As a result, the question of minimization of the decay rate is reduced to a four-dimensional problem of finding the zeroes of a function of two variables.Comment: 16 page