Pareto optimal structures producing resonances of minimal decay under L1L^1-type constraints

Optimization of resonances associated with 1-D wave equations in inhomogeneous media is studied under the constraint $\| B \|_1 <= m$ on the nonnegative function $B \in L^1 (0,\ell)$ that represents the medium's structure. From the Physics and Optimization points of view, it convenient to generalize the problem replacing $B$ by a nonnegative measure $d M$ and imposing on $d M$ the condition that its total mass is $<= m$. The problem is to design for a given frequency $\alpha \in R$ a medium that generates a resonance $\omega$ on the line $\alpha + i R$ with a minimal possible decay rate $| Im \omega |$. Such resonances are said to be of minimal decay and form a Pareto frontier. We show that corresponding optimal measures consist of finite number of point masses, and that this result yields non-existence of optimizers for the problem over the set of absolutely continuous measures $B(x) dx$. Then we derive restrictions on optimal point masses and their positions. These restrictions are strong enough to calculate optimal $d M$ if the optimal resonance $\omega$, the first point mass $m_1$, and one more geometric parameter are known. This reduces the original infinitely-dimensional problem to optimization over four real parameters. For small frequencies, we explicitly find the Pareto set and the corresponding optimal measures $d M$. The technique of the paper is based on the two-parameter perturbation method and the notion of local boundary point. The latter is introduced as a generalization of local extrema to vector optimization problems.Comment: 38 pages, the proof of Lemma 5.4 is corrected, typos are correcte

Optimization of quasi-normal eigenvalues for 1-D wave equations in inhomogeneous media; description of optimal structures

The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given $\alpha \in \R$ a medium that generates a resonance on the line \alpha + \i \R with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints $0\leq b_1 \leq B (x) \leq b_2$ with certain constants $b_{1,2}$. The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values $b_1$ and $b_2$. This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear eigenvalue problem.Comment: Typos are correcte

Indefinite Sturm-Liouville operators with the singular critical point zero

We present a new necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators. This condition is formulated in terms of Weyl-Titchmarsh $m$-functions. Also we obtain necessary conditions for regularity of the critical points 0 and $\infty$ of $J$-nonnegative Sturm-Liouville operators. Using this result, we construct several examples of operators with the singular critical point zero. In particular, it is shown that 0 is a singular critical point of the operator -\frac{(\sgn x)}{(3|x|+1)^{-4/3}} \frac{d^2}{dx^2} acting in the Hilbert space $L^2(\R, (3|x|+1)^{-4/3}dx)$ and therefore this operator is not similar to a self-adjoint one. Also we construct a J-nonnegative Sturm-Liouville operator of type (\sgn x)(-d^2/dx^2+q(x)) with the same properties.Comment: 24 pages, LaTeX2e <2003/12/01

Hard X-ray properties of NuSTAR blazars

Context. Investigation of the hard X-ray emission properties of blazars is key to the understanding of the central engine of the sources and associated jet process. In particular, simultaneous spectral and timing analyses of the intra-day hard X-ray observations provide us a means to peer into the compact innermost blazar regions, not accessible to our current instruments. Aims. The primary objective of the work is to associate the observed hard X-ray variability properties in the blazars to their flux and spectral states, thereby, based on the correlation among them, extract the details about the emission regions and the processes occurring near the central engine. Methods. We carried out timing, spectral and cross-correlation analysis of 31 NuSTAR observations of 13 blazars. We investigated the spectral shapes of the sources using single power-law, broken power-law and log-parabola models. We also studies the co-relation between the soft and the hard emission using z-transformed discrete correlation function. Results. We found that for most of the sources the hard X-ray emission can be well represented by log-parabola model; and that the spectral slopes for different blazar sub-classes are consistent with so called blazar sequence. We noted a close connection between the flux and spectral slope within the source sub-class in the sense that high flux and/or flux states tend to be harder in spectra. In BL Lacertae objects, assuming particle acceleration by diffusive shocks and synchrotron cooling as the dominant processes governing the observed flux variability, we constrain the magnetic field of the emission region to be a few gauss; whereas in flat-spectrum radio quasars, using external Compton models, we estimate the energy of the lower end of the injected electrons to be a few Lorentz factors.Comment: 12 figures, 21 pages, A&A accepte