Fréchet differentiability for a damped sine-Gordon equation

We consider a damped sine-Gordon equation with a variable diffusion coefficient. The goal is to derive necessary conditions for the optimal set of parameters minimizing the objective function J. First, we show that the solution map is continuous under a weak assumption on the topology of the admissible set P. Then the solution map is shown to be weakly Gâteux differentiable on P, implying the Gâteux differentiability of the objective function. Finally we show the Fréchet differentiability of J. The optimal set of parameters is shown to satisfy a bang–bang control law.     

Novel carboxylated pyrrole- and carbazole-based monomers. Synthesis and electro-oxidation features

Carboxylated pyrrole (Pyr, a index)- and carbazole (Cbz, b index)-containing monomers 6-7a/b and 9a/b have been readily synthesized from the monobenzyl ester of L-glutamic acid and triamine 2 using Clauson-Kaas and amide coupling reactions. In contrast to Pyr-containing compounds 6-7a, and 9a, the three Cbz-containing monomers 6-7b, and 9b have been found electroactive and were successfully electropolymerized on a Pt electrode resulting in the deposition of corresponding insoluble electroconducting polyCOOH polyCbz-films poly(6-7b) and poly(9b).     

Frequency conversion in novel materials and its application to high resolution gas sensing

We have constructed a compact room-temperature mid-infrared spectrometer and gas sensor, based on quasi-phase matched difference-frequency generation in periodically poled ferroelectric crystals of the KTiOPO₄ family, namely: KTiOPO₄, KTiOAsO₄ and RbTiOAsO₄. The wide tunabilty of the spectrometer (3.1–3.75 μm) enables us to cover an entire vibrational band of gases such as methane and nitrous oxide. The high spectral resolution (1 MHz) is used to investigate the spectral profile of the hyperfine components of a single rotational transition. The sensitivity of the described spectrometer is 75 parts per million. Applications of this technology include the detection of polluting or toxic gases, biomedical sensing, atmospheric research, volcanic monitoring and industrial process control.     

Monotonicity testing and shortest-path routing on the cube

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions $f:\{ 0,1\} ^n \to \mathcal{R}$ can be tested with O(n/∈) queries, for any totally ordered range $\mathcal{R}$ . More generally, if at least a µ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(∈µ(n))). We construct a family of instances of Ω(2 ⁿ ) pairs in n-dimensional hypercubes such that no more than roughly a $\frac{1} {{\sqrt n }}$ fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [16], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n ^3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture about routing on the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not $n^{\tfrac{1} {2} - \delta }$ -realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/∈) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.     

Spectral and temporal holograms with nonlinear optics

In this Letter we show how encoding techniques for computer-generated holograms may be used to arbitrarily shape a nonlinearly generated spectrum and consequently the temporal shape by modulating the quadratic nonlinear coefficient. We give examples of a modulation pattern and a simple setup that can generate high-order Hermite-Gauss and Airy functions through difference-frequency generation from a transform-limited Gaussian pulse, under practical fabrication considerations.     

Modeling the power of an optical parametric generator by discrete mode summation

An analytical expression for calculating the signal output power of an optical parametric generator (OPG) configuration was developed. The model is based on Heisenberg equations in space and radiation mode theory. A simple analytical expression can be obtained by assuming that all modes within the gain bandwidth of the nonlinear crystal have the same gain and the same interaction length. Another derivation considers the gain and interaction length of each individual mode. The model predictions are in excellent agreement with the measured signal power of OPGs based on 25- and 35-mm periodically poled LiNbO₃ crystal (with effective quadratic nonlinearity of ∼14.5 pm/V) and 47-mm periodically poled LiTaO₃ crystal (with effective quadratic nonlinearity of ∼9 pm/V). In addition, a comparison was made between the summation over discrete modes approach and the traditional approach of continuous integration over the beam parameters and pump frequency. We have found that the first approach, which is developed in this paper, predicts more accurately the output power of the OPG.     

X-ray source based on a compact electron linear accelerator with a thin anode

An experiment confirming the possibility of extracting a narrow energy band from a wide spectrum produced by a compact electron linear accelerator with a thin transmission anode is described. A flat crystal of pyrolytic graphite was used as a monochromator.