NUMERICAL METHODS FOR SOLVING PROBLEMS WITH CONTRAST STRUCTURES

In this paper, we investigate the features of the numerical solution of Cauchy problems for nonlinear differential equations with contrast structures (interior layers). Similar problems arise in the modeling of certain problems of hydrodynamics, chemical kinetics, combustion theory, computational geometry. Analytical solution of problems with contrast structures can be obtained only in particular cases. The numerical solution is also difficult to obtain. This is due to the ill conditionality of the equations in the neighborhood of the interior and boundary layers. To achieve an acceptable accuracy of the numerical solution, it is necessary to significantly reduce the step size, which leads to an increase of a computational complexity. The disadvantages of using the traditional explicit Euler method and fourth-order Runge-Kutta method, as well as the implicit Euler method with constant and variable step sizes are shown on the example of one test problem with two boundaries and one interior layers. Two approaches have been proposed to eliminate the computational disadvantages of traditional methods. As the first method, the best parametrization is applied. This method consists in passing to a new argument measured in the tangent direction along the integral curve of the considered Cauchy problem. The best parametrization allows obtaining the best conditioned Cauchy problem and eliminating the computational difficulties arising in the neighborhood of the interior and boundary layers. The second approach for solving the Cauchy problem is a semi-analytical method developed in the works of Alexander N. Vasilyev and Dmitry A. Tarkhov their apprentice and followers. This method allows obtaining a multilayered functional solution, which can be considered as a type of nonlinear asymptotic. Even at high rigidity, a semi-analytical method allows obtaining acceptable accuracy solution of problems with contrast structures. The analysis of the methods used is carried out. The obtained results are compared with the analytical solution of the considered test problem, as well as with the results of other authors

The Use of the Raabe Aspirator in Intraoperative neurophysiological Monitoring during Decompression and Stabilization Interventions for Degenerative Diseases and Injuries of the Lumbar Spine

Background. Raabe probe is a suction device with monopolar motor fibers mapping capabilities. A number of technical characteristics make it possible to use it for intraoperative neurophysiological monitoring during posterior lumbar fusion surgery.The aim of this study was to analyze our experience of Raabe probe using for intraoperative neurophysiological monitoring during posterior lumbar fusion surgery.Methods. Ninety-eight patients (55 women and 43 men) with degenerative changes and injuries of the lumbar spine were included into the study, mean age – 56.3 ± 12.8 years. Patients underwent the following operations: 85 cases (86.7 %) – spinal roots decompression with fusion by dorsal and ventral implants, 12 cases (12.2 %) – decompression with only dorsal fusion, 1 case (1.0 %) – dorsal fusion without decompression. In all cases intraoperative neurophysiological monitoring control by B. Calancie method with Raabe probe using was performed.Results. With a critical current threshold of 12 mA, the sensitivity of the method is 94 %, the specificity is 97 %. Comparing the thresholds of the M-response at the stage of screw stimulation, no statistically significant differences were found between the groups of true-positive and false-positive results, both for interested (p = 0.09) and intact (p = 0,16) screws. At the stage of the impactor stimulation, the threshold of the M-response in the true-positive group made11.39 ± 7.97 mA, and in the false-positive group – 24.16 ± 8.85 mA (p < 0.05).Conclusion. Raabe probe application for intraoperative neurophysiological monitoring during posterior lumbar fusion surgery show the high sensitivity and specificity. The most reliable sign of pedicle wall breach is a threshold below than 12 mA at the stage of the impactor stimulation

High Speed and High Efficiency Travelling Wave Single-Photon Detectors Embedded in Nanophotonic Circuits

Ultrafast, high quantum efficiency single photon detectors are among the most sought-after elements in modern quantum optics and quantum communication. High photon detection efficiency is essential for scalable measurement-based quantum computation, quantum key distribution, and loophole-free Bell experiments. However, imperfect modal matching and finite photon absorption rates have usually limited the maximum attainable detection efficiency of single photon detectors. Here we demonstrate a superconducting nanowire detector atop nanophotonic waveguides which allows us to drastically increase the absorption length for incoming photons. When operating the detectors close to the critical current we achieve high on-chip single photon detection efficiency up to 91% at telecom wavelengths, with uncertainty dictated by the variation of the waveguide photon flux. We also observe remarkably low dark count rates without significant compromise of detection efficiency. Furthermore, our detectors are fully embedded in a scalable silicon photonic circuit and provide ultrashort timing jitter of 18ps. Exploiting this high temporal resolution we demonstrate ballistic photon transport in silicon ring resonators. The direct implementation of such a detector with high quantum efficiency, high detection speed and low jitter time on chip overcomes a major barrier in integrated quantum photonics