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Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let In another article to appear, we will prove that when

This paper analyzes how distortion in the primary current waveform affects the magnetic flux density in the magnetic core of an inductive current transformer and its transformation accuracy. Keeping the primary current’s RMS value constant, it studies the impact of changes in the RMS values and phase angles of low-order harmonics on the core’s flux density and the values of current error and phase displacement of their transformation. The distorted current waveforms, resulting flux density, and hysteresis loops are examined to identify the operating conditions of the inductive current transformer. This study also highlights the strong influence of low-order harmonics and the diminishing effect of higher-frequency harmonics on the magnetic flux density in its magnetic core, e.g., third, fifth, and seventh higher harmonics may cause an increase in magnetic flux density in the magnetic core of the inductive current transformer in relation to that obtained for a sinusoidal current with a frequency of 50 Hz by about 8.5%, while with additional second, fourth, and sixth harmonics, the increase may reach about 23%. Therefore, the testing procedure should consider not only the load impedance and the RMS values of the primary current but also its harmonic content, including the RMS values of individual harmonics and their phase angles.

Material nonlinearity is explored for the assessment of structural integrity. Crack–wave interactions are of particular interest. The major focus is on higher-order harmonics, generated in propagating shear horizontal (SH) waves. These harmonics are generated due to global material nonlinearity and local effects such as fatigue cracks. The theoretical background of the proposed method is explained. The method is examined using numerical simulations and experimental tests. The former involves the Local Interaction Simulation Approach (LISA), implemented for the nonlinear shear horizontal wavefield. The latter is based on a high-frequency shear excitation approach. Experimental tests are conducted using a series of beam specimens with fatigue cracks. Low-profile, surface-bonded piezoceramic shear actuators are used for excitation. The excitation frequency is selected to minimize the number of generated modes in the examined specimens. Nonlinear ultrasonic responses are collected using a non-contact laser vibrometer. The results show that higher-order harmonic generation—based on shear horizontal wave propagation—can be used for crack detection in the presence of global material nonlinearity.

In the presented paper, an analytical approach based on the Harmonic Balance method is used to evaluate amplitudes of higher-order harmonics generated due to the interaction of a propagating shear wave with surfaces in frictional contact. The motion between surfaces is initially assumed to be of symmetric hysteretic stick-and-slip type. Experimental investigation is performed on steel samples with prepared contact surfaces, in order to validate the introduced solution approach. It shows that the hysteretic behavior obtained from the experiment exhibits asymmetric characteristic features. Consequently, a new asymmetric description of the hysteretic stress–strain relation is developed and used to confirm these findings. As a result, both qualitative and quantitative agreement between considered investigation methods are achieved. Finally, the identification study shows the impact of higher-order harmonics on particular features of the stiffness hysteresis.

Most of the basic control methods of the grid-connected converter (GCC) are defined to work with a sine wave grid voltage. In that case if the grid voltage is distorted by higher harmonics, the grid current may be distorted too, which, in consequence, may increase the value of the THD of the grid voltage. The paper deals with an improved finite control set model predictive control (FCS-MPC) method of an LCL-filtered GCC operating under distorted grid conditions. The proposed method utilizes supplementary grid current feedback to calculate the reference converter current. The introduced signal allows to effectively improve the operation when the grid is subject to harmonic distortion. The paper shows a simulation analysis of the proposed control scheme operating with and without additional feedback under grid distortions. To validate the practical feasibility of the proposed method an algorithm was implemented on a 32-bit microcontroller STM32F7 with a floating point unit to control a 10 kW GCC. The laboratory test setup provided experimental results showing properties of the introduced control scheme.

The effect of zero-sequence current of the fundamental and higher harmonics on the functioning of a low-voltage electric network has been estimated. To reduce the negative effect of zero-sequence current, it has been proposed to use special technical means: zero-sequence current compensators. The principles of designing such devices in the form of longitudinal and transverse inclusion of various combinations of dual electromagnetic devices with one or two windings that are similar to harmonic filters of low or high frequencies have been shown. A mathematical model of one of the possible circuits of a zero-sequence current compensator of the fundamental and higher harmonics and an experimental estimate of the operability of such a device has been provided. The efficiency of these devices for reducing zero-sequence currents has been shown.

This paper presents an investigation of self-focusing of a q-Gaussian laser beam and its effect on harmonic generation in a preformed collisionless parabolic plasma channel. In the presence of a q-Gaussian laser beam, the carriers get redistributed from high field region to low field region on account of ponderomotive force as a result of which a transverse density gradient is produced in the channel which in turn generates plasma wave at pump frequency. Generated plasma wave interacts with the incident laser beam and generate higher harmonics of the incident laser beam. Moment theory has been used to derive differential equation for the spot size of laser beam propagating through the channel. The differential equation so obtained has been solved numerically. The effect of the intensity of laser beam, deviation of intensity distribution of laser beam along its wave front from Gaussian distribution, plasma density and depth of channel on beam width of laser beam and harmonic yield has been investigated. The effect of order of higher harmonic on harmonic yield has been also investigated.

Plasmonic antennas can enhance the intensity of a nanojoule laser pulse by localizing the electric field in their proximity. It has been proposed that the field can become strong enough to convert the fundamental laser frequency into high-order harmonics through an extremely nonlinear interaction with gas atoms that occupy the nanoscopic volume surrounding the antennas. However, the small number of gas atoms that can occupy this volume limits the generation of high harmonics. Here we use an array of monopole nano-antennas to demonstrate plasmon-assisted high-harmonic generation directly from the supporting crystalline silicon substrate. The high density of the substrate compared with a gas allows macroscopic buildup of harmonic emission. Despite the sparse coverage of antennas on the surface, harmonic emission is ten times brighter than without antennas. Imaging the high-harmonic radiation will allow nanometre and attosecond measurement of the plasmonic field8 thereby enabling more sensitive plasmon sensors while opening a new path to extreme-ultraviolet-frequency combs.

The article presents a solution to the problem of optimizing the structure of filter-compensating devices (FCD) when installed in high-voltage mine networks with powerful nonlinear electrical receivers. The urgency of the problem of choosing a rational structure of the FCD. The problem of choosing the design and installation location of the FCD is presented. The main technical means of compensation of higher harmonics of currents and voltages in high-voltage networks with powerful nonlinear electrical receivers are considered. Analysis of different types of passive filters (PF) and their frequency properties showed that the choice of specific types of PF refers to the multi-criteria optimization problem. The main methods of optimization of FCD design are considered. The variant of FCD construction based on the solution of multi-criteria optimization problem with the use of fuzzy sets is proposed and justified. To this end, the calculation of PF parameters and frequency characteristics of equivalent systems of the \"filter-external network\" type for four possible combinations of PF is performed. The optimal is a FCD with two resonant PF tuned to the 11th and 13th harmonics, and a second-order broadband PF tuned to compensate harmonics starting from the 23rd and above. The analysis of simulation results showed effective compensation of higher harmonic currents and voltages.

Subwavelength optical resonators made of high-index dielectric materials provide efficient ways to manipulate light at the nanoscale through mode interferences and enhancement of both electric and magnetic fields. Such Mie-resonant dielectric structures have low absorption, and their functionalities are limited predominantly by radiative losses. We implement a new physical mechanism for suppressing radiative losses of individual nanoscale resonators to engineer special modes with high quality factors: optical bound states in the continuum (BICs). We demonstrate that an individual subwavelength dielectric resonator hosting a BIC mode can boost nonlinear effects increasing second-harmonic generation efficiency. Our work suggests a route to use subwavelength high-index dielectric resonators for a strong enhancement of light–matter interactions with applications to nonlinear optics, nanoscale lasers, quantum photonics, and sensors.