Nonlinear Analog Research Articles

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ 2(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ 2 among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.

Weyl geometry and the nonlinear mechanics of distributed point defects

The residual stress field of a nonlinear elastic solid with a spherically symmetric distribution of point defects is obtained explicitly using methods from differential geometry. The material manifold of a solid with distributed point defects—where the body is stress-free—is a flat Weyl manifold, i.e. a manifold with an affine connection that has non-metricity with vanishing traceless part, but both its torsion and curvature tensors vanish. Given a spherically symmetric point defect distribution, we construct its Weyl material manifold using the method of Cartan's moving frames. Having the material manifold, the anelasticity problem is transformed to a nonlinear elasticity problem and reduces the problem of computing the residual stresses to finding an embedding into the Euclidean ambient space. In the case of incompressible neo-Hookean solids, we calculate explicitly this residual stress field. We consider the example of a finite ball and a point defect distribution uniform in a smaller ball and vanishing elsewhere. We show that the residual stress field inside the smaller ball is uniform and hydrostatic. We also prove a nonlinear analogue of Eshelby's celebrated inclusion problem for a spherical inclusion in an isotropic incompressible nonlinear solid.

Parametric Fault Testing of Non-Linear Analog Circuits Based on Polynomial and V-Transform Coefficients

This paper is an exposition of recent advances made in polynomial coefficient and V-transform coefficient based testing of parametric faults in linear and non-linear analog circuits. V-transform is a non-linear transform that increases the sensitivity of polynomial coefficients with respect to circuit component variations by three to five times. In addition, it makes the original polynomial coefficients monotonic. Using simulation, the proposed test method is shown to uncover most parametric faults in the range of 5–15 % on a low noise amplifier (LNA) and an elliptic filter benchmark. Diagnosis of parametric faults clearly illustrates the effect of enhanced sensitivity through V-transform. Finally, we report an experimental validation of the polynomial coefficient based test scheme, with and without V-transform, using the National Instruments’ ELVIS bench-top testbed. The result demonstrates the benefit of V-transform.

Nonlinear Optical Spin Hall Effect and Long-Range Spin Transport in Polariton Lasers

We report on the experimental observation of the nonlinear analogue of the optical spin Hall effect under highly nonresonant circularly polarized excitation of an exciton-polariton condensate in a GaAs/AlGaAs microcavity. The circularly polarized polariton condensates propagate over macroscopic distances, while the collective condensate spins coherently precess around an effective magnetic field in the sample plane performing up to four complete revolutions.

Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos

AbstractWe propose a general strategy for determining the minimal finite amplitude disturbance that triggers transition to turbulence in shear flows. This involves constructing a variational problem that searches over all disturbances of fixed initial amplitude which respect the boundary conditions, incompressibility and the Navier–Stokes equations, to maximize a chosen functional over an asymptotically long time period. The functional must be selected such that it identifies turbulent velocity fields by taking significantly enhanced values compared to those for laminar fields. We illustrate this approach using the ratio of the final to initial perturbation kinetic energies (energy growth) as the functional and the energy norm to measure amplitudes in the context of pipe flow. Our results indicate that the variational problem yields a smooth converged solution provided that the initial amplitude is below the threshold for transition. This optimal is the nonlinear analogue of the well-studied (linear) transient growth optimal. At the critical threshold, the optimization seeks out a disturbance that is on the ‘edge’ of turbulence during the period. Above this threshold, when disturbances trigger turbulence by the end of the period, convergence is then practically impossible. The first disturbance found to trigger turbulence as the amplitude is increased identifies the ‘minimal seed’ for the given geometry and forcing (Reynolds number). We conjecture that it may be possible to select a functional such that the converged optimal below threshold smoothly converges to the minimal seed at threshold. Our choice of the energy growth functional is shown to come close to this for the pipe flow geometry investigated here.

Fully efficient adiabatic frequency conversion of broadband Ti:sapphire oscillator pulses

By adiabatic difference-frequency generation in an aperiodically poled nonlinear crystal-a nonlinear optical analog of rapid adiabatic passage in a two-level atomic system-we demonstrate the conversion of a 110 nm band from an octave-spanning Ti:sapphire oscillator to the infrared, spanning 1550 to 2450 nm, with near-100% internal conversion efficiency. The experiment proves the principle of complete Landau-Zener adiabatic transfer in nonlinear optical wave mixing. Our implementation is a practical approach to the seeding of high-energy ultrabroadband optical parametric chirped pulse amplifiers.

An Approach to Locate Parametric Faults in Nonlinear Analog Circuits

Aiming at the problem to locate parametric faults in nonlinear analog circuits, a new approach based on the subband decomposition combined with coherence functions is proposed. First, the Volterra series of the circuit under test decomposed by wavelet packets are used to detect the parametric faults. Then, the Volterra series in subbands are used to calculate the coherence functions. By comparison with the fault signatures, different states of the parametric faulty circuits are identified, and the faults are located. Simulations show the effectiveness of the method of the fault diagnosis in nonlinear circuits.

Diagnosis of Incipient Faults in Nonlinear Analog Circuits

Diagnosis of Incipient Faults in Nonlinear Analog Circuits Considering the problem to diagnose incipient faults in nonlinear analog circuits, a novel approach based on fractional correlation is proposed and the application of the subband Volterra series is used in this paper. Firstly, the subband Volterra series is calculated from the input and output sequences of the circuit under test (CUT). Then the fractional correlation functions between the fault-free case and the incipient faulty cases of the CUT are derived. Using the feature vectors extracted from the fractional correlation functions, the hidden Markov model (HMM) is trained. Finally, the well-trained HMM is used to accomplish the incipient fault diagnosis. The simulations illustrate the proposed method and show its effectiveness in the incipient fault recognition capability.

The Research on Fault Model in Nonlinear Analog and Mixed-Signal Circuits

Fault modeling is the key and the difficulty of analog and mixed-signal circuits. The existing fault model include resistance model, perfect switch model, MOS transistor width-length ratio model and slope fault model, these methods have succeed in a certain extent, but also existed much limitation. In this paper, on the basis of research into slope fault model, this paper presents a new fault model is improved slope fault model. The fault model utilizes piecewise linear function to approximate the nonlinear analog circuit, and solve the problem of tolerance by using eulerian length. The experimental result is given to clarify our approach and to show its efficiency. This research provides theoretical foundation for the research of analog and mixed-signal circuits.

Co-crystal structure selection of nonlinear optical analogue polyenes

We demonstrate co-crystallization of analogous phenolic polyene molecules that form non-isomorphic single-component crystal structures for nonlinear optical applications. The analogous polyene molecules form similar main supramolecular hydrogen-bond interactions with neighboring molecules resulting in polar linear molecular chains, but form different crystal space groups in single component crystals, which have a different orientation of these molecular chains due to subtle variations of weak intermolecular interactions. The resulting co-crystal structure is isomorphous with one of the single-component crystals. In co-crystallization with competition between two analogous crystal structures, the solubility plays an important role to select the equilibrium crystal structure of co-crystals. The phenolic polyene co-crystals retain optimal molecular ordering with large macroscopic nonlinearity of single component crystals for nonlinear optical applications.

Noise-Induced Phase Synchronization among Simple Digital Counters

We demonstrate noise-induced phase synchronization among simple digital counters, aiming at its practical application to phase synchronization among isolated digital systems (driven by precise clock sources) accepting common noisy pulse sequences. We employ a simple digital up-counter where i) relaxation oscillations in nonlinear analog oscillators are emulated by linear counting and carry overflow in a low-bit digital counter and ii) the counter value, which is equivalent to the phase of the digitally emulated oscillator, is increased by the precise base clocks, as well as by external noisy pulses, and the autonomous increase of the value (by the base clock) is inhibited by the noisy pulses, depending on the phase of the counter at which the pulse is given. Through extensive Verilog simulations, we demonstrate that the proposed digital counters can be synchronized by applying a common noise sequence, even if initial values of both the counters and phases of the counter's base clocks have different values.

Approximate Solution and Application of the Survival Probability Diffusion Equation

While stochastic neutron transport theories have been developed in rigorous detail, many applications have historically been investigated using the point-kinetics formulation. In this work we develop a space-dependent model using the diffusion approximation to the Pál-Bell probability generating function equation, resulting in a nonlinear analog of the conventional time-dependent neutron diffusion equation. We investigate a variety of approximate solutions for the time- and space-dependent survival probability in one-dimensional symmetric, one-speed, isotropic, delayed neutron precursor-free systems, and compare them to counterpart point-kinetics results. Following the theoretical developments, we apply the new results in the context of a criticality accident scenario, from which the importance of spatial effects is revealed.

Compressed Sensing With Nonlinear Analog Mapping in a Noisy Environment

We propose a low delay and low complexity sensor system based on the combination of Shannon-Kotel'nikov mapping and compressed sensing (CS). The proposed system uses nonlinear analog mappings on the CS measurements to increase their immunity against channel noise. Numerical results show that the proposed purely-analog system outperforms the state-of-the-art purely CS systems in terms of signal-to-distortion ratio. In addition to sparsity knowledge, we use a statistical characterization of the observed signal to further improve system performance.

Von Mises’ potential flow wave equation and nonlinear analog gravity

Von Mises' nonlinear second-order wave equation for the velocity potential in the case of an irrotational, compressible and barotropic classical perfect fluid is connected to the analog gravity formalism. It is shown that while the standard acoustic metric, built on a certain background, remains central in the higher-order perturbative scheme, at full nonlinear level it plays a less relevant role. Possible applications both for simulations and experiments are discussed.

Empirical‐statistical downscaling and error correction of daily precipitation from regional climate models

AbstractAlthough regional climate models (RCMs) are powerful tools for describing regional and even smaller scale climate conditions, they still feature severe systematic errors. In order to provide optimized climate scenarios for climate change impact research, this study merges linear and nonlinear empirical‐statistical downscaling techniques with bias correction methods and investigates their ability for reducing RCM error characteristics. An ensemble of seven empirical‐statistical downscaling and error correction methods (DECMs) is applied to post‐process daily precipitation sums of a high‐resolution regional climate hindcast simulation over the Alpine region, their error characteristics are analysed and compared to the raw RCM results.Drastic reductions in error characteristics due to application of DECMs are demonstrated. Direct point‐wise methods like quantile mapping and local intensity scaling as well as indirect spatial methods as nonlinear analogue methods yield systematic improvements in median, variance, frequency, intensity and extremes of daily precipitation. Multiple linear regression methods, even if optimized by predictor selection, transformation and randomization, exhibit significant shortcomings for modelling daily precipitation due to their linear framework. Comparing the well‐performing methods to each other, quantile mapping shows the best performance, particularly at high quantiles, which is advantageous for applications related to extreme precipitation events. The improvements are obtained regardless of season and region, which indicates the potential transferability of these methods to other regions. Copyright © 2010 Royal Meteorological Society

Oscillons: An encounter with dynamical chaos in 1953?

We discuss the works of one of electronic art pioneers, Ben F. Laposky (1914–2000), and argue that he might have been the first to create a family of essentially nonlinear analog circuits that allowed him to observe chaotic attractors.

Pseudorandom Test of Nonlinear Analog and Mixed-Signal Circuits Based on a Volterra Series Model

This paper presents new test methods for nonlinear Analog and Mixed-Signal (AMS) circuits which use a pseudorandom signal to test multiple Devices Under Test (DUTs) accurately. The goal of the studies presented in this paper is to understand the behaviors of nonlinear AMS circuits in a low-cost test environment and to develop the algorithm to extract the performance information of the DUTs using simple test measurements. The extracted information is then used to estimate the various specifications of DUTs. In order to achieve this goal, we analyze the behaviors of AMS circuits using a Volterra series model, and investigate the stochastic properties of the pseudorandom signals to develop the efficient performance characterization algorithms. The mathematical theory and experimental results are presented to validate the presented test methods.

Long-term dynamics for well productivity index for nonlinear flows in porous media

Motivated by the reservoir engineering concept of the well productivity index (PI) we study a time dependent functional for general nonlinear Forchheimer equation. The PI of the well characterizes the well capacity with respect to drainage area of the well. Unlike the linear case for which this concept is well developed, there are only a few recent publications dedicated to the PI for nonlinear case. In this paper the PI is comprehensively studied both theoretically and numerically. The impact of the nonlinearity of the flow filtration on the value of the PI is analyzed. Exact formula for the so called “skin factor” in radial case is derived depending on the rate of the flow, the order of nonlinearity and the geometric parameters. Dynamics of the PI for the class of boundary conditions is studied and its convergence to the specific value of steady state PI was justified. Developed framework is applied to obtain nonlinear analog of Peaceman formula for the well-block pressure in unstructured grid. Numerical simulations sustain theoretical results.

Поліноміальні повільно зростаючі розподіли

In the article, polynomial (nonlinear) analogue of tempered Schwartz distributions is constructed. Generalized operation of differentiation in the space of polynomial generalized functions as well as Fourier-Laplace transformation of such distributions are considered. Some examples are given.

Fault Diagnosis for Nonlinear Analog Circuit Based on Harmonic Decomposition and Coherent Measurement

This paper, a fault diagnosis approach for nonlinear dynamic circuit is presented based on harmonic decomposition and coherent measurement. According to the separability of Volterra spectral in weakly nonlinear circuit under AM stimulation, the Volterra response can decompose into linear sub-circuits based on ARMA harmonic decomposition firstly. In linear sub-circuits, the mapping relationship between the fault characteristics and fault states will become more clearly. Then, the dynamic linear sub-circuit is transformed into static circuit by coherent measurement, and the diagnosis equation is created with circuit node-equation. The fault diagnosis can be implemented by calculating the nodes’ fault-current. Finally, with an example to illuminate the proposed is available.