Perturbations Of Coefficients Research Articles

ALMA Measurement of 10 kpc Scale Lensing-power Spectra toward the Lensed Quasar MG J0414+0534

The lensing power spectra for gravitational potential, astrometric shift, and convergence perturbations are powerful probes to investigate dark matter structures on small scales. We report the first lower and upper bounds of these lensing power spectra on angular scale ∼1″ toward the anomalous quadruply lensed quasar MG J0414+0534 at a redshift z = 2.639. To obtain the spectra, we conducted observations of MG J0414+0534 using the Atacama Large Millimeter/submillimeter Array with high angular resolution (0.″02–0.″05). We developed a new partially nonparametric method in which Fourier coefficients of potential perturbation are adjusted to minimize the difference between linear combinations of weighted mean de-lensed images. Using positions of radio-jet components, extended dust emission on scales >1 kpc, and mid-infrared flux ratios, the range of measured convergence, astrometric shift, and potential powers at an angular scale of ∼1.″1 (corresponding to an angular wavenumber of l = 1.2 × 106 or ∼9 kpc in the primary lens plane) within 1σ are Δ κ = 0.021–0.028, Δ α = 7–9 mas, and Δ ψ = 1.2–1.6 mas2, respectively. Our result is consistent with the predicted abundance of halos in the line of sight and subhalos in cold dark matter models. Our partially nonparametric lens models suggest the presence of a clump in the vicinity of object Y, a possible dusty dwarf galaxy, and some small clumps in the vicinity of other lensed quadruple images. Although much fainter than the previous report, we detected weak continuum emission possibly from object Y with a peak flux of ∼100 μJy beam−1 at the ∼4σ level.

Homogenization for operators with arbitrary perturbations in coefficients

We consider a general second order matrix operator in a multi-dimensional domain subject to a classical boundary condition. We perturbed it by a first order differential operator arbitrarily depending on a small multi-dimensional parameter and we study the existence of a limiting (homogenized) operator in the sense of the norm resolvent convergence. The first part of our main results states that the norm resolvent convergence is equivalent to the convergence of the coefficients in the perturbing operator in certain space of multipliers. The second part of our results says that the convergence in the mentioned spaces of multipliers is equivalent to the convergence of certain local mean values over small pieces of the considered domains. These results are supported by series of examples. We also provide a series of ways of generating new non-periodically oscillating perturbations for which our results are applicable.

CFD 기반 인공신경망을 이용한 로터 블레이드 에어포일 공력 예측 모델

Rotor blade analysis and design process typically use airfoil aerodynamic tables with two-dimensional quasi-steady flow assumption. Thus, its performance highly depends on the accuracy of the airfoil table. However, constructing an airfoil table is not trivial for airfoil shape design problems where the number of design parameters significantly increases. In this study, a surrogate model based on artificial neural networks is developed for airfoil aerodynamic tables in rotorcraft applications. The shape of airfoils is parameterized through Class-Shape Transformation(CST) using coefficients of a Bernstein polynomial expansion. The training data is comprised of 530 shapes sampled from perturbations of the CST coefficients around ten baseline airfoils. The aerodynamic coefficients for each shape are calculated using fully-turbulent flow simulations in the subsonic and transonic regimes. The range of angles of attack also includes stall angle to include any nonlinearities in model prediction. Overall, the trained model accurately predicts aerodynamic coefficients of non-trained airfoil shapes. The variation of model accuracy is also analyzed between interpolated and extrapolated spaces.

Diverse lateral shifts of beams in non-Hermitian waveguide arrays.

Non-Hermitian systems have attracted considerable attention in optics due to the rich physics introduced by the existence of real spectra and exceptional points (EPs), which is exploited in lasers, optical sensors and on-chip manipulations of light. Here, focusing on the dynamics of beams in non-Hermitian waveguide arrays supporting a ring of EPs (exceptional ring) and 3rd-order EPs, we theoretically demonstrate that the center of energy of a beam prepared around an eigenstate of the waveguide array near EPs could exhibit non-zero shifts in the lateral direction during its propagation. When the initial state of the beam prepared around an eigenstate inside (outside) the exceptional ring with the imaginary (real) eigenvalue, the lateral shifts of the beams are manifested by the non-oscillating (Zitterbewegung-like) motions, which are robust to the perturbations of coupling coefficients between waveguides. Remarkably, the amplitude of the non-oscillating shift is dependent on a non-Hermitian Berry connection (U(1) gauge invariance). It contradicts the conventional wisdom that the Berry connection cannot induce the dynamic effect. Furthermore, near the high-order EPs, the initial-state-dependent lateral shifts of the beams present diversity, such as multifrequencies and destructive interferences. The counterintuitive lateral shifts of the beams stem from the non-orthogonal nature of eigenstate of the non-Hermitian systems, which may open a gateway towards the non-Hermitian beam dynamics and manipulations of beams.

Perturbations of Modal Attenuation Coefficients and Group Velocities due to Depth Variations along a Shallow-Water Acoustic Track

Perturbations of Modal Attenuation Coefficients and Group Velocities due to Depth Variations along a Shallow-Water Acoustic Track

Robust Finite-Time Zeroing Neural Networks With Fixed and Varying Parameters for Solving Dynamic Generalized Lyapunov Equation

For solving dynamic generalized Lyapunov equation, two robust finite-time zeroing neural network (RFTZNN) models with stationary and nonstationary parameters are generated through the usage of an improved sign-bi-power (SBP) activation function (AF). Taking differential errors and model implementation errors into account, two corresponding perturbed RFTZNN models are derived to facilitate the analyses of robustness on the two RFTZNN models. Theoretical analysis gives the quantitatively estimated upper bounds for the convergence time (UBs-CT) of the two derived models, implying a superiority of the convergence that varying parameter RFTZNN (VP-RFTZNN) possesses over the fixed parameter RFTZNN (FP-RFTZNN). When the coefficient matrices and perturbation matrices are uniformly bounded, residual error of FP-RFTZNN is bounded, whereas that of VP-RFTZNN monotonically decreases at a super-exponential rate after a finite time, and eventually converges to 0. When these matrices are bounded but not uniform, residual error of FP-RFTZNN is no longer bounded, but that of VP-RFTZNN still converges. These superiorities of VP-RFTZNN are illustrated by abundant comparative experiments, and its application value is further proved by an application to robot.

Temporal Evolution of Refractive Index Induced by Short Laser Pulses Accounting for Both Photoacoustic and Photothermal Effects

Materials such as silicon, copper, gold, and aluminum exhibit strong absorption and scattering characterization under short-pulsed laser irradiation. Due to the photoelastic effect and thermoelastic relaxation, the focal area may induce a local modulation in the refractive index, which can be detected with the intensity reflection coefficient perturbation. Normally, the thermal effect causes a weak refractive index change and is negligible, compared with the pressure-induced effect in most photoacoustic analytical systems. In this study, we present a theoretical model with the whole process of absorbed energy conversion analysis for the refractive index perturbation induced by both thermal effect and photoacoustic pressure. In this model, data analysis was carried out on the transformation of the energy absorbed by the sample into heat and stress. To prove the feasibility of this model, numerical simulation was performed for the photothermal and photoacoustic effects under different incident intensities using the finite element method. Experiment results on silicon and carbon fiber verified that the refractive index change induced by the photothermal effect can be detected and be incorporated with pressure-induced refractive index change. The simulation results showed very good agreement with the results of the experiments. The main aim of this study was to further understand the absorption and conversion process of short-pulsed light energy and the resulting photothermal and photoacoustic effects.

An Effective AUSM-Type Scheme for Both Cases of Low Mach Number and High Mach Number

A new scheme called AUSMAS (Advection Upstream Splitting Method for All Speeds) is proposed for both high speed and low speed simulation cases. For the cases of low speed, it controls the checkerboard decoupling by keeping the coefficient of the pressure difference to the order of O(Ma−1) in the mass flux. Furthermore, it is able to guarantee a high level of accuracy by keeping the coefficients of the dissipation terms to the order of O(Ma0) in the momentum flux. For the cases of high speeds, especially at supersonic and hypersonic speeds, it is able to avoid the appearance of the shock anomaly by controlling the coefficients of the density perturbation in the mass flux. AUSMAS is testified to have the following attractive properties according to various numerical tests: (1) robustness against the abnormal shock; (2) high resolution in discontinuity; (3) the appearance of the unphysical expansion shock is avoided; (4) high resolution and low dissipation at low speeds; (5) independent of any tuning coefficient. These properties determined that AUSMAS has great promise in efficiently and accurately simulating flows of all speeds.

The Impact of Digestive Dynamics on the Bioequivalence of Amino Acids in Broiler Chickens.

The purpose of this review is to consider the distinct possibility that dietary non-bound and protein-bound amino acids are not bioequivalent in broiler chickens. Usually, with conservative inclusions of a limited number of non-bound (synthetic, crystalline, feed-grade) amino acids in standard broiler diets, bioequivalency would not be an issue. However, reduced-crude protein (CP) broiler diets demand substantial inclusions of an extended range of non-bound amino acids to meet amino acid requirements. A standard diet may contain 5.0 g/kg non-bound amino acids, but a reduced-CP diet may contain up to 50 g/kg and this relative abundance skews the balance of non-bound to protein-bound amino acids and substantial proportions of certain amino acids are present in diets as non-bound entities. Importantly, tangible reductions in dietary CP, for example from 210 to 160 g/kg, usually both compromise broiler growth performance and increase fat deposition. Compromised growth performance is more evident in wheat- than maize-based diets but, paradoxically, fat deposition is more apparent in maize-based diets. The inability of birds to accommodate tangible dietary CP reductions appears to stem partially from the lack of bioequivalency between non-bound and protein-bound amino acids because of the differentials in intestinal uptake rates. Also, reduced-CP broiler diets generate perturbations in apparent amino acid digestibility coefficients which compound the fact that intestinal uptakes of non-bound acid acids are more rapid, and occur more anteriorly in the small intestine, than protein-bound amino acids. The likelihood is that greater proportions of non-bound amino acids transit the enterocytes of the gut mucosa without entering anabolic and/or catabolic pathways to gain entry to the portal circulation. This culminates in post-enteral amino acid imbalances and postprandial oxidation of surplus amino acids which involves deamination of amino acids and elevations in plasma ammonia (NH3) concentrations, but NH3 is inherently toxic and demands detoxification. Excessive deamination coupled with inadequate detoxification could result in 'ammonia overload' which would be expected to compromise growth performance. Thus, the hypothesis is that non-bound and protein-bound amino acids are not bioequivalent; moreover, it may be argued that this distinction is being overlooked and is thwarting the development and acceptance of reduced-CP broiler diets.

Approximate methods for solving amplitude-phase problem for discrete signals

Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.

On the minimum value of the condition number of polynomials

Abstract The condition number of a polynomial is a natural measure of the sensitivity of the roots under small perturbations of the polynomial coefficients. In 1993 Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In Beltrán et al. (2021, A sequence of polynomials with optimal condition number. J. Amer. Math. Soc., 34, 219–244) it was proved that the optimal value of the condition number is of the form $\mathcal{O}(\sqrt {N})$, and the sequence demanded by Shub and Smale was described by a closed formula for large enough $N\geqslant N_0$ with $N_0$ unknown, and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $\mathcal{O}(\sqrt {N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer.

Reconstructing small perturbations of an obstacle for acoustic waves from boundary measurements on the perturbed shape itself

We derive relationships between the shape deformation of an impenetrable obstacle and boundary measurements of scattering fields on the perturbed shape itself. Our derivation is rigorous by using a systematic way, based on layer potential techniques and the field expansion (FE) method (formal derivation). We extend these techniques to derive asymptotic expansions of the Dirichlet‐to‐Neumann (DNO) and Neumann‐to‐Dirichlet (NDO) operators in terms of the small perturbations of the obstacle as well as relationships between the shape deformation of an obstacle and boundary measurements of DNO or NDO on the perturbed shape itself. All relationships lead us to very effective algorithms for determining lower order Fourier coefficients of the shape perturbation of the obstacle.

Remark on Justification of Asymptotics of Spectra of Cylindrical Waveguides with Periodic Singular Perturbations of Boundary and Coefficients

To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max–min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes.

Stabilization and uncertainty analysis of a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation

AbstractIn this paper, we consider a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation (PDE), where the time‐fractional reaction diffusion equation possesses space‐dependent diffusivity and the time‐fractional hyperbolic equation acts as a boundary source for the time‐fractional reaction diffusion equation. Then, the control input is imposed at the boundary of the time‐fractional hyperbolic PDE. Two problems are investigated, namely, the stabilization by state feedback and the stability robustness of the closed‐loop system against small perturbations in the diffusion and reaction coefficients. By the backstepping transformation, we propose a boundary controller and show that this controller solves the boundary stabilization problem using the fractional Lyapunov method. Robustness to small perturbations in diffusion and reaction coefficients is proved. Finally, a numerical example is provided to test the effectiveness of the proposed synthesis for the stabilization problem when the kernel equations have not an explicit solution.

Recovering a perturbation of a matrix polynomial from a perturbation of its first companion linearization

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric differential Riccati equation are extended to the nonsymmetric case. Applying the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principle, two perturbation bounds are derived: the first one is based on the integral form of the solution and the second one considers the equivalent solution to the initial value problem of the associated differential system. The first bound is derived for the nonsymmetric differential Riccati equation in its general form. The perturbation bound based on the sensitivity analysis of the associated linear differential system is formulated for the low-dimensional approximate solution to the large-scale nonsymmetric differential Riccati equation. The two bounds exploit the existing sensitivity estimates for the matrix exponential and are alternative.

Trans-dimensional Bayesian inversion for airborne EM data in sparse domain

The conventional trans-dimensional Bayesian inversion uses Monte Carlo method to search the model space for a solution that satisfies both the acceptance probability and data fitting. With this method one can get the inverse model based on the maximum probability and determine the model uncertainty. However, because the search space is too big, the conventional Bayesian method can only invert simple models, which means that it may not be well applied to the inversion of actual geophysical survey data. In this paper, we propose a new inversion framework based on sampling via a deformed binary-tree structure and take the wavelet coefficients for sparsely constrained Bayesian inversions. This method has the advantages that in the inversion process, the change of the number of wavelet coefficients results in the change of the number of layers, while a slight perturbation of the wavelet coefficients can lead to a big change to the model in the space-domain, so that we avoid updating each individual model parameter, and the inversion process is largely speeded up. We validate our method with both synthetic data and survey data.

Multiple cost coefficients sensitivity theorems of integer linear optimization

In practical integer optimization applications, perturbations of multiple cost coefficients often occur simultaneously. One important example is the perturbation of the probability distribution estimates for scenarios of a stochastic integer optimization cost function formulation. This study aims to develop multiple cost coefficients sensitivity theorems, which indicate that the optimal solution remains the same if the perturbation amount of cost coefficients is greater than or equal to the derived bounds. We developed two sensitivity theorems. One depends on the underlying algorithm, the iterative dual method proposed by Bell and Shapiro , and the other directly uses the optimal integer output that can be obtained by any algorithms. We carried out numerical experiments using the two proposed theorems separately to compare their effectiveness and computational tractability. These theorems are useful especially when dealing with the problems where the cost coefficients change frequently.

Approximate solution of hypersingular integral equations on the number axis

In the paper we investigate approximate methods for solving linear and nonlinear hypersingular integral equations defined on the number axis. We study equations with the second-order singularities because such equations are widely used in problems of natural science and technology. Three computational schemes are proposed for solving linear hypersingular integral equations. The first one is based on the mechanical quadrature method. We used rational functions as the basic ones. The second computational scheme is based on the spline-collocation method with the first-order splines. The third computational scheme uses the zero-order splines. Continuous method for solving operator equations has been used for justification and implementation of the proposed schemes. The application of the method allows to weaken the requirements imposed on the original equation. It is sufficient to require solvability for a given right-hand side. The continuous operator method is based on Lyapunov's stability for solutions of systems of ordinary differential equations. Thus it is stable for perturbations of coefficients and of right-hand sides. Approximate methods for solving nonlinear hypersingular integral equations are presented by the example of the Peierls - Naborro equation of dislocation theory. By analogy with linear hypersingular integral equations, three computational schemes have been constructed to solve this equation. The justification and implementation are based on continuous method for solving operator equations. The effectiveness of the proposed schemes is shown on solving the Peierls - Naborro equation.

Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media

This paper is concerned with the existence and stability of steady states of a reaction-diffusion-ODE system arising from the theory of biological pattern formation. We are interested in spontaneous emergence of patterns from spatially heterogeneous environments, hence assume that all coefficients in the equations can depend on the spatial variable. We give some sufficient conditions on the coefficients which guarantee the existence of far-from-the-equilibrium patterns with jump discontinuity and then verify their stability in a weak sense. Our conditions cover the case where the number of equilibria of the kinetic system (i.e., without diffusion) changes from one to three in the spatial interval, which is not obtained by a small perturbation of constant coefficients. Moreover, we consider the asymptotic behavior of steady states as the diffusion coefficient tends to infinity. Some examples and numerical simulations are given to illustrate the theoretical results.