Order Coefficients Research Articles

Lorentzian Calderón problem under curvature bounds

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calderón problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior region of double null cones. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.

Hyperbolic Equation with Rapidly Oscillating Data: Reconstruction of the Small Lowest Order Coefficient and the Right-Hand Side from Partial Asymptotics of the Solution

Hyperbolic Equation with Rapidly Oscillating Data: Reconstruction of the Small Lowest Order Coefficient and the Right-Hand Side from Partial Asymptotics of the Solution

Non-local tug-of-war with noise for the geometric

This paper concerns the fractional ${\mathrm{\bf p}}$-Laplace operator $\Delta_{\mathrm{\bf p}}^s$ in non-divergence form, which has been introduced in [2]. For any ${\mathrm{\bf p}}\in [2,\infty)$ and $s\in (\frac{1}{2},1)$, we first define two families of non-local, non-linear averaging operators, parametrized by $\epsilon$ and defined for all bounded, Borel functions $u:\mathbb R^N\to \mathbb R$. We prove that $\Delta_{\mathrm{\bf p}}^s u(x)$ emerges as the $\epsilon^{2s}$-order coefficient in the expansion of the deviation of each $\epsilon$-average from the value $u(x)$, in the limit of the domain of averaging exhausting an appropriate cone in $\mathbb R^N$ at the rate $\epsilon\to 0$. Second, we consider the $\epsilon$-dynamic programming principles modeled on the first average, and show that their solutions converge uniformly as $\epsilon\to 0$, to viscosity solutions of the homogeneous non-local Dirichlet problem for $\Delta_{\mathrm{\bf p}}^s$, posed in a domain ${\mathcal{D}}$ satisfying the external cone condition and subject to bounded, uniformly continuous data on $\mathbb R^N\setminus {\mathcal{D}}$. Finally, we interpret such $\epsilon$-approximating solutions as values to the non-local Tug-of-War game with noise. In this game, players choose directions while the game position is updated randomly within the infinite cone that aligns with the specified direction, whose aperture angle depends on ${\mathrm{\bf p}}$ and $N$, and whose $\epsilon$-tip has been removed.

Variable order step size method for solving orbital problems with periodic solutions

Existing variable order step size numerical techniques for solving a system of higher-order ordinary differential equations (ODEs) requires direct calculating the integration coefficients at each step change. In this study, a variable order step size is presented for direct solving higher-order orbital equations. The proposed algorithm calculates the integration coefficients only once at the beginning and, if necessary, once at the end. The accuracy of the numerical approximation is validated with well-known orbital differential equations. To reduce computational costs, we obtain the relationship for the predictor-corrector algorithm between integration coefficients of various orders. The efficiency of the proposed method is substantiated by the graphical representation of accuracy at the total evaluation steps.

Research on Scattering Transform of Urban Sound Events Detection Based on Self-Attention Mechanism

Urban sound event detection can automatically preload relevant information for the robot to ensure that it can be competent for various scene activity tasks. Aiming at the limitations of timbre similarity and scene recognition limited by audio collection devices, a fusion model based on self-attention mechanism is proposed in this paper. The model consists of scattering transform and self-attention model. Scattering transform computes modulation spectrum coefficients of multiple orders, through cascades of wavelet convolutions and modulus operators, and it is learnable compared with Mel-scale Frequency Cepstral Coefficients (MFCC), and can be used to better restore the semantic features of some sound scenes with similar timbres. Transformer has an outstanding effect on Natural Language Processing (NLP) with its self-attention mechanism. In this paper, the self-attention mechanism in its encoder is used in the model, mainly to make the feature granularity consistent to refine the features. in addition, Focal Loss function is adopted in the model to curb the problem of sample distribution imbalance. The datasets Google-Command and ESC-50 are used to supplement the scene categories of dataset UrbanSound8K. The model parameters of the learnable filters that performed well on the dataset UrbanSound8K were preserved to fine-tune the other two datasets with insufficient data volume and more target categories. The length of slice duration are further explored the on the model. Experimental results show that the model can achieve better performance in a large range of scene models.

Polynomial fitting based on least squares approximates for first-order Tracy-Widom distribution

Tracy-Widom distribution can primely describe the limit distribution of the largest eigenvalue of noise matrix, so it is widely used in the field of signal processing and wireless communication. The exact expression of this distribution is very complex and difficult to be applied in practice. Therefore, an approximate method of Tracy-Widom distribution based on least squares is proposed, in which the optimal fitting order and optimal fitting coefficient are determined by minimizing the average fitting uncertainty. The simulation results show that the fitting results can accurately predict the largest eigenvalue distribution of the noise matrix, which proves the effectiveness of the polynomial fitting method to approximate the first-order Tracy-Widom distribution.

A piecewise heat equation with constant and variable order coefficients: A new approach to capture crossover behaviors in heat diffusion

<abstract><p>While the equation depicting the heat flow within homogeneous has been applied in several fields with some success, it has also faced several difficulties to depict heat diffusion in some non-homogeneous media. For particular behaviours adequate differential operators have been applied, for instance, a long-range behaviour has been depicted using operators based on power law kernel; stochastic behaviours have been included into mathematical equation using random function, some well-defined crossover behaviours have been depicted using the differential operators based on the generalized Mittag-Leffler kernel. Nevertheless, complex crossovers behaviours have not been modelled efficiently due to limitation of existing theories. Nevertheless, very recently piecewise calculus was proposed and applied in some complex world problems with great success. In this paper, heat equation with constant and variable coefficients will be subjected to piecewise numerical analysis. Several cases are considered, and their numerical simulation depicted.</p></abstract>

Estimates of the Fundamental Solution for Elliptic Equation in Divergence Form with Drift

For a linear divergent second-order elliptic equation with uniformly elliptic measurable matrix of the principal coefficients and drift, a condition on the low order coefficients is found, which guarantees the existence of classical two-sided estimates for the fundamental solution.

Holographic interferometry experiments and numerical analyses of the stress field on the Williams series expansion: higher – order terms

The study is aimed at experimental and numeric determination of the higher order coefficients of the Williams power expansion for the near crack-tip fields in isotropic linear elastic materials by two approaches. The first one is based on the experimental method of holographic interferometry, the second one is relied on the finite element analysis. The main problem and difficulty are accurate extracting the coefficients from the whole set of experimental data. To obtain the higher order coefficients of the Williams expansion the program tool which is designed for digital image processing of the experimental fringe patterns. As the first step the plate with the horizontal central crack has been chosen since this cracked geometry is thoroughly studied and the coefficients of the higher order terms are known. The stress intensity factors, T-stress and higher order coefficients are extracted and compared with the theoretical values. Then the mixed mode loading is considered. Using holographic interferometry a lot of tests with the inclined cracks were performed. It is shown that the digital holographic method and the developed programming tool allow us to find the coefficients with high accuracy. The results of digital image processing of the experimental data have been verified and confirmed by the finite element analysis.

New Approximations for the Higher Order Coefficients in an Asymptotic Expansion for the Barnes G-Function

In this paper, we provide new formulas for determining the coefficients appearing in the asymptotic expansion for the Barnes G-function as n tends to infinity for certain classes of asymptotic expansion for the Barnes G-function. We remark that our formulas can be used to approximate the coefficients appearing in an asymptotic expansion of the “random matrix factor” from the Keathing-Snaith conjecture and the coefficients appearing in an asymptotic expansion of the “Lévy-Khintchine type representation of the reciprocal of the Barnes G-function”.

On the Joint Moments of the Characteristic Polynomials of Random Unitary Matrices

Abstract We establish the asymptotics of the joint moments of the characteristic polynomial of a random unitary matrix and its derivative for general real values of the exponents, proving a conjecture made by Hughes [ 40] in 2001. Moreover, we give a probabilistic representation for the leading order coefficient in the asymptotic in terms of a real-valued random variable that plays an important role in the ergodic decomposition of the Hua–Pickrell measures. This enables us to establish connections between the characteristic function of this random variable and the $\sigma $-Painlevé III’ equation.

Resource Allocation in NOMA-Assisted Ambient Backscatter Communication System

This paper considers a non-orthogonal multiple access (NOMA)-assisted ambient backscatter communication (AmBC) system. To maximize the achievable sum rate (ASR) of the AmBC system, a joint optimization problem over a backscatter device (BD) grouping strategy, reflection coefficients, and decoding order is formulated, where the BD grouping strategy contains the number of BD groups and the BD allocation strategy. The BD grouping strategy, the reflection coefficients, and the decoding order are all intertwined, and the global search is extremely complex. As a result, we propose a four-step optimization algorithm. First, we give the closed-form optimal solution of the BD decoding order and reflection coefficient for a given grouping strategy. Then, for a given number of BD groups, we propose a low-complexity BD allocation strategy based on the complexity–performance trade-off. Finally, the number of BD groups with the largest ASR is selected as the global optimal number of BD groups. The simulation results show that the proposed four-step optimization algorithm is better than the benchmark solution.

Research on Population Development Trend in Huizhou of China Forecast Based on Optimal Weighted Combination Method and Fractional Grey Model

In this paper, the optimal weighted combination model and fractional grey model are constructed. The coefficients of the optimal weighted combination model are determined by minimizing the sum of squares of resists of each model. On the other hand, the optimal conformable fractional order and dynamic background value coefficient are determined by the quantum inspired evolutionary algorithm (QIEA). Taking the resident population from 2008 to 2018 as the research object, the optimal weighted combination model and fractional grey model were used to study the estimated and predicted values. The results are compared and analyzed. The results show that the fractional grey model is better than the optimal weighted combination model in the estimation of the values. The optimal weighted combination model is better than the fractional grey model in predicting. Meanwhile, the fractional grey model is found to be very suitable for the data values that are large, and the changes between the data are relatively small. The research results expand the application of the fractional grey model and have important implications for the policy implementation activities of Huizhou government according to the population growth trend in Huizhou.

On Symmetric Solutions to Linear Matrix Time-Varying Differential Equations

In this paper, we discuss when the solution to the initial value problem for a linear matrix time-varying differential equation is symmetric on a given interval. By symmetry, we mean that the solution does not change when transposed. Throughout the paper, we assume that the equation has coefficients of finite order of smoothness. We demonstrate that, in order to verify whether the solution to the initial value problem is symmetric on a given interval, it can be useful to construct two matrix sequences associated to the equation. Using these sequences, we prove a sufficient condition for the solution symmetry on a given interval. Assuming that the initial value problem for a linear matrix time-varying differential equation satisfies this condition, we derive a formula for a symmetric solution to this problem.

Transparency order versus confusion coefficient: a case study of NIST lightweight cryptography S-Boxes

Side-channel resistance is nowadays widely accepted as a crucial factor in deciding the security assurance level of cryptographic implementations. In most cases, non-linear components (e.g. S-Boxes) of cryptographic algorithms will be chosen as primary targets of side-channel attacks (SCAs). In order to measure side-channel resistance of S-Boxes, three theoretical metrics are proposed and they are reVisited transparency order (VTO), confusion coefficients variance (CCV), and minimum confusion coefficient (MCC), respectively. However, the practical effectiveness of these metrics remains still unclear. Taking the 4-bit and 8-bit S-Boxes used in NIST Lightweight Cryptography candidates as concrete examples, this paper takes a comprehensive study of the applicability of these metrics. First of all, we empirically investigate the relations among three metrics for targeted S-boxes, and find that CCV is almost linearly correlated with VTO, while MCC is inconsistent with the other two. Furthermore, in order to verify which metric is more effective in which scenarios, we perform simulated and practical experiments on nine 4-bit S-Boxes under the non-profiled attacks and profiled attacks, respectively. The experiments show that for quantifying side-channel resistance of S-Boxes under non-profiled attacks, VTO and CCV are more reliable while MCC fails. We also obtain an interesting observation that none of these three metrics is suitable for measuring the resistance of S-Boxes against profiled SCAs. Finally, we try to verify whether these metrics can be applied to compare the resistance of S-Boxes with different sizes. Unfortunately, all of them are invalid in this scenario.

Application of a novel grey Bernoulli model to predict the global consumption of renewable energy

Accurate prediction of renewable energy can provide an important basis for national energy security and the government to formulate policies. Therefore, according to the non-linearity of energy prediction system, this paper proposes a new nonlinear grey Bernoulli optimization model by the grey Bernoulli extended model, which is a nonlinear grey prediction model, and studies the properties of the new model. The order and nonlinear coefficient of the new model are optimized by Particle Swarm Optimization. Then, through the consumption of global renewable energy, such as solar, wind and hydropower as empirical analyses, the results of the four evaluation indicators show that the new model works better than the original model, which has higher prediction accuracy than before and makes the prediction model more applicable. At the same time, the model results were compared with the weighted grey model, Verhulst and the discrete grey model, and the new model has the highest accuracy. Finally, the new model is used to forecast the global consumption of wind, solar and hydropower energy in 2019–2023. The results will provide important forecasting information for global energy conservation and emission reduction policies.

Determination of Angstrom Coefficients for Masvingo and Makoholi Stations, Zimbabwe

Coefficients of the first order and second order Angstrom model were obtained for Masvingo and Makoholi stations in Zimbabwe. Long term average ground measurements of sunshine hours and global horizontal solar radiation from the meteorological services were used as input data. Monthly average values of the clearness index, and the fraction of sunshine hours, were determined. Regression analysis was used to obtain the first and second order coefficients of the Angstrom model for the two locations. The estimates from the developed models were tested against measured values and results obtained were in agreement to within +/- 5%. The results also showed that the models performed better with these coefficients compared with coefficients suggested in previous researches. Also the first order model was found to perform better than the second order model for Makoholi station while the second order model works well for Masvingo.

An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator

This article offers a study of the Calderón type inverse problem of determining up to second order coefficients of higher order elliptic operators. Here we show that it is possible to determine an anisotropic second order perturbation given by a symmetric matrix, along with a first order perturbation given by a vector field and a zero-th order potential function inside a bounded domain, by measuring the Dirichlet to Neumann map of the perturbed biharmonic operator on the boundary of that domain.

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation −div(A∇u+bu)+c∇u+du=−divf+g, assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions b,f∈Ln,1, d,g∈Ln2,1 and c∈Ln,q for q≤∞, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with “good” constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on b,d or c,d, we show a maximum principle and Moser's estimate for subsolutions with “good” constants. We also show the reverse Moser estimate for nonnegative supersolutions with “good” constants, under no smallness assumptions when q<∞, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.

Fundamental-ray aberration analysis of off-axial optical systems: analytical formulae of first-order aberrations.

In our previous paper [Appl. Opt.59, 4466 (2020)10.1364/AO.389600], we presented a fundamental-ray aberration analysis that extends ray matrix analysis to the third-order aberration region. The analysis results shown in that paper were applicable to coaxial rotationally symmetric optical systems. This time, we have extended the fundamental-ray aberration analysis so that it can be applied to off-axial optical systems. Here we present new analysis formulae for fundamental-ray aberration analysis of the first-order aberration region. In addition, we newly present first-order aberration expansion formulae for four-element fundamental-ray aberrations and calculation formulae for the fundamental-ray aberration coefficients of the first order, which are necessary for this extension.