THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Getting pre-stack time migration travel times from the single square root operator

Improving the focusing capability of pre-stack time migration allows the imaged section to reflect structural characteristics, depth, and interface shape and it is a key step for the preparation of the initial depth migration velocity model. The traditional symmetrical travel time equation is derived based on the assumption of a layered model. It is difficult to achieve the desired effect of focusing in media with strong lateral variation. The nonsymmetrical travel time equation based on Lie algebra and a pseudo-differential operator contains a lateral velocity derivative which can improve the focusing capability even in strongly lateral variable media and also the computation precision of the weight coefficients for relative amplitude preservation. Compared with the symmetrical methods, the nonsymmetrical method is more effective. In this paper, we describe several key steps of nonsymmetric pre-stack travel time calculation and present some test results using synthetic and real data.

Riemann Boundary Value Problem with Square Roots on an Open Arc

The Riemann boundary value problem with square roots in class h0 when the jumping curve is an open arc in the complex plane is considered. It is solved by reducing it to a classical Riemann boundary value problem so that its solutions are obtained in closed form. In certain cases, some auxiliary function ω（z）is introduced. With different choices of ω（z）＇s, some interesting examples are illustrated.

Logarithm Versus Square Root： Comparing Quantum Fisher Information

In classical statistics, the Fisher information is unique in the sense that it is essentially the only monotone Riemannian metric on the space of probability densities. In quantum theory, this uniqueness breaks down, and there are many natural quantum analogues of the Fisher information, among which two particular versions distinguish themselves by their intuitive and informational significance： The first has its origin in the skew information introduced by Wigner and Yanase in 1963 in the context of quantum measurement, and is defined via the square root of the density operator. The second arises from Helstrom＇s study of quantum detection in 1967, and is defined via the symmetric logarithmic derivative. The aim of this paper Js to compare these two versions of quantum Fisher information, and to establish two informational inequalities relating them.

WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON R^(n)

Let L=-div(A▽) be a second order divergence form elliptic operator with bounded measurable coefficients in R^(n).We establish weighted L^(p) norm inequalities for commutators generated by √L and Lipschitz functions,where the range of p is different from(1,∞),and we isolate the right class of weights introduced by Auscher and Martell.In this work,we use good-λ inequality with two parameters through the weighted boundedness of Riesz transforms ▽L^(-1/2).Our result recovers,in some sense,a previous result of Hofmann.

THE SQUARE ROOT OF H-MATRIX

1 IntroductionLet A∈Cn×n, B∈Cn×n.We say B is a square root of A if A=B×B i.e.A=B2.It is well-known that any symmetric positive definite matrix exists one and only onesquare root which is a symmetric positive definite matrix,too（e.g.see[5]）.Higham[4]studied carefully the relation of a real nonsingular matrix between its real square rootsand its eigenvalues.Alefeld and Schneider[1]pointed out that for any nonsingular M-ma-trix there is one and only one M-matrix as its square root.In this paper,we study on

The Commutator of the Kato Square Root for Second Order Elliptic Operators on R^n

Let L = -div（AV） be a second order divergence form elliptic operator, and A be an accretive, n × n matrix with bounded measurable complex coefficients in R^n. We obtain the Lp bounds for the commutator generated by the Kato square root v/L and a Lipschitz function, which recovers a previous result of Calderon, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.

H^p Spaces Associated with Square Root Operator

We characterize Hp spaces associated with square root operator by using maximal functions.

L^p-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on R^n

Let L=-div（A▽） be a second-order divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L1/2]of the Kato square root L1/2 and b with ▽b∈Ln（Rn）（n> 2）,is bounded from the homogenous Sobolev space L1p（Rn） to Lp（Rn）（p-（L） <p<p+（L））.

Assimilation of Doppler Radar Observations with an Ensemble Square Root Filter:A Squall Line Case Study

The effectiveness of using an Ensemble Square Root Filter（EnSRF） to assimilate real Doppler radar observations on convective scale is investigated by applying the technique to a case of squall line on 12July 2005 in midwest Shandong Province using the Weather Research and Forecasting（WRF） model.The experimental results show that：（1） The EnSRF system has the potential to initiate a squall line accurately by assimilation of real Doppler radar data.The convective-scale information has been added into the WRF model through radar data assimilation and thus the analyzed fields are improved noticeably.The model spin-up time has been shortened,and the precipitation forecast is improved accordingly.（2） Compared with the control run,the deterministic forecast initiated with the ensemble mean analysis of EnSRF produces more accurate prediction of microphysical fields.The predicted wind and thermal fields are reasonable and in accordance with the characteristics of convective storms.（3） The propagation direction of the squall line from the ensemble mean analysis is consistent with that of the observation,but the propagation speed is larger than the observed.The effective forecast period for this squall line is about 5-6 h,probably because of the nonlinear development of the convective storm.

Dynamics of one-prey two-predator system with square root functional response and time lag

Animals grouping together is one of the most interesting phenomena in population dynamics and different functional responses as a result of prey-predator forming groups have been considered by many authors in their models. In the present paper we have considered a model for one prey and two competing predator populations with time lag and square root functional response on account of herd formation by prey. It is shown that due to the inclusion of another competing predator, the underlying system without delay becomes more stable and limit cycles do not occur naturally. However, after considering the effect of time lag in the basic system, limit cycles appear in the case of all equilibrium points when delay time crosses some critical value. From the numerical simulation, it is observed that the length of delay is minimum when only prey population survives and it is maximum when all the populations coexist.

基于均方根容积卡尔曼滤波的船舶操纵运动响应模型参数辨识

为了解决扩展卡尔曼滤波(extended Kalman filter,EKF)算法在船舶操纵运动模型参数辨识中存在辨识精度低、稳定性差和泛化能力弱的问题,提出了一种基于均方根容积卡尔曼滤波(square root cubature Kalman filter,SRCKF)的辨识算法。在CKF框架下将方差矩阵的均方根代替原始方差矩阵,使用三角分解对其进行预测和更新以提高辨识的稳定性。将EKF作为对比算法,利用四阶龙格库塔法解算的数值仿真数据,对舵角符合舵机伺服机构变化的船舶二阶非线性响应模型参数进行辨识,并将得到的辨识模型开展泛化能力验证试验。结果表明:SRCKF算法具有比EKF算法更高的辨识精度、稳定性和泛化能力。

Manipulator tracking technology based on FSRUKF

Aiming at the shortcoming that the traditional industrial manipulator using off-line programming cannot change along with the change of external environment,the key technologies such as machine vision and manipulator control are studied,and a complete manipulator vision tracking system is designed.Firstly,Denavit-Hartenberg(D-H)parameters method is used to construct the model of the manipulator and analyze the forward and inverse kinematics equations of the manipulator.At the same time,a binocular camera is used to obtain the threedimensional position of the target.Secondly,in order to make the manipulator track the target more accurately,the fuzzy adaptive square root unscented Kalman filter(FSRUKF)is proposed to estimate the target state.Finally,the manipulator tracking system is built by using the position-based visual servo.The simulation experiments show that FSRUKF converges faster and with less error than the square root unscented Kalman filter(SRUKF),which meets the application requirements of the manipulator tracking system,and basically meets the application requirements of the manipulator tracking system in the practical experiments.

A New Multiobjective Particle Swarm Optimization Using Local Displacement and Local Guides

This paper introduces a novel variant of particle swarm optimization that leverages local displacements through attractors for addressing multiobjective optimization problems. The method incorporates a square root distance mechanism into the external archives to enhance the diversity. We evaluate the performance of the proposed approach on a set of constrained and unconstrained multiobjective test functions, establishing a benchmark for comparison. In order to gauge its effectiveness relative to established techniques, we conduct a comprehensive comparison with well-known approaches such as SMPSO, NSGA2 and SPEA2. The numerical results demonstrate that our method not only achieves efficiency but also exhibits competitiveness when compared to evolutionary algorithms. Particularly noteworthy is its superior performance in terms of convergence and diversification, surpassing the capabilities of its predecessors.

Machine acceleration time series prediction for dimensional accuracy of 3D printed parts

This study explores the influence of infill patterns on machine acceleration prediction in the realm of three-dimensional(3D)printing,particularly focusing on extrusion technology.Our primary objective was to develop a long short-term memory(LSTM)network capable of assessing this impact.We conducted an extensive analysis involving 12 distinct infill patterns,collecting time-series data to examine their effects on the acceleration of the printer’s bed.The LSTM network was trained using acceleration data from the adaptive cubic infill pattern,while the Archimedean chords infill pattern provided data for evaluating the network’s prediction accuracy.This involved utilizing offline time-series acceleration data as the training and testing datasets for the LSTM model.Specifically,the LSTM model was devised to predict the acceleration of a fused deposition modeling(FDM)printer using data from the adaptive cubic infill pattern.Rigorous testing yielded a root mean square error(RMSE)of 0.007144,reflecting the model’s precision.Further refinement and testing of the LSTM model were conducted using acceleration data from the Archimedean chords infill pattern,resulting in an RMSE of 0.007328.Notably,the developed LSTM model demonstrated superior performance compared to an optimized recurrent neural network(RNN)in predicting machine acceleration data.The empirical findings highlight that the adaptive cubic infill pattern considerably influences the dimensional accuracy of parts printed using FDM technology.

Prediction of Time Series Empowered with a Novel SREKRLS Algorithm

For the unforced dynamical non-linear state–space model,a new Q1 and efficient square root extended kernel recursive least square estimation algorithm is developed in this article.The proposed algorithm lends itself towards the parallel implementation as in the FPGA systems.With the help of an ortho-normal triangularization method,which relies on numerically stable givens rotation,matrix inversion causes a computational burden,is reduced.Matrix computation possesses many excellent numerical properties such as singularity,symmetry,skew symmetry,and triangularity is achieved by using this algorithm.The proposed method is validated for the prediction of stationary and non-stationary Mackey–Glass Time Series,along with that a component in the x-direction of the Lorenz Times Series is also predicted to illustrate its usefulness.By the learning curves regarding mean square error(MSE)are witnessed for demonstration with prediction performance of the proposed algorithm from where it’s concluded that the proposed algorithm performs better than EKRLS.This new SREKRLS based design positively offers an innovative era towards non-linear systolic arrays,which is efficient in developing very-large-scale integration(VLSI)applications with non-linear input data.Multiple experiments are carried out to validate the reliability,effectiveness,and applicability of the proposed algorithm and with different noise levels compared to the Extended kernel recursive least-squares(EKRLS)algorithm.

A simple approach for the estimation of CO_2 penetration depth into a caprock layer

Caprock is a water-saturated formation with a sufficient entry capillary pressure to prevent the upward migration of a buoyant fluid. When the entry capillary pressure of caprock is smaller than the pressure exerted by the buoyant CO2plume, CO2gradually penetrates into the caprock. The CO2penetration depth into a caprock layer can be used to measure the caprock sealing efficiency and becomes the key issue to the assessment of caprock sealing efficiency. On the other hand, our numerical simulations on a caprock layer have revealed that a square root law for time and pore pressure exists for the CO2penetration into the caprock layer. Based on this finding, this study proposes a simple approach to estimate the CO2penetration depth into a caprock layer. This simple approach is initially developed to consider the speed of CO2invading front. It explicitly expresses the penetration depth with pressuring time, pressure difference and pressure magnitude. This simple approach is then used to fit three sets of experimental data and good fittings are observed regardless of pressures, strengths of porous media, and pore fluids(water,hydrochloric acid, and carbonic acid). Finally, theoretical analyses are conducted to explore those factors affecting CO2penetration depth. The effects of capillary pressure, gas sorption induced swelling, and fluid property are then included in this simple approach. These results show that this simple approach can predict the penetration depth into a caprock layer with sufficient accuracy, even if complicated interactions in penetration process are not explicitly expressed in this simple formula.

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation , t 6 L, are considered, where L is a closed contour in the complex plane, b ≠ 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.

An Efficient Algorithm for Low Rank Matrix Restoration Problem with Unknown Noise Level

Recovering an unknown high dimensional low rank matrix from a small set of entries is widely spread in the fields of machine learning,system identification and image restoration,etc.In many practical applications,the few observations are always corrupted by noise and the noise level is also unknown.A novel model with nuclear norm and square root type estimator has been proposed,which does not rely on the knowledge or on an estimation of the standard deviation of the noise.In this paper,we firstly reformulate the problem to an equivalent variable separated form by introducing an auxiliary variable.Then we propose an efficient alternating direction method of multipliers(ADMM)for solving it.Both of resulting subproblems admit an explicit solution,which makes our algorithm have a cheap computing.Finally,the numerical results show the benefits of the model and the efficiency of the proposed method.

Inverse-square-root-acceleration method for predicting the failure time of landslides

Predicting the failure time of unstable slopes is one of the most pivotal issues. In this paper, the inverse square root acceleration(INSRA) method was proposed to estimate the time-of-failure(TOF) of landslides. Four collapsed slopes were presented in the three open-pit mines, two of them were probed by ground-based radar, and two of them were obtained from previous scientific papers. The inverse velocity(INV) method and INSRA method were adopted to analyze these four landslides and one slope which had great deformation but did not reach failure. Compared with the traditional INV method, the INSRA method can promote the forecasting effectiveness and has the advantage of higher accuracy.