Cauchy Kernel Research Articles

Event-Driven Maximum Correntropy Filter Based on Cauchy Kernel for Spatial Orientation Using Gyros/Star Sensor Integration

Gyros/star sensor integration provides a potential method to obtain high-accuracy spatial orientation for turntable structures. However, it is subjected to the problem of accuracy loss when the measurement noises become non-Gaussian due to the complex spatial environment. This paper presents an event-driven maximum correntropy filter based on Cauchy kernel to handle the above problem. In this method, a direct installation mode of gyros/star sensor integration is established and the associated mathematical model is derived to improve the turntable’s control stability. Based on this, a Cauchy kernel-based maximum correntropy filter is developed to curb the influence of non-Gaussian measurement noise for enhancing the gyros/star sensor integration’s robustness. Subsequently, an event-driven mechanism is constructed based on the filter’s innovation information for further reducing the unnecessary computational cost to optimize the real-time performance. The effectiveness of the proposed method has been validated by simulations of the gyros/star sensor integration for spatial orientation. This shows that the proposed filtering methodology not only has strong robustness to deal with the influence of non-Gaussian measurement noise but can also achieve superior real-time spatial applications with a small computational cost, leading to enhanced performance for the turntable’s spatial orientation using gyros/star sensor integration.

Calculating a perturbation of a plasma layer by an electric field

The paper presents the results of solving a boundary value problem for a system of two integro-differential equations that simulate the action of an external electric field on a plasma layer. This system is an implication of the Boltzmann–Maxwell equations, and the physical meaning of the sought functions is the strength of a self-consistent electric field and perturbation of the electron distribution density. The solution of the problem is constructed using the theories of Fourier transform of generalized functions and singular integral equations with the Cauchy kernel. The dependence of the solution on the frequency of the external field is studied.

Unsupervised domain adaptation bearing fault diagnosis method based on joint feature alignment

In this paper, the issue of cross-condition fault diagnosis of bearing is studied. During actual operation, the conditions of bearing vary due to changes in factors such as rotation speed and load, and the data distribution between different working conditions varies. Deep learning models that perform well in one condition are not ideal when applied to another condition directly. To address this problem, a novel unsupervised domain adaptation fault diagnosis method based on joint feature alignment is proposed in this paper. 1D-CNN is used as a weight-shared feature extractor to extract the features from both the source and target domains. The discrepancies in marginal and conditional distributions between the source and target domains are comprehensively considered by multi-layer multi-bandwidth Cauchy kernel maximum mean discrepancy (MB-CMMD) and mutual information (MI). The domain drift is reduced by aligning the feature representations of source and target domains. The network after feature alignment demonstrates a notable enhancement in the diagnostic accuracy of unlabeled samples within the target domain. The experimental results demonstrate that, in comparison to other domain adaptation approaches, The proposed approach can significantly enhance the accuracy of fault diagnosis while realizing feature alignment.

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Studies on the Marchenko–Pastur Law

In free probability, the theory of Cauchy–Stieltjes Kernel (CSK) families has recently been introduced. This theory is about a set of probability measures defined using the Cauchy kernel similarly to natural exponential families in classical probability that are defined by means of the exponential kernel. Within the context of CSK families, this article presents certain features of the Marchenko–Pastur law based on the Fermi convolution and the t-deformed free convolution. The Marchenko–Pastur law holds significant theoretical and practical implications in various fields, particularly in the analysis of random matrices and their applications in statistics, signal processing, and machine learning. In the specific context of CSK families, our study of the Marchenko–Pastur law is summarized as follows: Let K+(μ)={Qmμ(dx);m∈(m0μ,m+μ)} be the CSK family generated by a non-degenerate probability measure μ with support bounded from above. Denote by Qmμ•s the Fermi convolution power of order s>0 of the measure Qmμ. We prove that if Qmμ•s∈K+(μ), then μ is of the Marchenko–Pastur type law. The same result is obtained if we replace the Fermi convolution • with the t-deformed free convolution t.

Improved mixture correntropy cubature Kalman filter for attitude estimation

Abstract This work proposes an algorithm based on improved mixture correntropy cubature Kalman filtering (IMC-CKF) to address the issues of low accuracy and susceptibility to complex noise and outlier interference in inertial navigation attitude estimation. First, a combination of Gaussian kernel and Cauchy kernel is suggested to construct mixture correntropy to tackle the problem of single kernel-based correntropy being insufficient to confronted with complex noise. Second, the model fitting loss based on mean square error and measurement fitting loss based on mixture correntropy are used to establish the objective function. The maximum correntropy criterion (MCC) replaces the minimum mean square error (MMSE) criterion, and the fixed-point iteration method is used to solve the objective function, deriving the mixture correntropy matrix to adjust the measurement noise variance. Finally, the semi-trapezoidal membership function is used to determine the mixture correntropy coefficient. Accordingly, the algorithm can adaptively select the proportions of each kernel function according to the respective noise interference situation. Simulations and dynamic–static experiments have been conducted, and the algorithm has been compared with single kernel-based correntropy algorithms and other robust algorithms to confirm its superior precision and stability under complex noise and outlier interference.

Robust colored point cloud alignment based on L*a*b* guided and Cauchy kernel

AbstractPrecision agriculture benefits from point set registration, which can monitor plant health and growth in real time, promote the precise application of fertilizers and pesticides, and provide technical support for achieving sustainable development of agriculture. In this work, we propose a robust point set registration method for precision agriculture based on L*a*b* color guidance, bidirectional search and Cauchy distribution. First, the L*a*b* color guidance is applied to establish accurate correspondences between agricultural RGB‐D data. Second, the bidirectional nearest neighbor search strategy between point sets improves the reliability of establishing correspondences and broadens the convergence domain of the algorithm. Third, Cauchy distribution is utilized as an energy function for noise suppression, which further improves the robustness of the algorithm in dealing with complex vegetation scenes. Finally, results of ablation and simulation experiments indicate that the proposed registration algorithm can achieve more accurate and robust alignment results than other classic and state‐of‐the‐art point cloud registration algorithms to achieve monitoring and comparison of plant growth.

Performance enhancement of PPP/SINS tightly coupled navigation based on improved robust maximum correntropy kalman filtering

Extended kalman filter (EKF) is widely used to integrated navigation system of global navigation satellite system (GNSS) and strapdown inertial navigation system (SINS). However, non-Gaussian noise and the uncertainty of measurement noise can seriously reduce the performance of EKF, so it is difficult to obtain an optimal GNSS/INS integration solution. At present, non-Gaussian noise processing is still a difficult issue in filter research. In this paper, an adaptive and robust maximum correntropy extended kalman filter (MCEKF) method based on Cauchy kernel function is proposed to solve the above problem. Thanks to the excellent properties of Cauchy kernel function, the proposed method can effectively avoid filter faults and has better stability. However, the performance of MCEKF depends on the select of kernel bandwidth parameter, which is a difficult problem in the engineering application of MCEKF. In this paper, the switching kernel bandwidth algorithm is employed to adaptively estimate the kernel bandwidth parameter to solve the tradeoff between convergence rate and steady-state misalignment in the MCEKF algorithm under constant kernel bandwidth. Finally, the performance of the proposed algorithm is verified by two sets of vehicle-mounted precise point positioning (PPP) and SINS tightly coupled integration experiments in urban environment. The results show that compared with EKF, the proposed method can significantly improve the positioning accuracy, that is, the PPP/SINS positioning accuracy based on GE, GR and GRE system is improved by 25.1 %, 2.5 % and 17.0 % in D196 experiment, and 23.5 %, 16.5 % and 31.8 % in D107 experiment, respectively. The velocity and attitude accuracy of the two methods are similar. The velocity errors of all schemes can reach cm/s level. The roll, pitch and heading errors of navigation-grade inertial sensor are all less than 0.12 degree. The roll and pitch errors of MEMS are less than 1.0 degree, the heading error of the G, GE, GR, and GRE systems are 1.719, 1.464, 1.676, and 1.475 degree, respectively.

Orthonormal expansions for translation-invariant kernels

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of ℒ2(R). This allows us to derive explicit expansions on the real line for (i) Matérn kernels of all half-integer orders in terms of associated Laguerre functions, (ii) the Cauchy kernel in terms of rational functions, and (iii) the Gaussian kernel in terms of Hermite functions.

Tricomi problem and integral equations

Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.

Boundary Value Problems for the Perturbed Dirac Equation

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F˜λ.

Theoretical analysis of the interaction between blasting stress wave and linear interface crack under high in-situ stress in deep rock mass

This study theoretically investigates the scattering characteristics of blasting stress waves, treaded as elastic waves, at a linear interface crack in a deep rock mass with an even compressive in-situ stress field. First, the basic equations of the deep rock mass theoretically treaded as a homogeneous isotropic medium with an initial stress field are derived based on the linear elastic, finite deformation, plane strain, isotropic, and small deformation hypotheses, and incremental stress theory. The linear interface crack is theoretically treaded as two displacement-free boundaries, of which the interpolation displacements are described by introducing two dislocation density functions. Next, the mathematical model of the blasting stress wave scattering at the linear interface crack is established through a set of singular integral equations of the first type for these two dislocation density functions with Cauchy kernels. Finally, the dislocating and opening displacements of the linear interface crack and the displacement and effective stress fields of the scattered stress wave are numerically calculated. The principal stress field of the scattered stress wave is compared with the experimental data to qualitatively verify the rationality and correctness of the mathematical model. Theoretically, it is found that even compressive in-situ stress with an amplitude of tens of megapascals significantly affects the displacement and effective stress fields of the scattered stress wave near the linear interface crack. In conclusion, considering even compressive in-situ stress is greatly significant when studying the wave and scattering behaviors of deep rock masses. This mathematical model provides a theoretical guidance for the study of stress wave energy transformation and crack propagation in the process of rock blasting and mineral mining.

New Approach of Normal and Shear Stress Components for Multiple Curvilinear Holes Which Weakened a Flexible Plate

In this article, a thin infinite flexible plate weakened by multiple curvilinear holes is considered. The strength shapes are mapped outside a unit circle with the assistance of particular conformal mapping under certain conditions. The mathematical model that governs the rounded forces of the current physical problem is the boundary value problem of elastic media. This study is applicable to many phenomena throughout nature, like tunnels, caves, and excavations in soil or rock. The Cauchy method for complex variables is used to get the closed forms of Gaursat functions and change the problem to a second-type integrodifferential equation with a Cauchy kernel, which is used for a large area of the contact problems. Then, the normal and shear stress components that act on the model are derived. Afterward, some of the physical applications are studied, and different stress components at specific values in each application are calculated and plotted using Maple 2023.

Cauchy kernel minimum error entropy centralized fusion filter

With the help of Information Theory Learning (ITL) theory, a kind of increasingly popular non-Gaussian filtering method has been designed in the sense of the maximum correntropy (MC) criterion or minimum error entropy (MEE) criterion. In MC or MEE criterion, Gaussian kernel function is usually chosen as the kernel function. Compared to the Gaussian kernel function, the Cauchy kernel function has a wider range of action and is insensitive to the kernel parameters. In this paper, a new Cauchy kernel MEE criterion is defined. In the sense of the new defined criterion, two kinds of non-Gaussian centralized fusion filtering algorithms are proposed with the help of fixed-point theory, which are Cauchy kernel minimum error entropy parallel centralized fusion filter and Cauchy kernel minimum error entropy sequential centralized fusion filter. The propagation equation is used to calculate the prior estimates of the state and covariance, and then the fixed-point iteration is used to calculate the posterior estimates of the state and covariance. The convergence of the iteration process included in the two proposed methods is analyzed by using Banach fixed-point theorem. The final two experimental simulations verify the superior performance of the two proposed fusion filtering algorithms.

Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel

The integro-differential equation with the Cauchy kernel is used in many different technical problems, such as in circuit analysis or gas infrared radiation studies. Therefore, it is important to be able to solve this type of equation, even in an approximate way. This article compares two approaches for solving this type of equation. One of the considered methods is based on the application of the differential Taylor series, while the second approach uses selected heuristic algorithms inspired by the behavior of animals. Due to the problem domain, which is symmetric, and taking into account the form of the function appearing in this equation, we can use this symmetry in some cases. The paper also presents numerical examples illustrating how each method works and comparing the discussed approaches.

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g⩾2 with the explicit parametrization in terms of the Tyurin data. The ‘non-abelian’ theta divisor consists of bundles such that h1⩾1 . On the complement of this divisor we construct a non-abelian (i.e. matrix) Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which also depends holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1.

Weighted composition operators on spaces generated by fractional Cauchy kernels of the unit ball

Weighted composition operators on spaces generated by fractional Cauchy kernels of the unit ball

An analytical model for thermoelastic crack problems in a strip incorporating convective heat transfer between lateral surfaces and ambient environment

Most existing thermoelastic crack models commonly overlook the convective heat exchange occurring on the lateral surfaces of the structures being studied. Consequently, there is often a concern regarding the accurate estimation of temperature fields and the resulting thermal stresses. To address this concern, this paper presents a two-dimensional (2D) thermoelastic model for a cracked strip that incorporates convective heat transfer between the lateral surfaces and the ambient environment. The crack problem is reduced to a set of singular integral equations with Cauchy kernels which are solved numerically. Analytical solutions for temperature distribution, stress intensity factors (SIFs) and energy release rate at crack tips are obtained. The influence of the ratios of convective heat transfer coefficient to thermal conductivity and thickness to height of the strip on the key fracture parameters is investigated through the numerical results. It has been discovered that classical thermoelastic crack models often exhibit a significant underestimation of the SIFs and energy release rates for cracks located near the hot boundary of the strip. The analytical model proposed in this paper has significant potential to advance our understanding of thermoelastic fracture mechanics.

Cauchy DMP: Improving 3C industrial assembly quality with the Cauchy kernel and singular value decomposition

AbstractAlthough Dynamic Movement Primitives (DMP) is an effective tool for robotic arm trajectory generalisation, the application of DMP in the 3C (Computer, Communication, Consumer Electronics) industry still faces challenges such as low precision and high‐time consumption. To address this problem, we propose a novel Cauchy DMP framework. The main improvements and advantages of Cauchy DMP, compared to the original DMP, are (1) since the Cauchy distribution has a simpler model and wider shape, using the Cauchy distribution instead of the Gaussian distribution in the original DMP reduces the complexity of the algorithm and saves time. (2) Singular Value Decomposition (SVD) can effectively model the error. To reduce the interference of the rounding and human error on the trajectory, SVD can be used to obtain the weight of each basis function. The proposed Cauchy DMP framework combines the above two points and is validated on a real UR5 robotic arm. The results show that Cauchy DMP retains the learnability of the original DMP and has the advantages of short time consumption and low error rate.

New Monotonicity and Infinite Divisibility Properties for the Mittag-Leffler Function and for Stable Distributions

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions Eα,α∈(0,2], enjoy the HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable α and complex numbers z, the real and imaginary part of the functions x↦Eαzx, are tightly linked to the stable distributions and to the generalized Cauchy kernel.