Cauchy Method Research Articles

A semismooth conjugate gradients method – theoretical analysis

In large-scale applications, deterministic and stochastic variants of Cauchy's steepest descent method are widely used for the minimization of objectives that are only piecewise smooth. In this paper, we analyse a deterministic descent method based on the generalization of rescaled conjugate gradients proposed by Philip Wolfe in 1975 for objectives that are convex. Without this assumption, the new method exploits semismoothness to obtain pairs of directionally active generalized gradients such that it can only converge to Clarke stationary points. Numerical results illustrate the theoretical findings.

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.

Heavy-Tailed Distribution for Combining Dependent P-Values With Asymptotic Robustness

The issue of combining individual $p$-values to aggregate multiple small effects is prevalent in many scientific investigations and is a long-standing statistical topic. Many classical methods are designed for combining independent and frequent signals in a traditional meta-analysis sense using the sum of transformed $p$-values with the transformation of light-tailed distributions, in which Fisher's method and Stouffer's method are the most well-known. Since the early 2000, advances in big data promoted methods to aggregate independent, sparse and weak signals, such as the renowned higher criticism and Berk-Jones tests. Recently, Liu and Xie(2020) and Wilson(2019) independently proposed Cauchy and harmonic mean combination tests to robustly combine $p$-values under "arbitrary" dependency structure, where a notable application is to combine $p$-values from a set of often correlated SNPs in genome-wide association studies. The proposed tests are the transformation of heavy-tailed distributions for improved power with the sparse signal. It calls for a natural question to investigate heavy-tailed distribution transformation, to understand the connection among existing methods, and to explore the conditions for a method to possess robustness to dependency. In this paper, we investigate the regularly varying distribution, which is a rich family of heavy-tailed distribution and includes Pareto distribution as a special case. We show that only an equivalent class of Cauchy and harmonic mean tests have sufficient robustness to dependency in a practical sense. We also show an issue caused by large negative penalty in the Cauchy method and propose a simple, yet practical modification. Finally, we present simulations and apply to a neuroticism GWAS application to verify the discovered theoretical insights and provide practical guidance.

On a Fifth-Order Method for Multiple Roots of Nonlinear Equations

There are several measures for comparing methods for solving a single nonlinear equation. The first is the order of convergence, then the cost to achieve such rate. This cost is measured by counting the number of functions (and derivatives) evaluated at each step. After that, efficiency is defined as a function of the order of convergence and cost. Lately, the idea of basin of attraction is used. This shows how far one can start and still converge to the root. It also shows the symmetry/asymmetry of the method. It was shown that even methods that show symmetry when solving polynomial equations are not so when solving nonpolynomial ones. We will see here that the Euler–Cauchy method (a member of the Laguerre family of methods for multiple roots) is best in the sense that the boundaries of the basins have no lobes. The symmetry in solving a polynomial equation having two roots at ±1 with any multiplicity is obvious. In fact, the Euler–Cauchy method converges very fast in this case. We compare one member of a family of fifth-order methods for multiple roots with several well-known lower-order and efficient methods. We will show using a basin of attraction that the fifth-order method cannot compete with most of those lower-order methods.

Opt-RNN-DBFSVM: Optimal recurrent neural network density based fuzzy support vector machine

Two major problems are encountered when using fuzzy SVM: (a) the number of local minima increases exponentially with the number of samples and (b) the quantity of required computer storage, required for a regular quadratic programming solver, increases by an exponential magnitude as the problem size expands. The Kernel-Adatron family of algorithms gaining attention lately which has allowed to handle very large classification and regression problems. However, these methods treat different types of samples (Noise, border, and core) with the same manner, which causes searches in unpromising areas and increases the number of iterations. In this work, we introduce a hybrid method to overcome these shortcoming, namely Optimal Recurrent Neural Network Density Based fuzzy Support Vector Machine (Opt-RNN-DBFSVM). This method consists of four steps: (a) characterization of different samples, (b) elimination of samples with a low probability of being a support vector, (c) construction of an appropriate recurrent neural network based on an original energy function, and (d) solution of the system of differential equations, managing the dynamics of the RNN, using the Euler–Cauchy method involving an optimal time step. Thanks to its recurrent architecture, the RNN remembers the regions explored during the search process. We demonstrated that RNN-FSVM converges to feasible support vectors and Opt-RNN-DBFSVM has a very low time complexity compared to RNN-FSVM with constant time step, and KAs-FSVM. Several experiments were performed on academic data sets. We used several classification performance measures to compare Opt-RNN-DBFSVM to different classification methods and the results obtained show the good performance of the proposed method.

OPT-RNN-DBSVM: OPTimal Recurrent Neural Network and Density-Based Support Vector Machine

When implementing SVMs, two major problems are encountered: (a) the number of local minima of dual-SVM increases exponentially with the number of samples and (b) the computer storage memory required for a regular quadratic programming solver increases exponentially as the problem size expands. The Kernel-Adatron family of algorithms, gaining attention recently, has allowed us to handle very large classification and regression problems. However, these methods treat different types of samples (i.e., noise, border, and core) in the same manner, which makes these algorithms search in unpromising areas and increases the number of iterations as well. This paper introduces a hybrid method to overcome such shortcomings, called the Optimal Recurrent Neural Network and Density-Based Support Vector Machine (Opt-RNN-DBSVM). This method consists of four steps: (a) the characterization of different samples, (b) the elimination of samples with a low probability of being a support vector, (c) the construction of an appropriate recurrent neural network to solve the dual-DBSVM based on an original energy function, and (d) finding the solution to the system of differential equations that govern the dynamics of the RNN, using the Euler–Cauchy method involving an optimal time step. Density-based preprocessing reduces the number of local minima in the dual-SVM. The RNN’s recurring architecture avoids the need to explore recently visited areas. With the optimal time step, the search moves from the current vectors to the best neighboring support vectors. It is demonstrated that RNN-SVM converges to feasible support vectors and Opt-RNN-DBSVM has very low time complexity compared to the RNN-SVM with a constant time step and the Kernel-Adatron algorithm–SVM. Several classification performance measures are used to compare Opt-RNN-DBSVM with different classification methods and the results obtained show the good performance of the proposed method.

Hardware Suitability of Complex Natural Resonances Extraction Algorithms in Backscattered Radar Signals

Complex natural resonances (CNRs) extraction methods such as matrix pencil method (MPM), Cauchy, vector-fitting Cauchy method (VCM), or Prony’s method decompose a signal in terms of frequency components and damping factors based on Baum’s singularity expansion method (SEM) either in the time or frequency domain. The validation of these CNRs is accomplished through a reconstruction of the signal based on these complex poles and residues and a comparison with the input signal. Here, we perform quantitative performance metrics in order to have an evaluation of each method’s hardware suitability factor before selecting a hardware platform using benchmark signals, simulations of backscattering scenarios, and experiments.

Multi-Objective Optimization for Controlling the Dynamics of the Diabetic Population

To limit the adverse effects of diabetes, a personalized and long-term management strategy that includes appropriate medication, exercise and diet has become of paramount importance and necessity. Compartment-based mathematical control models for diabetes usually result in objective functions whose terms are conflicting, preventing the use of single-objective-based models for obtaining appropriate personalized strategies. Taking into account the conflicting aspects when controlling the diabetic population dynamics, this paper introduces a multi-objective approach consisting of four steps: (a) modeling the problem of controlling the diabetic population dynamics using a multi-objective mathematical model, (b) discretizing the model using the trapezoidal rule and the Euler–Cauchy method, (c) using swarm-intelligence-based optimizers to solve the model and (d) structuring the set of controls using soft clustering methods, known for their flexibility. In contrast to single-objective approaches, experimental results show that the multi-objective approach obtains appropriate personalized controls, where the control associated with the compartment of diabetics without complications is totally different from that associated with the compartment of diabetics with complications. Moreover, these controls enable a significant reduction in the number of diabetics with and without complications, and the multi-objective strategy saves up to 4% of the resources needed for the control of diabetes without complications and up to 18% of resources for the control of diabetes with complications.

Rounding error analysis of linear recurrences using generating series

We develop a toolbox for the error analysis of linear recurrences with constant or polynomial coefficients, based on generating series, Cauchy's method of majorants, and simple results from analytic combinatorics. We illustrate the power of the approach by several nontrivial application examples. Among these examples are a new worst-case analysis of an algorithm for computing Bernoulli numbers, and a new algorithm for evaluating differentially finite functions in interval arithmetic while avoiding interval blow-up.

Time-adaptive partitioned method for fluid-structure interaction problems with thick structures

In this paper, we present a partitioned numerical scheme for solving fluid-structure interaction (FSI) problems based on the adaptive time-stepping. The viscous, incompressible fluid is described using the Navier-Stokes equations expressed in an Arbitrary Lagrangian Eulerian (ALE) form while the elastic structure is modeled using elastodynamic equations. We implement a partitioned scheme based on the Robin-Robin coupling conditions at the interface, combined with the refactorization of Cauchy's one-legged θ-like method with adaptive time-stepping. The method is unconditionally stable, and for θ=12, it corresponds to the midpoint rule, which is conservative and second-order convergent in time. The focus of this paper is to study the time-adaptivity properties of the proposed method, and to explore the parameters used in the variable time-stepping. The adaptive process is based on the local truncation error (LTE), for computation of which we consider two methods: Milne's device using a modified Adams-Bashforth two-step method, and Taylor's method. The performance of the method is explored in numerical examples, where the adaptive approach is compared to the one where a fixed time step is used. We present an example based on the method of manufactured solutions, where the effect of different parameters is studied, followed by a classical benchmark problem of a flow around a rigid cylinder attached to a nonlinearly elastic bar inside a two-dimensional channel. Finally, we present a three-dimensional, simplified example of blood flow in a compliant artery.

Vector Fitting–Cauchy Method for the Extraction of Complex Natural Resonances in Ground Penetrating Radar Operations

In this paper, we obtain the Complex Natural Resonances of an object from the backscattered response in the frequency domain with a novel rational function approximation method based on both Vector Fitting and Cauchy methods. We determine the system order and an initial set of poles, which are used as a basis for a rational function approximation. The results from the simulations and experiments show an improvement in the reconstructed signals and the accuracy of the CNRs calculated, with an increased tolerance to the critical Signal-to-Noise Ratio. This is being used in the problem of GPR landmine humanitarian detection in Colombia.

Optimal control of an epidemic by vaccination using dynamic programming approach

In this paper, we are interested in solving a SIR epidemic model which can be reformulated as a control problem. We use some recent developments of the dynamics programming method to obtain a rigorous solution of the optimal control problem formulated in Trélat (2019). We use some refinement of Cauchy's method of characteristics for stratified Hamilton-Jacobi equations to describe a large set of admissible trajectories and identify a domain on which the value function exists and is generated by some admissible controlsm, and their optimality is justified by using one of the well-known verification theorems as an argument for sufficient optimality conditions.

The Cauchy method of analytical regularisation in the modelling of plane wave scattering by a flat pre‐fractal system of impedance strips

The Cauchy method of analytical regularisation, first in full explanation, is used for the modelling of the plane wave scattering by a flat pre-fractal system of impedance strips. It is based on a modification of the singular integral equation technique and the correct type of inversion formula of the simplest Cauchy singular integral equation. The approach is effective for the plane H-polarised electromagnetic (or sonic) wave scattering problems. The principle of strict mathematical orderings of the pre-fractal strip grating deals with a five-parts class of perfect fractal sets, which is characterised by a variable positive fractal dimension. Considerable attention is devoted to the asymptotic model of the plane wave scattering by a sparsely filled grating, which has an explicit solution. This result is very important for verification of the mathematical models presented and testing of direct numerical methods. The scattered electromagnetic field in the far zone is successfully analysed with the aid of the asymptotic solution derived for electrically narrow strips.

Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions

Using a small number of mathematical transformations, this article examines the nature of fractional models described by fractional differential equations or pseudo state space descriptions. Computation of the impulse response of a fractional model using the Cauchy method shows that they exhibit infinitely small and high time constants. This impulse response can be rewritten as a diffusive representation whose Fourier transform permits a representation of a fractional model by a diffusion equation in an infinite space domain. Fractional models can thus be viewed as doubly infinite dimensional models: infinite as distributed with a distribution in an infinite domain. This infinite domain or the infinitely large time constants of the impulse response reveal a property intrinsic to fractional models: their infinite memory. Solutions to generate fractional behaviors without infinite memory are finally proposed.

Self‐learning tuning of coaxial cavity filter by using the poles and residues of admittance function

To solve the complexity and blindness of the tuning process in the manufacture of microwave cavity filters, this study proposes a self-learning tuning method based on the poles and residues of the admittance function. First, the improved Cauchy's method based on the differential evolution algorithm is used to extract the poles and residues of the admittance parameters (Y-parameters) in a non-ideal environment, and the effect of different port phase shifts and cavity losses on the accuracy of parameter extraction is overcome. Second, a parametric model based on the experience and data fusion via the fuzzy neural network method is established according to the collected non-linear relation data. Furthermore, problems such as poor data reliability, low modelling accuracy and weak generalisation ability are solved. On this basis, an adaptive optimisation tuning of microwave cavity filters using an implicit space-mapping algorithm is proposed and problems such as convergence difficulty and dependence on the initial value are solved. The results of the online simulation show that the proposed method has a high tuning accuracy and fast tuning ability.

An elementary proof of Euler's formula using Cauchy's method

The use of Cauchy's method to prove Euler's well-known formula is an object of many controversies. The purpose of this paper is to prove that Cauchy's method applies for convex polyhedra and not only for them, but also for surfaces such as the torus, the projective plane, the Klein bottle and the pinched torus.

A generalized cauchy method for remaining useful life prediction of wind turbine gearboxes

The accurate estimate of the Remaining Useful Life (RUL) of mechanical tools is a fundamental problem in Engineering. This prediction often implies the knowledge and application of sophisticated mathematical methods based on fractal and Long-Range Dependence (LRD) stochastic processes. However, the existing RUL prediction methods based on stochastic model cannot simultaneously consider the fractal and LRD characteristics of the equipment degradation process. This paper describes a new RUL prediction model based on the Generalized Cauchy (GC) process, which is a stochastic process with independent parameters. That is, the GC process uses the fractal dimension D and Hurst index H to describe the fractal and LRD characteristics of the degradation sequence, respectively. Then, the GC process is taken as the diffusion term, describing the uncertainty of the degradation sequence, to establish the GC degradation model, and the power law and exponential forms are used to describe the nonlinear drift of the degradation sequence. The stochastic volatility of the degradation sequence causes the equipment RUL unable to be predicted for a long time. This article uses the largest Lyapunov index to reveal the maximum prediction range of RUL. The analysis of actual equipment degradation verifies the effectiveness of the degradation model based on power law drift and GC process. The prediction results of the comparative case show that the prediction performance of the GC degradation model is better than Brownian motion, fractional Brownian motion, and long short-term memory neural network.

Modelling of the process of vybromechanical activation of plant raw material hydrolysis for pectin extraction

Centrifugal and vibrational technological effects are among the main approaches to intensify the process of plant raw materials hydrolysis for pectin extraction. With the impulse intensification of such a process, it is possible not only to increase its efficiency, but also to achieve the compactness of the equipment, reduce the cost of electricity and improve the quality of the product of hydrolysis. The hypothesis is confirmed, according to which the vibro-centrifugal intensification of hydrolysis increases the driving force of the process by not only activating the material flows of raw materials and reagents, but also by reducing the resistance in the technological environment. Graphical and analytical dependencies of the power and energy parameters of the oscillatory system were obtained, which proved the overcoming of the flow resistance of the liquid medium in the entire speed range of the drive shaft with the potential to intensify the process at a power consumption of 2.0 – 3.0 kW and or by the force of 2.3 – 2.5 kN using the Lagrange and Cauchy methods for composing and solving the equations of motion of the moving components of the tested hydrolyser with vibrating activators, and the methods of mathematical analysis and their processing in the MathCAD. The analysis of the presented parameters of the studied process of mixing the pectin-containing mass in the hydrolyser allowed us to determine the rational mode parameters of processing, which correspond to the angular velocity of the drive shaft rad/s at the power consumption of 500 – 600 watts.

Optical properties and microstructure of two Ge-Sb-Se thin films

Multipoint Cauchy method (MCM) is presented to investigate the refractive index and dispersion for each of Ge<sub>20</sub>Sb<sub>15</sub>Se<sub>65</sub> and Ge<sub>28</sub>Sb<sub>12</sub>Se<sub>60</sub> chalcogenide thin films at any wavelength in the transmission spectrum based on the regional approach method and Cauchy fitting. We theoretically calculate and compare the refractive index and dispersion curves obtained by using six different models. The results show that the most accurate results are obtained by the MCM. Two Ge—Sb—Se films are prepared by magnetron sputtering experimentally, and transmission spectrum curves are measured by Fourier infrared spectrometer, the noise is removed by segmental filtering and then the refractive index, dispersion, absorption coefficient, and optical band gap of the two films ina range of 500–2500 nm are obtained by the MCM. The results show that the refractive index of Ge<sub>28</sub>Sb<sub>12</sub>Se<sub>60</sub> film is larger than that of Ge<sub>20</sub>Sb<sub>15</sub>Se<sub>65</sub> film, which is caused by the higher polarizability and density of the former. The refractive indexes of both films decrease with wavelength increasing, so the long waves travel faster than short waves in the two films. The optical band gap of Ge<sub>28</sub>Sb<sub>12</sub>Se<sub>60</sub> film (1.675 eV) is smaller than that of Ge<sub>20</sub>Sb<sub>15</sub>Se<sub>65</sub> film (1.729 eV), and the corresponding wavelengths of the two are 740.3 nm and 717.2 nm. Finally, the microstructures of the two films are characterized by Raman spectra, and the reasons why the two chalcogenide films have different optical properties are explained from the bonding properties between the atoms.

An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical behavior

Abstract In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Padé approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.