Symmetric Kernel Research Articles

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

An Algorithm for the Solution of Integro-Fractional Differential Equations with a Generalized Symmetric Singular Kernel

Dispersion relation from Lorentzian inversion in 1d CFT

Starting from the Lorentzian inversion formula, we derive a dispersion relation which computes a four-point function in 1d CFTs as an integral over its double discontinuity. The crossing symmetric kernel of the integral is given explicitly for the case of identical operators with integer or half-integer scaling dimension. This derivation complements the one that uses analytic functionals. We use the dispersion relation to evaluate holographic correlators defined on the half-BPS Wilson line of planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills, reproducing results up to fourth order in an expansion at large t’Hooft coupling.

Research on satellite structural health monitoring based on ultrashort femtosecond grating array and artificial neural network

The safety of spacecraft and satellite in orbit is very important, and structural health monitoring is needed. At present, the existing technology is limited by load and difficult to realize. In this paper, we propose a feasible method to detect and locate the damage of satellite by combining ultrashort femtosecond grating array inscribed on oxide-doped fiber with multilayer artificial neural network. An oxide-doped fiber with high robustness is designed, and ultrashort grating arrays are fabricated on the fiber by femtosecond laser point-by-point writing technology. The effects of impactor velocity and angle on impact response was investigated by numerical simulations and physical experiments. Subsequently, repeated impact experiments were conducted on the satellite to obtain the training dataset and testing dataset for two-dimensional convolutional neural network. The network with symmetric convention kernels has an 88.12% localization accuracy and a better performance in boundary region, and the network architecture with asymmetric convention kernels has a 90.31% accuracy and a better performance in middle region.

Persistence Symmetric Kernels for Classification: A Comparative Study

Persistence Symmetric Kernels for Classification: A Comparative Study

Stability Analysis of the Solution for the Mixed Integral Equation with Symmetric Kernel in Position and Time with Its Applications

Stability Analysis of the Solution for the Mixed Integral Equation with Symmetric Kernel in Position and Time with Its Applications

Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots.

Kernel density estimation of Tsalli’s entropy with applications in adaptive system training

Information theoretic learning plays a very important role in adaption learning systems. Many non-parametric entropy estimators have been proposed by the researchers. This work explores kernel density estimation based on Tsallis entropy. Firstly, it has been proved that for linearly independent samples and for equal samples, Tsallis-estimator is consistent for the PDF and minimum respectively. Also, it is investigated that Tsallis-estimator is smooth for differentiable, symmetric, and unimodal kernel function. Further, important properties of Tsallis-estimator such as scaling and invariance for both single and joint entropy estimation have been proved. The objective of the work is to understand the mathematics behind the underlying concept.

On Multiple-Type Wave Solutions for the Nonlinear Coupled Time-Fractional Schrödinger Model

Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as the Sardar-subequation technique (SSET) and the improved generalized tanh-function technique (IGTHFT). Numerous novel soliton solutions are computed using different formats, such as periodic, bell-shaped, dark, and combination single bright along with kink, periodic, and single soliton solutions. Additionally, single solitary wave, multi-wave, and periodic kink combined solutions are evaluated. The behavioral traits of the retrieved solutions are illustrated by certain distinctive two-dimensional, three-dimensional, and contour graphs. The results are encouraging, since they show that the suggested methods are trustworthy, consistent, and efficient in finding accurate solutions to the various challenging nonlinear problems that have recently surfaced in applied sciences, engineering, and nonlinear optics.

Application of the Green's function method for static analysis of nonlocal stress-driven and strain gradient elastic nanobeams

This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory.

A Symmetric Kernel Smoothing Estimation of the Time-Varying Coefficient for Medical Costs

In longitudinal studies, subjects are repeatedly observed at a set of distinct time points until the terminal event time. The time-varying coefficient model extends the parametric method and captures the dynamic trajectories of time-dependent covariate effects, thus enabling it to describe the potential relationship between the longitudinal variable and the observed time points. In this study, we propose a novel approach to the estimation of medical costs using a symmetric kernel smoothing method in the time-varying coefficient joint model. A smooth function of medical costs is derived by weighting the values of longitudinal data at all distinct observed time points via the combination of the kernel method and the inverse probability weighting method. For the simulation study, we first set up the true functions of time-varying coefficients; we then generated random samples for covariates and censored survival times. Subsequently, the longitudinal data of response variables could be produced. Further, numerical simulation experiments were conducted by using the proposed method and applying R code to the generated data. The estimated results for the parameters and non-parametric functions were compared with different settings. The numerical results illustrate that as the sample size increases, the bias and model-based standard errors decrease, and the performance improves with larger sample sizes. The estimates of functions in the model almost coincide with the true functions, as shown in the figures of the simulation study. Furthermore, the consistency of the obtained estimator is demonstrated via theoretical analysis, and a numerical simulation is performed to illustrate the performance of the proposed estimators. The proposed model is applied to a real-world data set acquired from a multicenter automatic defibrillator implantation trial (MADIT).

Incorporating the neutrosophic framework into kernel regression for predictive mean estimation

In traditional statistics, all research endeavors revolve around utilizing precise, crisp data for the predictive estimation of population mean in survey sampling, when the supplementary information is accessible. However, these types of estimates often suffer from bias. The major aim is to uncover the most accurate estimates for the unknown value of the population mean while minimizing the mean square error (MSE). We have employed the neutrosophic approach, which is the extension of classical statistics that deals with the uncertain, vague, and indeterminate information, and proposed a neutrosophic predictive estimator of finite population mean using the kernel regression. The proposed estimator does not yield a single numerical value but instead provides an interval range within which the population parameter is likely to exist. This approach enhances the efficiency of the estimators by offering an estimated interval that encompasses the unknown value of the population mean with the least possible mean squared error (MSE). The simulation-based efficiency of the proposed estimator is discussed using the Sine, Bump and real-time temperature data set of Islamabad by using symmetric (Gaussian) kernel. The proposed non-parametric neutrosophic estimator has shown more effective results under the various bandwidth selectors than the adapted neutrosophic estimators.

Learning with Asymmetric Kernels: Least Squares and Feature Interpretation.

Asymmetric kernels naturally exist in real life, e.g., for conditional probability and directed graphs. However, most of the existing kernel-based learning methods require kernels to be symmetric, which prevents the use of asymmetric kernels. This paper addresses the asymmetric kernel-based learning in the framework of the least squares support vector machine named AsK-LS, resulting in the first classification method that can utilize asymmetric kernels directly. We will show that AsK-LS can learn with asymmetric features, namely source and target features, while the kernel trick remains applicable, i.e., the source and target features exist but are not necessarily known. Besides, the computational burden of AsK-LS is as cheap as dealing with symmetric kernels. Experimental results on various tasks, including Corel, PASCAL VOC, Satellite, directed graphs, and UCI database, all show that in the case asymmetric information is crucial, the proposed AsK-LS can learn with asymmetric kernels and performs much better than the existing kernel methods that rely on symmetrization to accommodate asymmetric kernels.

Analysis of the Recursive Locally-Adaptive Filtration of 3D Tensor Images

This work is focused on the computational complexity (CC) reduction of the locally-adaptive processing of 3D tensor images, based on recursive approaches. As a basis, a local averaging operation is used, implemented as a sliding mean 3D filter (SM3DF) with a central symmetric 3D kernel. Symmetry plays a very important role in constructing the working window. The presented theoretical approach could be adopted in various algorithms for locally-adaptive processing, such as additive noise reduction, sharpness enhancement, texture segmentation, etc. The basic characteristics of the recursive SM3DF are analyzed, together with the main features of the adaptive algorithms for filtration of Gaussian noises and unsharp masking where the filter is aimed at. In the paper, the ability of SM3DF implementation through recursive sliding mean 1D filters, sequentially bonded together, is also introduced. The computational complexity of the algorithms is evaluated for the recursive and non-recursive mode. The recursive SM3DF also suits the 3D convolutional neural networks which comprise sliding locally-adaptive 3D filtration in their layers. As a result of the lower CC, a promising opportunity is opened for higher efficiency of the 3D image processing through tensor neural networks.

Hybrid Functions Approach via Nonlinear Integral Equations with Symmetric and Nonsymmetrical Kernel in Two Dimensions

The second kind of two-dimensional nonlinear integral equation (NIE) with symmetric and nonsymmetrical kernel is solved in the Banach space L2[0,1]×L2[0,1]. Here, the NIE’s existence and singular solution are described in this passage. Additionally, we use a numerical strategy that uses hybrid and block-pulse functions to obtain the approximate solution of the NIE in a two-dimensional problem. For this aim, the two-dimensional NIE will be reduced to a system of nonlinear algebraic equations (SNAEs). Then, the SNAEs can be solved numerically. This study focuses on showing the convergence analysis for the numerical approach and generating an estimate of the error. Examples are presented to prove the efficiency of the approach.

Computational Techniques for Solving Mixed (1 + 1) Dimensional Integral Equations with Strongly Symmetric Singular Kernel

This paper describes an effective strategy based on Lerch polynomial method for solving mixed integral equations (MIE) in position and time with a strongly symmetric singular kernel in the space L2(−1,1)×C[0,T],(T<1). The Quadratic numerical method (QNM) was applied to obtain a system of Fredholm integral equations (SFIE), then the Lerch polynomials method (LPM) was applied to transform SFIE into a system of linear algebraic equations (SLAE). The existence and uniqueness of the integral equation’s solution are discussed using Banach’s fixed point theory. Also, the convergence and stability of the solution and the stability of the error are discussed. Several examples are given to illustrate the applicability of the presented method. The Maple program obtains all the results. A numerical simulation is carried out to determine the efficacy of the methodology, and the results are given in symmetrical forms. From the numerical results, it is noted that there is a symmetry utterly identical to the kernel used when replacing each x with y.

On one problem for a fourth-order mixed-type equation that degenerates inside and on the boundary of a domain

In the article, a nonlocal boundary value problem has been investigated for a fourth-order mixed-type equation degenerating inside and on the boundary of a domain. Applying the method of separation of variables to the problem under study, the spectral problem for an ordinary differential equation is obtained. The Green function of the last problem is constructed, with the help of which it is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and the system of eigenfunctions for the spectral problem. The theorem of expansion of a given function into a uniformly convergent series with respect to the system of eigenfunctions is proved. Using the found integral equation and Mercer's theorem, a uniform convergence of some bilinear series depending on the found eigenfunctions is proved. The order of the Fourier coefficients is established. The solution of the problem under study is written as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. An estimate for the problem's solution is obtained, from which its continuous dependence on the given functions follows.

Fast D M,M calculation in LDR brachytherapy using deep learning methods

Objective. The Monte Carlo (MC) method provides a complete solution to the tissue heterogeneity effects in low-energy low-dose rate (LDR) brachytherapy. However, long computation times limit the clinical implementation of MC-based treatment planning solutions. This work aims to apply deep learning (DL) methods, specifically a model trained with MC simulations, to predict accurate dose to medium in medium (D M,M) distributions in LDR prostate brachytherapy. Approach. To train the DL model, 2369 single-seed configurations, corresponding to 44 prostate patient plans, were used. These patients underwent LDR brachytherapy treatments in which 125I SelectSeed sources were implanted. For each seed configuration, the patient geometry, the MC dose volume and the single-seed plan volume were used to train a 3D Unet convolutional neural network. Previous knowledge was included in the network as an r 2 kernel related to the first-order dose dependency in brachytherapy. MC and DL dose distributions were compared through the dose maps, isodose lines, and dose-volume histograms. Features enclosed in the model were visualized. Main results. Model features started from the symmetrical kernel and finalized with an anisotropic representation that considered the patient organs and their interfaces, the source position, and the low- and high-dose regions. For a full prostate patient, small differences were seen below the 20% isodose line. When comparing DL-based and MC-based calculations, the predicted CTV D 90 metric had an average difference of −0.1%. Average differences for OARs were −1.3%, 0.07%, and 4.9% for the rectum D 2cc, the bladder D 2cc, and the urethra D 0.1cc. The model took 1.8 ms to predict a complete 3D D M,M volume (1.18 M voxels). Significance. The proposed DL model stands for a simple and fast engine which includes prior physics knowledge of the problem. Such an engine considers the anisotropy of a brachytherapy source and the patient tissue composition.

Swarming: hydrodynamic alignment with pressure

We study the swarming behavior of hydrodynamic alignment. Alignment reflects steering toward a weighted average heading. We consider the class of so-called p p -alignment hydrodynamics, based on 2 p 2p -Laplacians and weighted by a general family of symmetric communication kernels. The main new aspect here is the long-time emergence behavior for a general class of pressure tensors without a closure assumption, beyond the mere requirement that they form an energy dissipative process. We refer to such pressure laws as “entropic”, and prove the flocking of p p -alignment hydrodynamics, driven by singular kernels with a general class of entropic pressure tensors. These results indicate the rigidity of alignment in driving long-time flocking behavior despite the lack of thermodynamic closure.

Нелокальная начально-граничная задача для вырождающиегося уравнения четвертого порядка с дробной производной Герасимова-Капуто

В последнее время интенсивно изучаются начально – граничные задачи в прямоугольной области для дифференциальных уравнений в частных производных как четного, так и нечетного порядка. При этом в качестве объекта исследования, в основном, берется не вырождающееся уравнение или уравнение, вырождающееся на одной стороне четырехугольника. Начально – граничные задачи (как локальные, так и нелокальные) для уравнений с двумя или тремя линиями вырождения остаются неизученными. В данной работе в прямоугольной области рассмотрено уравнение четвёртого порядка, вырождающееся на трех сторонах четырехугольника и содержащее оператор дробного дифференцирования Герасимова –Капуто. Для этого уравнения сформулирована и исследована одна начально – граничная задача с нелокальными условиями, связывающими значения искомой функции и её производных до третьего порядка (включительно), принимаемых на боковых сторонах прямоугольника. Сначала методом интегралов энергии доказана единственность решения поставленной задачи. Затем, исследована спектральная задача, возникающая при применении метода Фурье, основанном на разделении переменных, к поставленной начально – граничной задаче. Построена функция Грина спектральной задачи, с помощью чего она эквивалентно сведена к интегральному уравнению Фредгольма второго рода с симметричным ядром, откуда следует существование счетного числа собственных значений и собственных функций спектральной задачи. Доказана теорема разложения заданной функции в равномерно сходящийся ряд по системе собственных функций. С помощью найденного интегрального уравнения и теоремы Мерсера доказана равномерная сходимость некоторых билинейных рядов, зависящих от найденных собственных функций. Установлен порядок коэффициентов Фурье. Решение изучаемой задачи выписано в виде суммы ряда Фурье по системе собственных функций спектральной задачи. Исследована равномерная сходимость этого ряда и рядов, полученных из него почленным дифференцированием. Получена оценка для решения задачи, откуда следует его непрерывная зависимость от заданных функций. Recently, initial-boundary problems in a rectangular domain for differential equations in partial derivatives of both even and odd order have been intensively studied. In this case, non-degenerate equations or equations that degenerate on one side of the quadrilateral are taken as the object of study. But initialboundary problems (both local and non-local) for equations with two or three lines of degeneracy remain unexplored. In this paper, in a rectangular domain, a fourth-order equation degene-rating on three sides of the rectangular and contains the Gerasimov-Caputo fractional diffe-rentiation operator has been considered. For this equation, an initial-boundary problem is formulated and investigated, with non-local conditions connecting the values of the desired function and its derivatives up to the third order (inclusive), taken on the sides of the rectangle. From the beginning, the uniqueness of the solution of the formulated problem was proved by the method of energy integrals. Then, the spectral problem that arises when applying the Fourier method based on the separation of variables to the considered initial-boundary problem has been investigated. The Green’s function of the spectral problem was constructed, with the help of which it is equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel, which implies the existence of a countable number of eigenvalues and eigenfunctions of the spectral problem. A theorem is proved for expanding a given function into a uniformly convergent series in terms of a system of eigenfunctions. Using the found integral equation and Mercer’s theorem, we prove the uniform convergence of some bilinear series depending on the found eigenfunctions. The order of the Fourier coeffi-cients have been established. The solution of the considered is written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is studied. An estimate for solution to problem is obtained, from which follows its continuous dependence on the given functions.

Robust kernel-based feature representation for 3D point cloud analysis via circular convolutional network

Feature descriptors of point clouds are used in several applications, such as registration and part segmentation of 3D point clouds. Learning representations of local geometric features is unquestionably the most important task for accurate point cloud analyses. However, it is challenging to develop rotation or scale-invariant descriptors. Most previous studies have either ignored rotations or empirically studied optimal scale parameters, which hinders the applicability of the methods for real-world datasets. In this paper, we present a new local feature description method that is robust to rotation and scale variations. Moreover, we improve representations based on a global aggregation method. First, we place kernels aligned around each point in the normal direction. To avoid the sign problem of the normal vector, we use a symmetric kernel point distribution in the tangential plane. From each kernel point, we first project the points from the spatial space to the feature space, which is robust to multiple scales and rotation, based on angles and distances. Subsequently, we perform convolutions by considering local kernel point structures and long-range global context, obtained by a global aggregation method. We experimented with our proposed descriptors on benchmark datasets (i.e., ModelNet40 and ShapeNetPart) to evaluate the performance of registration, classification, and part segmentation on 3D point clouds. Our method showed superior performances when compared to the state-of-the-art methods by reducing 70% of the rotation and translation errors in the registration task. Our method also showed comparable performance in the classification and part-segmentation tasks without any external data augmentation.