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Статті в журналах з теми "Algebraic kernel"

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Автор: Grafiati

Опубліковано: 8 червня 2024

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1

Klos, Basia. "Kernels of Algebraic Curvature Tensors of Symmetric and Skew-Symmetric Builds." PUMP Journal of Undergraduate Research 6 (August 26, 2023): 301–16. http://dx.doi.org/10.46787/pump.v6i0.3456.

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The kernel of an algebraic curvature tensor is a fundamental subspace that can be used to distinguish between different algebraic curvature tensors. Kernels of algebraic curvature tensors built only of canonical algebraic curvature tensors of a single build have been studied in detail. We consider the kernel of an algebraic curvature tensor R that is a sum of canonical algebraic curvature tensors of symmetric and skew-symmetric build. An obvious way to ensure that the kernel of R is nontrivial is to choose the involved bilinear forms such that the intersection of their kernels is nontrivial. We present a construction wherein this intersection is trivial but the kernel of R is nontrivial. We also show how many bilinear forms satisfying certain conditions are needed in order for R to have a kernel of any allowable dimension.

2

Chen, Hung-Yuan. "Kernel Inclusions of Algebraic Automorphisms." Communications in Algebra 39, no. 4 (March 21, 2011): 1365–71. http://dx.doi.org/10.1080/00927871003705567.

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3

Altschuler, Jason M., and Pablo A. Parrilo. "Kernel Approximation on Algebraic Varieties." SIAM Journal on Applied Algebra and Geometry 7, no. 1 (February 17, 2023): 1–28. http://dx.doi.org/10.1137/21m1425050.

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4

Hulle, Marc M. Van. "Edgeworth-Expanded Gaussian Mixture Density Modeling." Neural Computation 17, no. 8 (August 1, 2005): 1706–14. http://dx.doi.org/10.1162/0899766054026657.

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Instead of increasing the order of the Edgeworth expansion of a single gaussian kernel, we suggest using mixtures of Edgeworth-expanded gaussian kernels of moderate order. We introduce a simple closed-form solution for estimating the kernel parameters based on weighted moment matching. Furthermore, we formulate the extension to the multivariate case, which is not always feasible with algebraic density approximation procedures.

5

Altürk, Ahmet. "Application of the Bernstein polynomials for solving Volterra integral equations with convolution kernels." Filomat 30, no. 4 (2016): 1045–52. http://dx.doi.org/10.2298/fil1604045a.

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In this article, we consider the second-type linear Volterra integral equations whose kernels based upon the difference of the arguments. The aim is to convert the integral equation to an algebraic one. This is achieved by approximating functions appearing in the integral equation with the Bernstein polynomials. Since the kernel is of convolution type, the integral is represented as a convolution product. Taylor expansion of kernel along with the properties of convolution are used to represent the integral in terms of the Bernstein polynomials so that a set of algebraic equations is obtained. This set of algebraic equations is solved and approximate solution is obtained. We also provide a simple algorithm which depends both on the degree of the Bernstein polynomials and that of monomials. Illustrative examples are provided to show the validity and applicability of the method.

6

Kim, Hee Sik, Choonkil Park, and Eun Hwa Shim. "Function kernels and divisible groupoids." AIMS Mathematics 7, no. 7 (2022): 13563–72. http://dx.doi.org/10.3934/math.2022749.

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<abstract><p>In this paper, we introduce the notion of a function kernel which was motivated from the kernel in group theory, and we apply this notion to several algebraic structures, e.g., groups, groupoids, $ BCK $-algebras, semigroups, leftoids. Using the notions of left and right cosets in groupoids, we investigate some relations with function kernels. Moreover, the notion of an idenfunction in groupoids is introduced, which is a generalization of an identity axiom in algebras by functions, and we discuss it with function kernels.</p></abstract>

7

Hasso, Mohammad Shami. "Algebraic Kernel Method for Solving Fredholm Integral Equations." International Frontier Science Letters 7 (March 2016): 25–33. http://dx.doi.org/10.18052/www.scipress.com/ifsl.7.25.

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In this paper, we study the exact solution of linear Fredholm integral equations using some classical methods including degenerate kernel method and Fredholm determinants method. We propose an analytical method for solving such integral equations. This work has some goals related to suggested technique for solving Fredholm integral equations. The primary goal gives analytical solutions of such equations with minimum steps. Another goal is to compare the suggested method used in this study with classical methods. The final goal is that the propose method is an explicit formula that can be studied in detail for non-algebraic function kernels by using Taylor series expansion and for system of Fredholm integral equations.

8

Wirsing, Martin. "Structured algebraic specifications: A Kernel language." Theoretical Computer Science 42 (1986): 123–249. http://dx.doi.org/10.1016/0304-3975(86)90051-4.

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9

Jan, A. R. "An Asymptotic Model for Solving Mixed Integral Equation in Position and Time." Journal of Mathematics 2022 (August 30, 2022): 1–11. http://dx.doi.org/10.1155/2022/8063971.

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In this paper, we considered a mixed integral equation (MIE) of the second kind in the space L 2 − b , b × C 0 , T , T < 1. The kernel of position has a singularity and takes some different famous forms, while the kernels of time are positive and continuous. Using an asymptotic method of separating the variables, we have a Fredholm integral equation (FIE) in position with variable parameters in time. Then, using the Toeplitz matrix method (TMM), we obtain a linear algebraic system (LAS) that can be solved numerically. Some applications with the aid of the maple 18 program are discussed when the kernel takes Coleman function, Cauchy kernel, Hilbert kernel, and a generalized logarithmic function. Also the error estimate, in each case, is computed.

10

Miana, Pedro J., Juan J. Royo, and Luis Sánchez-Lajusticia. "Convolution Algebraic Structures Defined by Hardy-Type Operators." Journal of Function Spaces and Applications 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/212465.

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The main aim of this paper is to show that certain Banach spaces, defined via integral kernel operators, are Banach modules (with respect to some known Banach algebras and convolution products onℝ+). To do this, we consider some suitable kernels such that the Hardy-type operator is bounded in weighted Lebesgue spacesLωpℝ+forp≥1. We also show new inequalities in these weighted Lebesgue spaces. These results are applied to several concrete function spaces, for example, weighted Sobolev spaces and fractional Sobolev spaces defined by Weyl fractional derivation.

11

Usmonov, Botir. "A Numerical Solution of Hereditary Equations with a Weakly Singular Kernel for Vibration Analysis of Viscoelastic Systems / Vienâdojumu Ar Vâjo Singulâro Kodolu Skaitliskais Risinâjums Iedzimto Viskoelastîgo Sistçmu Vibrâciju Analîzei." Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences. 69, no. 6 (December 1, 2015): 326–30. http://dx.doi.org/10.1515/prolas-2015-0048.

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Abstract Viscoelastic, or composite materials that are hereditary deformable, have been characterised by exponential and weakly singular kernels in a hereditary equation. An exponential kernel is easy to be numerically implemented, but does not well describe complex vibratory behaviour of a hereditary deformable system. On the other hand, a weakly singular kernel is known to describe the complex vibratory behaviour, but is nontrivial to be numerically implemented. This study presents a numerical formulation for solving a hereditary equation with a weakly singular kernel. Recursive algebraic equations, which are numerically solvable, are formulated by using the Galerkin method enhanced by a numerical integration and elimination of weak singularity. Numerical experiments showed that the present approach with a weakly singular kernel is well fitted into a realistic vibratory behaviour of a hereditary deformable system under dynamic loads, as compared to the same approach with an exponential kernel.

12

Martínez, Jorge. "Unit and kernel systems in algebraic frames." Algebra universalis 55, no. 1 (January 2006): 13–43. http://dx.doi.org/10.1007/s00012-006-1965-1.

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13

THǍŃG, NGUYÊÑ QUÔĆ. "ON THE RATIONALITY OF ALMOST SIMPLE ALGEBRAIC GROUPS." International Journal of Mathematics 10, no. 05 (August 1999): 643–65. http://dx.doi.org/10.1142/s0129167x99000252.

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We prove the stable rationality of almost simple adjoint algebraic groups, the connected components of the Dynkin diagram of anisotropic kernel of which contain at most two vertices. The (stable) rationality of many isotropic almost simple groups with small anisotropic kernel and some related results in weak approximation over arbitrary fields are discussed.

14

Gross, Leonard. "Some Norms on Universal Enveloping Algebras." Canadian Journal of Mathematics 50, no. 2 (April 1, 1998): 356–77. http://dx.doi.org/10.4153/cjm-1998-019-4.

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AbstractThe universal enveloping algebra, U(𝔤), of a Lie algebra 𝔤 supports some norms and seminorms that have arisen naturally in the context of heat kernel analysis on Lie groups. These norms and seminorms are investigated here from an algebraic viewpoint. It is shown that the norms corresponding to heat kernels on the associated Lie groups decompose as product norms under the natural isomorphism . The seminorms corresponding to Green's functions are examined at a purely Lie algebra level for sl (2, ℂ). It is also shown that the algebraic dual space U′ is spanned by its finite rank elements if and only if 𝔤 is nilpotent.

15

Aomoto, Kazuhiko. "Algebraic equations for Green kernel on a tree." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 4 (1988): 123–25. http://dx.doi.org/10.3792/pjaa.64.123.

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16

Jeong, Moon-Ja, and Masahiko Taniguchi. "ALGEBRAIC KERNEL FUNCTIONS AND REPRESENTATION OF PLANAR DOMAINS." Journal of the Korean Mathematical Society 40, no. 3 (May 6, 2003): 447–60. http://dx.doi.org/10.4134/jkms.2003.40.3.447.

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17

Prasad, Gopal, and Andrei S. Rapinchuk. "On the congruence kernel for simple algebraic groups." Proceedings of the Steklov Institute of Mathematics 292, no. 1 (January 2016): 216–46. http://dx.doi.org/10.1134/s0081543816010144.

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18

Wang, Q. "Algebraic structure and kernel of the Schrodinger equation." Journal of Physics A: Mathematical and General 20, no. 15 (October 21, 1987): 5041–44. http://dx.doi.org/10.1088/0305-4470/20/15/019.

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19

Prasad, Gopal, and Andrei S. Rapinchuk. "On the Congruence Kernel for Simple Algebraic Groups." Труды математического института им. Стеклова 292, no. 01 (2016): 224–54. http://dx.doi.org/10.1134/s0371968516010143.

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20

Kirby, Jonathan, Angus Macintyre, and Alf Onshuus. "The algebraic numbers definable in various exponential fields." Journal of the Institute of Mathematics of Jussieu 11, no. 4 (April 2, 2012): 825–34. http://dx.doi.org/10.1017/s1474748012000047.

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AbstractWe prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: for the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.

21

Zakarya, M., Ghada AlNemer, H. A. Abd El-Hamid, Roqia Butush, and H. M. Rezk. "On Refinements of Multidimensional Inequalities of Hardy-Type via Superquadratic and Subquadratic Functions." Journal of Mathematics 2022 (November 24, 2022): 1–17. http://dx.doi.org/10.1155/2022/7668860.

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By utilizing the peculiarities of superquadratic and subquadratic functions, we give the extensions for multidimensional inequalities of Hardy-type with general kernel. We use some algebraic inequalities such as the Minkowski inequality, the refined Jensen inequality, and the Bernoulli inequality to prove the essential results in this paper. The performance of the superquadratic functions is reliable and effective to obtain new dynamic inequalities on time scales. By utilizing special kernels, we also acquire numerous examples and implementations of the related inequalities.

22

Bordelon, Blake, and Cengiz Pehlevan. "Self-consistent dynamical field theory of kernel evolution in wide neural networks ^*." Journal of Statistical Mechanics: Theory and Experiment 2023, no. 11 (November 1, 2023): 114009. http://dx.doi.org/10.1088/1742-5468/ad01b0.

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Abstract We analyze feature learning in infinite-width neural networks trained with gradient flow through a self-consistent dynamical field theory. We construct a collection of deterministic dynamical order parameters which are inner-product kernels for hidden unit activations and gradients in each layer at pairs of time points, providing a reduced description of network activity through training. These kernel order parameters collectively define the hidden layer activation distribution, the evolution of the neural tangent kernel (NTK), and consequently, output predictions. We show that the field theory derivation recovers the recursive stochastic process of infinite-width feature learning networks obtained by Yang and Hu with tensor programs. For deep linear networks, these kernels satisfy a set of algebraic matrix equations. For nonlinear networks, we provide an alternating sampling procedure to self-consistently solve for the kernel order parameters. We provide comparisons of the self-consistent solution to various approximation schemes including the static NTK approximation, gradient independence assumption, and leading order perturbation theory, showing that each of these approximations can break down in regimes where general self-consistent solutions still provide an accurate description. Lastly, we provide experiments in more realistic settings which demonstrate that the loss and kernel dynamics of convolutional neural networks at fixed feature learning strength are preserved across different widths on a image classification task.

23

Karvonen, Toni, Chris Oates, and Mark Girolami. "Integration in reproducing kernel Hilbert spaces of Gaussian kernels." Mathematics of Computation 90, no. 331 (June 18, 2021): 2209–33. http://dx.doi.org/10.1090/mcom/3659.

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The Gaussian kernel plays a central role in machine learning, uncertainty quantification and scattered data approximation, but has received relatively little attention from a numerical analysis standpoint. The basic problem of finding an algorithm for efficient numerical integration of functions reproduced by Gaussian kernels has not been fully solved. In this article we construct two classes of algorithms that use N N evaluations to integrate d d -variate functions reproduced by Gaussian kernels and prove the exponential or super-algebraic decay of their worst-case errors. In contrast to earlier work, no constraints are placed on the length-scale parameter of the Gaussian kernel. The first class of algorithms is obtained via an appropriate scaling of the classical Gauss–Hermite rules. For these algorithms we derive lower and upper bounds on the worst-case error of the forms exp ⁡ ( − c 1 N 1 / d ) N 1 / ( 4 d ) \exp (-c_1 N^{1/d}) N^{1/(4d)} and exp ⁡ ( − c 2 N 1 / d ) N − 1 / ( 4 d ) \exp (-c_2 N^{1/d}) N^{-1/(4d)} , respectively, for positive constants c 1 > c 2 c_1 > c_2 . The second class of algorithms we construct is more flexible and uses worst-case optimal weights for points that may be taken as a nested sequence. For these algorithms we derive upper bounds of the form exp ⁡ ( − c 3 N 1 / ( 2 d ) ) \exp (-c_3 N^{1/(2d)}) for a positive constant c 3 c_3 .

24

Khakpour, F., and G. Ardeshir. "A novel algebraic method for kernel-based object tracking." Computers & Electrical Engineering 40, no. 5 (July 2014): 1482–97. http://dx.doi.org/10.1016/j.compeleceng.2014.02.006.

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25

Li, Fengling, Fengchun Lei, and Jie Wu. "3-Manifold invariants derived from the intersecting kernels." International Journal of Mathematics 27, no. 13 (December 2016): 1650109. http://dx.doi.org/10.1142/s0129167x16501093.

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The intersecting kernel of a Heegaard splitting [Formula: see text] for a compact orientable 3-manifold [Formula: see text] is the subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is the homomorphism induced by the inclusion [Formula: see text], [Formula: see text]. In the paper, we obtain some invariants of 3-manifolds [Formula: see text] from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.

26

Alqhtani, Manal, and Khaled M. Saad. "Numerical solutions of space-fractional diffusion equations via the exponential decay kernel." AIMS Mathematics 7, no. 4 (2022): 6535–49. http://dx.doi.org/10.3934/math.2022364.

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<abstract><p>The main object of this paper is to investigate the spectral collocation method for three new models of space fractional Fisher equations based on the exponential decay kernel, for which properties of Chebyshev polynomials are utilized to reduce these models to a set of differential equations. We then numerically solve these differential equations using finite differences, with the resulting algebraic equations solved using Newton 's method. The accuracy of the numerical solution is verified by computing the residual error function. Additionally, the numerical results are compared with other results obtained using the power law kernel and the Mittag-Leffler kernel. The advantage of the present work stems from the use of spectral methods, which have high accuracy and exponential convergence for problems with smooth solutions. The numerical solutions based on Chebyshev polynomials are in remarkably good agreement with numerical solutions obtained using the power law and the Mittag-Leffler kernels. Mathematica was used to obtain the numerical solutions.</p></abstract>

27

Nazarova, K. "ON ONE METHOD FOR OBTAINING UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR AN INTEGRO-DIFFERENTIAL EQUATION." Q A Iasaýı atyndaǵy Halyqaralyq qazaq-túrіk ýnıversıtetіnіń habarlary (fızıka matematıka ınformatıka serııasy), no. 1 (March 15, 2022): 42–54. http://dx.doi.org/10.47526/2022-2/2524-0080.04.

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The modified method of parametrization is used to study a linear Fredholm integro-differential equation with a degenerate kernel. Using the fundamental matrix, the conditions are established for the existence of a solution to the special Cauchy problem for the Fredholm integro-differential equation with a degenerate kernel. A system of linear algebraic equations is constructed with respect to the introduced additional parameters. Conditions for the unique solvability of a linear boundary value problem for the Fredholm integro-differential equation with a degenerate kernel are obtained.

28

Zhang, Tong. "Learning Bounds for Kernel Regression Using Effective Data Dimensionality." Neural Computation 17, no. 9 (September 1, 2005): 2077–98. http://dx.doi.org/10.1162/0899766054323008.

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Kernel methods can embed finite-dimensional data into infinite-dimensional feature spaces. In spite of the large underlying feature dimensionality, kernel methods can achieve good generalization ability. This observation is often wrongly interpreted, and it has been used to argue that kernel learning can magically avoid the “curse-of-dimensionality” phenomenon encountered in statistical estimation problems. This letter shows that although using kernel representation, one can embed data into an infinite-dimensional feature space; the effective dimensionality of this embedding, which determines the learning complexity of the underlying kernel machine, is usually small. In particular, we introduce an algebraic definition of a scale-sensitive effective dimension associated with a kernel representation. Based on this quantity, we derive upper bounds on the generalization performance of some kernel regression methods. Moreover, we show that the resulting convergent rates are optimal under various circumstances.

29

Chang, Yu-Lung, Ying-Te Lee, Li-Jie Jiang, and Jeng-Tzong Chen. "Green’s Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equation." International Journal of Computational Methods 13, no. 05 (August 31, 2016): 1650020. http://dx.doi.org/10.1142/s0219876216500201.

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A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.

30

Alalyani, Ahmad, M. A. Abdou, and M. Basseem. "On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method." Journal of Applied Mathematics 2023 (January 12, 2023): 1–9. http://dx.doi.org/10.1155/2023/5163398.

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This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space L 2 − 1 , 1 × C 0 , T , T < 1 . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

31

Kamoh, Nathaniel Mahwash, Geoffrey Kumlengand, and Joshua Sunday. "Matrix Approach To The Direct Computation Method For The Solution of Fredholm Integro-Differential Equations of The Second Kind With Degenerate Kernels." CAUCHY 6, no. 3 (November 19, 2020): 100–108. http://dx.doi.org/10.18860/ca.v6i3.8960.

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In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.

32

Raad, Sameeha Ali, and Mariam Mohammed Al-Atawi. "Nyström Method to Solve Two-Dimensional Volterra Integral Equation with Discontinuous Kernel." Journal of Computational and Theoretical Nanoscience 18, no. 4 (April 1, 2021): 1177–84. http://dx.doi.org/10.1166/jctn.2021.9718.

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In this paper, a linear two-dimensional Volterra integral equation of the second kind with the discontinuous kernel is considered. The conditions for ensuring the existence of a unique continuous solution are mentioned. The product Nystrom method, as a well-known method of solving singular integral equations, is presented. Therefore, the Nystrom method is applied to the linear Volterra integral equation with the discontinuous kernel to convert it to a linear algebraic system. Some formulas are expanded in two dimensions. Weights’ functions of the Nystrom method are obtained for kernels of logarithmic and Carleman types. Some numerical applications are presented to show the efficiency and accuracy of the proposed method. Maple18 is used to compute numerical solutions. The estimated error is calculated in each case. The Nystrom method is useful and effective in treating the two-dimensional singular Volterra integral equation. Finally, we conclude that the time factor and the parameter v have a clear effect on the results.

33

Litvinov, G. L., and G. B. Shpiz. "Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach." Journal of Mathematical Sciences 141, no. 4 (March 2007): 1417–28. http://dx.doi.org/10.1007/s10958-007-0049-x.

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34

Dinuzzo, Francesco, and Giuseppe De Nicolao. "An algebraic characterization of the optimum of regularized kernel methods." Machine Learning 74, no. 3 (January 10, 2009): 315–45. http://dx.doi.org/10.1007/s10994-008-5095-1.

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35

Basseem, M., and Ahmad Alalyani. "On the Solution of Quadratic Nonlinear Integral Equation with Different Singular Kernels." Mathematical Problems in Engineering 2020 (November 19, 2020): 1–7. http://dx.doi.org/10.1155/2020/7856207.

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All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.

36

Zhang, Huaiqing, Yu Chen, and Xin Nie. "Fast Hankel Transforms Algorithm Based on Kernel Function Interpolation with Exponential Functions." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/105469.

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The Pravin method for Hankel transforms is based on the decomposition of kernel function with exponential function. The defect of such method is the difficulty in its parameters determination and lack of adaptability to kernel function especially using monotonically decreasing functions to approximate the convex ones. This thesis proposed an improved scheme by adding new base function in interpolation procedure. The improved method maintains the merit of Pravin method which can convert the Hankel integral to algebraic calculation. The simulation results reveal that the improved method has high precision, high efficiency, and good adaptability to kernel function. It can be applied to zero-order and first-order Hankel transforms.

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Emil Sobhi Shoukralla, Nermin Abdelsatar Saber, and Ahmed Yehia Sayed. "The Numerical Solutions of weakly singular Fredholm integral equations of the Second kind Using Chebyshev Polynomials of the Second Kind." Journal of Advanced Research in Applied Sciences and Engineering Technology 44, no. 1 (April 26, 2024): 22–30. http://dx.doi.org/10.37934/araset.44.1.2230.

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In this study, the second kind Chebyshev Polynomials were utilized to acquire interpolated solutions for the second kind Fredholm integral equations with weakly singular kernel. To accomplish this, the data, unknown, and kernel functions were converted into matrix form, and consequently we completely isolated the singularity of the kernel. The primary benefit of this method is the ability to change the form of integral equation to an equivalent algebraic system, which is easier to solve. The effectiveness of our technique was evaluated by applying it to three illustrated examples, and it was observed that the solutions obtained exhibit strong convergence towards the exact solutions.

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Barik, Prasanta Kumar, Ankik Kumar Giri, and Philippe Laurençot. "Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 4 (February 19, 2019): 1805–25. http://dx.doi.org/10.1017/prm.2018.158.

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AbstractGlobal weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

39

AVRAMIDI, I. G. "COVARIANT TECHNIQUES FOR COMPUTATION OF THE HEAT KERNEL." Reviews in Mathematical Physics 11, no. 08 (September 1999): 947–80. http://dx.doi.org/10.1142/s0129055x99000295.

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The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard–Minakshisundaram–De Witt–Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.

40

Duits, Remco, Erik J. Bekkers, and Alexey Mashtakov. "Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs." Entropy 21, no. 1 (January 8, 2019): 38. http://dx.doi.org/10.3390/e21010038.

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Fokker–Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on S E ( 2 ) . Here, we extend these approaches to 3D using Fourier transform on the Lie group S E ( 3 ) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations R 3 ⋊ S 2 : = S E ( 3 ) / ( { 0 } × S O ( 2 ) ) as the quotient in S E ( 3 ) . In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker–Planck PDEs of α -stable Lévy processes on R 3 ⋊ S 2 . This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for α = 1 (the diffusion kernel) to the kernel for α = 1 2 (the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.

41

YILMAZ, Koray. "A Note on 4-Dimensional 2-Crossed Modules." Journal of New Theory, no. 42 (March 31, 2023): 86–93. http://dx.doi.org/10.53570/jnt.1208633.

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The study presents the direct product of two objects in the category of 4-dimensional 2-crossed modules. The structures of the domain, kernel, image, and codomain can be related using isomorphism theorems by defining the kernel and image of a morphism in a category. It then establishes the kernel and image of a morphism in the category of 4-dimensional 2-crossed modules to apply isomorphism theorems. These isomorphism theorems provide a powerful tool to understand the properties of this category. Moreover, isomorphism theorems in 4-dimensional 2-crossed modules allow us to establish connections between different algebraic structures and simplify complicated computations. Lastly, the present research inquires whether additional studies should be conducted.

42

Kalimbetov, B. T., V. F. Safonov, and O. D. Tuychiev. "Systems of integral equations with a degenerate kernel and an algorithm for their solution using the Maple program." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 108, no. 4 (December 30, 2022): 60–75. http://dx.doi.org/10.31489/2022m4/60-75.

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In the mathematical literature, a scalar integral equation with a degenerate kernel is well described (see below (1)), where all the written functions are scalar quantities). The authors are not aware of publications where systems of integral equations of (1) type with kernels in the form of a product of matrices would be considered in detail. It is usually said that the technique for solving such systems is easily transferred from the scalar case to the vector one (for example, in the monograph A.L. Kalashnikov "Methods for the approximate solution of integral equations of the second kind" (Nizhny Novgorod: Nizhny Novgorod State University, 2017), a brief description of systems of equations with degenerate kernels is given, where the role of degenerate kernels is played by products of scalar rather than matrix functions). However, as the simplest examples show, the generalization of the ideas of the scalar case to the case of integral systems with kernels in the form of a sum of products of matrix functions is rather unclear, although in this case the idea of reducing an integral equation to an algebraic system is also used. At the same time, the process of obtaining the conditions for the solvability of an integral system in the form of orthogonality conditions, based on the conditions for the solvability of the corresponding algebraic system, as it seems to us, has not been previously described. Bearing in mind the wide applications of the theory of integral equations in applied problems, the authors considered it necessary to give a detailed scheme for solving integral systems with degenerate kernels in the multidimensional case and to implement this scheme in the Maple program. Note that only scalar integral equations are solved in Maple using the intsolve procedure. The authors did not find a similar procedure for solving systems of integral equations, so they developed their own procedure.

43

Galanin, M. P., and D. L. Sorokin. "Simulation of Quasistationary Electromagnetic Fields in Regions Containing Disconnected Conducting Subregions." Herald of the Bauman Moscow State Technical University. Series Natural Sciences, no. 82 (2019): 4–15. http://dx.doi.org/10.18698/1812-3368-2019-1-4-15.

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Methods for a numerical solution of Maxwell's equations in the quasistationary aproximation in a region with multiply connected conducting subregions were discussed. The case of nontrivial operator kernel was explored. The methods for finding the solution of the linear algebraic equations system were analyzed. The method of introducing a "fictional armature" was offered as alternative method for searching [retrieving] a normal solution of linear algebraic equations. Results of computational experiments were presented. The study was carried out on the example of calculation for electrodynamic acceleration process in the railgun channel

44

SAIRA, Wen-Xiu Ma, and Guidong Liu. "A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations." Fractal and Fractional 8, no. 2 (January 26, 2024): 80. http://dx.doi.org/10.3390/fractalfract8020080.

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The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.

45

Ahmed, Shazad Shawki, and Shokhan Ahmed Hamasalih. "Solving a System of Caputo Fractional-Order Volterra Integro-Differential Equations with Variable Coefficients Based on the Finite Difference Approximation via the Block-by-Block Method." Symmetry 15, no. 3 (February 27, 2023): 607. http://dx.doi.org/10.3390/sym15030607.

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This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order lies in (0,1] using two and three order block-by-block approach with explicit finite difference approximation. With this method, we aim to use an appropriate process to transform our problem into an analogous piecewise iterative linear algebraic system. Moreover, algorithms for treating LSVIFDEs using the above process have been developed, in order to express these solutions. In addition, numerical examples for our scheme are presented based on various kernels, including symmetry kernel and other sorts of separate kernels, are used to illustrate the validity, effectiveness and applicability of the suggested approach. Consequently, comparisons are performed with exact results using this technique, to communicate these approaches most general programs are written in Python V.3.8.8. software (2021).

46

Li, Peter, and Gang Tian. "On the heat kernel of the Bergmann metric on algebraic varieties." Journal of the American Mathematical Society 8, no. 4 (1995): 857. http://dx.doi.org/10.1090/s0894-0347-1995-1320155-0.

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47

Avramidi, I. G. "Covariant algebraic method for calculation of the low‐energy heat kernel." Journal of Mathematical Physics 36, no. 9 (September 1995): 5055–70. http://dx.doi.org/10.1063/1.531371.

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48

Bresar, M. "The range and kernel inclusion of algebraic derivations and commuting maps." Quarterly Journal of Mathematics 56, no. 1 (March 1, 2005): 31–41. http://dx.doi.org/10.1093/qmath/hah019.

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49

Jan, A. R., M. A. Abdou, and M. Basseem. "A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method." Fractal and Fractional 7, no. 9 (August 31, 2023): 656. http://dx.doi.org/10.3390/fractalfract7090656.

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In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space L2Ω×C0,T, T<1. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS).

50

Shahoodh, Mohd. "On BDM-Algebras." Wasit Journal of Computer and Mathematics Science 2, no. 1 (March 31, 2023): 125–34. http://dx.doi.org/10.31185/wjcm.101.

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Abstract algebra is one of the influential branches in the field of pure Mathematics. This field concentrate on the study of the algebraic structures and discussed the relationships among them. Many studies have been presented various types of algebraic structures some of which independently and some others have been constructed via extending form other algebraic structure in order to investigate some of their properties. In this paper, we established an algebraic structure namely BDM-Algebras and studied some of its properties. Furthermore, we presented the 0-commtativity, sub-algebra and normal sub-algebra of a BDM-Algebras. In addition, we provided BDM-homomorphism and the kernel of BDM-homomorphism with some properties of them. Moreover, we introduced the quotient BDM-Algebras by using the notation of normal ideal of BDM-Algebras. Finally, we introduced the concept of the direct product of BDM-Algebras and some of its properties have been discussed. Some examples are given to illustrated the results.