Journal articles: 'Cauchy-type integrals'

1

Kapustin, V. V. "Cauchy-type integrals and singular measures." St. Petersburg Mathematical Journal 24, no. 5 (July 24, 2013): 743–57. http://dx.doi.org/10.1090/s1061-0022-2013-01263-5.

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Pakovich, F., N. Roytvarf, and Y. Yomdin. "Cauchy-type integrals of algebraic functions." Israel Journal of Mathematics 144, no. 2 (September 2004): 221–91. http://dx.doi.org/10.1007/bf02916714.

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3

Abdulkawi, M. "Bounded solution of Cauchy type singular integral equation of the first kind using differential transform method." JOURNAL OF ADVANCES IN MATHEMATICS 14, no. 1 (April 30, 2018): 7580–95. http://dx.doi.org/10.24297/jam.v14i1.7049.

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In this paper, an efficient approximate solution for solving the Cauchy type singular integral equation of the first kind is presented. Bounded solution of the Cauchy type singular Integral equation is discussed. Two type of kernel, separable and convolution, are considered. The differential transform method is used in the solution. New theorems for transformation of Cauchy singular integrals are given with proofs. Approximate results areshown to illustrate the efficiency and accuracy of the approximate solution.

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4

Lin, Feng. "On error for Cauchy principal integrals and Cauchy-type integrals under perturbation of integral curve and their applications." Complex Variables and Elliptic Equations 60, no. 11 (March 26, 2015): 1457–74. http://dx.doi.org/10.1080/17476933.2015.1022167.

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Lanzani, Loredana, and Elias M. Stein. "Cauchy-type integrals in several complex variables." Bulletin of Mathematical Sciences 3, no. 2 (June 9, 2013): 241–85. http://dx.doi.org/10.1007/s13373-013-0038-y.

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Vinogradov, S. A. "Continuity of perturbations of integral operators, Cauchy-type integrals, maximal operators." Journal of Soviet Mathematics 34, no. 6 (September 1986): 2033–39. http://dx.doi.org/10.1007/bf01741577.

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Cai, Hongzhu, and Michael Zhdanov. "Application of Cauchy-type integrals in developing effective methods for depth-to-basement inversion of gravity and gravity gradiometry data." GEOPHYSICS 80, no. 2 (March 1, 2015): G81—G94. http://dx.doi.org/10.1190/geo2014-0332.1.

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One of the most important applications of gravity surveys in regional geophysical studies is determining the depth to basement. Conventional methods of solving this problem are based on the spectrum and/or Euler deconvolution analysis of the gravity field and on parameterization of the earth’s subsurface into prismatic cells. We have developed a new method of solving this problem based on 3D Cauchy-type integral representation of the potential fields. Traditionally, potential fields have been calculated using volume integrals over the domains occupied by anomalous masses subdivided into prismatic cells. This discretization can be computationally expensive, especially in a 3D case. The technique of Cauchy-type integrals made it possible to represent the gravity field and its gradients as surface integrals. In this approach, only the density contrast surface between sediment and basement needed to be discretized for the calculation of gravity field. This was especially significant in the modeling and inversion of gravity data for determining the depth to the basement. Another important result was developing a novel method of inversion of gravity data to recover the depth to basement, based on the 3D Cauchy-type integral representation. Our numerical studies determined that the new method is much faster than conventional volume discretization method to compute the gravity response. Our synthetic model studies also showed that the developed inversion algorithm based on Cauchy-type integral is capable of recovering the geometry and depth of the sedimentary basin effectively with a complex density profile in the vertical direction.

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SAIRA and Shuhuang Xiang. "Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels." Symmetry 11, no. 6 (May 28, 2019): 728. http://dx.doi.org/10.3390/sym11060728.

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In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, ⨍ − 1 1 f ( x ) log ( x − α ) e i k x x − t d x , t ∉ ( − 1 , 1 ) , α ∈ [ − 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples.

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9

Conceição, Ana C., and Jéssica C. Pires. "Symbolic Computation Applied to Cauchy Type Singular Integrals." Mathematical and Computational Applications 27, no. 1 (December 31, 2021): 3. http://dx.doi.org/10.3390/mca27010003.

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The development of operator theory is stimulated by the need to solve problems emerging from several fields in mathematics and physics. At the present time, this theory has wide applications in the study of non-linear differential equations, in linear transport theory, in the theory of diffraction of acoustic and electromagnetic waves, in the theory of scattering and of inverse scattering, among others. In our work, we use the computer algebra system Mathematica to implement, for the first time on a computer, analytical algorithms developed by us and others within operator theory. The main goal of this paper is to present new operator theory algorithms related to Cauchy type singular integrals, defined in the unit circle. The design of these algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. Several nontrivial examples computed with the algorithms are presented. The corresponding source code of the algorithms has been made available as a supplement to the online edition of this article.

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10

Caballero, J., and K. Sadarangani. "A Cauchy–Schwarz type inequality for fuzzy integrals." Nonlinear Analysis: Theory, Methods & Applications 73, no. 10 (November 2010): 3329–35. http://dx.doi.org/10.1016/j.na.2010.07.013.

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11

Stepanets, A. I. "Approximation of cauchy-type integrals in Jordan domains." Ukrainian Mathematical Journal 45, no. 6 (June 1993): 890–917. http://dx.doi.org/10.1007/bf01061441.

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12

Stenger, Frank. "Sinc approximation of Cauchy-type integrals over arcs." ANZIAM Journal 42, no. 1 (July 2000): 87–97. http://dx.doi.org/10.1017/s1446181100011627.

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AbstractIn 1984, Elliott and Stenger wrote a joint paper on the approximation of Hilbert transforms over analytic arcs. In the present paper we sharpen the previously obtained results of Elliott and Stenger, and we also obtain formulas for approximating Cauchy integrals over analytic arcs.

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13

Tsamasphyros, G., and G. Dimou. "A Gauss quadrature formula for Cauchy type integrals." Computational Mechanics 4, no. 2 (March 1988): 137–48. http://dx.doi.org/10.1007/bf00282416.

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14

Morkunas, V. I. "Cauchy-type integrals in domains with smooth boundary." Journal of Soviet Mathematics 44, no. 6 (March 1989): 866–67. http://dx.doi.org/10.1007/bf01463198.

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15

A. K, Saha, Hota M. K., and Mohanty P. K. "Non-Classical Quadrature Schemes for the Approximation of Cauchy Type Oscillatory and Singular Integrals in Complex Plane." Malaysian Journal of Mathematical Sciences 16, no. 1 (January 31, 2022): 11–23. http://dx.doi.org/10.47836/mjms.16.1.02.

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In this paper, non-classical numerical schemes are proposed for the approximation of Cauchy type oscillatory and strongly singular integrals in complex plane. The schemes are developed by incorporating classical quadrature rule meant for the Cauchy type complex singular integrals over a line segment in complex plane with a quasi exact quadrature method meant for the numerical integration of complex definite integrals with an oscillatory weight function. The error bounds are established and the schemes are numerically validated using a set of standard test integrals. Numerical results show that these schemes are efficient.

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16

Amirjanyan, H. A., A. V. Sahakyan, and A. K. Kukudzhanov. "Quadrature formulas for Cauchy-type integrals with the Cauchy kernel to a integer power and a Jacobi weight function with complex exponents." Journal of Physics: Conference Series 2231, no. 1 (April 1, 2022): 012020. http://dx.doi.org/10.1088/1742-6596/2231/1/012020.

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Abstract The paper presents quadrature formulas for hypersingular integrals of various orders. It is assumed that the density of these integrals is represented as a product of a function that satisfies the Hölder condition and a weight function of Jacobi orthogonal polynomials. In this case, the exponents of the weight function can be complex numbers, the real part of which is greater than -1. Numerical analysis of the dependence of the root-mean-square deviation of the quadrature formula of order 8 on the value of the hypersingular integral calculated using standard software packages is carried out for various complex values of the weight function exponents. For hypersingular integrals up to the fourth order inclusive, a numerical analysis of the convergence of quadrature formulas is carried out for certain complex values of the exponents of the weight function.

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17

Král, Josef, and Dagmar Medková. "Angular limits of the integrals of the Cauchy type." Czechoslovak Mathematical Journal 47, no. 4 (December 1997): 593–617. http://dx.doi.org/10.1023/a:1022810416360.

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18

Kim, Philsu, and U. Jin Choi. "A quadrature rule of interpolatory type for Cauchy integrals." Journal of Computational and Applied Mathematics 126, no. 1-2 (December 2000): 207–20. http://dx.doi.org/10.1016/s0377-0427(99)00354-4.

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19

Jinyuan, Du, Xu Na, and Zhang Zhongxiang. "Boundary behavior of Cauchy-type integrals in Clifford analysis." Acta Mathematica Scientia 29, no. 1 (January 2009): 210–24. http://dx.doi.org/10.1016/s0252-9602(09)60022-9.

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20

Aliev, R. A. "$ N^\pm$-integrals and boundary values of Cauchy-type integrals of finite measures." Sbornik: Mathematics 205, no. 7 (July 31, 2014): 913–35. http://dx.doi.org/10.1070/sm2014v205n07abeh004403.

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21

Guiggiani, M., and A. Gigante. "A General Algorithm for Multidimensional Cauchy Principal Value Integrals in the Boundary Element Method." Journal of Applied Mechanics 57, no. 4 (December 1, 1990): 906–15. http://dx.doi.org/10.1115/1.2897660.

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This paper presents a new general method for the direct evaluation of Cauchy principal value integrals in several dimensions, which is an issue of major concern in any boundary element method analysis in applied mechanics. It is shown that the original Cauchy principal value integral can be transformed into an element-by-element sum of regular integrals, each one expressed in terms of intrinsic (local) coordinates. The actual computation can be performed by standard quadrature formulae and can be easily included in any existing computer code. The numerical results demonstrate the accuracy and efficiency of the method, along with its insensitivity to the mesh pattern. This new method has full generality and, therefore, can be applied in any field of applied mechanics. Moreover, there are no restrictions on the numerical implementation, as the singular integrals may be defined on surface elements or internal cells of any order and type.

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22

Sulym, Heorhiy, Iaroslav Pasternak, Mariia Smal, and Andrii Vasylyshyn. "Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities." Acta Mechanica et Automatica 13, no. 4 (December 1, 2019): 238–44. http://dx.doi.org/10.2478/ama-2019-0032.

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Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

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23

Petrosyan, A. I. "On the Derivatives of Cauchy-Type Integrals in the Polydisk." Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) 55, no. 5 (September 2020): 303–6. http://dx.doi.org/10.3103/s1068362320050040.

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24

Xie, Yonghong. "Boundary properties of hypergenic-Cauchy type integrals in Clifford analysis." Complex Variables and Elliptic Equations 59, no. 5 (January 10, 2013): 599–615. http://dx.doi.org/10.1080/17476933.2012.744403.

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25

Blaya, Ricardo Abreu, Dixan Peña Peña†, and Juan Bory Reyes‡. "Conjugate hyperharmonic functions and cauchy type integrals in douglis analysis." Complex Variables, Theory and Application: An International Journal 48, no. 12 (December 2003): 1023–39. http://dx.doi.org/10.1080/02781070310001634548.

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26

Eshkuvatov, Z. K., N. M. A. Nik Long, and M. Abdulkawi. "Numerical evaluation for Cauchy type singular integrals on the interval." Journal of Computational and Applied Mathematics 233, no. 8 (February 2010): 1995–2001. http://dx.doi.org/10.1016/j.cam.2009.09.034.

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27

Akhmedov, D. M., A. R. Hayotov, and Kh M. Shadimetov. "Optimal quadrature formulas with derivatives for Cauchy type singular integrals." Applied Mathematics and Computation 317 (January 2018): 150–59. http://dx.doi.org/10.1016/j.amc.2017.09.009.

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28

Drobek, Jaroslav. "Approximations by the Cauchy-type integrals with piecewise linear densities." Applications of Mathematics 57, no. 6 (December 2012): 627–40. http://dx.doi.org/10.1007/s10492-012-0038-3.

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29

Varlamov, V. V. "A method for the asymptotic expansion of Cauchy-type integrals." Mathematical Notes of the Academy of Sciences of the USSR 42, no. 5 (November 1987): 875–77. http://dx.doi.org/10.1007/bf01137431.

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30

Edmunds, D. E., V. Kokilashvili, and A. Meskhi. "Two-Weight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type." Canadian Journal of Mathematics 52, no. 3 (June 1, 2000): 468–502. http://dx.doi.org/10.4153/cjm-2000-022-5.

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AbstractTwo-weight inequalities of strong and weak type are obtained in the context of spaces of homogeneous type. Various applications are given, in particular to Cauchy singular integrals on regular curves.

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31

Tvrdá, Katarína, and Mária Minárová. "Computation of Definite Integral Over Repeated Integral." Tatra Mountains Mathematical Publications 72, no. 1 (December 1, 2018): 141–54. http://dx.doi.org/10.2478/tmmp-2018-0026.

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Abstract The tasks involving repeated integral occur from time to time in technical practice. This paper introduces the research of authors in the field of repeated integrals within the required class of functions. Authors focus on the definite integral over repeated integral and they develop a tool for its computation. It involves two principal steps, analytical and numerical step. In the analytical step, the definite integral over a repeated integral is decomposed into n integrals and then the Cauchy form is used for further rearrangement. Numerical step involves Gauss type integration slightly modified by the authors. Several examples illustrating the operation of both analytical and numerical steps of the method are provided in the paper.

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32

Anastassiou, George A., and Sorin G. Gal. "Global smoothness preservation by multivariate singular integrals." Bulletin of the Australian Mathematical Society 61, no. 3 (June 2000): 489–506. http://dx.doi.org/10.1017/s0004972700022516.

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By using various kinds of moduli of smoothness, it is established that the multivariate variants of the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson-type generalisations satisfy the “global smoothness preservation” property. The results are extensions of those proved by the authors for the univariate case.

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33

Kokilashvili, Vakhtang, and Stefan Samko. "Singular integrals and potentials in some Banach function spaces with variable exponent." Journal of Function Spaces and Applications 1, no. 1 (2003): 45–59. http://dx.doi.org/10.1155/2003/932158.

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We introduce a new Banach function space - a Lorentz type space with variable exponent. In this space the boundedness of singular integral and potential type operators is established, including the weighted case. The variable exponentp(t)is assumed to satisfy the logarithmic Dini condition and the exponentβof the power weightω(t)=|t|βis related only to the valuep(0). The mapping properties of Cauchy singular integrals defined on Lyapunov curves and on curves of bounded rotation are also investigated within the framework of the introduced spaces.

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34

Ioakimidis, N. I. "A new interpretation of Cauchy type singular integrals with an application to singular integral equations." Journal of Computational and Applied Mathematics 14, no. 3 (March 1986): 271–78. http://dx.doi.org/10.1016/0377-0427(86)90065-8.

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35

Khan, Suliman, Sakhi Zaman, and Siraj-ul -Islam. "Approximation of Cauchy-type singular integrals with high frequency Fourier kernel." Engineering Analysis with Boundary Elements 130 (September 2021): 209–19. http://dx.doi.org/10.1016/j.enganabound.2021.05.017.

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36

Aliev, Rashid A. "On Taylor coefficients of Cauchy-type integrals of finite complex measures." Complex Variables and Elliptic Equations 60, no. 12 (June 29, 2015): 1727–38. http://dx.doi.org/10.1080/17476933.2015.1047833.

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37

Ackermann, Nils. "A Cauchy-Schwarz type inequality for bilinear integrals on positive measures." Proceedings of the American Mathematical Society 133, no. 9 (April 15, 2005): 2647–56. http://dx.doi.org/10.1090/s0002-9939-05-08082-2.

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38

Okecha, G. E. "Evaluating singular integrals of cauchy type using a modified gaussian rule." International Journal of Computer Mathematics 19, no. 1 (January 1986): 85–92. http://dx.doi.org/10.1080/00207168608803506.

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39

Conceição, Ana C., Viktor G. Kravchenko, and José C. Pereira. "Computing some classes of Cauchy type singular integrals with Mathematica software." Advances in Computational Mathematics 39, no. 2 (September 4, 2012): 273–88. http://dx.doi.org/10.1007/s10444-012-9279-7.

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40

Shadimetov, Kh M., A. R. Hayotov, and D. M. Akhmedov. "Optimal quadrature formulas for Cauchy type singular integrals in Sobolev space." Applied Mathematics and Computation 263 (July 2015): 302–14. http://dx.doi.org/10.1016/j.amc.2015.04.066.

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41

Ioakimidis, N. J. "Hypersingular cauchy-type integrals in crack problems with hypersingular stress fields." International Journal of Fracture 42, no. 2 (February 1990): R33—R38. http://dx.doi.org/10.1007/bf00018389.

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42

Hong, Dug Hun, and Jae Duck Kim. "Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations." Advances in Fuzzy Systems 2019 (January 3, 2019): 1–10. http://dx.doi.org/10.1155/2019/5080723.

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The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.

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43

Iqbal, Sajid, Kristina Krulić, and Josip Pečarić. "Improvement of an inequality of G. H. Hardy." Tamkang Journal of Mathematics 43, no. 3 (September 30, 2012): 399–416. http://dx.doi.org/10.5556/j.tkjm.43.2012.834.

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44

SHAMAROV, N. N. "POISSON–MASLOV TYPE FORMULAS FOR SCHRÖDINGER EQUATIONS WITH MATRIX-VALUED POTENTIALS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 10, no. 04 (December 2007): 641–49. http://dx.doi.org/10.1142/s0219025707002877.

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Cauchy problems for Schrödinger equations with matrix-valued potentials are explicitly solved under following assumptions:. — equations are written in momentum form;. — the potentials are Fourier transformed matrix-valued measures with, in general, noncommuting values;. — initial Cauchy data are good enough. The solutions at time t are presented in form of integrals over some spaces of piecewise continuous mappings of the segment [0, t] to a finite-dimensional momentum space, and measures of the integration are countably additive but matrix-valued (resulting in matrices of ordinary Lebesgue integrals). Known results gave solutions either when the measure had commuting (complex) values, or when the integration over infinite dimensional spaces was quite symbolic, or when such integrating was of chronological type and hence more complicated. The method used below is based on technique of matrix-valued transition amplitudes.

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45

MARSHALL, J. S. "Solutions of a free boundary problem in a doubly connected domain via a circular-arc polygon." European Journal of Applied Mathematics 25, no. 5 (June 6, 2014): 579–94. http://dx.doi.org/10.1017/s0956792514000151.

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This paper addresses a free boundary problem for a steady, uniform patch of vorticity surrounding a single flat plate of zero thickness and finite length. Exact solutions to this problem have previously been found in terms of conformal maps represented by Cauchy-type integrals. Here, however, it is demonstrated how, by considering an associated circular-arc polygon and using ideas from automorphic function theory, these maps can be expressed in a simple non-integral form.

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46

Soldatov, Alexandre P. "Singular Integral Operators and Elliptic Boundary-Value Problems. I." Contemporary Mathematics. Fundamental Directions 63, no. 1 (December 15, 2017): 1–189. http://dx.doi.org/10.22363/2413-3639-2017-63-1-1-189.

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The book consists of three Parts I-III and Part I is presented here. In this book, we develop a new approach mainly based on the author’s papers. Many results are published here for the ﬁrst time. Chapter 1 is introductory. The necessary background from functional analysis is given there for completeness. In this book, we mostly use weighted Ho¨lder spaces, and they are considered in Ch. 2. Chapter 3 plays the main role: in weighted Ho¨lder spaces we consider there estimates of integral operators with homogeneous diﬀerence kernels, which cover potential-type integrals and singular integrals as well as Cauchy-type integrals and double layer potentials. In Ch. 4, analogous estimates are established in weighted Lebesgue spaces. Integrals with homogeneous diﬀerence kernels will play an important role in Part III of the monograph, which will be devoted to elliptic boundary-value problems. They naturally arise in integral representations of solutions of ﬁrst-order elliptic systems in terms of fundamental matrices or their parametrixes. Investigation of boundary-value problems for second-order and higher-order elliptic equations or systems is reduced to ﬁrst-order elliptic systems.

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47

Gray, John, and Andrew Vogt. "Means as Improper Integrals." Mathematics 7, no. 3 (March 20, 2019): 284. http://dx.doi.org/10.3390/math7030284.

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The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

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48

Qiao, Yuying, Yongzhi Xu, and Heju Yang. "Poincaré–Bertrand transformation formula of Cauchy-type singular integrals in Clifford analysis." Complex Variables and Elliptic Equations 57, no. 2-4 (February 2012): 197–217. http://dx.doi.org/10.1080/17476933.2011.593098.

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49

Kim, Philsu, and Beong In Yun. "On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals." Journal of Computational and Applied Mathematics 149, no. 2 (December 2002): 381–95. http://dx.doi.org/10.1016/s0377-0427(02)00481-8.

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50

Nicolò-Amati, L. Gori, and E. Santi. "On the convergence of Turán type rules for Cauchy principal value integrals." Calcolo 28, no. 1-2 (March 1991): 21–35. http://dx.doi.org/10.1007/bf02575867.

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Journal articles on the topic 'Cauchy-type integrals'

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