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Journal articles on the topic 'Multiple integrals'

Author: Grafiati

Published: 4 June 2021

Last updated: 28 July 2024

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1

Bandyrskii, B., L. Hoshko, I. Lazurchak, and M. Melnyk. "Optimal algorithms for computing multiple integrals." Mathematical Modeling and Computing 4, no. 1 (July 1, 2017): 1–9. http://dx.doi.org/10.23939/mmc2017.01.001.

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2

Greaves, G. R. H., R. R. Hall, M. N. Huxley, and J. C. Wilson. "Multiple Franel integrals." Mathematika 40, no. 1 (June 1993): 51–70. http://dx.doi.org/10.1112/s0025579300013711.

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3

Dynkin, E. B. "Multiple path integrals." Advances in Applied Mathematics 7, no. 2 (June 1986): 205–19. http://dx.doi.org/10.1016/0196-8858(86)90032-1.

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4

Dasgupta, A., and G. Kallianpur. "Multiple fractional integrals." Probability Theory and Related Fields 115, no. 4 (November 1, 1999): 505–25. http://dx.doi.org/10.1007/s004400050247.

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5

Papp, F. J. "Expressing certain multiple integrals as single integrals." International Journal of Mathematical Education in Science and Technology 21, no. 1 (March 1990): 137–39. http://dx.doi.org/10.1080/0020739900210120.

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6

Benabidallah, A., Y. Cherruault, and Y. Tourbier. "Approximation of multiple integrals by simple integrals." Kybernetes 30, no. 9/10 (December 2001): 1223–39. http://dx.doi.org/10.1108/03684920110405836.

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7

Malyutin, V. B., and B. O. Nurjanov. "The semiclassical approximation of multiple functional integrals." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 59, no. 4 (January 5, 2024): 302–7. http://dx.doi.org/10.29235/1561-2430-2023-59-4-302-307.

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In this paper, we study the semiclassical approximation of multiple functional integrals. The integrals are defined through the Lagrangian and the action. Of all possible trajectories, the greatest contribution to the integral is given by the classical trajectory x̅cl for which the action S takes an extremal value. The classical trajectory is found as a solution of the multidimensional Euler – Lagrange equation. To calculate the functional integrals, the expansion of the action with respect to the classical trajectory is used, which can be interpreted as an expansion in powers of Planck’s constant. The numerical results for the semiclassical approximation of double functional integrals are given.

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8

Greaves, G. R. H., R. R. Hall, M. N. Huxley, and J. C. Wilson. "Multiple Franel integrals: Corrigendum." Mathematika 41, no. 2 (December 1994): 401. http://dx.doi.org/10.1112/s0025579300007476.

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9

Grosjean, C. C. "Two Trigonometric Multiple Integrals." SIAM Review 33, no. 1 (March 1991): 114. http://dx.doi.org/10.1137/1033008.

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10

Podol’skii, A. A. "Identities for multiple integrals." Mathematical Notes 98, no. 3-4 (September 2015): 624–30. http://dx.doi.org/10.1134/s0001434615090291.

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11

Bardina, Xavier, and Carles Rovira. "On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals." Publicacions Matemàtiques 65 (July 1, 2021): 859–76. http://dx.doi.org/10.5565/publmat6522114.

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12

Li, Haibin, Yangtian Li, and Shangjie Li. "Dual Neural Network Method for Solving Multiple Definite Integrals." Neural Computation 31, no. 1 (January 2019): 208–32. http://dx.doi.org/10.1162/neco_a_01145.

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This study, which examines a calculation method on the basis of a dual neural network for solving multiple definite integrals, addresses the problems of inefficiency, inaccuracy, and difficulty in finding solutions. First, the method offers a dual neural network method to construct a primitive function of the integral problem; it can approximate the primitive function of any given integrand with any precision. On this basis, a neural network calculation method that can solve multiple definite integrals whose upper and lower bounds are arbitrarily given is obtained with repeated applications of the dual neural network to construction of the primitive function. Example simulations indicate that compared with traditional methods, the proposed algorithm is more efficient and precise in obtaining solutions for multiple integrals with unknown integrand, except for the finite input-output data points. The advantages of the proposed method include the following: (1) integral multiplicity shows no influence and restriction on the employment of the method; (2) only a finite set of known sample points is required without the need to know the exact analytical expression of the integrand; and (3) high calculation accuracy is obtained for multiple definite integrals whose integrand is expressed by sample data points.

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13

Benabidallah, A., Y. Cherruault, and Y. Tourbier. "Approximation method error of multiple integrals by simple integrals." Kybernetes 32, no. 3 (April 2003): 343–53. http://dx.doi.org/10.1108/03684920310458575.

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14

Bruggeman, Roelof, and Youngju Choie. "Multiple period integrals and cohomology." Algebra & Number Theory 10, no. 3 (June 12, 2016): 645–64. http://dx.doi.org/10.2140/ant.2016.10.645.

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15

Yu, Chii-Huei, and Bing-Huei Chen. "Solving Multiple Integrals Using Maple." World Journal of Computer Application and Technology 2, no. 4 (April 2014): 83–88. http://dx.doi.org/10.13189/wjcat.2014.020401.

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16

Yu, Chii-Huei, and Bing-Huei Chen. "Evaluating Multiple Integrals Using Maple." Mathematics and Statistics 2, no. 4 (April 2014): 155–61. http://dx.doi.org/10.13189/ms.2014.020401.

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17

Trif, Tiberiu. "Multiple Integrals of Symmetric Functions." American Mathematical Monthly 104, no. 7 (August 1997): 605. http://dx.doi.org/10.2307/2975053.

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18

Blankenbecler, Richard. "Multiple scattering and functional integrals." Physical Review D 55, no. 4 (February 15, 1997): 2441–48. http://dx.doi.org/10.1103/physrevd.55.2441.

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19

Tocino, A. "Multiple stochastic integrals with Mathematica." Mathematics and Computers in Simulation 79, no. 5 (January 2009): 1658–67. http://dx.doi.org/10.1016/j.matcom.2008.08.005.

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20

Zelikin, M. I. "Field theories for multiple integrals." Journal of Mathematical Sciences 177, no. 2 (August 2011): 270–98. http://dx.doi.org/10.1007/s10958-011-0457-9.

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21

Nourdin, Ivan, and Giovanni Peccati. "Noncentral convergence of multiple integrals." Annals of Probability 37, no. 4 (July 2009): 1412–26. http://dx.doi.org/10.1214/08-aop435.

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22

Trif, Tiberiu. "Multiple Integrals of Symmetric Functions." American Mathematical Monthly 104, no. 7 (August 1997): 605–8. http://dx.doi.org/10.1080/00029890.1997.11990688.

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23

Peccati, Giovanni. "Gaussian Approximations of Multiple Integrals." Electronic Communications in Probability 12 (2007): 350–64. http://dx.doi.org/10.1214/ecp.v12-1322.

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24

Mehta, M. L., and J.-M. Normand. "On some definite multiple integrals." Journal of Physics A: Mathematical and General 30, no. 24 (December 21, 1997): 8671–84. http://dx.doi.org/10.1088/0305-4470/30/24/026.

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25

Haddad, Roudy El. "Repeated integration and explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\)." Open Journal of Mathematical Sciences 6, no. 1 (June 10, 2022): 51–75. http://dx.doi.org/10.30538/oms2022.0178.

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Repeated integration is a major topic of integral calculus. In this article, we study repeated integration. In particular, we study repeated integrals and recurrent integrals. For each of these integrals, we develop reduction formulae for both the definite as well as indefinite form. These reduction formulae express these repetitive integrals in terms of single integrals. We also derive a generalization of the fundamental theorem of calculus that expresses a definite integral in terms of an indefinite integral for repeated and recurrent integrals. From the recurrent integral formulae, we derive some partition identities. Then we provide an explicit formula for the \(n\)-th integral of \(x^m(\ln x)^{m'}\) in terms of a shifted multiple harmonic star sum. Additionally, we use this integral to derive new expressions for the harmonic sum and repeated harmonic sum.

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26

Ernst, Thomas. "Further results on multiple q-Eulerian integrals for various q-hypergeometric functions." Publications de l'Institut Math?matique (Belgrade) 108, no. 122 (2020): 63–77. http://dx.doi.org/10.2298/pim2022063e.

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We continue the study of single and multiple q-Eulerian integrals in the spirit of Exton, Driver, Johnston, Pandey, Saran and Erd?lyi. The method of proof is often the q-beta integral method with the correct q-power together with the q-binomial theorem. By the Totov method we can prove summation theorems as special cases of multiple q-Eulerian integrals. The Srivastava ? notation for q-hypergeometric functions is used to enable the shortest possible form of the long formulas. The various q-Eulerian integrals are in fact meromorphic continuations of the various multiple q-functions, suitable for numerical computations. In the end of the paper a generalization of the q-binomial theorem is used to find q-analogues of a multiple integral formulas for q-Kamp? de F?riet functions.

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27

Shao, Zijia, Shuohao Wang, and Hetian Yu. "Application of the Residue Theorem to Euler Integral, Gaussian Integral, and Beyond." Highlights in Science, Engineering and Technology 38 (March 16, 2023): 311–16. http://dx.doi.org/10.54097/hset.v38i.5821.

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The paper is divided into three different parts, which use the residue theorem to solve several different integrals, namely, the Euler integral, the Gaussian integral, the Fresnel integral, and so forth. The process of using the resiude theorem to determine these integrals is to first turn the integrals into convenient forms of complex integrals, and then find integral perimeters so that any integral on one of the curves is the required integral, through the drawing observation of the contour to write the original integral into the form of multiple integral. By studying the resiude theorem to solve the problem of complex integrals, it is demonstrated that the resiude theorem is actually a process that makes the calculation easier. These solved integrals have a wide range of applications including the study of the refraction of light, analytics, probability theory, combinatorial mathematics, and unification of the continuous Fourier transform.

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28

Benabidallah, A., Y. Cherruault, and G. Mora. "Approximation of multiple integrals by simple integrals involving periodic functions." Kybernetes 33, no. 9/10 (October 2004): 1472–90. http://dx.doi.org/10.1108/03684920410534470.

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29

Bajic, Tatjana. "On relation between one multiple and a corresponding one-dimensional integral with applications." Yugoslav Journal of Operations Research 28, no. 1 (2018): 79–92. http://dx.doi.org/10.2298/yjor160916020b.

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For a given finite positive measure on an interval I ? R, a multiple stochastic integral of a Volterra kernel with respect to a product of a corresponding Gaussian orthogonal stochastic measure is introduced. The Volterra kernel is taken such that the multiple stochastic integral is a multiple iterated stochastic integral related to a parameterized Hermite polynomial, where parameter depends on Gaussian distribution of an underlying one-dimensional stochastic integral. Considering that there exists a connection between stochastic and deterministic integrals, we expose some properties of parameterized Hermite polynomials of Gaussian random variable in order to prove that one multiple integral can be expressed by a corresponding one-dimensional integral. Having in mind the obtained result, we show that a system of multiple integrals, as well as a collection of conditional expectations can be calculated exactly by generalized Gaussian quadrature rule.

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30

Gupta, A. K., and D. G. Kabe. "On a zonal polynomial integral." Journal of Applied Mathematics 2003, no. 11 (2003): 569–73. http://dx.doi.org/10.1155/s1110757x03209074.

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A certain multiple integral occurring in the studies of Beherens-Fisher multivariate problem has been evaluated by Mathai et al. (1995) in terms of invariant polynomials. However, this paper explicitly evaluates the context integral in terms of zonal polynomials, thus establishing a relationship between zonal polynomial integrals and invariant polynomial integrals.

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31

Pranevich, Andrei, Alexander Grin, and Yanka Musafirov. "Multiple partial integrals of polynomial Hamiltonian systems." Acta et commentationes: Ştiinţe Exacte şi ale Naturii 12, no. 2 (February 2022): 33–42. http://dx.doi.org/10.36120/2587-3644.v12i2.33-42.

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We consider an autonomous real polynomial Hamiltonian ordinary differential system. Sufficient conditions for the construction of additional first integrals on polynomial partial integrals and multiple polynomial partial integrals are obtained. Classes of autonomous polynomial Hamiltonian ordinary differential systems with first integrals which analytically expressed by multiple polynomial partial integrals are identified. Also we present examples that illustrate the theoretical results.

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32

Jang, Lee-Chae. "On Multiple Generalizedw-Genocchi Polynomials and Their Applications." Mathematical Problems in Engineering 2010 (2010): 1–8. http://dx.doi.org/10.1155/2010/316870.

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We define the multiple generalizedw-Genocchi polynomials. By using fermionicp-adic invariant integrals, we derive some identities on these generalizedw-Genocchi polynomials, for example, fermionicp-adic integral representation, Witt's type formula, explicit formula, multiplication formula, and recurrence formula for thesew-Genocchi polynomials.

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33

Rybakov, Konstantin. "Features of the Expansion of Multiple Stochastic Stratonovich Integrals Using Walsh and Haar Functions." Differential Equations and Control Processes, no. 1 (2023): 137–50. http://dx.doi.org/10.21638/11701/spbu35.2023.109.

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The problem of the root-mean-square convergence for approximations of multiple stochastic Stratonovich integrals based on the generalized multiple Fourier series method using Walsh and Haar functions is considered. It is shown that when they are chosen to expand multiple stochastic integrals, the proof of the root-mean-square convergence of a subsequence of series partial sums, which is formed in a way that is quite natural for these functions, does not require the explicit fulfillment of any additional conditions, except for the condition of the existence of the multiple stochastic Stratonovich integral.

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34

Ciesielski, Mariusz, and Tomasz Blaszczyk. "The multiple composition of the left and right fractional Riemann-Liouville integrals - analytical and numerical calculations." Filomat 31, no. 19 (2017): 6087–99. http://dx.doi.org/10.2298/fil1719087c.

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New fractional integral operators of order ? ? R+ are introduced. These operators are defined as the composition of the left and right (or the right and left) Riemann-Liouville fractional order integrals. Some of their properties are studied. Analytical results of fractional integrals of several functions are presented. For a numerical calculation of fractional order integrals, two numerical procedures are given. In the final part of this paper, examples of numerical evaluations of these operators of three different functions are shown in plots and the comparison of the numerical accuracy was analyzed in tables.

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35

Lax, Peter D. "Change of Variables in Multiple Integrals." American Mathematical Monthly 106, no. 6 (June 1999): 497. http://dx.doi.org/10.2307/2589462.

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36

Kałamajska, Agnieszka. "On lower semicontinuity of multiple integrals." Colloquium Mathematicum 74, no. 1 (1997): 71–78. http://dx.doi.org/10.4064/cm-74-1-71-78.

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37

Kolomoitsev, Yurii, and Elijah Liflyand. "Absolute convergence of multiple Fourier integrals." Studia Mathematica 214, no. 1 (2013): 17–35. http://dx.doi.org/10.4064/sm214-1-2.

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38

Tudor, Constantin, and Maria Tudor. "Power variation of multiple fractional integrals." Central European Journal of Mathematics 5, no. 2 (January 26, 2007): 358–72. http://dx.doi.org/10.2478/s11533-007-0001-9.

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39

Gunawardena, K. L. D. "On the Evaluation of Multiple Integrals." Missouri Journal of Mathematical Sciences 6, no. 1 (February 1994): 29–33. http://dx.doi.org/10.35834/1994/0601029.

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40

Seel, Alexander, Frank Göhmann, and Andreas Klümper. "From Multiple Integrals to Fredholm Determinants." Progress of Theoretical Physics Supplement 176 (2008): 375–83. http://dx.doi.org/10.1143/ptps.176.375.

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41

Loiseau, Jean-Francis, Jean-Pierre Codaccioni, and R{égis Caboz. "Hyperelliptic integrals and multiple hypergeometric series." Mathematics of Computation 50, no. 182 (May 1, 1988): 501. http://dx.doi.org/10.1090/s0025-5718-1988-0929548-0.

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42

Koyama, Shin-ya, and Nobushige Kurokawa. "Euler's integrals and multiple sine functions." Proceedings of the American Mathematical Society 133, no. 5 (May 1, 2005): 1257–65. http://dx.doi.org/10.1090/s0002-9939-04-07863-3.

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43

Tudor, Constantin, and Maria Tudor. "Approximation of Multiple Stratonovich Fractional Integrals." Stochastic Analysis and Applications 25, no. 4 (June 26, 2007): 781–99. http://dx.doi.org/10.1080/07362990701419979.

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44

Kloeden, P. E., E. Platen, and I. W. Wright. "The approximation of multiple stochastic integrals." Stochastic Analysis and Applications 10, no. 4 (January 1992): 431–41. http://dx.doi.org/10.1080/07362999208809281.

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45

Miroshin, R. N. "On multiple integrals of special form." Mathematical Notes 82, no. 3-4 (October 2007): 357–65. http://dx.doi.org/10.1134/s000143460709009x.

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46

Abel, Ulrich, and Vitaliy Kushnirevych. "Reducing Multiple Integrals of Beukers’s Type." American Mathematical Monthly 127, no. 10 (November 25, 2020): 918–26. http://dx.doi.org/10.1080/00029890.2020.1815477.

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47

Jones, Edwin R., and Richard L. Childers. "Helping students to recognize multiple integrals." American Journal of Physics 58, no. 10 (October 1990): 904–5. http://dx.doi.org/10.1119/1.16294.

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48

Peller, V. V. "Multiple operator integrals in perturbation theory." Bulletin of Mathematical Sciences 6, no. 1 (October 24, 2015): 15–88. http://dx.doi.org/10.1007/s13373-015-0073-y.

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49

Carbonell, F., J. C. Jímenez, and L. M. Pedroso. "Computing multiple integrals involving matrix exponentials." Journal of Computational and Applied Mathematics 213, no. 1 (March 2008): 300–305. http://dx.doi.org/10.1016/j.cam.2007.01.007.

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50

Fox, Robert, and Murad S. Taqqu. "Multiple stochastic integrals with dependent integrators." Journal of Multivariate Analysis 21, no. 1 (February 1987): 105–27. http://dx.doi.org/10.1016/0047-259x(87)90101-1.

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