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Journal articles on the topic 'Multiples integrals'

Author: Grafiati

Published: 2 November 2024

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1

Song, Jiang Yong. "An Elliptic Integral Solution to the Multiple Inflections Large Deflection Beams in Compliant Mechanisms." Advanced Materials Research 971-973 (June 2014): 349–52. http://dx.doi.org/10.4028/www.scientific.net/amr.971-973.349.

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In this paper, a solution based on the elliptic integrals is proposed for solving multiples inflection points large deflection. Application of the Bernoulli Euler equations of compliant mechanisms with large deflection equation of beam is obtained ,there is no inflection point and inflection points in two cases respectively. The elliptic integral solution which is the most accurate method at present for analyzing large deflections of cantilever beams in compliant mechanisms.

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2

van der Neut, Joost, and Kees Wapenaar. "Adaptive overburden elimination with the multidimensional Marchenko equation." GEOPHYSICS 81, no. 5 (September 2016): T265—T284. http://dx.doi.org/10.1190/geo2016-0024.1.

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Iterative substitution of the multidimensional Marchenko equation has been introduced recently to integrate internal multiple reflections in the seismic imaging process. In so-called Marchenko imaging, a macro velocity model of the subsurface is required to meet this objective. The model is used to back-propagate the data during the first iteration and to truncate integrals in time during all successive iterations. In case of an erroneous model, the image will be blurred (akin to conventional imaging) and artifacts may arise from inaccurate integral truncations. However, the scheme is still successful in removing artifacts from internal multiple reflections. Inspired by these observations, we rewrote the Marchenko equation, such that it can be applied early in a processing flow, without the need of a macro velocity model. Instead, we have required an estimate of the two-way traveltime surface of a selected horizon in the subsurface. We have introduced an approximation, such that adaptive subtraction can be applied. As a solution, we obtained a new data set, in which all interactions (primaries and multiples) with the part of the medium above the picked horizon had been eliminated. Unlike various other internal multiple elimination algorithms, the method can be applied at any specified target horizon, without having to resolve for internal multiples from shallower horizons. We successfully applied the method on synthetic data, where limitations were reported due to thin layers, diffraction-like discontinuities, and a finite acquisition aperture. A field data test was also performed, in which the kinematics of the predicted updates were demonstrated to match with internal multiples in the recorded data, but it appeared difficult to subtract them.

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3

Saouter, Yannick. "New pancake series for π." Mathematical Gazette 104, no. 560 (June 18, 2020): 296–303. http://dx.doi.org/10.1017/mag.2020.53.

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In [1], Dalzell proved that $\pi = \frac{{22}}{7} - \int_0^1 {\frac{{{t^4}{{(1 - t)}^4}}}{{1 + {t^2}}}}$ . He then used this equation to derive a new series converging to π. In [2], Backhouse studied the general case of integrals of the form $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}dt}$ and derived conditions on m and n so that they could be used to evaluate π. As a sequel, he derived accurate rational approximations of π. This work was extended in [3] where new rational approximations of π are obtained. Some related integrals of the forms $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}P(t)\,dt}$ and $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{\sqrt {1 - {t^2}} }}P(t)dt}$ with P(t) being of polynomial form are also investigated. In [4] the author gives more new approximations and new series for the case m = n = 4k. In [5] new series for π are obtained with the integral $\int_0^a {\frac{{{t^{12m}}{{(a - t)}^{12m}}}}{{1 + {t^2}}}dt}$ where $a = 2 - \sqrt 3$ . The general problem of improving the convergence speed of the arctan series by transformation of the argument has also been considered in [6, 7]. In the present work the author considers an alternative form for the denominators in integrals. As a result, new series are obtained for multiples of π by some algebraic numbers.

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4

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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5

Fleury, Clement, and Ivan Vasconcelos. "Imaging condition for nonlinear scattering-based imaging: Estimate of power loss in scattering." GEOPHYSICS 77, no. 1 (January 2012): S1—S18. http://dx.doi.org/10.1190/geo2011-0135.1.

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Imaging highly complex subsurface structures is a challenging problem because it must ultimately deal with nonlinear multiple-scattering effects (e.g., migration of multiples, wavefield extrapolation with nonlinear model interactions, amplitude-preserving migration) to overcome the limitations of linear imaging. Most of the current migration techniques rely on the linear single-scattering assumption, and therefore, fail to handle these complex scattering effects. Recently, seismic imaging has been related to scattering-based image-domain interferometry to address the fully nonlinear imaging problem. Building on this connection between imaging and interferometry, we define the seismic image as a locally scattered wavefield and introduce a new imaging condition that is suitable and practical for nonlinear imaging. A previous formulation of nonlinear scattering-based imaging requires the evaluation of volume integrals that cannot easily be incorporated in current imaging algorithms. Our method consisted of adapting the conventional crosscorrelation imaging condition to account for the interference mechanisms that ensure power conservation in the scattering of wavefields. To do so, we added the zero-lag autocorrelation of scattered wavefields to the zero-lag crosscorrelation of reference and scattered wavefields. In our development, we demonstrated that this autocorrelation of scattered fields fully replaces the volume scattering term required by the previous formulation. We also found that this replacement follows from the application of the generalized optical theorem. The resulting imaging condition accounts for nonlinear multiple-scattering effects, reduces imaging artifacts and improves amplitude preservation and illumination in the images. We addressed the principles of our nonlinear imaging condition and demonstrated its importance in ideal nonlinear imaging experiments, i.e., we presented synthetic data examples assuming ideal scattered wavefield extrapolation and studied the influence of different scattering regimes and aperture limitation.

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6

Král, Josef. "Note on generalized multiple Perron integral." Časopis pro pěstování matematiky 110, no. 4 (1985): 371–74. http://dx.doi.org/10.21136/cpm.1985.118252.

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7

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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8

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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9

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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10

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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11

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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12

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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13

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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14

Lee, Tuo-Yeong. "Multipliers for generalized Riemann integrals in the real line." Mathematica Bohemica 131, no. 2 (2006): 161–66. http://dx.doi.org/10.21136/mb.2006.134090.

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15

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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16

Tassaddiq, Asifa, Rekha Srivastava, Rabab Alharbi, Ruhaila Md Kasmani, and Sania Qureshi. "New Inequalities Using Multiple Erdélyi–Kober Fractional Integral Operators." Fractal and Fractional 8, no. 4 (March 22, 2024): 180. http://dx.doi.org/10.3390/fractalfract8040180.

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The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them.

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17

LAPORTA, S. "ANALYTICAL EXPRESSIONS OF THREE- AND FOUR-LOOP SUNRISE FEYNMAN INTEGRALS AND FOUR-DIMENSIONAL LATTICE INTEGRALS." International Journal of Modern Physics A 23, no. 31 (December 20, 2008): 5007–20. http://dx.doi.org/10.1142/s0217751x08042869.

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In this paper we continue the work began in 2002 on the identification of the analytical expressions of Feynman integrals which require the evaluation of multiple elliptic integrals. We rewrite and simplify the analytical expression of the three-loop self-mass integral with three equal masses and on-shell external momentum. We collect and analyze a number of results on double and triple elliptic integrals. By using very high-precision numerical fits, for the first time we are able to identify a very compact analytical expression for the four-loop on-shell self-mass integral with four equal masses, that is one of the master integrals of the four-loop electron g-2. Moreover, we fit the analytical expressions of some integrals which appear in lattice perturbation theory, and in particular the four-dimensional generalized Watson integral.

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18

Garg, O. P., and Virendra Kumar. "Certain multiple integral relations involving multivariable $ H $-function and general polynomials." Tamkang Journal of Mathematics 32, no. 4 (December 31, 2001): 259–69. http://dx.doi.org/10.5556/j.tkjm.32.2001.340.

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In this paper, first we obtain a finite integral involving multivariable $ H $-function and general classes of polynomials. Next, with the application of this and a lemma due to Srivastava et al. (1981) we obtain two general multiple integral relations involving the multivariable $ H $-function, general classes of polynomials and two arbitrary function $ f $ and $ g $. Again, by suitably specializing the functions $ f $ and $ g $ occurring in the main integral relations, we have also evaluated multiple integrals which are new and quite general in nature.

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19

Mukha, V. S., and N. F. Kako. "Integrals and integral transformations related to the vector Gaussian distribution." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 4 (January 7, 2020): 457–66. http://dx.doi.org/10.29235/1561-2430-2019-55-4-457-466.

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This paper is dedicated to the integrals and integral transformations related to the probability density function of the vector Gaussian distribution and arising in probability applications. Herein, we present three integrals that permit to calculate the moments of the multivariate Gaussian distribution. Moreover, the total probability formula and Bayes formula for the vector Gaussian distribution are given. The obtained results are proven. The deduction of the integrals is performed on the basis of the Gauss elimination method. The total probability formula and Bayes formula are obtained on the basis of the proven integrals. These integrals and integral transformations could be used, for example, in the statistical decision theory, particularly, in the dual control theory, and as table integrals in various areas of research. On the basis of the obtained results, Bayesian estimations of the coefficients of the multiple regression function are calculated.

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20

Rahal, Mohamed, and Djaouida Guettal. "Efficient α-Dense Curve Strategies for Multiple Integrals over Hyper-rectangle Regions." Annals of West University of Timisoara - Mathematics and Computer Science 60, no. 1 (January 1, 2024): 85–97. http://dx.doi.org/10.2478/awutm-2024-0006.

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Abstract In this paper, we propose an approximation technique to compute multiple integrals of a non-negative real continuous function over a hyper-rectangle Ω of ℝ n . The main idea is to use a reducing transformation procedure obtained by using α-dense curves. First, the region Ω f whose measure represents the value of the integral, is densified by a specific curve ℓ α(t) of finite length. Therefore, the multiple integral can be approached by a simple integral corresponding to ℓ α (t). Some numerical examples are given.

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21

Mao, Zhong-Xuan, Ya-Ru Zhu, Jun-Ping Hou, Chun-Ping Ma, and Shi-Pu Liu. "Multiple Diamond-Alpha Integral in General Form and Their Properties, Applications." Mathematics 9, no. 10 (May 15, 2021): 1123. http://dx.doi.org/10.3390/math9101123.

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In this paper, we introduce the concept of n-dimensional Diamond-Alpha integral on time scales. In particular, it transforms into multiple Delta, Nabla and mixed integrals by taking different values of alpha. Some of its properties are explored, and the relationship between it and the multiple mixed integral is provided. As an application, we establish some weighted Ostrowski type inequalities through the new integral. These new inequalities expand some known inequalities in the monographs and papers, and in addition, furnish some other interesting inequalities. Examples of Ostrowski type inequalities are posed in detail at the end of the paper.

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22

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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23

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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24

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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25

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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26

Arnautovski-Toseva, Vesna, and Leonid Grcev. "Electromagnetic analysis of single/multiple grounding rods." Facta universitatis - series: Electronics and Energetics 31, no. 3 (2018): 487–500. http://dx.doi.org/10.2298/fuee1803487a.

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This paper presents electromagnetic modeling of multiple driven grounding rods in homogeneous/two-layer soil. The mathematical model is formulated by mixed potential integral equation (MPIE) on the basis of Sommerfield integrals. Several configurations of multiple driven rods located in homogeneous or two-layer soil are analyzed. The authors are focused on the calculation of the current density along the rods in wide frequency range from 100Hz to 1MHz.

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27

Móricz, Ferenc. "On the regular convergence of multiple integrals of locally Lebesgue integrable functions over." Comptes Rendus Mathematique 350, no. 9-10 (May 2012): 459–64. http://dx.doi.org/10.1016/j.crma.2012.03.010.

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28

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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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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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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31

Zhang, Ch, and J. D. Achenbach. "Scattering by Multiple Crack Configurations." Journal of Applied Mechanics 55, no. 1 (March 1, 1988): 104–10. http://dx.doi.org/10.1115/1.3173614.

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A system of boundary integral equations is presented which governs the crack-opening displacements for two-crack configurations. The integral equations are highly singular and they cannot be solved directly by numerical methods. By the approach of this paper the higher order singularities are, however, reduced to integrable singularities, and the integral equations are subsequently discretized and solved numerically. For several configurations numerical results have been obtained for scattered fields and for elastodynamic stress intensity factors. The scattered-field results are interpreted to apply for a partially closed crack as well as for two separate but neighboring cracks. The stress-intensity factors are intended to apply only to the case of separate cracks. The scattered-field results have relevance to the problem of detection and characterization of cracks in the field of quantitative nondestructive evaluation.

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Mukha, V. S., and N. F. Kako. "The integrals and integral transformations connected with the joint vector Gaussian distribution." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 57, no. 2 (July 16, 2021): 206–16. http://dx.doi.org/10.29235/1561-2430-2021-57-2-206-216.

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In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.

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Mathai, A. M. "Evaluation of matrix-variate gamma and beta integrals as multiple integrals and Koberfractional integral operators in the complex matrix variate case." Applied Mathematics and Computation 247 (November 2014): 312–18. http://dx.doi.org/10.1016/j.amc.2014.08.097.

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34

LIU, WEI, GUOJU YE, and DAFANG ZHAO. "Multiple existence of solutions for a coupled system involving the distributional Henstock-Kurzweil integral." Carpathian Journal of Mathematics 34, no. 1 (2018): 77–84. http://dx.doi.org/10.37193/cjm.2018.01.08.

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This paper deals with a coupled system in the sense of distributions (generalized functions). Our main goal is to get the basic multiple existence results via some degree theory arguments. Differently from the literatures, the proof is based on the concept of a general integral named distributional Henstock-Kurzweil integral, which includes the Lebesgue and Henstock-Kurzweil integrals as special cases. Finally, an example is given to illustrate that the presented abstract theory contains some previous results as special cases.

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35

Treanţă, Savin, and Omar Mutab Alsalami. "Sufficient Efficiency Criteria for New Classes of Non-Convex Optimization Models." Axioms 13, no. 9 (August 23, 2024): 572. http://dx.doi.org/10.3390/axioms13090572.

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In this paper, we introduce and study a new class of minimization models driven by multiple integrals as cost functionals. Concretely, we formulate and establish some sufficient efficiency criteria for a feasible point in the considered optimization problem. To this end, we introduce and define the concepts of (Γ,ψ)-invexity and generalized (Γ,ψ)-invexity for the involved real-valued controlled multiple integral-type functionals. More precisely, we extend the notion of (generalized) (Γ,ψ)-invexity to the multiple objective control models driven by multiple integral functionals. In addition, innovative proofs are considered for the principal results derived in the paper.

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36

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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37

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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38

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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39

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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40

Sun, Yan, Hai-Tao Yang, and Feng Qi. "Some Inequalities for Multiple Integrals on then-Dimensional Ellipsoid, Spherical Shell, and Ball." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/904721.

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The authors establish some new inequalities of Pólya type for multiple integrals on then-dimensional ellipsoid, spherical shell, and ball, in terms of bounds of the higher order derivatives of the integrands. These results generalize the main result in the paper by Feng Qi, Inequalities for a multiple integral,Acta Mathematica Hungarica(1999).

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Yan, Kai, Christoph Lassner, Brian Budge, Zhao Dong, and Shuang Zhao. "Efficient estimation of boundary integrals for path-space differentiable rendering." ACM Transactions on Graphics 41, no. 4 (July 2022): 1–13. http://dx.doi.org/10.1145/3528223.3530080.

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Boundary integrals are unique to physics-based differentiable rendering and crucial for differentiating with respect to object geometry. Under the differential path integral framework---which has enabled the development of sophisticated differentiable rendering algorithms---the boundary components are themselves path integrals. Previously, although the mathematical formulation of boundary path integrals have been established, efficient estimation of these integrals remains challenging. In this paper, we introduce a new technique to efficiently estimate boundary path integrals. A key component of our technique is a primary-sample-space guiding step for importance sampling of boundary segments. Additionally, we show multiple importance sampling can be used to combine multiple guided samplings. Lastly, we introduce an optional edge sorting step to further improve the runtime performance. We evaluate the effectiveness of our method using several differentiable-rendering and inverse-rendering examples and provide comparisons with existing methods for reconstruction as well as gradient quality.

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Tan, Ziqian. "Cauchy Residue Theorem’s Application in Improper integrals." Highlights in Science, Engineering and Technology 49 (May 21, 2023): 331–35. http://dx.doi.org/10.54097/hset.v49i.8528.

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Definite integrals are an essential tool for understanding and calculating many aspects of the natural world. An improper integral, one type of definite integral, has either an infinite interval or an integrand that is not defined at one or more points within the interval of integration. In this research, improper integrals are the main concepts that will be discussed. The function can be expressed in terms of its Laurent series expansion–a general form or representation for analytic functions that includes both negative and positive power series of (z – singularity)–about each of its isolated singularities within the contour. And then, finding the singularities that are inside the contour based on the contour function. Calculating the residue of the singularities. Then, substituting the residue with the calculated value, adding all the residue together. Lastly, the result multiplies by, which is the result, namely the integral of the original functions. With the help of the other two methods, Keyhole Contour, and principal values, it is possible to evaluate the integrals and determine the domain of the function. Keyhole Contour separates the function’s domain and evaluates real integrals. And principal values can be used for determining the specific range of the function, so a single value can be chosen.

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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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Móricz, Ferenc. "On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over ℝ̄ + m." Analysis Mathematica 39, no. 2 (June 2013): 135–52. http://dx.doi.org/10.1007/s10476-013-0204-1.

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oʻgʻli, Maxmudov Baxodirjon Baxromjon. "PEDAGOGICAL FOUNDATIONS OF TEACHING MULTIPLE INTEGRALS IN THE FORMATION OF SPATIAL IMAGINATION AND MATHEMATICAL THINKING." International Journal of Pedagogics 4, no. 8 (August 1, 2024): 106–10. http://dx.doi.org/10.37547/ijp/volume04issue08-21.

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The teaching of multiple integrals plays a crucial role in the development of spatial imagination and mathematical thinking in students. This article explores the pedagogical foundations necessary for effective instruction in multiple integrals, focusing on strategies that enhance students' ability to visualize and manipulate complex spatial relationships. By integrating modern teaching methodologies with traditional mathematical instruction, educators can significantly improve students' conceptual understanding and problem-solving abilities in this area.

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Albalawi, Wedad. "Time-Scale Integral Inequalities of Copson with Steklov Operator in High Dimension." Journal of Mathematics 2022 (October 12, 2022): 1–11. http://dx.doi.org/10.1155/2022/2771854.

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The paper derives some new time-scale (TS) dynamic inequalities for multiple integrals. The obtained inequalities are special cases of Copson integral using Steklov operator in (TS) version with high dimension. We prove the inequalities with several formulas for the operator and in different cases m > μ + 1 and m < μ + 1 for every μ ≥ 1 , using time-scales (TSs) setting for integral properties, chain rules, Fubini’s theorem, and Hölder’s inequality.

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Ball, J. M., and K. W. Zhang. "Lower semicontinuity of multiple integrals and the Biting Lemma." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 114, no. 3-4 (1990): 367–79. http://dx.doi.org/10.1017/s0308210500024483.

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SynopsisWeak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

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Kokilashvili, Vakhtang, and Alexander Meskhi. "Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces." Fractional Calculus and Applied Analysis 23, no. 5 (October 1, 2020): 1452–71. http://dx.doi.org/10.1515/fca-2020-0072.

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Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

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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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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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