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Journal articles on the topic 'Summation method'

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Published: 10 December 2022
Last updated: 28 January 2023

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1

Terry, Elaine A. "Gauss's Method for Summation." Mathematics Teacher 109, no. 6 (February 2016): 480. http://dx.doi.org/10.5951/mathteacher.109.6.0480.

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2

Babich, V. M., and M. M. Popov. "Gaussian summation method (review)." Radiophysics and Quantum Electronics 32, no. 12 (December 1989): 1063–81. http://dx.doi.org/10.1007/bf01038632.

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3

Mu, Yan-Ping, and Zhi-Wei Sun. "Telescoping method and congruences for double sums." International Journal of Number Theory 14, no. 01 (November 21, 2017): 143–65. http://dx.doi.org/10.1142/s1793042118500100.

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In recent years, Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double summations of hypergeometric terms. Using the telescoping method and certain mathematical software packages, we transform such a double summation into a single sum. With this new approach, we confirm several open conjectures of Sun.
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4

Euler, Russell, and Jawad Sadek. "93.51 A ‘Sterling’ summation method." Mathematical Gazette 93, no. 528 (November 2009): 504–8. http://dx.doi.org/10.1017/s0025557200185304.

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5

Salles, Rafael S., Maise N. S. Silva, and Paulo F. Ribeiro. "Observations on Harmonics Summation in Transmission Systems: Alternative Aggregation Estimation." International Transactions on Electrical Energy Systems 2022 (November 3, 2022): 1–13. http://dx.doi.org/10.1155/2022/5313417.

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The aggregation of harmonic components from different sources is one of the critical and challenging assessments in electric power systems. Harmonic summation analysis and estimation is not a simple task since there will be variations because of the grid complexity, nonlinear sources, and unpredictable behaviour of harmonic currents that affect the results. An evaluation of harmonic summation using alternative methods to calculate the harmonic composition at any network point is suggested. A typical arrangement of transmission grids was modelled and used to simulate the results. This paper aims to highlight the results obtained by these alternative methods of harmonic summation and show the role of this type of analysis in transmission systems planning. The contributions are (a) illustrate how alternative methods of harmonic summation can be applied to investigate harmonic aggregation from different sources; (b) provide a case study that also discusses the harmonic aggregation effects with different locations of sources and component phase angle shifting; (c) show comparison and correlation between those alternative summations calculations with a standardized and firmly adopted method (proposed by IEC 61000-3-6). The software MATLAB/Simulink performs simulation and analysis. Finally, the work discusses the findings.
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6

Fujikawa, T., R. Yanagisawa, and N. Yiwata. "Partial Summation Method for XANES Calculation." Le Journal de Physique IV 7, no. C2 (April 1997): C2–99—C2–102. http://dx.doi.org/10.1051/jp4/1997044.

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7

Jenkins, Lynne D., and A. M. Cohen. "A method for double series summation." International Journal of Computer Mathematics 29, no. 2-4 (January 1989): 195–200. http://dx.doi.org/10.1080/00207168908803759.

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8

Beex, L. A. A., R. H. J. Peerlings, and M. G. D. Geers. "Central summation in the quasicontinuum method." Journal of the Mechanics and Physics of Solids 70 (October 2014): 242–61. http://dx.doi.org/10.1016/j.jmps.2014.05.019.

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9

Belokurov, V. V., Yu P. Solov'ev, and E. T. Shavgulidze. "A summation method for divergent series." Russian Mathematical Surveys 54, no. 3 (June 30, 1999): 626–27. http://dx.doi.org/10.1070/rm1999v054n03abeh000156.

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10

Kosmus, W. "Summation Method for Monitoring Nitrogen Oxides." International Journal of Environmental Analytical Chemistry 22, no. 3-4 (October 1985): 269–79. http://dx.doi.org/10.1080/03067318508076426.

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11

Duan, Zhong-Hui, and Robert Krasny. "An Ewald summation based multipole method." Journal of Chemical Physics 113, no. 9 (September 2000): 3492–95. http://dx.doi.org/10.1063/1.1289918.

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12

Ivasyk, Halyna. "The system of powers of conformal mappings and biorthogonal to them systems of the functions." Physico-mathematical modelling and informational technologies, no. 26 (December 30, 2017): 31–44. http://dx.doi.org/10.15407/fmmit2017.26.031.

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In this article we review the methods of power summation of factors. The degree of factors which are arbitrary powers of summation indices are considered. We show that using the Poisson-Abel method only those series can be summarized the order of member increase of which is proportional to the exponent depending on the summation index. At the same time the Gauss-Weierstrass method and other power factors methods can be also applied to the series the terms of which increase in proportion to the exponential dependence of the indices summation.
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13

Wu, Shaojiang, Yibo Wang, Fei Xie, and Xu Chang. "Crosscorrelation migration of microseismic source locations with hybrid imaging condition." GEOPHYSICS 87, no. 1 (December 1, 2021): KS17—KS31. http://dx.doi.org/10.1190/geo2020-0896.1.

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Locating microseismic sources is critical to monitoring the hydraulic fractures that occur during fluid extraction/injection in unconventional oil/gas exploration, geothermal operations, and CO2 sequestration. Waveform-based seismic location methods can reliably and automatically image weak microseismic source locations without phase picking. Among them, the crosscorrelation migration (CCM) method can avoid excitation time scanning by generating virtual gathers. We have adopted a CCM location method based on the hybrid imaging condition (HIC). There are four main steps in the implementation of this method: (1) selection of receivers with good azimuthal coverage, (2) generation of virtual gathers by correlating the reference receiver with the rest of the receivers, (3) summation of back projections in the virtual gathers, and (4) multiplication of all summations. The CCM-HIC method is tested on synthetic and field data sets, and the results are compared with those obtained by the conventional summation imaging condition and multiplication imaging condition. The comparison results determine that CCM-HIC is sufficiently robust to obtain a source image with better stability and higher spatial resolution, despite the presence of strong noise.
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14

CHU, WENCHANG, and CANGZHI JIA. "ABEL'S METHOD ON SUMMATION BY PARTS FOR ELLIPTIC HYPERGEOMETRIC SERIES." Communications in Contemporary Mathematics 11, no. 03 (June 2009): 337–53. http://dx.doi.org/10.1142/s0219199709003387.

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By means of Abel's lemma on summation by parts, a new approach for evaluating elliptic hypergeometric series is presented. Some well-known terminating series identities are reviewed and several new transformation and summation formulae are established.
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15

Hou, Qing-Hu, and Yarong Wei. "Telescoping method, summation formulas, and inversion pairs." Electronic Research Archive 29, no. 4 (2021): 2657. http://dx.doi.org/10.3934/era.2021007.

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16

Hardy, David J., Zhe Wu, James C. Phillips, John E. Stone, Robert D. Skeel, and Klaus Schulten. "Multilevel Summation Method for Electrostatic Force Evaluation." Journal of Chemical Theory and Computation 11, no. 2 (January 8, 2015): 766–79. http://dx.doi.org/10.1021/ct5009075.

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17

Gordunovsky, Victor. "A Summation Constraint Method for Linear Programming." Procedia Computer Science 55 (2015): 246–50. http://dx.doi.org/10.1016/j.procs.2015.07.039.

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18

Oyamatsu, Kazuhiro. "Reactor Kinetics Calculated in the Summation Method." Journal of Nuclear Science and Technology 39, sup2 (August 2002): 1109–11. http://dx.doi.org/10.1080/00223131.2002.10875296.

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19

Wu, Zhe, David J. Hardy, James C. Phillips, John E. Stone, Robert D. Skeel, and Klaus Schulten. "Multilevel Summation Method for Electrostatic Force Evaluation." Biophysical Journal 108, no. 2 (January 2015): 183a. http://dx.doi.org/10.1016/j.bpj.2014.11.1013.

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20

Burlyai, M. F. "Logarithmic method for the summation of integrals." Ukrainian Mathematical Journal 41, no. 11 (November 1989): 1343–46. http://dx.doi.org/10.1007/bf01056506.

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21

Song, Jiming, and Sidharath Jain. "Midpoint Summation: A Method for Accurate and Efficient Summation of Series Appearing in Electromagnetics." IEEE Antennas and Wireless Propagation Letters 9 (2010): 1084–87. http://dx.doi.org/10.1109/lawp.2010.2091488.

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22

Leung, Shingyu, Jianliang Qian, and Robert Burridge. "Eulerian Gaussian beams for high-frequency wave propagation." GEOPHYSICS 72, no. 5 (September 2007): SM61—SM76. http://dx.doi.org/10.1190/1.2752136.

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We design an Eulerian Gaussian beam summation method for solving Helmholtz equations in the high-frequency regime. The traditional Gaussian beam summation method is based on Lagrangian ray tracing and local ray-centered coordinates. We propose a new Eulerian formulation of Gaussian beam theory which adopts global Cartesian coordinates, level sets, and Liouville equations, yielding uniformly distributed Eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources. The time harmonic wavefield can be constructed by summing up Gaussian beams with ingredients provided by the new Eulerian formulation. The conventional Gaussian beam summation method can be derived from the proposed method. There are three advantages of the new method: (1) We have uniform resolution of ray distribution. (2) We can obtain wavefields excited at different sources by varying only source locations in the summation formula. (3) We can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula. Numerical experiments indicate that the Gaussian beam summation method yields accurate asymptotic wavefields even at caustics. The new method may be used for seismic modeling and migration.
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23

Chu, Wenchang. "Abel’s method on summation by parts and balanced q-series identities." Applicable Analysis and Discrete Mathematics 4, no. 1 (2010): 54–65. http://dx.doi.org/10.2298/aadm1000006c.

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The Abel method on summation by parts is reformulated to present new and elementary proofs of several classical identities of terminating balanced basic hypergeometric series. The examples strengthen our conviction that as traditional analytical instrument, the revised Abel method on summation by parts is indeed a very natural choice for working with basic hypergeometric series.
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24

Chu, Chunlei, and Paul L. Stoffa. "Efficient 3D frequency response modeling with spectral accuracy by the rapid expansion method." GEOPHYSICS 77, no. 4 (July 1, 2012): T117—T123. http://dx.doi.org/10.1190/geo2011-0415.1.

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Frequency responses of seismic wave propagation can be obtained either by directly solving the frequency domain wave equations or by transforming the time domain wavefields using the Fourier transform. The former approach requires solving systems of linear equations, which becomes progressively difficult to tackle for larger scale models and for higher frequency components. On the contrary, the latter approach can be efficiently implemented using explicit time integration methods in conjunction with running summations as the computation progresses. Commonly used explicit time integration methods correspond to the truncated Taylor series approximations that can cause significant errors for large time steps. The rapid expansion method (REM) uses the Chebyshev expansion and offers an optimal solution to the second-order-in-time wave equations. When applying the Fourier transform to the time domain wavefield solution computed by the REM, we can derive a frequency response modeling formula that has the same form as the original time domain REM equation but with different summation coefficients. In particular, the summation coefficients for the frequency response modeling formula corresponds to the Fourier transform of those for the time domain modeling equation. As a result, we can directly compute frequency responses from the Chebyshev expansion polynomials rather than the time domain wavefield snapshots as do other time domain frequency response modeling methods. When combined with the pseudospectral method in space, this new frequency response modeling method can produce spectrally accurate results with high efficiency.
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25

Hamdan, Hamdan, Nikos Economou, Antonis Vafidis, Maksim Bano, and Jose Ortega-Ramirez. "A New Approach for Adaptive GPR Diffraction Focusing." Remote Sensing 14, no. 11 (May 26, 2022): 2547. http://dx.doi.org/10.3390/rs14112547.

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Several researchers have utilized multipath summation to manage the common problem of scattered energy within GPR sections. Such energy results in degrading the lateral resolution and continuity of reflectors. If detailed velocity models are known, then it is fairly easy to focus the scattered energy by means of conventional migration methods. However, this is rarely the case in GPR sections, as the common-offset antenna array is mostly used, and therefore cannot provide velocity models. This gives an important advantage for the multipath summation method, which has proved to be successful in focusing such diffractions, without the need to build a detailed migration velocity field model. This multipath summation method is based on stacking (summation) of constant velocity migrated sections (weighted or not) over a predefined velocity range. The main drawback of this technique is the high computational cost and the need for user interference to select the appropriate stacking weights. We developed an improved implementation of the weighted multipath summation method that reduces both the computational cost, and the user interference in stacking weights selections. This data adaptive methodology can expedite the migration process, suppress the need for a detailed velocity model, and reduce the user subjectivity. Moreover, a data adaptive spectral scaling scheme was developed. This is applied on the output of the multipath summation process to reduce the expected blurriness in the resulting GPR sections.
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26

Chagas, Jocemar Q., José A. Tenreiro Machado, and António M. Lopes. "Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums." Mathematics 9, no. 22 (November 20, 2021): 2963. http://dx.doi.org/10.3390/math9222963.

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This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.
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27

Guseinov, Gusein Sh. "Spectral method for deriving multivariate Poisson summation formulae." Communications on Pure and Applied Analysis 12, no. 1 (September 2012): 359–73. http://dx.doi.org/10.3934/cpaa.2013.12.359.

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28

Bakhshandeh, Amin, and Yan Levin. "Widom insertion method in simulations with Ewald summation." Journal of Chemical Physics 156, no. 13 (April 7, 2022): 134110. http://dx.doi.org/10.1063/5.0085527.

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We discuss the application of the Widom insertion method for calculation of the chemical potential of individual ions in computer simulations with Ewald summation. Two approaches are considered. In the first approach, an individual ion is inserted into a periodically replicated overall charge neutral system representing an electrolyte solution. In the second approach, an inserted ion is also periodically replicated, leading to the violation of the overall charge neutrality. This requires the introduction of an additional neutralizing background. We find that the second approach leads to a much better agreement with the results of grand canonical Monte Carlo simulation for the total chemical potential of a neutral ionic cluster.
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29

ROSS, KALLA, and KALLA. "THE METHOD OF FRACTIONAL OPERATORS APPLIED TO SUMMATION." Real Analysis Exchange 11, no. 1 (1985): 271. http://dx.doi.org/10.2307/44151747.

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30

Popov, Mikhail M., Nikolay M. Semtchenok, Peter M. Popov, and Arie R. Verdel. "Depth migration by the Gaussian beam summation method." GEOPHYSICS 75, no. 2 (March 2010): S81—S93. http://dx.doi.org/10.1190/1.3361651.

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Seismic depth migration aims to produce an image of seismic reflection interfaces. Ray methods are suitable for subsurface target-oriented imaging and are less costly compared to two-way wave-equation-based migration, but break down in cases when a complex velocity structure gives rise to the appearance of caustics. Ray methods also have difficulties in correctly handling the different branches of the wavefront that result from wave propagation through a caustic. On the other hand, migration methods based on the two-way wave equation, referred to as reverse-time migration, are known to be capable of dealing with these problems. However, they are very expensive, especially in the 3D case. It can be prohibitive if many iterations are needed, such as for velocity-model building. Our method relies on the calculation of the Green functions for the classical wave equation by per-forming a summation of Gaussian beams for the direct and back-propagated wavefields. The subsurface image is obtained by cal-culating the coherence between the direct and backpropagated wavefields. To a large extent, our method combines the advantages of the high computational speed of ray-based migration with the high accuracy of reverse-time wave-equation migration because it can overcome problems with caustics, handle all arrivals, yield good images of steep flanks, and is readily extendible to target-oriented implementation. We have demonstrated the quality of our method with several state-of-the-art benchmark subsurface models, which have velocity variations up to a high degree of complexity. Our algorithm is especially suited for efficient imaging of selected subsurface subdomains, which is a large advantage particularly for 3D imaging and velocity-model refinement applications such as subsalt velocity-model improvement. Because our method is also capable of providing highly accurate migration results in structurally complex subsurface settings, we have also included the concept of true-amplitude imaging in our migration technique.
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31

White, B. S., A. Norris, A. Bayliss, and R. Burridge. "Some remarks on the Gaussian beam summation method." Geophysical Journal International 89, no. 2 (May 1, 1987): 579–636. http://dx.doi.org/10.1111/j.1365-246x.1987.tb05184.x.

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32

Zacharovas, Vytas. "A Tauberian theorem for the Ingham summation method." Acta Arithmetica 148, no. 1 (2011): 31–54. http://dx.doi.org/10.4064/aa148-1-3.

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33

Bai, Min, Xiaohong Chen, Juan Wu, Guochang Liu, Yangkang Chen, Hanming Chen, and Qingqing Li. "Q-compensated migration by Gaussian beam summation method." Journal of Geophysics and Engineering 13, no. 1 (January 15, 2016): 35–48. http://dx.doi.org/10.1088/1742-2132/13/1/35.

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34

Denlinger, Ryan, Zydrunas Gimbutas, Leslie Greengard, and Vladimir Rokhlin. "A fast summation method for oscillatory lattice sums." Journal of Mathematical Physics 58, no. 2 (February 2017): 023511. http://dx.doi.org/10.1063/1.4976499.

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35

Kwee, B. "On the (J, pn, qn) method of summation." Proceedings of the Edinburgh Mathematical Society 28, no. 1 (February 1985): 59–66. http://dx.doi.org/10.1017/s0013091500003199.

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In the following discussion we shall assume that pn≧0, qn≧0 for all n and that qn + 1 > qn → ∞. The (J, pn, qn) method of summation is defined as follows.The series with the partial sum sn, is called summable (J, pn, qn) to s, and we write if the seriesand converge to the sum functions p*(x) and p(s)(x) respectively for 0<x<1 and if τ(x) = p(s)(x)/p*(x)→s as x→1–0.
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36

Krivec, R., and V. B. Mandelzweig. "Quasilinearization method and summation of the WKB series." Physics Letters A 337, no. 4-6 (April 2005): 354–59. http://dx.doi.org/10.1016/j.physleta.2005.01.072.

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37

Eichelsbacher, Peter, and Matthias Löwe. "Lindeberg’s Method for Moderate Deviations and Random Summation." Journal of Theoretical Probability 32, no. 2 (February 4, 2019): 872–97. http://dx.doi.org/10.1007/s10959-019-00881-5.

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38

Mur, V. D. "Summation of Divergent Series and Zeldovich's Regularization Method." Physics of Atomic Nuclei 68, no. 4 (2005): 677. http://dx.doi.org/10.1134/1.1903096.

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39

Janusauskas, A. "A method of summation of multiple trigonometric series." Lithuanian Mathematical Journal 29, no. 3 (1990): 297–300. http://dx.doi.org/10.1007/bf00966635.

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40

Woźny, Paweł, and Rafał Nowak. "Method of summation of some slowly convergent series." Applied Mathematics and Computation 215, no. 4 (October 2009): 1622–45. http://dx.doi.org/10.1016/j.amc.2009.07.016.

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41

Imiya, Atsushi, Toshio Sasaki, and Hidemitsu Ogawa. "Signal smoothing by summation–generalized fejer method and generalized lanczos method." Electronics and Communications in Japan (Part I: Communications) 69, no. 12 (1986): 21–28. http://dx.doi.org/10.1002/ecja.4410691203.

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42

Li, Chun Lan, Song Huai Du, Zhai Shi, and Juan Su. "Study on Detection Method of Electrical Shock Current Based on Sliding Window and Wavelet Transform." Advanced Materials Research 860-863 (December 2013): 2035–39. http://dx.doi.org/10.4028/www.scientific.net/amr.860-863.2035.

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With the popularization of the rural electricity utilization, the rural electric safety was still an immediate problem to be solved. It was difficult to exactly detect and judge electric shock signals in the summation leakage current on the low-voltage electric power grid. A detection method of electrical contact signals based on sliding window and wavelet multi-resolution method was proposed. Under the different signal-to-noise ratio level, the summation leakage current was reconstructed by wavelet decomposition reconstruction algorithm. According to the characteristic of the slow change of the normal leakage current and the rapid change of the electric shock current within a short time, the electric shock current were extracted from the restructured summation leakage current signal by sliding window method. The mean square error and correlation analysis between the extracted signal and the actual testing results were studied. The analysis results show that the proposed method could identify electric shock current in the summation leakage current among noise, and its detection precision was superior to single sliding window method.
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43

Hu, Wenting, Yin Sheng, and Xianjun Zhu. "A Semantic Image Retrieval Method Based on Interest Selection." Wireless Communications and Mobile Computing 2022 (February 27, 2022): 1–6. http://dx.doi.org/10.1155/2022/3029866.

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There is a semantic gap between people’s understanding of images and the underlying visual features of images, which makes it difficult for image retrieval results to meet the needs of individual interests. To overcome the semantic gap in image retrieval, this paper proposes a semantic image retrieval method based on interest selection. This method analyses the interest points of individual selections and gives the weight of the interest selection in different regions of an image. By extracting the underlying visual features of different regions, this paper constructs two feature vector methods after users’ interest point weighting. The two methods are called interest weighted summation and interest weighting. Finally, this paper compares the accuracy of different image classification methods using a support vector machine classification algorithm. The experimental results show that the target classification accuracy of the classification algorithm based on interest weighted summation is higher than that of the traditional and interest weighted methods. The classification algorithm based on interest weighted summation has the highest overall effect on target object classification in the four experimental scenarios. Therefore, the interest point selection method can effectively improve the overall satisfaction of image recommendation and can be used as a novel solution to overcome the semantic gap.
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44

Merzlikin, Dmitrii, and Sergey Fomel. "Analytical path-summation imaging of seismic diffractions." GEOPHYSICS 82, no. 1 (January 1, 2017): S51—S59. http://dx.doi.org/10.1190/geo2016-0140.1.

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Diffraction imaging aims to emphasize small subsurface objects, such as faults, fracture swarms, and channels. Similar to classical reflection imaging, velocity analysis is crucially important for accurate diffraction imaging. Path-summation migration provides an imaging method that produces an image of the subsurface without picking a velocity model. Previous methods of path-summation imaging involve a discrete summation of the images corresponding to all possible migration velocity distributions within a predefined integration range and thus involve a significant computational cost. We have developed a direct analytical formula for path-summation imaging based on the continuous integration of the images along the velocity dimension, which reduces the cost to that of only two fast Fourier transforms. The analytic approach also enabled automatic migration velocity extraction from diffractions using a double path-summation migration framework. Synthetic and field data examples confirm the efficiency of the proposed techniques.
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45

Wang, W.-Z., H. Wang, Y.-C. Liu, Y.-Z. Hu, and D. Zhu. "A comparative study of the methods for calculation of surface elastic deformation." Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 217, no. 2 (February 1, 2003): 145–54. http://dx.doi.org/10.1243/13506500360603570.

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A fundamental issue of lubrication analysis is the calculation of surface deformation, which includes two major steps: determination of influence coefficients and multiplication and summation. There are various interpolation schemes, such as the bilinear interpolation, the piecewise constant function or Green's function, available for determining the influence coefficients, while the summation operation may be performed by using one of the following methods: direct summation (DS), multilevel multi-integration (MLMI) or the discrete convolution and fast Fourier transform (DC-FFT) method. To limit the periodical errors, the proper way to implement the DC-FFT method is described in detail. The computation efficiency and numerical accuracy are compared by applying the different methods to typical contact problems. The results show that the three methods can achieve comparable numerical accuracy, but the DC-FFT method shows much higher computation efficiency than the others, especially when a great number of grid points are involved. It is concluded that the DC-FFT method has great potential in applications to the numerical analysis of, for example, surface deformations and temperature rises.
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46

Daley, P. F., and F. Hron. "Practical numerical considerations for the Alekseev–Mikhailenko method." Canadian Journal of Earth Sciences 27, no. 8 (August 1, 1990): 1023–30. http://dx.doi.org/10.1139/e90-106.

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Programs that utilize the Alekseev–Mikhailenko method are becoming viable seismic interpretation aids because of the availability of a new generation of supercomputers. This method is highly numerically accurate, employing a combination of finite integral transforms and finite difference methods, for the solution of hyperbolic partial differential equations, to yield the total seismic wave field.In this paper two questions of a numerical nature are addressed. For coupled P–Sv wave propagation with radial symmetry, Hankel transforms of order 0 and 1 are required to cast the problem in a form suitable for solution by finite difference methods. The inverse series summations would normally require that the two sets of roots of the transcendental equations be employed, corresponding to the zeroes of the Bessel functions of order 0 and 1. This matter is clarified, and it is shown that both inverse series summations may be performed by considering only one set of roots.The second topic involves providing practical means of determining the lower and upper bounds of a truncated series that suitably approximates the infinite inverse series summation of the finite Hankel transform. It is shown that the number of terms in the truncated series generally decreases with increasing duration of the source pulse and that the truncated series may be further reduced if near-vertical-incidence seismic traces are avoided.
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47

Kang, Myeongseok, and Donghyun You. "A Low Dissipative and Stable Cell-Centered Finite Volume Method with the Simultaneous Approximation Term for Compressible Turbulent Flows." Mathematics 9, no. 11 (May 26, 2021): 1206. http://dx.doi.org/10.3390/math9111206.

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Abstract:
A simultaneous-approximation term is a non-reflecting boundary condition that is usually accompanied by summation-by-parts schemes for provable time stability. While a high-order convective flux based on reconstruction is often employed in a finite-volume method for compressible turbulent flow, finite-volume methods with the summation-by-parts property involve either equally weighted averaging or the second-order central flux for convective fluxes. In the present study, a cell-centered finite-volume method for compressible Naiver–Stokes equations was developed by combining a simultaneous-approximation term based on extrapolation and a low-dissipative discretization method without the summation-by-parts property. Direct numerical simulations and a large eddy simulation show that the resultant combination leads to comparable non-reflecting performance to that of the summation-by-parts scheme combined with the simultaneous-approximation term reported in the literature. Furthermore, a characteristic boundary condition was implemented for the present method, and its performance was compared with that of the simultaneous-approximation term for a direct numerical simulation and a large eddy simulation to show that the simultaneous-approximation term better maintained the average target pressure at the compressible flow outlet, which is useful for turbomachinery and aerodynamic applications, while the characteristic boundary condition better preserved the flow field near the outlet.
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48

Qureshi, M. I., J. Majid, and A. H. Bhat. "Summation of some infinite series by the methods of Hypergeometric functions and partial fractions." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 103, no. 3 (September 30, 2021): 87–95. http://dx.doi.org/10.31489/2021m3/87-95.

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In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].
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49

Deeb, Ahmad, Aziz Hamdouni, Erwan Liberge, and Dina Razafindralandy. "Borel-Laplace summation method used as time integration scheme." ESAIM: Proceedings and Surveys 45 (September 2014): 318–27. http://dx.doi.org/10.1051/proc/201445033.

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50

Hautman, Joseph, and Michael L. Klein. "An Ewald summation method for planar surfaces and interfaces." Molecular Physics 75, no. 2 (February 10, 1992): 379–95. http://dx.doi.org/10.1080/00268979200100301.

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