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Series, Taylor’s theorem and its uses
Published in Alan Jeffrey, Mathematics, 2004
The terms of series are not always of the same sign, and so it is useful to associate with the series ∑an the companion series ∑|an|. If this latter series is convergent, then the series ∑an is said to be absolutely convergent. It can happen that although ∑an is convergent, ∑|an| is divergent. When this occurs the series ∑an is said to be conditionally convergent. Now when terms of differing signs are involved, the sum of the absolute values of the terms of a series clearly exceeds the sum of the terms of the series, and so it seems reasonable to expect that absolute convergence implies convergence. Let us prove this fact.
Research on performance of ionization chamber used in medical laser proton accelerator based on Garfield++
Published in Radiation Effects and Defects in Solids, 2023
Xi-Cheng Xie, X.Q. Yan, K. Zhu, Hui-Lin Ge, Ke-Dong Wang
The finite element analysis convergence diagram of the electrostatic field is shown in Figure 6(b). The AMPS L2 in it is the convergence criteria curve for the current calculation, while the VOLT L2 is the convergence criteria curve for the voltage calculation. The AMPSCRIT is the convergence curve of the actual current calculation of the ionization chamber model, while the VOLTCRIT is the convergence curve of the actual voltage calculation of the ionization chamber model. The abscissa represents the number of iterations, and the ordinate represents the absolute convergence norm. If the calculated curve’s position is not lower than the convergence standard curve, then the calculated curve is conforming to convergence criteria, so as to prove the correctness of the ansys finite element electric field analysis of the gas ionization chamber, the accuracy of the gas ionization chamber ansys model established, the precisely of the voltage boundary set in the model and the reasonable of the grid division chosen in the model. It lays a reliable foundation for the subsequent import of the model into Garfield++ for simulation.
Resonance frequencies of arbitrarily shaped dielectric cylinders
Published in Applicable Analysis, 2023
One can perform direct calculation of CNs from (25) and (26) using Fourier expansions and the expressions and ready formulas of the theory of singular integral operators with the Cauchy and Hilbert kernels available e.g. in [44]. Assume to this end that the Fourier series is absolutely convergent and admits termwise differentiation. Next, according to [44], Calculating the Cauchy product , of series (28) and (29) (under the assumption of absolute convergence of the Fourier series leading to the convergence of the resulting Cauchy series product) separating the first term on the right-hand side of (29), using the orthogonality , , of complex exponents on , and integrating then the obtained product series over this interval, we get Substitution of (30) to (25) yields the sought formula for CN of operator pencil .
A boundary element implementation for fracture mechanics problems using generalised Westergaard stress functions
Published in European Journal of Computational Mechanics, 2018
Ney Augusto Dumont, Elvis Yuri Mamani, Marilene Lobato Cardoso
where is the semicrack length at the crack tip (it would be on the right in Figure 4) and and are the force parameters corresponding to the opening shapes – relative displacement and crack interface rotation – on the right of the same Figure 4. The results for 1, 4, 16, 64 and 256 elements are given on the left in Figure 11. Although the results have been shown to converge for stresses in the vicinity of the crack tip, as in the previous figures, absolute convergence cannot be observed in this case – and cannot be demonstrated by any theorem, since this is a localised behaviour (how the stress goes locally to infinity) for which no energy norm can be given (Dumont & Lopes, 2003). It is even surprising that the results for elliptic elements do converge better than in the case of the generally more accurate implementation with mixed elements with crack interface rotation (6% accuracy against 8%). Also, observe that, in the present case, the result for just one crack element trivially coincides with the analytic solution.