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Статті в журналах з теми "Quadratic estimates"

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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Golub, Gene H., and Zdeněk Strakoš. "Estimates in quadratic formulas." Numerical Algorithms 8, no. 2 (September 1994): 241–68. http://dx.doi.org/10.1007/bf02142693.

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2

Karel Pravda-Starov. "Subelliptic estimates for quadratic differential operators." American Journal of Mathematics 133, no. 1 (2011): 39–89. http://dx.doi.org/10.1353/ajm.2011.0003.

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3

Zioutas, G., L. Camarinopoulos, and E. Bora Senta. "Robust autoregressive estimates using quadratic programming." European Journal of Operational Research 101, no. 3 (September 1997): 486–98. http://dx.doi.org/10.1016/s0377-2217(96)00190-7.

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4

Petunin, Yu I., and N. P. Tupko. "Theory of quadratic estimates of variance." Ukrainian Mathematical Journal 51, no. 9 (September 1999): 1370–85. http://dx.doi.org/10.1007/bf02593004.

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5

Lieberman, Gary M. "Gradient estimates for semilinear elliptic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 100, no. 1-2 (1985): 11–17. http://dx.doi.org/10.1017/s0308210500013597.

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SynopsisEstimates on the gradient of solutions to the Dirichlet problem for a semilinear elliptic equation are given when the nonlinearity in the equation is quadratic with respect to the gradient of the solution. These estimates extend results of F. Tomi to less smooth boundary data and results of the author to the full quadratic growth.

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6

Severin, Valeriy P. "Automatic Control Systems Integral Quadratic Estimates Minimization. Part 1. Estimates computation." Journal of Automation and Information Sciences 36, no. 7 (2004): 1–11. http://dx.doi.org/10.1615/jautomatinfscien.v36.i7.10.

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7

Hitrik, Michael, Johannes Sjöstrand, and Joe Viola. "Resolvent estimates for elliptic quadratic differential operators." Analysis & PDE 6, no. 1 (June 1, 2013): 181–96. http://dx.doi.org/10.2140/apde.2013.6.181.

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8

Rotar’, V. I., and T. L. Shervashidze. "Some Estimates of Distributions of Quadratic Forms." Theory of Probability & Its Applications 30, no. 3 (September 1986): 585–90. http://dx.doi.org/10.1137/1130072.

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9

Vieu, Philippe. "Quadratic errors for nonparametric estimates under dependence." Journal of Multivariate Analysis 39, no. 2 (November 1991): 324–47. http://dx.doi.org/10.1016/0047-259x(91)90105-b.

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10

Henriot, Kevin, and Kevin Hughes. "On Restriction Estimates for Discrete Quadratic Surfaces." International Mathematics Research Notices 2019, no. 23 (February 3, 2018): 7139–59. http://dx.doi.org/10.1093/imrn/rnx255.

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Abstract We obtain truncated restriction estimates of an unexpected form for discrete surfaces $$\begin{align*}S_N = \{\, ( n_1 , \dots , n_d , R( n_1 , \dots, n_d ) ) \,,\, n_i \in [-N,N] \cap {\mathbb{Z}} \,\},\end{align*}$$ where $R$ is an indefinite quadratic form with integer matrix.

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11

SMITH, BLAKE WH. "Accuracy and Consistency of Quadratic Odds Estimates." Family Practice 8, no. 3 (1991): 269–75. http://dx.doi.org/10.1093/fampra/8.3.269.

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12

Blomer, Valentin, and Andrew Granville. "Estimates for representation numbers of quadratic forms." Duke Mathematical Journal 135, no. 2 (November 2006): 261–302. http://dx.doi.org/10.1215/s0012-7094-06-13522-6.

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13

Girko, V. L., and T. V. Pavlenko. "G-estimates of the quadratic discriminant function." Ukrainian Mathematical Journal 41, no. 12 (December 1989): 1469–73. http://dx.doi.org/10.1007/bf01056118.

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14

Bandara, Lashi. "Rough metrics on manifolds and quadratic estimates." Mathematische Zeitschrift 283, no. 3-4 (March 5, 2016): 1245–81. http://dx.doi.org/10.1007/s00209-016-1641-x.

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15

Beltracchi, T. J., and G. A. Gabriele. "A Recursive Quadratic Programming Based Method for Estimating Parameter Sensitivity Derivatives." Journal of Mechanical Design 113, no. 4 (December 1, 1991): 487–94. http://dx.doi.org/10.1115/1.2912809.

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Parameter sensitivity analysis is defined as the estimation of changes in the modeling functions and design point due to small changes in the fixed parameters of the formulation. There are currently several methods for estimating parameter sensitivities which either require second order information, or do not return reliable estimates for the derivatives. This paper presents a method based on the use of the recursive quadratic programming method in conjunction with differencing formulas to estimate parameter sensitivity derivatives without the need to calculate second order information. In addition, a modified variable metric method for estimating the Hessian of the Lagrangian function is presented that is used to increase the accuracy of the sensitivity derivatives. Testing is performed on a set of problems with Hessians obtained analytically, and on a set of engineering related problems whose derivatives must be estimated numerically. The results indicate that the method provides good estimates of the parameter sensitivity derivatives on both test sets.

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16

Brydges, David, and Thomas Spencer. "Fluctuation estimates for sub-quadratic gradient field actions." Journal of Mathematical Physics 53, no. 9 (September 2012): 095216. http://dx.doi.org/10.1063/1.4747194.

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17

Almada Monter, Sergio Angel. "Quadratic covariation estimates in non-smooth stochastic calculus." Stochastic Processes and their Applications 125, no. 1 (January 2015): 343–61. http://dx.doi.org/10.1016/j.spa.2014.09.005.

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18

Doukhan, Paul, and Elisabeth Gassiat. "Quadratic deviation of penalized mean squares regression estimates." Journal of Multivariate Analysis 41, no. 1 (April 1992): 89–101. http://dx.doi.org/10.1016/0047-259x(92)90059-o.

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19

Hitrik, Michael, Karel Pravda-Starov, and Joe Viola. "From semigroups to subelliptic estimates for quadratic operators." Transactions of the American Mathematical Society 370, no. 10 (May 17, 2018): 7391–415. http://dx.doi.org/10.1090/tran/7251.

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20

Shor, N. Z. "Dual quadratic estimates in polynomial and Boolean programming." Annals of Operations Research 25, no. 1 (December 1990): 163–68. http://dx.doi.org/10.1007/bf02283692.

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21

Berezovskii, O. A., and P. I. Stetsyuk. "An approach to determining Shor’s dual quadratic estimates." Cybernetics and Systems Analysis 44, no. 2 (March 2008): 225–33. http://dx.doi.org/10.1007/s10559-008-0022-9.

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22

McKillop, William, and Gao Liu. "Modelling disaggregated lumber demand and supply by constrained estimation techniques." Canadian Journal of Forest Research 20, no. 6 (June 1, 1990): 781–89. http://dx.doi.org/10.1139/x90-103.

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Анотація:

Constrained estimation techniques were used to estimate a 12-equation demand and supply system for Douglas-fir and hemlock–fir (true fir) lumber by clear and common grades. Such techniques combine sample information with nonsample information and produce estimates with smaller variances than those based only on sample data. Conventional econometric estimation was compared with a quadratic programming technique with regression coefficients constrained to their a priori correct signs. A goal programming technique that minimized the sum of the absolute deviations was rejected because of its substantially different results and lack of information on the statistical properties of its estimates. The quadratic programming technique had the advantages of statistical efficiency, objectivity, and speed. The conventional estimation technique excluded fewer variables from the system and thus was less susceptible to omission of variables bias. Elasticity estimates for most key variables were similar. Quadratic programming versus conventional estimates of demand elasticity were, respectively, −0.95 and −0.88 for Douglas-fir clears, −2.83 and −2.91 for Douglas-fir commons, and −2.13 and −2.27 for all Douglas-fir; whereas supply elasticities for hemlock–fir commons were 1.35 and 1.37, and for all hemlock–fir, −1.76 and −1.42.

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23

Kato, Keiichi, Masaharu Kobayashi, and Shingo Ito. "Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials." Journal of Functional Analysis 266, no. 2 (January 2014): 733–53. http://dx.doi.org/10.1016/j.jfa.2013.08.017.

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24

Hafizoglu, Cavit, Irena Lasiecka, Tijana Levajković, Hermann Mena, and Amjad Tuffaha. "The Stochastic Linear Quadratic Control Problem with Singular Estimates." SIAM Journal on Control and Optimization 55, no. 2 (January 2017): 595–626. http://dx.doi.org/10.1137/16m1056183.

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25

Axelsson, Andreas, Stephen Keith, and Alan McIntosh. "Quadratic estimates and functional calculi of perturbed Dirac operators." Inventiones mathematicae 163, no. 3 (October 25, 2005): 455–97. http://dx.doi.org/10.1007/s00222-005-0464-x.

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26

McGonagle, Matt, Chong Song, and Yu Yuan. "Hessian estimates for convex solutions to quadratic Hessian equation." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 36, no. 2 (March 2019): 451–54. http://dx.doi.org/10.1016/j.anihpc.2018.07.001.

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27

Auscher, P., A. McIntosh, and A. Nahmod. "Holomorphic functional calculi of operators, quadratic estimates and interpolation." Indiana University Mathematics Journal 46, no. 2 (1997): 0. http://dx.doi.org/10.1512/iumj.1997.46.1180.

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28

HEJUN, SUN, and QI XUERONG. "EIGENVALUE ESTIMATES FOR QUADRATIC POLYNOMIAL OPERATOR OF THE LAPLACIAN." Glasgow Mathematical Journal 53, no. 2 (December 8, 2010): 321–32. http://dx.doi.org/10.1017/s0017089510000728.

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AbstractFor a bounded domain Ω in a complete Riemannian manifold M, we investigate the Dirichlet weighted eigenvalue problem of quadratic polynomial operator Δ2 − aΔ + b of the Laplacian Δ, where a and b are the nonnegative constants. We obtain an inequality for eigenvalues which contains a constant that only depends on the mean curvature of M. It yields an upper bound of the (k + 1)th eigenvalue Λk + 1. As their applications, some inequalities and bounds of eigenvalues on a complete minimal submanifold in a Euclidean space and a unit sphere are obtained.

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29

Cianci, P., G. R. Cirmi, S. D’Asero, and S. Leonardi. "Morrey estimates for solutions of singular quadratic nonlinear equations." Annali di Matematica Pura ed Applicata (1923 -) 196, no. 5 (February 8, 2017): 1739–58. http://dx.doi.org/10.1007/s10231-017-0636-5.

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30

Mitrouli, Marilena, Athanasios Polychronou, Paraskevi Roupa, and Ondřej Turek. "Estimating the Quadratic Form xTA−mx for Symmetric Matrices: Further Progress and Numerical Computations." Mathematics 9, no. 12 (June 19, 2021): 1432. http://dx.doi.org/10.3390/math9121432.

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In this paper, we study estimates for quadratic forms of the type xTA−mx, m∈N, for symmetric matrices. We derive a general approach for estimating this type of quadratic form and we present some upper bounds for the corresponding absolute error. Specifically, we consider three different approaches for estimating the quadratic form xTA−mx. The first approach is based on a projection method, the second is a minimization procedure, and the last approach is heuristic. Numerical examples showing the effectiveness of the estimates are presented. Furthermore, we compare the behavior of the proposed estimates with other methods that are derived in the literature.

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31

Gogishvili, Guram. "New Estimates of the Singular Series Corresponding to Positive Quaternary Quadratic Forms." gmj 13, no. 4 (December 2006): 687–91. http://dx.doi.org/10.1515/gmj.2006.687.

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Abstract Let 𝑚 ∈ ℕ, 𝑓 be a positive definite, integral, primitive, quaternary quadratic form of the determinant 𝑑 and let ρ(𝑓,𝑚) be the corresponding singular series. When studying the best estimates for ρ(𝑓,𝑚) with respect to 𝑑 and 𝑚 we proved in [Gogishvili, Trudy Tbiliss. Univ. 346: 72–77, 2004] that where 𝑏(𝑘) is the product of distinct prime factors of 16𝑘 if 𝑘 ≠ 1 and 𝑏(𝑘) = 3 if 𝑘 = 1. The present paper proves a more precise estimate where 𝑑 = 𝑑0𝑑1, if 𝑝 > 2; 𝑕(2) ⩾ –4. The last estimate for ρ(𝑓,𝑚) as a general result for quaternary quadratic forms of the above-mentioned type is unimprovable in a certain sense.

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32

Gaddam, Sharat, and Thirupathi Gudi. "Bubbles Enriched Quadratic Finite Element Method for the 3D-Elliptic Obstacle Problem." Computational Methods in Applied Mathematics 18, no. 2 (April 1, 2018): 223–36. http://dx.doi.org/10.1515/cmam-2017-0018.

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AbstractAn optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in [14]. There was no analogue of a quadratic finite element method on tetrahedron meshes for the three-dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three-dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.

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33

Aceves-Lara, C. A., E. Aguilar-Garnica, V. Alcaraz-González, O. González-Reynoso, J. P. Steyer, J. L. Dominguez-Beltran, and V. González-Álvarez. "Kinetic parameters estimation in an anaerobic digestion process using successive quadratic programming." Water Science and Technology 52, no. 1-2 (July 1, 2005): 419–26. http://dx.doi.org/10.2166/wst.2005.0548.

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In this work, an optimization method is implemented in an anaerobic digestion model to estimate its kinetic parameters and yield coefficients. This method combines the use of advanced state estimation schemes and powerful nonlinear programming techniques to yield fast and accurate estimates of the aforementioned parameters. In this method, we first implement an asymptotic observer to provide estimates of the non-measured variables (such as biomass concentration) and good guesses for the initial conditions of the parameter estimation algorithm. These results are then used by the successive quadratic programming (SQP) technique to calculate the kinetic parameters and yield coefficients of the anaerobic digestion process. The model, provided with the estimated parameters, is tested with experimental data from a pilot-scale fixed bed reactor treating raw industrial wine distillery wastewater. It is shown that SQP reaches a fast and accurate estimation of the kinetic parameters despite highly noise corrupted experimental data and time varying inputs variables. A statistical analysis is also performed to validate the combined estimation method. Finally, a comparison between the proposed method and the traditional Marquardt technique shows that both yield similar results; however, the calculation time of the traditional technique is considerable higher than that of the proposed method.

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34

Madenjian, Charles P., Gary L. Rogers, and Arlo W. Fast. "Estimation of Whole Pond Respiration Rate." Canadian Journal of Fisheries and Aquatic Sciences 47, no. 4 (April 1, 1990): 682–86. http://dx.doi.org/10.1139/f90-075.

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A regression application of the whole pond respiration–diffusion (WPRD) model to overnight observations of dissolved oxygen concentration (DO) was introduced as a new method to estimate whole pond respiration rate. A new modification of the Welch procedure was also presented. These two techniques were compared with the tangent method for whole pond respiration rate estimation. The tangent method required fitting a quadratic curve to observed DO; and then calculating the slope of the tangent line to the curve at the time the curve reached 100% saturation of DO. When the quadratic was fitted to DO observed during both day and light, the tangent method often yielded inaccurate estimates of whole pond respiration rate. Estimates by the tangent method were sensitive to the time at which 100% saturation of DO occurred. The WPRD model regression and the modified Welch procedure were recommended to estimate whole pond respiration rate.

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35

Campione, Nicolás E. "Extrapolating body masses in large terrestrial vertebrates." Paleobiology 43, no. 4 (June 30, 2017): 693–99. http://dx.doi.org/10.1017/pab.2017.9.

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AbstractDespite more than a century of interest, body-mass estimation in the fossil record remains contentious, particularly when estimating the body mass of taxa outside the size scope of living animals. One estimation approach uses humeral and femoral (stylopodial) circumferences collected from extant (living) terrestrial vertebrates to infer the body masses of extinct tetrapods through scaling models. When applied to very large extinct taxa, extant-based scaling approaches incur obvious methodological extrapolations leading some to suggest that they may overestimate the body masses of large terrestrial vertebrates. Here, I test the implicit assumption of such assertions: that a quadratic model provides a better fit to the combined humeral and femoral circumferences-to-body mass relationship. I then examine the extrapolation potential of these models through a series of subsetting exercises in which lower body-mass sets are used to estimate larger sets. Model fitting recovered greater support for the original linear model, and a nonsignificant second-degree term indicates that the quadratic relationship is statistically linear. Nevertheless, some statistical support was obtained for the quadratic model, and application of the quadratic model to a series of dinosaurs provides lower mass estimates at larger sizes that are more consistent with recent estimates using a minimum convex-hull (MCH) approach. Given this consistency, a quadratic model may be preferred at this time. Still, caution is advised; extrapolations of quadratic functions are unpredictable compared with linear functions. Further research testing the MCH approach (e.g., the use of a universal upscaling factor) may shed light on the linear versus quadratic nature of the relationship between the combined femoral and humeral circumferences and body mass.

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36

Treviño, Enrique. "The Burgess inequality and the least kth power non-residue." International Journal of Number Theory 11, no. 05 (August 2015): 1653–78. http://dx.doi.org/10.1142/s1793042115400163.

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The Burgess inequality is the best upper bound we have for incomplete character sums of Dirichlet characters. In 2006, Booker gave an explicit estimate for quadratic Dirichlet characters which he used to calculate the class number of a 32-digit discriminant. McGown used an explicit estimate to show that there are no norm-Euclidean Galois cubic fields with discriminant greater than 10140. Both of their explicit estimates are on restricted ranges. In this paper, we prove an explicit estimate that works for any range. We also improve McGown's estimates in a slightly narrower range, getting explicit estimates for characters of any order. We apply the estimates to the question of how large must a prime p be to ensure that there is a kth power non-residue less than p1/6.

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37

Franks, Edwin, and Alan McIntosh. "Discrete quadratic estimates and holomorphic functional calculi in Banach spaces." Bulletin of the Australian Mathematical Society 58, no. 2 (October 1998): 271–90. http://dx.doi.org/10.1017/s000497270003224x.

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We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.

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38

Yazici, Ozcan. "Dynamical estimates on a class of quadratic polynomial automorphisms of." Complex Variables and Elliptic Equations 62, no. 2 (August 9, 2016): 230–40. http://dx.doi.org/10.1080/17476933.2016.1218852.

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39

Balandin, D. V., and M. M. Kogan. "Design of Pareto-Optimal Linear Quadratic Estimates, Filters and Controllers." Automation and Remote Control 79, no. 1 (January 2018): 24–38. http://dx.doi.org/10.1134/s0005117918010034.

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40

KHARLAMPOVICH, OLGA, and ALINA VDOVINA. "LINEAR ESTIMATES FOR SOLUTIONS OF QUADRATIC EQUATIONS IN FREE GROUPS." International Journal of Algebra and Computation 22, no. 01 (February 2012): 1250004. http://dx.doi.org/10.1142/s0218196711006704.

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We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by 2s for an orientable equation and by 12s4 for a non-orientable equation, where s is the sum of the lengths of the coefficients.

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41

Kishimoto, Nobu. "Low-regularity bilinear estimates for a quadratic nonlinear Schrödinger equation." Journal of Differential Equations 247, no. 5 (September 2009): 1397–439. http://dx.doi.org/10.1016/j.jde.2009.06.009.

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42

Ding, Xiaqi. "Estimates on the Finite Element Approximations of A Quadratic Functional." Acta Mathematica Scientia 32, no. 1 (January 2012): 209–18. http://dx.doi.org/10.1016/s0252-9602(12)60013-7.

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43

Rösch, A. "Error estimates for linear-quadratic control problems with control constraints." Optimization Methods and Software 21, no. 1 (February 2006): 121–34. http://dx.doi.org/10.1080/10556780500094945.

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44

Gould, Nicholas I. M., and Valeria Simoncini. "Error estimates for iterative algorithms for minimizing regularized quadratic subproblems." Optimization Methods and Software 35, no. 2 (October 7, 2019): 304–28. http://dx.doi.org/10.1080/10556788.2019.1670177.

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45

Menne, Ulrich. "Decay Estimates for the Quadratic Tilt-Excess of Integral Varifolds." Archive for Rational Mechanics and Analysis 204, no. 1 (December 21, 2011): 1–83. http://dx.doi.org/10.1007/s00205-011-0468-1.

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46

Davydov, V. F. "Majorant estimates for solutions of quadratic difference systems with lag." Ukrainian Mathematical Journal 47, no. 4 (April 1995): 629–33. http://dx.doi.org/10.1007/bf01056050.

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47

Karaki, Zeinab. "Study of the Kramers–Fokker–Planck quadratic operator with a constant magnetic field." Journal of Mathematical Physics 63, no. 8 (August 1, 2022): 081503. http://dx.doi.org/10.1063/5.0090025.

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Анотація:

We study the quadratic Kramers–Fokker–Planck operator with a constant magnetic field and a quadratic potential. We describe the exact expression of the norm of the semi-group associated with the operator near the equilibrium. At this level, explicit and accurate estimates of this norm are shown in small and long times and uniform-in-time estimates are shown when the magnetic parameter b tends to infinity.

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48

ZUMALACÁRREGUI, ANA. "CONCENTRATION OF POINTS ON MODULAR QUADRATIC FORMS." International Journal of Number Theory 07, no. 07 (November 2011): 1835–39. http://dx.doi.org/10.1142/s1793042111004897.

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Анотація:

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ ( mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ( mod p).

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49

Sarma, Rajan, and Labananda Choudhury. "A new model for estimating district life expectancy at birth in India, with special reference to Assam state." Canadian Studies in Population 41, no. 1-2 (June 11, 2014): 180. http://dx.doi.org/10.25336/p6ms5d.

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Анотація:

Life expectancy at birth (e0) is considered as an important indicator of the mortality level of a population. In India, direct estimation of e0 is not possible due to incomplete death registration. The Sample Registration System (SRS) of India provides information on e0 only for the 16 major states. Estimates of e0 for the districts are not available. Using data from the Coale-Demeny West model life tables, United Nations South Asian model life tables, and SRS life tables of India and its major states, the paper shows that the relationship between life expectancy at age one (e0) and the probability of surviving to age one (l1) is linear, and the relationship between e0 and l1 is quadratic. From the quadratic relationship between e0 and l1, an attempt is made to estimate e0 for some selected districts of India for 2001 and 2010, using estimated l1 from 2001 census data and Annual Health Survey (2010–11) data.

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50

Liu, Jing Hong, and Gui Hu. "Maximum Norm Error Estimates for Quadratic Block Finite Elements with Twenty-Six Degrees of Freedom." Key Engineering Materials 480-481 (June 2011): 1388–92. http://dx.doi.org/10.4028/www.scientific.net/kem.480-481.1388.

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Анотація:

For a model elliptic boundary value problem in three dimensions, we give the weak estimates for the quadratic block finite element with twenty-six degrees of freedom (QBFETSDF). Combined with the estimates for discrete Green’s function, we get maximum norm error estimates for the QBFETSDF.

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