scholarly journals FRACTIONAL TREND OF THE VARIANCE IN CAVALIERI SAMPLING

2011 ◽  
Vol 19 (2) ◽  
pp. 71 ◽  
Author(s):  
Marta García-Fiñana ◽  
Luis M Cruz-Orive
Keyword(s):  
Fractional Derivatives ◽  
Real Data ◽  
Current Theory ◽  
Continuous Derivative ◽  
General Representation ◽  
Higher Order Terms ◽  
Measurement Function ◽  
Constant Distance ◽  
Derivatives Of ◽  
Maclaurin Formula

Cavalieri sampling is often used to estimate the volume of an object with systematic sections a constant distance T apart. The variance of the corresponding estimator can be expressed as the sum of the extension term (which gives the overall trend of the variance and is used to estimate it), the 'Zitterbewegung' (which oscillates about zero) and higher order terms. The extension term is of order T2m+2 for small T, where m is the order of the first non-continuous derivative of the measurement function f, (namely of the area function if the target is the volume). A key condition is that the jumps of the mth derivative f (m) of f are finite. When this is not the case, then the variance exhibits a fractional trend, and the current theory fails. Indeed, in practice the mentioned trend is often of order T2q+2, typically with 0 <q <1. We obtain a general representation of the variance, and thereby of the extension term, by means of a new Euler-MacLaurin formula involving fractional derivatives of f. We also present a new and general estimator of the variance, see Eq. 26a, b, and apply it to real data (white matter of a human brain).

Finite Heisenberg Quantum Spin Chain

2019 ◽  
pp. 667-686
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Conformal Symmetry ◽  
Finite Size ◽  
Quantum Spin ◽  
Ansatz Method ◽  
Finite Sums ◽  
Mathematical Techniques ◽  
Higher Order Corrections ◽  
Finite Size Corrections ◽  
Maclaurin Formula

The Bethe ansatz genuinely considers a finite system. The extraction of finite-size results from the Bethe ansatz equations is of genuine interest, especially against the background of the results of finite-size scaling and conformal symmetry in finite geometries. The mathematical techniques introduced in chapter 19 permit a systematic treatment in this chapter of finite-size corrections as corrections to the thermodynamic limit of the system. The application of the Euler-Maclaurin formula transforming finite sums into integrals and finite-size corrections transforms the Bethe ansatz equations into Wiener–Hopf integral equations with inhomogeneities representing the finite-size corrections solvable using the Wiener–Hopf technique. The results can be compared to results for finite systems obtained from other approaches that are independent of the Bethe ansatz method. It briefly discusses higher-order corrections and offers a general assessment of the finite-size method.

Contour integrals of analytic functions given on a grid in the complex plane

2020 ◽  
Author(s):  
Bengt Fornberg
Keyword(s):  
Complex Plane ◽  
Line Segment ◽  
Analytic Functions ◽  
Trapezoidal Rule ◽  
Contour Integrals ◽  
Grid Points ◽  
Maclaurin Formula

Abstract Ability to evaluate contour integrals is central to both the theory and the utilization of analytic functions. We present here a complex plane realization of the Euler–Maclaurin formula that includes weights also at some grid points adjacent to each end of a line segment (made up of equispaced grid points, along which we use the trapezoidal rule). For example, with a $5\times 5$ ‘correction stencil’ (with weights about two orders of magnitude smaller than those of the trapezoidal rule), the accuracy is increased from $2$nd to $26$th order.

A generalization of the Euler-Maclaurin formula

1997 ◽  
Vol 61 (2) ◽  
pp. 253-257 ◽  
Author(s):  
R. M. Trigub
Keyword(s):  
Maclaurin Formula

Reactivity calculation using the Euler–Maclaurin formula

2013 ◽  
Vol 53 ◽  
pp. 104-108 ◽  
Author(s):  
Daniel Suescún-Díaz ◽  
Jesús A. Rodríguez-Sarasty ◽  
Jorge H. Figueroa-Jiménez
Keyword(s):  
Maclaurin Formula

scholarly journals Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

2012 ◽  
Vol 62 (2) ◽  
pp. 821-858 ◽  
Author(s):  
Arzu Boysal ◽  
Michèle Vergne
Keyword(s):  
Maclaurin Formula

scholarly journals Approximations for von Neumann and Renyi entropies of graphs using the Euler-Maclaurin formula

2018 ◽  
Vol 48 ◽  
pp. 227-242
Author(s):  
Natália Bebiano ◽  
Susana Furtado ◽  
João da Providência ◽  
Wei-Ru Xu ◽  
João P. da Providência
Keyword(s):  
Von Neumann ◽  
Rényi Entropies ◽  
Maclaurin Formula ◽  
Renyi Entropies
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