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arxiv: 2509.26274 · v2 · submitted 2025-09-30 · 🧮 math.NA · cond-mat.str-el· cs.NA

Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics

Andreas A. Buchheit , Jonathan K. Busse , Torsten Ke{\ss}ler , Filipp N. Rybakov This is my paper

Pith reviewed 2026-05-18 12:07 UTC · model grok-4.3

classification 🧮 math.NA cond-mat.str-elcs.NA
keywords periodic boundary conditionslong-range interactionszeta functionsmicromagneticsRiesz potentialsdipolar interactionsnumerical summationdemagnetization field
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The pith

Infinite lattice sums for dipolar and power-law interactions can be computed exactly using a small direct sum plus zeta function derivative corrections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to compute exact infinite sums for long-range interactions like dipolar forces and Riesz potentials between bodies in periodic arrays without truncating the lattice. By adding a correction based on derivatives of generalized zeta functions to a small direct sum, the method achieves machine precision efficiently. This addresses a key challenge in micromagnetics simulations under periodic boundary conditions, where truncation errors have been common. A superexponentially convergent algorithm for evaluating the zeta functions and related special functions like incomplete Bessel functions makes this practical.

Core claim

For general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. The resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes.

What carries the argument

zeta expansion correction term involving derivatives of generalized zeta functions

Load-bearing premise

The required derivatives of generalized zeta functions admit a superexponentially convergent algorithm that remains practical and stable for the geometries and derivative orders needed in micromagnetics.

What would settle it

Direct comparison of the zeta-corrected sum against an extremely large truncated summation for a specific periodic cuboidal geometry, verifying agreement to machine precision.

Figures

Figures reproduced from arXiv: 2509.26274 by Andreas A. Buchheit, Filipp N. Rybakov, Jonathan K. Busse, Torsten Ke{\ss}ler.

Figure 1
Figure 1. Figure 1: Sketch illustrating a 3D system of of Nx × Ny × Nz = 5 × 3 × 4 interacting cuboids repeated infinitely in x direction (a), and both in x and y direction (b). The green cuboid is influenced by the red cuboid indicated by the arrow, as well as by the total influence of all opaque red cuboids complex shapes by representing them as composed of simplices such as tetrahedrons. At the same time, it is challenging… view at source ↗
Figure 2
Figure 2. Figure 2: Relative error between the three-dimensional potential U (0,0,2)(r) centered at r = (1, 1, 1)/2 for the lattice at Z2 × {0} as obtained by the zeta representation in Theorem 2.3 in comparison with direct summation over the truncated grid {−Ncut, . . . , Ncut} 2× {0} as dots a function of the the truncation length integers 1 ≤ Ncut ≤ 10. The error scaling CνN −ν cut for some fitted parameter Cν is shown as … view at source ↗
Figure 3
Figure 3. Figure 3: Cuboid Ω = [1/(2N), 1/(2N)]3 and its infinite repeti￾tions z + Ω along the two-dimensional lattice embedded in three dimensions L = Z2 × {0} for N = 2 (a), N = 5 (b) and N = 10 (c). On the other hand, for N → ∞, the cuboid is small compared to the lattice spacing, such that it interacts with its repetitions as a point dipole. The magnetic field contribution from the copies is expressed through a lattice su… view at source ↗
Figure 4
Figure 4. Figure 4: Difference between demagnetization factor Dz for 3D cuboids Ω = [−1/(2N), 1/(2N)]3 interacting with a lattice Λ = Z2 , computed from our method, and the known asymptotic formula. The asymptotic behavior is correctly reproduced, as well as the corner cases (Dz = 1 for N = 1 and Dz → 1/3 for N → ∞). The next order correction from the known asymptotics as N → ∞ scales as N −7 (obtained by a fit). 4. Computati… view at source ↗
Figure 5
Figure 5. Figure 5: In what follows, we discuss the procedures employed in Algorithm 1 and [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Regions of Algorithm 1. The white line indicates the split, where above the dashed white line we employ the reciprocal values. The purple region where the upper bound of the incomplete Bessel function is smaller than 10−16 is shown for ν = 0. and the statement follows from the symmetry Ks(z) = K−s(z) [31, Sec. 7.2.2, Eq. (14)]. □ For small vector arguments, we may now use the series expansion in the second… view at source ↗
Figure 6
Figure 6. Figure 6: Cuboid Ω centered in the Cartesian coordinate sys￾tem (x, y, z). We consider a cuboid-shaped source (domain Ω) of size a × b × c placed in the center of the Cartesian coordinate system, see [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
read the original abstract

We address the efficient computation of power-law-based interaction potentials of homogeneous $d$-dimensional bodies with an infinite $n$-dimensional array of copies, including their higher-order derivatives. This problem forms a serious challenge in micromagnetics with periodic boundary conditions and related fields. Nowadays, it is common practice to truncate the associated infinite lattice sum to a finite number of images, introducing uncontrolled errors. We show that, for general interacting geometries, the exact infinite sum for both dipolar interactions and generalized Riesz power-law potentials can be obtained by complementing a small direct sum by a correction term that involves efficiently computable derivatives of generalized zeta functions. We show that the resulting representation converges exponentially in the derivative order, reaching machine precision at a computational cost no greater than that of truncated summation schemes. In order to compute the generalized zeta functions efficiently, we provide a superexponentially convergent algorithm for their evaluation, as well as for all required special functions, such as incomplete Bessel functions. Magnetic fields can thus be evaluated to machine precision in arbitrary cuboidal domains periodically extended along one or two dimensions. We benchmark our method against known formulas for magnetic interactions and against direct summation for Riesz potentials with large exponents, consistently achieving full precision. In addition, we identify new corrections to the asymptotic limit of the demagnetization field and tabulate high-precision benchmark values that can be used as a reliable reference for micromagnetic solvers. The techniques developed are broadly applicable, with direct impact in other areas such as molecular dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a zeta-expansion technique for computing exact infinite lattice sums of power-law (Riesz) and dipolar interactions for homogeneous bodies under 1D or 2D periodic boundary conditions. The central construction supplements a small direct sum over nearby images with a correction assembled from derivatives of generalized zeta functions and incomplete Bessel functions; a superexponentially convergent algorithm is supplied for these special functions. The resulting scheme is asserted to reach machine precision at a cost comparable to truncation, with benchmarks against analytic formulas and direct summation for large exponents, plus new corrections to the demagnetization-field asymptotics and tabulated high-precision reference values for micromagnetic solvers.

Significance. If the convergence and numerical-stability claims hold, the work supplies a practical, controllable-precision alternative to truncation for long-range periodic sums in micromagnetics and molecular dynamics. The explicit identification of new asymptotic corrections and the release of high-precision benchmark tables constitute concrete, reusable contributions to the field. The generality to arbitrary cuboids and to both dipolar and Riesz potentials broadens the potential impact beyond the immediate application.

major comments (2)
  1. [§3] §3 (algorithm for generalized zeta derivatives): the claim that the representation converges exponentially in derivative order and reaches machine precision is load-bearing for the central result, yet no a-priori bounds on conditioning, round-off propagation, or stability of the recurrence relations are given for derivative orders 10–30 in 3D cuboids with 1D/2D periodicity; this directly addresses the weakest assumption identified in the stress test.
  2. [§5] §5 (numerical benchmarks): while full precision is reported against analytic formulas and direct summation, the tables do not list the derivative orders employed, the lattice spacings tested, or any observed condition numbers, preventing independent verification that the method remains stable for the geometries and Riesz exponents required in micromagnetics.
minor comments (2)
  1. The notation for the incomplete Bessel functions and the precise definition of the generalized zeta function could be collected in a single preliminary subsection to improve readability for readers outside the immediate special-functions community.
  2. Figure captions for the convergence plots should explicitly state the Riesz exponent, periodicity dimensions, and derivative order used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding numerical stability analysis and the completeness of benchmark reporting are well taken, and we address them point by point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [§3] §3 (algorithm for generalized zeta derivatives): the claim that the representation converges exponentially in derivative order and reaches machine precision is load-bearing for the central result, yet no a-priori bounds on conditioning, round-off propagation, or stability of the recurrence relations are given for derivative orders 10–30 in 3D cuboids with 1D/2D periodicity; this directly addresses the weakest assumption identified in the stress test.

    Authors: We acknowledge that the manuscript does not supply a-priori bounds on conditioning or round-off propagation for the recurrence relations at high derivative orders. The central construction relies on the observed superexponential convergence, which in practice limits the required orders to moderate values (typically below 20) for machine precision. Extensive numerical experiments confirm stability in double precision across the relevant geometries and periodicity settings. In the revised manuscript we will add a dedicated paragraph in §3 summarizing these stability observations, including representative condition-number estimates for derivative orders up to 30, while noting that a complete theoretical error analysis lies beyond the present scope. revision: partial

  2. Referee: [§5] §5 (numerical benchmarks): while full precision is reported against analytic formulas and direct summation, the tables do not list the derivative orders employed, the lattice spacings tested, or any observed condition numbers, preventing independent verification that the method remains stable for the geometries and Riesz exponents required in micromagnetics.

    Authors: We agree that the benchmark tables would benefit from explicit reporting of the derivative orders, lattice spacings, and observed condition numbers. This information will be added to all tables in §5, together with a short note on the parameter ranges chosen to match typical micromagnetics applications. These updates will allow readers to reproduce and verify the stability claims directly. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via standard zeta-function identities

full rationale

The paper obtains the exact periodic lattice sum by writing it as a small direct sum plus a correction built from derivatives of generalized zeta functions (and incomplete Bessel functions). This representation follows from classical Poisson-summation or Mellin-transform techniques applied to power-law kernels; the zeta functions themselves are independent special functions whose properties are not defined in terms of the target micromagnetic sum. No parameter is fitted to the output quantity, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The claimed superexponential convergence of the special-function algorithm is an independent numerical claim that does not presuppose the final result. Consequently the derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and efficient computability of derivatives of generalized zeta functions together with the exponential convergence of the resulting series for the geometries considered.

axioms (1)
  • standard math Generalized zeta functions and their derivatives admit a superexponentially convergent evaluation algorithm for the required arguments and orders.
    Invoked to justify the correction term and its practical cost; stated in the abstract as the basis for machine-precision results.

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Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    magnum.fe: A micromagnetic finite-element simulation code based on FEniCS

    Claas Abert et al. “magnum.fe: A micromagnetic finite-element simulation code based on FEniCS”. In:Journal of Magnetism and Magnetic Materials 345 (2013), pp. 29–35.issn: 0304-8853.doi: https://doi.org/10.1016/ j.jmmm.2013.05.051 .url: https://www.sciencedirect.com/science/ article/pii/S0304885313004022

    work page 2013

  2. [2]

    Demagnetizing factors for rectangular ferromagnetic prisms

    Amikam Aharoni. “Demagnetizing factors for rectangular ferromagnetic prisms”. In:Journal of Applied Physics83.6 (Mar. 1998), pp. 3432–3434.issn: 0021- 8979.doi:10.1063/1.367113.url:https://doi.org/10.1063/1.367113

    work page doi:10.1063/1.367113.url:https://doi.org/10.1063/1.367113 1998

  3. [3]

    Some Recent Developments in Micromagnetics at the Weizmann Institute of Science

    Amikam Aharoni. “Some Recent Developments in Micromagnetics at the Weizmann Institute of Science”. In:Journal of Applied Physics30.4 (1959), S70–S78.doi: 10.1063/1.2185971 . eprint: https://doi.org/10.1063/1. 2185971.url:https://doi.org/10.1063/1.2185971

    work page doi:10.1063/1.2185971 1959

  4. [4]

    Fundamentals of Ferromagnetism

    A. Arrott and J. E. Goldman. “Fundamentals of Ferromagnetism”. In:Elec- trical Manufacturing63 (3 Mar. 1959), pp. 109–140.url: https://archive. org/details/sim_electro-technology-newsletter_1959-03_63_3

    work page 1959

  5. [5]

    On the field Lagrangians in micromagnetics

    P. Asselin and A. Thiele. “On the field Lagrangians in micromagnetics”. In: IEEE Transactions on Magnetics22.6 (1986), pp. 1876–1880.doi: 10.1109/ TMAG.1986.1064664.url: https://doi.org/10.1109/TMAG.1986.1064664

    work page doi:10.1109/tmag.1986.1064664 1986

  6. [6]

    Approximation of Boundary Element Matrices

    M. Bebendorf. “Approximation of Boundary Element Matrices”. In:Numer. Math.86 (2000), pp. 565–589. 24 REFERENCES

    work page 2000

  7. [7]

    Adaptive Low-Rank Approximation of Collocation Matrices

    M. Bebendorf and S. Rjasanow. “Adaptive Low-Rank Approximation of Collocation Matrices”. In:Computing70 (2003), pp. 1–24

    work page 2003

  8. [8]

    Asymp- totically exact formulas for the stripe domain period in ultrathin ferromagnetic films with out-of-plane anisotropy

    Anne Bernand-Mantel, Valeriy V. Slastikov, and Cyrill B. Muratov. “Asymp- totically exact formulas for the stripe domain period in ultrathin ferromagnetic films with out-of-plane anisotropy”. In:Phys. Rev. B111 (18 May 2025), p. 184423.doi: 10.1103/PhysRevB.111.184423 .url: https://link.aps. org/doi/10.1103/PhysRevB.111.184423

    work page doi:10.1103/physrevb.111.184423 2025

  9. [9]

    B¨ orm.Efficient Numerical Methods for Non-local Operators

    S. B¨ orm.Efficient Numerical Methods for Non-local Operators. European Mathematical Society, 2010

    work page 2010

  10. [10]

    Borwein et al.Lattice Sums Then and Now

    J. Borwein et al.Lattice Sums Then and Now. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2013.url: https://doi. org/10.1017/CBO9781139626804

    work page doi:10.1017/cbo9781139626804 2013

  11. [11]

    Borwein and Peter B

    Jonathan M. Borwein and Peter B. Borwein.Pi and the AGM: a study in the analytic number theory and computational complexity. Wiley-Interscience, United States, 1987

    work page 1987

  12. [12]

    W. F. Brown.Magnetostatic Principles in Ferromagnetism. Selected Topics in Solid State Physics, Vol. 1. North-Holland, Amsterdam, 1962

    work page 1962

  13. [13]

    W. F. Brown.Micromagnetics. Interscience Tracts on Physics and Astronomy, Vol. 18. Interscience Publishers, Wiley & Sons, 1963

    work page 1963

  14. [14]

    Micromagnetics: Successor to domain theory?

    William Fuller Brown. “Micromagnetics: Successor to domain theory?” In:J. Phys. Radium20 (1959), pp. 101–104.doi: 10.1051/jphysrad:01959002002- 3010100.url: https : / / doi . org / 10 . 1051 / jphysrad : 01959002002 - 3010100

    work page doi:10.1051/jphysrad:01959002002- 1959

  15. [15]

    Bruckner, S

    Florian Bruckner et al. “magnum.np: a PyTorch based GPU enhanced finite difference micromagnetic simulation framework for high level development and inverse design”. In:Scientific Reports13.1 (July 2023), p. 12054.issn: 2045-2322.doi: 10.1038/s41598-023-39192-5 .url: https://doi.org/10. 1038/s41598-023-39192-5

    work page doi:10.1038/s41598-023-39192-5 2023

  16. [16]

    On the computation of lattice sums without translational invariance

    Andreas Buchheit, Torsten Keßler, and Kirill Serkh. “On the computation of lattice sums without translational invariance”. In:Mathematics of Computation 94.355 (2025), pp. 2533–2574.url:https://doi.org/10.1090/mcom/4024

    work page doi:10.1090/mcom/4024 2025

  17. [17]

    Computation and properties of the epstein zeta function with high-performance implementation in epsteinlib

    Andreas A Buchheit, Jonathan Busse, and Ruben Gutendorf. “Computation and properties of the Epstein zeta function with high-performance implemen- tation in EpsteinLib”. In:arXiv preprint arXiv:2412.16317(2024)

    work page arXiv 2024

  18. [18]

    Numerical Bifurcation Analysis of PDEs From Lattice Boltzmann Model Simulations: A Parsimonious Machine Learning Approach

    Andreas A Buchheit and Torsten Keßler. “On the Efficient Computation of Large Scale Singular Sums with Applications to Long-Range Forces in Crystal Lattices”. In:J. Sci. Comput.90.1 (2022), pp. 1–20.doi: 10.1007/s10915- 021-01731-5.url:https://doi.org/10.1007/s10915-021-01731-5

    work page doi:10.1007/s10915- 2022

  19. [19]

    Exact continuum representation of long-range interacting systems and emerging exotic phases in unconventional supercon- ductors

    Andreas A Buchheit et al. “Exact continuum representation of long-range interacting systems and emerging exotic phases in unconventional supercon- ductors”. In:Physical Review Research5.4 (2023), p. 043065.doi: https: //doi.org/10.1103/PhysRevResearch.5.043065

    work page doi:10.1103/physrevresearch.5.043065 2023

  20. [20]

    Computation and properties of the epstein zeta function with high-performance implementation in epsteinlib

    Andreas A. Buchheit, Jonathan Busse, and Ruben Gutendorf.Computation and Properties of the Epstein Zeta Function with High-Performance Implemen- tation in EpsteinLib. Dec. 2024.doi: 10.48550/arXiv.2412.16317 . arXiv: 2412.16317 [math]. (Visited on 03/13/2025). REFERENCES 25

    work page doi:10.48550/arxiv.2412.16317 2024

  21. [21]

    Buchheit and Jonathan K

    Andreas A. Buchheit and Jonathan K. Busse.Epstein Zeta Method for Many- Body Lattice Sums. 2025.doi: 10.48550/ARXIV.2504.11989 . (Visited on 04/28/2025)

    work page doi:10.48550/arxiv.2504.11989 2025

  22. [22]

    A Fast Adaptive Multipole Al- gorithm in Three Dimensions

    H. Cheng, L. Greengard, and V. Rokhlin. “A Fast Adaptive Multipole Al- gorithm in Three Dimensions”. In:J. Comput. Phys.155 (1999), pp. 468– 498

    work page 1999

  23. [23]

    D. J. Craik and R. S. Tebble.Ferromagnetism and Ferromagnetic Domains. Selected Topics in Solid State Physics, Vol. 4. North-Holland, Amsterdam, 1965

    work page 1965

  24. [24]

    Unified algorithms for polylogarithm, L-series, and zeta variants

    R. Crandall. “Unified algorithms for polylogarithm, L-series, and zeta variants”. In:Algorithmic Reflections: Selected Works. PSIpress, 2012

    work page 2012

  25. [25]

    Recent Analytical Developments in Micromagnetics

    Antonio DeSimone et al. “Recent Analytical Developments in Micromagnetics”. In:The Science of Hysteresis. Ed. by Giorgio Bertotti and Isaak D. Mayergoyz. Oxford: Academic Press, 2006, pp. 269–381.isbn: 978-0-12-480874-4.doi: 10.1016/B978-012480874-4/50015-4 .url: https://www.sciencedirect. com/science/article/pii/B9780124808744500154

    work page doi:10.1016/b978-012480874-4/50015-4 2006

  26. [26]

    Variational Principles of Micromagnetics Revisited

    Giovanni Di Fratta et al. “Variational Principles of Micromagnetics Revisited”. In:SIAM Journal on Mathematical Analysis52.4 (2020), pp. 3580–3599.doi: 10.1137/19M1261365.url:https://doi.org/10.1137/19M1261365

    work page doi:10.1137/19m1261365.url:https://doi.org/10.1137/19m1261365 2020

  27. [27]

    M. J. Donahue and D. Porter.OOMMF Software. http://math.nist.gov/ oommf/

    work page

  28. [28]

    Near-Optimal Perfectly Matched Layers for Indefinite Helmholtz Problems

    Vladimir Druskin, Stefan G¨ uttel, and Leonid Knizhnerman. “Near-Optimal Perfectly Matched Layers for Indefinite Helmholtz Problems”. In:SIAM Review58.1 (2016), pp. 90–116.doi: 10 . 1137 / 140966927.url: https : //doi.org/10.1137/140966927

    work page doi:10.1137/140966927 2016

  29. [29]

    Zur Theorie allgemeiner Zetafunctionen

    P. Epstein. “Zur Theorie allgemeiner Zetafunctionen”. In:Math. Ann.56 (1903), pp. 615–644.url:https://doi.org/10.1007/BF01444309

    work page doi:10.1007/bf01444309 1903

  30. [30]

    Zur Theorie allgemeiner Zetafunktionen. II

    P. Epstein. “Zur Theorie allgemeiner Zetafunktionen. II”. In:Math. Ann.63 (1906), pp. 205–216.url:https://doi.org/10.1007/BF01449900

    work page doi:10.1007/bf01449900 1906

  31. [31]

    Erdeley, ed.Higher Transcendental Functions, Volume 2

    A. Erdeley, ed.Higher Transcendental Functions, Volume 2. McGraw-Hill, 1953

    work page 1953

  32. [32]

    Oxford University Press, Feb

    Olle Eriksson et al.Atomistic Spin Dynamics: Foundations and Applications. Oxford University Press, Feb. 2017.isbn: 9780198788669.doi: 10 . 1093 / oso / 9780198788669 . 001 . 0001.url: https : / / doi . org / 10 . 1093 / oso / 9780198788669.001.0001

    work page 2017

  33. [33]

    Hertz vector potentials of electromagnetic theory

    E. A. Essex. “Hertz vector potentials of electromagnetic theory”. In:American Journal of Physics45.11 (Nov. 1977), pp. 1099–1101.issn: 0002-9505.doi: 10.1119/1.10955.url:https://doi.org/10.1119/1.10955

    work page doi:10.1119/1.10955.url:https://doi.org/10.1119/1.10955 1977

  34. [34]

    Die Berechnung optischer und elektrostatischer Gitterpoten- tiale

    P. P. Ewald. “Die Berechnung optischer und elektrostatischer Gitterpoten- tiale”. In:Ann. d. Physik369 (3 1921), pp. 253–287.doi: 10.1002/andp. 19213690304

    work page doi:10.1002/andp 1921

  35. [35]

    Preconditioned nonlinear conjugate gradient method for micromagnetic energy minimization

    Lukas Exl et al. “Preconditioned nonlinear conjugate gradient method for micromagnetic energy minimization”. In:Computer Physics Communications 235 (2019), pp. 179–186.issn: 0010-4655.doi: https://doi.org/10.1016/ j.cpc.2018.09.004 .url: https://www.sciencedirect.com/science/ article/pii/S0010465518303187. 26 REFERENCES

    work page 2019

  36. [36]

    Vision for unified micromagnetic modeling (UMM) with Ubermag

    Hans Fangohr et al. “Vision for unified micromagnetic modeling (UMM) with Ubermag”. In:AIP Advances14.1 (Jan. 2024), p. 015138.issn: 2158-3226. doi:10.1063/9.0000661.url:https://doi.org/10.1063/9.0000661

    work page doi:10.1063/9.0000661.url:https://doi.org/10.1063/9.0000661 2024

  37. [37]

    I. M. Gel’fand and G. E. Shilov.Generalized Functions. Volume I. Properties and Operations. Academic Press, 1964

    work page 1964

  38. [38]

    Phase Transitions Induced by Nanoconfinement in Liquid Water

    Nicolas Giovambattista, Peter J. Rossky, and Pablo G. Debenedetti. “Phase Transitions Induced by Nanoconfinement in Liquid Water”. In:Phys. Rev. Lett.102 (5 Feb. 2009), p. 050603.doi: 10.1103/PhysRevLett.102.050603. url:https://link.aps.org/doi/10.1103/PhysRevLett.102.050603

    work page doi:10.1103/physrevlett.102.050603 2009

  39. [39]

    Computing Binomial Coefficients

    P. Goetgheluck. “Computing Binomial Coefficients”. In:The American Math- ematical Monthly94.4 (Apr. 1987), pp. 360–365.issn: 0002-9890, 1930-0972. doi:10.1080/00029890.1987.12000648. (Visited on 05/12/2025)

    work page doi:10.1080/00029890.1987.12000648 1987

  40. [40]

    A new version of the Fast Multipole Method for the Laplace equation in three dimensions

    L. Greengard and V. Rokhlin. “A new version of the Fast Multipole Method for the Laplace equation in three dimensions”. In:Acta Numer.6 (1997), pp. 229–269

    work page 1997

  41. [41]

    Incomplete Bessel, Generalized Incomplete Gamma, or Leaky Aquifer Functions

    Frank E. Harris. “Incomplete Bessel, Generalized Incomplete Gamma, or Leaky Aquifer Functions”. In:Journal of Computational and Applied Mathematics 215.1 (May 2008), pp. 260–269.issn: 03770427.doi: 10.1016/j.cam.2007. 04.008. (Visited on 05/09/2025)

    work page doi:10.1016/j.cam.2007 2008

  42. [42]

    Guided Spin Waves

    Riccardo Hertel. “Guided Spin Waves”. In:Handbook of Magnetism and Ad- vanced Magnetic Materials. John Wiley & Sons, Ltd, 2007.isbn: 9780470022184. doi: https : / / doi . org / 10 . 1002 / 9780470022184 . hmm212.url: https : //onlinelibrary.wiley.com/doi/abs/10.1002/9780470022184.hmm212

    work page doi:10.1002/9780470022184.hmm212 2007

  43. [43]

    Springer Monographs in Mathematics

    Alex Hubert and Rudolf Sch¨ afer.Magnetic Domains: The Analysis of Magnetic Microstructures. Springer, Berlin, Heidelberg, 1998.doi: 10.1007/978- 3- 540-85054-0.url:https://doi.org/10.1007/978-3-540-85054-0

    work page doi:10.1007/978- 1998

  44. [44]

    Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

    David Ingerman, Vladimir Druskin, and Leonid Knizhnerman. “Optimal finite difference grids and rational approximations of the square root I. Elliptic problems”. In:Communications on Pure and Applied Mathematics53.8 (2000), pp. 1039–1066.doi: 10.1002/1097-0312(200008)53:8<1039::AID-CPA4> 3.0.CO;2- I .url: https://doi.org/10.1002/1097- 0312(200008)53: 8%3C1...

    work page doi:10.1002/1097-0312(200008)53:8 2000

  45. [45]

    The role of magnetic dipolar interactions in skyrmion lattices

    Elizabeth M. Jefremovas et al. “The role of magnetic dipolar interactions in skyrmion lattices”. In:Newton1.2 (2025), p. 100036.issn: 2950-6360. doi: https : / / doi . org / 10 . 1016 / j . newton . 2025 . 100036.url: https : //www.sciencedirect.com/science/article/pii/S2950636025000283

    work page 2025

  46. [46]

    Ferromagnetic Domain Theory

    C. Kittel and J. K. Galt. “Ferromagnetic Domain Theory”. In: ed. by Fred- erick Seitz and David Turnbull. Vol. 3. Solid State Physics. Academic Press, 1956, pp. 437–564.doi: https : / / doi . org / 10 . 1016 / S0081 - 1947(08 ) 60136-8.url: https://www.sciencedirect.com/science/article/pii/ S0081194708601368

    work page 1956

  47. [47]

    Theory of the dispersion of magnetic permeability in ferromagnetic bodies

    L. D. Landau and E. M. Lifshitz. “Theory of the dispersion of magnetic permeability in ferromagnetic bodies”. In:Phys. Z. Sowietunion8 (1935), pp. 153–169

    work page 1935

  48. [48]

    Periodic boundary con- ditions for demagnetization interactions in micromagnetic simulations

    K M Lebecki, M J Donahue, and M W Gutowski. “Periodic boundary con- ditions for demagnetization interactions in micromagnetic simulations”. In: Journal of Physics D: Applied Physics41.17 (Aug. 2008), p. 175005.doi: REFERENCES 27 10.1088/0022-3727/41/17/175005 .url: https://dx.doi.org/10.1088/ 0022-3727/41/17/175005

    work page doi:10.1088/0022-3727/41/17/175005 2008

  49. [49]

    Boris computational spintronics—High performance multi- mesh magnetic and spin transport modeling software

    Serban Lepadatu. “Boris computational spintronics—High performance multi- mesh magnetic and spin transport modeling software”. In:Journal of Applied Physics128.24 (Dec. 2020), p. 243902.issn: 0021-8979.doi: 10 . 1063 / 5 . 0024382. eprint: https://pubs.aip.org/aip/jap/article-pdf/doi/10. 1063/5.0024382/15257104/243902_1_online.pdf .url: https://doi. org/1...

    work page doi:10.1063/5.0024382 2020

  50. [50]

    Topological foundations of ferroelectricity

    Igor A. Lukyanchuk et al. “Topological foundations of ferroelectricity”. In: Physics Reports1110 (2025). Topological foundations of ferroelectricity, pp. 1– 56.issn: 0370-1573.doi: https://doi.org/10.1016/j.physrep.2025. 01.002.url: https://www.sciencedirect.com/science/article/pii/ S0370157325000225

    work page doi:10.1016/j.physrep.2025 2025

  51. [51]

    W. D. MacMillan.The theory of the potential. McGraw-Hill, New York, 1930. url:https://catalog.hathitrust.org/Record/000584021

    work page arXiv 1930

  52. [52]

    A. P. Malozemoff and J. C. Slonczewski.Magnetic Domain Walls in Bubble Materials. Academic Press, New York, 1979.isbn: 0120029510

    work page 1979

  53. [53]

    Discretization of Dirac delta functions in level set methods.Journal of Computational Physics, 207(1):28–51, July 2005.doi:10.1016/j.jcp

    C.B. Muratov and V.V. Osipov. “Optimal grid-based methods for thin film micromagnetics simulations”. In:Journal of Computational Physics216.2 (2006), pp. 637–653.issn: 0021-9991.doi: https://doi.org/10.1016/j.jcp. 2005.12.018.url: https://www.sciencedirect.com/science/article/ pii/S0021999106000040

    work page doi:10.1016/j.jcp 2006

  54. [54]

    Domain-wall engineering and topological defects in fer- roelectric and ferroelastic materials

    G. F. Nataf et al. “Domain-wall engineering and topological defects in fer- roelectric and ferroelastic materials”. In:Nature Reviews Physics2.11 (Nov. 2020), pp. 634–648.issn: 2522-5820.doi: 10.1038/s42254-020-0235-z .url: https://doi.org/10.1038/s42254-020-0235-z

    work page doi:10.1038/s42254-020-0235-z 2020

  55. [55]

    A generalization of the demagnetizing tensor for nonuniform magnetization

    Andrew J. Newell, Wyn Williams, and David J. Dunlop. “A generalization of the demagnetizing tensor for nonuniform magnetization”. In:Journal of Geophysical Research: Solid Earth98.B6 (1993), pp. 9551–9555.doi: https: //doi.org/10.1029/93JB00694 .url: https://agupubs.onlinelibrary. wiley.com/doi/abs/10.1029/93JB00694

    work page doi:10.1029/93jb00694 1993

  56. [56]

    Calculation of Demagnetizing Field Distribution Based on Fast Fourier Transform of Convolution

    Nobuo Hayashi Nobuo Hayashi, Koji Saito Koji Saito, and Yoshinobu Nakatani Yoshinobu Nakatani. “Calculation of Demagnetizing Field Distribution Based on Fast Fourier Transform of Convolution”. In:Japanese Journal of Applied Physics35.12R (Dec. 1996), p. 6065.doi: 10 . 1143 / JJAP . 35 . 6065.url: https://dx.doi.org/10.1143/JJAP.35.6065

    work page doi:10.1143/jjap.35.6065 1996

  57. [57]

    Collective antiskyrmion-mediated phase transition and defect-induced melting in chiral magnetic films

    L. Pierobon et al. “Collective antiskyrmion-mediated phase transition and defect-induced melting in chiral magnetic films”. In:Scientific Reports8.1 (Nov. 2018), p. 16675.issn: 2045-2322.doi: 10.1038/s41598-018-34526-0 . url:https://doi.org/10.1038/s41598-018-34526-0

    work page doi:10.1038/s41598-018-34526-0 2018

  58. [58]

    Demagnetising Energies of Uniformly Magnetised Rectangular Blocks

    P. Rhodes and G. Rowlands. “Demagnetising Energies of Uniformly Magnetised Rectangular Blocks”. In:Proc. Leeds Phil. Liter. Soc6 (1954), pp. 191–210

    work page 1954

  59. [59]

    Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod–Teller–Muto term applied to cuboidal phase transitions

    Andres Robles-Navarro et al. “Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod–Teller–Muto term applied to cuboidal phase transitions”. In:The Journal of Chemical Physics163.9 (Sept. 2025), p. 094104.issn: 0021-9606.doi: 10.1063/5.0276677. eprint: https: / / pubs . aip . org / aip / jcp / article - pdf / doi / 10 . 106...

    work page doi:10.1063/5.0276677 2025

  60. [60]

    Magnetostatic interaction fields for a three- dimensional array of ferromagnetic cubes

    M. Schabes and A. Aharoni. “Magnetostatic interaction fields for a three- dimensional array of ferromagnetic cubes”. In:IEEE Transactions on Mag- netics23.6 (1987), pp. 3882–3888.doi:10.1109/TMAG.1987.1065775

    work page doi:10.1109/tmag.1987.1065775 1987

  61. [61]

    Numerical Methods in Micromagnetics (Finite Element Method)

    Thomas Schrefl et al. “Numerical Methods in Micromagnetics (Finite Element Method)”. In:Handbook of Magnetism and Advanced Magnetic Materials. John Wiley & Sons, Ltd, 2007.isbn: 9780470022184.doi: https://doi.org/ 10 . 1002 / 9780470022184 . hmm203.url: https : / / onlinelibrary . wiley . com/doi/abs/10.1002/9780470022184.hmm203

    work page doi:10.1002/9780470022184.hmm203 2007

  62. [62]

    E., & Bacon, D

    K. Shibata et al. “Temperature and Magnetic Field Dependence of the Internal and Lattice Structures of Skyrmions by Off-Axis Electron Holography”. In: Phys. Rev. Lett.118 (8 Feb. 2017), p. 087202.doi: 10.1103/PhysRevLett. 118.087202.url: https://link.aps.org/doi/10.1103/PhysRevLett.118. 087202

    work page doi:10.1103/physrevlett 2017

  63. [63]

    A Recursive Algorithm for an Efficient and Accurate Computation of Incomplete Bessel Functions

    Richard M. Slevinsky and Hassan Safouhi. “A Recursive Algorithm for an Efficient and Accurate Computation of Incomplete Bessel Functions”. In: Numerical Algorithms92.1 (Jan. 2023), pp. 973–983.issn: 1017-1398, 1572- 9265.doi:10.1007/s11075-022-01438-0. (Visited on 05/07/2025)

    work page doi:10.1007/s11075-022-01438-0 2023

  64. [64]

    K. H. Stewart.Ferromagnetic Domains. Cambridge University Press, 1954

    work page 1954

  65. [65]

    Benjamin, Suguru Endo, William J

    Attila Szilva et al. “Quantitative theory of magnetic interactions in solids”. In:Rev. Mod. Phys.95 (3 Sept. 2023), p. 035004.doi: 10.1103/RevModPhys. 95.035004.url: https://link.aps.org/doi/10.1103/RevModPhys.95. 035004

    work page doi:10.1103/revmodphys 2023

  66. [66]

    Investigating skyrmion stability and core polarity reversal in NdMn2Ge2

    Samuel K. Treves et al. “Investigating skyrmion stability and core polarity reversal in NdMn2Ge2”. In:Scientific Reports15.1 (Jan. 2025), p. 461.issn: 2045-2322.doi: 10.1038/s41598-024-82114-2 .url: https://doi.org/10. 1038/s41598-024-82114-2

    work page doi:10.1038/s41598-024-82114-2 2025

  67. [67]

    The design and verification of MuMax3

    Arne Vansteenkiste et al. “The design and verification of MuMax3”. In:AIP Advances4.10 (Oct. 2014), p. 107133.issn: 2158-3226.doi: 10 . 1063 / 1 . 4899186

    work page 2014

  68. [68]

    The Newtonian potential of a homogeneous cube

    J¨ org Waldvogel. “The Newtonian potential of a homogeneous cube”. In: Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP27.6 (Nov. 1976), pp. 867–871.issn: 1420-9039.doi: 10.1007/BF01595137.url: https://doi. org/10.1007/BF01595137

    work page doi:10.1007/bf01595137.url: 1976

  69. [69]

    Observation of unprecedented fractional magnetization plateaus in a new Shastry-Sutherland Ising compound

    Lalit Yadav et al. “Observation of unprecedented fractional magnetization plateaus in a new Shastry-Sutherland Ising compound”. In:arXiv preprint arXiv:2405.12405(2024)

    work page arXiv 2024

  70. [70]

    Fast adaptive algorithms for micromagnetics

    S.W. Yuan and H.N. Bertram. “Fast adaptive algorithms for micromagnetics”. In:IEEE Transactions on Magnetics28.5 (1992), pp. 2031–2036.doi: 10. 1109/20.179394

    work page 1992

  71. [71]

    The Exact Evaluation of Some New Lattice Sums

    I.J. Zucker. “The Exact Evaluation of Some New Lattice Sums”. In:Symmetry 9.12 (2017), p. 314.url:https://doi.org/10.3390/sym9120314. REFERENCES 29 Department of Mathematics, Saarland University, 66123 Saarbr ¨ucken, Germany Department of Mathematics, Saarland University, 66123 Saarbr ¨ucken, Germany, German Aerospace Center (DLR), 51147 Cologne, Germany ...

    work page doi:10.3390/sym9120314 2017