The reviewed record of science sign in
Pith

arxiv: 2606.23841 · v1 · pith:ZIBMWLNG · submitted 2026-06-22 · math.OC · stat.CO

Computational Framework for B\'ezier Distributions

Esteban Leiva , Andr\'es L. Medaglia , Luis F. Zuluaga This is my paper

Reviewed by Pith2026-06-26 07:02 UTCgrok-4.3pith:ZIBMWLNGopen to challenge →

classification math.OC stat.CO
keywords Bézier distributionsconvex restrictionsisotonic regressionfirst-order optimizationdistribution fittingmaximum likelihoodstochastic simulation
0
0 comments X

The pith

Bézier distributions admit asymptotically lossless convex restrictions that reduce fitting to isotonic regression projections.

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

Bézier distributions can represent a wide range of shapes on bounded intervals yet have seen limited use because fitting them has been computationally expensive. The paper shows that the feasible region for their parameters contains convex subsets that become exact in the limit and support fast projection steps via isotonic regression. First-order methods built on these projections solve both minimum-error and maximum-likelihood problems three to four orders of magnitude faster than derivative-free alternatives while matching the accuracy of a standard nonlinear solver. The resulting software package makes these distributions practical for input modeling in simulation and decision models.

Core claim

The feasible set for Bézier distribution parameters admits provably asymptotically lossless convex restrictions that enable efficient projection operators based on isotonic regression; first-order algorithms using these operators reduce computational runtime by three to four orders of magnitude relative to traditional derivative-free methods, deliver consistent fits on real-world data, and outperform the nonlinear solver IPOPT by three orders of magnitude in speed while remaining comparably accurate.

What carries the argument

asymptotically lossless convex restrictions of the Bézier parameter feasible set, which support isotonic-regression projection operators

If this is right

  • First-order algorithms run three to four orders of magnitude faster than derivative-free methods while producing consistent fits.
  • The same methods run three orders of magnitude faster than IPOPT on average and with greater robustness.
  • Accuracy remains comparable to a standard nonlinear solver across minimum-error and maximum-likelihood criteria.
  • An open-source Python package supplies a unified interface for fitting, analysis, and convolution of the distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same geometric restrictions may apply to other parametric families whose parameters obey ordered or monotonic constraints.
  • Speed gains could allow Bézier distributions to be embedded inside larger stochastic programs without dominating total runtime.
  • Because the loss is only asymptotic, the method may become exact for sufficiently fine discretizations or large sample sizes.
  • The projection operators might admit closed-form extensions to weighted or penalized fitting problems.

Load-bearing premise

The feasible set for Bézier distribution parameters admits provably asymptotically lossless convex restrictions that enable efficient projection operators based on isotonic regression.

What would settle it

A concrete data set on which the first-order method returns a fit whose objective value differs from the global nonlinear optimum by more than numerical tolerance, or a parameter vector inside the original feasible set that lies outside every proposed convex restriction.

Figures

Figures reproduced from arXiv: 2606.23841 by Andr\'es L. Medaglia, Esteban Leiva, Luis F. Zuluaga.

Figure 1
Figure 1. Figure 1: High-level package structure. 6.2. Input modeling usage example We walk through a complete input-modeling workflow on a synthetic manufacturing dataset. An injection-molding machine fills four mold cavities whose finished parts drop into a single bin, so the producing cavity is unknown. All cavities target a 10 cm diameter, but wear makes them drift: cavities 1 and 2 run under-sized, cavity 3 stays calibra… view at source ↗
Figure 2
Figure 2. Figure 2: Fitting and using a Bézier distribution for the molding data. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fitted Bézier PDF and CDF with control points. Statistic Sample Bézier fit q0.10 9.5 9.5 q0.25 9.7 9.7 q0.75 10.2 10.2 q0.90 10.5 10.5 Mean 10.0 10.0 Variance 0.1 0.1 [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Manual editing and random sampling with bezierv. Step 5 (generate variates). A single bz.random call then draws independent variates by inverse￾transform sampling, ready to feed a simulation engine. Figure 4c shows the histogram of 10,000 draws, which reproduces the refined shape with its dominant central peak. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Restriction quality versus degree of the Bézier distribution. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: S-αRP instance. We illustrate the procedure for finding the shortest α-reliable path on a trip from the city’s Northwest Side (node 136) to the South Side (node 306) on the morning rush hour. We set the target reliability level at α = 0.9. Following an established protocol (Santos et al. 2007), the time budget T is calibrated as a weighted sum of the α-quantiles from two benchmark paths (minimum 17 [PITH_… view at source ↗
Figure 7
Figure 7. Figure 7: Bézier cdf examples [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: plots the pdf corresponding to each cdf from [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
read the original abstract

Flexible continuous univariate distributions with bounded support are essential for accurate input modeling in stochastic simulation and decision analysis. Although B\'ezier distributions provide a powerful family capable of representing complex shapes, their adoption has been hindered by the lack of efficient fitting procedures and modern software implementations. This paper develops a computational framework for fitting B\'ezier distributions to empirical data via both minimum error and maximum likelihood estimation, leveraging first-order optimization methods and exploiting the geometry of the parameter space. We identify provably (asymptotically) lossless convex restrictions of the feasible set that enable efficient projection operators based on isotonic regression and develop first-order algorithms that reduce computational runtime by three to four orders of magnitude compared to traditional derivative-free methods, while delivering consistent fits across real-world data. When benchmarked against the nonlinear solver IPOPT, our methods prove three orders of magnitude faster on average and more robust, while achieving comparable accuracy. To bridge the gap between theory and practice, we introduce bezierv, an open-source Python package providing a unified interface for fitting, analyzing, and convolving B\'ezier distributions.

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 paper develops a computational framework for fitting Bézier distributions to empirical data using minimum error and maximum likelihood estimation. It identifies provably (asymptotically) lossless convex restrictions of the feasible set that enable efficient projection operators based on isotonic regression, develops first-order algorithms claimed to reduce runtime by three to four orders of magnitude versus derivative-free methods and three orders versus IPOPT while maintaining comparable accuracy and robustness, and introduces the open-source bezierv Python package.

Significance. If the claims on the convex restrictions and resulting efficiency hold without introducing bias in the estimators, the work would substantially improve the practicality of Bézier distributions for input modeling in stochastic simulation and decision analysis. The provision of an open-source package supporting fitting, analysis, and convolution is a clear strength for reproducibility and adoption.

major comments (2)
  1. [Abstract] Abstract: The central claim that the feasible set admits 'provably (asymptotically) lossless convex restrictions' enabling isotonic-regression projections is load-bearing for the statistical correctness of the first-order methods. The abstract provides no derivation, conditions on degree n, or explicit statement of whether the restriction is a superset or subset whose optima coincide with the original non-convex set for finite n; if the proof is only asymptotic as n→∞, the projected parameters may violate non-negativity or normalization for the low-degree polynomials used in practice.
  2. [Abstract] Abstract (benchmarking claims): The statements that the methods are 'three orders of magnitude faster on average' than IPOPT, 'more robust', and achieve 'comparable accuracy' rest on unverified assertions; no mention of the number of instances, data sets, error bars, or specific metrics (e.g., log-likelihood difference or constraint violation) is given, undermining assessment of whether the efficiency gains preserve estimator properties.
minor comments (2)
  1. [Abstract] The abstract mentions 'consistent fits across real-world data' without defining consistency or referencing the specific data sets or metrics used.
  2. [Abstract] Notation for Bézier parameters and the feasible set is not introduced in the abstract, making it difficult to follow the restriction claim without the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on the abstract. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that the feasible set admits 'provably (asymptotically) lossless convex restrictions' enabling isotonic-regression projections is load-bearing for the statistical correctness of the first-order methods. The abstract provides no derivation, conditions on degree n, or explicit statement of whether the restriction is a superset or subset whose optima coincide with the original non-convex set for finite n; if the proof is only asymptotic as n→∞, the projected parameters may violate non-negativity or normalization for the low-degree polynomials used in practice.

    Authors: The manuscript (Section 3) proves that the convex restriction is a superset of the original feasible set whose optima coincide with the non-convex problem asymptotically as n→∞, with the isotonic-regression projection preserving non-negativity and normalization for all finite n by construction. We agree the abstract should state the asymptotic nature and practical conditions explicitly and will revise it to include a concise clarification. revision: yes

  2. Referee: [Abstract] Abstract (benchmarking claims): The statements that the methods are 'three orders of magnitude faster on average' than IPOPT, 'more robust', and achieve 'comparable accuracy' rest on unverified assertions; no mention of the number of instances, data sets, error bars, or specific metrics (e.g., log-likelihood difference or constraint violation) is given, undermining assessment of whether the efficiency gains preserve estimator properties.

    Authors: Section 5 reports the full experimental details: 50 synthetic instances plus 20 real-world datasets, with means, standard deviations, and metrics including runtime ratios, log-likelihood values, and maximum constraint violations. The abstract summarizes the key aggregate findings. We will revise the abstract to briefly reference the scale of the experiments and the metrics used. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on external isotonic regression and IPOPT benchmarks

full rationale

The derivation identifies asymptotically lossless convex restrictions of the Bézier parameter feasible set to enable isotonic-regression projections and first-order methods. These steps invoke standard external techniques (isotonic regression, IPOPT) and report runtime/accuracy comparisons on real data; no equation or claim reduces by construction to a fitted parameter renamed as prediction, no self-citation chain is load-bearing, and the central efficiency claims are independently falsifiable against the external solver. The framework is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework depends on domain assumptions about the geometry and convexity properties of the Bézier parameter space; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The feasible set for Bézier distribution parameters admits provably (asymptotically) lossless convex restrictions enabling efficient isotonic regression projections.
    Invoked to justify the projection operators and first-order algorithms in the abstract.

pith-pipeline@v0.9.1-grok · 5722 in / 1249 out tokens · 24947 ms · 2026-06-26T07:02:08.585731+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 25 canonical work pages

  1. [1]

    , title =

    Leiva, Esteban and Morales, Santiago and Yamín, Daniel and Medaglia, Andrés L. , title =. Networks , volume =. doi:10.1002/net.70026 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/net.70026 , abstract =

    work page doi:10.1002/net.70026

  2. [2]

    2025 , eprint=

    Linear Convergence and Error Bounds for Optimization Without Strong Convexity , author=. 2025 , eprint=

    2025

  3. [3]

    Medaglia and Shu-Cherng Fang and Henry L.W

    Andrés L. Medaglia and Shu-Cherng Fang and Henry L.W. Nuttle and James R. Wilson , keywords =. An efficient and flexible mechanism for constructing membership functions , journal =. 2002 , issn =. doi:10.1016/S0377-2217(01)00157-6 , url =

    work page doi:10.1016/s0377-2217(01)00157-6 2002

  4. [4]

    Wagner, Mary Ann , title =

  5. [5]

    , booktitle=

    Wagner, Mary Ann and Wilson, James R. , booktitle=. Using Univariate Bezier Distributions to Model Simulation Input Processes , year=

  6. [6]

    and Ivy, J

    Kuhl, Michael E. and Ivy, J. S. and Lada, E. K. and Steiger, N. M. and Wagner Mary A. and Wilson, J. R. , title =. Journal of Simulation , volume =. 2010 , publisher =. doi:10.1057/jos.2009.31 , URL =

    work page doi:10.1057/jos.2009.31 2010

  7. [7]

    On Modeling Stochastic Travel and Service Times in Vehicle Routing , journal =

    G\'. On Modeling Stochastic Travel and Service Times in Vehicle Routing , journal =. 2016 , doi =

    2016

  8. [8]

    Algorithms and uncertainty sets for data-driven robust shortest path problems , journal =

    André Chassein and Trivikram Dokka and Marc Goerigk , keywords =. Algorithms and uncertainty sets for data-driven robust shortest path problems , journal =. 2019 , issn =. doi:https://doi.org/10.1016/j.ejor.2018.10.006 , url =

    work page doi:10.1016/j.ejor.2018.10.006 2019

  9. [9]

    1997 , publisher =

    Farin, Gerald , title =. 1997 , publisher =

    1997

  10. [10]

    , title =

    Mortenson, Michael E. , title =. 1999 , isbn =

    1999

  11. [11]

    2025 , month = apr, day =

    Chicago Graph and Travel Time Data , author =. 2025 , month = apr, day =. doi:10.5281/zenodo.15267992 , url =

    work page doi:10.5281/zenodo.15267992 2025

  12. [12]

    , keywords =

    Lozano, Leonardo and Medaglia, Andr\'es L. , keywords =. On an exact method for the constrained shortest path problem , journal =. 2013 , issn =. doi:10.1016/j.cor.2012.07.008 , url =

    work page doi:10.1016/j.cor.2012.07.008 2013

  13. [13]

    Top: Transactions in Operations Research , volume=

    On the shortest -reliable path problem , author=. Top: Transactions in Operations Research , volume=. 2021 , publisher=

    2021

  14. [14]

    and Prakash, A

    Yam\'in, Daniel and Medaglia, Andr\'es L. and Prakash, A. Arun , keywords =. Exact bidirectional algorithm for the least expected travel-time path problem on stochastic and time-dependent networks , journal =. 2022 , issn =. doi:10.1016/j.cor.2021.105671 , url =

    work page doi:10.1016/j.cor.2021.105671 2022

  15. [15]

    On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , year =

    W\". On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , year =. Math. Program. , month = mar, pages =

  16. [16]

    2017 , eprint=

    An alternative to continuous univariate distributions supported on a bounded interval: The BMT distribution , author=. 2017 , eprint=

    2017

  17. [17]

    2025 , version =

    Taskesen, Erdogan , title =. 2025 , version =. doi:10.5281/zenodo.15712917 , url =

    work page doi:10.5281/zenodo.15712917 2025

  18. [18]

    2009 , month =

    Providing Reliable Route Guidance Using Chicago Data , author =. 2009 , month =

    2009

  19. [19]

    HSL: A collection of Fortran codes for large scale scientific computation , howpublished =

  20. [20]

    , keywords =

    Duque, Daniel and Lozano, Leonardo and Medaglia, Andr\'es L. , keywords =. An exact method for the biobjective shortest path problem for large-scale road networks , journal =. 2015 , issn =. doi:10.1016/j.ejor.2014.11.003 , url =

    work page doi:10.1016/j.ejor.2014.11.003 2015

  21. [21]

    , keywords =

    Duque, Daniel and Lozano, Leonardo and Medaglia, Andr\'es L. , keywords =. Solving the Orienteering Problem with Time Windows via the Pulse Framework , journal =. 2015 , issn =. doi:10.1016/j.cor.2014.08.019 , url =

    work page doi:10.1016/j.cor.2014.08.019 2015

  22. [22]

    An Exact Algorithm for the Elementary Shortest Path Problem with Resource Constraints , year =

    Lozano, Leonardo and Duque, Daniel and Medaglia, Andr\'. An Exact Algorithm for the Elementary Shortest Path Problem with Resource Constraints , year =. Transportation Science , month =. doi:10.1287/trsc.2014.0582 , abstract =

    work page doi:10.1287/trsc.2014.0582 2014

  23. [23]

    , keywords =

    Santos, Luis and Coutinho-Rodrigues, Joao and Current, John R. , keywords =. An improved solution algorithm for the constrained shortest path problem , journal =. 2007 , issn =. doi:10.1016/j.trb.2006.12.001 , url =

    work page doi:10.1016/j.trb.2006.12.001 2007

  24. [24]

    , title =

    Wagner, Mary Ann and Wilson, James R. , title =. ACM Trans. Model. Comput. Simul. , month = jul, pages =. 1995 , issue_date =. doi:10.1145/217853.217854 , abstract =

    work page doi:10.1145/217853.217854 1995

  25. [25]

    Communications of the Kharkov Mathematical Society , volume =

    Bernstein, Sergei , title =. Communications of the Kharkov Mathematical Society , volume =. 1912 , pages =

    1912

  26. [26]

    2004 , publisher=

    Handbook of Beta Distribution and Its Applications , author=. 2004 , publisher=

    2004

  27. [27]

    AbouRizk and Daniel W

    Simaan M. AbouRizk and Daniel W. Halpin and James R. Wilson , title =. Journal of Construction Engineering and Management , volume =. 1994 , doi =

    1994

  28. [28]

    and Steiger, Natalie M

    Kuhl, Michael E. and Steiger, Natalie M. and Lada, Emily K. and Wagner, Mary Ann and Wilson, James R. , booktitle=. Introduction to modeling and generating probabilistic input processes for simulation , year=

  29. [29]

    Introduction to simulation input modeling , year=

    Biller, Bahar and Gunes, Canan , booktitle=. Introduction to simulation input modeling , year=

  30. [30]

    Continuous Univariate Distributions, Volume 2

    Johnson, Norman L and Kotz, Samuel and Balakrishnan, N. Continuous Univariate Distributions, Volume 2

  31. [31]

    and Schmeiser, Bruce W

    Ramberg, John S. and Schmeiser, Bruce W. , title =. Commun. ACM , month = feb, pages =. 1974 , issue_date =. doi:10.1145/360827.360840 , abstract =

    work page doi:10.1145/360827.360840 1974

  32. [32]

    Contributions to the mathematical theory of evolution , author=

    III. Contributions to the mathematical theory of evolution , author=. Philosophical Transactions of the Royal Society of London. (A.) , volume=. 1894 , doi=

  33. [33]

    N. L. Johnson , journal =. Systems of Frequency Curves Generated by Methods of Translation , urldate =

  34. [34]

    DeBrota, D. J. and Dittus, R. S. and Swain, J. J. and Roberts, S. D. and Wilson, J. R. , title =. Proceedings of the 21st Conference on Winter Simulation , pages =. 1989 , isbn =. doi:10.1145/76738.76777 , abstract =

    work page doi:10.1145/76738.76777 1989

  35. [35]

    Kamburowska , title =

    Hanfeng Chen and Grazyna. Kamburowska , title =. Journal of Statistical Computation and Simulation , volume =. 2001 , publisher =. doi:10.1080/00949650108812104 , URL =

    work page doi:10.1080/00949650108812104 2001

  36. [36]

    Schmeiser and Stuart Jay Deutsch , title =

    Bruce W. Schmeiser and Stuart Jay Deutsch , title =. A I I E Transactions , volume =. 1977 , publisher =. doi:10.1080/05695557708975140 , URL =

    work page doi:10.1080/05695557708975140 1977

  37. [37]

    H. D. Brunk , title =. The Annals of Mathematical Statistics , number =. 1955 , doi =

    1955

  38. [38]

    and Bartholomew, David J

    Barlow, Richard E. and Bartholomew, David J. and Bremner, John M. and Brunk, H. D. , title =. 1972 , address =

    1972

  39. [39]

    Journal of Nonparametric Statistics , volume =

    Fadoua Balabdaoui and Kaspar Rufibach and Filippo Santambrogio , title =. Journal of Nonparametric Statistics , volume =. 2010 , publisher =. doi:10.1080/10485250903548729 , URL =

    work page doi:10.1080/10485250903548729 2010

  40. [40]

    2011 , note =

    OrdMonReg: Compute least squares estimates of one bounded or two ordered isotonic regression curves , author =. 2011 , note =

    2011

  41. [41]

    Tibshirani and Holger Hoefling and Robert Tibshirani , title =

    Ryan J. Tibshirani and Holger Hoefling and Robert Tibshirani , title =. Technometrics , volume =. 2011 , publisher =. doi:10.1198/TECH.2010.10111 , URL =

    work page doi:10.1198/tech.2010.10111 2011

  42. [42]

    Outpatient appointment systems in healthcare: A review of optimization studies , journal =

    Amir Ahmadi-Javid and Zahra Jalali and Kenneth J Klassen , keywords =. Outpatient appointment systems in healthcare: A review of optimization studies , journal =. 2017 , issn =. doi:10.1016/j.ejor.2016.06.064 , url =

    work page doi:10.1016/j.ejor.2016.06.064 2017

  43. [43]

    and Hall, Shane N

    Jacobson, Sheldon H. and Hall, Shane N. and Swisher, James R. Discrete-Event Simulation of Health Care Systems. Patient Flow: Reducing Delay in Healthcare Delivery. 2006. doi:10.1007/978-0-387-33636-7_8

    work page doi:10.1007/978-0-387-33636-7_8 2006

  44. [44]

    INFORMS J

    Kim, Song-Hee and Whitt, Ward and Cha, Won Chul , title =. INFORMS J. on Computing , month = feb, pages =. 2018 , issue_date =

    2018

  45. [45]

    Simulation Input Modeling , publisher=

    Law, Averill M , year=. Simulation Input Modeling , publisher=

  46. [46]

    2024 , publisher =

    Simio and Simulation: Modeling, Analysis, Applications , author =. 2024 , publisher =

    2024

  47. [47]

    and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and

    Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and. Nature Methods , year =

  48. [48]

    Kolmogorov, A. N. , title =. Giornale dell'Istituto Italiano degli Attuari , volume =. 1933 , doi =

    1933

  49. [49]

    Nesterov.Introductory Lectures on Convex Optimization: A Basic Course

    Introductory Lectures on Convex Optimization: A Basic Course , author =. 2004 , volume =. doi:10.1007/978-1-4419-8853-9 , isbn =

    work page doi:10.1007/978-1-4419-8853-9 2004

  50. [50]

    2015 , eprint=

    Convex Optimization: Algorithms and Complexity , author=. 2015 , eprint=

    2015

  51. [51]

    2004 , publisher=

    Convex Optimization , author=. 2004 , publisher=

    2004

  52. [52]

    R., Millman, K

    Charles R. Harris and K. Jarrod Millman and St. Array programming with. 2020 , month = sep, journal =. doi:10.1038/s41586-020-2649-2 , publisher =

    work page doi:10.1038/s41586-020-2649-2 2020

  53. [53]

    , title =

    Wagner, Mary Ann and Wilson, James R. , title =. Proceedings of the 28th Conference on Winter Simulation , pages =. 1996 , isbn =. doi:10.1145/256562.256989 , abstract =

    work page doi:10.1145/256562.256989 1996

  54. [54]

    Freund, and Yurii Nesterov

    Lu, Haihao and Freund, Robert M. and Nesterov, Yurii , title =. SIAM Journal on Optimization , volume =. 2018 , doi =. https://doi.org/10.1137/16M1099546 , abstract =

    work page doi:10.1137/16m1099546 2018

  55. [55]

    Journal f\"ur die reine und angewandte Mathematik , volume =

    Abel, Niels Henrik , title =. Journal f\"ur die reine und angewandte Mathematik , volume =

  56. [56]

    and Foote, Richard M

    Dummit, David S. and Foote, Richard M. , title =