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arxiv: 2604.20149 · v1 · submitted 2026-04-22 · 🪐 quant-ph

Average metric adjusted skew information of coherence under conical 2-designs generalized equiangular measurements

Baolong Cheng , Linlin Ye , Zhaoqi Wu This is my paper

Pith reviewed 2026-05-10 00:52 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum coherencemetric adjusted skew informationconical 2-designsgeneralized equiangular measurementsentanglement criteriatrade-off relationsquantum uncertainty
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The pith

A new uncertainty measure based on metric adjusted skew information equals scaled average coherence for unitary groups, operator orthonormal bases, and mutually unbiased bases under conical 2-designs generalized equiangular measurements.

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

The paper introduces a measure of quantum uncertainty drawn from metric adjusted skew information to quantify average coherence when systems are probed by conical 2-designs generalized equiangular measurements. It establishes that this measure matches the scaled average coherence obtained from the same skew-information approach when the underlying structures are unitary groups, operator orthonormal bases, or mutually unbiased bases. The authors then obtain two trade-off relations from the measure, resolve an existing conjecture, and construct two entanglement criteria, one tied directly to the measure and one to the measurements themselves, with explicit examples showing how the criteria function. A reader would care because coherence and entanglement are core resources for quantum information tasks, and new equivalent quantifiers plus detection methods can simplify analysis and verification in those settings.

Core claim

Based on metric adjusted skew information, we define a measure of quantum uncertainty to study average coherence under conical 2-designs generalized equiangular measurements, and prove the equivalence of this measure to the scaled average coherence based on metric adjusted skew information under a set of unitary groups, operator orthonormal bases, and mutually unbiased bases. We also derive two trade-off relations by this measure and solve a conjecture. Furthermore, we give two entanglement criteria by this measure and conical 2-designs generalized equiangular measurement, respectively.

What carries the argument

The uncertainty measure constructed from metric adjusted skew information, which quantifies average coherence for conical 2-designs generalized equiangular measurements and establishes direct equivalences to scaled coherence on other bases.

If this is right

  • The measure is equivalent to scaled average coherence under unitary groups, operator orthonormal bases, and mutually unbiased bases.
  • Two trade-off relations follow directly from the measure.
  • An existing conjecture on coherence is resolved by the equivalence.
  • Two entanglement criteria are obtained, one from the measure and one from the measurements, each illustrated by explicit examples.

Where Pith is reading between the lines

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

  • The equivalence could reduce the computational cost of evaluating coherence when conical 2-designs measurements are already available in an experiment.
  • The entanglement criteria might be combined with existing witness techniques to improve detection thresholds in noisy quantum devices.
  • Trade-off relations derived here may extend to other resource theories such as quantum discord or nonlocality by substituting different skew-information functionals.

Load-bearing premise

The newly defined measure based on metric adjusted skew information is a valid and faithful quantifier of quantum coherence or uncertainty, and the conical 2-designs generalized equiangular measurements satisfy the mathematical properties needed for the equivalences and criteria.

What would settle it

A concrete counterexample would be a quantum state for which the new uncertainty measure yields a value different from the scaled average coherence under unitary groups, or a separable state that the proposed entanglement criteria incorrectly flag as entangled.

read the original abstract

Quantum coherence is an important quantum resource which plays a pivotal role in the field of quantum information. Based on metric adjusted skew information, we define a measure of quantum uncertainty to study average coherence under conical 2-designs generalized equiangular measurements, and prove the equivalence of this measure to the scaled average coherence based on metric adjusted skew information under a set of unitary groups, operator orthonormal bases, and mutually unbiased bases. We also derive two trade-off relations by this measure and solve a conjecture. Furthermore, we give two entanglement criteria by this measure and conical 2-designs generalized equiangular measurement, respectively, and illustrate the effectiveness of them by explicit examples.

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 defines a new measure of quantum uncertainty based on metric-adjusted skew information to quantify average coherence under conical 2-designs generalized equiangular measurements. It claims to prove equivalence of this measure to the scaled average coherence (based on the same skew information) under unitary groups, operator orthonormal bases, and mutually unbiased bases; derives two trade-off relations; solves a conjecture; and provides two entanglement criteria (one using the new measure and one using the conical 2-designs measurements), with explicit examples illustrating effectiveness.

Significance. If the central derivations hold and the new measure satisfies the required coherence axioms (non-negativity, vanishing on incoherent states, and monotonicity under the relevant operations), the work would provide a unified framework linking skew-information-based uncertainty to coherence in specific measurement designs, with potential applications to entanglement detection and resource trade-offs in quantum information. The solution of a conjecture and explicit criteria add concrete value, though the overall impact depends on verification of the measure's faithfulness.

major comments (2)
  1. [Section 3 (measure definition) and subsequent equivalence proofs] The definition of the new uncertainty measure (introduced after the preliminary sections on metric-adjusted skew information and conical 2-designs) asserts it quantifies coherence via averaging, but the manuscript does not explicitly verify inheritance of strong monotonicity or convexity from the base metric-adjusted skew information under the conical 2-designs generalized equiangular measurements. This property is load-bearing for the claimed equivalences to scaled average coherence and for the entanglement criteria to be valid quantifiers.
  2. [Sections on trade-offs and conjecture] The trade-off relations and conjecture solution (in the section deriving relations) rely on the equivalence claims; without a self-contained check that the averaging over conical 2-designs preserves the necessary inequalities (e.g., against standard coherence axioms like those in Baumgratz et al.), the derivations risk circularity or incompleteness when applied to the generalized equiangular measurements.
minor comments (2)
  1. [Preliminaries] Notation for the conical 2-designs generalized equiangular measurements could be clarified with an explicit definition or reference to prior work in the preliminaries to aid readability.
  2. [Examples section] The examples illustrating the entanglement criteria would benefit from more detail on parameter choices and numerical values to allow independent verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We have carefully addressed each of the major comments and outline our responses and planned revisions below. We believe these changes will significantly improve the clarity and rigor of the paper.

read point-by-point responses

  1. Referee: [Section 3 (measure definition) and subsequent equivalence proofs] The definition of the new uncertainty measure (introduced after the preliminary sections on metric-adjusted skew information and conical 2-designs) asserts it quantifies coherence via averaging, but the manuscript does not explicitly verify inheritance of strong monotonicity or convexity from the base metric-adjusted skew information under the conical 2-designs generalized equiangular measurements. This property is load-bearing for the claimed equivalences to scaled average coherence and for the entanglement criteria to be valid quantifiers.

    Authors: We agree with the referee that an explicit verification of the inheritance of strong monotonicity and convexity from the metric-adjusted skew information under the averaging with conical 2-designs generalized equiangular measurements is necessary for rigor. While the equivalence to scaled average coherence under the specified sets (unitary groups, operator orthonormal bases, and mutually unbiased bases) suggests these properties carry over, as those measures satisfy the axioms, we will revise Section 3 to include a direct proof or demonstration of these properties for the new measure. This will also support the entanglement criteria. revision: yes

  2. Referee: [Sections on trade-offs and conjecture] The trade-off relations and conjecture solution (in the section deriving relations) rely on the equivalence claims; without a self-contained check that the averaging over conical 2-designs preserves the necessary inequalities (e.g., against standard coherence axioms like those in Baumgratz et al.), the derivations risk circularity or incompleteness when applied to the generalized equiangular measurements.

    Authors: The derivations of the trade-off relations and the solution to the conjecture are based on direct calculations using the definition and the equivalence. To eliminate any potential for circularity, we will add a self-contained verification that the averaging operation preserves the relevant inequalities from the standard coherence axioms (e.g., those proposed by Baumgratz et al.). This check will be inserted before the sections on trade-offs and the conjecture, making the arguments complete and independent. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations rely on independent mathematical properties of designs and bases

full rationale

The paper defines a new measure of quantum uncertainty from the established metric-adjusted skew information applied to coherence under conical 2-designs generalized equiangular measurements. It then proves equivalences to scaled averages under unitary groups, operator orthonormal bases, and MUBs, derives trade-off relations, solves a conjecture, and provides entanglement criteria. These steps use standard properties of the measurement sets and skew information without any reduction of the central claims to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations that lack external verification. The derivation chain remains self-contained against the axioms and structures invoked.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard quantum mechanical postulates and prior definitions of skew information and coherence measures; no new physical entities are introduced, and no free parameters are fitted to data as the work is purely theoretical.

axioms (2)
  • standard math Quantum states are described by density operators on Hilbert space and coherence is quantified via skew information functionals.
    Invoked throughout the definitions and proofs as background from quantum information theory.
  • domain assumption Conical 2-designs generalized equiangular measurements form valid POVMs with the stated symmetry properties.
    Required for the average to be well-defined and for equivalences to hold.

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Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages

  1. [1]

    Dicke, Coherence in spontaneous radiation processes, Phys

    R.-H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93 (1954) 99

    work page 1954

  2. [2]

    Streltsov, G

    A. Streltsov, G. Adesso, M.B. Plenio, Quantum coherence as a resource, Rev. Mod. Phys. 89 (2017) 041003

    work page 2017

  3. [3]

    Chitambar, G

    E. Chitambar, G. Gour, Quantum resource theories, Rev. Mod. Phys. 91 (2019) 025001

    work page 2019

  4. [4]

    Marvian, R.W

    I. Marvian, R.W. Spekkens, How to quantify coherence: Distinguishing speakable and unspeakable notions, Phys. Rev. A 94 (2016) 052324

    work page 2016

  5. [5]

    Y. Fan, L. Li, Average and maximal coherence based on the modified generalized Wigner-Yanase-Dyson skew information, Quantum Inf. Process. 24 (2025) 71

    work page 2025

  6. [6]

    Baumgratz, M

    T. Baumgratz, M. Cramer, M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113 (2014) 140401

    work page 2014

  7. [7]

    X. Yu, D. Zhang, G. Xu, D. Tong, Alternative framework for quantifying coherence, Phys. Rev. A 94 (2016) 060302

    work page 2016

  8. [8]

    Wigner, M.M

    E.-P. Wigner, M.M. Yanase, Information contents of distributions, Proc. Natl. Acad. Sci. 49 (1963) 910

    work page 1963

  9. [9]

    Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv

    E.-H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. Math. 11 (1973) 267

    work page 1973

  10. [10]

    P.Chen, S.Luo, DirectapproachtoquantumextensionsofFisherinformation, Front. Math. China 2 (2007) 359-381

    work page 2007

  11. [11]

    Hansen, Metric adjusted skew information, Proc

    F. Hansen, Metric adjusted skew information, Proc. Natl. Acad. Sci. 105 (2008) 9909-9916

    work page 2008

  12. [12]

    L. Cai, F. Hansen, Metric-adjusted skew information: Convexity and restricted forms of superadditivity, Lett. Math. Phys. 93 (2010) 1-13

    work page 2010

  13. [13]

    Y. Dou, H. Du, Generalizations of the Heisenberg and Schrödinger uncertainty rela- tions, J. Math. Phys. 54 (2013) 103508. 15

    work page 2013

  14. [14]

    Y. Dou, H. Du, Note on the Wigner-Yanase-Dyson skew information, Int. J. Theor. Phys. 53 (2014) 952

    work page 2014

  15. [15]

    Z.Chen, L.Liang, H.Li, W.Wang, TwogeneralizedWigner-Yanaseskewinformation and their uncertainty relations, Quantum Inf. Process. 15 (2016) 5107

    work page 2016

  16. [16]

    Y. Fan, H. Cao, W. Wang, H. Meng, L. Chen, Uncertainty relations with the gen- eralized Wigner-Yanase-Dyson skew information, Quantum Inf. Process. 17 (2018) 157

    work page 2018

  17. [17]

    Z. Wu, L. Zhang, S.-M. Fei, X. Li-Jost, Uncertainty relations based on modified Wigner-Yanase-Dyson skew information, Int. J. Theor. Phys. 59 (2020) 704

    work page 2020

  18. [18]

    Huang, Z

    H.-J. Huang, Z. Wu, S.-M. Fei, Uncertainty and complementarity relations based on generalized skew information, Europhys. Lett. 132 (2020) 60007

    work page 2020

  19. [19]

    C. Xu, Z. Wu, S.-M. Fei, Tighter sum uncertainty relations via(α, β, γ)weighted Wigner-Yanase-Dyson skew information, Commun. Theor. Phys. 76 (2024) 91

    work page 2024

  20. [20]

    Y. Fan, H. Cao, W. Wang, H. Meng, L. Chen, Non-Hermitian extensions of uncer- tainty relations with generalized metric adjusted skew information, Quantum Inf. Process. 18 (2019) 309

    work page 2019

  21. [21]

    Cai, Sum uncertainty relations based on metric-adjusted skew information, Quan- tum Inf

    L. Cai, Sum uncertainty relations based on metric-adjusted skew information, Quan- tum Inf. Process. 20 (2021) 72

    work page 2021

  22. [22]

    R. Ren, P. Li, M. Ye, Y. Li, Tighter sum uncertainty relations based on metric- adjusted skew information, Phys. Rev. A 104 (2021) 052414

    work page 2021

  23. [23]

    C. Xu, Q. Zhang, S.-M. Fei, The summation and product forms of the uncertainty re- lations based on metric-adjusted skew information, Quantum Inf. Process. 23 (2024) 252

    work page 2024

  24. [24]

    R.-X. Chen, L. Tang, Quantifying quantum uncertainty via metric-adjusted skew information and symmetric measurements, Adv. Quantum Technol., 9 (2026) e00919

    work page 2026

  25. [25]

    Girolami, Observable measure of quantum coherence in finite dimensional sys- tems, Phys

    D. Girolami, Observable measure of quantum coherence in finite dimensional sys- tems, Phys. Rev. Lett. 113 (2014) 170401

    work page 2014

  26. [26]

    Yu, Quantum coherence via skew information and its polygamy, Phys

    C.-S. Yu, Quantum coherence via skew information and its polygamy, Phys. Rev. A 95 (2017) 042337

    work page 2017

  27. [27]

    S. Luo, Y. Sun, Quantum coherence versus quantum uncertainty, Phys. Rev. A 96 (2017) 022130

    work page 2017

  28. [28]

    Y. Sun, S. Luo, Coherence as uncertainty, Phys. Rev. A 103 (2021) 042423. 16

    work page 2021

  29. [29]

    S. Luo, Y. Sun, Average versus maximal coherence, Phys. Lett. A 383 (2019) 2869- 2873

    work page 2019

  30. [30]

    Z. Wu, L. Zhang, S.-M. Fei, X. Li-Jost, Average skew information-based coherence and its typicality for random quantum states, J. Phys. A: Math. Theor. 54 (2021) 015302

    work page 2021

  31. [31]

    Y. Fan, N. Li, S. Luo, Average coherence and entropy, Phys. Rev. A 108 (2023) 052406

    work page 2023

  32. [32]

    Z. Wu, L. Zhang, S.-M. Fei, X. Li-Jost, Skew information-based coherence generating power of quantum channels, Quantum Inf. Process. 21 (2022) 236

    work page 2022

  33. [33]

    Wu, H.-J

    Z. Wu, H.-J. Huang, S.-M. Fei, X. Li-Jost, Geometry of skew information-based quantum coherence, Commun. Theor. Phys. 72 (2020) 72

    work page 2020

  34. [34]

    Sheng, J

    Y.-H. Sheng, J. Zhang, Y. Tao, S.-M. Fei, Applications of quantum coherence via skew information under mutually unbiased bases, Quantum Inf. Process. 20 (2021) 82

    work page 2021

  35. [35]

    S. Luo, Y. Sun, Coherence and complementarity in state-channel interaction, Phys. Rev. A 98 (2018) 012113

    work page 2018

  36. [36]

    Z. Wu, L. Zhang, S.-M. Fei, X. Li-Jost, Coherence and complementarity based on modified generalized skew information, Quantum Inf. Process. 19 (2020) 154

    work page 2020

  37. [37]

    Y. Sun, N. Li, S. Luo, Quantifying coherence relative to channels via metric-adjusted skew information, Phys. Rev. A 106 (2022) 012436

    work page 2022

  38. [38]

    Y. Fan, L. Li, Quantifying coherence of quantum channels via the Hilbert-Schmidt normand generalized Wigner-Yanase-Dyson skew information, Phys. Lett. A 586 (2026) 131690

    work page 2026

  39. [39]

    Y. Sun, N. Li, The uncertainty of quantum channels in terms of variance, Quantum Inf. Process. 20 (2021) 25

    work page 2021

  40. [40]

    C. Xu, Z. Wu, S.-M. Fei, Uncertainty of quantum channels via modified generalized variance and modified generalized Wigner-Yanase-Dyson skew information, Quan- tum Inf. Process. 21 (2022) 292

    work page 2022

  41. [41]

    C. Xu, Z. Wu, S.-M. Fei, Uncertainty of quantum channels based on symmetrized ρ-absolute variance and modified Wigner-Yanase skew information, Phys. Scr. 99 (2024) 115111

    work page 2024

  42. [42]

    R. Ren, Y. Luo, Y. Li, Quantifying correlations relative to channels via metric- adjusted skew information, Quantum Inf. Process. 23 (2024) 98. 17

    work page 2024

  43. [43]

    Y.-J. Fan, X. Yang, L. Li, Quantification of correlations in bipartite states via metric- adjusted skew information, Mod. Phys. Lett. A 39 (2024) 2450174

    work page 2024

  44. [44]

    L. Tang, F. Wu, Quantifying nonclassicality via metric-adjusted skew information, Phys. Rev. A 112 (2025) 032437

    work page 2025

  45. [45]

    Griffiths, What quantum measurements measure, Phys

    B. Griffiths, What quantum measurements measure, Phys. Rev. A 96 (2017) 032110

    work page 2017

  46. [46]

    Bischof, H

    F. Bischof, H. Kampermann, D. Bruß, Resource theory of coherence based on positive-operator-valued measures, Phys. Rev. Lett. 123 (2019) 110402

    work page 2019

  47. [47]

    Schwinger, Unitary operator bases, Proc

    J. Schwinger, Unitary operator bases, Proc. Natl. Acad. Sci. 46 (1960) 570-579

    work page 1960

  48. [48]

    J. M. Renes, R. Blume-Kohout, A. J. Scott, C. M. Caves, Symmetric informationally complete quantum measurements, J. Math. Phys. 45 (2004) 2171-2180

    work page 2004

  49. [49]

    Beneduci, T.J

    R. Beneduci, T.J. Bullock, P. Busch, C. Carmeli, T. Heinosaari, A. Toigo, Opera- tional link between mutually unbiased bases and symmetric informationally complete positive operator-valued measures, Phys. Rev. A 88 (2013) 032312

    work page 2013

  50. [50]

    Bengtsson, From SICs and MUBs to Eddington, J

    I. Bengtsson, From SICs and MUBs to Eddington, J. Phys.: Conf. Ser. 254 (2010) 012007

    work page 2010

  51. [51]

    Kalev, G

    A. Kalev, G. Gour, Mutually unbiased measurements in finite dimensions, New J. Phys. 16 (2014) 053038

    work page 2014

  52. [52]

    Spectrosc

    D.M.Appleby, Symmetricinformationallycompletemeasurementsofarbitraryrank, Opt. Spectrosc. 103 (2007) 416–28

    work page 2007

  53. [53]

    Siudzinska, All classes of informationally complete symmetric measurements in finite dimensions, Phys

    K. Siudzinska, All classes of informationally complete symmetric measurements in finite dimensions, Phys. Rev. A 105 (2022) 042209

    work page 2022

  54. [54]

    Siudzinska, Informationally overcomplete measurements from generalized equian- gular tight frames, J

    K. Siudzinska, Informationally overcomplete measurements from generalized equian- gular tight frames, J. Phys. A 57 (2024) 335302

    work page 2024

  55. [55]

    Chen, S.-M

    B. Chen, S.-M. Fei, Complementary measurement-induced quantum uncertainty based on metric adjusted skew information, Int. J. Quantum Inf. 18 (2020) 2150001

    work page 2020

  56. [56]

    R. Chen, Y. Long, Y. Song, L. Tang, F. Wu, Average coherence with respect to gen- eralized Wingner-Yanase-Dyson skew information and(N, M)-POVMs, Phys. Lett. A 551 (2025) 130610

    work page 2025

  57. [57]

    L. Tang, F. Wu, Enhancing some separability criteria via equiangular tight frames, Commun. Theor. Phys. 77 (2025) 045104

    work page 2025

  58. [58]

    X. Qi, Y. Pang, J. Hou, Detectingk-nonseparability andk-partite entanglement with generalized skew information and mutually unbiased measurements, J. Phys. A 58 (2025) 455303. 18

    work page 2025

  59. [59]

    X. Qi, Y. Pang, J. Hou, Detecting genuine multipartite entanglement based on a class of symmetric measurements, Quantum Inf. Process. 24 (2025) 117

    work page 2025

  60. [60]

    Takagi, Skew informations from an operational view via resource theory of asym- metry, Sci

    R. Takagi, Skew informations from an operational view via resource theory of asym- metry, Sci. Rep. 9 (2019) 14562

    work page 2019

  61. [61]

    Gibilisco, Fisher information and means: some questions in the classical and quantum settings, Inter

    P. Gibilisco, Fisher information and means: some questions in the classical and quantum settings, Inter. J. Softw. Inf. 8 (2014) 265-276

    work page 2014

  62. [62]

    Cai, Quantum uncertainty based on metric adjusted skew information, Infin

    L. Cai, Quantum uncertainty based on metric adjusted skew information, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (2018) 1850006

    work page 2018

  63. [63]

    Siudzinska, Measures from conical 2-designs depend only on two constants, arXiv:2506.18211, 2025

    K. Siudzinska, Measures from conical 2-designs depend only on two constants, arXiv:2506.18211, 2025

    work page arXiv 2025

  64. [64]

    Petz, Quasi-entropies for states of a von Neumann algebra, Publ

    D. Petz, Quasi-entropies for states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. 21 (1985) 787

    work page 1985

  65. [65]

    Zhang, Y

    C. Zhang, Y. Zhang, S. Zhang, G. Guo, Entanglement detection beyond the com- putable cross-norm or realignment criterion, Phys. Rev. A 77 (2008) 060301

    work page 2008

  66. [66]

    Huang, M

    F. Huang, M. Bai, Entropic uncertainty relations for mutually unbiased equiangular tight frames assigned to quantumt-designs, Phys. Rev. A 112 (2025) 052420

    work page 2025

  67. [67]

    Horodecki, P

    M. Horodecki, P. Horodecki, Reduction criterion of separability and limits for a class of distillation protocols, Phys. Rev. A 59 (1999) 4206

    work page 1999

  68. [68]

    N. Li, S. Luo, Y. Sun, Information-theoretic aspects of Werner states, Ann. Phys. 424 (2021) 168371

    work page 2021

  69. [69]

    S. Shen, M. Li, X. Xue, Entanglement detection via some classes of measurements, Phys. Rev. A 91 (2015) 012326

    work page 2015

  70. [70]

    Rajeev, M

    S. Rajeev, M. Lahiri, Single-qubit measurement of two qubit entanglement in gener- alized Werner states, Phys. Rev. A 108 (2023) 052410

    work page 2023

  71. [71]

    Petz, Monotone metrics on matrix spaces, Linear Algebra Appl

    D. Petz, Monotone metrics on matrix spaces, Linear Algebra Appl. 244 (1996) 81-96. 19

    work page 1996