Publications

2024

• A holographic uniqueness theorem
  
  • Novikov Roman
  Proceedings of the Steklov Institute of Mathematics, MAIK Nauka/Interperiodica, 2024, 325, pp.218–223. We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in R^3. We consider a ray in this region, such that its direction is different from the propagation direction of the plane wave. We show that the restriction of the radiation solution to this ray is uniquely determined by the intensity of the total solution on an interval of this ray. As a corollary, we also obtain that the restriction of the radiation solution to any plane in the exterior region is uniquely determined by the intensity of the total solution on an open domain in this plane. In particular, these results solve one of old mathematical questions of holography. (10.1134/S0081543824020123)
  DOI : 10.1134/S0081543824020123
• BAYESIAN CALIBRATION WITH ADAPTIVE MODEL DISCREPANCY
  
  • Leoni Nicolas
  • Le Maitre Olivier
  • Rodio Maria-Giovanna
  • Congedo Pietro Marco
  International Journal for Uncertainty Quantification, Begell House Publishers, 2024, 14 (1), pp.19-41. We investigate a computer model calibration technique inspired by the well-known Bayesian framework of Kennedy and O'Hagan (KOH). We tackle the full Bayesian formulation where model parameter and model discrepancy hyperparameters are estimated jointly and reduce the problem dimensionality by introducing a functional relationship that we call the full maximum a posteriori (FMP) method. This method also eliminates the need for a true value of model parameters that caused identifiability issues in the KOH formulation. When the joint posterior is approximated as a mixture of Gaussians, the FMP calibration is proven to avoid some pitfalls of the KOH calibration, namely missing some probability regions and underestimating the posterior variance. We then illustrate two numerical examples where both model error and measurement uncertainty are estimated together. Using the solution to the full Bayesian problem as a reference, we show that the FMP results are accurate and robust, and avoid the need for high-dimensional Markov chains for sampling. (10.1615/Int.J.UncertaintyQuantification.2023046331)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2023046331
• Activity report ciroquo research & industry consortium
  
  • Blanchet-Scalliet Christophette
  • Helbert Céline
  • Sinoquet Delphine
  • Munoz Zuniga Miguel Munoz
  • Le Riche Rodolphe
  • Rullière Didier
  • Prieur Clémentine
  • Zahm Olivier
  • Garnier Josselin
  • Brockhoff Dimo
  • Roustant Olivier
  • Bachoc François
  • Pronzato Luc
  • Richet Yann
  • Rohmer Jérémy
  • Huguet Frédéric
  • Gaudrie David
  • Marrel Amandine
  • Kerleguer Baptiste
  • Perrillat-Bottonet Thomas
  • Durantin Cedric
  , 2024, pp.1-11. This project emulated the major scientific challenges of the discipline, such as code transposition/calibration/validation, modeling of complex environments and stochastic codes. In each of these scientific fields, particular attention has been paid to large-scale problems. Real-life problems sometimes involve dozens or hundreds of inputs. Methodological advances have been proposed to take account of this additional difficulty. Developments in mathematical research are distinguished from the dominant work in machine learning by the exploration of costly numerical simulations, i.e., by the need to be thrifty in terms of the quantity of data available.
• Neural network time-series classifiers for gravitational-wave searches in single-detector periods
  
  • Trovato A
  • Chassande-Mottin É
  • Bejger M
  • Flamary R
  • Courty N
  Classical and Quantum Gravity, IOP Publishing, 2024, 41 (12), pp.125003. The search for gravitational-wave signals is limited by non-Gaussian transient noises that mimic astrophysical signals. Temporal coincidence between two or more detectors is used to mitigate contamination by these instrumental glitches. However, when a single detector is in operation, coincidence is impossible, and other strategies have to be used. We explore the possibility of using neural network classifiers and present the results obtained with three types of architectures: convolutional neural network, temporal convolutional network, and inception time. The last two architectures are specifically designed to process time-series data. The classifiers are trained on a month of data from the LIGO Livingston detector during the first observing run (O1) to identify data segments that include the signature of a binary black hole merger. Their performances are assessed and compared. We then apply trained classifiers to the remaining three months of O1 data, focusing specifically on single-detector times. The most promising candidate from our search is 2016-01-04 12:24:17 UTC. Although we are not able to constrain the significance of this event to the level conventionally followed in gravitational-wave searches, we show that the signal is compatible with the merger of two black holes with masses $m_1 = 50.7^{+10.4}_{-8.9}\,M_{\odot}$ and $m_2 = 24.4^{+20.2}_{-9.3}\,M_{\odot}$ at the luminosity distance of $d_L = 564^{+812}_{-338}\,\mathrm{Mpc}$. (10.1088/1361-6382/ad40f0)
  DOI : 10.1088/1361-6382/ad40f0
• Uncertainty quantification and global sensitivity analysis of seismic fragility curves using kriging
  
  • Gauchy Clement
  • Feau Cyril
  • Garnier Josselin
  International Journal for Uncertainty Quantification, Begell House Publishers, 2024, 14 (4), pp.39-63. Seismic fragility curves have been introduced as key components of seismic probabilistic risk assessment studies. They express the probability of failure of mechanical structures conditional to a seismic intensity measure and must take into account various sources of uncertainties, the so-called epistemic uncertainties (i.e., coming from the uncertainty on the mechanical parameters of the structure) and the aleatory uncertainties (i.e., coming from the randomness of the seismic ground motions). For simulation-based approaches we propose a methodology to build and calibrate a Gaussian process surrogate model to estimate a family of nonparametric seismic fragility curves for a mechanical structure by propagating both the surrogate model uncertainty and the epistemic ones. Gaussian processes have indeed the main advantage to propose both a predictor and an assessment of the uncertainty of its predictions. In addition, we extend this methodology to sensitivity analysis. Global sensitivity indices such as aggregated Sobol' indices and kernel-based indices are proposed to know how the uncertainty on the seismic fragility curves is apportioned according to each uncertain mechanical parameter. This comprehensive uncertainty quantification framework is finally applied to an industrial test case consisting of a part of a piping system of a pressurized water reactor. (10.1615/Int.J.UncertaintyQuantification.2023046480)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2023046480
• Hommage à Elisabeta Vergu
  
  • Bansaye Vincent
  • Boëlle Pierre-Yves
  • Cauchemez Simon
  • Cazelles Bernard
  • Crepey Pascal
  • Cristancho-Fajardo Lina
  • Dhersin Jean-Stéphane
  • Dumitrescu Sorin
  • Ezanno Pauline
  • Flahault Antoine
  • Golmard Jean-Louis
  • Hoscheit Patrick
  • Mermoz Kouye Henri
  • Kubasch Madeleine
  • Laredo Catherine
  • Mallet Alain
  • Opatowski Lulla
  • Prieur Clémentine
  • Véber Amandine
  , 2024. Ce texte rend hommage à Elisabeta Vergu, directrice de recherche INRAE au sein de l'unité MaIAGE du centre de Jouy en Josas, décédée en mai 2023 à l'âge de 49 ans. Née à Tulcea en Roumanie, elle est arrivée en France en 1992 grâce à une bourse lui permettant de se former à l'INSA de Lyon. Après avoir obtenu son diplôme d'ingénieur en 1997 puis un doctorat en biomathématiques de l'Université Pierre et Marie Curie en 2003, elle a été recrutée à l'INRAE en tant que chargée de recherche en 2005 et y sera promue directrice de recherche en 2018. Le texte retrace ses nombreuses contributions au domaine des mathématiques appliquées à l'épidémiologie (sans prétention d'exhaustivité), tant du point de vue des résultats qu'elle a obtenus au travers d'une cinquantaine d'articles et qui s'inscrivent dans un spectre très large de questions théoriques et d'applications, que du point de vue de son engagement sans faille pour la communauté. Par son enthousiasme communicatif, elle aura permis le rapprochement de chercheuses et chercheurs d'horizons très divers pour répondre à des problématiques pluridisciplinaires d'une indéniable actualité.
• On unitarity of the scattering operator in non-Hermitian quantum mechanics
  
  • Novikov Roman
  • Taimanov Iskander
  Annales de l'Institut Henri Poincaré (A). Physique Theorique, Birkhäuser, 2024, 25, pp.3899–3909. We consider the Schrödinger operator with regular short range complex-valued potential in dimension d ≥ 1. We show that, for d ≥ 2, the unitarity of scattering operator for this Hamiltonian at high energies implies the reality of potential (that is Hermiticity of Hamiltonian). In contrast, for d = 1, we present complex-valued exponentially localized soliton potentials with unitary scattering operator for all positive energies and with unbroken PT symmetry. We also present examples of complex-valued regular short range potentials with real spectrum for d = 3. Some directions for further research are formulated. (10.1007/s00023-024-01414-5)
  DOI : 10.1007/s00023-024-01414-5
• High-order adaptive time discretisation of one-dimensional low-Mach reacting flows: a case study of solid propellant combustion
  
  • François Laurent
  • Dupays Joël
  • Davidenko Dmitry
  • Massot Marc
  Journal of Computational and Applied Mathematics, Elsevier, 2024, pp.115758. Solving the reactive low-Mach Navier-Stokes equations with high-order adaptive methods in time is still a challenging problem, in particular due to the handling of the algebraic variables involved in the mass constraint. We focus on the one-dimensional configuration, where this challenge has long existed in the combustion community. We consider a model of solid propellant combustion, which possesses the characteristic difficulties encountered in the homogeneous or spray combustion cases, with the added complication of an active interface. The system obtained after semi-discretisation in space is shown to be differential-algebraic of index 1. A numerical strategy relying on stiffly accurate Runge-Kutta methods is introduced, with a specific discretisation of the algebraic constraints and time adaptation. High order is shown to be reached on all variables, while handling the constraints properly. Three challenging test cases are investigated: ignition, limit cycle, and unsteady response with detailed gas-phase kinetics. We show that the time integration method can greatly affect the ability to predict the dynamics of the system. The proposed numerical strategy exhibits high efficiency and accuracy for all cases compared to traditional schemes used in the combustion literature. (10.1016/j.cam.2024.115758)
  DOI : 10.1016/j.cam.2024.115758
• Long-time derivation at equilibrium of the fluctuating Boltzmann equation
  
  • Bodineau Thierry
  • Gallagher Isabelle
  • Saint-Raymond Laure
  • Simonella Sergio
  The Annals of Probability, Institute of Mathematical Statistics, 2024, 52 (1). We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7]. (10.1214/23-AOP1656)
  DOI : 10.1214/23-AOP1656
• A Multi-Level Fast-Marching Method For The Minimum Time Problem
  
  • Akian Marianne
  • Gaubert Stéphane
  • Liu Shanqing
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2024, 62 (6), pp.2963-2991. We introduce a new numerical method to approximate the solutions of a class of stationary Hamilton-Jacobi (HJ) partial differential equations arising from minimum time optimal control problems. We rely on several grid approximations, and look for the optimal trajectories by using the coarse grid approximations to reduce the search space in fine grids. This may be thought of as an infinitesimal version of the ''highway hierarchy'' method which has been developed to solve shortest path problems (with discrete time and discrete state). We obtain, for each level, an approximate value function on a sub-domain of the state space. We show that the sequence obtained in this way does converge to the viscosity solution of the HJ equation. Moreover, for our multi-level algorithm, if $0<\gamma \leq 1$ is the convergence rate of the classical numerical scheme, then the number of arithmetic operations needed to obtain an error in $O(\varepsilon)$ is in $\widetilde{O}(\varepsilon^{-\frac{\theta d}{\gamma}})$, to be compared with $\widetilde{O}(\varepsilon^{-\frac{d}{\gamma}})$ for ordinary grid-based methods. Here $d$ is the dimension of the problem, and $\theta <1$ depends on $d$ and on the ''stiffness" of the value function around optimal trajectories, and the notation $\widetilde{O}$ ignores logarithmic factors. When $\gamma =1$ and the stiffness is high, this complexity becomes in $\widetilde{O}(\varepsilon^{-1})$. We illustrate the approach by solving HJ equations of eikonal type up to dimension 7. (10.1137/23M1563657)
  DOI : 10.1137/23M1563657
• Estimation of extreme quantiles from heavy-tailed distributions with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  Statistics and Computing, Springer Verlag (Germany), 2024, 34 (12), pp.1-35. We propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log-quantiles and their neural network approximation is established. The finite sample performances of the non-conditional neural network estimator are compared to other bias-reduced extreme-value competitors on simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail. The source code is available at https://github.com/michael-allouche/nn-quantile-extrapolation.git. Finally, the conditional neural network estimators are implemented to investigate the behaviour of extreme rainfalls as functions of their geographical location in the southern part of France. (10.1007/s11222-023-10331-2)
  DOI : 10.1007/s11222-023-10331-2
• Fully algebraic domain decomposition preconditioners with adaptive spectral bounds
  
  • Gouarin Loïc
  • Spillane Nicole
  Electronic Transactions on Numerical Analysis, Kent State University Library, 2024, 60, pp.169–196. In this article a new family of preconditioners is introduced for symmetric positive definite linear systems. The new preconditioners, called the AWG preconditioners (for Algebraic- Woodbury-GenEO) are constructed algebraically. By this, we mean that only the knowledge of the matrix A for which the linear system is being solved is required. Thanks to the GenEO spectral coarse space technique, the condition number of the preconditioned operator is bounded theoretically from above. This upper bound can be made smaller by enriching the coarse space with more spectral modes. The novelty is that, unlike in previous work on the GenEO coarse spaces, no knowledge of a partially non-assembled form of A is required. Indeed, the spectral coarse space technique is not applied directly to A but to a low-rank modification of A of which a suitable non- assembled form is known by construction. The extra cost is a second (and to this day rather expensive) coarse solve in the preconditioner. One of the AWG preconditioners has already been presented in the short preprint [38]. This article is the first full presentation of the larger family of AWG preconditioners. It includes proofs of the spectral bounds as well as numerical illustrations.
• New Convergence Analysis of GMRES with Weighted Norms, Preconditioning and Deflation, Leading to a New Deflation Space
  
  • Spillane Nicole
  • Szyld Daniel B
  SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2024, 45 (4), pp.1721-1745. New convergence bounds are presented for weighted, preconditioned, and deflated GMRES for the solution of large, sparse, non-Hermitian linear systems. These bounds are given for the case when the Hermitian part of the coefficient matrix is positive definite, the preconditioner is Hermitian positive definite, and the weight is equal to the preconditioner. The new bounds are a novel contribution in and of themselves. In addition, they are sufficiently explicit to indicate how to choose the preconditioner and the deflation space to accelerate the convergence. One such choice of deflating space is presented, and numerical experiments illustrate the effectiveness of such space.
• Lipschitz continuity in the Hurst parameter of functionals of stochastic differential equations driven by a fractional Brownian motion
  
  • Richard Alexandre
  • Talay Denis
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2024, 29 (none). Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this paper we consider solutions $\{X^H_t\}_{t\in \R_+}$ to stochastic differential equations driven by frac{t}ional Brownian motions. We develop two innovative sensitivity analyses when the Hurst parameter~$H$ of the noise tends to the critical Brownian parameter $H=\tfrac{1}{2}$ from above or from below. First, we examine expected smooth functions of $X^H$ at a fixed time horizon~$T$. Second, we examine Laplace transforms of functionals which are irregular with regard to Malliavin calculus, namely, first passage times of $X^H$ at a given threshold. In both cases we exhibit the Lipschitz continuity w.r.t.~$H$ around the value $\tfrac{1}{2}$. Therefore, our results show that the Markov Brownian model is a good proxy model as long as the Hurst parameter remains close to~$\tfrac{1}{2}$. (10.1214/24-EJP1191)
  DOI : 10.1214/24-EJP1191
• Change of measure for Bayesian field inversion with hierarchical hyperparameters sampling
  
  • Polette Nadège
  • Le Maître Olivier
  • Sochala Pierre
  • Gesret Alexandrine
  Journal of Computational Physics, Elsevier, 2024, 529, pp.113888. This paper proposes an effective treatment of hyperparameters in the Bayesian inference of a scalar field from indirect observations. Obtaining the joint posterior distribution of the field and its hyperparameters is challenging. The infinite dimensionality of the field requires a finite parametrization that usually involves hyperparameters to reflect the limited prior knowledge. In the present work, we consider a Karhunen-Loève (KL) decomposition for the random field and hyperparameters to account for the lack of prior knowledge of its autocovariance function. The hyperparameters must be inferred. To efficiently sample jointly the KL coordinates of the field and the autocovariance hyperparameters, we introduce a change of measure to reformulate the joint posterior distribution into a hierarchical Bayesian form. The likelihood depends only on the field's coordinates in a fixed KL basis, with a prior conditioned on the hyperparameters. We exploit this structure to derive an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial Chaos expansions) of the forward model predictions to further accelerate the MCMC sampling. A first application to a transient diffusion problem shows that our method is consistent with other approaches based on a change of coordinates (Sraj et al., 2016). A second application to a seismic traveltime tomography highlights the importance of inferring the hyperparameters. A third application to a 2D anisotropic groundwater flow problem illustrates the method on a more complex geometry. (10.2139/ssrn.4799579)
  DOI : 10.2139/ssrn.4799579
• Analysis of the vanishing discount limit for optimal control problems in continuous and discrete time
  
  • Cannarsa Piermarco
  • Gaubert Stéphane
  • Mendico Cristian
  • Quincampoix Marc
  Mathematical Control and Related Fields, AIMS, 2024, 14 (4), pp.1275-1305. A classical problem in ergodic continuous time control consists of studying the limit behavior of the optimal value of a discounted cost functional with infinite horizon as the discount factor $\lambda$ tends to zero. In the literature, this problem has been addressed under various controllability or ergodicity conditions ensuring that the rescaled value function converges uniformly to a constant limit. In this case the limit can be characterized as the unique constant such that a suitable Hamilton-Jacobi equation has at least one continuous viscosity solution. In this paper, we study this problem without such conditions, so that the aforementioned limit needs not be constant. Our main result characterizes the uniform limit (when it exists) as the maximal subsolution of a system of Hamilton-Jacobi equations. Moreover, when such a subsolution is a viscosity solution, we obtain the convergence of optimal values as well as a rate of convergence. This mirrors the analysis of the discrete time case, where we characterize the uniform limit as the supremum over a set of sub-invariant half-lines of the dynamic programming operator. The emerging structure in both discrete and continuous time models shows that the supremum over sub-invariato half-lines with respect to the Lax-Oleinik semigroup/dynamic programming operator, captures the behavior of the limit cost as discount vanishes. (10.3934/mcrf.2024010)
  DOI : 10.3934/mcrf.2024010
• Coarse Grained Molecular Dynamics with Normalizing Flows
  
  • Tamagnone Samuel
  • Laio Alessandro
  • Gabrié Marylou
  Journal of Chemical Theory and Computation, American Chemical Society, 2024, 20 (18), pp.7796–7805. We propose a sampling algorithm relying on a collective variable (CV) of mid-size dimension modelled by a normalizing flow and using non-equilibrium dynamics to propose full configurational moves from the proposition of a refreshed value of the CV made by the flow. The algorithm takes the form of a Markov chain with non-local updates, allowing jumps through energy barriers across metastable states. The flow is trained throughout the algorithm to reproduce the free energy landscape of the CV. The output of the algorithm are a sample of thermalized configurations and the trained network that can be used to efficiently produce more configurations. We show the functioning of the algorithm first on a test case with a mixture of Gaussians. Then we successfully test it on a higher dimensional system consisting in a polymer in solution with a compact and an extended stable state separated by a high free energy barrier. (10.1021/acs.jctc.4c00700)
  DOI : 10.1021/acs.jctc.4c00700
• A bi-directional low-Reynolds-number swimmer with passive elastic arms
  
  • Levillain Jessie
  • Alouges François
  • Desimone Antonio
  • Choudhary Akash
  • Nambiar Sankalp
  • Bochert Ida
  ESAIM: Proceedings and Surveys, EDP Sciences, 2024. It has been recently shown that it is possible to design simple artificial swimmers at low Reynolds number that possess only one degree of freedom and, nevertheless, can overcome Purcell’s celebrated scallop theorem. One of the few examples is given by Montino and DeSimone, Eur. Phys. J. E, vol. 38, 2015, who consider the three-sphere Swimmer of Najafi and Golestanian, replacing one active arm with a passive elastic spring. We further generalize this idea by increasing the number of springs and show that it is possible to invert the swimming direction using the frequency of the single actuated arm.
• The linear sampling method for data generated by small random scatterers
  
  • Garnier Josselin
  • Haddar Houssem
  • Montanelli Hadrien
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2024, 17 (4), pp.2142-2173. (10.1137/24M1650417)
  DOI : 10.1137/24M1650417
• Uncertainty quantification and global sensitivity analysis of seismic fragility curves using kriging
  
  • Gauchy Clement
  • Feau Cyril
  • Garnier Josselin
  International Journal for Uncertainty Quantification, Begell House Publishers, 2024, 4 (4). Seismic fragility curves have been introduced as key components of seismic probabilistic risk assessment studies. They express the probability of failure of mechanical structures conditional to a seismic intensity measure and must take into account various sources of uncertainties, the so-called epistemic uncertainties (i.e., coming from the uncertainty on the mechanical parameters of the structure) and the aleatory uncertainties (i.e., coming from the randomness of the seismic ground motions). For simulation-based approaches we propose a methodology to build and calibrate a Gaussian process surrogate model to estimate a family of nonparametric seismic fragility curves for a mechanical structure by propagating both the surrogate model uncertainty and the epistemic ones. Gaussian processes have indeed the main advantage to propose both a predictor and an assessment of the uncertainty of its predictions. In addition, we extend this methodology to sensitivity analysis. Global sensitivity indices such as aggregated Sobol' indices and kernel-based indices are proposed to know how the uncertainty on the seismic fragility curves is apportioned according to each uncertain mechanical parameter. This comprehensive uncertainty quantification framework is finally applied to an industrial test case consisting of a part of a piping system of a pressurized water reactor. (10.1615/Int.J.UncertaintyQuantification.2023046480)
  DOI : 10.1615/Int.J.UncertaintyQuantification.2023046480
• Large-eddy simulation of solid/fluid heat and mass transfer applied to the thermal degradation of composite material
  
  • Grenouilloux Adrien
  • Letournel Roxane
  • Dellinger Nicolas
  • Bioche Kévin
  • Moureau Vincent
  , 2024.
• Dimension results for extremal-generic polynomial systems over complete toric varieties
  
  • Bender Matías R
  • Spaenlehauer Pierre-Jean
  Journal of Algebra, Elsevier, 2024, 646, pp.156-182. We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial formulas for the dimensions of their associated subvarieties under genericity assumptions on the coefficients of the polynomials. Using these formulas, we identify at which degrees generic systems in polytopal algebras form regular sequences. Our motivation comes from sparse elimination theory, where knowing the expected dimension of these subvarieties leads to specialized algorithms and to large speed-ups for solving sparse polynomial systems. As a special case, we classify the degrees at which regular sequences defined by weighted homogeneous polynomials can be found, answering an open question in the Gr\"obner bases literature. We also show that deciding whether a sparse system is generically a regular sequence in a polytopal algebra is hard from the point of view of theoretical computational complexity. (10.1016/j.jalgebra.2024.01.029)
  DOI : 10.1016/j.jalgebra.2024.01.029