[{"day":"21","article_processing_charge":"No","scopus_import":"1","keyword":["Atomic and Molecular Physics","and Optics"],"date_published":"2023-07-21T00:00:00Z","publication":"Optics Letters","citation":{"ista":"Mishra U, Li V, Wald S, Agafonova S, Diorico FR, Hosten O. 2023. Monitoring and active stabilization of laser injection locking using beam ellipticity. Optics Letters. 48(15), 3973–3976.","apa":"Mishra, U., Li, V., Wald, S., Agafonova, S., Diorico, F. R., & Hosten, O. (2023). Monitoring and active stabilization of laser injection locking using beam ellipticity. Optics Letters. Optica Publishing Group. https://doi.org/10.1364/ol.495553","ieee":"U. Mishra, V. Li, S. Wald, S. Agafonova, F. R. Diorico, and O. Hosten, “Monitoring and active stabilization of laser injection locking using beam ellipticity,” Optics Letters, vol. 48, no. 15. Optica Publishing Group, pp. 3973–3976, 2023.","ama":"Mishra U, Li V, Wald S, Agafonova S, Diorico FR, Hosten O. Monitoring and active stabilization of laser injection locking using beam ellipticity. Optics Letters. 2023;48(15):3973-3976. doi:10.1364/ol.495553","chicago":"Mishra, Umang, Vyacheslav Li, Sebastian Wald, Sofya Agafonova, Fritz R Diorico, and Onur Hosten. “Monitoring and Active Stabilization of Laser Injection Locking Using Beam Ellipticity.” Optics Letters. Optica Publishing Group, 2023. https://doi.org/10.1364/ol.495553.","mla":"Mishra, Umang, et al. “Monitoring and Active Stabilization of Laser Injection Locking Using Beam Ellipticity.” Optics Letters, vol. 48, no. 15, Optica Publishing Group, 2023, pp. 3973–76, doi:10.1364/ol.495553.","short":"U. Mishra, V. Li, S. Wald, S. Agafonova, F.R. Diorico, O. Hosten, Optics Letters 48 (2023) 3973–3976."},"article_type":"original","page":"3973-3976","abstract":[{"text":"We unveil a powerful method for the stabilization of laser injection locking based on sensing variations in the output beam ellipticity of an optically seeded laser. The effect arises due to an interference between the seeding beam and the injected laser output. We demonstrate the method for a commercial semiconductor laser without the need for any internal changes to the readily operational injection locked laser system that was used. The method can also be used to increase the mode-hop free tuning range of lasers, and has the potential to fill a void in the low-noise laser industry.","lang":"eng"}],"issue":"15","type":"journal_article","oa_version":"Preprint","_id":"14749","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","title":"Monitoring and active stabilization of laser injection locking using beam ellipticity","intvolume":" 48","month":"07","publication_identifier":{"eissn":["1539-4794"],"issn":["0146-9592"]},"doi":"10.1364/ol.495553","language":[{"iso":"eng"}],"external_id":{"arxiv":["2212.01266"]},"quality_controlled":"1","author":[{"id":"4328fa4c-f128-11eb-9611-c107b0fe4d51","first_name":"Umang","last_name":"Mishra","full_name":"Mishra, Umang"},{"id":"3A4FAA92-F248-11E8-B48F-1D18A9856A87","last_name":"Li","first_name":"Vyacheslav","full_name":"Li, Vyacheslav"},{"last_name":"Wald","first_name":"Sebastian","id":"133F200A-B015-11E9-AD41-0EDAE5697425","full_name":"Wald, Sebastian"},{"full_name":"Agafonova, Sofya","first_name":"Sofya","last_name":"Agafonova","id":"09501ff6-dca7-11ea-a8ae-b3e0b9166e80","orcid":"0000-0003-0582-2946"},{"id":"2E054C4C-F248-11E8-B48F-1D18A9856A87","last_name":"Diorico","first_name":"Fritz R","full_name":"Diorico, Fritz R"},{"full_name":"Hosten, Onur","orcid":"0000-0002-2031-204X","id":"4C02D85E-F248-11E8-B48F-1D18A9856A87","last_name":"Hosten","first_name":"Onur"}],"date_created":"2024-01-08T13:01:46Z","date_updated":"2024-01-09T08:09:32Z","volume":48,"year":"2023","publication_status":"published","publisher":"Optica Publishing Group","department":[{"_id":"OnHo"}]},{"date_updated":"2024-01-09T08:54:03Z","date_created":"2024-01-08T13:07:49Z","volume":4,"author":[{"last_name":"Fildier","first_name":"B.","full_name":"Fildier, B."},{"full_name":"Muller, Caroline J","last_name":"Muller","first_name":"Caroline J","orcid":"0000-0001-5836-5350","id":"f978ccb0-3f7f-11eb-b193-b0e2bd13182b"},{"last_name":"Pincus","first_name":"R.","full_name":"Pincus, R."},{"last_name":"Fueglistaler","first_name":"S.","full_name":"Fueglistaler, S."}],"publication_status":"published","publisher":"American Geophysical Union","department":[{"_id":"CaMu"}],"acknowledgement":"The authors would like to thank two anonymous reviews and gratefully acknowledge diverse funding agencies and resources used for this work. B.F. and C.M. thank funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Project CLUSTER, grant agreement no. 805041), and the EUREC4A campaign organizers for giving the opportunity to take part to the campaign and use the data early on. R. P. was supported by the US National Science Foundation (award AGS 19–16908), by the National Oceanic and Atmospheric Administration (award NA200AR4310375), and the Vetlesen Foundation.","year":"2023","file_date_updated":"2024-01-09T08:51:25Z","ec_funded":1,"article_number":"e2023AV000880","language":[{"iso":"eng"}],"doi":"10.1029/2023av000880","quality_controlled":"1","project":[{"name":"organization of CLoUdS, and implications of Tropical cyclones and for the Energetics of the tropics, in current and waRming climate","call_identifier":"H2020","grant_number":"805041","_id":"629205d8-2b32-11ec-9570-e1356ff73576"}],"tmp":{"name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","short":"CC BY-NC-ND (4.0)","image":"/images/cc_by_nc_nd.png"},"oa":1,"month":"06","publication_identifier":{"eissn":["2576-604X"]},"oa_version":"Published Version","file":[{"access_level":"open_access","file_name":"2023_AGUAdvances_Fildier.pdf","file_size":24149551,"content_type":"application/pdf","creator":"dernst","relation":"main_file","file_id":"14761","checksum":"af773220a9fa194c61a8dc2fae092c16","success":1,"date_updated":"2024-01-09T08:51:25Z","date_created":"2024-01-09T08:51:25Z"}],"status":"public","ddc":["550"],"title":"How moisture shapes low‐level radiative cooling in subsidence regimes","intvolume":" 4","_id":"14752","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"Radiative cooling of the lowest atmospheric levels is of strong importance for modulating atmospheric circulations and organizing convection, but detailed observations and a robust theoretical understanding are lacking. Here we use unprecedented observational constraints from subsidence regimes in the tropical Atlantic to develop a theory for the shape and magnitude of low‐level longwave radiative cooling in clear‐sky, showing peaks larger than 5–10 K/day at the top of the boundary layer. A suite of novel scaling approximations is first developed from simplified spectral theory, in close agreement with the measurements. The radiative cooling peak height is set by the maximum lapse rate in water vapor path, and its magnitude is mainly controlled by the ratio of column relative humidity above and below the peak. We emphasize how elevated intrusions of moist air can reduce low‐level cooling, by sporadically shading the spectral range which effectively cools to space. The efficiency of this spectral shading depends both on water content and altitude of moist intrusions; its height dependence cannot be explained by the temperature difference between the emitting and absorbing layers, but by the decrease of water vapor extinction with altitude. This analytical work can help to narrow the search for low‐level cloud patterns sensitive to radiative‐convective feedbacks: the most organized patterns with largest cloud fractions occur in atmospheres below 10% relative humidity and feel the strongest low‐level cooling. This motivates further assessment of favorable conditions for radiative‐convective feedbacks and a robust quantification of corresponding shallow cloud dynamics in current and warmer climates.","lang":"eng"}],"issue":"3","type":"journal_article","date_published":"2023-06-01T00:00:00Z","article_type":"original","publication":"AGU Advances","citation":{"short":"B. Fildier, C.J. Muller, R. Pincus, S. Fueglistaler, AGU Advances 4 (2023).","mla":"Fildier, B., et al. “How Moisture Shapes Low‐level Radiative Cooling in Subsidence Regimes.” AGU Advances, vol. 4, no. 3, e2023AV000880, American Geophysical Union, 2023, doi:10.1029/2023av000880.","chicago":"Fildier, B., Caroline J Muller, R. Pincus, and S. Fueglistaler. “How Moisture Shapes Low‐level Radiative Cooling in Subsidence Regimes.” AGU Advances. American Geophysical Union, 2023. https://doi.org/10.1029/2023av000880.","ama":"Fildier B, Muller CJ, Pincus R, Fueglistaler S. How moisture shapes low‐level radiative cooling in subsidence regimes. AGU Advances. 2023;4(3). doi:10.1029/2023av000880","apa":"Fildier, B., Muller, C. J., Pincus, R., & Fueglistaler, S. (2023). How moisture shapes low‐level radiative cooling in subsidence regimes. AGU Advances. American Geophysical Union. https://doi.org/10.1029/2023av000880","ieee":"B. Fildier, C. J. Muller, R. Pincus, and S. Fueglistaler, “How moisture shapes low‐level radiative cooling in subsidence regimes,” AGU Advances, vol. 4, no. 3. American Geophysical Union, 2023.","ista":"Fildier B, Muller CJ, Pincus R, Fueglistaler S. 2023. How moisture shapes low‐level radiative cooling in subsidence regimes. AGU Advances. 4(3), e2023AV000880."},"day":"01","article_processing_charge":"Yes","has_accepted_license":"1","keyword":["General Earth and Planetary Sciences"],"scopus_import":"1"},{"month":"05","publication_identifier":{"eissn":["1471-2962"],"issn":["1364-503X"]},"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"external_id":{"pmid":["36907214"]},"oa":1,"quality_controlled":"1","doi":"10.1098/rsta.2022.0112","language":[{"iso":"eng"}],"article_number":"0112","file_date_updated":"2024-01-09T09:13:53Z","year":"2023","acknowledgement":"K.D.’s research was supported by Australian Research Council Discovery Early Career Researcher Award (DE170100171). B.W., R.A., F.M. and A.M. research was supported by the Spanish Ministerio de Economía y Competitividad (grant nos. FIS2016-77849-R and FIS2017-85794-P) and Ministerio de Ciencia e Innovación (grant no. PID2020-114043GB-I00) and the Generalitat de Catalunya (grant no. 2017-SGR-785). B.W.’s research was also supported by the Chinese Scholarship Council (grant CSC no. 201806440152). F.M. is a Serra-Húnter Fellow.","pmid":1,"publication_status":"published","department":[{"_id":"BjHo"}],"publisher":"The Royal Society","author":[{"first_name":"B.","last_name":"Wang","full_name":"Wang, B."},{"full_name":"Mellibovsky, F.","last_name":"Mellibovsky","first_name":"F."},{"full_name":"Ayats López, Roger","first_name":"Roger","last_name":"Ayats López","id":"ab77522d-073b-11ed-8aff-e71b39258362","orcid":"0000-0001-6572-0621"},{"full_name":"Deguchi, K.","last_name":"Deguchi","first_name":"K."},{"first_name":"A.","last_name":"Meseguer","full_name":"Meseguer, A."}],"date_created":"2024-01-08T13:11:45Z","date_updated":"2024-01-09T09:15:29Z","volume":381,"scopus_import":"1","keyword":["General Physics and Astronomy","General Engineering","General Mathematics"],"day":"01","has_accepted_license":"1","article_processing_charge":"No","publication":"Philosophical Transactions of the Royal Society A","citation":{"ama":"Wang B, Mellibovsky F, Ayats López R, Deguchi K, Meseguer A. Mean structure of the supercritical turbulent spiral in Taylor–Couette flow. Philosophical Transactions of the Royal Society A. 2023;381(2246). doi:10.1098/rsta.2022.0112","ista":"Wang B, Mellibovsky F, Ayats López R, Deguchi K, Meseguer A. 2023. Mean structure of the supercritical turbulent spiral in Taylor–Couette flow. Philosophical Transactions of the Royal Society A. 381(2246), 0112.","apa":"Wang, B., Mellibovsky, F., Ayats López, R., Deguchi, K., & Meseguer, A. (2023). Mean structure of the supercritical turbulent spiral in Taylor–Couette flow. Philosophical Transactions of the Royal Society A. The Royal Society. https://doi.org/10.1098/rsta.2022.0112","ieee":"B. Wang, F. Mellibovsky, R. Ayats López, K. Deguchi, and A. Meseguer, “Mean structure of the supercritical turbulent spiral in Taylor–Couette flow,” Philosophical Transactions of the Royal Society A, vol. 381, no. 2246. The Royal Society, 2023.","mla":"Wang, B., et al. “Mean Structure of the Supercritical Turbulent Spiral in Taylor–Couette Flow.” Philosophical Transactions of the Royal Society A, vol. 381, no. 2246, 0112, The Royal Society, 2023, doi:10.1098/rsta.2022.0112.","short":"B. Wang, F. Mellibovsky, R. Ayats López, K. Deguchi, A. Meseguer, Philosophical Transactions of the Royal Society A 381 (2023).","chicago":"Wang, B., F. Mellibovsky, Roger Ayats López, K. Deguchi, and A. Meseguer. “Mean Structure of the Supercritical Turbulent Spiral in Taylor–Couette Flow.” Philosophical Transactions of the Royal Society A. The Royal Society, 2023. https://doi.org/10.1098/rsta.2022.0112."},"article_type":"original","date_published":"2023-05-01T00:00:00Z","type":"journal_article","abstract":[{"text":"The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor–Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without notably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a co-rotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary role.","lang":"eng"}],"issue":"2246","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"14754","status":"public","ddc":["530"],"title":"Mean structure of the supercritical turbulent spiral in Taylor–Couette flow","intvolume":" 381","oa_version":"Submitted Version","file":[{"date_created":"2024-01-09T09:13:53Z","date_updated":"2024-01-09T09:13:53Z","success":1,"checksum":"1978d126c0ce2f47c22ac20107cc0106","file_id":"14763","relation":"main_file","creator":"dernst","content_type":"application/pdf","file_size":6421086,"file_name":"2023_PhilTransactionsA_Wang_accepted.pdf","access_level":"open_access"}]},{"language":[{"iso":"eng"}],"doi":"10.1016/j.physletb.2023.137871","quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"oa":1,"publication_identifier":{"eissn":["1873-2445"],"issn":["0370-2693"]},"month":"05","volume":840,"date_updated":"2024-01-09T09:02:22Z","date_created":"2024-01-08T13:09:17Z","author":[{"full_name":"Faccioli, Pietro","last_name":"Faccioli","first_name":"Pietro"},{"orcid":"0000-0002-5636-9259","id":"30d4014e-7753-11eb-b44b-db6d61112e73","last_name":"Krätschmer","first_name":"Ilse","full_name":"Krätschmer, Ilse"},{"full_name":"Lourenço, Carlos","last_name":"Lourenço","first_name":"Carlos"}],"department":[{"_id":"MaRo"}],"publisher":"Elsevier","publication_status":"published","year":"2023","acknowledgement":"P.F. and C.L. acknowledge support from Fundação para a Ciência e a Tecnologia, Portugal, under contract CERN/FIS-PAR/0010/2019.\r\nOpen Access funded by SCOAP3.","file_date_updated":"2024-01-09T08:59:24Z","article_number":"137871","date_published":"2023-05-10T00:00:00Z","article_type":"original","citation":{"ieee":"P. Faccioli, I. Krätschmer, and C. Lourenço, “Low-pT quarkonium polarization measurements: Challenges and opportunities,” Physics Letters B, vol. 840. Elsevier, 2023.","apa":"Faccioli, P., Krätschmer, I., & Lourenço, C. (2023). Low-pT quarkonium polarization measurements: Challenges and opportunities. Physics Letters B. Elsevier. https://doi.org/10.1016/j.physletb.2023.137871","ista":"Faccioli P, Krätschmer I, Lourenço C. 2023. Low-pT quarkonium polarization measurements: Challenges and opportunities. Physics Letters B. 840, 137871.","ama":"Faccioli P, Krätschmer I, Lourenço C. Low-pT quarkonium polarization measurements: Challenges and opportunities. Physics Letters B. 2023;840. doi:10.1016/j.physletb.2023.137871","chicago":"Faccioli, Pietro, Ilse Krätschmer, and Carlos Lourenço. “Low-PT Quarkonium Polarization Measurements: Challenges and Opportunities.” Physics Letters B. Elsevier, 2023. https://doi.org/10.1016/j.physletb.2023.137871.","short":"P. Faccioli, I. Krätschmer, C. Lourenço, Physics Letters B 840 (2023).","mla":"Faccioli, Pietro, et al. “Low-PT Quarkonium Polarization Measurements: Challenges and Opportunities.” Physics Letters B, vol. 840, 137871, Elsevier, 2023, doi:10.1016/j.physletb.2023.137871."},"publication":"Physics Letters B","article_processing_charge":"Yes (via OA deal)","has_accepted_license":"1","day":"10","keyword":["Nuclear and High Energy Physics"],"scopus_import":"1","file":[{"date_updated":"2024-01-09T08:59:24Z","date_created":"2024-01-09T08:59:24Z","checksum":"02dec160dbc81d95985e755869d8afbf","success":1,"relation":"main_file","file_id":"14762","content_type":"application/pdf","file_size":855494,"creator":"dernst","file_name":"2023_PhysicsLettersB_Faccioli.pdf","access_level":"open_access"}],"oa_version":"Published Version","intvolume":" 840","ddc":["530"],"title":"Low-pT quarkonium polarization measurements: Challenges and opportunities","status":"public","_id":"14753","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"lang":"eng","text":"Several fixed-target experiments reported J/ψ and ϒ polarizations, as functions of Feynman x (xF) and transverse momentum (PT), in three different frames, using different combinations of beam particles, target nuclei, and collision energies. Despite the diverse and heterogeneous picture formed by these measurements, a detailed look allows us to discern qualitative physical patterns that inspire a simple empirical model. This data-driven scenario offers a good quantitative description of the J/ψ and ϒ(1S) polarizations measured in proton- and pion-nucleus collisions, in the xF 0.5 domain: more than 80 data points (not statistically independent) are well reproduced with only one free parameter. This study sets the context for future low-PT\r\n quarkonium polarization measurements in proton- and pion-nucleus collisions, such as those to be made by the AMBER experiment, and shows that such measurements provide significant constraints on the poorly-known parton distribution functions of the pion."}],"type":"journal_article"},{"doi":"10.1214/22-aap1882","language":[{"iso":"eng"}],"external_id":{"arxiv":["2010.16083"]},"oa":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2010.16083"}],"project":[{"_id":"62796744-2b32-11ec-9570-940b20777f1d","grant_number":"101020331","name":"Random matrices beyond Wigner-Dyson-Mehta","call_identifier":"H2020"}],"quality_controlled":"1","publication_identifier":{"issn":["1050-5164"]},"month":"08","author":[{"last_name":"Ding","first_name":"Xiucai","full_name":"Ding, Xiucai"},{"full_name":"Ji, Hong Chang","id":"dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d","last_name":"Ji","first_name":"Hong Chang"}],"volume":33,"date_created":"2024-01-08T13:03:18Z","date_updated":"2024-01-09T08:16:41Z","year":"2023","acknowledgement":"The first author is partially supported by NSF Grant DMS-2113489 and grateful for the AMS-SIMONS travel grant (2020–2023). The second author is supported by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nThe authors would like to thank the Editor, Associate Editor and an anonymous referee for their many critical suggestions which have significantly improved the paper. We also want to thank Zhigang Bao and Ji Oon Lee for many helpful discussions and comments.","department":[{"_id":"LaEr"}],"publisher":"Institute of Mathematical Statistics","publication_status":"published","ec_funded":1,"date_published":"2023-08-01T00:00:00Z","citation":{"apa":"Ding, X., & Ji, H. C. (2023). Local laws for multiplication of random matrices. The Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1882","ieee":"X. Ding and H. C. Ji, “Local laws for multiplication of random matrices,” The Annals of Applied Probability, vol. 33, no. 4. Institute of Mathematical Statistics, pp. 2981–3009, 2023.","ista":"Ding X, Ji HC. 2023. Local laws for multiplication of random matrices. The Annals of Applied Probability. 33(4), 2981–3009.","ama":"Ding X, Ji HC. Local laws for multiplication of random matrices. The Annals of Applied Probability. 2023;33(4):2981-3009. doi:10.1214/22-aap1882","chicago":"Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.” The Annals of Applied Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1882.","short":"X. Ding, H.C. Ji, The Annals of Applied Probability 33 (2023) 2981–3009.","mla":"Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.” The Annals of Applied Probability, vol. 33, no. 4, Institute of Mathematical Statistics, 2023, pp. 2981–3009, doi:10.1214/22-aap1882."},"publication":"The Annals of Applied Probability","page":"2981-3009","article_type":"original","article_processing_charge":"No","day":"01","scopus_import":"1","keyword":["Statistics","Probability and Uncertainty","Statistics and Probability"],"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"14750","intvolume":" 33","title":"Local laws for multiplication of random matrices","status":"public","issue":"4","abstract":[{"text":"Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N × N deterministic matrices and U is either an N × N Haar unitary or orthogonal random matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991) 201–220), the limiting empirical spectral distribution (ESD) of the above model is given by the free multiplicative convolution\r\nof the limiting ESDs of A and B, denoted as μα \u0002 μβ, where μα and μβ are the limiting ESDs of A and B, respectively. In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues and eigenvectors statistics. We prove that both the density of μA \u0002μB, where μA and μB are the ESDs of A and B, respectively and the associated subordination functions\r\nhave a regular behavior near the edges. Moreover, we establish the local laws near the edges on the optimal scale. In particular, we prove that the entries of the resolvent are close to some functionals depending only on the eigenvalues of A, B and the subordination functions with optimal convergence rates. Our proofs and calculations are based on the techniques developed for the additive model A+UBU∗ in (J. Funct. Anal. 271 (2016) 672–719; Comm. Math.\r\nPhys. 349 (2017) 947–990; Adv. Math. 319 (2017) 251–291; J. Funct. Anal. 279 (2020) 108639), and our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020) 108639) for the multiplicative model. ","lang":"eng"}],"type":"journal_article"}]