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The two main approaches to the study of irreducible representations of orders (via traces and Poisson orders) have so far been applied in a completely independent fashion. We define and study a natural compatibility relation between the two approaches leading to the notion of Poisson trace orders. It is proved that all regular and reduced traces are always compatible with any Poisson order structure. The modified discriminant ideals of all Poisson trace orders are proved to be Poisson ideals and the zero loci of discriminant ideals are shown to be unions of symplectic cores, under natural assumptions (maximal orders and Cayley–Hamilton algebras). A base change theorem for Poisson trace orders is proved. A broad range of Poisson trace orders are constructed based on the proved theorems: quantized universal enveloping algebras, quantum Schubert cell algebras and quantum function algebras at roots of unity, symplectic reflection algebras, 3D and 4D Sklyanin algebras, Drinfeld doubles of pre-Nichols algebras of diagonal type, and root of unity quantum cluster algebras.

Let $R$ be a commutative local $k$-algebra of Krull dimension one, where $k$ is a field. Let $\alpha$ be a $k$-algebra automorphism of $R$, and define $S$ to be the skew polynomial algebra $R[\theta; \alpha]$. We offer, under some additional assumptions on $R$, a criterion for $S$ to have injective hulls of all simple $S$-modules locally Artinian - that is, for $S$ to satisfy property $(\diamond)$. It is easy and well known that if $\alpha$ is of finite order, then $S$ has this property, but in order to get the criterion when $\alpha$ has infinite order we found it necessary to classify all cyclic (Krull) critical $S$-modules in this case, a result which may be of independent interest. With the help of the above we show that $\hat{S}=k[[X]][\theta, \alpha]$ satisfies $(\diamond)$ for all $k$-algebra automorphisms $\alpha$ of $k[[X]]$., Comment: 26 pages; comments welcome

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Full list of authors: Akiyama, Kazunori; Alberdi, Antxon; Alef, Walter; Algaba, Juan Carlos; Anantua, Richard; Asada, Keiichi; Azulay, Rebecca; Bach, Uwe; Baczko, Anne-Kathrin; Ball, David; Balokovic, Mislav; Barrett, John; Baubock, Michi; Benson, Bradford A.; Bintley, Dan; Blackburn, Lindy; Blundell, Raymond; Bouman, Katherine L.; Bower, Geoffrey C.; Boyce, Hope; Bremer, Michael; Brinkerink, Christiaan D.; Brissenden, Roger; Britzen, Silke; Broderick, Avery E.; Broguiere, Dominique; Bronzwaer, Thomas; Bustamante, Sandra; Byun, Do-Young; Carlstrom, John E.; Ceccobello, Chiara; Chael, Andrew; Chan, Chi-kwan; Chatterjee, Koushik; Chatterjee, Shami; Chen, Ming-Tang; Chen, Yongjun; Cheng, Xiaopeng; Cho, Ilje; Christian, Pierre; Conroy, Nicholas S.; Conway, John E.; Cordes, James M.; Crawford, Thomas M.; Crew, Geoffrey B.; Cruz-Osorio, Alejandro; Cui, Yuzhu; Davelaar, Jordy; De Laurentis, Mariafelicia; Deane, Roger; Dempsey, Jessica; Desvignes, Gregory; Dexter, Jason; Dhruv, Vedant; Doeleman, Sheperd S.; Dougal, Sean; Dzib, Sergio A.; Eatough, Ralph P.; Emami, Razieh; Falcke, Heino; Farah, Joseph; Fish, Vincent L.; Fomalont, Ed; Ford, H. Alyson; Fraga-Encinas, Raquel; Freeman, William T.; Friberg, Per; Fromm, Christian M.; Fuentes, Antonio; Galison, Peter; Gammie, Charles F.; Garcia, Roberto; Gentaz, Olivier; Georgiev, Boris; Goddi, Ciriaco; Gold, Roman; Gomez-Ruiz, Arturo, I; Gomez, Jose L.; Gu, Minfeng; Gurwell, Mark; Hada, Kazuhiro; Haggard, Daryl; Haworth, Kari; Hecht, Michael H.; Hesper, Ronald; Heumann, Dirk; Ho, Luis C.; Ho, Paul; Honma, Mareki; Huang, Chih-Wei L.; Huang, Lei; Hughes, David H.; Ikeda, Shiro; Impellizzeri, C. M. Violette; Inoue, Makoto; Issaoun, Sara; James, David J.; Jannuzi, Buell T.; Janssen, Michael; Jeter, Britton; Jiang, Wu; Jimenez-Rosales, Alejandra; Johnson, Michael D.; Jorstad, Svetlana; Joshi, Abhishek, V; Jung, Taehyun; Karami, Mansour; Karuppusamy, Ramesh; Kawashima, Tomohisa; Keating, Garrett K.; Kettenis, Mark; Kim, Dong-Jin; Kim, Jae-Young; Kim, Jongsoo; Kim, Junhan; Kino, Motoki; Koay, Jun Yi; Kocherlakota, Prashant; Kofuji, Yutaro; Koch, Patrick M.; Koyama, Shoko; Kramer, Carsten; Kramer, Michael; Krichbaum, Thomas P.; Kuo, Cheng-Yu; La Bella, Noemi; Lauer, Tod R.; Lee, Daeyoung; Lee, Sang-Sung; Leung, Po Kin; Levis, Aviad; Li, Zhiyuan; Lico, Rocco; Lindahl, Greg; Lindqvist, Michael; Lisakov, Mikhail; Liu, Jun; Liu, Kuo; Liuzzo, Elisabetta; Lo, Wen-Ping; Lobanov, Andrei P.; Loinard, Laurent; Lonsdale, Colin J.; Lu, Ru-Sen; Mao, Jirong; Marchili, Nicola; Markoff, Sera; Marrone, Daniel P.; Marscher, Alan P.; Marti-Vidal, Ivan; Matsushita, Satoki; Matthews, Lynn D.; Medeiros, Lia; Menten, Karl M.; Michalik, Daniel; Mizuno, Izumi; Mizuno, Yosuke; Moran, James M.; Moriyama, Kotaro; Moscibrodzka, Monika; Muller, Cornelia; Mus, Alejandro; Musoke, Gibwa; Myserlis, Ioannis; Nadolski, Andrew; Nagai, Hiroshi; Nagar, Neil M.; Nakamura, Masanori; Narayan, Ramesh; Narayanan, Gopal; Natarajan, Iniyan; Nathanail, Antonios; Fuentes, Santiago Navarro; Neilsen, Joey; Neri, Roberto; Ni, Chunchong; Noutsos, Aristeidis; Nowak, Michael A.; Oh, Junghwan; Okino, Hiroki; Olivares, Hector; Ortiz-Leon, Gisela N.; Oyama, Tomoaki; Palumbo, Daniel C. M.; Paraschos, Georgios Filippos; Park, Jongho; Parsons, Harriet; Patel, Nimesh; Pen, Ue-Li; Pesce, Dominic W.; Pietu, Vincent; Plambeck, Richard; PopStefanija, Aleksandar; Porth, Oliver; Potzl, Felix M.; Prather, Ben; Preciado-Lopez, Jorge A.; Pu, Hung-Yi; Ramakrishnan, Venkatessh; Rao, Ramprasad; Rawlings, Mark G.; Raymond, Alexander W.; Rezzolla, Luciano; Ricarte, Angelo; Ripperda, Bart; Roelofs, Freek; Rogers, Alan; Ros, Eduardo; Romero-Canizales, Cristina; Roshanineshat, Arash; Rottmann, Helge; Roy, Alan L.; Ruiz, Ignacio; Ruszczyk, Chet; Rygl, Kazi L. J.; Sanchez, Salvador; Sanchez-Arguelles, David; Sanchez-Portal, Miguel; Sasada, Mahito; Satapathy, Kaushik; Savolainen, Tuomas; Schloerb, F. Peter; Schonfeld, Jonathan; Schuster, Karl-Friedrich; Shao, Lijing; Shen, Zhiqiang; Small, Des; Sohn, Bong Won; SooHoo, Jason; Souccar, Kamal; Sun, He; Tazaki, Fumie; Tetarenko, Alexandra J.; Tiede, Paul; Tilanus, Remo P. J.; Titus, Michael; Torne, Pablo; Traianou, Efthalia; Trent, Tyler; Trippe, Sascha; Turk, Matthew; van Bemmel, Ilse; van Langevelde, Huib Jan; van Rossum, Daniel R.; Vos, Jesse; Wagner, Jan; Ward-Thompson, Derek; Wardle, John; Weintroub, Jonathan; Wex, Norbert; Wharton, Robert; Wielgus, Maciek; Wiik, Kaj; Witzel, Gunther; Wondrak, Michael F.; Wong, George N.; Wu, Qingwen; Yamaguchi, Paul; Yoon, Doosoo; Young, Andre; Young, Ken; Younsi, Ziri; Yuan, Feng; Yuan, Ye-Fei; Zensus, J. Anton; Zhang, Shuo; Zhao, Guang-Yao; Zhao, Shan-Shan; Ozel, Feryal; Agurto, Claudio; Allardi, Alexander; Amestica, Rodrigo; Araneda, Juan Pablo; Arriagada, Oriel; Berghuis, Jennie L.; Bertarini, Alessandra; Berthold, Ryan; Blanchard, Jay; Brown, Ken; Cardenas, Mauricio; Cantzler, Michael; Caro, Patricio; Castillo-Dominguez, Edgar; Chan, Tin Lok; Chang, Chih-Cheng; Chang, Dominic O.; Chang, Shu-Hao; Chang, Song-Chu; Chen, Chung-Chen; Chilson, Ryan; Chuter, Tim C.; Ciechanowicz, Miroslaw; Colin-Beltran, Edgar; Coulson, Iain M.; Crowley, Joseph; Degenaar, Nathalie; Dornbusch, Sven; Duran, Carlos A.; Everett, Wendeline B.; Faber, Aaron; Forster, Karl; Fuchs, Miriam M.; Gale, David M.; Geertsema, Gertie; Gonzalez, Edouard; Graham, Dave; Gueth, Frederic; Halverson, Nils W.; Han, Chih-Chiang; Han, Kuo-Chang; Hasegawa, Yutaka; Hernandez-Rebollar, Jose Luis; Herrera, Cristian; Herrero-Illana, Ruben; Heyminck, Stefan; Hirota, Akihiko; Hoge, James; Schimpf, Shelbi R. Hostler; Howie, Ryan E.; Huang, Yau-De; Jiang, Homin; Jinchi, Hao; John, David; Kimura, Kimihiro; Klein, Thomas; Kubo, Derek; Kuroda, John; Kwon, Caleb; Lacasse, Richard; Laing, Robert; Leitch, Erik M.; Li, Chao-Te; Liu, Ching-Tang; Liu, Kuan-Yu; Lin, Lupin C-C; Lu, Li-Ming; Mac-Auliffe, Felipe; Martin-Cocher, Pierre; Matulonis, Callie; Maute, John K.; Messias, Hugo; Meyer-Zhao, Zheng; Montana, Alfredo; Montenegro-Montes, Francisco; Montgomerie, William; Nolasco, Marcos Emir Moreno; Muders, Dirk; Nishioka, Hiroaki; Norton, Timothy J.; Nystrom, George; Ogawa, Hideo; Olivares, Rodrigo; Oshiro, Peter; Perez-Beaupuits, Juan Pablo; Parra, Rodrigo; Phillips, Neil M.; Poirier, Michael; Pradel, Nicolas; Qiu, Richard; Raffin, Philippe A.; Rahlin, Alexandra S.; Ramirez, Jorge; Ressler, Sean; Reynolds, Mark; Rodriguez-Montoya, Ivan; Saez-Madain, Alejandro F.; Santana, Jorge; Shaw, Paul; Shirkey, Leslie E., Jr.; Silva, Kevin M.; Snow, William; Sousa, Don; Sridharan, T. K.; Stahm, William; Stark, Anthony A.; Test, John; Torstensson, Karl; Venegas, Paulina; Walther, Craig; Wei, Ta-Shun; White, Chris; Wieching, Gundolf; Wijnands, Rudy; Wouterloot, Jan G. A.; Yu, Chen-Yu; Yu, Wei; Zeballos, Milagros; Event Horizon Telescope Collaboration., We present the first Event Horizon Telescope (EHT) observations of Sagittarius A* (Sgr A*), the Galactic center source associated with a supermassive black hole. These observations were conducted in 2017 using a global interferometric array of eight telescopes operating at a wavelength of λ = 1.3 mm. The EHT data resolve a compact emission region with intrahour variability. A variety of imaging and modeling analyses all support an image that is dominated by a bright, thick ring with a diameter of 51.8 ± 2.3 μas (68% credible interval). The ring has modest azimuthal brightness asymmetry and a comparatively dim interior. Using a large suite of numerical simulations, we demonstrate that the EHT images of Sgr A* are consistent with the expected appearance of a Kerr black hole with mass ∼4 × 106 M⊙, which is inferred to exist at this location based on previous infrared observations of individual stellar orbits, as well as maser proper-motion studies. Our model comparisons disfavor scenarios where the black hole is viewed at high inclination (i > 50°), as well as nonspinning black holes and those with retrograde accretion disks. Our results provide direct evidence for the presence of a supermassive black hole at the center of the Milky Way, and for the first time we connect the predictions from dynamical measurements of stellar orbits on scales of 103–105 gravitational radii to event-horizon-scale images and variability. Furthermore, a comparison with the EHT results for the supermassive black hole M87* shows consistency with the predictions of general relativity spanning over three orders of magnitude in central mass. © 2022. The Author(s). Published by the American Astronomical Society., The Event Horizon Telescope Collaboration thanks the following organizations and programs: the Academia Sinica; the Academy of Finland (projects 274477, 284495, 312496, 315721); the Agencia Nacional de Investigación y Desarrollo (ANID), Chile via NCN19_058 (TITANs) and Fondecyt 1221421, the Alexander von Humboldt Stiftung; an Alfred P. Sloan Research Fellowship; Allegro, the European ALMA Regional Centre node in the Netherlands, the NL astronomy research network NOVA and the astronomy institutes of the University of Amsterdam, Leiden University and Radboud University; the ALMA North America Development Fund; the Black Hole Initiative, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation (although the opinions expressed in this work are those of the author(s) and do not necessarily reflect the views of these Foundations); Chandra DD7-18089X and TM6-17006X; the China Scholarship Council; China Postdoctoral Science Foundation fellowship (2020M671266); Consejo Nacional de Ciencia y Tecnología (CONACYT, Mexico, projects U0004-246083, U0004-259839, F0003-272050, M0037-279006, F0003-281692, 104497, 275201, 263356); the Consejería de Economía, Conocimiento, Empresas y Universidad of the Junta de Andalucía (grant P18-FR-1769), the Consejo Superior de Investigaciones Científicas (grant 2019AEP112); the Delaney Family via the Delaney Family John A. Wheeler Chair at Perimeter Institute; Dirección General de Asuntos del Personal Académico-Universidad Nacional Autónoma de México (DGAPA-UNAM, projects IN112417 and IN112820); the Dutch Organization for Scientific Research (NWO) VICI award (grant 639.043.513) and grant OCENW.KLEIN.113; the Dutch National Supercomputers, Cartesius and Snellius (NWO Grant 2021.013); the EACOA Fellowship awarded by the East Asia Core Observatories Association, which consists of the Academia Sinica Institute of Astronomy and Astrophysics, the National Astronomical Observatory of Japan, Center for Astronomical Mega-Science, Chinese Academy of Sciences, and the Korea Astronomy and Space Science Institute; the European Research Council (ERC) Synergy Grant "BlackHoleCam: Imaging the Event Horizon of Black Holes" (grant 610058); the European Union Horizon 2020 research and innovation programme under grant agreements RadioNet (No 730562) and M2FINDERS (No 101018682); the Generalitat Valenciana postdoctoral grant APOSTD/2018/177 and GenT Program (project CIDEGENT/2018/021); MICINN Research Project PID2019-108995GB-C22; the European Research Council for advanced grant "JETSET: Launching, propagation and emission of relativistic jets from binary mergers and across mass scales" (Grant No. 884631); the Institute for Advanced Study; the Istituto Nazionale di Fisica Nucleare (INFN) sezione di Napoli, iniziative specifiche TEONGRAV; the International Max Planck Research School for Astronomy and Astrophysics at the Universities of Bonn and Cologne; DFG research grant "Jet physics on horizon scales and beyond" (Grant No. FR 4069/2-1); Joint Princeton/Flatiron and Joint Columbia/Flatiron Postdoctoral Fellowships, research at the Flatiron Institute is supported by the Simons Foundation; the Japan Ministry of Education, Culture, Sports, Science and Technology (MEXT; grant JPMXP1020200109); the Japanese Government (Monbukagakusho: MEXT) Scholarship; the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for JSPS Research Fellowship (JP17J08829); the Joint Institute for Computational Fundamental Science, Japan; the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS, grants QYZDJ-SSW-SLH057, QYZDJSSW-SYS008, ZDBS-LY-SLH011); the Leverhulme Trust Early Career Research Fellowship; the Max-Planck-Gesellschaft (MPG); the Max Planck Partner Group of the MPG and the CAS; the MEXT/JSPS KAKENHI (grants 18KK0090, JP21H01137, JP18H03721, JP18K13594, 18K03709, JP19K14761, 18H01245, 25120007); the Malaysian Fundamental Research Grant Scheme (FRGS) FRGS/1/2019/STG02/UM/02/6; the MIT International Science and Technology Initiatives (MISTI) Funds; the Ministry of Science and Technology (MOST) of Taiwan (103-2119-M-001-010-MY2, 105-2112-M-001-025-MY3, 105-2119-M-001-042, 106-2112-M-001-011, 106-2119-M-001-013, 106-2119-M-001-027, 106-2923-M-001-005, 107-2119-M-001-017, 107-2119-M-001-020, 107-2119-M-001-041, 107-2119-M-110-005, 107-2923-M-001-009, 108-2112-M-001-048, 108-2112-M-001-051, 108-2923-M-001-002, 109-2112-M-001-025, 109-2124-M-001-005, 109-2923-M-001-001, 110-2112-M-003-007-MY2, 110-2112-M-001-033, 110-2124-M-001-007, and 110-2923-M-001-001); the Ministry of Education (MoE) of Taiwan Yushan Young Scholar Program; the Physics Division, National Center for Theoretical Sciences of Taiwan; the National Aeronautics and Space Administration (NASA, Fermi Guest Investigator grant 80NSSC20K1567, NASA Astrophysics Theory Program grant 80NSSC20K0527, NASA NuSTAR award 80NSSC20K0645); NASA Hubble Fellowship grant HST-HF2-51431.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555; the National Institute of Natural Sciences (NINS) of Japan; the National Key Research and Development Program of China (grant 2016YFA0400704, 2017YFA0402703, 2016YFA0400702); the National Science Foundation (NSF, grants AST-0096454, AST-0352953, AST-0521233, AST-0705062, AST-0905844, AST-0922984, AST-1126433, AST-1140030, DGE-1144085, AST-1207704, AST-1207730, AST-1207752, MRI-1228509, OPP-1248097, AST-1310896, AST-1440254, AST-1555365, AST-1614868, AST-1615796, AST-1715061, AST-1716327, AST-1716536, OISE-1743747, AST-1816420, AST-1935980, AST-2034306); NSF Astronomy and Astrophysics Postdoctoral Fellowship (AST-1903847); the Natural Science Foundation of China (grants 11650110427, 10625314, 11721303, 11725312, 11873028, 11933007, 11991052, 11991053, 12192220, 12192223); the Natural Sciences and Engineering Research Council of Canada (NSERC, including a Discovery Grant and the NSERC Alexander Graham Bell Canada Graduate Scholarships-Doctoral Program); the National Youth Thousand Talents Program of China; the National Research Foundation of Korea (the Global PhD Fellowship Grant: grants NRF-2015H1A2A1033752, the Korea Research Fellowship Program: NRF-2015H1D3A1066561, Brain Pool Program: 2019H1D3A1A01102564, Basic Research Support Grant 2019R1F1A1059721, 2021R1A6A3A01086420, 2022R1C1C1005255); Netherlands Research School for Astronomy (NOVA) Virtual Institute of Accretion (VIA) postdoctoral fellowships; Onsala Space Observatory (OSO) national infrastructure, for the provisioning of its facilities/observational support (OSO receives funding through the Swedish Research Council under grant 2017-00648); the Perimeter Institute for Theoretical Physics (research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science); the Spanish Ministerio de Ciencia e Innovación (grants PGC2018-098915-B-C21, AYA2016-80889-P, PID2019-108995GB-C21, PID2020-117404GB-C21); the University of Pretoria for financial aid in the provision of the new Cluster Server nodes and SuperMicro (USA) for a SEEDING GRANT approved towards these nodes in 2020; the Shanghai Pilot Program for Basic Research, Chinese Academy of Science, Shanghai Branch (JCYJ-SHFY-2021-013); the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award for the Instituto de Astrofísica de Andalucía (SEV-2017- 0709); the Spinoza Prize SPI 78-409; the South African Research Chairs Initiative, through the South African Radio Astronomy Observatory (SARAO, grant ID 77948), which is a facility of the National Research Foundation (NRF), an agency of the Department of Science and Innovation (DSI) of South Africa; the Toray Science Foundation; Swedish Research Council (VR); the US Department of Energy (USDOE) through the Los Alamos National Laboratory (operated by Triad National Security, LLC, for the National Nuclear Security Administration of the USDOE (Contract 89233218CNA000001); and the YCAA Prize Postdoctoral Fellowship. APEX is a collaboration between the Max-Planck-Institut für Radioastronomie (Germany), ESO, and the Onsala Space Observatory (Sweden). The SMA is a joint project between the SAO and ASIAA and is funded by the Smithsonian Institution and the Academia Sinica. The JCMT is operated by the East Asian Observatory on behalf of the NAOJ, ASIAA, and KASI, as well as the Ministry of Finance of China, Chinese Academy of Sciences, and the National Key Research and Development Program (No. 2017YFA0402700) of China and Natural Science Foundation of China grant 11873028. Additional funding support for the JCMT is provided by the Science and Technologies Facility Council (UK) and participating universities in the UK and Canada. The LMT is a project operated by the Instituto Nacional de Astrófisica, Óptica, y Electrónica (Mexico) and the University of Massachusetts at Amherst (USA). The IRAM 30-m telescope on Pico Veleta, Spain is operated by IRAM and supported by CNRS (Centre National de la Recherche Scientifique, France), MPG (Max-Planck-Gesellschaft, Germany) and IGN (Instituto Geográfico Nacional, Spain). The SMT is operated by the Arizona Radio Observatory, a part of the Steward Observatory of the University of Arizona, with financial support of operations from the State of Arizona and financial support for instrumentation development from the NSF. Support for SPT participation in the EHT is provided by the National Science Foundation through award OPP-1852617 to the University of Chicago. Partial support is also provided by the Kavli Institute of Cosmological Physics at the University of Chicago. The SPT hydrogen maser was provided on loan from the GLT, courtesy of ASIAA. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), supported by NSF grant ACI-1548562, and CyVerse, supported by NSF grants DBI-0735191, DBI-1265383, and DBI-1743442. XSEDE Stampede2 resource at TACC was allocated through TG-AST170024 and TG-AST080026N. XSEDE JetStream resource at PTI and TACC was allocated through AST170028. This research is part of the Frontera computing project at the Texas Advanced Computing Center through the Frontera Large-Scale Community Partnerships allocation AST20023. Frontera is made possible by National Science Foundation award OAC-1818253. This research was carried out using resources provided by the Open Science Grid, which is supported by the National Science Foundation and the U.S. Department of Energy Office of Science. Additional work used ABACUS2.0, which is part of the eScience center at Southern Denmark University. Simulations were also performed on the SuperMUC cluster at the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, on the HazelHen cluster at the HLRS in Stuttgart, and on the Pi2.0 and Siyuan Mark-I at Shanghai Jiao Tong University. The computer resources of the Finnish IT Center for Science (CSC) and the Finnish Computing Competence Infrastructure (FCCI) project are acknowledged. This research was enabled in part by support provided by Compute Ontario (http://computeontario.ca), Calcul Quebec (http://www.calculquebec.ca) and Compute Canada (http://www.computecanada.ca).

Although Mathematics is a pillar which many of the subjects of any Engineering degree are based on, and in spite of being used as a tool in almost all of such subjects, it is sometimes difficult for students to assimilate the mathematical concepts. Since this relationship between Engineering and Mathematics is unavoidable, a consortium of universities from different countries is working on a project, EngiMath, to, on the one hand, make it easier for teachers to teach Mathematics in the first years of Engineering degrees; and, on the other hand, to promote study from a “student-centric” point of view. As a previous step to the development of educational material that fulfills these purposes, a study of the students' feelings about Mathematics and their experience in studying them has been carried out.

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.

Six Higher Education Institutions (HEI), from different European countries (Estonia, Ireland, Poland, Portugal, Romania, and Spain) joined forces in 2018 to launch a shared Erasmus+ Project under the Key Action - Cooperation for innovation and the exchange of good practices. The project's “heart and soul” was to provide an online course on fundamental mathematical themes that would serve as a base for the specialized domain of engineering mathematics. Upon its approval and funding, partner institutions have developed a common platform for students and professors interested in Fundamentals of Linear Algebra and engineering applications. A mathematics on-line learning model was developed from the scratch and is now available in seven different languages. As presented in the project proposal, the student competition activity was introduced as a Learning/Teaching/Training activity (LTT) and it was established inside the online course setting and framework to add an international dimension to the studies of the enrolled students. The students’ competition in EngiMath project, connected learners from different countries through common tasks. It has settled an open international space, where students could deal with assessment “stresses” in an indirect and ludic manner, promoting, in a gamification way, their self-confidence when dealing with on-line tasks, tight schedules or even “against the clock”, motivating them to complete the course and to avoid drop out behaviour. Despite all of the pandemic limitations that all partner members had to cope with, the students' competition can be considered one of the Project's main triumphs. Peer Reviewed Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat::4.3 - Per a 2030, assegurar l’accés en condicions d’igualtat per a tots els homes i dones a una formació tècnica, professional i superior de qualitat, inclòs l’ensenyament universitari Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat::4.4 - Per a 2030, augmentar substancialment el nombre de joves i persones adultes que tenen les competències necessàries, en particular tècniques i professionals, per a accedir a l’ocupació, el treball digne i l’emprenedoria

According to the New UNESCO global survey studying the effect of Covid-19 on higher education (2021) the pandemic has had an impact on higher education systems in terms of access and quality of teaching and learning. Covid-19 has caused the suspension and cancellation of teaching activities and its major impact on teaching and learning is the increase in online education. The Engimath - Mathematics on-line learning model in engineering education - was successfully carried out at the very right moment, as in recent two years there has been a significant increase in demand for a 100% online mathematics course with interactive learning materials and largescale practicing opportunities as well as for a mathematics online assessment model. This project has successfully developed a high-level online course as a basis for offering output-oriented education in engineering mathematics. Some of development stages, that led this project – EngiMath - to an outstanding successful end, will be presented in this paper. These go from the educational needs’ analysis to its implementation and use, even allowing an open and live test. Peer Reviewed Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat::4.3 - Per a 2030, assegurar l’accés en condicions d’igualtat per a tots els homes i dones a una formació tècnica, professional i superior de qualitat, inclòs l’ensenyament universitari Objectius de Desenvolupament Sostenible::4 - Educació de Qualitat::4.4 - Per a 2030, augmentar substancialment el nombre de joves i persones adultes que tenen les competències necessàries, en particular tècniques i professionals, per a accedir a l’ocupació, el treball digne i l’emprenedoria

PURPOSE The primary aim of this phase III randomized trial was to test whether the addition of vincristine, topotecan, and cyclophosphamide (VTC) to interval compressed chemotherapy improved survival outcomes for patients with previously untreated nonmetastatic Ewing sarcoma. METHODS Patients were randomly assigned to receive standard five-drug interval compressed chemotherapy (regimen A) for 17 cycles or experimental therapy with five cycles of VTC within the 17 cycles (regimen B). Patients were stratified by age at diagnosis (< 18 years and ≥18 years) and tumor site (pelvic bone, nonpelvic bone, and extraosseous). Tumor volume at diagnosis was categorized as < 200 mL or ≥ 200 mL. Local control occurred following six cycles. Histologic response was categorized as no viable or any viable tumor. Event-free survival (EFS) and overall survival (OS) were compared between randomized groups with stratified log-rank tests. RESULTS Of 642 enrolled patients, 309 eligible patients received standard and 320 received experimental therapy. The 5-year EFS and OS were 78% and 87%, respectively. There was no difference in survival outcomes between randomized groups (5-year EFS regimen A v regimen B, 78% v 79%; P = .192; 5-year OS 86% v 88%; P = .159). Age and primary site did not affect the risk of an EFS event. However, age ≥ 18 years was associated with an increased risk of death at 5 years (hazard ratio 1.84; 95% CI, 1.15 to 2.96; P = .009). The 5-year EFS rates for patients with pelvic, nonpelvic bone, and extraosseous primary tumors were 75%, 78%, and 85%, respectively. Tumor volume ≥ 200 mL was significantly associated with lower EFS. CONCLUSION While VTC added to five-drug interval compressed chemotherapy did not improve survival, these outcomes represent the best survival estimates to date for patients with previously untreated nonmetastatic Ewing sarcoma.

Job 18 depicts the destruction of the wicked as a kind of ambush by “the firstborn of death.” Much of the discussion of this passage has centered on this figure’s identification, and whether one should look primarily to Ugaritic or Mesopotamian mythological traditions for its background. Yet the passage as a whole concludes with a reference to a single “God,” knowledge of whom is determinative for human fate. This raises a basic question concerning the relation between “God” and the “firstborn of death.” Through a close comparison with the Ugaritic Baal Cycle and the Neo-Assyrian Underworld Vision on the one hand, and Job 5 and Deuteronomy 32 on the other, this paper argues that “the firstborn of death” most likely does represent a chthonic deity, but that such powers have been subordinated to the one God whom Bildad presumes to bear sole authority over life and death.

The goal of this paper is to introduce a technique of creating self-tests that has allowed to actively incorporate university students into the learning process. The study was conducted within the framework of the Erasmus+ Project EngiMath. Partners’ peer reviews, the survey results and the students’ comments in forums and test results were used to conduct the research. The students’ overall satisfaction was in a high level. However, opportunities for some technical improvement has been emerged like the formulation of the tasks needs to be very clear and the time required to perform the tests must be limited. The following conclusions can be drawn from the study. The use of self-tests at all stages of training has intensified the assimilation of the material, i.e. increased understanding of theoretical material and developed computational skills. By completing a series of such assignments on each topic of the course, students had mastered the methodology of studying the topic and mastered specific teaching material on this topic. Feedback made, taking into account typical errors, has allowed the students to analyse their knowledge. A large number of variations for such tasks has allowed students to be involved in the process of active independent and individualized self-study.

The fast evolution of distance learning tools such as Open Educational Resources (OER)) is an evidence of a shift in the way teaching and learning are understood. Several Higher Education Institutions are trying to increase their efficiency, competitiveness and expand their potential public by investing in the development of online courses, which can offer more interaction and support and be accessible to a larger number of students from a wider sort of backgrounds. The adaptation of a face-to face course into an online one is not simple, as it is not just about uploading lecture videos, sets of notes or lessons, it is necessary to consider the student’s needs, the instructional design and the best digital educational tools in order to support learning and teaching process. A partnership between Higher Education Institutions from six European countries, connecting more than twenty lecturers from distinct knowledge areas, is working on a European Erasmus+ Project, EngiMath, Mathematics online learning model in engineering education. These are developing a shared online platform for teachers to teach Mathematics in the first years of Engineering degrees and, on the other hand, to support student-centric learning and to encourage students to actively engage in the learning process to construct their own learning, addressing the recent requirements for an open, independent, competitive and innovative education. In this context, several new methodological and development issues of the online course will be presented, as well as students’ perceptions and feedback about the pilot course conducted by each institution partner of the project.

Innovation has become the central engine of economic well-being, competitiveness, employment and long-term income growth for most economies. As the European Union Innovation Scoreboard shows, Romania is part of the modest innovators group. The situation of innovation in Romania has worsened so much that since 2015 it occupies the last place in the EU. The deterioration of the innovation condition in Romania is confirmed by The Global Competitiveness Report. There is a direct link between state measures and performance in innovation, as shown by the "Contributors and Detractors: Ranking Countries’ Impact on Global Innovation” report. The process of innovation itself involves a concerted effort of economic agents and the state, the latter having the role of facilitating, consolidating and supporting innovative activities. The innovative potential of an enterprise can be strengthened by the human factor through the use of creative methods and techniques.

Engineering and Mathematics are closely related, the latter being a fundamental tool and being also at the base of many, if not all, of the processes employed in the former one. Engineering in turn is a source of application for those mathematical concepts that are, sometimes, difficult to understand by undergraduate students in universities. Precisely, so that this relationship can be more fluid and comfortable for students in Higher Education, a group of European universities have teamed up around a project called EngiMath -Mathematics on-line learning model in Engineering education.

There has been a rapid growth of massive open online courses (MOOCs) in the global education market in the last decade. Online learning technologies are becoming increasingly widespread in the non-formal education sector and in higher and supplementary vocational education. The use of MOOCs in Russia to support the delivery of educational programmes at university level opens opportunities in terms of expanding the educational choice for students, the development of virtual academic mobility, reduction in the cost of educational services, and improvement in the accessibility of education. However, the effectiveness of using different online learning technologies at university level, and the consequences of their widespread adoption, has not been sufficiently explored. In this research study, a comparative analysis is made of the effects of different online learning models on student educational outcomes in a university setting. A study was undertaken in which different groups of students at the Ural Federal University, Russia, were encouraged to study technical and humanities disciplines using a framework of blended learning, and online learning with tutoring support. The results of the study were compared with the results of a reference (control) group of students who studied the same disciplines in a traditionally taught model. It was found that both models (blended and online) of MOOC implementation demonstrated greater learning gains, in comparison with the traditional model. For engineering and technical disciplines, there was no statistically significant difference between blended or online learning technologies. For the humanities discipline, where the communicative component of the learning process was significant, the blended learning technology produced better results. Conclusions of this empirical research may be useful for heads of educational organizations and teachers in helping them to make strategic decisions about the modernization of university courses by increasing the effectiveness of the implementation of new educational technologies. The results of this research project will be used for implementing the State Priority Project, ‘The Modern Digital Educational Environment of the Russian Federation’. © 2018, © The Author(s) 2018.

Some students perceive that online assessment does not provide for a true reflection of their work effort. This article reports on a collaborative international project between two higher education institutions with the aim of researching issues relating to engineering student perceptions with respect to online assessment of mathematics. It provides a comparison between students of similar educational standing in Finland and Ireland. The students undertook to complete questionnaires and a sample of students was selected to participate in several group discussion interviews. Evidence from the data suggests that many of the students demonstrate low levels of confidence, do not display knowledge of continuous assessment processes and perceive many barriers when confronted with online assessment in their first semester. Alternative perspectives were sought from lecturers by means of individual interviews. The research indicates that perceptions of effort and reward as seen by students are at variance with those held by lecturers. The study offers a brief insight into the thinking of students in the first year of their engineering mathematics course. It may be suggested that alternative approaches to curriculum and pedagogical design are necessary to alleviate student concerns.

This paper reflects on some of the themes emerging from a consideration of recent research at the nexus of technologies, learning and culture. The authors comment on the expansive nature of the concept of learning spaces in papers featuring an investigation of technology enhanced learning (TEL) and communication design studios in the UK and Australia, the use of interdisciplinary research collaborations to develop novel implementations of TEL learning spaces, and the challenges of developing an e-university in Malawi. They also examine a comparative study focused on classroom-based learning spaces augmented by computer-based assessment technologies, and the role of TEL both within and in response to protests at universities in South Africa. Massive open online courses are then considered as distinctive educational designs that may offer diverse student experiences, either formal or informal. The next emerging theme considers the sources of tension and richness arising from the widely divergent values that can be embedded in TEL. This is followed by consideration of infrastructural issues and the technologies–learning–culture nexus, followed by the use of theory in TEL work, leading to interdisciplinary theory-informed TEL projects that may be beneficial in the wider project of reimagining higher education for work and study. Finally, the paper examines the theme of mobile TEL and the hegemonic issues surrounding the building of sustainable and authentic foundations for learning with mobiles in the globalised South. The theme points to the methodologically challenging and problematic aspects of this hegemonic analysis and considers how the arguments may be further developed.

At the end of the 2018, a new multinational project saw its approval as a European Erasmus+ Project. This project, named EngiMath - Mathematics on-line learning model in Engineering education, is closely related to Mathematics Teaching in Engineering Degrees. With its end planned for 2021, the steps needed to kick-start the development of this multinational project have not been easy in general logistic terms and even in more specific and particular issues. One of the first steps in its development was connected to the identification of common themes/subjects on which this project should be based in theoretical and general terms. This paper will describe, in a detailed way, all the educational needs analysis development. Its inclusion within this programme of work was due to the development, testing and deployment of a novel paradigm in the technology enhanced learning mediated environment in six geographically, socially, and culturally separate, engineering institutions. This was not a trivial task and took many resource hours to complete. Prior research to the submission of the project found no literature guiding educational needs analysis for joint-shared initiatives relating to socially, culturally, or geographically, separate locations. Therefore, this paper hopes to mark a complete list of objectives and tasks developed. It will point out several important issues to consider when designing multinational educational projects as well as other international connections made in education.

Brief life case-based games are referred to as sprint games. The short games strengthened with predefined gaming construct and gaming components are used under conditions of limited time. Gaming components can include, for example, competition and struggle for the prize. Introduction of these components into learning environment help bridging game with reality, creating business-like atmosphere of competition and maximizing gains within the game. Sprint games help forming efficient management skills. There are several types of sprint games that differ in terms of content and approaches taken. In our practice we use several types of short games depending on the tasks that we pursue. We are ready to share experience, but the key point is: the one that would understand potential clients better and faster, and present the product the customers are ready to pay for-wins. Such experience of change management is important and actual for a modern entrepreneur, helping to adopt company strategy with regard to internal and external challenges. © The Authors, 2019. All Rights Reserved.

Traditional didactic pedagogies employed within the culture of the Russian higher education system precluded students’ engagement with problems which were described as generating dissonances in learning cognition. Addressing issues of dissonance within the higher education learning sphere requires re-imagining the educational culture. Re-imagining provides an opportunity to promote new approaches to learning through alternative affordances; one such affordance is technology mediated learning. Pedagogical re-design within an alternative learning paradigm requires deep understanding of the problems associated with the previous paradigm. Re-imagined pedagogical scope for exploration of the professional, learning, cultural, institutional and technical aspects expand the knowledge base beyond the didactic towards an engaging student-centered ethos using open education and gamification. To address issues of learning, culture, technology, and institution, a convergent mixed methods design using student questionnaires and academic interviews alongside performance observations was employed. The research study examined the re-imagining of the educational culture to promote new approaches to learning through the affordances of technology mediated learning within a constructivist, critical realism epistemology using thematic analysis. The re-imagined pedagogical design within a technology mediated learning environment demonstrates a cultural shift towards an engaging and supportive educational experience. The lessons learned may be applied in other higher educational contexts.

The COVID pandemic has touched many aspects of everyone’s life. Education is one of the fields greatly affected by it, as students and teachers were forced to move online and quickly adapt to the online environment. Assessment is a crucial part of education, especially in STEM fields. A gap analysis was performed by expert groups in the frame of an Erasmus+ project looking at the practices of six European countries. Specialists teaching university-grade mathematics in seven European institutions were asked about their perception of gaps in the assessment of students both before (2019) and during (2021) the pandemic. This qualitative study looks at the difference in perception of such gaps after almost one year of online teaching. The analysis of their responses showed that some gaps were present before the pandemic, as well as others that are specific to it. Some gaps, such as the lack of IT infrastructure and the need to adapt materials to an online environment, have been exacerbated by the outbreak., This research was co-funded by the Erasmus+ Programme of the European Union, project name EngiMath, ref. no. 2018-1-EE01-KA203-047098.