{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T14:47:52Z","timestamp":1775054872809,"version":"3.50.1"},"reference-count":32,"publisher":"Pleiades Publishing Ltd","issue":"1","license":[{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Program Comput Soft"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1134\/s0361768823010048","type":"journal-article","created":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T21:02:18Z","timestamp":1679864538000},"page":"42-48","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Implementation of Geometric Algebra in Computer Algebra Systems"],"prefix":"10.1134","volume":"49","author":[{"given":"M. N.","family":"Gevorkyan","sequence":"first","affiliation":[]},{"given":"A. V.","family":"Korol\u2019kova","sequence":"additional","affiliation":[]},{"given":"D. S.","family":"Kulyabov","sequence":"additional","affiliation":[]},{"given":"A. V.","family":"Demidova","sequence":"additional","affiliation":[]},{"given":"T. R.","family":"Velieva","sequence":"additional","affiliation":[]}],"member":"137","published-online":{"date-parts":[[2023,3,27]]},"reference":[{"key":"3719_CR1","doi-asserted-by":"publisher","first-page":"222","DOI":"10.1007\/BF01442659","volume":"12","author":"H.G. Grassmann","year":"1877","unstructured":"Grassmann, H.G., Die mechanik nach den principien der ausdehnungslehre, Math. Ann., 1877, vol. 12, pp.\u00a0222\u2013240.","journal-title":"Math. Ann."},{"key":"3719_CR2","doi-asserted-by":"publisher","DOI":"10.1515\/9780691211701","volume-title":"Quaternions and Rotation sequences","author":"J.B. Kuipers","year":"1999","unstructured":"Kuipers, J.B., Quaternions and Rotation sequences, Princeton Univ. Press, 1999."},{"key":"3719_CR3","doi-asserted-by":"publisher","first-page":"350","DOI":"10.2307\/2369379","volume":"1","author":"W.K. Clifford","year":"1878","unstructured":"Clifford, W.K., Applications of Grassmann\u2019s extensive algebra, Am. J. Math., 1878, vol. 1, no. 4, pp. 350\u2013358.","journal-title":"Am. J. Math."},{"key":"3719_CR4","volume-title":"L\u2019Alg\u00e8bre Vectorielle","author":"G. Casanova","year":"1976","unstructured":"Casanova, G., L\u2019Alg\u00e8bre Vectorielle, Presse Iniversitaires de France, 1976."},{"key":"3719_CR5","volume-title":"Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics","author":"D. Hestenes","year":"1987","unstructured":"Hestenes, D. and Sobczyk, G., Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Fundamental Theories of Physics, Netherlands: Springer, 1987."},{"key":"3719_CR6","volume-title":"Clifford Algebra and Spinor-Valued Functions. Mathematics and Its Applications","author":"R. Delanghe","year":"1992","unstructured":"Delanghe, R., Sommen, F., and Soucek, V., Clifford Algebra and Spinor-Valued Functions. Mathematics and Its Applications, Kluwer, 1992."},{"key":"3719_CR7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511807497","volume-title":"Geometric Algebra for Physicists","author":"C. Doran","year":"2003","unstructured":"Doran, C. and Lasenby, A., Geometric Algebra for Physicists, Morgan Kaufmann, 2003."},{"key":"3719_CR8","volume-title":"Geometric Algebra for Computer Science. The Morgan Kaufmann Series in Computer Graphics","author":"L. Dorst","year":"2007","unstructured":"Dorst, L., Fontijne, D., and Mann, S., Geometric Algebra for Computer Science. The Morgan Kaufmann Series in Computer Graphics, Morgan Kaufmann, 2007."},{"key":"3719_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-84628-997-2","volume-title":"Geometric Algebra for Computer Graphics","author":"J. Vince","year":"2008","unstructured":"Vince, J., Geometric Algebra for Computer Graphics, Springer, 2008."},{"key":"3719_CR10","volume-title":"Mathematics","author":"E. Lengyel","year":"2016","unstructured":"Lengyel, E., Mathematics, Lincoln, Calif.: Terathon Software LLC, 2016, vol. 1."},{"key":"3719_CR11","doi-asserted-by":"publisher","DOI":"10.1201\/b18273","volume-title":"Understanding Geometric Algebra","author":"K. Kanatani","year":"2015","unstructured":"Kanatani, K., Understanding Geometric Algebra, Taylor and Francis Group\/CRC, 2015."},{"key":"3719_CR12","unstructured":"ten Bosch, M., Let\u2019s remove quaternions from every 3D engine. https:\/\/marctenbosch.com\/quaternions\/."},{"key":"3719_CR13","unstructured":"Perwa, C.B.U., Geometric Algebra with Applications in Engineering. Geometry and Computing, Berkin: Springer, 2009."},{"key":"3719_CR14","volume-title":"Geometric Algebra for Electrical Engineers: Multivector Electromagnetism","author":"P. Joot","year":"2019","unstructured":"Joot, P., Geometric Algebra for Electrical Engineers: Multivector Electromagnetism, CreateSpace Independent Publishing Platform, 2019."},{"key":"3719_CR15","unstructured":"Winitzki, S., Linear Algebra via Exterior Products, 2020.\nhttps:\/\/github.com\/winitzki\/linear-algebra-book."},{"key":"3719_CR16","doi-asserted-by":"publisher","first-page":"1340","DOI":"10.1109\/JPROC.2014.2339299","volume":"102","author":"J.M. Chappell","year":"2014","unstructured":"Chappell, J.M., Drake, S.P., Seidel, C.L., et al. Geometric algebra for electrical and electronic engineers, Proc. IEEE, 2014, vol. 102, no. 9, pp. 1340\u20131363.","journal-title":"Proc. IEEE"},{"key":"3719_CR17","unstructured":"Galgebra \u2013 symbolic geometric algebra\/calculus package for sympy. 2022. https:\/\/galgebra.readthedocs.io\/en\/latest\/index.html."},{"key":"3719_CR18","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1134\/S0361768822010054","volume":"48","author":"M.N. Gevorkyan","year":"2022","unstructured":"Gevorkyan, M.N., Demidova, A.V., Velieva, T.R., Korol\u2019kova, A.V., and Kulyabov, D.S., Analytical-numerical implementation of polyvector algebra in Julia, Program. Comput. Software, 2022, vol. 48, pp. 49\u201358.","journal-title":"Program. Comput. Software"},{"key":"3719_CR19","unstructured":"Sympy, 2022. http:\/\/www.sympy.org\/ru\/index.html."},{"key":"3719_CR20","volume-title":"Linear Algebra","author":"A.I. Kostrikin","year":"2009","unstructured":"Kostrikin, A.I., Linear Algebra, Moscow: Mosk. Tsentr Nepreryvnogo Matematicheskogo Obrazovaniya, 2009), vol. 2 [in Russain]."},{"key":"3719_CR21","unstructured":"Bivector.net: Geometric algebra resources, 2022. https:\/\/bivector.net\/index.html."},{"key":"3719_CR22","unstructured":"Hadfield, H., E. Wieser, A. Arsenovic, et al., 2022. pygae\/clifford."},{"key":"3719_CR23","unstructured":"De Keninck, S., 2020. ganja.js."},{"key":"3719_CR24","unstructured":"Grassmann.jl, 2022. https:\/\/github.com\/chakravala\/Grassmann.jl."},{"key":"3719_CR25","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/s00006-019-0987-7","volume":"29","author":"S. Breuils","year":"2019","unstructured":"Breuils, S., Nozick, V., and Fuchs, L., Garamon: A geometric algebra library generator, Adv. Appl. Clifford Algebras, 2019, vol. 29, no. 4, p. 69.","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"3719_CR26","doi-asserted-by":"crossref","unstructured":"Gunn, C.G. and Keninck, S.D., Geometric algebra and computer graphics, ACM SIGGRAPH 2019 Courses, ACM, 2019, no. 7.","DOI":"10.1145\/3305366.3328099"},{"key":"3719_CR27","unstructured":"Colapinto, P., Versor: Spatial computing with conformal geometric algebra, 2011. \nhttp:\/\/versor.mat.ucsb.edu."},{"key":"3719_CR28","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1134\/S0361768821020079","volume":"47","author":"D.S. Kulyabov","year":"2021","unstructured":"Kulyabov, D.S. and Korol\u2019kova, A.V., Computer algebra in JULIA, Program. Comput. Software, 2021, vol. 47, no. 2, pp. 133\u2013138.","journal-title":"Program. Comput. Software"},{"key":"3719_CR29","unstructured":"Gevorkyan, M.N., Kulyabov, D.S., Korolkova, A.V., et\u00a0al., Symbolic implementation of multivector algebra in Julia language, Computer algebra: 4th International Conference Materials, LCC MAKS, 2021, no. 5. pp. 57\u201360."},{"key":"3719_CR30","doi-asserted-by":"crossref","unstructured":"Kulyabov, D.S., Korolkova, A.V., and Sevastianov, L.A., Complex numbers for relativistic operations, MDPI AG, 2021. no. 12.","DOI":"10.20944\/preprints202112.0094.v1"},{"key":"3719_CR31","volume-title":"Quantum Field Theory in a Nutshell","author":"A. Zee","year":"2010","unstructured":"A. Zee, Quantum Field Theory in a Nutshell (Princeton Univ. Press, 2010)."},{"key":"3719_CR32","doi-asserted-by":"crossref","unstructured":"Kulyabov, D.S., Korolkova, A.V., and Gevorkyan, M.N., Hyperbolic numbers as Einstein numbers, J. Phys., Conf. Ser., 2020, vol. 1557, no. 5, p. 012027.","DOI":"10.1088\/1742-6596\/1557\/1\/012027"}],"container-title":["Programming and Computer Software"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1134\/S0361768823010048.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1134\/S0361768823010048","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1134\/S0361768823010048.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T02:54:54Z","timestamp":1775012094000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1134\/S0361768823010048"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["3719"],"URL":"https:\/\/doi.org\/10.1134\/s0361768823010048","relation":{},"ISSN":["0361-7688","1608-3261"],"issn-type":[{"value":"0361-7688","type":"print"},{"value":"1608-3261","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2]]},"assertion":[{"value":"1 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 September 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 March 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflicts of interest.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"CONFLICT OF INTEREST"}}]}}