{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,17]],"date-time":"2024-07-17T18:40:16Z","timestamp":1721241616143},"reference-count":14,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,6,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Estimating hyper-volumes of convex and non-convex sets are of interest in a number of areas. In this article we develop further a simple geometric Monte Carlo method, known also as the sample-mean method, which transforms the domain to an equivalent hyper-sphere with the same volume. We first examine the performance of the method to compute the volumes of star-convex unit balls and show that it gives accurate estimates of their volumes. We then examine the use of this method for computing the volumes of nonstar-shaped domains. In particular, we develop two algorithms, which couple the sample-mean method with algebraic and geometric techniques, to generate and compute the volumes of low-dimensional stability domains in parameter space.<\/jats:p>","DOI":"10.1515\/mcma-2019-2034","type":"journal-article","created":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T09:03:03Z","timestamp":1557219783000},"page":"163-176","source":"Crossref","is-referenced-by-count":3,"title":["On the sample-mean method for computing hyper-volumes"],"prefix":"10.1515","volume":"25","author":[{"given":"Nima","family":"Rabiei","sequence":"first","affiliation":[{"name":"Department of Mathematics , The American University of Iraq , Sulaimani , Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elias G.","family":"Saleeby","sequence":"additional","affiliation":[{"name":"Mount Lebanon , Beirut , Lebanon"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2019,5,7]]},"reference":[{"key":"2023040101340310724_j_mcma-2019-2034_ref_001_w2aab3b7b2b1b6b1ab1b5b1Aa","doi-asserted-by":"crossref","unstructured":"S. Ahmed and E. G. Saleeby, On volumes of hyper-ellipsoids, Math. Mag. 9 (2018), 43\u201350.","DOI":"10.1080\/0025570X.2018.1404834"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_002_w2aab3b7b2b1b6b1ab1b5b2Aa","unstructured":"K. Ball, An elementary introduction to modern convex geometry, Flavors in Geometry, Math. Sci. Res. Inst. Publ. 31, Cambridge University, Cambridge (1997), 1\u201358."},{"key":"2023040101340310724_j_mcma-2019-2034_ref_003_w2aab3b7b2b1b6b1ab1b5b3Aa","unstructured":"J. Edwards, A Treatise on the Integral Calculus. Vol. II, Chelsea, New York, 1922."},{"key":"2023040101340310724_j_mcma-2019-2034_ref_004_w2aab3b7b2b1b6b1ab1b5b4Aa","unstructured":"A. T. Fam, The volume of the coefficient space stability domain of monic polynomials, IEEE International Symposium on Circuits and Systems (Portalnd 1989), IEEE Press, Piscataway (1989), 1780\u20131783."},{"key":"2023040101340310724_j_mcma-2019-2034_ref_005_w2aab3b7b2b1b6b1ab1b5b5Aa","doi-asserted-by":"crossref","unstructured":"A. T. Fam and J. S. Meditch, A canonical parameter space for linear systems design, IEEE Trans. Automat. Control 23 (1978), 454\u2013458. 10.1109\/TAC.1978.1101744","DOI":"10.1109\/TAC.1978.1101744"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_006_w2aab3b7b2b1b6b1ab1b5b6Aa","doi-asserted-by":"crossref","unstructured":"D. S. K. Fok and D. Crevier, Volume estimation by monte carlo methods, J. Stat. Comput. Simul. 34 (1989), 223\u2013235.","DOI":"10.1080\/00949658908811145"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_007_w2aab3b7b2b1b6b1ab1b5b7Aa","unstructured":"J. Hopcroft and R. Kannan, Foundation of Data Science, in preparation 11\/4\/2014."},{"key":"2023040101340310724_j_mcma-2019-2034_ref_008_w2aab3b7b2b1b6b1ab1b5b8Aa","doi-asserted-by":"crossref","unstructured":"U. Jaekel, A Monte Calro method for high-dimensional volume estimation and application to polytopes, Procedia Comp. Sci. 4 (2011), 1403\u20131411. 10.1016\/j.procs.2011.04.151","DOI":"10.1016\/j.procs.2011.04.151"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_009_w2aab3b7b2b1b6b1ab1b5b9Aa","doi-asserted-by":"crossref","unstructured":"B. Kawohl, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math. 1150, Springer, Berlin, 1985.","DOI":"10.1007\/BFb0075060"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_010_w2aab3b7b2b1b6b1ab1b5c10Aa","doi-asserted-by":"crossref","unstructured":"M. E. Muller, A note on a method for generating points uniformly on n-dimensional spheres, Commun. ACM 2 (1959), 19\u201320. 10.1145\/377939.377946","DOI":"10.1145\/377939.377946"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_011_w2aab3b7b2b1b6b1ab1b5c11Aa","doi-asserted-by":"crossref","unstructured":"Y. P. Nikolaev, The multidimensional asymptotic stability domain of linear discrete systems: Its symmetry and other properties, Autom. Remote Control 62 (2001), 109\u2013120.","DOI":"10.1023\/A:1012794324554"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_012_w2aab3b7b2b1b6b1ab1b5c12Aa","doi-asserted-by":"crossref","unstructured":"L. A. Santalo, Integral Geometry and Geometric Probability, 2nd ed., Cambridge University, Cambridge, 2004.","DOI":"10.1017\/CBO9780511617331"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_013_w2aab3b7b2b1b6b1ab1b5c13Aa","doi-asserted-by":"crossref","unstructured":"R. J. Solomonoff and E. G. Saleeby, On the application of algorithmic probability to autoregreessive models, Algorithmic Probability and Friends. Bayesian Prediction and Artificial Intelligence, Lecture Notes in Comput. Sci. 7070, Springer, Berlin (2013), 366\u2013385.","DOI":"10.1007\/978-3-642-44958-1_29"},{"key":"2023040101340310724_j_mcma-2019-2034_ref_014_w2aab3b7b2b1b6b1ab1b5c14Aa","doi-asserted-by":"crossref","unstructured":"X. Wang, Volumes of generalized unit balls, Math. Mag. 78 (2005), 390\u2013395. 10.2307\/30044198","DOI":"10.2307\/30044198"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.degruyter.com\/view\/j\/mcma.2019.25.issue-2\/mcma-2019-2034\/mcma-2019-2034.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2019-2034\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2019-2034\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,17]],"date-time":"2024-07-17T17:50:25Z","timestamp":1721238625000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/mcma-2019-2034\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,7]]},"references-count":14,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,5,16]]},"published-print":{"date-parts":[[2019,6,1]]}},"alternative-id":["10.1515\/mcma-2019-2034"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2019-2034","relation":{},"ISSN":["1569-3961","0929-9629"],"issn-type":[{"value":"1569-3961","type":"electronic"},{"value":"0929-9629","type":"print"}],"subject":[],"published":{"date-parts":[[2019,5,7]]}}}