{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,12]],"date-time":"2026-03-12T00:58:25Z","timestamp":1773277105781,"version":"3.50.1"},"reference-count":16,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11271144"],"award-info":[{"award-number":["11271144"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1021203"],"award-info":[{"award-number":["1021203"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Stochastic matrices play an important role in the study of probability theory\nand statistics, and are often used in a variety of modeling problems in\neconomics, biology and operation research. Recently, the study\nof tensors and\ntheir applications\nbecame a hot topic in numerical analysis and\noptimization.\nIn this paper, we focus on studying stochastic tensors and, in\nparticular, we study the extreme points of a set of multi-stochastic tensors. Two\nnecessary and sufficient conditions for a multi-stochastic tensor to be an\nextreme point are established.\nThese conditions characterize the \u201cgenerators\u201d of multi-stochastic tensors.\nAn algorithm to search the convex combination of extreme points for an arbitrary given\nmulti-stochastic tensor is developed. Based on our obtained results, some expression\nproperties for third-order and <jats:italic>n<\/jats:italic>-dimensional multi-stochastic tensors\n(<jats:inline-formula id=\"j_cmam-2016-0005_eq_9999_w2aab3b7d324b1b6b1aab1c13b1b3Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>n<\/m:mi>\n                              <m:mo>=<\/m:mo>\n                              <m:mn>3<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${n=3}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and 4) are derived, and all extreme points of\n3-dimensional and 4-dimensional triply-stochastic tensors can be produced in a simple\nway. As an application, a new approach for the\npartially filled square problem under the framework of multi-stochastic\ntensors is given.<\/jats:p>","DOI":"10.1515\/cmam-2016-0005","type":"journal-article","created":{"date-parts":[[2016,1,29]],"date-time":"2016-01-29T11:57:01Z","timestamp":1454068621000},"page":"459-474","source":"Crossref","is-referenced-by-count":12,"title":["Characterization of Extreme Points of Multi-Stochastic Tensors"],"prefix":"10.1515","volume":"16","author":[{"given":"Rihuan","family":"Ke","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.\u2009R. China"}]},{"given":"Wen","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.\u2009R. China"}]},{"given":"Mingqing","family":"Xiao","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, USA"}]}],"member":"374","published-online":{"date-parts":[[2016,1,29]]},"reference":[{"key":"2023033112444839998_j_cmam-2016-0005_ref_001_w2aab3b7d324b1b6b1ab2b2b1Aa","doi-asserted-by":"crossref","unstructured":"Barry R. A. and Humblet P. A.,\nLatin routers, design and implementation,\nIEEE\/OSA J. Lightwave Technol. 11 (1993), 891\u2013899.","DOI":"10.1109\/50.233253"},{"key":"2023033112444839998_j_cmam-2016-0005_ref_002_w2aab3b7d324b1b6b1ab2b2b2Aa","doi-asserted-by":"crossref","unstructured":"Chang K. and Zhang T.,\nOn the uniqueness and non-uniqueness of the positive Z-eigenvector for transition probability tensors,\nJ. Math. Anal. 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