{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:39:08Z","timestamp":1760236748931,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T00:00:00Z","timestamp":1640044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this survey article, we look into some recent results concerning summability matrices, both regular as well as those which are not regular (called semi-regular) and generated matrix ideals as the overall view of the inter relationship between the notions of ideal convergence and summability methods by regular summability matrices.<\/jats:p>","DOI":"10.3390\/axioms11010001","type":"journal-article","created":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T09:50:43Z","timestamp":1640080243000},"page":"1","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Ideals, Nonnegative Summability Matrices and Corresponding Convergence Notions: A Short Survey of Recent Advancements"],"prefix":"10.3390","volume":"11","author":[{"given":"Pratulananda","family":"Das","sequence":"first","affiliation":[{"name":"Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Dutta, H., and Rhoades, B.E. (2016). Summability and Convergence Using Ideals, Current Topics in Summability Theory and Applications, Springer.","DOI":"10.1007\/978-981-10-0913-6"},{"key":"ref_2","unstructured":"Filipczak, M., and Wagner-Bojakowska, E. (2013). Ideal Convergence, a Chapter in the Monograph Traditional and Present-Day Topics in Real Analysis, L\u00f3d\u017a University Press."},{"key":"ref_3","unstructured":"Banach, S. (1987). Theory of Linear Operations, North-Holland Mathematical Library, Elsevier. (In English)."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1007\/s00025-018-0876-8","article-title":"Continuous projections onto ideal convergent sequences","volume":"73","author":"Leonetti","year":"2018","journal-title":"Results Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"101","DOI":"10.4064\/sm170413-12-12","article-title":"Ideal convergence versus matrix summability","volume":"245","author":"Filipow","year":"2019","journal-title":"Studia Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"123760","DOI":"10.1016\/j.jmaa.2019.123760","article-title":"Representation of ideal convergence as a union and intersection of matrix summability methods","volume":"484","author":"Filipow","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Daniel Mauldin, R. (2015). The Scottish Book: Mathematics from The Scottish Caf\u00e9, with Selected Problems from The New Scottish Book, Springer.","DOI":"10.1007\/978-3-319-22897-6"},{"key":"ref_8","first-page":"xvi+177","article-title":"Analytic quotients: Theory of liftings for quotients over analytic ideals on the integers","volume":"148","author":"Farah","year":"2000","journal-title":"Mem. Am. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/S0168-0072(98)00051-7","article-title":"Analytic ideals and their applications","volume":"99","author":"Solecki","year":"1999","journal-title":"Ann. Pure Appl. Log."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/0003-4843(77)90006-7","article-title":"Happy families","volume":"12","author":"Mathias","year":"1977","journal-title":"Ann. Math. Log."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"560","DOI":"10.2307\/2371785","article-title":"The measure theoretic approach to density","volume":"68","author":"Buck","year":"1946","journal-title":"Am. J. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"241","DOI":"10.4064\/cm-2-3-4-241-244","article-title":"Sur la convergence statistique","volume":"2","author":"Fast","year":"1951","journal-title":"Colloq. Math."},{"key":"ref_13","first-page":"73","article-title":"Sur la convergence ordinaire et la convergence asymptotique","volume":"2","author":"Steinhaus","year":"1951","journal-title":"Colloq. Math."},{"key":"ref_14","first-page":"139","article-title":"On statistically convergent sequences of real numbers","volume":"30","year":"1980","journal-title":"Math. Slovaca"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1524\/anly.1988.8.12.47","article-title":"The statistical and strong p-Cesaro convergence of sequences","volume":"8","author":"Connor","year":"1988","journal-title":"Analysis"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"194","DOI":"10.4153\/CMB-1989-029-3","article-title":"On strong matrix summability with respect to a modulus and statistical convergence","volume":"32","author":"Connor","year":"1989","journal-title":"Canad. Math. Bull."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1524\/anly.1990.10.4.373","article-title":"Two valued measures and summability","volume":"10","author":"Connor","year":"1990","journal-title":"Analysis"},{"key":"ref_18","first-page":"319","article-title":"R-type summability methods, Cauchy criteria, P-sets and statistical convergence","volume":"115","author":"Connor","year":"1992","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1524\/anly.1985.5.4.301","article-title":"On statistical convergence","volume":"5","author":"Fridy","year":"1985","journal-title":"Analysis"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1090\/S0002-9939-1993-1181163-6","article-title":"Statistical limit points","volume":"118","author":"Fridy","year":"1993","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"669","DOI":"10.2307\/44154069","article-title":"I-convergence","volume":"26","author":"Kostyrko","year":"2001","journal-title":"Real Anal. Exch."},{"key":"ref_22","first-page":"595","article-title":"Th\u00e9orie des filtres","volume":"205","author":"Cartan","year":"1937","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"293","DOI":"10.2140\/pjm.1981.95.293","article-title":"Densities and summability","volume":"9","author":"Freedman","year":"1981","journal-title":"Pacific J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1080\/00029890.1959.11989303","article-title":"The integrability of certain functions and related summability methods","volume":"66","author":"Schoenberg","year":"1959","journal-title":"Am. Math. Mon."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2621","DOI":"10.1016\/j.topol.2012.04.007","article-title":"Some further results on ideal convergence in topological spaces","volume":"159","author":"Das","year":"2019","journal-title":"Topol. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"153","DOI":"10.21136\/MB.2005.134133","article-title":"I and I*-convergence in topological spaces","volume":"130","author":"Lahiri","year":"2015","journal-title":"Math. Bohem."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2647","DOI":"10.1090\/S0002-9939-00-05891-3","article-title":"On statistical limit points","volume":"129","author":"Kostyrko","year":"2001","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.topol.2008.01.015","article-title":"Statistical convergence in topology","volume":"156","year":"2008","journal-title":"Topol. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/j.topol.2018.11.016","article-title":"Invariance of ideal limit points","volume":"252","author":"Leonetti","year":"2019","journal-title":"Topol. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.topol.2018.11.022","article-title":"On the relationship between ideal cluster points and ideal limit points","volume":"252","author":"Balcerzak","year":"2019","journal-title":"Topol. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"39","DOI":"10.4064\/fm203-1-3","article-title":"Ideal limits of sequences of continuous functions","volume":"203","author":"Laczkovich","year":"2009","journal-title":"Fund. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/j.jmaa.2008.08.032","article-title":"Ideal version of Egorov\u2019s Theorem for analytic P-ideal","volume":"349","author":"Mrozek","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Boos, J. (2000). Classical and Modern Methods in Summability, Oxford University Press. Oxford Mathematical Monographs.","DOI":"10.1093\/oso\/9780198501657.001.0001"},{"key":"ref_34","unstructured":"Cooke, R.G. (1965). Infinite Matrices and Sequence Spaces, Dover Publ."},{"key":"ref_35","unstructured":"Silverman, L.L. (1913). On the Definition of the Sum of a Divergent Series, University of Missouri."},{"key":"ref_36","first-page":"113","article-title":"\u00dcber allgemeine lineare Mittelbildungen","volume":"22","author":"Toeplitz","year":"1913","journal-title":"Prace Matematyczno-Fizyczne"},{"key":"ref_37","unstructured":"Daniel Mauldin, R. (1981). The Scottish Book, Birkh\u00e1user. Mathematics from the Scottish Cafe."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/j.jmaa.2007.01.084","article-title":"Matrix characterization of A-statistical convergence","volume":"335","author":"Khan","year":"2007","journal-title":"J. Math. Anal. Appl."},{"key":"ref_39","unstructured":"Leonetti, P. (2012). Tauberian Theorems for Ordinary Convergence. arXiv, Available online: https:\/\/arxiv.org\/pdf\/2012.03311.pdf."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1524\/anly.1991.11.1.59","article-title":"A matrix characterization of statistical convergence","volume":"11","author":"Fridy","year":"1991","journal-title":"Analysis"},{"key":"ref_41","first-page":"27","article-title":"On Ic(q)-convergence","volume":"38","author":"Gogola","year":"2011","journal-title":"Ann. Math. Inform."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"125374","DOI":"10.1016\/j.jmaa.2021.125374","article-title":"A characterization of (I,J)-regular matrices","volume":"504","author":"Connor","year":"2021","journal-title":"J. Math. Anal. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1090\/conm\/533\/10503","article-title":"Combinatorics of filters and ideals","volume":"533","year":"2011","journal-title":"Contemp. Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s10474-015-0510-0","article-title":"Generalized kinds of density and the associated ideals","volume":"147","author":"Balcerzak","year":"2015","journal-title":"Acta Math. Hungar."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s10474-011-0122-2","article-title":"Restricting statistical convergence","volume":"134","author":"Bhunia","year":"2012","journal-title":"Acta Math. Hungar."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1016\/j.jmaa.2015.05.042","article-title":"A note on a new ideal","volume":"430","author":"Kwela","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1016\/j.jmaa.2019.04.046","article-title":"Properties of simple density ideals","volume":"447","author":"Kwela","year":"2019","journal-title":"J. Math. Anal. Appl."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"29","DOI":"10.4064\/cm7632-12-2018","article-title":"A note on nonregular matrices and ideals associated with them","volume":"159","author":"Das","year":"2020","journal-title":"Colloq. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.laa.2015.08.031","article-title":"On matrix summability of spliced sequences and A-density of points","volume":"487","author":"Bartoszewich","year":"2015","journal-title":"Linear Algebra Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/1\/1\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:50:23Z","timestamp":1760169023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/1\/1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,21]]},"references-count":49,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["axioms11010001"],"URL":"https:\/\/doi.org\/10.3390\/axioms11010001","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2021,12,21]]}}}