Secant Distributions of Unitals
| dc.authorid | Gezek, Mustafa/0000-0001-5488-9341 | |
| dc.contributor.author | Gezek, Mustafa | |
| dc.date.accessioned | 2024-10-29T17:58:17Z | |
| dc.date.available | 2024-10-29T17:58:17Z | |
| dc.date.issued | 2024 | |
| dc.department | Tekirdağ Namık Kemal Üniversitesi | |
| dc.description.abstract | Let U be a unital embedded in a projective plane Pi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document} of order q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>2$$\end{document}. For R is an element of U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\in U$$\end{document}, let sR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_R$$\end{document} and tR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_R$$\end{document} be a secant line through R and the tangent line to U at point R, respectively. If the tangent lines to U, passing through the points in sR boolean AND U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_R\cap U$$\end{document}, intersect at a single point on tR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_R$$\end{document}, then sR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_R$$\end{document} is referred to as a secant line satisfying the desired property. If ni\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_i$$\end{document} of the points of U have exactly mi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_i$$\end{document} secant lines satisfying the desired property, then m1n1,m2n2,& ctdot;\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m_1<^>{n_1}, m_2<^>{n_2}, \cdots \end{aligned}$$\end{document}is called the secant distribution of U, where & sum;ni=q3+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum n_i=q<^>3+1$$\end{document}, and 0 <= mi <= q2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le m_i\le q<^>2$$\end{document}. In this article, we show that collinear pedal sets of a unital U plays an important role in the secant distribution of U. Formulas for secant distributions of unitals having 0,1,q2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0,1,q<^>2,$$\end{document} or q2+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>2+q$$\end{document} special points are provided. Statistics regarding to secant distributions of unitals embedded in planes of orders q2 <= 25\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>2\le 25$$\end{document} are presented. Some open problems related to secant distributions of unitals having specific number of collinear pedal sets are discussed. | |
| dc.identifier.doi | 10.1007/s00025-024-02261-w | |
| dc.identifier.issn | 1422-6383 | |
| dc.identifier.issn | 1420-9012 | |
| dc.identifier.issue | 6 | en_US |
| dc.identifier.scopus | 2-s2.0-85202024977 | |
| dc.identifier.scopusquality | Q2 | |
| dc.identifier.uri | https://doi.org/10.1007/s00025-024-02261-w | |
| dc.identifier.uri | https://hdl.handle.net/20.500.11776/14193 | |
| dc.identifier.volume | 79 | |
| dc.identifier.wos | WOS:001298794100002 | |
| dc.identifier.wosquality | N/A | |
| dc.indekslendigikaynak | Web of Science | |
| dc.indekslendigikaynak | Scopus | |
| dc.language.iso | en | |
| dc.publisher | Springer Basel Ag | |
| dc.relation.ispartof | Results in Mathematics | |
| dc.relation.publicationcategory | Makale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanı | en_US |
| dc.rights | info:eu-repo/semantics/closedAccess | |
| dc.subject | Pedal sets | |
| dc.subject | projective plane | |
| dc.subject | steiner designs | |
| dc.subject | unitals | |
| dc.title | Secant Distributions of Unitals | |
| dc.type | Article |
