[Turkmath:9435] GYTE Weekly Seminars

[İrem YAMAN]

Sayın liste üyeleri,

GYTE Matematik haftalık seminerler kapsamında, Şule Yazıcı (Koç Üniversitesi)
25 Aralık 2013, saat 14.00'da " A polynomial embedding of pairs of orthogonal partial latin squares"
başlıklı bir seminer verecektir.

Seminerin detayları ekte (ve aşağıda) olup tüm ilgilenenler davetlidir.

İyi çalışmalar.

İrem Yaman
Gebze Yüksek Teknoloji Enstitüsü
Matematik bölümü

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Dear all,

Gebze Institute of Technology (GYTE) Mathematics department continues its weekly seminars
on December 25th, 2013, at 14.00h with Şule Yazıcı from Koç University.
Her talk is entitled " A polynomial embedding of pairs of orthogonal partial latin squares".

Everyone who is interested is cordially invited.
Please find attached (and below) a more detailed information.

Sincerely
İrem Yaman
Gebze Institute of Technology
Department of Mathematics

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Abstract:

Let $N$ represent a set of $n$ distinct elements. A non-empty subset $P$ of $N\times N\times N$ is said to be a {\em partial latin square}, of order $n$, if for all $(x_1,x_2,x_3),(y_1,y_2,y_3)\in P$ and for all distinct $i,j,k\in \{1,2,3\}$, \begin{align*} x_i=y_i\mbox{ and }x_j=y_j\mbox{ implies }x_k=y_k. \end{align*} If $|P|=n^2$, then we say that $P$ is a {\em latin square}, of order $n$.

Two partial latin squares $P$ and $Q$, of the same order are said to be {\em orthogonal} if they have the same non-empty cells and for all $r_1,c_1,r_2,c_2,x,y\in N$ \begin{align*} \{(r_1,c_1,x),(r_2,c_2,x)\}\subseteq P\mbox{ implies }\{(r_1,c_1,y),(r_2,c_2,y)\}\not\subseteq Q. \end{align*}

In 1960 Evans proved that a partial latin square of order $n$ can always be embedded in some latin square of order $t$ for every $t\geq 2n$. In the same paper Evans raised the question as to whether a pair of finite partial latin squares which are orthogonal can be embedded in a pair of finite orthogonal latin squares. We show that a pair of orthogonal partial latin squares of order $t$ can be embedded in a pair of orthogonal latin squares of order at most $16t^4$ and all orders greater than or equal to $48t^4$. This is the first polynomial embedding result of its kind.

Place:
GIT, Department of Mathematics, Building I, Seminar Room



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