User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Theorem
Let $(X, \mathcal B)$ be a Definition:Pairwise Balanced Design.

That is, let $(X,\mathcal B)$ be a Design, with $|X| \ge 2$, and the number of occurences of each pair of distinct points in $\mathcal B$ be $\lambda$ for some $\lambda > 0$ constant.

Then the set union of all the elements in $\mathcal B$ is precisely $X$.

Proof
Let $X = \{x_1, x_2, \ldots, x_v\}$

Let $\mathcal B = \{\!\{y_1,y_2,\ldots, y_b\}\!\}$.

Let $\displaystyle \bigcup_{i=1}^{b} y_i = Y$.

We shall show that $Y \subseteq X$ and $X \subseteq Y$.

$Y \subseteq X$:
By definition, $\mathcal B$ is a multiset of subsets of $X$.

By Union of Subsets is Subset, $Y$ itself is a subset of $\mathcal B$

$X \subseteq Y$:
Let $x_i \in X$ be arbitrary.

Choose any $x_j \in X$ such that $i\ne j$. Such an $x_j$ necessarily exists because by hypothesis $|X|\ge 2$.

Then $\{x_i,x_j\} \subseteq Y$ by the definition of Pairwise Balanced Design.

Hence the result, as $x_i$ was arbitrary.

Block design
Any idea how to cite this? I emailed the professor and asked him how he would like it cited, waiting for a response. --GFauxPas (talk)

Response:



Those are the first week's lecture notes for a graduate course that I had been teaching for about 20 years that I have called Combinatorial Structures. Every so often I would tweak the notes depending on the text I used (if I was using one). The tweaks in this case were based on Doug Stinson's book, Combinatorial Designs. So while you could say that this was my own work, there are certainly some proofs that are Doug's. Since these were lecture notes that I had no intention of formally publishing, I didn't think it was necessary to reference Doug, being that everyone had the text. The course number had been changed from 6406 to 7410 due to some administrator's whim, but the course remained the same. I am retired now, and how you reference these notes doesn't make a lot of difference to me. Feel free to do it in any way that makes sense to you. Bill


 * " --GFauxPas (talk) 19:41, 16 November 2017 (EST)

To do

 * $\digamma$

Eventually
User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

User:GFauxPas/Sandbox/NumberTheory