User:Robinh/sandbox

{{About|generalized metric spaces|society|dog}}

{{short description|Generalization of metric spaces}}

In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced by Bryant and Tupper in 2012{{cite journal|last1=Bryant|first1=David|last2=Tupper|first2=Paul|journal=Advances in Mathematics|volume=231|pages=3172-3198|year=2012|title=Hyperconvexity and tight-span theory for diversities}}. A diversity is a pair $(X,\backslash delta)$ where X is a set and $\backslash delta$ is a function from the finite subsets of X to the non-negative reals satisfying

• (D1) $\backslash delta(A)\backslash geq\; 0$ with $\backslash delta(A)=0$ if and only if $\backslash left|A\backslash right|\backslash leq\; 1$.
• (D2) If $B\backslash neq\backslash emptyset$ then $\backslash delta(A\backslash cup\; C)\backslash leq\backslash delta(A\backslash cup\; B)\; +\; \backslash delta(B\; \backslash cup\; C)$.

Bryant and Tupper prove that these axioms imply monotonicity, that is, $A\backslash subseteq\; B\backslash longrightarrow\backslash delta(A)\backslash leq\backslash delta(B)$, and state that the term "diversity" comes from the appearance of a special case of their definition in work on phylogenetic and ecological diversities. They point out that diversities crop up in a broad range of contexts and give the following examples:

Let $(X,d)$ be a metric space. Denoting finite subsets of X by $\backslash wp\_\backslash mbox\{fin\}(X)$, defining $\backslash delta(A)=\backslash max\_\{a,b\backslash in\; A\}\; d(a,b)=\backslash operatorname\{diam\}(A)$ for all $A\backslash in\backslash wp\_\backslash mbox\{fin\}(X)$ is a diversity.

For all finite $A\backslash subseteq\backslash mathbb\{R\}^n$ if we define $\backslash delta(A)=\backslash sum\_i\backslash max\_\{a,b\}\backslash left\backslash \{\backslash left|\; a\_i-b\_i\backslash right|\backslash colon\; a,b\backslash in\; A\backslash right\backslash \}$ then $(\backslash mathbb\{R\}^n,\backslash delta)$ is a diversity.

If T is a phylogenetic tree with taxon set X. For each finite $A\backslash subseteq\; X$, define

$\backslash delta(A)$ as the length of the smallest subtree of T connecting taxa in A. Then $(X,\; \backslash delta)$ is a (phylogenetic) diversity.

Let $(X,\; d)$ be a metric space. For each finite $A\backslash subseteq\; X$, let $\backslash delta(A)$ denote

the minimum length of a Steiner tree within X connecting elements in A. Then $(X,\backslash delta)$ is a

diversity.

Let $(X,\backslash delta)$ be a diversity. For all $A\backslash in\backslash wp\_\backslash mbox\{fin\}(X)$ define

$\backslash delta^\{(k)\}(A)\; =\; \backslash max\backslash left\backslash \{\backslash delta(B)\backslash colon\; |B|\backslash leq\; k,\; B\backslash subseteq\; A\backslash right\backslash \}$. Then if $k\backslash geq\; 2$, $(X,\backslash delta^\{(k)\})$ is a diversity.

If $(X,E)$ is a graph, and $\backslash delta(A)$ is defined for any finite A as the largest clique of A, then $(X,\backslash delta)$ is a diversity.

I hate music. To me, music is like an alarm clock or a (moderately distant) pneumatic drill: it annoys me and I just want it to stop. There is no circumstance where I would prefer to listen to music rather than to turn it off. Whenever I mention this to anyone, they say "Oh, that's because you haven't heard William Shatner/The Ramones/Schonberg/Justin Bieber/The Birdy Song (insert random music here). You'd *love* that!". I have encountered an enormous amount of disbelief and hostility over this issue but I have simply had enough of pretending I like music, when actually I hate it, and have done all my life (I'm 45 yo). People seem quite threatened by my not liking music, for some reason. No-one minds someone saying "I don't like sport or computer games (or whatever)"; but music seems to be different. No-one has a problem with different people having different tastes in music (AFAICS) but a person who likes no music *at all* must be some sort of deviant threat to society. "Don't be silly! *Everyone* likes music!".

I have three questions: (1). I think this an example of the Abilene paradox, because there might be other people like me who don't like music but do not speak up because they don't want to be spoilsports. Is this valid? (2) Is there a word for the phenomenon where someone says "I like X. This guy does not like X. There must be something wrong with him and I'm gonna educate him"? (3) *is* music different in this respect from other things?

$x^6$}

In statistics, the multiplicative binomial distribution is a generalization of the binomial distribution which can account for both overdispersion and underdispersion. It was introduced in 1978{{cite journal| last=Altham| first=P.M.E.| title=Two Generalizations of the Binomial distribution| journal=Journal of the Royal Statistical Society Series C (Applied Statistics)| year=1978| volume=27|issue=2 |pages=162–167| postscript=.}}

The distribution may be viewed as a generalization of the Binomial distribution $B(n,p)$ that incorporates a new parameter $\backslash theta$ which accounts for overdispersion. The distribution has probability mass function

:$Pr(Z=k)\; =\; C^\{-1\}p^k(1-p)^\{n-k\}\backslash theta^\{k(n-k)\}$

where 'n' is the size of the binomial distribution, and $\backslash theta>0$ is the new parameter.

Here $0\backslash leqslant\; k\backslash leqslant\; n$ is an integer, and $C=C(n,p,\backslash theta)\; =\; \backslash sum\_\{k=0\}^np^k(1-p)^\{n-k\}\backslash theta^\{n(n-k)\}$ is a normalization constant. The distribution is thus of exponential family form.

The Typo Eradication Advancement League is a term used in connection with Jeff Michael Deck and Benjamin Douglas Herson who were reported in British and American news reports to have altered a sign post in the Grand Canyon, correcting some grammatical errors. They covered up an erroneous apostrophe, replaced it in its correct place, and added a comma.{{cite news|url=http://www.nytimes.com/2008/08/24/us/24signs.html?_r=1&adxnnl=1&oref=slogin&partner=rssnyt&emc=rss&adxnnlx=1219832000-9wbPxweR/egXcGVvvqo8Ww|title=Not Thanks, but Probation for Fixing Typos on a Sign|date=2008-08-23|work=New York Times|accessdate=2008-08-27}}

The story was reported on BBC Radio 4 and appears on the Language Log.

{{reflist}}

• [http://www.jeffdeck.com/teal/ TEAL home page]
• [http://www.azcentral.com/news/articles/2008/08/22/20080822grammarcops0822.html Typo vigilantes answer to letter of the law] at AZcentral.com
• [http://languagelog.ldc.upenn.edu/nll/?p=522 "Grammar vigilantes" brought to justice] on Language Log