User:Robinh/sandbox


 * 1) Diversity

Diversity
In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced by Bryant and Tupper in 2012. A diversity is a pair $$(X,\delta)$$ where X is a set and $$\delta$$ is a function from the finite subsets of X to the non-negative reals satisfying


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

Bryant and Tupper prove that these axioms imply monotonicity, that is, $$A\subseteq B\longrightarrow\delta(A)\leq\delta(B)$$, 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:

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

$$L_1$$ diversity
For all finite $$A\subseteq\mathbb{R}^n$$ if we define $$\delta(A)=\sum_i\max_{a,b}\left\{\left| a_i-b_i\right|\colon a,b\in A\right\}$$ then $$(\mathbb{R}^n,\delta)$$ is a diversity.

Phylogenetic diversity
If T is a phylogenetic tree with taxon set X. For each finite $$A\subseteq X$$, define $$\delta(A)$$ as the length of the smallest subtree of T connecting taxa in A. Then $$(X, \delta)$$ is a (phylogenetic) diversity.

Steiner diversity
Let $$(X, d)$$ be a metric space. For each finite $$A\subseteq X$$, let $$\delta(A)$$ denote the minimum length of a Steiner tree within X connecting elements in A. Then $$(X,\delta)$$ is a diversity.

Truncated diversity
Let $$(X,\delta)$$ be a diversity. For all $$A\in\wp_\mbox{fin}(X)$$ define $$\delta^{(k)}(A) = \max\left\{\delta(B)\colon |B|\leq k, B\subseteq A\right\}$$. Then if $$k\geq 2$$, $$(X,\delta^{(k)})$$ is a diversity.

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

music
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

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


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

where 'n' is the size of the binomial distribution, and $$\theta>0$$ is the new parameter. Here $$0\leqslant k\leqslant n$$ is an integer, and $$C=C(n,p,\theta) = \sum_{k=0}^np^k(1-p)^{n-k}\theta^{n(n-k)}$$ 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.

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