link to a related concept

]]>Aesthetically it seems weird for an encyclopedic reference to include the special case when it’s not a simplification of the more general case.

As a more practical note, I’ve actually found this a usability issue on the nLab from time to time where pages pay attention to a theorem written for a special case, leading me to completely miss that more general statements are available. Or when I do notice, to wind up spending a lot of time trying to understand what’s different about the special case that it would be needed addition to what is actually a strictly more general theorem – especially if there’s some restatement involved.

But, maybe it’s more a phrasing issue. I had taken as a given that we’d eventually want to remove the restatement to the special case – so I’m thinking of the question more as how to reorganize the interesting contents of the proof we’d like to retain as “here’s more interesting information!”

]]>But on re-reading I still found it a little weird, so I took the liberty of adding this line:

]]>this proof was written in 2011 when no comparable statement seemed to be available in the literature

brief `category:people`

-entry for hyperlinking references at *Cayley distance*, *Mallows kernel* etc.

brief `category:people`

-entry for hyperlinking references at *Cayley distance* and *Mallows kernel* etc.

also added pointer to

- Michael A. Fligner, Joseph S. Verducci (eds.),
*Probability Models and Statistical Analyses for Ranking Data*(doi:10.1007/978-1-4612-2738-0)

which mentions the Cayley-distance kernel at least in the forword as the “only reasonable bi-invariant distance”.

]]>I have merged the two subsections. Added a lead-over sentence: “We spell out a proof for the special case that $\mathcal{C}$ carries the extra structure of a simplicial model category:”

]]>Thanks for your addition. But why would any of the proof offered on the page need to be removed. There is no harm in spelling out a proof of a special case of theorem that is proven more generally elsewhere. On the contrary. Unless I am missing something in your question?

]]>added pointer to:

- M. A. Fligner, J. S. Verducci, Section 4 of:
*Distance Based Ranking Models*, Journal of the Royal Statistical Society. Series B (Methodological) Vol. 48, No. 3 (1986), pp. 359-369 (jstor:2345433)

(one of the few references that I found so far which do at least consider the “Cayley distance kernel” $\exp(-d_C(-,-,))$ )

]]>I’ve cited a theorem in Cisinksi’s paper that proves the slice construction with a fibrant object is correct for any model category.

This makes the theorem proved in the section on derived hom-spaces redundant, and can be removed. Is there any content in the proof that should be retained on the page?

]]>added statement and proof of Cayley’s formula

$d_C(\sigma_1, \sigma_2) \;=\; n - \left\vert Cycles\big( \sigma_1 \circ \sigma_2^{-1} \big) \right\vert$ ]]>Added the example of a permutation cycle.

]]>a minimum, just for completeness

]]>giving *cycle of a permutation* its own hyperlink

made explicit the example of the symmetric group:

The symmetric group $Sym(n)$ may be generated from

all transposition permutations – the corresponding Cayley graph distance is the original

*Cayley distance*;the

*adjacent*transpositions – the corresponding Cayley graph distance is known as the*Kendall tau distance*.

]]>

added also pointer to:

- Persi Diaconis, Chapter 6 of:
*Group Representations in Probability and Statistics*, Lecture Notes - Monographs Series, Institute of Mathematical Statistics 1988 (pdf)

for completeness, to go alongside *Kendall distance*

brief `category:people`

-entry for hyperlinking references at *Mallows kernel*

added also pointer to

- C. L. Mallows,
*Non-Null Ranking Models. I*, Biometrika Vol. 44, No. 1/2 (Jun., 1957), pp. 114-130 (jstor:2333244)

which allegedly defines the Mallows kernel, though I haven’t actually spotted it yet…

]]>expanded out first name in title

]]>brief `category:people`

-entry for hyperlinking references at *Kendall tau distance*

added pointer to the original:

- M. G. Kendall,
*A new measure of rank correlation*Biometrika, Volume 30, Issue 1-2, June 1938, Pages 81–93 (doi:10.1093/biomet/30.1-2.81)

(Now that’s a different style of doing maths…)

]]>Yes, today we finally met to come back to this. So I thought I’d bring more nLab material into place.

]]>Mallows kernels came up before. Ah yes, back in this discussion.

]]>