User:Michael Hardy/scratchwork


 * Pr(X ∈ g−1(F) )
 * Pr(X ∈ g−1(F) )
 * I prefer TeX myself, but for a CSS approach, span style="padding-top: 4px; border-top:1px solid" will give a better result:
 * Pr(X ∈ g−1(F) )
 * -- 20:05, 30 March 2019 (UTC)


 * $$ \oint . $$

siℓℓy &Sigma; &Sigma;  a≪ b $\sqrt{1 − ζ^{2}}$

$ or \textstyle

gmail

 * Just something to check - in Chrome (with the censored e-mails), click on the cog button at the top right, choose Settings, and then along the top go to the Inbox settings. The last setting on that screen is "Filtered mail" - try disabling that (Override filters) and see if it changes anything. (I haven't actually noticed that setting before today, but I'd be really interested if it helps.) OrganicsLRO 09:50, 12 February 2014 (UTC)

copied from User_talk:Constant314:

 * I don't know of an article that uses numbered equations, but the templates EquationNote and EquationRef can be used for that purpose. You may want to ask at WT:WikiProject Mathematics on whether equations should be numbered - personally I'd try to avoid numbered equations if at all possible. Huon (talk) 22:50, 9 February 2014 (UTC)


 * There's also NumBlk which lets you layout a line containing a formula and number. -- JohnBlackburne wordsdeeds 23:49, 9 February 2014 (UTC)


 * To see this all in action, take a look at Poisson summation formula.--LutzL (talk) 00:16, 10 February 2014 (UTC)


 * Equation numbering is described in Help:Formula. --Mark viking (talk) 01:09, 10 February 2014 (UTC)

section





 * $$\mathbf{u} − $$ +

So, we get the following relation:

Now, we can easily observe the generating function relation for ($$) to be:

next
{{legend|#c0c0ff|0–1}} {{legend|#8080ff|1–2}} {{legend|#5a5aff|2–5}} {{legend|#0000c0|5–10}} {{legend|#000080|10–20}} {{legend|#00005c|> 20}}


 * Database reports/WikiProject watchers
 * Database reports/WikiProjects by changes
 * User:Topbanana/WPtalk.

WikiProject watchers

Tychonian orrery


 * \digamma \,
 * \koppa \,
 * \stigma \,
 * \coppa \,

&#x3DA; and &#x3DB;





redirect ambiguities
I think it was a couple of years ago that someone explained on some Wikipedia discussion page---maybe it was this very page---how to find a comprehensive list of instances of the following kind of thing.
 * "Xmith's hypothesis" redirects to A
 * "Xmith's Hypothesis" redirects to B
 * "the Xmithian Hypothesis" redirects to C

The three are synonymous but A, B, and C are _different_ articles, and it's absurd that synonymous terms with minor spelling differences should redirect differently.

From time to time I come across that situation. After that discussion two or three years ago, I clumsily failed to save for future reference the means by which the comprehensive list was found. Can anyone identify it? Michael Hardy (talk) 18:00, 13 October 2012 (UTC)


 * A toolserver query could identify some potential duplicates by listing redirect pages with different destinations where the titles are nearly identical (apart from upper/lower case or punctuation), as in your first two examples. But this would not identify "the Xmithian Hypothesis" in your third example.
 * Toolserver database queries can be requested at Wikipedia talk:Database reports.
 * But I don't see how a more comprehensive list could be compiled automatically.
 * — Richardguk (talk) 21:58, 13 October 2012 (UTC)
 * Thank you, Richardguk. Michael Hardy (talk) 23:00, 14 October 2012 (UTC)

proved by and ,

. Bateman defined it by



WikiProjects ranked by liveliness (copied from Wikipedia_talk:Database_reports)
I would like a list of discussion pages of WikiProjects (so I'm talking about pages called "Wikipedia talk:WikiProject Whatever") ranked by the frequency with which they are edited---in effect the most active WikiProjects listed first. Can that be done? Michael Hardy (talk) 03:46, 24 October 2012 (UTC)


 * There are a couple of relevant reports already being produced. Do either of:


 * Database reports/WikiProject watchers
 * Database reports/WikiProjects by changes
 * closely enough match your needs? - TB (talk) 07:01, 24 October 2012 (UTC)


 * I've spoken with Michael Hardy previously at the WikiProject Council page and recommended he come here. In trying to measure a project's activity or "liveliness", I can see the limits of the two reports you've listed. The current report on how many people are watching gives an impression that some projects are busy when in reality there has been little discussion in years (the dead WikiProject Contents tops the list). The other list provides the number of edits for articles under a project's scope, not necessarily edits made by the project's members. I think Michael Hardy is looking to see how many times the project's talk page is edited in a particular timeframe (maybe six months or a year) to gauge which projects are active places for discussion. –Mabeenot (talk) 15:44, 26 October 2012 (UTC)


 * Okay, initial report at User:Topbanana/WPtalk. It seems to be a good measure of activity for some projects, and an awful one for others.  I suspect the parameters need tweaked to make it more useful before we make it a regular thing. - TB (talk) 17:25, 26 October 2012 (UTC)

These look as if they may be useful.

Thank you. Michael Hardy (talk) 18:29, 26 October 2012 (UTC)

Error bounds

 * Polynomial_interpolation
 * Euler_method
 * Numerical error
 * Trapezoidal rule
 * Simpson's rule

(copied from the mathematics reference desk)
Is there a convex polyhedron whose vertices correspond to the 15 unordered pairs of elements of a 6-element set { a, b, c, d, e, f}, with an edge between two pairs precisely if they are intersecting subsets of { a, b, c, d, e, f} (thus there would be an edge between ab and ac but not between ab and cf)?

(Each vertex would thus have degree 8, so there would be 8 &times; 15/2 = 60 edges. Euler's formula V &minus; E + F = 2 would then imply
 * 15 &minus; 60 + F = 2,

so F = 47, i.e. there would be 47 faces.) Michael Hardy (talk) 23:55, 5 April 2011 (UTC)


 * The answer is no. The graph you describe is a convex polyhedron only if it is a planar graph.  But it's fairly straightforward to construct a subgraph that is a subdivision of the complete bipartite graph K3,3.  I can get one from the subgraph spanned by the eight vertices {ab, ac, cd, bd, ad, be, ef, df}.  Specifically, take the complete bipartite graph on the pair of vertex sets {ab,cd,ef} and {ac,bd,df}, and then insert the vertex ad into the (ab,df) edge, and the vertex be into the (bd,ef) edge.  The resulting graph appears as a subgraph of the graph in question.  Sławomir Biały  (talk) 01:18, 6 April 2011 (UTC)
 * Very nice. Your answer leaves me wondering why I didn't think of that.  I must be getting rusty in some things. Michael Hardy (talk) 01:56, 6 April 2011 (UTC)

Next question: Is there some visually nice way to display the graph I described? Michael Hardy (talk) 02:03, 6 April 2011 (UTC)
 * The graph you've described is the complement of the Kneser graph KG6,2, so it's quite closely related to the Petersen graph, which is KG5,2. I'm sure someone has made a nice drawing of KG6,2. —Bkell (talk) 03:04, 6 April 2011 (UTC)
 * I see that the complement of KG6,2 is called the (6,2)-Johnson graph; there's a bit about Johnson graphs in the Kneser graph article (in the #Related graphs section), but I didn't notice that before. So now you have a name for your graph, at least. Wolfram Alpha will draw a picture of it, but there's probably a more beautiful way to draw it. —Bkell (talk) 03:31, 6 April 2011 (UTC)

Interesting. Probably Wolfram Alpha's picture will be far better than what I could draw by hand. Thank you, Bkell and Sławomir Biały.

(BTW, Sławomir, am I right in suspecting that the first syllable of your first name is pronounced something like "suave"?) Michael Hardy (talk) 04:11, 6 April 2011 (UTC)


 * That's a bit strange, probably worth a look, Mathworld has Triangular Graph with $T_{6}$ corresponding to this whereas we have triangular graph redirecting to planar graph as composed of triangles. Dmcq (talk) 08:39, 6 April 2011 (UTC)
 * So that picture on Mathworld is it! Thank you. Michael Hardy (talk) 12:50, 6 April 2011 (UTC)
 * Another representation is given in where they remove the complete graphs of 5 points. Dmcq (talk) 13:38, 6 April 2011 (UTC)


 * I've used Sage_(mathematics_software) in the past to draw graphs. StatisticsMan (talk) 15:48, 6 April 2011 (UTC)

etc.
Jaynes on Wolf's dice data

CAT:CSD

watchers





\xrightarrow

ℝ

&lfloor;t/2&rfloor;

π $π$




 * x ↦ y

a ≡ b mod n

≫ ≪

a ≈ b

∩

A \ B

∈ ∉ ⊆ ⊇ ∅ ± ∞ &#8467; ∩

Unicode mathematical operators User:KSmrq/Chars

open-closed and closed-open

a bit of formatting by Hans Adler


$$

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The orbital speed was thought to be $$.

The orbital speed was thought to be 220±20.

The orbital speed was thought to be 220 ± 20.

section
siℓ

〈x,y〉instead of 


 * m = number of groups


 * n = size of each group





The problem of extinction of surnames

 * X0 = 1 = number of men initially named "Smith".


 * X1 = number of sons of the first man named "Smith".


 * X2 = number of grandsons of the first man named "Smith".

and so on.


 * Xn,i = number of sons of the ith man in the nth generation,

so that


 * $$ X_{n+1} = \sum_{i=1}^{X_n} X_{n,i}, $$

i.e. to get the number of (n + 1)th-generation "Smith"s, just sum the numbers of sons of men in the nth generation.

We assume (doubtless an oversimplification) that the probability distribution of the number of sons is the same for all men, and that the numbers of sons of different men are independent random variables.

Our question will be: What can we say about the probability of eventual extinction of the name "Smith"?

The theorem
That is the probability of the event that for some value of n we have Xn = 1. We have a theorem:


 * If the average number of a man's sons is less than 1, then the probability of eventual extinction of the family name is 1
 * If the average number of a man's sons is exactly 1, then the probability of eventual extinction of the family name is 1 except in the trivial case in which every man always has exactly one son.
 * If the average number of a man's sons is more than 1, then the probability of eventual extinction of the family name is less than 1.

Proof
The probability generating function of the number of any man's sons is


 * $$ g(s) = \sum_{i = 0}^\infty s^i\Pr(\text{number of sons} = i) = \operatorname{E}(s^{X_1}). $$

Concerning this generating function we will need three important facts:


 * $$ g(1) = 1,\, $$


 * $$ g(0) = \Pr(\text{the man has no sons}), $$

and the average number of sons is


 * $$ \sum_{i=1}^\infty i\cdot \Pr(\text{number of sons} = i)

= \sum_{i=1}^\infty i\cdot 1^{i-1}\cdot \Pr(\text{number of sons} = i) = g'(1). $$

(In this last sum we can start at i = 1 rather than 0, since the term with i = 0 vanishes.)

Lemma: The probability generating function of the number Xn of nth-generation "Smith"s is


 * $$ \operatorname{E}(s^{X_n}) = g^n(s) = (g\circ\cdots\circ g)(s).\, $$

The proof of the Lemma uses the law of total expectation and proceeds by mathematical induction. We have


 * $$ \operatorname{E}(s^{X_{n+1}}) = \operatorname{E}(\operatorname{E}(s^{X_{n+1}}\mid X_n ))

= \operatorname{E}\left(\operatorname{E}\left( \left. s^{\sum_{i=1}^{X_n} X_{n,i}} \,\right| X_n \right) \right) = \operatorname{E}\left( \operatorname{E}\left( \left.\prod_{i=1}^{X_n} s^{X_{n,i}} \,\right| X_n \right) \right)$$

Since the inner expectation is conditional on Xn, we can treat Xn as constant for purposes of evaluating it. Thus, relying on the independence of Xn,i, we get


 * $$ \operatorname{E}\left( \operatorname{E}\left( \left.\prod_{i=1}^{X_n} s^{X_{n,i}} \,\right| X_n \right) \right)

= \operatorname{E}\left( \prod_{i=1}^{X_n} \operatorname{E}\left( \left. s^{X_{n,i}} \,\right| X_n \right) \right). $$

Since the Xn,i are independent of Xn, this is just


 * $$ \operatorname{E}\left( \prod_{i=1}^{X_n} \operatorname{E}\left( s^{X_{n,i}} \right) \right). $$

Since the Xn,i are identically distributed, each factor in the product above is the same. Recalling the definition of the generating function g, we see that each factor is just g(s). Now our expectation is


 * $$ \operatorname{E}\left( \prod_{i=1}^{X_n} g(s) \right).\, $$

Then we get


 * $$ \operatorname{E}\left( \prod_{i=1}^{X_n} g(s) \right) = \operatorname{E}\left( g(s)^{X_n} \right) $$

Observe that that is just the probability-generating function of Xn, evaluated at g(s). The induction hypothesis tells us that the probability-generating function of Xn is


 * $$ (\,\underbrace{g\circ\cdots\circ g}_{n}\,). $$

Evaluating this at g(s) give us


 * $$ (\,\underbrace{g\circ\cdots\circ g}_{n+1}\,)(s), $$

and the proof of the Lemma is complete.

Now observe that g and all of its derivatives are nonnegative on the interval [0, 1], and all derivatives are strictly positive on ( 0, 1 ] except in the trivial case in which the number of a man's sons is always either 0 or 1. Since g and its first and second derivatives are positive on ( 0, 1 ] and g(1) = 1, elementary calculus tells us that, besides the fixed point that g has at 1,


 * $$ g\text{ has } \begin{cases}

\text{exactly one fixed point }a \in (0,1),\text{ which is attractive} & \text{if }g'(1) > 1, \\ \text{no fixed points in }(0,1)\text{ and an attractive fixed point at 1} & \text{if } g'(1) \le 1. \end{cases}$$

Consequently for s < 1,


 * $$ \lim_{n\to\infty}(\,\underbrace{g\circ\cdots\circ g}_{n}\,)(s)

=\begin{cases} 1 & \text{if }g'(1) \le 1, \\ a < 1 & \text{if }g'(1) > 1. \end{cases} $$

[a continuity result is needed]

Proposing a new "sister project"?
[copied from village pump (proposals)]

Is there a standard procedure for proposing a new "sister project" (Wiktionary, Wikiquote, WikiBooks, WikeSource, WikiNews, and a number of others are "sister projects" of Wikipedia, run by the Wikimedia Foundation)? I've created this proposal, but I don't know if that's the sort of proposal being sought there, and as far as I know, no one's noticed it. What would be the best way to inform online communities of the proposal and invite participation? (Besides maybe a brief notice on the talk pages of interested WikiProjects?)

Briefly the idea is this: Web sites like rate-my-prof (or whatever its called) are for soundbites only; they do not welcome serious substantive discussion. They have extremely small limits on lengths of comments and don't want to change that. I'd like to have a forum devoted to the same topic, differing from those in that it would allow and encourage serious discussion. Michael Hardy (talk) 23:34, 16 March 2011 (UTC)


 * I don't believe this is something that the Wikimedia Foundation would be interested in. All of the current projects revolve around making knowledge more accessible, an encyclopedia, a dictionary, a collection of primary source documents, a collection of multimedia files. Your idea seems more along the lines of a social network or a forum on living people. I would personally not support such an idea being created under the WMF banner.  S ven M anguard   Wha?  04:59, 17 March 2011 (UTC)


 * You may want to check out the Strategy wiki, it's a place where you can make these kinds of proposals. -- &oelig; &trade; 08:01, 17 March 2011 (UTC)
 * Without considering the merits of the proposal, Wikimedia does have a page for proposing new projects at Proposals for new projects. Qwyrxian (talk) 08:04, 17 March 2011 (UTC)

other uses templates

 * (disambiguous):

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This article is about USE1. For other articles with similar names, see Otheruses (disambiguation).



This article is about USE1. For USE2, see Otheruses (disambiguation).



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 * (fully-specified):

This article is about USE1. For USE2, see PAGE2. For USE3, see PAGE3.


 * (all parameters except second and third are optional; however, omitting both the first and fourth values creates ambiguity, so please avoid):

For USE2, see PAGE2. For other articles with similar names, see Otheruses (disambiguation).

euclid sequences


\begin{array}{lclclclc} \{5,7\} & \overset{-1}\mapsto & \{2,5,7,17\} & \overset{-1}\mapsto & \{2,5,7,17,29,41\} & \overset{+1}\mapsto & \{2,3,5,7,17,19,29,41,103,241\} \\ & & & & & \overset{+1}\mapsto & \{2,3,5,7,13,17,19,29,41,103,241,105727,1456561\} \end{array} $$



\begin{array}{lclclclc} \{5,11\} & \overset{-1}\mapsto & \{2,3,5,11\} & \overset{-1}\mapsto & \{2,3,5,7,11,47\} & \overset{-1}\mapsto & \{2,3,5,7,11,47,151,719\} \\ & & & & & \overset{-1}\mapsto & \{2,3,5,7,11,47,67,79,151,719,1249,1783\} \end{array} $$