User:Michael Hardy/scratchwork

Pr(X ∈ g⁻¹(F))

Pr(''X'' ∈ <span style="text-decoration:overline">''g''⁻¹(''F'')</span>)

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))

--{{u|Mark viking}} {Talk} 20:05, 30 March 2019 (UTC)

$\oint .$

siℓℓy Σ¹
₁ Σ{{supsub|1|1}} a≪ b √1 − ζ²

<math display="inline"> or <math>\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#Numbering_equations:

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^words_deeds 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#Equation numbering. --Mark viking (talk) 01:09, 10 February 2014 (UTC)

section

${\frac {\partial }{\partial t}}\iiint _{\scriptstyle V}\rho \mathbf {u} \,dV=-\,{}$ $_{\scriptstyle S}$ $(\rho \mathbf {u} \cdot d\mathbf {S} )\mathbf {u} -{}$ ${\scriptstyle S}$ ${}\,p\,d\mathbf {S}$ $\displaystyle {}+\iiint _{\scriptstyle V}\rho \mathbf {f} _{\text{body}}\,dV+\mathbf {F} _{\text{surf}}$

${\partial \over \partial t}\iiint _{\scriptstyle V}\rho \mathbf {u} \,dV=-\,{}$ $\mathbf {u} -{}$ ${\scriptstyle S}$ $\displaystyle {}+\iiint _{\scriptstyle V}\rho \mathbf {f} _{\text{body}}\,dV+\mathbf {F} _{\text{surf}}$

${\partial \over \partial t}\iiint _{\scriptstyle V}\rho \mathbf {u} dV=$− ${\scriptstyle S}$ $(\rho \mathbf {u} \cdot d\mathbf {S} )$$\mathbf {u}$ − ${\scriptstyle S}$ $pd\mathbf {S}$ +$\iiint _{\scriptstyle V}\rho \mathbf {f} _{body}dV+\mathbf {F} _{surf}$

---

So, we get the following relation:

${\begin{aligned}f(n,k)&=\sum _{r}g(n',r)\times f(n-n',k-r)\end{aligned}}$ (4)

Now, we can easily observe the generating function relation for (4) to be:

next

  0–1

  1–2

  2–5

  5–10

  10–20

  > 20

---

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

WikiProject watchers

Tychonian orrery

$\digamma \,$

$\mathrm {\koppa} \,$

$\mathrm {\stigma} \,$

$\mathrm {\coppa} \,$

Ϛ and ϛ

$2001971723011=13\times 105727\times 1456561\,$

$\mathbb {E} {\bigl [}X{\bigr ]}\,$

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 Oskar Perron (1907) and Georg Frobenius (1912),

---

Bateman (1931). Bateman defined it by

$\displaystyle k_{n}(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan(\theta )-n\theta )\,d\theta$

References

• Bateman, H. (1931), "The k-function, a particular case of the confluent hypergeometric function", Transactions of the American Mathematical Society, 33 (4): 817–831, ISSN 0002-9947, MR 1501618

WikiProjects ranked by liveliness (copied from Wikipedia_talk:Database_reports#WikiProjects_ranked_by_liveliness)

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:

• Wikipedia:Database reports/WikiProject watchers
• Wikipedia: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#Interpolation_error
• Euler_method#Error_bound
• Numerical error
• Trapezoidal rule
• Simpson's rule

Polyhedron whose vertices correspond to unordered pairs

(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 × 15/2 = 60 edges. Euler's formula V − E + F = 2 would then imply

15 − 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 K_3,3. 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 KG_6,2, so it's quite closely related to the Petersen graph, which is KG_5,2. I'm sure someone has made a nice drawing of KG_6,2. —Bkell (talk) 03:04, 6 April 2011 (UTC)

I see that the complement of KG_6,2 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 T₆ 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 [1] 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

{{DISPLAYTITLE:''n''-sphere}}

CAT:CSD#Pages in category

watchers

$\underbrace {1+\cdots +1} _{n\ {\mbox{terms}}}\,$

$\underbrace {1+\cdots +1} _{n{\text{ terms}}}\,$

\xrightarrow

ℝ

⌊t/2⌋

π π

$A{\xrightarrow {\varphi }}B\,$

x ↦ y

a ≡ b mod n

{{foreignchars|ü and ß|Guenter Kiessling}}

≫ ≪

a ≈ b

x

∩

A \ B

√−1

∈ ∉ ⊆ ⊇ ∅ ± ∞ ℓ ∩

Unicode mathematical operators
User:KSmrq/Chars

$-{\frac {d}{dx}}\left[p(x){\frac {dy}{dx}}\right]+q(x)y=\lambda w(x)y,$ (1)

{{open-closed}} and {{closed-open}}

a bit of formatting by Hans Adler

${\begin{aligned}{\frac {1}{N_{e}^{(F)}}}={\frac {1}{4T}}\left\{{\frac {1}{N_{0}^{f}}}+{\frac {1}{N_{0}^{m}}}+\sum _{i}\left(\ell _{i+1}^{f}\right)^{2}\left(v_{i+1}^{f}\right)^{2}\left({\frac {1}{\ell _{i+1}^{f}}}-{\frac {1}{\ell _{i}^{f}}}\right)\right.\;\;\,&\\\left.{}+\sum _{i}\left(\ell _{i+1}^{m}\right)^{2}\left(v_{i+1}^{m}\right)^{2}\left({\frac {1}{\ell _{i+1}^{m}}}-{\frac {1}{\ell _{i}^{m}}}\right)\right\}.&\end{aligned}}$

120±12 in

220±20 km/s

The orbital speed was thought to be 220±20 km/s.

The orbital speed was thought to be 220±20.

The orbital speed was thought to be 220 ± 20.

Wolfgang Doeblin on Kolmogorov's equation

$X_{t}=x+\beta _{\left(\int _{0}^{t}\sigma ^{2}(s,X_{s})\,ds\right)}+\int _{0}^{t}a(s,X_{s})\,ds.$

section

siℓ

〈x,y〉instead of <x,y>

m = number of groups

n = size of each group

$SSW=\sum _{i=1}^{m}\sum _{j=1}^{n}(y_{ij}-{\overline {y}}_{i\bullet })^{2}\,$

$SSB=n\sum _{i=1}^{m}({\overline {y}}_{i\bullet }-{\overline {y}}_{\bullet \bullet })^{2}\,$

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.

X_n,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 X_n = 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 X_n 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 X_n, we can treat X_n as constant for purposes of evaluating it. Thus, relying on the independence of X_n,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 X_n,i are independent of X_n, this is just

$\operatorname {E} \left(\prod _{i=1}^{X_{n}}\operatorname {E} \left(s^{X_{n,i}}\right)\right).$

Since the X_n,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 X_n, evaluated at g(s). The induction hypothesis tells us that the probability-generating function of X_n 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,

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

Consequently for s < 1,

$\lim _{n\to \infty }(\,\underbrace {g\circ \cdots \circ g} _{n}\,)(s)={\begin{cases}1&{\text{if }}g'(1)\leq 1,\\a<1&{\text{if }}g'(1)>1.\end{cases}}$

[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. Sven Manguard 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. -- œ^™ 08:01, 17 March 2011 (UTC)

Without considering the merits of the proposal, Wikimedia does have a page for proposing new projects at meta:Proposals for new projects. Qwyrxian (talk) 08:04, 17 March 2011 (UTC)

other uses templates

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euclid sequences

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

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