text:
Talk:Non-analytic smooth function

What is section 1.4 About?
Section 1.4 about the transition function seems to be missing some details / explanation as to its relevance. There is nothing to explain its relationship to the previous section and, even after re-reading several times, I personally cannot figure out what it has to do with the topic or why it is relevant in any way, to this page. For reference, I have a basic applied-maths background at the undergrad level and landed on this page looking for some clarification. Perhaps I'm not the target-audience, and it would be clear to others, but personally, I would appreciate clarification, if possible, as it certainly looks interesting. 69.168.163.21 (talk) 18:09, 28 June 2019 (UTC)

How it is ill-behaved : added formula for n-th derivative with proof
I just wonder if formalizing the proof is really a good thing, or rather a liability. I sort of liked the older approach, which was driving the point home while keeping the discussion informal. Is the new proof a bit less approachable for non-mathematicians? Oleg Alexandrov 17:50, 18 Apr 2005 (UTC)

 * Sorry for the delay in reply. Maybe you're right (I was somehow afraid when I saw how it grew big...) But in fact I wanted to have the formula written down for further reference (I didn't remember that it was on x^3n, e.g.). Could we avoid complete annihilation by moving it to somewhere else (subpage /proof ?).  — MFH: Talk 16:30, 21 Apr 2005 (UTC)

 * Ugh - three problems with that. No need to divide the page; the use of subpages A/X has been deprecated for several years. What was the third? Doesn't really matter ... might have been that the copy-pasted page makes little sense. Charles Matthews 17:03, 21 Apr 2005 (UTC)
 * Actually we have a smooth function page as it is, and I think this page was only made to link to from list of mathematical examples. That's OK - the result is very important. But no need to chop everything so fine. Why not put proofs on smooth function, and leave this less formal - Oleg's point is good. Charles Matthews 17:06, 21 Apr 2005 (UTC)

I came across the subpage and immediately began the process of merging it back into the main article, since subpages are long since obsoleted in the article namespace (Subpages). However, now that I see this discussion here, it might not be appropriate for me to just paste it right in since there's dispute on the subject. So I'm putting it here into talk:, below. Bryan 05:56, 5 Jun 2005 (UTC)

f is smooth on R
Let us prove in the sequel that the function
 * $$f: x\mapsto e^{-1/x^2} ~ (x\ne0); f(0) = 0$$

admits continuous derivatives of any order $$n\in\mathbb{N}$$ in all points of $$\mathbb{R}$$, given by

 * $$f^{(n)}(x) = \left\{\begin{matrix} R_n(x)\,f(x) & (x \ne 0) \\ 0 & (x = 0)\end{matrix}\right.,$$

where $$ R_n $$ is a rational function of the form $$ R_n(x) = P_n(x)/x^{3n} $$, with $$ P_n $$ a polynomial, such that $$R_n$$ is well-defined on $$\mathbb{R}^*=\mathbb{R}\setminus\{0\}$$.

continuity of f and f(n)
Any function of this form is indeed continuous on $$\mathbb{R}$$:
 * on $$\mathbb{R}^*$$, it is a product of continuous functions,
 * in $$x = 0$$, $$ R_n $$ is "at worst" equivalent to $$c/x^{3n}$$, and for any (also negative) integer $$m$$,
 * $$\lim_{x \to 0} x^m e^{-1/x^2} = 0,$$
 * such that $$\lim_{x \to 0} f^{(n)}(x) = 0 = f^{(n)}(0)$$, i.e. continuity of $$f^{(n)}$$ also in $$x = 0$$.

proof of the formula for the n-th derivative
For $$n = 0$$, we do have $$f(x) = f^{(0)}(x)$$ as in the above formula, with $$R_0 = 1 (=P_0)$$. In order to complete the proof by induction, it remains to show that if $$f^{(n)}$$ is of the above form for some $$n$$, then its derivative is again of the form $$f^{(n+1)}$$.

Of course, $$f^{ (n)} = R_n f$$ implies that $$f^{ (n+1)} = R_n' f + R_n f' = R_{n+1} f,$$ with (using f' =f×(+2/x3))
 * $$R_{n+1}(x) = R_n'(x) + 2 R_n(x)/x^3 = P_{n+1}(x)/x^{3n+3}$$

for all $$x \ne 0$$, where (for $$R_n = P_n/x^{3n}$$) $$P_{n+1} = x^3 P_n' - 3 n x^2 P_n + 2 P_n.$$

Thus it remains to consider the difference quotient in $$x = 0$$, which has the form

 * $$\lim_{x \to 0} \frac{f^{(n)}(x) - f^{(n)}(0)}{x} = \lim_{x \to 0} \frac{P_n(x)}{x^{3n+1}}\,e^{-1/x^2} = 0$$

(using the limit already mentioned previously), i.e. $$f^{(n + 1)}(0)=0$$.

Page title
Isn't the page title a little long? Enochlau 17:21, 21 Apr 2005 (UTC)
 * Yes, but not infinitely. Charles Matthews 17:36, 21 Apr 2005 (UTC)

I also find it long. Of course not inifinitely, since this is an imaginary concept and does not exist in reality. It also is of no practical use, e.g. everything is always smooth up to the measuring precision.

But to have proofs on a separate page where they could conveniently be edited would be much more useful imho... why not leave it there, even if it is not considered as part of the encycopedia? are the Talk pages considered as part of the encyclopedia? — MFH: Talk 4 July 2005 13:49 (UTC)
 * It is not a good idea to have subpages to a page. Yesterday I cleaned up all the subpages from the list of mathematical topics. You could put your proof from the subpage as a separate article a rigurous proof that an infinitely differentiable function that is not analytic (which becomes even longer :) Otherwise you can just paste it here on the talk page, but then you cannot put a link to it from the main article. Cheers, Oleg Alexandrov 4 July 2005 15:26 (UTC)
 * Maybe I don't understand what your point is. Are you trying to say that the schetched proof from this article should be on a separate page? Oleg Alexandrov 4 July 2005 15:28 (UTC)

Sheaf of smooth functions flasque?
In point 3.2 it is stated that the sheaf of smooth functions is flasque. This is not true (example: you cannot extend 1/x from the half line to the whole line). But my defintion of flasque comes from wikipedia, so i dont know if it is the right definition. Maybe what the author meant was that with holomorphic functions the restriction operation is injective (i.e. extension of holomorphic functions are unique) in contrast to smooth ones.

correlation with Stone-Weierstrass?
By Stone-Weierstrass, there must be some sequence of polynomials in x^n converging uniformly to the non-analytic smooth function on [-1, 1]. What does this sequence look like? (IT can't approach a Taylor polynomial). I forgot to log in; username nnn9245129.2.56.113 18:04, 16 February 2007 (UTC)

 * That sequence will look absolutely and completely different than Taylor's polynomial. That two functions are close to each other says nothing about whether their first derivatives are close to each other, same for higher order derivatives. And the Taylor polynomial is about all derivatives. Oleg Alexandrov (talk) 03:22, 17 February 2007 (UTC)

Requesting a clarification
I arrived here purely by chance and am not a mathematician. The opening sentence says that there are two important types of function. The second sentence states that "One can prove that...". Does this mean that it is possible to prove, or that one of the two function types can be used to prove? I assume it is the former, but the wording is unclear enough that I wish to raise the question. —Preceding unsigned comment added by 87.113.135.124 (talk) 17:38, 30 June 2009 (UTC)

Doubts about first proof (Which seem to be unjustified)
I have noticed that the proof considers affinely extended real numbers and not real numbers. (I refer to the non-analytic function in the article (the one on top).) exp(-1/x) for x > 0 can lead to strange things when considering just real numbers. (non necessarily correct)

 * I have been interchanging $$\lim x \downarrow 0$$ and $$1 / \infty$$. There is a subtle difference. This is the difference between arbitrary small and positive (i.e. negligible) and something that is more difficult to imagine.

Why? This is a set-theoretic issue.

Let +0 be 1 / infinity and let +0 be an element of the real set. (Which is not the case......)

When considering this page: https://en.wikipedia.org/wiki/Real_number

It seems that there is a strict lexicographical ordering on the reals. (By quickly looking at it.)

if not a < b and not b = b and b >= a meaning that a == b.

I guess that a + (+0) > a. Thus if you are very strict, a + (+0) <> a. In calculus, the == operator is overridden for practical purposes.

In the couter example:

first -1 / x is computed.

Let's take x = +0

We get -1 / x == negative infinity, which is not a real number.

Now suppose that affinely extended real numbers are used. Then exp(negative infinity) = 0, according to mathworld.

Using affinely extended real numbers seems to make things really complicated.

I think that exp(-1 /x) is already non analytic for x = +0.

My understanding of non-analytic smooth functions is that the constants of the power series do not shrink fast enough. As i (term index) gets larger.

As I was told, and what is illustrated on Wikipedia, is that the natrual logarithm seems to diverge in the domain x >= 2. However, this part of the function seems to be smooth. Perhaps the natural logarithm is a better example. — Preceding unsigned comment added by Wouterbu (talk • contribs) 03:45, 30 April 2013 (UTC)

 * why do you need to consider x=0+ ? the definition only requires to compute exp(-1/x) for x > 0 ; if x > 0 then 1 / x is a real positive number and exp(-1/x) is a real positive number . No need to use affinely extended real numbers anywhere. Mennucc (talk) 09:20, 30 April 2013 (UTC)

Your reasoning is correct. However, as I view it, rounding the infinitesimals is inappropriate here.

Let there be infinitely many reals within the domain [0,+0]. (By the definition of the real set.) It says that x is a positive real number. However, +0 is an element of R^+ (that's what I think).

According to Wikipedia, there is a total ordering on the reals. I find this very difficult, so let me just repeat what I read.

a <= b or b <= a, where a and b are in R Both sides of the disjunction can be true yielding equality.

My guess is that equality means set theoretic equality.

https://en.wikipedia.org/wiki/Real_number Not having read the full article, I do not think it answers our question.

Next we can try this:

https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals

(Not having read the full article either.)

Again 0 < x, then according to the axioms there exists an y in between 0 and x. Next there exists an y' in between 0 and y. a y, y' etc. This is a recusion.

What we get is a positive infinitesimal that is > 0. To be precise, by applying axiom 2 an exhaustively you get something like:

x -> x / 2, which is a definition of an infinitesimal. It is still debateble wheter it is allowed to apply the aximom non-finitely many times. Also the axiom does not state that the intermediate value is in the middle.

However, this may just about be the difference between reals and rationals. With rational numbers you are allowed to use finitely many steps, with real numbers, this need not be the case.

Thus, assuming that +0 is a positive real number, (like it is in the IEEE floating point standard), the sign bit is set to 0, indicating positive, then -1/+0 is not a real number (it should give -infinity, which can also be represented by IEEE floating point)

Though using this same IEEE standard, you may get the same solution as described on the page associated to this talk (or NaN). However, I don't think that Taylor's theorem is defined on affinely extended real numbers. — Preceding unsigned comment added by Wouterbu (talk • contribs) 11:03, 30 April 2013 (UTC)

 * You should read http://en.wikipedia.org/wiki/Archimedean_property where it says "In the axiomatic theory of real numbers, the non-existence of nonzero infinitesimal real numbers is implied by the least upper bound property". Usually in calculus +0 does not exists. +0 exists in "non standard analysis" but that is quite esoteric, never taught in calculus 101 . Moreover IEEE standard is not a good reference to understand how the mathematical real line works. Mennucc (talk) 11:59, 30 April 2013 (UTC)

Actually, +0 is not a positvive number in the IEEE floating point standard. +0 > 0 (0 is either +0 or -0) is false.

Indeed, it is not a good reference anyway. I think that I will read the article about the Archimedian property when I have time.

Every infinitesimal is 0 in span. However, there are different kinds of 0. Not all infinitesimals have the same span. This is because there can be a ratio difference right. dy/dx for example may be unequal to 1. Both dx and dy are 0 spanned infintesimals.

Currently I am under the impression that I do understand. Yesterday I found this:

http://en.wikipedia.org/wiki/Ordinal_number#Ordinals_extend_the_natural_numbers (Again without having read everything.)

Is it a good idea to look at ``unit infinitesimal`` as $$\frac{1}{\omega}$$? According to another wikipedia page, Leibniz used the definition $$\frac{1}{\infty}$$. — Preceding unsigned comment added by 62.131.60.88 (talk) 12:08, 1 May 2013 (UTC)

Having read (part of the article), I am now convinced that my critisicm was wrong. The Archimedean property makes sense. No need to use big infinite like numbers. I have misinterpreted the axiomatization of Tarski. It is only allowed to apply the a rule finitely many times and not infinite. Just as $\infty$ is not a member of the real set, neither is $$\frac{1}{\infty}$$. The real set is a field. If $$1/\infty$$ would be an element of the real set and if $$\frac{1}{\infty} \neq 0$$, then $$\infty$$ would also be an element of the real set, thus leading to a contradiction. — Preceding unsigned comment added by Wouterbu (talk • contribs)

What is with the searrows?
In the detailed proof of smoothness drop section, the limits use sea-arrows and not normal arrows? Should it not be like this:

$$\lim_{x \to 0}$$

and not:

$$\lim_{x \searrow 0}$$

What is the purpose of this? --MrMineHeads (talk) 01:01, 6 January 2019 (UTC)

 * It means limit from above, that is, it's an alternate notation for $$\lim_{x \to 0^+}$$. It might be nice to actually mention this in the article, though!  --JBL (talk) 15:34, 6 January 2019 (UTC)
 * I have now added a link to one-sided limit to clarify. --JBL (talk) 17:51, 6 January 2019 (UTC)

 * Alright thanks for the clarification. --MrMineHeads (talk) 23:52, 6 January 2019 (UTC)

Is the last edit right?
Is this edit correct? (And separately, that section is in really bad need of cleanup. Is there a source for that construction?!) --JBL (talk) 23:44, 26 May 2020 (UTC)

Is this the same k or not?
In the section A smooth function which is nowhere real analytic, a caption to an illustration of an approximation to the smooth but nowhere analytic function reads:

"Approximation of the smooth-everywhere, but nowhere-analytic function mentioned here. This partial sum is taken from k=20 to 2500."

But it is not at all clear that this is the same k used in the definition of the function:

"Define for all $$x \in \mathbb{R}$$
 * $$F(x):=\sum_{k\in \mathbb{N}} e^{-\sqrt{2^k}}\cos(2^k x)\ .$$"

Especially since 2(2 500) >  2(10 150)  >  10(10 149)  >  googolplex seems well beyond current computational ability.

If it is not, then I hope someone knowledgeable about this subject will fix this ambiguity.

In any case I hope this will be clarified. (If it is not the same k in the caption as in the text, maybe the picture caption should describe its index by using the same k as in the text.) — Preceding unsigned comment added by 2601:200:c082:2ea0:d8ea:383e:f1c6:7024 (talk) 01:31, 30 March 2024 (UTC)

title: Talk:Non-analytic smooth function


text:
Victorinox

Victorinox is a knife manufacturer and watchmaker based in the town of Ibach, in the Canton of Schwyz, Switzerland. It is known for its Swiss Army knives. Since its acquisition of rival Wenger in 2005, it has become the sole supplier of multi-purpose knives to the Swiss army. It is the world's biggest manufacturer of pocket knives; additionally, the company licenses its logo for watches, apparel, and travel gear.

Founding

The company was founded in 1884 as Messerfabrik Carl Elsener, the workshop of Karl Elsener in Ibach. In 1891, Elsener co-founded Schweizerischer Messerschmiedverband an association of Swiss knife manufacturers. This began their relationship delivering knives to the Swiss army. In 1892, most of Elsener's knife craftsmen left knives manufacturing as it became apparent that Solingen manufacturers could deliver at a lower price. Elsener persisted in competing with Solingen, hence incurring severe financial losses.

Elsener's Schweizer Offiziers- und Sportmesser was patented in 1897, and was later marketed internationally as the Original Swiss Army Knife. This knife was not adopted by the Swiss Armed Forces as ordnance, but its commercial success allowed the company to recover financially.

Upon the death of his mother in 1909, Elsener named the brand "Victoria" in her honor. The same year, the Swiss coat of arms was introduced as the company logo. In 1921, the company changed its trade name to the present "Victorinox", a portmanteau of "Victoria" and "inox", an abbreviation for acier inoxydable, French for stainless steel. Its legal name was officially changed in 1979.

Original versus Genuine
From 1908 to 2005, the delivery of knives to the Swiss Armed Forces was shared with Wenger. A compromise between the two companies stated that Victorinox would market their knives as the "Original Swiss Army Knives", while Wenger would market theirs as "Genuine Swiss Army Knives". Wenger was acquired by Victorinox in 2005. Knives actually made for the army (as opposed to the generic "Swiss Army" trademark) are known as Soldatenmesser ("soldier knives"). These were produced in five generations, known by the date of their introduction as models 1890, 1908, 1951, 1961 and "08" (with minor variants within each generation). Model 1890 was originally produced by Wester & Co, Solingen, and produced by Elsenser in competition with the Solingen manufacturer. Models 1908 and 1951 were also produced by several companies in Germany and Switzerland. Model 1961 was produced exclusively by Victorinox and Wenger, and Model 08, introduced after the acquisition of Wenger, is exclusively manufactured by Victorinox.

Swiss army issued knives: 1890–2008
As issued by the Swiss Army.

Victorinox in North America
In 1972, the Forschner Butcher Scale Company of New Britain, Connecticut became the exclusive Victorinox distributor for the United States. In 1981, the company went public, and Charles Elsener, president of Victorinox, acquired a significant percentage of its shares. In 1983, it was renamed the Forschner Group, Inc. In the 1980s, Forschner registered the Swiss Army name as a trademark in the USA. In 1992, Precise Imports Corp., a U.S. and Canadian importer of Wenger knives, sued and Forschner retained the rights to use the trademark on its compasses, timepieces, and sunglasses, while Precise could use it in marketing other non-knife items. In the mid-1990s, Forschner changed its name to Swiss Army Brands, Inc. (SABI).

In 2001, Victorinox teamed up with SABI to establish an international watch company, Victorinox Swiss Army Watch AG. In August 2002, Victorinox acquired all remaining shares of SABI to gain control of the Swiss Army trademark. Previously, SABI had sold the Swiss Army branded watch in North America and—under the license—the Victorinox branded watch outside North America. But afterwards, the combined Victorinox Swiss Army brand has been marketed worldwide.

Wenger acquisition
On 26 April 2005, Victorinox acquired Wenger, the other official supplier of the Swiss Army knife, announcing that it intended to keep both brands intact, only separate entities.

In 2006, the company had a workforce of 900 employees. Its daily output was about $34,000$ Swiss Army knives, $38,000$ multi-tools, and $30,000$ household kitchen knives. Approximately 90% of its production is exported to more than 100 countries. To avoid this, they set aside profits during boom periods to supplement recessionary periods as well as temporarily contracting employees to other companies as outsourced labor during recessions.

On 30 January 2013, Victorinox announced that the company was merging Wenger's knife product lines with the Victorinox brand to strengthen its competitive position internationally.

Bushcraft usage
The Farmer model was adopted by the Bushcraft community as their folding knife (circa 2005).

Luggage and clothing
In 2014, Victorinox acquired the TRG Group from Centric Group. For several years, TRG Group was the Victorinox licensee for the manufacturing of luggage and travel-oriented products. Victorinox integrated TRG Group in the company as the Victorinox Travel Gear division.

In 2017, Victorinox decided to close the apparel division with the purpose of focusing in other core product lines.

Swiss Army knives

The Swiss Army knife is the best-known product by Victorinox. Originally the sole supplier, Victorinox has shared the contract with Wenger since 1908. A compromise between the two companies gave Victorinox the right to advertise as the Original Swiss Army Knife, while Wenger laid claim to the title of Genuine Swiss Army Knife. Victorinox took over Wenger in 2005.

Swiss Army knives are widely used outside the army (and civil sales represent most of the turnover ). They are multi-functional tools, and many sizes and functional combinations are produced. NASA astronauts each have a Victorinox knife at their disposal as standard equipment. Victorinox knives have also been taken to Mount Everest and the Arctic as well. The "Champion", Victorinox's flagship model, prior to the introduction of the "SwissChamp" in 1986, is displayed in the New York Museum of Modern Art's Permanent Design Collection.

SwissCard
The SwissCard is roughly the size of a business card, typically with a small pair of scissors, a short non-folding knife, a small file with a screwdriver point, a plastic toothpick, tweezers, a slim ballpoint pen, and a straight pin, housed in a hard plastic case of 82 × 54.5 x 4.5 mm in size, with an inch ruler on one side and metric measurements on the other. Victorinox produces three types of SwissCards, the Classic, the Quattro and the Lite model. All three models differ in the number of functions they provide, ranging from 10 (Classic) to 13 functions (Quattro and Lite).

Cutlery
Victorinox has long produced other kitchen cutlery under their own name and the Forschner brand name. In 2011 Victorinox stopped using the Forschner name and has since manufactured the same knives only with the Victorinox name.

Professional knives
Victorinox make a range of household and professional (NSF certified) food preparation knives. They offer chef's knives, bread, carving, filleting, deboning, paring, Santoku and specialized knives such as a cheese knife. Victorinox models are made with molded plastic handles, wooden handles or riveted handles. Most models have stamped blades, but one series has a forged bolster. They also have no ricasso – the flat section of the blade located at the junction of the blade and the heel. Most models use stainless steel blades and one type is also available with ceramic blades.

Bayonets
Victorinox also produces the Sturmgewehr 90 bayonet for the Swiss Sturmgewehr 90 assault rifle. The bayonet has an overall length of 310 mm and a muzzle ring diameter of 22 mm. The 177 mm long blade is single-edged and it has no fuller. The bayonets were manufactured exclusively for the Swiss Army by both Wenger and Victorinox before the two companies merged.

Timepieces
In 1989, Victorinox entered the timepiece business in the United States under the brand name "Swiss Army".

Victorinox has various collections of watches which range from luxury dress watches to rugged diving watches. They feature mechanical and quartz movements. Collections include Infantry, Heritage, Divemaster, Airboss and I.N.O.X. The timepieces are all produced in Switzerland.

In 2014, Victorinox introduced the I.N.O.X. range of watches which it claims are resistant to high levels of shock, including being run over by a tank.

Other
Victorinox produced a fashion line up until the late 1990s, but retired from this project in 2017.

In 2012, Victorinox licensed its name to luggage manufacturer TRG Accessories to produce branded travel gear and business accessories. Victorinox acquired TRG Accessories from Centric Group in 2014.

A number of Emissive Energy Corps products have been redesigned and rebranded as Victorinox LED flashlights. Most are available with a knurled aluminium body, similar to Swiss Army knives. Similarly, SwissTools are Victorinox's multi-tools, i.e. a pair of pliers with other tools folded into the grips. In 2018, Victorinox marketed a perfume under the name "Swiss Army", and registered "Swiss Army" and "Swiss Military" as trademarks under US law. In October 2018, the Swiss Federal Office for Defence Procurement (Armasuisse) sued Victorinox, demanding the removal of these trademarks from the marketplace. The context of the lawsuit is a 2012 parliamentary motion by state councillor Thomas Minder requesting the active protection of trademarks such as "Swiss Army", "Swiss Military" or "Swiss Air Force" on the part of the federal authorities. Victorinox argued that it does own the (English-language) trademarks "Swiss Army" and "Swiss Military".

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Victorinox


text:
Ásgrímur Jónsson

Ásgrímur Jónsson (March 4, 1876 – April 5, 1958) was an Icelandic painter, and one of the first in the country to make art a professional living.

He studied at the Royal Academy in Copenhagen between 1900 and 1903 and traveled widely after graduation. The subjects of his pictures are mostly the landscapes of his home country, particularly mountains. His painting style is similar to the French impressionists like Corot. Some of his pictures also illustrate Icelandic sagas and folk tales.

He was also noted for his murals in various churches in Iceland. A number of his works are on display in the National Gallery of Iceland. Ásgrímur influenced many artists in Iceland including Jóhannes Sveinsson Kjarval. A short time before he died he donated his house at No. 74, Bergstaðastræti, Reykjavík to the Icelandic Government along with all those paintings which were at that time in his possession. These consisted of 192 oil paintings and 277 water colours together with a great number of unfinished pictures dating from various periods in his life. His house is now the Ásgrímur Jónsson Collection, part of the National Gallery of Iceland.

During his lifetime Ásgrímur was honoured in many ways. He was made honorary professor at the University of Iceland and, in 1933 he was made Grand Knight of the Icelandic Order of the Falcon. He was an honorary member of the Royal Swedish Academy of Arts and Knight of Dannebrog, first class.

He died in 1958 and was buried on the cemetery of Gaulverjabæjar Church in Flóahreppur.

title: Ásgrímur Jónsson


text:
Rotational symmetry

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.

Formal treatment
Formally the rotational symmetry is symmetry with respect to some or all rotations in $m$-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of $E^{ +}(m)$ (see Euclidean group).

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole $E(m)$. With the modified notion of symmetry for vector fields the symmetry group can also be $E^{ +}(m)$.

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the special orthogonal group $SO(m)$, the group of $m × m$ orthogonal matrices with determinant 1. For $m = 3$ this is the rotation group $SO(3)$.

In another definition of the word, the rotation group of an object is the symmetry group within $E^{ +}(n)$, the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, the rotational symmetry of a physical system is equivalent to the angular momentum conservation law.

Discrete rotational symmetry
Rotational symmetry of order $n$, also called $n$-fold rotational symmetry, or discrete rotational symmetry of the $n$th order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of $\tfrac{360^\circ}{n}$ (180°, 120°, 90°, 72°, 60°, 51 $3/7$°, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all objects look alike after a rotation of 360°).

The notation for $n$-fold symmetry is $C_{n}$ or simply $n$. The actual symmetry group is specified by the point or axis of symmetry, together with the $n$. For each point or axis of symmetry, the abstract group type is cyclic group of order $n$, $Z_{n}$. Although for the latter also the notation $C_{n}$ is used, the geometric and abstract $C_{n}$ should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see cyclic symmetry groups in 3D.

The fundamental domain is a sector of $\tfrac{360^\circ}{n}.$

Examples without additional reflection symmetry:
 * $n = 2$, 180°: the dyad; letters Z, N, S; the outlines, albeit not the colors, of the yin and yang symbol; the Union Flag (as divided along the flag's diagonal and rotated about the flag's center point)
 * $n = 3$, 120°: triad, triskelion, Borromean rings; sometimes the term trilateral symmetry is used;
 * $n = 4$, 90°: tetrad, swastika
 * $n = 6$, 60°: hexad, Star of David (this one has additional reflection symmetry)
 * $n = 8$, 45°: octad, Octagonal muqarnas, computer-generated (CG), ceiling

$C_{n}$ is the rotation group of a regular $n$-sided polygon in 2D and of a regular $n$-sided pyramid in 3D.

If there is e.g. rotational symmetry with respect to an angle of 100°, then also with respect to one of 20°, the greatest common divisor of 100° and 360°.

A typical 3D object with rotational symmetry (possibly also with perpendicular axes) but no mirror symmetry is a propeller.

Multiple symmetry axes through the same point
For discrete symmetry with multiple symmetry axes through the same point, there are the following possibilities:
 * In addition to an $n$-fold axis, $n$ perpendicular 2-fold axes: the dihedral groups $D_{n}$ of order $C2$ ($C3$). This is the rotation group of a regular prism, or regular bipyramid. Although the same notation is used, the geometric and abstract $D_{n}$ should be distinguished: there are other symmetry groups of the same abstract group type which are geometrically different, see dihedral symmetry groups in 3D.
 * 4×3-fold and 3×2-fold axes: the rotation group $T$ of order 12 of a regular tetrahedron. The group is isomorphic to alternating group $C4$.
 * 3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cube and a regular octahedron. The group is isomorphic to symmetric group $C5$.
 * 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group $I$ of order 60 of a dodecahedron and an icosahedron. The group is isomorphic to alternating group $C6$. The group contains 10 versions of $2n$ and 6 versions of $n ≥ 2$ (rotational symmetries like prisms and antiprisms).

In the case of the Platonic solids, the 2-fold axes are through the midpoints of opposite edges, and the number of them is half the number of edges. The other axes are through opposite vertices and through centers of opposite faces, except in the case of the tetrahedron, where the 3-fold axes are each through one vertex and the center of one face.

Rotational symmetry with respect to any angle
Rotational symmetry with respect to any angle is, in two dimensions, circular symmetry. The fundamental domain is a half-line.

In three dimensions we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). That is, no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates. The fundamental domain is a half-plane through the axis, and a radial half-line, respectively. Axisymmetric and axisymmetrical are adjectives which refer to an object having cylindrical symmetry, or axisymmetry (i.e. rotational symmetry with respect to a central axis) like a doughnut (torus). An example of approximate spherical symmetry is the Earth (with respect to density and other physical and chemical properties).

In 4D, continuous or discrete rotational symmetry about a plane corresponds to corresponding 2D rotational symmetry in every perpendicular plane, about the point of intersection. An object can also have rotational symmetry about two perpendicular planes, e.g. if it is the Cartesian product of two rotationally symmetry 2D figures, as in the case of e.g. the duocylinder and various regular duoprisms.

Rotational symmetry with translational symmetry
2-fold rotational symmetry together with single translational symmetry is one of the Frieze groups. A rotocenter is the fixed, or invariant, point of a rotation. There are two rotocenters per primitive cell.

Together with double translational symmetry the rotation groups are the following wallpaper groups, with axes per primitive cell:
 * p2 (2222): 4×2-fold; rotation group of a parallelogrammic, rectangular, and rhombic lattice.
 * p3 (333): 3×3-fold; not the rotation group of any lattice (every lattice is upside-down the same, but that does not apply for this symmetry); it is e.g. the rotation group of the regular triangular tiling with the equilateral triangles alternatingly colored.
 * p4 (442): 2×4-fold, 2×2-fold; rotation group of a square lattice.
 * p6 (632): 1×6-fold, 2×3-fold, 3×2-fold; rotation group of a hexagonal lattice.
 * 2-fold rotocenters (including possible 4-fold and 6-fold), if present at all, form the translate of a lattice equal to the translational lattice, scaled by a factor 1/2. In the case translational symmetry in one dimension, a similar property applies, though the term "lattice" does not apply.
 * 3-fold rotocenters (including possible 6-fold), if present at all, form a regular hexagonal lattice equal to the translational lattice, rotated by 30° (or equivalently 90°), and scaled by a factor $$\tfrac{1}{3} \sqrt {3}$$
 * 4-fold rotocenters, if present at all, form a regular square lattice equal to the translational lattice, rotated by 45°, and scaled by a factor $$\tfrac{1}{2} \sqrt {2}$$
 * 6-fold rotocenters, if present at all, form a regular hexagonal lattice which is the translate of the translational lattice.

Scaling of a lattice divides the number of points per unit area by the square of the scale factor. Therefore, the number of 2-, 3-, 4-, and 6-fold rotocenters per primitive cell is 4, 3, 2, and 1, respectively, again including 4-fold as a special case of 2-fold, etc.

3-fold rotational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, i.e. double translational symmetry and 6-fold rotational symmetry at some point (or, in 3D, parallel axis). The translation distance for the symmetry generated by one such pair of rotocenters is $$2\sqrt {3}$$ times their distance.

title: Rotational symmetry


text: Category:Lunar deities

title: Category:Lunar deities


text:
Luttinger liquid

A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting electrons (or other fermions) in a one-dimensional conductor (e.g. quantum wires such as carbon nanotubes). Such a model is necessary as the commonly used Fermi liquid model breaks down for one dimension.

The Tomonaga–Luttinger's liquid was first proposed by Sin-Itiro Tomonaga in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963, J.M. Luttinger reformulated the theory in terms of Bloch sound waves and showed that the constraints proposed by Tomonaga were unnecessary in order to treat the second-order perturbations as bosons. But his solution of the model was incorrect; the correct solution was given by Daniel C. Mattis and Elliot H. Lieb 1965.

Theory
Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron Hamiltonian:

$$H = \sum_{k} \epsilon_k c_k^{\dagger} c_k$$

is separated into left and right moving electrons and undergoes linearization with the approximation $$\epsilon_k \approx \pm v_{\rm F}(k-k_{\rm F})$$ over the range $$\Lambda$$:

$$H = \sum_{k = k_{\rm F} -\Lambda}^{ k_{\rm F} +\Lambda} v_{\rm F} k \left(c_k^{\mathrm R \dagger } c_k^{\mathrm R } - c_k^{\mathrm L \dagger }c_k^{\mathrm L}\right)$$

Expressions for bosons in terms of fermions are used to represent the Hamiltonian as a product of two boson operators in a Bogoliubov transformation.

The completed bosonization can then be used to predict spin-charge separation. Electron-electron interactions can be treated to calculate correlation functions.

Features
Among the hallmark features of a Luttinger liquid are the following:

 * The response of the charge (or particle) density to some external perturbation are waves ("plasmons" - or charge density waves) propagating at a velocity that is determined by the strength of the interaction and the average density. For a non-interacting system, this wave velocity is equal to the Fermi velocity, while it is higher (lower) for repulsive (attractive) interactions among the fermions.
 * Likewise, there are spin density waves (whose velocity, to lowest approximation, is equal to the unperturbed Fermi velocity). These propagate independently from the charge density waves. This fact is known as spin-charge separation.
 * Charge and spin waves are the elementary excitations of the Luttinger liquid, unlike the quasiparticles of the Fermi liquid (which carry both spin and charge). The mathematical description becomes very simple in terms of these waves (solving the one-dimensional wave equation), and most of the work consists in transforming back to obtain the properties of the particles themselves (or treating impurities and other situations where 'backscattering' is important). See bosonization for one technique used.
 * Even at zero temperature, the particles' momentum distribution function does not display a sharp jump, in contrast to the Fermi liquid (where this jump indicates the Fermi surface).
 * There is no 'quasiparticle peak' in the momentum-dependent spectral function (i.e. no peak whose width becomes much smaller than the excitation energy above the Fermi level, as is the case for the Fermi liquid). Instead, there is a power-law singularity, with a 'non-universal' exponent that depends on the interaction strength.
 * Around impurities, there are the usual Friedel oscillations in the charge density, at a wavevector of $$2 k_\text{F}$$. However, in contrast to the Fermi liquid, their decay at large distances is governed by yet another interaction-dependent exponent.
 * At small temperatures, the scattering of these Friedel oscillations becomes so efficient that the effective strength of the impurity is renormalized to infinity, 'pinching off' the quantum wire. More precisely, the conductance becomes zero as temperature and transport voltage go to zero (and rises like a power law in voltage and temperature, with an interaction-dependent exponent).
 * Likewise, the tunneling rate into a Luttinger liquid is suppressed to zero at low voltages and temperatures, as a power law.

The Luttinger model is thought to describe the universal low-frequency/long-wavelength behaviour of any one-dimensional system of interacting fermions (that has not undergone a phase transition into some other state).

Physical systems
Attempts to demonstrate Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in condensed matter physics.

Among the physical systems believed to be described by the Luttinger model are:

 * artificial 'quantum wires' (one-dimensional strips of electrons) defined by applying gate voltages to a two-dimensional electron gas, or by other means (lithography, AFM, etc.)
 * electrons in carbon nanotubes
 * electrons moving along edge states in the fractional Quantum Hall Effect or integer Quantum Hall Effect although the latter is often considered a more trivial example.
 * electrons hopping along one-dimensional chains of molecules (e.g. certain organic molecular crystals)
 * fermionic atoms in quasi-one-dimensional atomic traps
 * a 1D 'chain' of half-odd-integer spins described by the Heisenberg model (the Luttinger liquid model also works for integer spins in a large enough magnetic field)
 * electrons in Lithium molybdenum purple bronze.

title: Luttinger liquid


text:
Translational symmetry

In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation). Discrete translational symmetry is invariant under discrete translation.

Analogously, an operator $A$ on functions is said to be translationally invariant with respect to a translation operator $$T_\delta$$ if the result after applying $A$ doesn't change if the argument function is translated. More precisely it must hold that $$\forall \delta \ A f = A (T_\delta f).$$

Laws of physics are translationally invariant under a spatial translation if they do not distinguish different points in space. According to Noether's theorem, space translational symmetry of a physical system is equivalent to the momentum conservation law.

Translational symmetry of an object means that a particular translation does not change the object. For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.

Geometry
Translational invariance implies that, at least in one direction, the object is infinite: for any given point p, the set of points with the same properties due to the translational symmetry form the infinite discrete set ${p + na | n ∈ Z} = p + Z a$. Fundamental domains are e.g. $H + [0, 1] a$ for any hyperplane H for which a has an independent direction. This is in 1D a line segment, in 2D an infinite strip, and in 3D a slab, such that the vector starting at one side ends at the other side. Note that the strip and slab need not be perpendicular to the vector, hence can be narrower or thinner than the length of the vector.

In spaces with dimension higher than 1, there may be multiple translational symmetry. For each set of k independent translation vectors, the symmetry group is isomorphic with Zk. In particular, the multiplicity may be equal to the dimension. This implies that the object is infinite in all directions. In this case, the set of all translations forms a lattice. Different bases of translation vectors generate the same lattice if and only if one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1. The absolute value of the determinant of the matrix formed by a set of translation vectors is the hypervolume of the n-dimensional parallelepiped the set subtends (also called the covolume of the lattice). This parallelepiped is a fundamental region of the symmetry: any pattern on or in it is possible, and this defines the whole object. See also lattice (group).

E.g. in 2D, instead of a and b we can also take a and $a − b$, etc. In general in 2D, we can take $pa + qb$ and $ra + sb$ for integers p, q, r, and s such that $ps − qr$ is 1 or −1. This ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair. Each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental domain. The vectors a and b can be represented by complex numbers. For two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group, see lattice (group).

Alternatively, e.g. a rectangle may define the whole object, even if the translation vectors are not perpendicular, if it has two sides parallel to one translation vector, while the other translation vector starting at one side of the rectangle ends at the opposite side.

For example, consider a tiling with equal rectangular tiles with an asymmetric pattern on them, all oriented the same, in rows, with for each row a shift of a fraction, not one half, of a tile, always the same, then we have only translational symmetry, wallpaper group p1 (the same applies without shift). With rotational symmetry of order two of the pattern on the tile we have p2 (more symmetry of the pattern on the tile does not change that, because of the arrangement of the tiles). The rectangle is a more convenient unit to consider as fundamental domain (or set of two of them) than a parallelogram consisting of part of a tile and part of another one.

In 2D there may be translational symmetry in one direction for vectors of any length. One line, not in the same direction, fully defines the whole object. Similarly, in 3D there may be translational symmetry in one or two directions for vectors of any length. One plane (cross-section) or line, respectively, fully defines the whole object.

Examples

 * Frieze patterns all have translational symmetries, and sometimes other kinds.
 * The Fourier transform with subsequent computation of absolute values is a translation-invariant operator.
 * The mapping from a polynomial function to the polynomial degree is a translation-invariant functional.
 * The Lebesgue measure is a complete translation-invariant measure.

title: Translational symmetry


text:
User:Chrism/monobook.css

span.updatedmarker { background-color: transparent; color: #006400; } strong.mw-watched a { font-weight: normal; background: url(//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Pentagram_dis.svg/13px-Pentagram_dis.svg.png) no-repeat left; /* @noflip */ padding-left: 16px; }
 * 1) watchlist formatting

.mw-rollback-link { font-weight: bold; text-transform: uppercase; font-family: "Bank Gothic" }

 * 1) siteSub { display: none !important; }

/* "All-Color" v0.1 for the article message box templates */ table.ambox { background: #f8fcff; border: 1px solid #aaa; } table.ambox-delete { background: #efd2d2; } table.ambox-content { background: #fce6cc; } table.ambox-style { background: #fcf3d5; } table.ambox-move { background: #ead6f4; } table.ambox-notice { background: #d2e8ff; }

/*Hide category edit-notice*/
 * 1) category-namespace-editnotice { display: none !important; }

/*Hide footer*/ li#f-copyright, li#f-lastmod,
 * 1) footer {display: none; }
 * 1) editpage-copywarn { display: none; }
 * 2) editpage-copywarn2 { display: none; }
 * 3) editpage-copywarn3 { display: none; }

a:link { text-decoration: none; } a:visited { text-decoration: none; } a.new:link { text-decoration: none; } a.new:visited {text-decoration: none; } a.new:link:hover { text-decoration: bold; color: #CC2200; } a.new:visited:hover { text-decoration: bold; color: #CC2200; } a.extiw:link { color: #22447D; } a.extiw:link:hover { color: #22447D; } a.stub:link { text-decoration: none; } a.stub:link:visited { text-decoration: none; }

.watchlistredir { font-style: italic }

/* make a few corners round, only supported by moz/firefox/other gecko browsers for now */ -moz-border-radius-topleft: 1em; -moz-border-radius-topright: 1em; } -moz-border-radius-topleft: 1em; -moz-border-radius-bottomleft: 1em; } div.pBody { -moz-border-radius-topright: 1em; -moz-border-radius-bottomright: 1em; }
 * 1) p-cactions ul li, #p-cactions ul li a {
 * 1) content {

/* underline just the text that's different */ span.diffchange { text-decoration:underline; }

.diffchange { background: #fdd; }

/******************************************************************/ /* moving catlinks to the right                                  */ /******************************************************************/

position:absolute; z-index:1; border: 1px solid #ffd700; background: #0077aa; right:1em; top:-1em; max-width: 75%; float:right; margin: 0.2em; padding:0.2em; }
 * 1) catlinks {

.catlinks { color: #aaaaaa; font-family: Verdana,sans-serif; font-size:50%; line-height: 1.2em; text-align:left; text-indent:0; text-transform: none; text-decoration: none; white-space:normal; margin: 0.2em; }

.catlinks a { color:#ffffff; text-decoration: none; }

/******************************************************************/ /* Custom bottom tabs                                            */ /******************************************************************/

width: 100%; font-size: smaller; position: static; } padding: 0 0.4em; border-left: 1px solid Grey; }
 * 1) mytabs {
 * 1) mytabs h5 { display: none }
 * 2) mytabs ul { margin: 0; }
 * 3) mytabs li { display: inline; }
 * 4) mytabs li a {

/******************************************************************/ /* Custom table options                                          */ /******************************************************************/

table.popupBorderTable { background: #0000FF /* blue */ !important } table.popupTable { background: #f0ffff /* azure */ !important } font.popupFont { font-family: serif !important; color: #440044 !important; font-size: medium !important }

/******************************************************************/ /* Language colours                                              */ /******************************************************************/

span[lang] { color: green; }

/******************************************************************/ /* Popup configs                                              */ /******************************************************************/

.navpopup { background-color: #fdf7b9 !important; border: 1px solid #ffd700; }

title: User:Chrism/monobook.css


text:
User:Chrism/monobook.js

importScript('User:Cameltrader/Advisor.js'); importScript('User:Smith609/reftool.js');

// Edd edit tabs for other page if (mw.config.get('wgNamespaceNumber') >= 0) $(function{ var pCactions = document.getElementById('p-cactions'); if (!pCactions) return; var caMain = pCactions.getElementsByTagName('li')[0]; var caTalk = document.getElementById('ca-talk'); var caEdit = document.getElementById('ca-edit'); if (!caEdit) caEdit = document.getElementById('ca-viewsource'); var caHistory = document.getElementById('ca-history'); if (!caMain || !caTalk || !caEdit || !caHistory) return; var el_move, el_create, id2; if (mw.config.get('wgNamespaceNumber') % 2) { //talk space  el_move = caTalk; el_create = caMain; id2 = ''; } else {   el_move = caMain; el_create = caTalk; id2 = 'discussion'; } caEdit.firstChild.innerHTML = 'edit'; caHistory.firstChild.innerHTML = 'hist'; el_move.parentNode.insertBefore(caHistory, el_move.nextSibling); el_move.parentNode.insertBefore(caEdit, el_move.nextSibling); var href = el_create.firstChild.getAttribute('href', 2); if (el_create.className.indexOf('new') < 0){ mw.util.addPortletLink('p-cactions', href + '?action=history', 'Hist',    'ca-history-'+id2, id2+' history', '', el_create.nextSibling); mw.util.addPortletLink('p-cactions', href + '?action=edit', 'Edit',    'ca-edit-'+id2, 'Edit '+id2, '', el_create.nextSibling); } caMain.style.marginRight = '0.3em'; caTalk.style.marginRight = '0.3em'; caTalk.style.marginLeft = '1.6em'; }) //

function morelinks { var tabs = document.getElementById('p-cactions').cloneNode(true); // don't use the same ids twice- replace the p-cactions id and prepend 'mytabs-' to the li's tabs.id = 'mytabs'; // needs this to be set from js, it ignores the css width for some reason tabs.style.width = '100%'; var listitems = tabs.getElementsByTagName('LI'); for (i=0;i<listitems.length;i++) { if(listitems[i].id) listitems[i].id = 'mytabs-' + listitems[i].id; } // drop them at the bottom of the content area document.getElementById('content').appendChild(tabs); } if (window.addEventListener) window.addEventListener("load",morelinks,false); else if (window.attachEvent) window.attachEvent("onload",morelinks);

//Category tab at top function categoryToTop { var thebody = document.getElementById('contentTop'); var categories = document.getElementById('catlinks');

if (categories != null) { categories.parentNode.removeChild (categories); thebody.parentNode.insertBefore(categories, thebody); } }

//Popup menus config

window.popupStructure = 'menus'; window.popupTocLinks = false; window.popupRedlinkRemoval = true; window.popupHistoricalLinks = true; window.popupFixRedirs = true; window.popupFixDabs = true; window.popupShowGender = true; window.popupSimplifyMainLink = true; window.popupNewWindows = true;

// See changes since last edited

defaultdiffs = 1000;   // or whatever number, see script for details importScript('Wikipedia:WikiProject User scripts/Scripts/Changes since I last edited');

// Adds a "Changes since last load" link to your watchlist.

title: User:Chrism/monobook.js


text:
Randwick, New South Wales

Randwick is a suburb in the Eastern Suburbs of Sydney, in the state of New South Wales, Australia. Randwick is located 6 kilometres south-east of the Sydney central business district and is the administrative centre for the local government area of the City of Randwick. The postcode is 2031.

History
Randwick was named after the village of Randwick, Gloucestershire, England, birthplace of Simeon Henry Pearce, who became Mayor of Randwick six times. Simeon, who migrated to Australia in 1842, and his brother James who arrived in 1848, were responsible for the founding and early development of Randwick. Simeon built the first stone house in the area in 1848, called Blenheim House, which can still be seen in Blenheim Street. It was neglected for some time in the mid-1900s, but was eventually acquired by Randwick City Council and restored.

Proclaimed as a Municipality in February 1859, and as a City in 1990, Randwick has a rich history and a number of heritage buildings. Another Mayor of Randwick, George Kiss, lived in the house known as Ventnor in the 1870s. A two-storey sandstone house, Ventnor was built by Alderman Edward Dawson in 1859, and is situated on Avoca Street, overlooking Coogee. It is now owned by the nearby Our Lady of the Sacred Heart Church (built 1888). Other buildings of note include the St Jude's Church group, also on Avoca Street. Originally designed by Edmund Blacket, (who also designed St. Andrew's Cathedral, Sydney and the University of Sydney), the church was completed in 1865. It was modified by H.M. Robinson in 1889. The rectory next door was built in 1870. The Verger's Residence, designed by Thomas Rowe and completed in 1862, was the original Randwick Municipal Chambers. This distinctive building, with its Gothic touches, was followed by the present Randwick Town Hall, which was built further north in Avoca Street in 1881. The church group and Ventnor are listed on the Register of the National Estate.

Further south down Avoca Street is the sandstone building with a tower, built in 1859 as an investment property by William Ellis, one of Randwick's first six elected aldermen. After briefly being used as the 'Star and Garter' Inn by Mrs Elizabeth Shipway, it was later the home of Captain Thomas Watson, who was responsible for commissioning the 1874 statue of Captain James Cook, sculpted by Walter McGill, which still stands at the south side of the building. Captain Watson also heightened the square tower, thus ensuring another distinctive landmark feature of the area. Close by is another even more imposing building in Randwick's heartland - the Prince of Wales Hospital, which started life as the Randwick Asylum for Destitute Children. Opened in 1858, it was partly financed by the legacy of Dr. Cuthill, whose name is commemorated by the adjacent Cuthill Street. The southern wing was added in 1863, by which time over 800 children were being cared for. In 1915 the Asylum buildings were finally deemed unsuitable as premises in which to provide a realistic home life for children, and they were then adapted for WWI use as a military and repatriation hospital. In 1953 the complex was officially renamed as the Prince of Wales Hospital, which continues to grow as a renowned medical facility.

Other noteworthy buildings include private homes like Ilfracombe, Clovelly and Torquay in Avoca Street, and Venice, in Frenchmans Road. The latter was built 1884–84 on part of St Mark's Glebe. The land had been leased in 1880 for 99 years to S.Holmes and J.Parsons. The house features Tudor and Gothic elements and has stained glass windows. It has been described as a "comparatively rare and distinctive example of late Victorian Gothic architecture retaining most of its detailing intact." It has a New South Wales heritage listing.

Another notable home was Sandgate, located in Belmore Road. This sandstone house was built circa 1870 on Crown land purchased by Simeon Pearce in 1853. It was bought by the Federal Government in the 1920s; it was then used as a Red Cross facility for First World War veterans. It was saved from demolition in 1978 and restored by Randwick Council.

The last Gothic mansion left in Randwick is Nugal Hall, located in Milford Street. It was designed by Mortimer Lewis and the southern portion of the house was built in 1853 for Alexander McArthur, a shipping merchant. The house features a staircase with glass dome above; the staircase was built of Mauritius mahogany and cedar. The coach house building at the front was originally a lodge for horse-drawn vehicles. The northern portion of the house was completed by Dr Fred Tidswell who owned the Coogee Bay Hotel. The architect of the northern portion is thought to have been Mortimer Lewis's son, Oswald. The Tidswell family owned the house from c. 1880 – 1903. Frederick Squire Tidswell (1831–1898) and his wife Mary Ann (1836–1912) had nine children including the microbiologist Dr Frank Tidswell (1867–1941) and architect Thomas Tidswell (1870–1950).

Heritage listings
Randwick has a number of heritage-listed sites, including the following sites listed on the New South Wales State Heritage Register:
 * 124 Alison Road: Randwick Post Office
 * 162 Alison Road: Randwick Presbyterian Church
 * 102-108 Avoca Street: St Jude's Church, Randwick
 * 211-215 Avoca Street: Corana and Hygeia
 * 128 Belmore Road: Sandgate, Randwick
 * 60 Bundock Lane: Electricity Substation No. 341
 * 66 Frenchmans Road: Venice, Randwick
 * 17 Gilderthorpe Avenue: Hooper Cottage
 * 16-18 Milford Street: Nugal Hall
 * 2S Frances Street: Electricity Substation No. 349
 * 43 St Marks Road: Rathven, Randwick
 * 43 St Pauls Street: Ritz Cinema, Sydney
 * 18-20 Stanley Street: Emanuel School, Australia
 * 26-42 The Avenue: Avonmore Terrace
 * 29-39 Young Street: Big Stable Newmarket

The following structures are listed on the (now defunct) Register of the National Estate (unless stated otherwise), or the heritage schedule of Randwick Council's Local Environment Plan; the latter list includes approximately four hundred items.
 * Ventnor
 * Original Randwick Borough Chambers
 * Goldring House and Attached Residence
 * Prince of Wales Hospital Group
 * Our Lady of the Sacred Heart Church
 * The Park, Avoca and Cuthill Streets
 * Ilfracrombe, Avoca Street, (circa 1906)
 * Torquay, Avoca Street, (circa 1906)
 * Clovelly, Avoca Street, (circa 1906)
 * Newmarket House and The Big Stable, Barker and Young Streets
 * Captain James Cook Statue, corner Belmore Road and Avoca Street
 * Hannan's Butcher Shop (former Star and Garter Inn), Belmore Road
 * Former Randwick Police Station, Coogee Bay Road
 * Peckham, Cowper Street
 * Essex, Cuthill Street
 * House, 11 Cuthill Street
 * Royal Hotel, Cuthill Street
 * Shaldon, Darley Road
 * Swan Isle, Darley Road
 * Gordon Terrace, Gordon Street
 * Members' Stand, Randwick Racecourse
 * Busby Bore (Tunnel from Lachlan Swamp to Hyde Park, City)
 * Tayar, The Avenue

Demographics
In the 2021 Census, there were 28,943 people in Randwick, of which 52.9% were female and 47.1% were male. The most common ancestries in Randwick were 31.1% English, 22.2% Australian and 17.8% Irish. 73.5% of people only spoke English at home. Other languages spoken at home included 2.4% Mandarin, 1.9% Spanish, 1.5% Greek, 1.4% Portuguese and 1.3% French. The most common responses for religion were 41.0% No Religion, 26.0% Roman Catholicism, 8.1% Anglicanism and 4.9% Judaism. The median weekly household income in Randwick was $2,442, which was 28.5% higher than the national median.

Commercial area
Randwick is primarily a residential area. The Randwick Racecourse takes up a large portion in the north-west corner of the suburb. The Prince of Wales Hospital, Royal Hospital for Women, Sydney Children's Hospital and the Prince of Wales Private Hospital together occupy the Randwick Hospitals' Campus, a large contiguous campus bounded by Barker Street, Avoca Street, High Street and Hospital Road. A number of teaching and research facilities are also located on the campus. The campus is being expanded under a major project called the "Randwick Campus Redevelopment" project, which will see it expand west to the suburb boundary with Kensington and Kingsford. Prince Henry Hospital is also in Randwick.

The main commercial area is centred on Belmore Road. A range of retail stores are located at Belmore Road along with two shopping centres (Royal Randwick Shopping Centre and Randwick Plaza, which contains a Coles Supermarket) on eitherside of Belmore Road. Avoca Street is also quite popular for commerce, there is the 150-year-old Coach and Horses Hotel. Where the two busy streets meet is the gourmet Cookhouse, a heritage sandstone building with a statue of Captain Cook outside and High Cross Park, with a war memorial. Within one kilometre, Coogee is another retail precinct offering good food services while enjoying the great view of the beach. St Pauls Road junction also wins good fame for its popular restaurants, cafés and recreational facilities. Randwick's main entertainment district is The Spot, located in the suburb's south-east, which contains the Randwick Ritz cinema, a famous art deco cinema which dates from the 1930s and numerous cafes, bars and restaurants.

Churches

 * Our Lady of the Sacred Heart Catholic Church
 * St Jude's Anglican Church
 * East Sydney Community Christian Church,
 * Randwick Presbyterian Church,
 * Canvas Church
 * Rhema Church of God.

Sport and recreation

Just 6 kilometres east of Sydney's CBD, the City of Randwick, once known as the 'Sports Coast' is home to some 29 km of coastline, 10 beaches and bays and many major recreation facilities. Randwick City is one of the most visited tourist areas outside central Sydney, attracting some 13 million visits each year. The city is home to the Coastal Walkway, Randwick Racecourse, Coogee and Maroubra Beaches, Botany Bay National Park, Centennial Parklands (which includes Queens Park and Moore Park), several top class golf courses and major institutions including the University of New South Wales and the Prince of Wales Hospitals.

A number of well-known sporting teams represent the local area. Since 1882 the Randwick District Rugby Union Football Club nicknamed the 'Galloping Greens' or 'The Wicks' have represented the district in the Sydney premier grade rugby competition, the 'Shute Shield' and play their home games at the picturesque Coogee Oval. The local cricket team Randwick Petersham Cricket Club plays in the Sydney Grade Cricket domestic competition and was formed through a merger between Randwick and Petersham Cricket Clubs.

Randwick is represented in the National Rugby League by the Sydney Roosters, formerly the Eastern Suburbs District Rugby League Football Club. Junior rugby league teams include the Clovelly Crocodiles, the Coogee Dolphins and Coogee Randwick Wombats.

The former Randwick Post Office building in the centre of Randwick Junction (corner of Alison Road and Avoca Street) is known as Easts House. The building is owned by the Sydney Roosters (Eastern Suburbs Rugby League Club), which has donated the building for use by the Ted Noffs Foundation, for the purpose of providing assistance to troubled youth in the district. There are numerous community or corner parks like Alison Park, Randwick Peace Park, Frank Doyle Park, Fred Hollows Reserve and Writtle Park in the locality of Randwick used for recreational and sporting purposes.

Schools
Primary schools in the area include Our Lady of the Sacred Heart, Coogee Public School, Claremont College, Randwick Public School, Coogee Preparatory School, and Rainbow Street Primary. Secondary schools include two systemic Catholic schools, Brigidine College and Marcellin College, a Jewish day school Emanuel School and three state schools, Randwick Boys High School, Randwick Girls' High School, and the Open High School. Randwick North High School was closed in 2001 and the site was divided between Open High School and Randwick Public School.

There are branch campuses of the University of New South Wales and Sydney Institute of TAFE located on King Street, between the Racecourse and Centennial Park.

Transport
Until December 2019, Randwick was only served by buses. The main bus corridor is Belmore Road with buses to Bondi Junction, Coogee, Eastgardens, Maroubra, Sydney Airport and Sydney CBD.

On 13 December 2012, the NSW Government announced a commitment to build the CBD and South East Light Rail from Circular Quay down George Street to Central Station, then across to Moore Park and down Anzac Parade. South of Moore Park the line splits into two branches; one of which heads to Randwick via Alison Road. A bus/tram interchange will be established in Randwick and many of the bus routes that currently traverse Anzac Parade to access the city will be replaced by feeder routes connecting to the light rail. Construction commenced in 2015 and services on the Randwick branch commenced in December 2019.

Notable people
The following people are residents or former residents of Randwick.

 * Amy Vera Ackman - hospital administrator
 * Len Allmond - rugby league player
 * Doug Ayres - Australian rules footballer
 * Thomas Barkell - flying ace
 * John Baxter - author and journalist
 * Henry Bland - public servant
 * Kevin Brennan - actor
 * Henry Burrell - naturalist
 * Bruce Caldwell - rugby union player
 * Esma Cannon - actress
 * John Carrick - politician
 * Michael Cleary - rugby league and union player
 * Dylan Collard - soccer player
 * Bill Cosgrove - Australian rules footballer
 * Frederick Crouch, politician
 * Geoff Davies - cricketer
 * John Earnshaw - engineer and inventor
 * Reg Ellis - cricketer and Australian rules footballer
 * Lauryn Eagle - boxer and water skier
 * Reg Farnell - rugby league player
 * William Patrick Fitzgerald - Lord Mayor of Sydney
 * Doug Freeman - cricketer
 * Clifford Grant - singer
 * Sandy Harbutt - actor and director
 * Dean Hawkins - rugby league player
 * Mirrie Hill - composer
 * Jacques Hilling - actor
 * Fred Hollows - ophthalmologist
 * Bob Hornery - actor
 * Sarah Hynes - botanist and teacher
 * Peter Garrett - musician and politician
 * Syd Gregory - cricketer
 * Lauri Kennedy - cellist
 * Antony Kidman - psychologist and academic
 * Edward Hope Kirkby - watchmaker
 * Col Loughnan - musician
 * Raecene McGregor - rugby league player
 * Frederick Menkens - architect
 * Paulini - singer
 * John Peden - jurist and politician
 * Dick Persson - public servant
 * Oliver Puflett - soccer player
 * Tom Mead - politician
 * Patrick Rowe - cricketer
 * Jason Sangha - cricketer
 * Gabi Simpson - netballer
 * John Solomon - rugby union player
 * Ryan Teague - soccer player
 * Mary Tindale - botanist
 * Jessica Thornton - sprinter
 * Quinton Tidswell - artist
 * Frank Wall - doctor and politician
 * Colin Windon - rugby union player and soldier
 * Keith Windon - rugby union player
 * Mervyn Wood - rower and police officer
 * Tom Wright - rugby union player

title: Randwick, New South Wales


text:
Jóhannes Sveinsson Kjarval

Jóhannes Sveinsson Kjarval (15 October 1885 – 13 April 1972) was an Icelandic painter. He is considered one of the most important artists of Iceland. His son is interior designer Sveinn Kjarval.

Early life
Born into poverty, Kjarval was adopted and as a young man worked as a fisherman. He spent his spare time drawing and painting, and learned basic skills from artist Ásgrímur Jónsson. At age 27, with financial support from fishermen and the Icelandic Confederation of Labour, he passed an entrance examination and was admitted to the Royal Danish Academy of Fine Arts, where he completed his studies. During his time in Copenhagen, he became acquainted with various styles including Impressionism, Expressionism, and Cubism, as well as becoming an accomplished draughtsman. Later, he also took trips to France and Italy.

Style
Kjarval was a prolific painter, leaving thousands of drawings and paintings. The paintings vary greatly in style and frequently mix different styles. Although not surreal, some of his works include absurd and symbolist elements, mixing elves and myths into landscape. Many of his works include Icelandic landscapes and lava formations but many of his landscape paintings are partially "cubist" and abstract with his focus on the closest ground rather than the mountains in the background. Later in his life his art frequently also included abstract painting. In 1958, he was awarded the Prince Eugen Medal by the King of Sweden.

Legacy

 * In Reykjavík, one of three buildings belonging to the Reykjavík Art Museum is called Kjarvalsstaðir and presents Kjarval's works alongside temporary exhibitions.
 * He is depicted on the Icelandic 2000 króna banknote.
 * The 1977 debut album Björk by Björk includes an instrumental flute tribute to Kjarval, written and performed by Björk

title: Jóhannes Sveinsson Kjarval


text: Talk:Gadfly (philosophy and social science)


text:
Wikipedia talk:Half-million pool

Until when are we allowed to place our bets? Certainly you'll agree with me that placing a bet when we're at 499,999 shouldn't be allowed. — Timwi 10:49, 3 Jun 2004 (UTC)

 * Ahem. Look up my original statement- Two weeks from this writing, this page will be locked to prevent late entries. ;) →Raul654 13:16, Jun 3, 2004 (UTC)

What do we win? GUllman 21:21, 3 Jun 2004 (UTC)
 * The same thing you get for contributing here, only lots of it all at once. →Raul654 21:48, Jun 3, 2004 (UTC)

 * Lots of wikilove, perhaps? No, we have lots of time to draw some special star or banner we can award the skilled statistican (if that is a word). ✏ Sverdrup 21:50, 16 Jun 2004 (UTC)

Million article pool
So, when does the million article pool start? I was thinking the day we hit 500k, unless other people want an intermediate level pool. Not that we need a decision any time soon. -- Cyrius|✎ 21:17, 16 Jun 2004 (UTC)

 * Do you think there's really a million different things which deserve an interesting article having written about? pir 09:59, 17 Jun 2004 (UTC)
 * Let me count the ways: Every living species, currently in the millions, every town, currently in the hundreds of thousands, every politico, currently in the hundreds of thousands, etc. Yes there are probably about 2 or 3 billions things we could write about. Burgundavia 10:11, Jun 17, 2004 (UTC)

 * The Million pool might as well start now. The english Wikipedia's doubling time is between 10 and 13 months now, and its only 13 if you use among the lowest historical monthly growth rates and predict based on those.  The latest 5 month moving average rate is 7.6% growth per month and it's been relativley stable between 5% and 8%.  7.6% puts the 500,000 article milestone at early Febuary 2005 and the 1,000,000 in late October 2005. (I'll take October 25th in the pool). I'm surprised so many people guessed at dates that would require growth outside of any of the recent average growth rates.  I guess few people did the analysis.  Clearly most people voting in Febuary 2005 like Angela and I did. :)  Of course, as insightfully pointed out in Modelling Wikipedia's growth, the growth may be the early part of a logistic curve, but who knows where the exponential growth will stop and diminishing returns set in. I'm betting it's past 500,000, but may be before 1,000,000. - Taxman 12:38, Jun 17, 2004 (UTC)

Default time of day
I've noticed that many people didn't specify a time of day. We should decide on what time to assign those guesses before closing the pool. Otherwise, it's possible that the actual time will occur when different people would win depending on whether their guess counts for 0000 UTC or 1200 UTC. I suggest 1200 UTC. -- Yath 09:42, 17 Jun 2004 (UTC)

 * 1200 sounds good to me. Dysprosia 09:47, 17 Jun 2004 (UTC)
 * Since I don't see any date collisions here, I'll just assume that their vote is valid for the entire day. And for the record, all guesses are assumed to be UTC. →Raul654 18:23, Jun 17, 2004 (UTC)
 * Let me point out that making people's votes valid for a period of one day contradicts the contest page, which states that whoever comes closest wins. Presumably, someone gets to win even if nobody guessed the correct date?  Also, are you going to assign 24-hour periods straddling the guesses of people who specified the time of day? Assigning a specific instant to each entry would be simpler, IMHO. -- Yath 11:55, 19 Jun 2004 (UTC)
 * Making users votes valid for the entire day does not contradict the idea that the closest wins, it is just one way of defining the rules that removes the ambiguity of who wins. If you have the entire day, and the 500,000th article happens on that day, you are by definition, the closest.  Therefore the pronouncement that people have the entire day simply makes specific times of day irrelevant.  This make sense since few people chose a time of day, and the contest rules did not mention to choose one. Therefore a rule that does not consider the time of day is more consistent with the rules that were originally put forth. - Taxman 13:37, Jun 19, 2004 (UTC)
 * I hate to tell you, but if we assume 12:00 for each day, then we get *exactly* the same result. Let's say someone guesses May 22 and someone else May 23, and someone May 24. Let's say we hit 500,000 at at 00:01 May 23, then the person who guessed May 23 is correct. Let's say someone guessed May 22 and May 24. It happens at 12:01 May 23. Then the person who guessed the 24th is correct. The only slight issues is with the two people who guessed the same date, but each of them specificed a specific time, so we're still good. →Raul654 15:50, Jun 19, 2004 (UTC)

Damn it
Just my luck to miss this opportunity. Wikipedia has got its last 100,000 since January 4, so it should take another year from now (assuming the growth rate is constant and not exponential, which it likely will be). - Mark 02:22, 22 Jun 2004 (UTC)

 * That sounds like a vote for a 750,000 pool instead of waiting for the full million.

 * I just had a thought, we could open the next pool now, and leave it open until the half-million pool ends. -- Cyrius|✎ 03:32, 22 Jun 2004 (UTC)

 * No, just be patient - this pool probably won't take a year to finish - 14 months at the outside. Then we can start the next one. →Raul654 03:49, Jun 22, 2004 (UTC)

Please add to Category:Wikipedia statistics
Thanks. ··gracefool |☺ 02:36, 1 Sep 2004 (UTC)

 * Done. -- Cyrius|✎ 03:12, 1 Sep 2004 (UTC)

 * Sorry to be niggly, but could you give it a sort key (eg. )? ··gracefool |☺

 * Done.

 * Gracefool banned. (<-- joke, you "sysops are evil" idiots!) -- Cyrius|✎ 03:39, 1 Sep 2004 (UTC)
 * Lol. Cheers. ··gracefool |☺ 04:01, 1 Sep 2004 (UTC)

 * Aw C'mon, I want to ban someone. I might as well start with Gracefool -- nothing personal, Grace -- so should I need to ban someone in an actual emergency, I know what the steps are. -- llywrch 23:44, 4 Sep 2004 (UTC)
 * Look, an emerge 'n' see!!!!!11!!111!1111 ··gracefool |☺ 05:18, 5 Sep 2004 (UTC)

I Win!!
I just want to note that my entry was actually a prediction of the date we would hit one million articles in all versions - and is undoubtedly the most accurate on that basis. I assure you that this was my intention from the start and will be pleased to receive all the acclaim and cookies that this amazingly accurate prediction deserves. Thank you, thank you -- sannse (talk) 16:25, 14 Sep 2004 (UTC)

Congratulations! You won a tractor! ✏ Sverdrup 23:28, 23 Sep 2004 (UTC)

I shall treasure it always! -- sannse (talk) 17:40, 24 Sep 2004 (UTC)

Million article pool
I have started the Million pool, since Wikipedia just exceeded 400,000 articles. The pool will be closed when 500,000 is exceeded (i.e. when we find out who won this pool). Feel free to place your entries any time between now and that day. — David Remahl 11:04, 21 Nov 2004 (UTC)

My failed attempt
About 100 articles were submitted at nearly the same time to be the 500,000th. Of those, 50 were mine, all public domain American Civil War battle articles. (only took about half an hour to make them all). -- BRIAN 0918  21:46, 17 Mar 2005 (UTC)

Which article is it?
Which one? -- Toytoy 06:30, Mar 18, 2005 (UTC)

 * 04:54, Mar 18, 2005 Milon's Secret Castle (1163 bytes) . . 63.249.99.51
 * 04:54, Mar 18, 2005 Involuntary settlements in the Soviet Union (2071 bytes) . . Mikkalai
 * 04:54, Mar 18, 2005 Fire 'n Ice (340 bytes) . . 63.249.99.51
 * 04:54, Mar 18, 2005 Snow Brothers (265 bytes) . . 63.249.99.51
 * 04:54, Mar 18, 2005 Little Samson (273 bytes) . . 63.249.99.51
 * 04:54, Mar 18, 2005 Bionic Commando (607 bytes) . . 63.249.99.51
 * 04:54, Mar 18, 2005 Keshmirian (135 bytes) . . 69.227.0.149
 * 04:54, Mar 18, 2005 P. J. Abbott (322 bytes) . . Hedley

Anyone of them? -- Toytoy 06:35, Mar 18, 2005 (UTC)

 * We don't know, but people I've talked to have agreed that it's probably one of the top six or so there. Alterego went nuts with trying to get a snapshot as it happened, and wrote about it on his blog. -- Cyrius|✎ 10:38, 18 Mar 2005 (UTC)

 * They all now seem to be markerd as stubs. So they no longer really count.--Henrygb 14:22, 18 Mar 2005 (UTC)

 * We went with Involuntary settlements in the Soviet Union based on Alterego's calculation of the time, down to the second. If you assume that these were all spaced out equally over that minute, that one comes closest to being the one created at that precise time. The fact that it isn't about a Nintendo game is an added bonus. --Michael Snow 17:29, 18 Mar 2005 (UTC)

 * The reasoning that eliminated Milon's Secret Castle was incorrect. I'm pissed off about it because I'm quoted in said reasoning as saying something I didn't. -- Cyrius|✎ 20:10, 18 Mar 2005 (UTC)

 * Cyrius is referring to Alterego's explanation at Wikipedia talk:Press releases/March 2005. This assumed that certain articles weren't included in the count. My reasoning was not based on this erroneous assumption, so it shouldn't be affected by it. --Michael Snow 21:02, 18 Mar 2005 (UTC)

 * The line of reasoning you are using for this, then, is faulty as well. I will upload more data when I get home in a couple of hours. The articles are not spaced out evenly over a period of a minute - there was a huge rush of them. From memory (i'm not on the server I used), at the time of turning 500,000 there was a ten second gap before turning 500,000 where the server did not purge the cache. If a decision is to be made on a SINGLE article, it would have to be Milon's Secret Castle (being as it meets the requirements of an article). However, a more accurate representation will be a tie between all articles that were produced in that ten second period.  --Alterego 22:32, Mar 18, 2005 (UTC)

 * Just now browsing those that I have posted, consider this page produced ten seconds before turning 500,000 (and this is with a purged cache). The count is 499993. So in those ten seconds, seven articles were created - they are the top seven in toytoy's reproduced list above. --Alterego 22:32, Mar 18, 2005 (UTC)

 * Uh, how do we know they're the top seven rather than the bottom seven? I realize that it's quite possible the articles were posted more closely together in terms of time; my assumption about being evenly spaced was mostly because lots of people want to have a specific article they can point to. Calling it a tie isn't very satisfying in that sense, and this seems as good a way as any of breaking the tie. --Michael Snow 22:48, 18 Mar 2005 (UTC)

 * Further studying the data has presented a new problem - conflicting data. I assumed that there was no need to specifically request a non-cached page from New Pages, however, this page at 23 seconds and this page at 46 seconds are identical. This conflicts with this non-cached page at 36 seconds which says 499993, and this non-cached page at 46 seconds which says exactly 500000. Note that between 23 and 46 I received 12 copies of the exact same New Pages - all of which have the articles listed above. However, between 36 and 46 I received NO copies of User:JRM/Sandbox which contained the article count, because I was requesting a non-cached page, which take longer to produce.  So how do we analyse this data? Clearly there were seven new articles created between 36 and 46 seconds, however, all the articles listed above existed since 23 seconds.  This would on the face seem contradictory, unless we consider that the server timestamps are not accurate, as all articles listed after the ones posted above are stamped with 55 minutes. At this point, can someone intimately familiar with the software answer two questions? They are 1) Is New Pages cached for non-logged in users (appears yes) and 2) is there a delay, somewhere in the processing, and not necessarily in this order, between a) The article counter going up and b) the timestamp being placed on that article and showing up on RC or NP.  I understand that it may not be an intentional delay, but I have a sneaking suspicion that due to the heavy load, some of those articles marked as coming into existence at 55 minutes were actually from 54. Oh, how I could not have predicted this =) PS: I saved all pages from within a minute or so to backups.rar (250 KB) if anyone likes. --Alterego 00:54, Mar 19, 2005 (UTC)

 * Okay, so maybe it was Battle of Bean's Station after all? (See Wikipedia talk:Press releases/March 2005). Anyway, as you indicated earlier, it's a kind of precision that's really beyond our technical capacity. At this point the Involuntary settlements information has already gone out beyond hope of recall, and it's about as likely to be right as any other random article within those few minutes. Supposedly a Hebrew Wikipedia article on the Flag of Kazakhstan was the millionth article overall, I believe, but I have no idea how that's supposed to have been identified - probably just an equally arbitrary designation. --Michael Snow 01:08, 19 Mar 2005 (UTC)

What about the deleted articles? Do they decrement the counter? What about deleted and recreated articles? Mikkalai 05:11, 19 Mar 2005 (UTC)

title: Wikipedia talk:Half-million pool