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User:Supernerd11/Talk archive July 2014-January 2015

'These are archived messages from my talk page dating from August 2014 to January 2015 (I listed it as starting in July simply for continuity in the archive box, but my talk page was very quiet that month). If you wish to revive any of them, please move it back there with a sentence or two saying that it came from here, but don't edit it while it's here.'

Articles for deletion/GamesRadar
Hey there Supernerd, new sources have been found for this article - maybe you'd like to add a !vote? Thanks, Ansh666 04:48, 2 August 2014 (UTC)

 * Thanks, but as of now, I don't have the time to look over the new sources. If this debate carries on for a few more weeks, then I'll be happy to analyze 's list, but until then, I'll just stick with my little side comment. Supernerd11 Firemind ^_^ Pokedex 14:56, 5 August 2014 (UTC)

 * Aaaaand it's gone. Supernerd11 Firemind ^_^ Pokedex 21:00, 6 August 2014 (UTC)

Hey
I haven't seen you around lately. What's up? Tezero (talk) 21:42, 3 August 2014 (UTC)

 * Now that I'm not in school, my ability to use the internet has been a lot less than before. I still pop in when I can, and I should be a lot more active starting here in the next few weeks. Supernerd11 Firemind ^_^ Pokedex 14:43, 5 August 2014 (UTC)
 * A'ight. My ability to use the internet this summer, incidentally, is much greater because no school. Tezero (talk) 15:50, 5 August 2014 (UTC)

 * Actually, I just got my new laptop a lot sooner than I thought I would, so I'm back! (Getting ready for college is still top priority, but I can still put more time to Wikipedia) Supernerd11 Firemind ^_^ Pokedex 20:49, 6 August 2014 (UTC)
 * Cool. Hope to see you around these parts. Tezero (talk) 23:45, 6 August 2014 (UTC)

Polandball
Hi, I have recently expanded and rewritten the Polandball article at Draft:Polandball. I have also requested undeletion of the original article (that should never have been deleted in the first place) at Deletion_review/Log/2014_August_6. I am leaving you this message as you have the Polandball article in your userspace. <IP_ADDRESS> (talk) 07:19, 6 August 2014 (UTC)

 * Hm, don't think that's the reaction you were expecting. Ah well, I'll be working on my draft, and maybe someday it'll be good enough to go live. Supernerd11 Firemind ^_^ Pokedex 21:04, 6 August 2014 (UTC)
 * "Poor sourcing" - sounds like the AfDer may have been the nationalist-leaning one. I thought it looked fine. Tezero (talk) 23:44, 6 August 2014 (UTC)
 * Supernerd. Oh, I got the reaction I was expecting, believe you me :> You may like to look at this. the AfD was a load of nonsense; you'll see how Sandstein also topic banned me so that I couldn't dispute anything. Read what an independent Polish editor had to say on the sourcing of the original article -- Commons:User:Russavia/Sources. And I was also banned for 12 months because of the creation of this article. Nice huh? Cheers, <IP_ADDRESS> (talk) 09:05, 7 August 2014 (UTC)
 * That's ridiculous on the face of it, . (For future reference, do IPs get notifications when they're tagged? I don't know.) Was there anything else you were accused of that could've caused it? You may consider creating an account. Tezero (talk) 14:48, 7 August 2014 (UTC)
 * 80.109 was blocked as a sock of Russavia, so that'll remain a mystery. Polandball cannot into Wikipedia czar ♔   21:37, 7 August 2014 (UTC)
 * gib sources pl0x Tezero (talk) 22:16, 7 August 2014 (UTC)
 * I am accused of many, many things. None of which are particularly true; in most cases they are outright falsehoods. I just accept that I am the devil incarnate :>

 * What is funny is this. tr:Polandball was deleted by an admin on that project some time ago. The other day I put into my userspace tr:User:Russavia/Polandball. I left a message for the deleting admin pointing them to the draft, and they went ahead and undeleted the article based upon the clear evidence it is notable and the article on Turkish Wikipedia simply needs expansion and sourcing. Another editor then left a message on the article talk page noting the userspace draft and that it could be used for expansion of the article. Both editors were very cordial and it's a very nice example of interwiki collaboration.

 * Compare that to the deletion review request here on English Wikipedia. The likes of, and  continue to engage in utter douchebaggery, and it's obvious that their butthurt is stronk. So much so, that I really think they need some of this. :> Regards, <IP_ADDRESS> (talk) 01:24, 8 August 2014 (UTC)
 * Here's Theoden's diff, for the public record. It should be noted that not all Wikipedias have the same inclusion standards; I believe English is one of the strictest, and particularly for Internet phenomena and other new forms of media. (Have you ever noticed how many one-sentence pages we have on bridges, docks, tiny villages, and train stations in Europe with no sources to prove notability? Truly strange.)
 * That being said, I think if the Polandball sources' reliability can be asserted, there shouldn't be much of any dirt on the article to keep it down. The AfD looks to have been based mostly on ad hominems at you and the potential of the subject being controversial (which isn't supposed to even prevent articles from reaching FA status - see intelligent design, global warming, atheism, and Xenu) rather than sources, anyway. It's kinda nice that you have a changing IP; no meaningful banning. (And if they somehow got around that, I'm sure you could use proxies.) If you decide to get another account, though, I would recommend staying away from this kind of topic; that would be a sure tip-off that it's you. Tezero (talk) 02:16, 8 August 2014 (UTC)

 * I just added 's references to the bottom of my userspace draft (not ready to go live yet, even ignoring the deletion review and associated discussions), but since I don't know Polish, I can't really do much with them. I better log off for tonight, but tomorrow I can do a better check into the various arguments, etc. (I can also make sure to have a case full of the ointment for if and when it's needed) Supernerd11 Firemind ^_^ Pokedex 03:40, 8 August 2014 (UTC)

 * First off, and this is just my opinion, the original AfD should've been relisted; a lot of the Delete !votes are along the lines of "Wikipedia isn't knowyourmeme" or "Delete as an attack page". By this train of thought, the Deletion Review (which relied primarily on arguments about the outcome of the AfD) is rendered moot. It looks like the best plan here is to merge the various Polandball articles (the aforementioned drafts in the Deletion Review and my userspace (which in turn is from simple.wiki), as well as User:Babelia/Polandball) floating around, then presenting that. I'll go ahead and ping a couple other users who might be interested in this:  Supernerd11 Firemind ^_^ Pokedex 06:11, 9 August 2014 (UTC)
 * Addendum: Apparently, there was an ArbCom case about, among other things, Polandball. Supernerd11 Firemind ^_^ Pokedex 06:29, 9 August 2014 (UTC)

 * Uh, Russavia? Is your draft somewhere else? It's no longer on the Deletion Review.... Supernerd11 Firemind ^_^ Pokedex 06:45, 9 August 2014 (UTC)
 * You understand that Russavia was banned and won't be responding, right? czar ♔   15:10, 9 August 2014 (UTC)
 * Would emailing Russavia still work, or is that dependent on the account being registered? Tezero (talk) 15:25, 9 August 2014 (UTC)
 * Russavia is quite capable of responding despite his ban...FYI, the rolling assortment of worldwide IPs is him. That being said, his draft of the article is visible in the history of the DR, where he posted it . Revent talk 08:35, 10 August 2014 (UTC)

 * Ah, couldn't find it in the history before, thanks Revent. Supernerd11 Firemind ^_^ Pokedex 20:16, 10 August 2014 (UTC)

Hi peoples, this is "Russavia". I am linking this as it contains some further comment. Supernerd11, you are welcome to take my text and use it verbatim. All information is verified, and the quality of the sources are good. There is no danger in taking my text (CC-Zero) and using that. The thing that you would have to be mindful of is that some people will try to attack the sources. If you look at this you will see an admin removing information from an article using one of the sources, claiming it is not a reliable source. Another editor started a discussion here and in response to the admin I posted this (which was reverted of course). The other editor has provided other information which confirms the reliability of the source. This mindless and debunked attacking of the reliability of sources will be used by people such as FPaS. When they do this, here is something else one can mention....

At the end of 2013 there was a conference in Poland which was convened by the Institute of Literary Research (pl:Instytut_Badań_Literackich_PAN) (information in English on the Institute). The institute is a research institute and is under the aegis of the Polish Academy of Science (PAN). So there is no doubt as to the reliability of such institutes. If you refer to page 21 of the linked to PDF, you will see "Paweł Lewandowski (UWr), Polandball. Dlaczego memy śmieją się z polskiej historii?". Paweł gave a lecture at this conference on the Polandball meme and whether it accurately portrays Polish history and other issues -- such as those presented in the culture.pl case study. This lecture could also be used as a source, but it means getting a hold of video or audio and taking information from it. Such information could be used via Cite conference. I have also been in touch with Paweł and his resultant, more extensive, paper will also be published in the coming months in a scholarly Polish journal -- publication date unknown.

There was even information on Vesti FM in Russia about Polandall and how Russians and Poles perceive each other.

All requirements such as WP:N, WP:GNG, WP:RS, WP:V, etc are met with the text that I wrote, and the fact that there is continued coverage of the meme, even going so high as the Polish Academy of Science, would likely indicate there will be more in future.

If one sticks to discussing the sources and ignoring all the numbskullery, you'd be fine. Whether you want to take my text and use that at WP:DRV I'll leave that up to you. Cheers Russavia 07:13, 16 August 2014 (UTC)

This doesn't look like it needs to be an article to me.... How else can I help? ReginaldTQ (talk) 08:32, 27 August 2014 (UTC)

Incomplete DYK nomination
Hello! Your submission of Template:Did you know nominations/Nowhere Men (comic) at the Did You Know nominations page is not complete; see step 3 of the nomination procedure. If you do not want to continue with the nomination, tag the nomination page with db-g7, or ask a DYK admin. Thank you. DYKHousekeepingBot (talk) 11:28, 17 August 2014 (UTC)

DYK for Nowhere Men
— HJ Mitchell |  Penny for your thoughts?  12:02, 24 September 2014 (UTC)

Wikipedia talk:Articles for creation/John Cooney (Video game designer) concern
Hi there, I'm HasteurBot. I just wanted to let you know that Wikipedia talk:Articles for creation/John Cooney (Video game designer), a page you created, has not been edited in 6 months. The Articles for Creation space is not an indefinite storage location for content that is not appropriate for articlespace.

You may request Userfication of the content if it meets requirements.

Thank you for your attention. HasteurBot (talk) 01:30, 13 October 2014 (UTC)

 * I went ahead and nominated it for deletion since I've had it userfied for a long time now. Supernerd11 Firemind ^_^ Pokedex 20:00, 1 November 2014 (UTC)

Archiving URLs
Its never a bad time to archive, because the original link doesn't have to be replaced. swaps the links to make the archive more prominent if the link is dead. « Ryūkotsusei » 21:39, 10 November 2014 (UTC)

 * Oh, alright, thanks. I'll get on fixing that tonight. Supernerd11 Firemind ^_^ Pokedex 17:41, 11 November 2014 (UTC)

DYK for Jack Lumber
Cas Liber (talk · contribs) 22:03, 21 November 2014 (UTC)

Merry Christmas!

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Happy New Year!

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text:
User:William Ellison/sandbox 3

PROJECTIVE INCIDENCE STRUCTURES

Introduction
The object of this article is to describe the axiomatic foundations of the theory of projective incidence structures.

The main, and historically first, application is in the foundations of projective and affine geometry. However, the theory of such structures has applications in communication theory and in cryptography.

The aim is to develope the classification of projective incidence structures up to the point where one is lead to the study of specific algebraic structures such as : GL_n (R), where specific R is a division ring or specific ternary rings.

The first classifying facto is the dimension of the structure.
 * Structures of dimension greater than 2
 * Structures of dimension 2

For structures of dimension 2 the classifications is by the Desargues axiom
 * Desarguesian planes
 * Non-desarguesian planes

A projective incidence structure can be associated with an algebraic coordinate structure and the "isotopic" equivalence classes of these algebraic structures classify all the projective incidence structures.

Geometry :

 * Affine geometry
 * Affine space
 * Space (mathematics)
 * Affine geometry
 * Affine group
 * Affine transformation
 * Affine variety
 * Affine hull
 * Affine transformation
 * Heap (mathematics)
 * Equipollence (geometry)
 * Interval measurement, an affine observable in statistics
 * Exotic affine space
 * Complex affine space
 * Complex projective space
 * Real projective space
 * Projection (mathematics)
 * Projective geometry
 * Projective line
 * Projective plane
 * Projective space
 * Incidence
 * Cross-ratio
 * Möbius transformation
 * Projective transformation
 * Homogeneous coordinates
 * Duality (projective geometry)
 * Fundamental theorem of projective geometry
 * Projective configuration
 * Complete quadrangle
 * Desargues' theorem
 * Pasch's axiom
 * Pappus's hexagon theorem
 * Pascal's theorem
 * Projective line over a ring
 * Joseph Wedderburn
 * Grassmann–Cayley algebra
 * Planar ternary ring

Combinatorics

 * Cryptography
 * Error correction code
 * Kirkman's schoolgirl problem
 * Steiner system
 * Block design
 * Modular lattice
 * De Bruijn–Erdős theorem (incidence geometry)
 * Sylvester–Gallai theorem
 * Sylvester–Gallai configuration
 * Happy ending problem
 * Difference set

What we will describe

 * If N > 2, then any projective incidence structure of dimension N is isomorphic to GLN(R), where R is a division ring. It is isomorphic to GLN(F), where F is a field (finite or infinite) if and only if the Pappus theorem is true.
 * If N = 2, then a projective incidence structure is isomorphic to GL2(R), R a division ring, if and only Desargues' theorem holds.
 * If N = 2 and Desargues' theorem does not hold, then there is a rich family of Non-Desarguesian planes.

Vocabulary
The unique line k containg the distinct points A and B is called the line joining A and B. It will be noted as AB.

The unique point P contained in two distinct lines l and k will be called the intersection of l and k.

The axioms
A projective incidence structure is a set of objects called "points", denoted by $${Points}$$ and a set of distinguished sub-sets of $${Points}$$ called "lines", denoted by $${Lines}$$ which satisfy a certain number of simple axioms. We denote points by upper-case latin letters, lines by lower-case latin letters, and later, planes (to be defined later) by upper-case greek letters.

The pair $$\mathcal{P} = ( Points, Lines )$$ satisfying the following axioms is called a projective incidence structure.

 * A1. If $$A, B \in {Points}, A \neq B$$, then there is at least one $$\ell \in {Lines}$$ such that $$A \in \ell, B \in \ell$$.

 * A2. If $$A, B \in {Points}, A \neq B$$, then there is at most one $$\ell \in {Lines}$$ such that $$A \in \ell, B \in \ell$$.

 * A3. There are at least three points on any line.

 * A4. There is at least one point $$A$$ and at least one line $$l$$ such that $$A \notin l$$

 * A5. If $$A, B, C \in {Points}$$ are not all on the same line and $$D, E (D \neq E)$$ are such that $$D$$ is on the line $$AB$$ and $$E$$ on the line $$AC$$, then the lines $$BC$$ and $$DE$$ have a point in common.

 * Axiom 5 can be thought of as saying : If ABC is a triangle and a line intersects two sides of the triangle, then it intersects the third side.'

In the mathematical litterature there are many variations for the set of axioms to be used to define a projective incidence structure. However, they are all equivalent and given two sets of axioms it is a simple exercice to deduce one set from the other.

The following theorems, some of which are used as axioms in other expositions, are simple consequences of the above axioms.

 * Theorem 1 - Two distinct points are on one, and only one, line.
 * Theorem 2 - There are at least two distinct lines.
 * Theorem 3 - If $$C, D$$ are distinct points on the line $$AB$$, then $$A, B$$ are distinct points on the line $$CD$$.
 * Theorem 4 - Two distinct lines cannot have more than one common point.
 * Theorem 5 - There exists four points $$A, B, C, D$$ no three of which are collinear.

The following theorem is slightly less evident.

 * Theorem 6 - All $$\mathcal{l} \in Lines$$ have the same cardinality.

 * Corollary - If the set $$Points$$ is finite, then all lines in $$\mathcal{P}$$ have the same number of points.
 * Proof
 * If $$\mathcal{l_{1}}, \mathcal{l_{2}} \in Lines$$, we show that there exists a bijective map $$\phi : \mathcal{l_{1}} \mapsto \mathcal{l_{2}}$$ from the set of points on $$\mathcal{l_{1}}$$ to the set of ponts on $$\mathcal{l_{2}}$$. We can suppose that $$\mathcal{l_{1}} \neq \mathcal{l_{2}}$$.

 * Case 1 : The lines $$\mathcal{l_{1}}, \mathcal{l_{2}}$$ intersect at a point $$S$$.

 * Let $$P_{1} \in \mathcal{l_{1}}, P_{1} \neq S ; P_{2} \in \mathcal{l_{2}} , P_{1} \neq S$$. By axiom A3 there is a point $$Q$$ on the line $$ P_{1}P_{2} : Q \notin \mathcal{l_{1}} , Q \notin \mathcal{l_{2}}$$.

 * If $$X \in \mathcal{l_{1}}, X \neq S$$, then the line $$XQ$$ intersects $$\mathcal{l_{2}}$$ in a unique point $$\phi_{Q} (X)$$. That is, the map $$\phi_{Q}$$ defined by : $$\forall X \in \mathcal{l_{1}} , X \neq S : \phi_{Q} (X) \mapsto XQ \cap \mathcal{l_{2}}$$ and $$\phi_{Q} (S) = S$$ is an injection from $$\mathcal{l_{1}}$$ into $$\mathcal{l_{2}}$$. Similarly we can construct an injection $$\psi : \mathcal{l_{2}} \mapsto \mathcal{l_{1}}$$. By the Schröder-Bernstein theorem there exists a bijection between the sets $$\mathcal{l_{1}} , \mathcal{l_{2}}$$.

 * Case 2 : The lines $$\mathcal{l_{1}}, \mathcal{l_{2}}$$ do not intersect. Let $$ A \in \mathcal{l_{1}}, B \in \mathcal{l_{2}}$$ and $$\mathcal{h}$$ the line $$AB$$. By case 1 the bijections $$\phi_{1} (\mathcal{l_{1}} ) \mapsto \mathcal{h} , \phi_{2} (\mathcal{h} ) \mapsto \mathcal{l_{2}}$$ exist, and so the composite map $$\phi_{2} \circ \phi_{1}$$ is a bijection between $$\mathcal{l_{1}} , \mathcal{l_{2}}$$.

Dimension of an incidence structure
Definition : If $$P, Q, R$$ are three points not all on the same line and $$\mathcal{l}$$ is the line joining $$Q$$ and $$R$$, the class of all points on the lines joining $$P$$ to the points on the line $$\mathcal{l}$$l is called the plane determined by $$P$$ and $$\mathcal{l}$$. Small greek letters will be used to denote planes.

Theorem - Any two lines on the same plane $$\pi$$ have a common point.

Theorem - The plane $$\alpha$$ determined by a line $$\mathcal{l}$$ and a point $$P$$ is identical with the plane $$\beta$$ determined by a line $$\mathcal{m}$$ and a point $$Q$$, provided $$\mathcal{m}$$ and $$Q$$ are on $$\alpha$$.

Corollary - There is one and only one plane determined by three non-collinear points, or by a line and one point not on the line or by two intersecting lines.

Theorem - Two distinct points planes which have two distinct points $$A, B$$ in common contain all the points on the line $$AB$$ and have no other points in common.

Corollary - Two distinct planes cannot have more than one common line.

Points have dimension 0

Lines have dimension 1

Planes have dimension 2

If $$S(n-1)$$ is an incidence structure of dimension $$n-1$$ and $$P \in {Points}, P \notin S(n-1)$$. The set of all points on the lines joining $$P$$ to the points of $$S(n-1)$$ is an n-dimensional structure $$S(n)$$

The analogous theorems to the above are straightforward.

Definition : If all the points of $$Points$$ are in $$S(N)$$, the the incidence structure has dimension $$N$$.

Thus the dimension of an incidence structure is either a positive integer or infinity.

Infinite dimensional incidence structures exist.

The collineation group and some sub-groups
Depuis Félix Klein and his Erlanger programme, it is always a fruitful occupation in mathematics when studying a mathematical structure to examine in detail those properties which are conserved by symmetries, i.e. subgroups of the automorphism group.

In the case of projective incidence structures the exercise yields some beautiful mathematics and is still an active source of research.

An intuitive motivation for some of the formel definitions
Imagine two planes $$\Pi_{1}, \Pi_{2}$$ in Euclidean three-space, which intersect in a line $$\mathcal{l}$$ and two points $$P_{1} , P_{2}$$ not on either plane.

We project the point $$Q \in \Pi_{1}$$ onto the point $$R \in \Pi_{2}$$ by drawing the line $$P_{1}Q$$ and $$R$$ is the point of intersection with the plane $$\Pi_{2}$$. This procedure gives an isomorphisme between the points of $$\Pi_{1}$$ and $$\Pi_{2}$$ and lines in $$\Pi_{1}$$ are mapped into lines on $$\Pi_{2}$$. Call this isomorphism $$\phi_{P_{1}}$$.

We can define a similar isomorphism, $$\phi_{P_{2}}$$, from $$\Pi_{2} \mapsto \Pi_{1}$$.

The combined map $$\phi_{P_{1}} \circ \phi_{P_{2}}$$ is an automorphism of $$\Pi_{1}$$. This automorphism has several interesting features :
 * 1) Every point on the line $$\mathcal{l}$$ is mapped onto itself. The line is fixed by the automorphism.
 * 2) The point of intersection, $$C$$ of the line $$P_{1} P_{2}$$ with the plane $$\Pi_{1}$$ is also a fixed point.
 * 3) Any line in the plane $$\Pi_{1}$$ which passes through $$C$$ is mapped into itself (the points are not fixed).

Automorphisms with these properties, which arise from a simple and intuitive geometrical construction, will play a very important role in the study of projective incidence structures.

Definitions

 * A collineation is an automorphism $$\alpha$$ of $${Points}$$ which also maps $${Lines}$$ onto itself in the sense that if $$\mathcal{l} \in {Lines}$$, then the image of the set $$\{ P \in \mathcal{l} \}$$, that is $$m = \{ \alpha (P) \forall P \in \mathcal{l} \}$$ is also $$ \in {Lines}$$.

The set of collineations form a group under composition, $$\mathcal{Coll( \mathcal{P} )}$$.

Note : The identity is a collineation and there there exist incidence structures for which it is the ONLY collineation.

Lemma - If $$\alpha$$ is a collineation of $$\mathcal{P}$$, then for two distinct points $$X, Y \in \mathcal{P}$$ the image of the line $$XY, \alpha ( X Y )$$ is the line $$\alpha (X) \alpha (Y)$$.

 * If $$\alpha \in \mathcal{Coll( \mathcal{P} )}$$ and for $$P \in Points, \alpha (P) = P$$ then $$P$$ is a fixed point of $$\alpha$$.

A collineation maps $$Lines$$ onto $$Lines$$ and there are two possibilities for the notion of fixed line :
 * If $$\alpha \in \mathcal{Coll( \mathcal{P} )}$$ and $$\mathcal{l} \in {Lines} : \alpha (\mathcal{l} ) = \mathcal{l}$$ as sets, then we say $$\alpha$$ preserves the line $$\mathcal{l}$$.

 * If $$\alpha \in \mathcal{Coll( \mathcal{P} )}$$ and $$\mathcal{l} \in {Lines} : \forall P \in \mathcal{l}, \alpha (P) = P $$, then we say that $$\alpha$$ fixes the line $$\mathcal{l}$$.

If $$\Pi_{n}$$ is a hyperplane of dimension $$n$$, then we can define the collineations $$\alpha$$ which preserve or which fix $$\Pi_{n}$$ in the obvious way.

Simple consequences of the definitions
Theorem If $$\alpha \in Coll( \mathcal{P} )$$, then :(i) $$\alpha$$ has at least one fixed point. (ii) $$\alpha$$ preserves at least one line.

 * Proof - Since any $$\mathcal{P}$$ contains at least one plane, it suffices to prove the case for planes.

 * (i) Suppose that there is a line $$\mathcal{l}$$ that is not preserved by $$\alpha$$, then $$\alpha(\mathcal{l}) \ne \mathcal{l} \Rightarrow P = \mathcal{l} \cap \alpha (\mathcal{l} )$$, hence $$P$$ is a fixed point of $$\alpha$$.

 * Now suppose that all lines are preserved by $$\alpha$$. Then $$\exists \mathcal{l_{1}}, \mathcal{l_{2}}, \mathcal{l_{1}} \neq \mathcal{l_{2}} , \alpha (\mathcal{l_{1}} )= \mathcal{l_{1}} , \alpha (\mathcal{l_{2}} ) = \mathcal{l_{2}} $$ which implies that $$ P = \mathcal{l_{1}} \cap \mathcal{l_{2}} $$ is a fixed point.

 * (ii) Suppose that no line is preserved by $$\alpha$$. Let $$\mathcal{l_{1}}, \mathcal{l_{2}}$$ be distinct lines, hence $$\alpha (\mathcal{l_{1}} ) \ne \mathcal{l_{1}} , \alpha (\mathcal{l_{2}} ) \ne \mathcal{l_{2}} ; P = \mathcal{l_{1}} \cap \alpha (\mathcal{l_{1}} ) ; Q = \mathcal{l_{2}} \cap \alpha (\mathcal{l_{2}} )$$. The points $$P, Q$$ are fixed by $$\alpha$$.

 * If $$P \ne Q$$, the the line $$PQ$$ is preserved by $$\alpha$$. This is contrary to the hypotheses, hence $$P = Q$$ and so all the lines in the plane must pass through $$P$$.

 * Let $$A \in \mathcal{l_{1}}, B \in \alpha (\mathcal{l_{1}})$$ and different from $$P$$.
 * The line $$AB$$ must pass through $$P$$. The lines $$AB, \mathcal{l_{1}}$$ have $$A$$ in common. If they are distinct they cannot have $$P$$ in common, so they are not distinct, and this implies that $$\alpha (\mathcal{l_{1}}) = \mathcal{l_{1}}$$, a contradiction, since we are assuming that no lines are preserved.
 * Hence $$\alpha$$ preserves at least one line.

Theorem The set of $$\alpha \in {Coll( \mathcal{P} )}$$ which preserve a line $$\mathcal{l}$$ form a sub-group $${Coll}_{pres}(\mathcal{l}) \subseteq {Coll( \mathcal{P} )}$$.

Theorem The set of $$\alpha \in {Coll}_{pres}(\mathcal{l})$$ which fix a line $$\mathcal{l}$$ form a sub-group $${Coll}_{fix}(\mathcal{l}) \subseteq {Coll}_{pres}(\mathcal{l})$$.

The groups $${Coll}_{fix}(\mathcal{l})$$ are important ; they will give rise to affine geometry, the carcterisation of n-dimensional projective structures. We state and prove some simple theorems which will enable us to better understant the collineations which fix a given line.

Clarify the details of :
 * Given any two lines of P there is a collineation that maps L1 to L2
 * Given any line in P, then there is a collineation, different from the identity, which preserves the line.
 * This shows that the collineation group is not trivial and is 'large'.

Theorem - If $$\mathcal{l}, \mathcal{m} \in {Lines}, \mathcal{l} \neq \mathcal{m}$$, then $${Col}_{fix}(\mathcal{l}) \cap {Coll}_{fix}(\mathcal{m} ) = \iota$$, the identity collineation. i.e. A collineation different from the identity cannot fix more than one line.

 * Proof

 * Case 1 : The two lines have a point in common.

 * Proof.
 * Let $$Q$$ be the intersection of the lines $$\mathcal{l_{1}}, \mathcal{l_{2}}$$ fixed by $$\alpha$$.
 * Let $$P$$ be a point of the plane not on these lines.
 * Let $$R, S$$ be two points on $$\mathcal{l_{1}}$$ distinct from $$Q$$.
 * Suppose that the line $$PR$$ intersects $$\mathcal{l_{2}}$$ in $$T$$ and that the line $$PS$$ intersects $$\mathcal{l_{2}}$$ in $$U$$.

 * The points $$R, S, T, U$$, are fixed by $$\alpha$$, so the lines $$RT$$ and $$SU$$ are preserved by $$\alpha$$. Hence their intersection $$P$$ is fixed by $$\alpha$$. But $$P$$ is any point in the plane not on $$\mathcal{l_{1}}$$ or on $$\mathcal{l_{2}}$$. Thus all the points of the plane are fixed by $$\alpha$$. Hence, $$\alpha$$ is the identity collineation.

 * Case 2 : The two lines have no point in common.

 * Proof.

Theorem - If $$\alpha \in Coll_{fix}(\mathcal{l})$$, then there is at most one point $$ C \notin \mathcal{l} : \alpha (C)=C$$.

 * Proof
 * Lemma 2 : A collineation of a plane $$\Pi$$ which fixes a line $$\mathcal{l}$$ and two points $$P_{1}, P_{2}$$ not on $$\mathcal{l}$$ is the identity collineation

 * Proof Let $$P$$ be a point in the plane not on the line $$\mathcal{l}$$ and not on the line $$P_{1}P_{2}$$.

 * Let $$PP_{1}$$ intersect $$\mathcal{l}$$ at $$Q_{1}$$ and $$PP_{2}$$ at $$Q_{2}$$.

 * The lines $$P_{1}Q_{1}$$ and $$P_{2}Q_{2}$$ are distinct and $$P$$ is their unique point of intersection.

 * The points $$P_{1}, Q_{1}$$ are fixed by the collineation, hence the line $$P_{1}Q_{1}$$ is preserved by the collineation, likewise the line $$P_{2}Q_{2}$$ is preserved by the collineation.

 * This implies that the intersection of these two lines is fixed by the collineation. But $$P$$ is any point not on $$\mathcal{l}$$ and not on the line $$P_{1}P_{2}$$. This implies that the line $$P_{1}Q_{1}$$ is fixed by the collineation, and by lemma 1, the collineation must be the identity.

The next theorem proves that if a collineation has a fixed line, then it must have a unique fixed point with the property that all lines through the fixed point are preserved by the collineation.

Theorem - If $$\alpha \in Coll_{Fix}(\mathcal{l})$$, then $$\exist C_{\mathcal{l}} \in Points : \alpha (C_{\mathcal{l}})= C_{\mathcal{l}}$$ and any line $$\mathcal{m}$$ through $$C_{\mathcal{l}}$$ is preserved by $$\alpha$$. The point $$C_{\mathcal{l}}$$ is unique.

 * N.B. $$C_{\mathcal{l}}$$ may or may not be on the line $$\mathcal{l}$$.

 * Proof

 * By lemma 2 there is at most one point not on the fixed line which has the stated properties.

We now show that there cannot exist a point on the line L with these properties. Suppose that such a point, C1 exists. If Q is a point not on L and different from C, then the line C2.Q is preserved by alpha. The line CQ is preserved by alpha. The intersection of these two lines is then a fixed point of alpha, nameley Q, and Q is different from C. Thus alpha has twou fixed points not on L, hence alpha is the identity by lemma 2.

 * Similarly there cannot be two distinct fixed points C1, C2 on L which preseve all lines throug C1 and C2.

 * Thus, the point C, if it exists is unique. We now show that such a point does, in fact, exist.

 * Suppose that alpha has a fixed point C, C not on L. Any line m through C intersects L at a point Q. The points P and Q are fixed by alpha, hence the line m is preserved by alpha. Thus, any line through C is preserved by alpha.

 * Now suppose that alpha does not fix any point not on the line L.

 * Let P be any point, then $$\alpha (P) \neq P$$ and $$\alpha (P) \notin \mathcal{l}$$.

 * The line $$\mathcal{m} = P.P(\alpha )$$ intersects $$\mathcal{l} $$ in a point $$C \neq P$$.

 * The line m = P.C. and so alpha (m) = alpha (P).alpha (C) = PC. Thus m is preserved by alpha.

 * Let Q be a point not on L or m, then Q is on a preserved line n. The intersection of m and n must be a fixed point and by hypothesis there are no fixed points not on L. Hence the point of intersection must be the point C. Thus every line preserved by alpha must pass through C and every line through C is preserved.

The 'dual' theorem is also true.

Theorem - If $$\alpha \in Coll$$ and there exists a $$ C \in Points : \alpha (C) = C$$ and all lines through $$C$$ are preserved by $$\alpha$$, then there exists a line $$\mathcal{l_{C}}$$ that is fixed by $$\alpha$$. The line $$\mathcal{l_{C}}$$ is unique.

Lemma  Let $$\alpha$$ be a collineation of $$\mathcal{P}$$ and $$\Pi$$ a hyperplane such that each point is fixed by $$\alpha$$. Then :
 * (i) There exists a point $$C$$ such that each line through $$C$$ is preserved by $$\alpha$$.
 * (ii) If $$C \notin \mathcal{H}$$, then $$C$$ is unique.

 * Proof - If $$C \notin \mathcal{H}, \alpha (C) = C$$, the $$C$$ is a centre : for each line through $$C$$ is preserved (if $$\mathcal{l}$$ is on C, then $$\mathcal{l} \cap \mathcal{H} = P , \alpha (P) = P$$. The line $$CP = \mathcal{l}$$ is preserved.

 * Suppose now that no point outside $$\mathcal{H}$$ is fixed.

 * Let $$P \notin \mathcal{H} \Rightarrow \alpha (P) \neq P$$. Let $$C = P \alpha (P) \cap \mathcal{H} $$, then the lines : $$P \alpha (P) = C \alpha (P) = CP$$ and $$\alpha (P \alpha (P)) = \alpha (PC) = \alpha (P) \alpha (C) = \alpha (P) C = \alpha (P) P$$ i.e. the line $$ P \alpha (P)$$ is preserved.

 * We now show that any line through $$C$$ is preserved.

 * Let $$\mathcal{g} \in Lines, C \in \mathcal{g} , \mathcal{g} \notin \mathcal{H} ,Q \notin \mathcal{H} , Q \notin P \alpha (P)$$. The line $$Q\alpha (Q)$$ passes through $$C = P\alpha (P) \cap \mathcal{H}$$. For let $$S = PQ \cap \mathcal{H}$$, then $$S = \alpha (S) \in \alpha (PQ) = \alpha (P) \alpha (Q)$$

 * Hence the points $$S, P, Q, \alpha (P), \alpha (Q)$$ are contained in a common plane $$\Pi$$.

 * Therefore the lines $$Q\alpha (Q), P\alpha (P)$$ intersect at $$ X \in \Pi$$.

 * Since the lines $$Q\alpha (Q), P\alpha (P)$$ are preserved by $$\alpha , X$$ satisfies $$\alpha (X) = \alpha (P\alpha (P)) \cap \alpha (Q\alpha (Q)) = P\alpha (P) \cap Q\alpha (Q) = X$$.

 * Thedrefore $$X \in \mathcal{H}$$, thus it must coincide with $$P\alpha (P) \cap \mathcal{H} = C$$.

 * Hence all the lines of the form $$Q\alpha (Q)$$ pass through the point $$C$$. Thus each line trough $$C$$ is preserved.

Lemma  Let $$\alpha$$ be a collineation of $$\mathcal{P}$$ and $$C$$ a point such that each line through $$C$$ is preserved by $$\alpha$$. Then :
 * (i) There exists a hyperplane $$\mathcal{H}$$ which is fixed by $$\alpha$$
 * (ii) The hyperplane is unique.

 * Proof

 * Suppose that the line $$\mathcal{l}$$ does not pass by $$C$$ but is preserved by $$\alpha$$. If $$P \in \mathcal{l}$$, the the line $$CP$$ is preserved by definition. Hence the point of intersection of $$\mathcal{l}$$ and $$CP$$ is a fixed point. But $$P$$ is any point of $$\mathcal{l}$$, hence $$\mathcal{l}$$ is a fixed line.

 * Suppose now the \mathcal{l_{1}}, \mathcal{l_{2}} are two lines which are NOT preserved by \alpha , we will construct a line tha IS preserved by \alpha.

 * Let $$P = \mathcal{l_{1}} \cap \alpha ( \mathcal{l_{1}}, Q = \mathcal{l_{2}} \cap \alpha ( \mathcal{l_{2}})$$. The points $$P, Q$$ are fixed by $$\alpha $$, hence $$PQ$$ is preserved by $$\alpha$$.

 * Since there at least three lines in $$\mathcal{P}$$ there is at least one preserved line that does not pass by $$C$$ and so is fixed.

 * We now have to build up a fixed hyperplane.

The above theorems were proved on the assumption that a colleation with a fixed line exists. We investigate the conditions of their existance in the next section.

Central collineations
Definition : An $$\alpha \in Coll$$ with the above properties is called a Central collineation ; $$C$$ is the centre and $$\mathcal{l}$$ the axis of the collineation.

Definition : A collineation $$\alpha$$ of $$\mathcal{P}$$ is a central collineation if there exists a hyperplane $$\mathcal{H}$$ (the axis if $$\alpha$$ and a oint $$C$$, the centre of $$\alpha$$ such that :
 * - For every point $$ X \in \mathcal{H} \Rightarrow \alpha (X) = X$$ i.e. is a fixed point.
 * - For every line$$\mathcal{l}$$ through $$C$$ $$ \alpha ( \mathcal{l} ) = \mathcal{l}$$  i.e. the line is preserved.

Lemma - Let $$\mathcal{H}$$ be a hyperplane and $$C$$ a point of $$\mathcal{P}$$. The set of central collineations with axis $$\mathcal{H}$$ and centre $$C$$ form a group, $$\mathcal{Coll ( H, C )}$$ with respect to composition.

N.B. The group is not empty since it contains the identity collineation.

Lemma - Let $$\alpha \in \mathcal{Coll (H,C)}$$ and suppose that $$ P \neq C, P \notin H , \alpha (P) = P'$$. Then $$\alpha$$ is uniquely determined. In particular the image of any $$ X, X \notin \mathcal{H} , X \notin PP' ( = PC)$$ satisfies : $$\alpha (X) = CX \cap FP'$$, where $$ F = PX \cap X$$

Proof The image $$X'$$ of $$X$$ is subject to the following restrictions :
 * -The line $$CX$$ is mapped onto itself, so $$\alpha (X)$$ is on the line $$CX$$.
 * - Consider the point $$F = PX \cap \mathcal{H} $$ is on the axis of \alpha, so is fixed. X is on the line FP.

Since $$X$$ is not on $$PP', F$$ is not on $$CX$$ , thus $$X'$$ is on the intersection of two distinct lines and so uniquely determined. It follows that $$X'$$ is on the image of the line $$FP$$. The image of $$\alpha ( FP ) = F \alpha (P)$$.

Thus $$X$$ is the intersection of the lines $$CX, FP'$$ , hence uniquely determined.
 * We now show that any $$Y$$ on the line $$CP$$ is also uniquely determined. Replace the pair of points $$(P, P')$$ by any other pair of points $$(R, R')$$ with $$R \neq R', R \in CP$$, the repeat the above construction to determine $$\alpha (Y)$$ (need to quote axioms to say there exists a point $$R$$ not fixed and not on $$CP$$, then use the construction to determine $$\alpha (R) = R' )$$.

Corollary (Uniqueness of central collineations) - If $$\alpha \in \mathcal{Coll (H, C )} , \alpha \ne \iota$$ :
 * (a) If $$ P \ne C, P \notin \mathcal{H}$$, the P is not fixed by $$\alpha$$;
 * (b) $$\alpha$$ is uniquely determined by one pair of points $$P, \alpha (P), P \ne C$$.

Proof (a) Supose that Q is fixed by \alpha, then we will show that any P \in \mathcal{P} is fixed i.e. \alpha is the identity collineation.

Let X \notin CP, then \alpha (X) = CX \cap FP' = CX \cap FP = X, since X is on FP, so every point not on CP is fixed. We pick a pont X_{0} not on CP and repeat the argument to show that all points of CP are fixed. hence \alpha = \iota.

(b) follows directly from the lemma.

Note : We will be considering the group of central collineations with centre on $$\mathcal{H}$$ (the elations or translations). To show that two elations with the same centre are the same, it suffices to show that for just one point $$X \notin \mathcal{H}$$, that $$\alpha (X) = \beta (X)$$.

QUESTION : Is it always true that : A collination is the product of a fine number of central collineations.

Theorem - There is at most one central collineation $$\alpha$$ with given centre $$C_{0}$$, axis $$\mathcal{l}$$ and pair of points $$P_{0}, \alpha(P_{0})$$.
 * Proof
 * Lemma - A central collineation $$\alpha$$ of a plane is completely determined by its axis $$\mathcal{l}$$, centre $$C_{0}$$ and a pair of points $$P_{0}, \alpha (P_{0})$$.

 * Proof
 * Let $$Q$$ be any point in the plane. We will show how to determine $$\alpha (Q)$$.
 * The line $$P_{0} Q$$ intersects $$\mathcal{l}$$ at $$R$$, a fixed point. The line $$\alpha (P_{0}) R$$ intersects the line $$C_{0}Q$$ at $$X$$.
 * We claim that $$\alpha (Q) = X$$, because $$\alpha (Q) $$ must be on the line $$C_{0}Q$$, since $$\alpha$$ preserves all lines through $$C_{0}$$ ; $$\alpha (Q)$$ must also lie on the line $$R \alpha (P_{0})$$ and $$X$$ is the unique intersection of these lines.

Question : What if L, C, P, Q are not in the plane C, L ?

Theorem If N > 2 there exists a central collineation with any given centre, axis and (P, image(P) ).

Theorem Desargues theorem is true if and only if all the possible central collineations exist.

Cor For N > 2 Desargues theorem is true. For N = 2 we have to introduce as an axion that all possible central collineations exist.

N.B. Check out the little an big Desargues theorems to get the logical structure clear.

If $$\mathcal{P_{2}}$$ is non-Desarguenian, the there exist at least one line which is not fixed by any collineation of $$\mathcal{P_{2}}$$.

Definition :
 * Central collineations with centre on the axis are called elations.
 * A central collineation with centre NOT on the axis is called a homology/

N.B. In many papers Central collineations are called perspectivities.

Theorem - The homologies with centre C and axis L form an abelian group.

If $$\sigma_{1}, \sigma_{3}$$ are homologies with cetres $$C_{1}, C_{2}$$ and axes $$\mathcal{l_{1}}, \mathcal{l_{2}}$$, then $$\sigma_{1} * \sigma_{2}$$ is a collineation with centre $$\mathcal{l_{1}} \cap \mathcal{l_{2}}$$ and preserves the line $$C_{1} C_{2}$$. The line is not necessarily fixed.

Theorem - The elations with axis $$\mathcal{l}$$ and centre $$C_{i}$$ form a group $$El( \mathcal{l}, C_{i})$$

Theorem - The elations with axis $$\mathcal{l}$$ form a group $$El(\mathcal{l})$$, the $$El(\mathcal{l}, C_{i})$$ are subgroups.

Theorem - If $$El (\mathcal{l})$$ has at least two non-trivial subgroups $$El(\mathcal{l}, C_{1})$$ and $$El(\mathcal{l}, C_{2})$$, then $$El(\mathcal{l})$$ is an abelian group. The orders of the elements of $$El(\mathcal{l})$$ is either infinity or all are of finite order $$p$$, a fixed prime number.

There exists examples where the group $$El(\mathcal{l})$$ does not have two such sub-groups.

Question : What are the examples ?

Definition : The group generated by all the elations of $$\mathcal{P}$$ is often called the little projective group.

Question : Is this for different line or for one fixed line ?

Desarguesian and non-desarguesian projective incidence structures
The first major classification of projective incidence structures is a binary distinction :
 * - A desarguesian structure.
 * - A non-desarguesian structure.

A desarguesian projective structure satisfies an additional axiom, which we give in two forms (Our first theorem will be to show that the two forms of the axiom are equivalent.)

- Axiom 5a : - Axiom 5b :

Theorem - The two forms of axiom 5 are equivalent.

There exist projective incidence structures which satisfy axiom 5 and structures which do not satisfy axiom 5, so the concept of desarguesian structres is a useful one.

Example 1 : (A desarguesian plane)

Example 2 : (Moulton plane)

Example 3 : (Finite not-desarguesian plane)

Theorem A projective incidence structure of dimension N > 2 is desarguesian.

We will see in the section : Algebraic structures associated with projective incidence structures that a desarguesian projection structure of dimension n is isomorphic GLn(R), where R is a division ring.

For two dimensionnel projective incidence structures a great deal of research has centered on finding criteria (usually in termes of the collineation group) which ensure that the plane is desarguesian.

THE BARSOTTI CLASSIFICATION

Artin's axioms 4a, 4b 4b P Get the inter-relations clear !

Axiom 4a Given any two points p, q there is a translation that sends p to q.

Axiom 4b If tau1, tau2 are dfferent non-identity translations with the same traces, then there is a homomorphism that sends tau1 to tau2

Axiom 4b P - For a given P, then given Q, R such that P, Q, R are distict and PQR are collinear, then there is a dilatation which has P as fixed point and sends Q to R

Are thes axioms equivalent ?

Is axiom 4b P equivalent to Desargues theorem ,

Give the Moulton plane example of a non-Desarguesian plane (Refractive index)

QUESTIONS TO CLARIFY
 * 1) If N >2 are all collineations central, all collineations of a Desarguesian plane ?
 * 2) Does there exist at least one line that is fixed by by some alpha ?
 * 3) If alpha has a fixed point does it have a preserved line ?
 * 4) If alpha has a centre does it have a fixed line ?

Definitions

 * 1) A collineation of order 2 is an involution

 * 1) A affine plane is a projective plane with a distinguished lin $$\mathcal{l}_{\infty}$$

 * 1) An elation of an affine plane with axis $$\mathcal{l}_{\infty}$$ is  translation.

 * 1) The group generated by all the translations of a plane is the Translation group.
 * 2) It the translation group of an affine plane is transitive, then the plane is a 'Translation plane''.

 * 1) A projective plane is $$(C- \mathcal{l})$$ transitive with fixed point $$C$$ and fixed line $$\mathcal{l}$$ if for any pair of points $$Q_{1}, Q_{2}$$ with $$Q_{1}, Q_{2} \ni \mathcal{l} ; Q_{1}, Q_{2} \neq C ; CQ_{1} ne CQ_{2}$$, there exists a $$(C - \mathcal{l})$$ perspectivity $$\alpha : \alpha (Q_{1}) = Q_{2}$$.

Ternary algebraic structures
If $$\mathcal{G}$$ is a sub-group of $$\mathcal{Coll}$$ and $${Hom(} \mathcal{G}, \mathcal{G} {)}$$ the semi-group of homomorphisms of $$\mathcal{G} \rightarrow \mathcal{G}$$ we define two binary operations "$$\boxtimes$$" and $$\boxplus$$" as follows :
 * If $$\alpha, \beta \in {Hom(} \mathcal{G}, \mathcal{G} {)}$$ then $$\alpha \boxtimes \beta = \gamma \in {Hom(} \mathcal{G}, \mathcal{G} )$$ defined by $$\gamma (g) = \alpha ( \beta ( g )) \forall g \in \mathcal{G}$$
 * If $$\alpha, \beta \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$ then $$\alpha \boxplus \beta = \delta \in {Hom(} \mathcal{G}, \mathcal{G} )$$ defined by $$\gamma (g) = \alpha (g)* \beta ( g ) \forall g \in \mathcal{G}$$, where "$$*$$" is the group operation in $$\mathcal{G}$$

The binary operations $$\boxtimes, \boxplus$$ are a priori not assumed to be communtative, associative or distributative.

However, there always exists

 * A two sided 'multiplicative identity' denoted by $$\iota \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$such that $$\alpha \boxtimes \iota = \iota \boxtimes \alpha = \alpha$$ and defined $$ \forall g \in \mathcal{G}$$ by $$\iota (g) = {g}$$.
 * If $$\alpha \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$ then $$\exists \beta, \gamma \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$ such that $$ \alpha \boxtimes \beta = \iota$$ and $$\gamma \boxtimes \alpha = \iota$$ i.e. every element has a right inverse and a left inverse.
 * Thus, "$$\boxtimes$$" defines a loop.

 * An 'additive zero', denoted by $$o \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$ and defined $$ \forall g \in \mathcal{G}$$ by $$o (g) = {I}$$, the identity element of $$\mathcal{G}$$, such that $$\alpha \boxplus o = o \boxplus \alpha = \alpha$$ $$\forall \alpha \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$
 * Il est à noter que $$\iota \neq o$$.
 * An 'additive inverse' : the homomorphism $$inv \in {Aut(} \mathcal{G}, \mathcal{G} {)}$$ and defined by $$inv(g) \rightarrow g^{-1} \forall g \in \mathcal{G}$$ is such that $$\alpha \boxplus inv(\alpha) = o$$.
 * It is suggestive to denote $$\boxplus inv$$ by $$\boxminus$$ and write : $$\alpha \boxminus \alpha = o$$.

 * $$\boxplus$$ is associative : $$\alpha \boxplus (\beta \boxplus \gamma ) = (\alpha \boxplus \beta ) \boxplus \gamma$$
 * However, $$\boxplus$$ is commutative if and only if the group $$\mathcal{G}$$ is abelian.

By restricting attention to special groups $$\mathcal{G}$$ this ternary algebraic structure acquires more structure and can become a division ring or even a field. It will be used later to introduce coordinates into projective incidence structures and to characterise them.

Mathematiciens who contributed

 * Emil Artin
 * Reinhold Baer
 * Leonard Eugene Dickson
 * Marshall Hall
 * David Hilbert
 * Félix Klein
 * Donald Knuth
 * Oswald Veblen
 * Joseph Wedderburn
 * Alfred North Whitehead
 * John Wesley Young

Notes for WJE

 * Let $$\mathcal{P}$$ be a projective plane and $$\sigma$$ an involutory collineation, then either $$\sigma$$ is a perspectivity or $$\sigma$$ leaves fixed pointwise a proper subplane $$\mathcal{D} \subset \mathcal{P}$$.
 * Let $$\mathcal{P}$$ be a projective plane and $$\sigma$$ an involutory collineation, then either $$\sigma$$ is a perspectivity or $$\sigma$$ leaves fixed pointwise a proper subplane $$\mathcal{D} \subset \mathcal{P}$$.

 * Fundamental result : In a desaguesian plane there is a collineation that maps any quadrangle to any other quadrangle. The same is true for alternative planes. He examines the converse problem : what configurations existe in projective structures whose collineation group is transitive on quadrangles ? He proves that if the plane is finite, then it is desarguesian and conjectures that if it is infnite the plane is alternative.
 * Fundamental result : In a desaguesian plane there is a collineation that maps any quadrangle to any other quadrangle. The same is true for alternative planes. He examines the converse problem : what configurations existe in projective structures whose collineation group is transitive on quadrangles ? He proves that if the plane is finite, then it is desarguesian and conjectures that if it is infnite the plane is alternative.

Finite planes

 * Let $$\mathcal{P}$$ be a projective plane, $$\sigma_{1}, \sigma_{2}$$ two involutory homologies with centres $$C_{1}, C_{2}$$ and axes $$\mathcal{l}_{1} , \mathcal{l}_{2}$$. If $$C_{1} \in \mathcal{l}_{2} , C_{2} \in \mathcal{l}_{1}$$ then $$\sigma_{1} \sigma_{2}$$ is an involutory homology with centre $$\mathcal{l}_{1} \cap \mathcal{l}_{2}$$ and axis $$C_{1} C_{2}$$.
 * Let $$\mathcal{P}$$ be a projective plane, $$\sigma_{1}, \sigma_{2}$$ two involutory homologies with centres $$C_{1}, C_{2}$$ and axes $$\mathcal{l}_{1} , \mathcal{l}_{2}$$. If $$C_{1} \in \mathcal{l}_{2} , C_{2} \in \mathcal{l}_{1}$$ then $$\sigma_{1} \sigma_{2}$$ is an involutory homology with centre $$\mathcal{l}_{1} \cap \mathcal{l}_{2}$$ and axis $$C_{1} C_{2}$$.

 * THEOREM A. Let P be a finite projective plane admitting a collineation group doubly transitive on the points of P. Then, P is desarguesian.
 * THEOREM B. Let P be a finite affine plane admitting a collineation group doubly transitive on the a]fine, then P is a translation plane.
 * THEOREM B. Let P be a finite affine plane admitting a collineation group doubly transitive on the a]fine, then P is a translation plane.

Algebraic structure

Infinite dimension structures

When does a planar ternary ring uniquely coordinitise a projective plane?

https://mathoverflow.net/questions/106888/when-does-a-planar-ternary-ring-uniquely-coordinitise-a-projective-plane/160978

projective plane over algebraic structure

https://math.stackexchange.com/questions/734288/projective-plane-over-algebraic-structure

 * A. Wagner, ''Perspectivities and the little projective group, Algebraic and Topological Foundations of Geometryt, Proc. of a Coll., Utrecht, august 1959, 1962, pages 199-208.
 * The little projective group = Elation group, often a simple group.

 * T.G. Room, P.B. Kirkpatrick, "Miniquaternion geometry", Cambridge Univ. Press (1971)

 * W.M. Kantor, "Primitive permutation groups of odd degree, and an application to finite projective planes" J. Algebra, 106 (1987) pp. 15–45
 * G. Pickert, "Projective Ebenen", Springer (1975)
 * D.R. Hughes, F.C. Piper, "Projective planes", Springer (1973)
 * H. Lüneburg, "Translation planes", Springer (1979)
 * K.G.C. von Staudt, "Beiträge zur Geometrie der Lage", I , Korn , Nürnberg (1865)
 * G. Fano, "Sui postulati fondamentali della geometria proiettiva" Giornale di Mat., 30 (1892) pp. 106–132
 * I. Singer, "A theorem in finite projective geometry and some applications to number theory" Trans. Amer. Math. Soc., 43 (1938) pp. 377–385

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text:
List of Dhruva supporting characters

Super Commando Dhruva titles have featured a long list of secondary supporting characters over the years that includes friends, foes, family, love interests and at times other Raj comics superheroes in crossovers. The first major recurring supporting character to be introduced in Dhruva series was his foster father I. G. Rajan Mehra in Dhruva's debut title itself. A couple of issues later Chandika, arguably the most important and most popular side character not only in Dhruva series but in entire Raj comics universe, was introduced. At times, Dhruva has appeared alongside other superheroes of Raj comics universe in multi-hero crossovers. Dhruva also has a well known rogues gallery full of an assortment of supervillains including Grand Master Robo, Chandakaal and Mahamanav.

Family
This section enlists characters who are either related to Dhruva (blood relatives/foster relatives) or are very close to him.

Immediate family
Following is the list of Dhruva's blood relatives.
 * Shyam: Shyama was Dhruva's father. He worked as a trapeze artist in Jupiter circus. He was killed in front of Dhruva's eyes at the hands of rival Globe circus's strongman Jubisko. First appearance - GENL #74 'Pratishodh Ki Jwala' (1988)
 * Radha: Radha was Shyam's wife and Dhruva's mother. She too worked as a trapeze artist in the Jupiter circus where she met Shyam for the first time. She was killed on the same fateful day of her husband's death in the Jupiter circus carnage. First appearance - GENL #74 'Pratishodh Ki Jwala' (1988)

Foster family
After the Jupiter circus carnage, when a 14yrs old Dhruva had nowhere to go, Rajangar's SP Rajan Mehra adopted him as his son. Dhruva grew really fond and protective of his foster family to the extent that when he found out that he had an extended family living in France and his grand father requested him to stay back in France and take care of the family's vast estate business, he rejected the idea citing that he can't leave his family in Rajnagar alone. His foster family consists of-
 * IG Rajan Mehra: SP Rajan Mehra is a top ranked police office in Rajnagar police department. Initially an SP at the time of adopting Dhruva, he was later promoted to the post of IG. IG Rajan has been a constant ray of inspiration in Dhruva's vigil against criminals. First appearance - GENL #74 'Pratishodh Ki Jwala' (1988)
 * Mrs. Rajni Mehra: She is IG Rajan's wife and a simple housewife. First appearance - GENL #74 'Pratishodh Ki Jwala' (1988)
 * Shweta Mehra: Shweta is Dhruva's little foster sister, the only biological child of the Mehra's. She is a budding scientist and also functions as a masked heroine, Chandika, a secret even Dhruva doesn't know. First appearance - GENL #74 'Pratishodh Ki Jwala' (1988)

Love interests

 * Natasha: Natasha is Dhruva's primary love interest and the daughter of his enemy Grand Master Robo. Always torn up between her love for her superhero boyfriend and her criminal father, she keeps oscillating between the two worlds, the crime world of Robo and Dhruva's lawful world. In Nagayana, a futuristic parallel series set in alternate universe, Natasha has been shown to be Dhruva's lawfully wedded wife. First appearance - SPCL #2 'Grand Master Robo' (1991)
 * Richa: Known as Black cat. She has a team of trained cats.

Dhruva's lineage

 * Jalaj Son of Natasha
 * Rishi Son of Richa
 * Dhruvishya
 * Vitler: Vitler is Dhruva's descendant from a very distant future. Dhruva met Vitler when he time travelled to 25,496 AD. Unlike Dhruva and his other descendants, who were the protectors of humanity, Vitler was a tyrant dictator of his times who ruled half the world. After Dhruva's visit, Vitler has a change of heart resulting in bringing peace and harmony to the world. First appearance - GENL #230 'Udantashtari Ke Bandhak' (1991)

Friends and allies
Dhruva has a huge assortment of allies ranging from normal humans and masked vigilantes to ferals and superpowered beings. Apart from his own range of supporting characters, Dhruva has also interacted with other superheroes of Raj comics universe and often fought as a team with them against bigger threats. Following is the list of various allies of Dhruva:

Major allies

 * Chandika: Alter-Ego of Shweta. Chandika is the most important ally and most recurring character of Dhruva series. Chandika, a blonde superheroine who wears purple colored tights, blue boots and a blue domino mask, is the alter ego of Shweta, Dhruva's little sister. The only person who knows this secret is Natasha. Natasha herself has donned the identity of Chandika a couple of times to rescue Shweta from being identified. Chandika, like her brother, depends more on brains than brawns and to a large extent on her self-invented gadgets. In Nagayana, when Shweta was presumably dead, although later revived, Natasha kept Chandika alive by donning her costume. First appearance - GENL #96 'Swarg Ki Tabahi' (1989)
 * Dhananjay: Dhananjay is a warrior of Swarna-Nagri, a golden city hidden beneath the ocean, located at the ocean bed near Indian coast. It is the home to Swarna-Manavas, descendants of Gods. He carries a golden lasso and has the ability to make interdimensional transit window. He also provided Dhruva with the ability to breathe in water. First appearance - SPCL #2 'Grand Master Robo' (1991)
 * Jingaloo: Jingaloo is the prince of yetis residing in the Himalayas, and a very close friend of Dhruva. He is claimed to be descendant of Lord Hanuman, Hindu deity of Banar (monkey) race. First appearance - GENL #140 'Barf Ki Chita' (1990)
 * Vanputra: Vanputra is a forest dweller, a tribal who lives in National Forest, a forest on the outskirts of Rajnagar. He is a plant manipulator. He has the ability to communicate with plants. On his command, plants can grow in a matter of seconds, can move and can even kill someone. First appearance - GENL #255 'Vinash Ke Vriksha' (1991)

Jupiter Circus
Jupiter circus was where Dhruva was born and brought up in his childhood. Apart from Dhruva himself and his parents Shyam and Radha, following were the significant members of Jupiter circus:
 * Jacob: Jacob was the owner of Jupiter circus and loved Dhruva very much. He declared Dhruva a beneficiary in Jupiter circus insurance. The money he inherited from the insurance helped Dhruva to set up his own Commando Force after the Jupiter Circus carnage.
 * Pawan: A motorcycle stunt expert.
 * Hercules: Strongman of Jupiter circus.
 * Sulaiman: A sharp shooter.
 * Ranjan: Knife throwing expert.
 * Sherkhan: Ring-Master of Jupiter circus.
 * Anna: Used to run dolphin shows in Jupiter circus.

All the Jupiter Circus members made their first appearance in Dhruva's debut title GENL #74 'Pratishodh Ki Jwala' (1988) with the exception of Anna who made his first appearance in SPCL #2 'Grand Master Robo' (1991) in a flashback sequence. All the characters of Jupiter circus are believed to have died in the Jupiter circus carnage though, much later, Jacob was revealed to be the sole survivor when the character returned in SPCL #100 'Khooni Khandan' (1997) to help Dhruva find out the truth about his father's past.

Commando Force
Commando Force is a specially trained unit formed by Dhruva to aid him in his crime fighting sojourn. He used 50 lakh rupees that he received from Jupiter Circus insurance claim to establish it, while his foster father IG Rajan Mehra ensured he got all the necessary government approvals to establish the same. Initially, just 3 cadets strong with Rajnagar being their operative ground, commando force has expanded its presence throughout the country. Dhruva himself is the captain of the Commando Force. Apart from Dhruva, there are 3 prominent and named members in this team: All three of them made their first appearance in GENL #79 'Roman Hatyara' (1988). While Peter and Renu do most of the field work, Karim is primarily a communications expert. Apart from these 3 named cadets, commando force has added many unnamed cadets to its ranks since its inception.
 * Peter Macey
 * Renu
 * Karim Shah

Villains
Dhruva's rogues gallery is one of the most identifiable one in Indian comics genre. Grand Master Robo, Chandakaal and Mahamanav are some of the most well known and recognisable villains in Indian comics. Dhruva's enemies include normal criminals like Jubisko, Champion Killer; tricksters like Bauna Waman, Vidooshak; mad scientists like Dhwaniraj, Dr. Virus; demons like Chandakaal and even aliens, but Dhruva's most implacable foe is Grand Master Robo. Some of Dhruva's enemies possess a bit more ambiguous personalities and are known to sometimes fight against him while supporting him on other occasions. Classic examples include Nostredamas, Commander Natasha, Black Cat and Nakshatra.

title: List of Dhruva supporting characters


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Cauchy Theorem for a triangle

Theorem
Let $$D \subseteq \mathbb{C}$$ be a domain, $$f : D \to \mathbb{C}$$ a differentiable function. Let $$T$$ be a triangle such that $$\bar{T} \subset D$$. Then

$$\left| \int\limits_{\partial T} f \right| = 0$$

Proof
Assume

$$\left| \int\limits_{\partial T} f \right| = c \geq 0$$.

It will be shown that $$c = 0$$.

First, subdivide $$T$$ into four triangles, marked $$T^{1}$$, $$T^{2}$$, $$T^{3}$$, $$T^{4}$$ by joining the midpoints on the sides. Then it is true that

$$\int\limits_{\partial T} f = \sum\limits_{r=1}^{4} \left( \int\limits_{T^{r}} f \right)$$.

Giving that

$$c = \left| \int\limits_{\partial T} f \right| \leq \sum\limits_{r=1}^{4} \left| \int\limits_{T^{r}} f \right|$$

Choose $$r$$ such that

$$\left| \int\limits_{\partial T^{r}} f \right| \geq \frac{1}{4} c$$

Defining $$T^{r}$$ as $$T_{1}$$, then

$$\left| \int\limits_{\partial T_{1}} f \right| \geq \frac{1}{4} c$$ and $$L\left(\partial T_{1} \right) = \frac{1}{2} L\left(\partial T\right)$$

(where $$L\left( \gamma \right)$$ describes length of curve).

Repeat this process of subdivision to get a sequence of triangles

$$T \supset T_{1} \supset T_{2} \supset \ldots \supset \ldots T_{n} \supset \ldots$$

satisfying that

$$\left| \int\limits_{\partial T_{n}} f \right| \geq \left(\frac{1}{4}\right)^{n}c$$ and $$L\left(\partial T_{n} \right) = \left( \frac{1}{2} \right)^{n} L\left(\partial T\right)$$.

Claim: The nested sequence $$\bar{T} \supset \bar{T_{1}} \supset \bar{T_{2}} \supset \ldots \supset \bar{T_{n}} \supset \ldots$$ contains a point $$z_{0} \in \bigcap\limits_{n=1}^{\infty} \bar{T_{n}}$$. On each step choose a point $$z_{n} \in T_{n}$$. Then it is easy to show that $$\left( z_{n} \right)$$ is a Cauchy sequence. Then $$\left( z_{n} \right)$$ converges to a point $$z_{0} \in \bigcap\limits_{n=1}^{\infty} \bar{T_{n}}$$ since each of the $$\bar{T_{n}}$$s are closed, hence, proving the claim.

We can generate another estimate of $$c$$ using the fact that $$f$$ is differentiable. Since $$f$$ is differentiable at $$z_{0}$$, for a given $$\varepsilon > 0$$ there exists $$\delta > 0$$ such that

$$0 < \left| z - z_{0} \right| < \delta$$ implies $$\left| \frac{f\left(z\right) - f\left(z_{0}\right)}{z-z_{0}} - f'\left(z_{0}\right)\right| < \varepsilon$$

which can be rewritten as

$$0 < \left| z - z_{0} \right| < \delta$$ implies $$\left| f\left(z\right) - f\left(z_{0}\right) - f'\left(z_{0}\right)\left(z-z_{0}\right)\right| < \varepsilon\left|z-z_{0}\right|$$

For $$z \in T_{n}$$ we have $$\left| z-z_{0} \right| < L\left(\partial\right)$$, and so, by the Estimation Lemma we have that

$$\left| \int\limits_{\partial T_{n}} \left\{f\left(z\right) - \left(f\left(z_{0}\right) + f'\left(z_{0}\right)\left(z-z_{0}\right)\right)\right\} dz \right| \leq \varepsilon L^{2}\left(\partial\right)$$

As $$f\left(z_{0}\right) + f'\left(z_{0}\right)\left(z-z_{0}\right)$$ is of the form $$ \alpha z + \beta $$ it has an antiderivative in D, and so $$\int\limits_{\partial T_{n}} f\left(z_{0}\right) + f'\left(z_{0}\right)\left(z-z_{0}\right) = 0$$, and the above is then just

$$\left| \int\limits_{\partial T_{n}} f\left(z\right) dz \right| \leq \varepsilon L^{2}\left(\partial\right)$$

Notice that

$$\left(\frac{1}{4}\right)^{n}c \leq \left| \int\limits_{\partial T_{n}} f\left(z\right) dz \right| \leq \varepsilon L^{2}\left(\partial\right) = \left(\frac{1}{4}\right)^{n} \varepsilon L^{2}\left(\partial\right) $$

Giving

$$c \leq \varepsilon L^{2}\left(\partial\right)$$

Since $$\varepsilon > 0$$ can be chosen arbitrary small, then $$c = 0$$.

title: Cauchy Theorem for a triangle


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User:Adekiya Joscor

Am Adekiya joscor blogger at NaijaCloud. I am 19 years old born August 25 year 2001.

I started blogging when i finished my secondary schooling

title: User:Adekiya Joscor


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Glynn Lunney

Glynn Stephen Lunney (November 27, 1936 – March 19, 2021) was an American NASA engineer. An employee of NASA since its creation in 1958, Lunney was a flight director during the Gemini and Apollo programs, and was on duty during historic events such as the Apollo 11 lunar ascent and the pivotal hours of the Apollo 13 crisis. At the end of the Apollo program, he became manager of the Apollo–Soyuz Test Project, the first collaboration in spaceflight between the United States and the Soviet Union. Later, he served as manager of the Space Shuttle program before leaving NASA in 1985 and later becoming a vice president of the United Space Alliance.

Lunney was a key figure in the US human spaceflight program from Project Mercury through the coming of the Space Shuttle. He received numerous awards for his work, including the National Space Trophy, which he was given by the Rotary Club in 2005. Chris Kraft, NASA's first flight director, described Lunney as "a true hero of the space age", saying that he was "one of the outstanding contributors to the exploration of space of the last four decades".

Early life and NACA career
Glynn Stephen Lunney was born in the coal city of Old Forge, Lackawanna County, Pennsylvania, on November 27, 1936, the eldest son of William Lunney, a welder and former miner who encouraged his son to get an education and to find a job beyond the mines, and his wife Helen Glynn Lunney. That family's surname rhymes with "sunny". He graduated from the Scranton Preparatory School in 1953.

A childhood interest in model airplanes prompted Lunney to study engineering in college. After attending the University of Scranton (1953–1955), he transferred to the University of Detroit, where he enrolled in the cooperative training program run by the Lewis Research Center in Cleveland, Ohio. The center was a part of the National Advisory Committee for Aeronautics (NACA), a United States federal agency founded to promote aeronautical research. Cooperative students at NACA took part in a program that combined work and study, providing a way for them to fund their college degrees while gaining experience in aeronautics. Lunney graduated from college in June 1958, with a Bachelor of Science degree in aerospace engineering.

After graduation, Lunney remained with NACA. His first job was as a researcher in aerospace dynamics at Lewis Research Center, where he worked with a team studying the thermodynamics of vehicles during high-speed reentry. Using a B-57 Canberra bomber, the team sent small rockets high into the atmosphere in order to measure their heating profile.

Mercury
Only a month after Lunney graduated, President Eisenhower signed into existence the National Aeronautics and Space Administration (NASA), into which NACA was subsumed. His timing was perfect, for as Lunney later said, "there was no such thing as space flight until the month I got out of college". Lunney was soon transferred to Langley Research Center in Hampton, Virginia, where in September 1959 he became a member of the Space Task Group, which was the body given responsibility for the creation of NASA's human spaceflight program. Aged twenty-one, he was the youngest of the forty-five members of the group. His first assignment was with the Control Center Simulation Group, which planned the simulations used to train both flight controllers and astronauts for the as-yet unknown experience of human spaceflight.

A member of the Flight Operations Division, Lunney was one of the engineers responsible for planning and creating procedures for Project Mercury, America's first human spaceflight program. He took part in the writing of the first set of mission rules, the guidelines by which both flight controllers and astronauts operated. During Mercury, Lunney became, after Tecwyn Roberts, the second man to serve as the Flight Dynamics Officer (FIDO) in the Mercury Control Center, controlling the trajectory of the spacecraft and planning adjustments to it.

Lunney's colleague Gene Kranz described him as "the pioneer leader of trajectory operations, who turned his craft from an art practiced by a few into a pure science". It was during these years that Lunney became the protege of flight director Chris Kraft. He was sometimes referred to as "the son of Chris Kraft."

Lunney worked both in the Control Center and at remote sites; during the flight of John Glenn, America's first orbital spaceflight, he was serving as the FIDO in Bermuda. In September 1961, NASA's Space Task Group was reorganized into the Manned Spacecraft Center and moved to Houston, Texas, and Lunney moved with it. In Houston, he became head of the Mission Logic and Computer Hardware section, where he defined and oversaw the computing and display requirements of the flight dynamics division within the new Mission Control Center.

Gemini
Gemini was a step forward for NASA's human spaceflight program: the Gemini capsule was larger and more advanced than Mercury, capable of supporting two men for up to a two-week mission. Because of the longer mission durations, Mission Control began to be staffed in shifts. In 1964, Lunney and Kranz were selected by Kraft to join him and his deputy John Hodge as flight directors. Aged only twenty-eight, Lunney was the youngest of the four.

Lunney was stationed in Bermuda for the uncrewed Gemini 2 mission. He worked backup on Gemini 3, taking charge of the newly established Mission Control Center in Houston, at a time when flights were still controlled from Cape Canaveral in Florida. On Gemini 4, he again was working backup, this time in Florida, supporting the first mission that was controlled entirely from Houston. After spending some time on uncrewed testing for the Apollo program, he returned to work as a flight director on Gemini 9 and Gemini 11 and lead flight director on, Gemini 10 and Gemini 12.

Apollo
As with Project Mercury, Lunney was involved in Project Apollo right from the beginning. He took charge of the "boilerplate" tests of the Apollo abort escape system at White Sands, which took place during the Gemini program, and was flight director during the first uncrewed Saturn V test flight, SA-501.

Lunney was not scheduled to serve as a flight director on the first crewed Apollo mission, later known as Apollo 1. During the countdown demonstration test that resulted in the Apollo 1 fire, Lunney was at home having dinner with astronaut Bill Anders and his wife, and was called into Mission Control when the fire occurred. It was, as he recalled, "a tremendous punch in the stomach to all of us".

The aftermath of the fire, in which three astronauts were killed, left Lunney and his colleagues at NASA feeling that they had perhaps failed to recognize the risks they were running in their efforts to meet Kennedy's timetable of landing a man on the Moon by the end of the decade and bringing him safely back to Earth. "Maybe," said Lunney over thirty years later, "we had gotten a little overconfident".

Lunney attracted significant media attention in 1968, when he worked as lead flight director on Apollo 7, the first of the crewed Apollo flights. Coming as it did after the Apollo 1 fire, the mission was an important test for the Apollo program, and was stressful for astronauts and controllers alike. Lunney had primary responsibility for dealing with the mission commander, Wally Schirra, who repeatedly questioned orders from the ground. Although pressed by reporters in news conferences, Lunney stayed diplomatic and said nothing critical of Schirra.

Privately, however, he was exasperated, and later assured his team of young controllers that "manned spaceflight is usually better than this". He was diplomatic about Donn Eisele's sarcastic comment to the CAPCOM that he would "like to meet the man, or whomever it was, that dreamed up that little gem." The "gem" turned out to be Lunney's. As a flight director Lunney was known for his good memory and his unusually quick thought processes—traits that could sometimes prove problematic for his team of flight controllers. "Glynn would drive you crazy", said Jay Greene, a fellow controller, "because his mind would race so fast that he could churn out action items quicker than you could absorb, much less answer." He was the lead flight director again during the Apollo 10 mission, a dress rehearsal for the Apollo 11 Moon landing.

During the Apollo 13 crisis, Lunney played a key role. Coming on shift an hour after the oxygen tank explosion that put the crew's lives in jeopardy, Lunney and his team faced the unprecedented challenge of having to power up the Lunar Module on an extremely tight timeline, while transferring guidance and navigation data to it from the dying command module. His excellent memory and quick thinking were critical in the success of his team during the ensuing hours. Ken Mattingly, the astronaut who had been bumped from the Apollo 13 crew due to his exposure to German measles, later called Lunney's performance "the most magnificent display of personal leadership that I've ever seen."

On the day following the Apollo 13 splashdown, Lunney joined his fellow flight directors in accepting the Presidential Medal of Freedom as a member of the Apollo 13 mission operations team.

Apollo–Soyuz Test Program
In 1970, while still a flight director, Lunney was selected as one of the members of a NASA delegation to the Soviet Union, which was to discuss the possibility of cooperation between the two countries in the field of human spaceflight. "For me it was out of the clear blue sky", said Lunney, who was told of the plans while at a conference in early October. "I did not know anything about [the proposed talks] until that time."

The trip took place in late October. While in Moscow, Lunney gave a presentation to Soviet engineers on the techniques that NASA used for orbital rendezvous, and on the compromises that would have to be made in order to achieve a rendezvous between American and Soviet spacecraft. The technical agreement that he helped to draft laid the groundwork for the mission which was to become the Apollo–Soyuz Test Project (ASTP). It was intended to be a joint mission, whose highlight was to be a docking between an American Apollo spacecraft and a Soviet Soyuz.

Lunney was named technical director of the ASTP in the following year. As technical director, he made several more trips to the Soviet Union, helping to negotiate the seventeen-point agreement that would govern the conduct of the mission. He also took part in working groups in Houston that dealt with the technical details of the project. A New York Times profile reported that he was taking Russian lessons in order to be better prepared for the role.

On June 13, 1972, Lunney was given overall responsibility for the test project; henceforth he would be in charge not only of building a partnership with the Soviets, but also of mission planning and of negotiating with North American Rockwell, the spacecraft contractor. According to the official history of the ASTP, Lunney's performance during Apollo 13 and during the Soviet negotiations had recommended him to Chris Kraft, who was by then director of Johnson Space Center. In 1973, Lunney became manager of the Apollo Spacecraft Program Office, a position which gave him responsibility for the Apollo spacecraft used during Skylab missions, as well giving him more authority in his role as head of the ASTP.

The ASTP mission took place in July 1975. It was criticized by some journalists as a "costly space circus", who felt that it wasted NASA funds that could have been better spent on projects such as Skylab. However, Lunney supported the project, saying in a later interview that he did not believe the cooperation necessary to build the International Space Station would have been possible if ASTP had not laid the groundwork for it.

Space Shuttle
After the ASTP mission was completed, Lunney became manager of the Shuttle Payload Integration and Development Program. During this period, it was anticipated that NASA's space shuttle fleet would be flying very frequent missions, and carrying commercial payloads as well as flying missions for government organizations such as the Department of Defense and the Jet Propulsion Laboratory. The payload integration program was responsible for determining how the various demands of these customers could be satisfied, and how mixed payloads could best be physically accommodated within the cargo bay of the shuttle. During these years Lunney also spent time working at NASA Headquarters in Washington, D.C., as Deputy Associate Administrator for Space Flight and later as Acting Associate Administrator for Space Transportation Operations.

In 1981, Lunney became manager of the Space Shuttle program, a high-level position where Lunney found himself responsible for setting the agenda for the developing program. His responsibilities were broad ones; they included supervising program planning, budgeting and scheduling; systems engineering; and mission planning. During the earlier shuttle flights he was involved in determining whether the weather was suitable for launch, but in later years that responsibility was largely devolved to lower levels of the hierarchy.

Many of his colleagues had expected Lunney to succeed his mentor, Kraft, as director of Johnson Space Center; Neil Hutchinson, a fellow flight director, later commented that Lunney "was sort of the anointed one". However, when Kraft retired in 1982, former Apollo flight director Gerry Griffin was offered the position instead.

In 1985, Lunney decided to leave NASA, feeling that the Space Shuttle program had worn him out physically and mentally and that he was ready for a new type of challenge. Although he had retired from NASA the year before, he was called to testify before the U.S. House Committee on Science and Technology in the aftermath of the Challenger accident. While still manager of the shuttle program, he had signed the "Criticality 1" waiver that allowed Challenger to launch even though the joints of its solid rocket boosters had recently been redefined as non-redundant systems. His actions were not unusual in the context of NASA practice at the time, which allowed a "walk through" of such potentially controversial waivers if no debate was expected.

Career at Rockwell
Upon leaving NASA in 1985, Lunney took a position at Rockwell International, the contractor responsible for the construction, operation, and maintenance of the Space Shuttle. At first he worked in California, managing a Rockwell division that was building satellites for the Global Positioning System; this was his first experience with satellites. In 1990, he returned to Houston as President of the Rockwell Space Operations Company, which provided support for flight operations at Johnson Space Center and employed about 3,000 people. For Lunney, this represented a return to his roots in mission operations, which he had left twenty years before.

In 1995, Rockwell joined forces with its competitor Lockheed Martin to form the United Space Alliance, a jointly owned organization created to provide operations support for NASA, as well as to take over some of the functions previously performed by NASA employees. At this point, Lunney became Vice President and Program Manager of the United Space Alliance's spaceflight operations in Houston; he stayed in this position until his retirement in 1999.

Personal life
While at Lewis Research Center, Lunney met Marilyn Kurtz, who worked there as a nurse. They were married in 1960 and had four children: Jennifer, Glynn Jr., Shawn, and Bryan. Their youngest son Bryan also pursued a career at NASA, becoming a flight director in 2001 and retiring in 2011. Lunney and his son Bryan were the first multi-generational flight directors to have served NASA.

During his leisure hours, Lunney enjoyed sailing; during the 1960s the family owned a twenty-foot sailboat which they took out on Galveston Bay, and he occasionally dreamed of going with his wife and children on an ocean cruise lasting for months. In his retirement he enjoyed golf, saying that "I have come to realize that golf will not be mastered, but will continue to be humbling."

Described as "legendary" by NASA, Lunney died on March 19, 2021, at his home in Clear Lake, Texas, at the age of 84. He had been treated for leukemia for several years, but according to his family he succumbed to stomach cancer.

Awards and honors
Lunney was a Fellow of the American Astronomical Society and of the American Institute of Aeronautics and Astronautics. In 1971, he was awarded an honorary Doctorate from the University of Scranton. He received many awards from NASA, including three Group Achievement Awards, two Exceptional Service Medals and three Distinguished Service Medals.

In 2005, he received the National Space Trophy from the Rotary National Award for Space Achievement Foundation. The award is given to individuals who have made an outstanding and career-spanning contribution to America's space program. Previous winners have included Chris Kraft and Neil Armstrong. "Lunney's innovation and dedication to the U.S. space flight program", said the RNASA Advisor General, "has set a standard for current and future generations of space explorers. As a manager, he inspired his employees to do their best work and offered direction and encouragement to his team when challenges arose; as an explorer, he always looked toward the future and saw the endless possibilities and benefits of man's journey into space."

In 2008 he received the Elmer A. Sperry Award, jointly with Thomas P. Stafford, Alexey Leonov and Konstantin Bushuyev, for their work on the Apollo–Soyuz mission and the Apollo–Soyuz docking interface design.

In films
In the 1995 film Apollo 13, Glynn Lunney was portrayed by Marc McClure. McClure had a relatively minor role leading writer Charles Murray to lament that Lunney was "barely visible in the movie", being overshadowed by the focus on Lunney's fellow flight director Gene Kranz. "Without slighting Kranz's role", Murray commented, "the world should remember that it was Glynn Lunney ... who orchestrated a masterpiece of improvisation that moved the astronauts safely to the lunar module while sidestepping a dozen potential catastrophes that could have doomed them." "They didn't give me credit for any of the work that I did," Lunney said in 2019. "As a matter of fact, if you watch the movie, you'll see I'm sort of portrayed as a flunky."

In the 2020 television miniseries The Right Stuff, Lunney was played by Jackson Pace.

Select publications

 * Lunney, G. S. and K. C. Weston. (1959). "Heat-Transfer Measurements on an Air-Launched, Blunted Cone-Cylinder Rocket Vehicle to Mach 9.7". NASA-TM X-84. Cleveland, Ohio: NASA Lewis Research Center.
 * Lunney, G. S., L. C. Dunseith, and J. F. Dalby. (1960). "Project Mercury: Methods and Pertinent Data for Project Mercury Flight Computing Requirements". NASA-TM-X-69335. Hampton, Virginia: NASA Langley Research Center.
 * Lunney, G. S. (1964). "Launch-Phase Monitoring". In Manned Spacecraft: Engineering Design and Operation. Ed. Paul E. Purser, Maxime A. Faget, and Norman F. Smith. New York: Fairchild Publications, Inc.

title: Glynn Lunney


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Oktyabrskoye, Altai Krai

Oktyabrskoye (Октябрьское) is a rural locality (a selo) in Sibirsky Selsoviet, Pervomaysky District, Altai Krai, Russia. The population was 392 as of 2013. There are 7 streets.

Geography
Oktyabrskoye is located 24 km north of Novoaltaysk (the district's administrative centre) by road. Sibirsky is the nearest rural locality.

authors: Витёк Артамонов

title: Oktyabrskoye, Altai Krai


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Le Tigre (clothing brand)

Le Tigre is an American brand of apparel designed to rival Lacoste in styling. First offered in 1977, Le Tigre polos sported a leaping tiger in lieu of Lacoste's signature crocodile and Retro Fox's leaping fox. The brand made a comeback in 2003, after being out of production through the 1990s. Le Tigre had become popular during the later 1980s; celebrities from Wilt Chamberlain, LL Cool J, and even Ronald Reagan were seen wearing the tiger. The company is based in New York City. Le Tigre has been a subsidiary of Kenneth Cole Productions since 2007. The brand was then purchased by Infinity Lifestyle Brands and Hilco Brands in November 2015.

title: Le Tigre (clothing brand)


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Instructional design/Cognitive behaviors/Understanding Concepts

Back to Topic:Instructional Design > Cognitive behaviors > Understanding Concepts > Define >  Learn >  Teach >  Tactics >  Try It >  Example

Source: Understanding Understanding by Charles M. Reigeluth. Used by Permission.

Introduction
This lesson covers how to design instruction that includes meaningful learning, or understanding. After completing this lesson, you should be able to do the following:

 * Define Understanding
 * Design Meaningful Learning
 * Apply the Principals for Teaching Understanding

Topics include the following:

 * Define: Understanding: Defined
 * Learn: Understanding: The Importance of Understanding
 * Teach: Understanding: Principles for Learning Meaningful Knowledge
 * Tactics: Understanding: Principles for Teaching Understanding
 * Try It: Understanding: Skill Builder Practice
 * Example: Understanding: Example

Click Next to continue.

References and Resources for this Lesson
References
 * Ausubel, D.P. (1968). Educational Psychology: A Cognitive View. New York: Holt, Rinehart & Winston.

 * Ausubel, D.P., Hanesian, & Novak, (1978) Educational Psychology: A Cognitive View (2nd ed.). New York: Holt, Rinehart & Winston.

 * Bloom, B.S. (1976). Human Characteristics and School Learning. New York: McGraw-Hill.

 * Gagné, R.M. (1985). The Conditions of Learning and Theory of Instruction (4th ed.). New York: Holt, Rinehart & Winston.

 * Kaufman, R. (1979). Needs Assessment: Concept and Application. Englewood Cliffs, NJ: Educational Technology Publications.

 * Keller, J. (October 1987). Strategies for stimulating the motivation to learn. Performance and Instruction, 1-7.

 * Kulhavy, R. (1977). Feedback in written instruction. Review of Educational Research, 47, 211-232.

 * Miller, G. (1956). The magical number seven, plus or minus two: Some limits on our capacity to process information. Psychological Review, 63, 81-97.

 * Thiagarajan, S. (2004). Framegames by Thiagi. Bloomington, IN: Workshops by Thiagi.

 * Thorndike, E.M. (1913). Educational Psychology. Volume II. The Psychology of Learning. New York: Teachers College, Columbia University.

Resources
 * Wikipedia.com


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Molly Byrne

Molly Veronica Byrne (born 30 November 1928) is a former Australian politician who represented the South Australian House of Assembly seats of Barossa from 1965 to 1970, Tea Tree Gully from 1970 to 1977, and Todd from 1977 to 1979 for the Labor Party.

At the 1965 election Byrne was Labor's first woman elected to the Parliament of South Australia, the second woman elected to the House of Assembly, and the third woman elected to the Parliament of South Australia after the Liberal and Country League's Jessie Cooper and Joyce Steele both elected at the 1959 election.

In 1985 Byrne was awarded the Medal of the Order of Australia for "service to the community and parliamentary service".

title: Molly Byrne


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moral minimum

Noun

 * 1)  A standard or principle upheld as indispensable for moral conduct, whether within a particular context or in general.
 * 2)  The specific rule that one should do no intentional harm, often considered the bare minimum required for ethical behavior.
 * 1)  The specific rule that one should do no intentional harm, often considered the bare minimum required for ethical behavior.
 * 1)  The specific rule that one should do no intentional harm, often considered the bare minimum required for ethical behavior.
 * 1)  The specific rule that one should do no intentional harm, often considered the bare minimum required for ethical behavior.

title: moral minimum


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Patania harutai

Patania harutai is a species of moth in the family Crambidae. It was described by Inoue in 1955. It is found in Japan (Honshu), China and the Russian Far East (Amur).

The wingspan is 35–42 mm.

title: Patania harutai


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Clinical data repository

A Clinical Data Repository (CDR) or Clinical Data Warehouse (CDW) is a real time database that consolidates data from a variety of clinical sources to present a unified view of a single patient. It is optimized to allow clinicians to retrieve data for a single patient rather than to identify a population of patients with common characteristics or to facilitate the management of a specific clinical department. Typical data types which are often found within a CDR include: clinical laboratory test results, patient demographics, pharmacy information, radiology reports and images, pathology reports, hospital admission, discharge and transfer dates, ICD-9 codes, discharge summaries, and progress notes.

A Clinical Data Repository could be used in the hospital setting to track prescribing trends as well as for the monitoring of infectious diseases. One area CDR's could potentially be used is monitoring the prescribing of antibiotics in hospitals especially as the number of antibiotic-resistant bacteria is ever increasing. In 1995, a study at the Beth Israel Deaconess Medical Center conducted by the Harvard Medical School used a CDR to monitor vancomycin use and prescribing trends since vancomycin-resistant enterococci is a growing problem. They used the CDR to track the prescribing by linking the individual patient, medication, and the microbiology lab results which were all contained within the CDR. If the microbiology lab result did not support the use of vancomycin, it was suggested to change the medication to something appropriate as under the Center for Disease Control CDC guidelines. The use of CDR's could help monitor infectious diseases in the hospital and the appropriate prescribing based on lab results.

The use of Clinical Data Repositories could provide a wealth of knowledge about patients, their medical conditions, and their outcome. The database could serve as a way to study the relationship and potential patterns between disease progression and management. The term "Medical Data Mining" has been coined for this method of research. Past epidemiological studies may not have had as complete of information as that which is contained in a CDR, which could lead to inconclusive data/results. The use of medical data mining and correlative studies using the CDR could serve as a valuable resource helping the future of healthcare in all facets of medicine. The idea of data mining a CDW was used for screening variables that were associated with diabetes and poor glycemic control. It allowed for novel correlations that may have not been discovered without this method.

One potential use of a clinical data repository would be for clinical trials. This would allow for researchers to have all the information from a study in one place as well as let other researchers benefit from the data to further innovation. They would also be advantageous since they are digital and real-time. This would be easier to log data and keep it accurate since it would be digital rather than in paper form.

The clinical data repository is not without its weaknesses, however. Since they usually don't integrate with other non-clinical sources, following patient treatment across the care continuum becomes very difficult. In turn, tracking the true cost per case for each patient isn't feasible. IT teams spend most of their time gathering and compiling data instead of interpreting information and finding opportunities for cutting costs and improving patient care.

title: Clinical data repository


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Template:Did you know nominations/Beirut I (1960)

The result was: promoted by Allen3 talk 21:42, 2 March 2015 (UTC)

Beirut I (1960)

 * ... that the electoral district Beirut I had the largest concentration of Jewish voters in Lebanon?
 * Reviewed: Template:Did you know nominations/Yantaromyrmex

Created by Soman (talk). Self nominated at 20:10, 10 February 2015 (UTC).

 * Article was created on February 8, expanded to 3221 DYK-eligible characters on the same day, and nominated here on Feburary 10.
 * The hook fact relies on a book that's hard to find. Thus it is difficult to verify, but hardly implausible. The other source provided, listing figures by district, groups Jews and miscellaneous Christians as together "Minorities". Beirut I is said to contain far more of these than Beirut II and Beirut III. Other sources on the web confirm that the Jewish neighborhood(s) of Beiruit were located in the northeast part of the city, more or less within the area of Beirut I. (For example, Beirut's only synagogue is located in that area.) Finally, the author, User:Soman, has a long history of well-researched contributions to Wikipedia, as well as obvious familiarity with the demographics and politics of Beiruit. On the basis of all the above, I think we can accept the hook as given.
 * The article is neutral and well-footnoted. The majority of these footnotes refer to difficult-to-find books. Those parts of the article which correspond to online sources appear to report them well. For example, the description of Beirut I's votes in the 1968 elections match up well with the description given in the cited source.
 * Symbol confirmed.svg This article is ready to roll for DYK and furthermore a fine contribution to Wikipedia—a useful compilation of information which otherwise would be difficult for people to find. shalom, groupuscule (talk) 20:22, 2 March 2015 (UTC)


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User:Glowzx/Evaluate an Article

Evaluate an article
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 * Name of article: (link) Sigmund Freud
 * Briefly describe why you have chosen this article to evaluate
 * He is a well known figure in psychology and so looking at this wikipedia page will let me see how to frame psychological history in the way Wikipedia wants as well as see how information is organized on a wikipedia page since this page has lots of details/ information on Freud's life

Lead

 * Guiding questions

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 * Yes. See following "Sigmund Freud (/frɔɪd/ FROYD; German: [ˈziːkmʊnt ˈfʁɔʏt]; born Sigismund Schlomo Freud; 6 May 1856 – 23 September 1939) was an Austrian neurologist and the founder of psychoanalysis, a clinical method for treating psychopathology through dialogue between a patient and a psychoanalyst."
 * Does the Lead include a brief description of the article's major sections?
 * Yes it lists Freud's main contributions to psychology
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 * No
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 * concise

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Checking the talk page

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 * What kinds of conversations, if any, are going on behind the scenes about how to represent this topic?
 * Copyright violation in Life and Death Instincts section
 * How is the article rated? Is it a part of any WikiProjects?
 * level-3 vital article in People
 * B-Class
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 * How does the way Wikipedia discusses this topic differ from the way we've talked about it in class?

Overall impressions

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 * What is the article's overall status?
 * Sigmund Freud was a good article, but it was removed from the list as it no longer met the good article criteria at the time (2006)
 * What are the article's strengths?
 * very detailed
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 * more pictures?
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 * well-developed

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title: User:Glowzx/Evaluate an Article