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Offa's Dyke Path

Offa's Dyke Path (Llwybr Clawdd Offa) is a long-distance footpath loosely following the Wales–England border. Officially opened on 10 July 1971, by Lord Hunt, it is one of Britain's National Trails and draws walkers from throughout the world. About 60 mi of the 177 mi route either follows, or keeps close company with, the remnants of Offa's Dyke, an earthwork traditionally thought to have been constructed in the late 8th century on the orders of King Offa of Mercia.

Walking trail
Traveling south to north, starting by the Severn Estuary at Sedbury, near Chepstow, and finishing at Prestatyn on the north coast, the walk will take an average walker roughly 12 days to complete. Roughly following the border in parts, and elsewhere the ancient monument of Offa's Dyke, as well as natural features such as the Hatterrall Ridge, the Dyke Path passes through a variety of landscapes. The route traces the eastern edge of the Black Mountains, traverses Clun Forest, the Eglwyseg moors north of Llangollen and the Clwydian Range.

The route passes through the counties of Monmouthshire, Gloucestershire, Powys, Herefordshire, Shropshire, Wrexham, Denbighshire and Flintshire. The Welsh Marches (Marchia Wallie) is a term used to describe this border region between England and Wales, since it was recorded in the Domesday Book in 1086. It passes through, or close to, the towns of Chepstow, Monmouth, Hay-on-Wye, Kington, Presteigne, Knighton, Montgomery, Welshpool and Oswestry, then in and around the North Wales towns and villages of Llangollen, Llandegla, Bodfari and Dyserth.

The half-way point of the path is marked by the Offa's Dyke Centre in Knighton (52.3458°N, -3.0517°W). There used to be around 600 stiles along the route, but many of these have now been replaced by kissing gates.

a certain vigorous king called Offa......had a great dyke built between Wales and Mercia from sea to sea.

Route
Places on the route and highlights on or near the trail:

Chepstow to Monmouth

 * Sedbury Cliffs: Severn Estuary, Severn Bridge
 * Chepstow: Chepstow Castle, River Wye
 * View of Tintern Abbey from the Devil's Pulpit
 * Redbrook: Iron railway bridge
 * The Kymin naval temple

Monmouth to Hay-on-Wye

 * Monmouth: Monnow Bridge
 * White Castle
 * Llangattock Lingoed: St Cadoc's church
 * Pandy
 * Hatterrall Ridge is the highest point on the trail at 703 m
 * Black Mountains
 * Llanthony Priory

Hay-on-Wye to Knighton

 * Hay-on-Wye
 * Newchurch
 * Gladestry
 * Hergest Ridge with wild ponies, 425 m
 * Kington
 * Hawthorn Hill, 406 m

Knighton to Montgomery

 * Knighton: Offa's Dyke visitor centre
 * Panpunton Hill, 374 m
 * Cwm-Sanaham Hill 406 m
 * Llanfair Hill, highest point of the dyke at 430 m
 * Churchtown and Edenhope Hill

Montgomery to Llanymynech

 * Montgomery
 * Chirbury
 * Beacon Ring Iron Age hill fort (Caer Digoll)
 * Buttington
 * Alongside Montgomery Canal and dyke beside River Severn
 * Four Crosses

Llanymynech to Trevor

 * Llanymynech
 * Moelydd, 285 m
 * Trefonen
 * Oswestry old racecourse at Racecourse Common
 * Craignant
 * Chirk Castle
 * Llangollen Canal
 * Pontcysyllte Aqueduct (World Heritage Site) over the River Dee

Trevor to Prestatyn

 * Trevor
 * Llangollen Castle
 * Eglwyseg Crags
 * Llandegla Forest (with mountain bike trails)
 * Llandegla
 * Clwydian Range of hills:
 * Around Moel-y-Plas, 440 m, Moel Llanfair, 447 m, Moel Gyw, 467 m and Foel Fenlli, 511 m
 * Moel Famau, 555 m and Jubilee Tower at summit
 * Around Moel Dywyll, 472 m, Moel Llys-y-Coed, 465 m and Moel Arthur, 455 m
 * Penycloddiau hill fort at 440 m
 * Bodfari
 * Rhuallt
 * Prestatyn: Offa's Dyke Monument on the beach

Promotion and media
Various bodies on either side of the border are collaborating on a sustainable tourism partnership, a principal focus of which is Walking with Offa, both on the trail but also in what has been dubbed Offa's Country i.e. in a corridor along the border.

The path was the focus of an episode of the Channel 4 program Britain's Ancient Tracks with Tony Robinson.

In June 2021, during the footpath's 50th year, an Offa's Dyke Rescue Fund was launched to restore eroded and other damaged parts of the route and to buy parts of the path at risk of sustained damage or negligence from local land owners. The fund is working in consultation with Cadw and English Heritage and the National Trail Unit.

On 22 August 2021, the BBC's Countryfile programme celebrated 50 years of the path.

title: Offa's Dyke Path


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Talk:General linear group

Application to polynomials
If p is a polynomial in F[x] of degree n, then we can consider p as an element of the vector space with basis elements (1, x, x2, ..., xn).

''Late now... I'll finish this up tomorow...''

Here's an example of a general issue with the mathematics pages (and even any of the more esoteric topics) of Wikipedia which I'm struggling with.

By removing the section on "Motivations", Axel has condensed and clarified the article; which makes it easier to read - if one knows what's going on already in the topic.

But if someone is not quite sure what GL(n,F) is, or why it exists, (which is the reason that I presume they sought out this article) I find a sentence such as:

 * If the dimension of V is n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Once a basis has been chosen, every automorphism of V can be represented as an invertible n by n matrix, which establishes the isomorphism.

may not satisfy them; what is an automorphism of V? What is the relationship between V and n and F? What does it mean for an isomorphism to be canonical in this context? Exactly how do you apply an n by n matrix as an automorphism of V? How do you multiply a vector space by a matrix??? WILL THIS BE ON THE TEST!!????

Let me add that the above quoted statement is absolutely correct, and that I find it a clear explanation of the relationships I alluded to in the motivation section. (Also, there was an error in the original motivation (automorphisms of V should satisfy T(cv)=cT(v), not U(c)T(v)).

I already know the answers to all the above questions (and no, it will not be on the test :) ). My question is - does it serve the expected reader? Who is the expected reader, and how do we best serve them?

These thoughts came to mind in discussion with a friend of mine who is getting his masters in mathematics, and with whom I've been coaching regarding group theory in general, and linear groups as automorphisms of roots of polynomials in particular. While he might understand some of the mechanics of these issues, I find that where he needs the most assistance is in the acquisition of "gut" feelings or instincts about the role and use of some of these structures, and I've been writing (or at least attempting to write :) ) with his point of view in mind.

So I find myself somewhere between writing a textbook explanation of concepts on the one hand (too much info), and an encyclopedia/dictionary approach (assumes too little background for a naive user).

With any of the articles I write in this area, I am trying to satisfy all readers - the mathematically sophisticated, as well as those whoe are just starting the topic under consideration. So, for example, in an article regarding PSL(n,F) (affine transformations), I could see using language like the above; since once you're looking at PSL, you've got to be already pretty well versed in these issues; but GL is just general enough that I can assume knowledge of the terms automorphism, vector space, and so on; but not neccessarily a knowledge of how it all ties together.

I'll address this style question mainly to you, Axel; as I seem to be bumping into you editing away in the mathematics domain; but I'm curious as regards the philosophy overall of wikipedians. Chas zzz brown 20:57 Oct 20, 2002 (UTC)

Sorry, I was a bit rude in cutting the motivation section. There were a couple of things wrong with it; the given formula for T was not the general form of an automorphism and the field automorphism U was not really needed, especially in an introductory motivation paragraph. Then, the next section seemed to repeat that elements of GL are automorphisms, but talked about "automorphic space induced by M" which I couldn't make sense of. That the rows and columns of an invertible matrix form bases of Fn is certainly useful information, but should probably either go into matrix or basis (linear algebra) where more people will see it.

Generally speaking, I want the articles to give useful information to amateurs and experts alike. Amateurs will usually only be able to understand the first couple of sentences (which should give the proper "gut feeling" and intuition), before they run into terms they don't understand and have to click on a link. That's just fine.

I agree that the quoted paragraph above about the non-canonical isomorphisms is not clear. This should be expanded, with an explicit construction of the isomorphism. On the other hand, people who know what a finite-dimensional vector space is usually also know how to represent linear transformations as matrices. AxelBoldt 22:29 Oct 20, 2002 (UTC)

Thanks for responding; no offence taken at the edit. I agree there were many problems with the section you cut (read: wrongness!), and had been thinking about how to correct them when I woke up; then I saw the whole section was gone. I just wanted to get some feedback on how to fix it from the point of view of usefulness; i.e., whether it was deleted for lack of correctness or from inappropriate context (or both).

As regards the style question, I suppose I look at mathworld.com as a sort of minimal standard; it leans more to the dictionary end of things. I'd like to see Wikipedia stand somewhat closer to the explanatory end than that; not that it should become a collection of proofs, but that it should contain plenty of intuitions regarding why certain structures and concepts exist.

I happened to be looking at the EPR paradox page today and saw the recurring "expert article" style problem; although understanding bra-ket notation implies an ability to understand EPR, the converse isn't the case; I have only a vague notion of the mathematics on that page, but I think I understand why E, P, and R were so upset. (A better explanation of why faster than light communication would be considered paradoxical would help there, but that's a different thread...).

It's hard to recall what some of this stuff looked like before I got my head around it; and that makes striking the right tone a bit challenging (at least for me :). Chas zzz brown 23:07 Oct 20, 2002 (UTC)

I agree that we should have more explanations than mathworld (and less mistakes), and I also don't have anything against the occasional proof, as long as they are clearly marked and can be skipped. AxelBoldt 16:17 Oct 21, 2002 (UTC)

We should have a few words about the projective groups, no? Someone care to take this on? Revolver

Rings
Can't this be defined for matrices over commutative rings just as well as fields? Mac Lane's book has that usage... A5 11:44, 3 March 2006 (UTC)

 * You can take the unit group of a matrix ring over any ring. What do you do next, though? We don't strive for ultimate generality. Charles Matthews 12:01, 3 March 2006 (UTC)

 * I don't understand what you're getting at. What part of the GL definition depends on having a multiplicative inverse? As for "we don't strive for ultimate generality" - for instance, people talk about SL2(Z) a lot, are you suggesting Wikipedia just ignore such objects because Z is not a field, or ...? A5 14:24, 3 March 2006 (UTC)

 * Even Serge Lang's "Algebra" defines it over a ring. I don't know where Wikipedia's definition came from, but it is non-standard and I think we should change it to be standard. A5 14:45, 3 March 2006 (UTC)

What part of the GL definition depends on having a multiplicative inverse? It doesn't. On the other hand, if you want GLn(R) for R a general commutative ring, you might as well say it is a group scheme. I'm sure that was in the back of Mac Lane's mind. Since it works without the commutative law, are you not going to allow us to stop there? I'm sure you're wrong about the non-standard definition, though. General linear group over a field is a baic, useful definition. Charles Matthews 16:06, 3 March 2006 (UTC)

 * General linear group over a field is a basic, useful definition. Yes, and it is a special case of the correct definition, over a ring. I've explained what the costs of keeping the current definition are. The ring definition is used quite frequently - I've not seen any other - and it is used in many important special cases like SL2Z, and people will be confused if they look these up in Wikipedia and find that the Wikipedia definition uses fields instead. Please tell me why you object to using rings in Wikipedia's definition, or I'll go ahead and change it. What is the cost. A5 01:46, 4 March 2006 (UTC)

 * The definition over a field should be kept as the primary one. However, I think it would be appropriate to have a section (called Generalizations or similiar) on the ring-theoretic definition. -- Fropuff 07:15, 4 March 2006 (UTC)

Well, I certainly object, in all cases, to generality that makes it harder for the general reader to see what's going on. For GL of a ring there are numerous things that don't work. For example, PGL isn't so clear. So I agree with Fropuff. We can certainly speak somewhere about the GL that denotes a functor from rings to groups; but this is not appropriate for the entry definition. Charles Matthews 13:05, 4 March 2006 (UTC)

 * How about we say "GLn(R) is the group of invertible n by n matrices over a commutative ring R. In most cases R is a field such as R or C." That way, users who don't know what a ring is will still be in familiar territory. Additionally, we will not change the PGL entry to make any reference to rings, to keep Charles happy. Is that OK? A5 21:47, 4 March 2006 (UTC)

It's still the wrong way to introduce the topic. We do have style guides for lead sections; we do have conventions about using 'concentric' style, where you tell the reader the short version, then the expanded version, then the still fuller version. That may not be the way graduate-level mathematics texts are written. But there is no need at all for Wikipedia to imitate such books. It is doing something different. Charles Matthews 21:58, 4 March 2006 (UTC)

 * That is starting with the short version. Most uses are over R and C, I think you'll agree.

 * No, I don't agree. The case of a finite field is fundamental, to finite group theory. And I still think you are confused about what we are doing here. We are not followers of Bourbaki, this is not a classroom, this is not a textbook treatment, and we are constantly asked not to make the abstract gradient steeper than it absolutely has to be. Charles Matthews 15:59, 5 March 2006 (UTC)

 * Additionally, it keeps things simple by not giving the definition in terms of a mathematical object with more operations than are necessary (i.e. a field). Should we say, under the entry for group, that a group is something with an associative, commutative, invertible, binary operation, only to mention as an aside half-way through the article that the operation need not be commutative? No, it makes it harder for people to learn the correct definition, and it forces people who just wanted a quick summary to read the entire article. The same considerations apply here. This is crazy. Besides, I don't see why you think that people who know what a field is, won't also know what a ring is. A5 22:08, 4 March 2006 (UTC)

 * Gotta admit I find it rather shocking that this article doesn't mention rings. It definitely should, somewhere. That being said, I think the starting posture of the article should be about GL(field), especially over R or C, because that's the most accessible to the widest audience. Dmharvey 22:52, 4 March 2006 (UTC)

 * I think the starting posture of the article should be about GL(field) OK, sure. But my feeling is that GL(field) should be presented as an early example, not as the canonical definition. If it is presented as a definition, it should at least be preceeded by something that says "For a more general definition in terms of rings, see ... below". However, I think users will find it a bit silly when they scroll down to the "more general definition" and the only difference they notice is that the word "field" has been replaced by the word "ring". It's overkill. A5 23:13, 4 March 2006 (UTC)

 * I don't think it's overkill. If you prefer, we don't even need to give a general definition that happens to be the same sentence with "field" replaced by "ring"; you just say "An important generalisation is where we replace "fields" by "rings"." The point of the latter section is not to give a definition; the point should be to explore some important examples, for example R = Z, R = a p-adic ring, R = Z/n (ha you can tell I do number theory!), point out that forming GL is functorial, discuss the relation to lattices and who knows what else. The principle I work by in these cases is that: (1) someone who doesn't know much doesn't get scared away, and (2) someone who knows more is experienced enough to know that the good stuff is a few pages down. Maybe you want to start a new article, like General linear group (ring theory), and think of this one as General linear group (linear algebra)? Dmharvey 23:29, 4 March 2006 (UTC)

 * I'd start with not mentioning either rings or fields, but just by saying "the general linear group of degree n, written as GL(n), is the group of n×n invertible matrices" (perhaps don't even say that it's a group). Then make another section with a precise definition, where you mention that the fields of entries needs to be specified, and that it can also be a ring. -- Jitse Niesen (talk) 00:18, 5 March 2006 (UTC)

 * Jitse: If you had said "the rings of entries need to be specified" then I might agree. Dmharvey: I think it's much more clear to say "an important special case is fields" than "an important generalization is rings". If you say the latter then people will wonder what else the generalization consists of, that it was too difficult to introduce at the beginning. You'll have to add some explanation like "The generalization is actually trivial; we could have been using the word 'ring' all this time, but we didn't want to intimidate you." just to make it clear what's going on...

 * Also, since I've never gotten an answer: (a) why are rings argued to be so much more "intimidating" than fields - and especially, why do we think this is the case for people who have some reason to view this article; (b) why stop at fields, why not give our objects some other irrelevant properties like uncountability or completeness, just in case some readers are really really timid? It's not like someone will have heard of finite fields if they haven't heard of rings. A5 02:38, 5 March 2006 (UTC)

For what it's worth, I think I like Jitse's solution, with A5's caveats. Trying not to scare someone by avoiding the mention of "rings" in an article like this just seems silly. And I really hate the idea of lying to someone (telling them it must be field) rather than not telling them the whole story (that fields are 99+% of the usage). wnoise 03:03, 5 March 2006 (UTC)

 * Linear algebra is done over fields because it is easier, more important, and better understood. Show me an undergraduate linear algebra class that defines modules first then gives vector spaces as a special case. It would be perfectly logical. It has never been established Wikipedia practice to start an article with the most general definition. But by all means, lets have a section on the general linear group over rings. Those who understand modules vs. vector spaces, and rings vs. fields aren't going to be confused by this layout. -- Fropuff 03:09, 5 March 2006 (UTC)

 * Show me an undergraduate linear algebra class that defines modules first then gives vector spaces as a special case. It's not really necessary to define modules in a class about vector spaces, that's why they are not defined. But when vector spaces and modules are defined in the same class, in my experience modules always come first. A5 03:29, 5 March 2006 (UTC)

 * Let me expand on my previous point. We like to do fields first because of the connection with vector spaces. GL(n,K) can be identified with the set of all invertible linear transformations of a n-dimensional vector space over K. What about GL(n,R)? It is the set of all invertible linear transformations of a rank n free (right?) R-module. Far more people are going to scratch their heads over this statement as compared to the vector space version. -- Fropuff 03:17, 5 March 2006 (UTC)

 * For a ring R with identity, GL(n, R) is the group of invertible n by n matrices over R. This is the definition in Lang's Algebra, Mac Lane's "Category Theory...", and Mathworld. A5 03:29, 5 March 2006 (UTC)

(via several edit conflicts :-) Concerning (a). Yes, rings are "more intimidating" than fields in this context. Just because they take less axioms to define does not make them easier to understand. The group GL2(Z) is *much* harder to understand than the group GL2(R). For much the same reason that it's easier to learn about varieties over a field than about general schemes; for the same reason that the theory of eigenvalues/vectors is easier over an algebraically closed field than is the theory of decomposition of modules over an arbitrary ring; for the same reason that this article currently restricts itself to finite-dimensional vector spaces; for the same reason that the derivative article only considers the high-school version of the derivative and leaves more general ideas to other articles.

Let me try to be more specific. I think at least the first part of this article should be accessible to a first-year undergraduate student. Such a student almost certainly doesn't even know what a ring is, let alone a field. They may or may not have seen complex numbers before. What they know is that "numbers" can be added, subtracted, multiplied and divided (that's right, divided). Their knowledge of linear algebra is limited to solving two-by-two systems of linear equations, and it has never occurred to them that the division you need to perform during that calculation is anything special, because for them 7/4 means exactly the same thing as 1.75. Perhaps they got led to this article because they've just started learning about matrices and such things, and it's already a giant leap to collect together all of the invertible matrices into one object and give it a name. For such a person, discussing general linear groups over an arbitrary ring will in actual fact cause them to stop reading and go elsewhere.

Anyway, the question is, apart from a modification to the definition, what else do you want to add to the article? Perhaps you could start a draft on a subpage, listing some of the facts you think should be included, and then when we have a better idea of where the article could be going, we can do a merge. Dmharvey 03:27, 5 March 2006 (UTC)

 * I'm not willing to do that. I've wasted far too much time on this already. I think that your priorities are very misplaced. There are probably 2 people in the entire world who (1) know what a field is and what a group is but not what a ring is; and (2) are interested in learning about GL; and (3) are not willing to read through a sentence containing the mysterious word "ring" to get to the following sentence containing the more familiar "field" and R and C in order to do so. And when I add, who (4) are worth obfuscating the article for; it drops to 0 people. A5 03:40, 5 March 2006 (UTC)

 * Well I've tried your suggestion. How does it look to you? Dmharvey 04:01, 5 March 2006 (UTC)

 * Dmharvey: It's OK with me, actually. It's very pedantic, but one doesn't have to read far for the general definition, which was my complaint about the earlier proposals. A5 15:09, 5 March 2006 (UTC)

You don't need to know about rings and modules (or fields and vector spaces) to play around with SL2(Z) and lattices.--gwaihir 14:00, 5 March 2006 (UTC)

 * Current introduction looks good to me. Start with the easiest to explain example, then give the full generalisation, highlighting the importance of the Real and Complex cases. There could be a need for a section on the General linear group over a ring. There seems to be enough material in this discussion for this.

 * Also not sure about ordering of sections, seems like Special linear group and Other subgroups naturally follow on from the As a Lie group section. Maybe its better to move Over finite fields below these (are there interesting examples of these subgroups over finite fields or rings?). --Salix alba (talk) 13:29, 6 March 2006 (UTC)

 * FWIW, I agree with majority opinion here. When I was an undergrad, I had no no clue what a field or a ring was, and yet I was required to know a lot about GL_n and its subgroups. Like almost all physics students, I had a pretty good idea of what Lie groups were long before I'd heard of a module, and long before I stopped being scared of rings. One should assume that most readers will have the same general experience. linas 15:22, 6 March 2006 (UTC)

 * How about an analogy: many people have poor eyesight, and the default browser font size is too small for them. But these people can always make the font bigger. Sure, some people will be too lazy to do this, or maybe they won't know how - but although we shouldn't make it purposefully difficult for these people, their potential inability or unwillingness to configure their browsers is not a reason for Wikipedia to have a huge font by default.

 * Similarly, many people won't know what a ring is, and they'll be confused if a definition mentions it. To be sure, we shouldn't go out of our way to make things difficult for these people - we can say that a field is a special case of a ring, and that R is a special case of a field (and we have done so, and I'm happy with the result). Yet, as pointed out, this will not be enough for some users - perhaps they'll have such poor reading comprehension or poor motivation, that the first new word they encounter will make them give up, or lose interest.

 * But that's a personal issue which such users can and should deal with themselves - as long as they haven't addressed it, anything we do to try to work around it will only be marginally helpful to them. It certainly isn't a reason for Wikipedia to water itself down - and this is where my position differs from that of Charles Matthews, Fropuff, Dmharvey, etc. Aside from the fact that it would make Wikipedia useless as a reference work for experts, oversimplification also leads to misunderstandings and confusion among people such as myself who are still learning the field. Just as with the font size analogy, a policy of oversimplification would insult the vast majority of users, and an even larger fraction of contributors, for an extremely small gain. A5 15:16, 7 March 2006 (UTC)

 * We are not trying to "water things down". We're just trying to put things in an order which will be useful to audiences with a variety of backgrounds. Have you considered that the particular level of sophistication that you expect of the introduction to this article might be related to your own level of mathematical experience? By the way, if you don't mind me asking, what kind of mathematical background do you have? Dmharvey 15:44, 7 March 2006 (UTC)

 * Dmharvey: I don't think that in the above paragraph you are really confronting the points of the argument that I made. I feel that the things you say have been said and responded to already. Obviously I don't think that "watering down" is your primary motive. A5 02:03, 9 March 2006 (UTC)

 * In that case I don't think I understand your argument. Are you suggesting that the opening definition should always be the most general possible? Dmharvey 02:27, 9 March 2006 (UTC)

 * Yes, within reason. Note that, way at the top of this discussion, people such as Charles Matthews have argued that rings shouldn't be mentioned at all in this article; this I strongly disagree with - but I also disagreed with a suggestion that they go into a separate section. If there is a "standard" definition, then the introduction should give it; if there are multiple common definitions, then the introduction should mention them all. A few people seemed to suggest that giving general definitions in the introduction is a bad idea because some users are frightened or confused by the sight of things which they don't know yet - the argument seemed to be that we should help them out by omitting or secluding potentially frightening pieces of information. In my penultimate comment, I made an analogy between this group of users and another group of users, with the purpose of illustrating why "hiding" information for their benefit is the wrong policy in an encyclopedia. Clearly I am not opposed to making articles more accessible, but I think that this should be done by adding new information, not by obscuring or by oversimplifying. Also, you seem to be stuck on the idea that this is just about levels of education - it isn't. As I have tried to argue before, people who have good reading comprehension will have no trouble skipping over the mention of "ring", whether or not they know what a ring is, when it is pointed out to them that something they are familiar with is a type of ring. If we hide information, we do so not on behalf of those who are uneducated, but on behalf of those who are uneducated and have poor reading skills - a small population, who would be better assisted in their development by a personal instructor. Please let me know if I have misunderstood your position, as you seem to have taken my comment personally. A5 04:24, 9 March 2006 (UTC)

 * First, don't worry, none of this is taken personally. It's actually a very interesting discussion, and I think this is one of the deepest issues we face in working on wikipedia, which is why I'm devoting so much time to discussing it :-)
 * I still don't buy your argument. Why don't we start by saying "Given a vector space V over a field K, the general linear group GL(V) is the group of automorphisms of V", which frees us from the restriction of finite dimensionality (and incidentally from some non-canonical choice of basis)? How about: "Given a Banach space V over a field K, the general linear group GL(V) is the group of continuous automorphisms of V". Do we need to cover this case in the introduction too? Where do we stop? What is your basis for the (implied) claim that the definition in terms of matrices over a general ring (commutative? associative? unital?) is the "most common" or "standard"? Dmharvey 13:12, 9 March 2006 (UTC)

 * What is your basis for the (implied) claim that the definition in terms of matrices over a general ring (commutative? associative? unital?) is the "most common" or "standard"? First of all, it includes the definition over a field. The definition over a field doesn't actually make use of the field structure - it is no different from the definition over a ring - so it seems silly to say "field" instead of "ring". That's a very small and natural change. Furthermore, there are many uses of GL(n,R), even on Wikipedia. Finally, the ring definition was the only one I found when consulting several sources. All this has been said before. I don't know about these other ideas you have brought up, but they seem like straw men. A5 23:25, 9 March 2006 (UTC)

 * Serre's Linear Representations of Finite Groups begins by defining GL(V), for an arbitrary vector space V over C, as the group of isomorphisms from V to itself (as I suggested above, but for a general field). Later he specialises to the case of a finite-dimensional V, and points out that in this case you can define the group in terms of matrices. Do you think perhaps we are misleading our readers by not including this non-finite-dimensional case in the introduction? Why is this a "straw man"? Dmharvey 00:38, 10 March 2006 (UTC)

 * There is a big difference between a "generalization" that involves removing an unnecessary condition, and a generalization that involves a fundamental change of notation/perspective. A lot of the arguments that I made for GLn(F) vs. GLn(R) don't apply to GLn(F) vs. GL(V), which is why I suggested it was a straw man. But since GL(V) is discussed immediately following the introduction, I don't think it is necessary to mention it; otherwise I think it would be. Alternatively, the first section could be moved into the introduction, since it is short and defines a notation. As for continuous transformations in a Banach space, shouldn't it be enough to mention in the Banach space article that automorphisms are assumed to be continuous? A5 19:08, 10 March 2006 (UTC)

 * Hang on a second. When going from GL(n, F) to GL(n, R), we're basically dropping the field axiom that says that elements have to have multiplicative inverses. When going from GL(n, F) to GL(V), we're dropping the assumption that the vector space has to have finitely many dimensions. Can you explain why, as you seem to be suggesting, that dropping the first assumption is "removing an uneccessary condition", whereas dropping the second assumption "involves a fundamental change of notation/perspective"? (If it would make you feel more comfortable, let's write GL(V) as GL(n, F) where n is allowed to be an infinite cardinal; or for emphasis, perhaps I should say "a cardinal without the unnecessary restriction of finiteness".) Dmharvey 20:47, 10 March 2006 (UTC)

 * I don't know where we're going with this since GL(V) is already mentioned in the article and I said that I think it's an important definition. But from the article: If the dimension of V is n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Once a basis has been chosen, every automorphism of V can be represented as an invertible n by n matrix, which establishes the isomorphism. I think this "the isomorphism is not canonical" makes clear how GL(V) is a different viewpoint/notation. Also, I don't think that it is as simple as defining GL(V) as GL(n,F) for some possibly infinite cardinal n. I'm not an expert, but I think bases are usually defined to be things such that every element of the space can be expressed as a finite linear combination of the basis elements. So does that mean that your infinite matrices can only have finitely many non-zero elements in each row? What about each column? This isn't covered in the matrix article. And how do I go about choosing a canonical basis for, e.g., the space of sequences N->R (which can have infinitely many non-zero entries)? It raises a lot of issues. Maybe you can unify the two concepts but I hope you agree that doing so isn't as trivial as saying "hey this multiplicative inverse condition isn't really necessary". A5 17:53, 12 March 2006 (UTC)

The article intro looks more or less okay at the moment. I just wanted to add a comment in support of A5. Defining GLn(R) for an arbitrary (associative and unital) ring is perfectly natural and sensible and hardly more difficult or less accessible in any serious way than defining it for a field. It is probably safe to say that students interested in learning about GLn who have no idea what a ring is are due to spend ten minutes looking at the definition, seeing as how they've entered ' the algebra zone '. Of course, it's still good to introduce the concept via a well-known and important special case, as the article does. Incidentally, for an example of an undergraduate text that does define modules first and then introduces vector spaces as a special case (all in a quite sensible way), see Elements of Abstract and Linear Algebra. — merge 12:17, 8 March 2006 (UTC)

 * Do not confuse a pedagogical technique with an encylopeadia. If you have a class-full of undergrad math majors who signed up for a first course on algebra, that would be a great way to do it. If you have a someone who is *not* a math major, say, for example, an engineering student, who doesn't know what GL_n is, and needed to look it up on wikipedia, is self-motivated enough to go look it up, I think its safe to assume that they will have a rather poor math background. Don't screw over somebody who is motivated and wants to learn, by throwing "rings" at them early on. Learning what a ring is is a *lot* harder than changing the size of your fonts on the browser! linas 03:24, 9 March 2006 (UTC)

 * Linas: Please read my above reply to Dmharvey, just posted. It may make what I am saying clearer; you seem to be misinterpreting the analogy. A5 04:28, 9 March 2006 (UTC)

 * I am following your conversation with dmharvey. I think dmharvey is right. There are many strong advantages to a definition having GL(V) with V either a hilbert space or a banach space. Its easier to learn what a hilbert space is than to learn what a ring is; most students will know hilbert spaces pretty well long before before they hear of a ring. However, even that is too high a hurdle.

 * Look, why don't you read the conversation below, about determinants. I think it makes clear the pitfalls of starting with a ring-based definition. So, does the naive reader have to keep in mind that some of the results, but not all, hold only for commutative rings? Maybe some of the results and facts  hold only for integral domains. Which ones? Does the reader have to learn what a domain is before they can grok what GL(R^n) is?  How much of ring theory is needed? Does one need to know about ideals, or prime ideals?  If not, why not? linas 01:15, 10 March 2006 (UTC)

 * I wasn't the one who suggested that we generalize to non-commutative rings, I merely asked what the correct thing to do should be. I'm not even sure what the "standard" definition for matrix multiplication for matrices over non-commutative rings is. The person who extended the generalization to non-commutative rings was Charles Matthews. I think we should try to do it if it makes sense, but not if it doesn't. The definition of determinant is the same over a commutative ring and a field. So, does the naive reader have to keep in mind that some of the results, but not all, hold only for commutative rings? The naive reader should be expected to have the ability / common sense / motivation to ignore things which we indicate he can ignore, and not get derailed. If a result doesn't hold for rings, we should just say so. My suggestion for the current state of the article might be to go back to "commutative ring" in the intro and have a reference elsewhere (to a subsection, or another article) for non-commutative rings. Obviously generalizations have to be evaluated based on the complexity they add to the presentation. If they are in common use, then they will have to be included in Wikipedia at some point, the question is whether they overlap enough with what is in this article to be made a part of it, or whether they should be treated separately. But in no case should "it might scare the uneducated" be a reason to avoid mentioning the existence of something. A5 19:08, 10 March 2006 (UTC)

Umm... I'm just a senior undergraduate, but I think the layout of the vector spaces article is helpful for me. It talks about generalizations such as modules in a subsection. Would it be such a bad idea for this article to discuss what happens in generalizations of GL in a subsection also? Anon 05:44, 3 September 2009 (UTC)

What kind of rings?
We say "commutative ring", Mac Lane discusses GLn as a functor on commutative rings, Lang defines GL in a chapter dealing with commutative rings; but Mathworld only says "ring with identity". Obviously an identity is necessary. Is commutativity necessary (for a sensible definition of matrix multiplication)? Do we care? I don't have enough background to say... A5 15:09, 5 March 2006 (UTC)

 * I think Charles removed the "commutative" qualifier from the article. Where can I find a definition of invertible matrix for noncommutative rings? A5 15:19, 7 March 2006 (UTC)

Above. Like I said before, it would be the unit group in the matrix ring. Charles Matthews 15:50, 7 March 2006 (UTC)
 * Maybe it should be mentioned that the definition of SLn cannot be carried over in a similar way.--gwaihir 16:13, 7 March 2006 (UTC)

 * Can one even make sense of a determinant over a non-commutative ring? Dmharvey 16:29, 7 March 2006 (UTC)

 * Not really. To define the group SL(n,H) one embeds GL(n,H) in GL(4n,R) and uses the real determinant. -- Fropuff 17:42, 7 March 2006 (UTC)
 * But the same construction for C doesn't give SLn(C), so this is apparently not a generalization, but a different concept. (It is unclear to me how this should be done for a general ring, anyway.)--gwaihir 23:22, 7 March 2006 (UTC)

While this is all very interesting, it is seeming a bit beyond the scope of this article, maybe we need a seperate article General linear group over a ring, or possible expand the stub Matrix ring? --Salix alba (talk) 15:00, 9 March 2006 (UTC)

 * I wouldn't worry too much about it for now. So far we have zero content in this article pertaining to rings. If the material grows too large we can splice it off. I would very much like to see the stub at matrix ring expanded. Any takers? -- Fropuff 05:23, 10 March 2006 (UTC)

Warming to A5's position
What A5 is saying is making more and more sense. My argument appears to be going down in flames. I might go crazy with a rewrite. Let's see.... Dmharvey 19:36, 12 March 2006 (UTC)

 * Well, I'll be darned... Thank you, I've never had an argument end that way before. :) In any case, I quite like the new version. A5 20:53, 15 March 2006 (UTC)

Proof that SL(n,C) is simply connected?
Anyone know a reference for this?

 * This is essentially the same result as that the special unitary group is simply connected (which is mentioned in that article). That is because the two groups have the same homotopy type (which can be proved by Gram-Schmidt or the Iwasawa decomposition). By the homotopy long exact sequence this is the same as saying the unitary groups have fundamental group that is infinite cyclic. So we have all three results on the topology standing or falling together. I suspect that the special unitary case is easiest to prove from scratch, using induction and the action on complex projective spaces. Charles Matthews 18:03, 15 May 2006 (UTC)
 * To show that SU(n) is simply connected, use the fiber bundle SU(n-1) -> SU(n) -> S^(2n-1) gotten by looking at where a fixed vector goes. Then use the fact that F -> E -> B implies πn(E) = πn(F) x πn(B), that π1(S^{2n-1}) = 1 for n > 1, and induction. The base case is SU(1), which is a point; then SU(2) = S3, etc. [Following Hatcher.] Tesseran 05:09, 31 May 2007 (UTC)

As an aside, isn't SL(n, R) often also called SO(n,R)?

 * No. SO(n,R) is another group, the special orthogonal group. -- Jitse Niesen (talk) 09:02, 25 June 2006 (UTC)

Semidirect product structure
I've never edited a talk page before and have no idea where to put this. However, I noticed that someone claims GL_{n}(F) is the semidirect product of F with GL_{n}(F). While this is true in fields F where every element has a square root (also in R and probably many other fields, eg ordered fields where positive elements have square roots), I believe it isn't true in general for n \geq 2. In particular it isn't true for n = 2, F being the finite field with say 13 elements. In that case, GL_{n}(F) is (I think) the direct product along with a (nontrivial) group extension by Z_{2} (there should only be one of those possible). I'm sorry for not editing in the 'standard' way. I'm a long-time wikipedia user but have never been involved in article creation. My apologies as well if I in fact made an error here. —Preceding unsigned comment added by 171.64.38.45 (talk) 17:30, 24 October 2007 (UTC)

 * No problem. I added a heading to separate it from the previous discussion and removed a stray newline. I think the second time you mention GL, you mean SL, but I didn't want to change your text.

 * I can confirm that GL(2,13) is definitely a semidirect product of SL(2,13) and a subgroup D of GL(2,13) isomorphic to the group of units of the field with 13 elements. Specifically D is all diagonal matrices whose 1,1 entry is nonzero and whose 2,2 entry is 1.  This group has 13-1=12 elements, and multiplication is component-wise, so isomorphic to the multiplicative group of the field.  The intersection of D and SL(2,13) is all those diagonal matrices diag(d,1) whose determinant is 1, but the determinant of diag(d,1) is d*1=d, which is 1 if and only if d=1.  In other words, the intersection of D and SL(2,13) is the identity subgroup.  Furthermore, SL(2,13) is normal, being a kernel, and the order of GL(2,13) is the product of the order of D and the order of SL(2,13).  This can be verified directly, or by using the determinant homomorphism. Here is a GAP session to show a computer verification, if you want to test similar examples, though the definition of D and the verification that GL = D |x SL is the same for any finite field.  For an infinite field, you want to use the homomorphism idea more than the "just check the sizes" idea.
 * gap> G:=GL(2,13);
 * GL(2,13)
 * gap> S:=SL(2,13);
 * SL(2,13)
 * gap> D:=Group(Z(13),0],[0,1*One(GF(13)));
 * Group([ [ [ Z(13), 0*Z(13) ], [ 0*Z(13), Z(13)^0 ] ] ])
 * gap> Intersection(D,S);
 *
 * gap> IsNormal(G,S);
 * true
 * gap> Size(D)*Size(S)=Size(G);
 * true
 * That's enough to verify the definition for a finite group. JackSchmidt 18:15, 24 October 2007 (UTC)

C*
Since in almost every articles of research in mathematics the product group of C is denoted with exponent the star, I suggest to follow this notation, even if it should be less "naturally" than the notation with the times "x". Emc2fred83 (talk)

Preface contradicts first section
In the preface, the first part of the entire article, it says at the beggining "In mathematics, the general linear group of degree n is the set of n×n invertible matrices" And then in the first section called "General linear group of a vector space" it says this statement that obviously contradicts the former quoted, because matrix is different from linear transformation: "If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V" This is very confusing. I know the relationship between matrix and linear operators, but we need a better definition.Santropedro1 (talk) 18:34, 23 September 2015 (UTC)Santropedro

Topology of the general linear group
It would be great for this article to have a section on the topology of GL(n,R), the way the article on the special linear group does. — Preceding unsigned comment added by 90.207.21.248 (talk) 22:58, 20 June 2019 (UTC)

A section on history and application of the general linear group (and special linear group)
There is already a short remark on the history of the concept, stating that Galois defined the general linear group when investigating the Galois group of an equation. I think the article would benefit from extending this section (and maybe adding some remarks about the usefulness und applications of the concept). In particular, why is the general (special) linear group interesting? For what do we use it today? Why exactly was it introduced by Galois, why is it helpful when investigating the Galois group of an equation? Zaunlen (talk) 14:46, 10 November 2019 (UTC)

 * Go ahead and add what you see fit! As a general remark I think it is important to stick to reliable sources when it comes to mathematical history. For example, I would like to see a reference for the assertaion that Galois defined the general linear group if this assertion appears in the article. Jakob.scholbach (talk) 11:04, 11 November 2019 (UTC)

 * I myself don't have the knowledge to write this section, it just was a suggestion that maybe someone else can execute. Zaunlen (talk) 19:23, 14 November 2019 (UTC)

What about Linear group?
Should there really be two distinct articles, General linear group and Linear group? Both articles are about the general linear group and its subgroups. Should the other article be merged into this one? Sylvain Ribault (talk) 21:23, 1 December 2020 (UTC)

Invertible in R is not always invertible in F
The article mentions GL(3,2) as an example. Its elements are the 168 invertible 3×3 matrices with elements in F2. It may not be obvious that whether or not a matrix is invertible depends on the field. The binary matrix shown on the right is invertible, but not part of the group. I suppose that one would describe it as "invertible in R" but "singular in F2". Maybe this should be clarified in the article. Watchduck (quack) 12:20, 30 August 2022 (UTC)

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

title: Talk:General linear group


text:
Caravaggio (disambiguation)

Caravaggio (1571–1610) was a painter, famous for his dramatic use of lighting.

Caravaggio may also refer to:
 * Caravaggio, Lombardy, Italy, a comune in the province of Bergamo
 * Polidoro da Caravaggio (c. 1499 – 1543), painter best known for his now-vanished paintings on the facades of Roman houses (unrelated to the above)
 * Caravaggio (1941 film), an Italian film directed by Goffredo Alessandrini
 * Caravaggio (1986 film), a British film about the painter directed by Derek Jarman
 * Caravaggio (2007 film), a 2007 Italian television film
 * Caravaggio (restaurant), a New York City Italian restaurant
 * Caravaggio (horse), thoroughbred racehorse, winner of the 2017 Commonwealth Cup
 * Caravaggio (train), an electric train built by Hitachi Rail for Italian railways
 * "Caravaggio", a song by Claudia Faniello for the Malta Song for Europe 2008
 * David Caravaggio, a character from the novels In the Skin of a Lion and The English Patient
 * Caravaggio, a character in the Canadian science fiction television series Starhunter and Starhunter 2300
 * Caravaggio, a ballet by Italian choreographer Mauro Bigonzetti, with music by Bruno Moretti

title: Caravaggio (disambiguation)


text:
Tintin and Alph-Art

Tintin and Alph-Art (Tintin et l'Alph-Art) is the unfinished twenty-fourth and final volume of The Adventures of Tintin, the comics series by Belgian cartoonist Hergé. Left incomplete on Hergé's death, the manuscript was posthumously published in 1986. The story revolves around Brussels' modern art scene, where the young reporter Tintin discovers that a local art dealer has been murdered. Investigating further, he encounters a conspiracy of art forgery, masterminded by a religious teacher named Endaddine Akass.

Reflecting his own fascination for modern art, Hergé began work on Tintin and Alph-Art in 1978. However, it was left unfinished at the time of his death in March 1983. At this point it consisted of around 150 pages of pencil-drawn notes, outlines and sketches – not yet rendered in Hergé's trademark ligne claire drawing style – with no ending having been devised for the story. Hergé's colleague Bob de Moor offered to complete the story for publication, and while Hergé's widow Fanny Vlamynck initially agreed, she changed her decision, citing the fact that her late husband had not wanted anyone else to continue The Adventures of Tintin.

A selection of the original notes were collected together and published in book form by Casterman in 1986. Since that point, several other cartoonists, such as Yves Rodier, have produced their own finished, unauthorized versions of the story. Critical reception of the work has been mixed; some commentators on The Adventures of Tintin believe that if Tintin and Alph-Art had been completed, it would have been an improvement over the previous two volumes, while others have characterised such assessments as wishful thinking.

Synopsis
Tintin and Captain Haddock receive a phone call from their friend, the opera singer Bianca Castafiore, who informs them about a new spiritual leader whom she has begun following, Endaddine Akass, stating her intention to stay at his villa in Ischia. Later that day, Haddock enters the Fourcart Gallery in Brussels, where Jamaican avant-garde artist Ramó Nash convinces him to purchase one of his "Alph-Art" works, a perspex letter "H" ("Personalph-Art"). The gallery's owner, Henri Fourcart, arranges to meet Tintin, but is killed in a car accident while on his way to do so. Tintin begins to investigate, discovering that Fourcart's death was murder. Tintin and Haddock attend one of Akass' lectures; there, Tintin recognises Akass' voice, but is unsure why. Investigating further, he concludes that Akass was spying on Fourcart through a micro-transmitter hidden in a pendant worn by the latter's assistant, Martine Vandezande. Tintin soon faces a number of attacks designed to kill him, but survives both.

To further his investigations, Tintin decides to visit Akass' villa. Arriving at Ischia, Tintin and Haddock receive death threats warning them to leave. When Castafiore hears that they are on the island, she welcomes them to Akass' villa, informing them that Akass himself is in Rome. At the villa, they meet a number of acquaintances, the corrupt industrialists W.R. Gibbons (from The Blue Lotus) and R.W. Trickler (from The Broken Ear), Emir Ben Kalish Ezab (from Land of Black Gold), Luigi Randazzo (a singer), and Ramó Nash. Tintin and Haddock stay the night at the villa on Castafiore's insistence. During the night, Tintin witnesses men loading canvases into a van, and exploring the villa discovers a room full of faked paintings by prominent artists. He is discovered by Akass, who informs him that he uses Nash's "Alph-Art" as a front for his criminal forgery business. Akass declares his intention to kill Tintin by having him covered in liquid polyester and sold as a work of art by César Baldaccini.

Background
In 1976, a few months after the publication of Tintin and the Picaros, Hergé told the journalist Numa Sadoul that he was contemplating setting the next Adventure of Tintin in an airport lounge. This idea was dropped, and in 1978, he decided to set the story in the world of modern art. During his later years Hergé had grown increasingly interested in this subject, and decided to incorporate his love of avant-garde artwork into the new story. Being a regular at Brussels' art galleries and exhibitions, he was able to draw directly upon his experiences in the art scene when producing the story. Given his age, and the length of time that it was taking him to produce each Adventure, British literary critic Michael Farr suggested that Hergé likely knew that this would be his final installment in the series.

Hergé initially titled this new work Tintin et les Faussaires ("Tintin and the Forgers") before changing this working title to Tintin et L'Alph-Art. The story's main antagonist, Endaddine Akass, was based on a real-life art forger, Fernand Legros, whom Hergé had learned about through reading a biography of him. The Akass character was also influenced by an article about the Indian guru Rajneesh which Hergé had read in a December 1982 edition of the Paris Match. In Hergé's notes, he included the idea of revealing that Akass was Tintin's nemesis, Rastapopoulos, in disguise, with Farr believing that this plot twist would have been revealed had Hergé completed the story. Conversely, Harry Thompson suggested that Hergé had dropped Rastapopoulos from the story in 1980. Hergé's notes reveal that he considered various different names for the artist character, including Ramon Hasj and Ramo Nasj, before settling on Ramo Nash. Similarly, he had initially debated whether Nash should be a painter or a sculptor, before deciding on the latter.

Three months before he died, Hergé stated that "unfortunately I cannot say much about this forthcoming Tintin adventure because, though I started it three years ago, I have not had much time to work on it and still do not know how it will turn out. I know very roughly where I am going... I am continuing my research and I really do not know where this story will lead me". Upon his death, Hergé left around one hundred and fifty pages of pencil sketches for the story. The story itself had no ending, and is left on a cliffhanger, with Tintin's fate left unexplained.

Hergé's main assistant, Bob de Moor, showed an interest in completing the book following Hergé's death. As De Moor explained, "Personally I would have loved to finish Alph-Art. It would have been a tribute to Hergé. Fanny Remi asked me to finish it, and I began work on it, but after a few months she changed her mind. I didn't insist, but for me it was logical that there was a studio, there were artists in the studio, Casterman asked for it to be finished, there were twenty-three finished books, that one story was not finished; so I had to finish it". In the end, Fanny decided Hergé would not have approved and the book must remain unfinished. Hergé had made it clear that he did not want anyone else to continue The Adventures of Tintin after his death, informing Sadoul that "after me there will be no more Tintin. Tintin is my creation—my blood, my sweat, my guts".

Publication
Fanny ultimately decided that Hergé's unfinished sketches could be published in book form. To do so, the sketches were edited by a team of experts, including Benoît Peeters, Michel Bareau and Jean-Manuel Duvivier, with forty-four being selected for publication. The book produced devoted one half to reproducing the sketches, and the other to producing a transcript of Hergé's text for the story. Published on 8 October 1986 by Casterman, it proved to be a bestseller despite its high cover price. An English language translation was published in 1990.

Critical analysis
For Michael Farr, Tintin and Alph-Art provided "an almost perfect ending to more than fifty years of defying danger, threats to his life and a succession of villains". He believed that it was "full of a vigour and enthusiasm disappointingly absent from the two previous adventures", and that as a result "it promised to be Hergé's most accomplished Tintin story for twenty years". Jean-Marc Lofficier and Randy Lofficier stated that had the story been completed, it "may have turned out to be a smaller-scale, unpretentious yet far more exciting and true-to-life adventure" than various recent volumes. Hergé biographer Benoît Peeters felt that "despite its limitations, and perhaps on account of them, this unfinished story fits perfectly alongside the other 23 Tintin adventures. Full of promise, it ends the series admirably by leaving the last word to every reader's imagination". Philippe Goddin opined that Tintin and Alph-Art's opening "promised much, coming like new breath".

In a later publication, Peeters expressed a different view, stating that "one cannot help but feel disappointed" with Alph-Art. Similarly, Harry Thompson expressed the view that Hergé would never have completed the story, and that while "many critics like to think... Tintin and Alph-Art was set to be Hergé's last great masterpiece", he disagreed, describing this as "surely just wishful thinking". He added that it could be seen as "a nostalgic, or regressive book, depending on your point of view".

The literary critic Tom McCarthy suggested that some of the early scenes of the book "take the form almost of Platonic dialogues, dramatized theoretical discussions about art itself", although later in the book Hergé turns to revealing "the deeper and more essential Platonic truth of art: that it is fake, that its whole currency is fakeness". Discussing the final scene in the book, he opined that it "reads like a snuff movie: not of Tintin, ultimately, but of Hergé". Further, he argued that Tintin and Alph-Art reflected a number of themes that also appear in other Adventures of Tintin; these include the idea of Castafiore bridging the connection between the protagonist and the villains, and the concept of Haddock being confronted "with his own inauthenticity", in this case through purchasing a giant plastic 'H' artwork.

Adaptations and exhibitions

Other artists have produced a number of pastiche versions of the story, which provide their own endings. In 1987, a version was completed by an artist using the pseudonym of Ramó Nash. Several further versions were produced by the Canadian artist Yves Rodier during the 1990s. A further version appeared in the 1990s, produced by an individual known as Regric.

In honour of Hergé's legacy, the awards given at the Angoulême International Comics Festival were named Alph-Art Awards, having previously been named after Alain Saint-Ogan's character Alfred.

In November 2015, Somerset House in London held the exhibition "Tintin: Hergé's Masterpiece", which featured the final sketch from Tintin and Alph-Art as one of its displays.

title: Tintin and Alph-Art


text:
User:Zingarello

20-10-2002, created.

title: User:Zingarello


text:
Inbetweening

Inbetweening, also known as tweening, is a process in animation that involves creating intermediate frames, called inbetweens, between two keyframes. The intended result is to create the illusion of movement by smoothly transitioning one image into another.

Traditional animation
Traditional inbetweening involves the use of a light table to draw a set of pencil and paper drawings.

The process of inbetweening in traditional animation starts with a primary artist, who draws key frames to define movement. After the testing and approval of a rough animation, the scene is passed down to assistants, who perform clean-up and add necessary inbetweening. In large studios, assistants usually add breakdowns, which define the movement in more detail. The scene is then passed down to another assistant, the inbetweener, who completes the animation. In small animation teams, animators will often carry out the full inbetweening process themselves.

Dick Huemer developed this system in the 1920s, and it has become widely used due to its efficiency. Art Davis is said to be the first Inbetweener.

Frame frequency
Typically, the key animator does not make drawings for all 24 frames required for one second of film length. In large studios, a specialized inbetweener artist fills in the gaps between the key drawings. Only very fast movements require 24 drawings per second, which is referred to as animating "on ones". Most movements can be done with 12 drawings per second—called animating "on twos", drawing one out of every two frames. When the number of in-betweens is too few, such as four drawings per second, an animation may begin to lose the illusion of the movement altogether. Computer-generated animation is usually animated "on ones." Frame frequency often varies depending on animation style and is an artistic choice. Animation "on twos" has been used for over 100 years; Fantasmagorie (1908), widely considered the first fully animated movie, was animated on twos.

Modern animation uses various techniques to adapt frame rates. Slow movements may be animated on threes or fours. Different components of a shot might be animated at different frame ratesfor example, a character in a panning shot might be animated "on twos", while everything in the shot is shifted every frame ("on ones") to accomplish a panning effect. Optical effects such as motion blur may be used to simulate the appearance of a higher frame rate.

Digital animation
When animating in a digital context, the shortened term tweening is commonly used, and the resulting sequence of frames is called a tween. Sophisticated animation software enables the animator to specify objects in an image and to define how they should move and change during the tweening process. The software may be used to manually render or adjust transitional frames by hand or may be used to automatically render transitional frames using interpolation of graphic parameters.

Some of the earliest software that utilises automatic interpolation in the realm of digital animation includes Adobe Flash and Animo (developed by Cambridge Animation Systems) in the late 90s, and Tweenmaker, released around 2006. The free software program Synfig specializes in automated tweening.

"Ease-in" and "ease-out" in digital animation typically refer to a mechanism for defining the physics of the transition between two animation states, i.e., the linearity of a tween. For example, an ease-in transition would start the animation out slowly, and then progressively get faster as the animation continues. An ease-out transition does the opposite, where the animation starts out quickly but then slows down.

The use of computers for inbetweening was enhanced by Nestor Burtnyk and Marceli Wein at the National Research Council of Canada. They received a Technical Achievement Academy Award in 1997, for "pioneering work in the development of software techniques for computer-assisted keyframing for character animation".

title: Inbetweening


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Let's Take It to the Stage

Let's Take It to the Stage is the seventh album by American funk rock band Funkadelic. It was released in April 1975 on Westbound Records. The album charted at number 102 on the Billboard 200 and number 14 on the R&B Albums.

Music and lyrics
Let's Take It to the Stage is a funk rock album. The opening track, "Good to Your Earhole", features extensive guitar parts and a pronounced funk groove played by the rhythm section. According to Funkadelic bandleader George Clinton, the guitar solo for "Get Off Your Ass and Jam" was performed by "a smack addict" guitar player who had found his way into the studio. He asked Clinton if he could play for some cash and proceeded to play "like he was possessed", soloing over the entire duration of the song. He received $50 for his efforts without Clinton ever learning his name. In a July 2009 interview with Vintage Guitar, guitarist Paul Warren said that he played the solo.

The closing track "Atmosphere", which begins with a monologue by Clinton about "dicks and clits", appropriates an extended organ coda from Johann Sebastian Bach. "Get Off Your Ass and Jam" has been sampled on several hip hop hits, including Brand Nubian's "Slow Down", Public Enemy's "Bring the Noise", and N.W.A's "100 Miles and Runnin'".

Critical reception
Reviewing the album in 1975, Billboard magazine called it a collection of Funkadelic's "usual good mix of soul and jazz sounds, mixed in with singing and street raps", citing the title track and "Baby I Owe You Something Good" as highlights. In The Village Voice, Robert Christgau said Funkadelic finally does on record "what they've always promised to do in the hype—make the Ohio Players sound like the Mike Curb Congregation." Years later, he wrote in Christgau's Record Guide: Rock Albums of the Seventies (1981) that, despite the group's "disturbingly occultish bent", the music is "tough-minded, outlandish, very danceable, and finally, I think (and hope), liberating"; for Blender, he said it is their "tightest album ... all 10 tracks rock on."

In a retrospective review, AllMusic's Ned Raggett found Let's Take It to the Stage to be one of the band's most comical records with "more P-Funk all-time greats as well, making for a grand balance of the serious and silly." Sasha Frere-Jones, writing in The Rolling Stone Album Guide (2004), said it was "a summing-up of everything Funkadelic had done to date, and is still their most playable record." He felt that, although Clinton's "sexual politics weren't at their best" on tracks such as "No Head No Backstage Pass", the album is exemplary of the band's musicianship.

Track listing
"Good To Your Earhole" was sampled by Childish Gambino on the song "Riot".

Personnel

 * Vocals: 'Cool' Cal Simon, 'Bad Bosco' Bernie Worrell, C 'Boogie' Mosson, Garry 'Dowop' Shider
 * Bass Vocals: 'Sting' Ray Davis
 * Genie Vocals: 'Shady' Grady Thomas
 * Werewolf Vocals: Clarence "Fuzzy" Haskins
 * Maggot Overlord: George Clinton
 * Congas: Calvin Simon
 * Keyboards: Bernie Worrell
 * Bass: C Boogie Mosson
 * Percussion: R Tiki Fulwood
 * Guitar: Eddie Hazel, Michael Hampton, Garry Shider
 * Alumni Funkadelic: Bootsy Collins (vocals), Billy Bass, Eddie Hazel, Ron Bykowski
 * Guest Funkadelic: Paul Warren, Reggie McBride, Frosty [Barry Frost, drummer for Rare Earth], Mello Garcia, Honeys, Denise Hurd, Delores whats-her-name, Gary "Mudbone" Cooper, Parliament l, Mallia Franklin- uncredited

title: Let's Take It to the Stage


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Yiye Ávila

José Joaquín Ávila Portalatín (September 11, 1925 – June 28, 2013), better known as Yiye Ávila, was a Puerto Rican Pentecostal Evangelist and writer of Sephardic origins. His preaching and messages were characterized in proclaiming that the coming of Christ is imminent. He is considered one of the most influential Protestant preachers of the Spanish language. He was also the second runner-up for Mr. North America in 1953.

Early life
José Joaquín Ávila Portalatín was born in Camuy, Puerto Rico. His parents, Pablo Ávila and Herminia Portalatin, were both Sephardic Jew schoolteachers that had converted to Christianity, and he eventually followed in their footsteps. He enrolled at the Interamerican University of Puerto Rico San Germán campus, with plans to study medicine. There he graduated with a bachelor's degree in natural sciences and completed his pre-medical studies. He chose to return to his hometown of Camuy and teach chemistry and biology at the local high school, where he taught for the next 22 years. Eventually a change of careers would come about that would take him to over 100 countries.

Sporting life
He began playing baseball in the Double A level and training as a weight lifter. In 1952, he competed and won the Mr. Puerto Rico bodybuilding contest, and would go on to become the second runner-up for the title of Mr. North America in 1953.

While training for the summer Olympic Games of 1956, Ávila experienced terrible pain in parts of his body and, after various tests, was diagnosed with chronic arthritis by his doctors. This prevented him from participating in the Olympics or in any other future sporting event. The pain was so intense that Ávila suffered limited mobility and could not even perform everyday chores. He turned to Christianity and began praying, seeking divine intervention for his situation. According to him, he received a divine revelation through the reading and study of the Protestant Bible, where he received healing from Jesus Christ. From that time on, he dedicated himself to ministry.

Evangelism
Although Yiye Ávila was always deeply interested in Christianity, he reportedly was convinced to dedicate all his life and work to evangelism after watching a Christian television show hosted by Oral Roberts, an American televangelist, in the early 1960s. In 1967, Ávila retired after 22 years as a teacher and began to concentrate on Christian work instead.

He became a full-time preacher and, in 1972, he invited several friends, family, and neighbors to form the Escuadron Relampago Cristo Viene ("Christ Is Coming Lightning Squadron"), which congregated in a small room in his house to discuss the Bible, salvation, Christianity, and the Apocalypse. Eventually, his ministry and followers increased dramatically, and a new three-story building near the PR # 2 state road was built to accommodate the attending public. His preaching was followed by various church groups and many Latin American evangelists invited him to preach to their local communities and churches. This demand prompted Yiye to create a radio program and, eventually, a television network called Cadena Del Milagro (Miracle Network) so that his preaching could reach those same communities that he visited.

The network and his ministry has grown considerably over the years, expanding to magazines, books, audio tapes, videos and televised scheduled programming. The three-story building which he and his close followers originally built now serves as the headquarters for Yiye's ministry and his television network.

Beginning in the 1970s, Yiye became a laborious evangelist, visiting almost all Latin American countries, many North American cities, and parts of Europe. He appeared on a daily morning telecast preaching selected verses from the Bible and receiving prayer petitions from viewers. He also appeared every Saturday night on the network's flagship show, Campaña Por TV (Campaigning Through TV), where he preached selected verses of the New Testament, presented Evangelical Christian music groups, and invited viewers to convert to Evangelical Christianity.

In his shows, he encouraged viewers to congregate by going to Evangelical Christian churches and to form Evangelical Christian groups to discuss the Bible and current events. These shows and many other programs are broadcast in Puerto Rico and the Caribbean, and are transmitted via satellite to Latin America, the continental United States and certain parts of Europe.

In addition to his television shows and public preaching, Yiye Ávila was a prolific writer, publishing numerous works throughout his ministry years. Many of his public preachings have been recorded and published on tapes and compact discs.

Personal life and death
Yiye Ávila was married to Carmen Delia Talavera (better known as Yeya) and had three daughters: Noemí, Doris Myrna, and Carmen Ilia. In 1989, his daughter Carmen Ilia was murdered by her husband. On April 28, 2009, his daughter Noemi Ávila, an Evangelical pastor herself, died in a car accident in Freites, Venezuela.

Although his ministry has expanded across three continents and he has become a household name among Evangelical Christian families in Latin America, he preferred a humble life, reportedly living in the same house he owned and in the same conditions before his ministry began. In 2009, he suffered a debilitating stroke that limited his speech capabilities, and he retired from public preaching. On June 28, 2013, Yiye Ávila died from cardiac arrest at the age of 87. He was buried at Cementerio Municipal el Remanso de Paz in Camuy, Puerto Rico.

title: Yiye Ávila


text:
Second Kok cabinet

The second Kok cabinet, also called the second Purple cabinet, was the executive branch of the Dutch government from 3 August 1998 until 22 July 2002.

The cabinet was a continuation of the previous first Kok cabinet and was formed by the social-democratic Labour Party (PvdA), the conservative-liberal People's Party for Freedom and Democracy (VVD) and the social-liberal Democrats 66 after the election of 1998. The cabinet was a centrist grand coalition and had a substantial majority in the House of Representatives with Labour Leader Wim Kok serving as Prime Minister. Prominent Liberal politician Annemarie Jorritsma, the Minister of Transport and Water Management in the previous cabinet, served as Deputy Prime Minister and Minister of Economic Affairs, while former progressive-liberal leader Els Borst continued as Minister of Health, Welfare and Sport and also served as Deputy Prime Minister.

The cabinet served during the economic expansion of the late 1990s and early unstable 2000s. Domestically, it was able to implement several major social reforms such as legalising same-sex marriage, and had to deal with the assassination of Pim Fortuyn. Internationally, it had to deal with several crises such as the fallout of the Srebrenica massacre and the response to September 11 attacks. The cabinet suffered several major internal and external conflicts including multiple cabinet resignations; the cabinet itself resigned prematurely on two occasions: first following a major political crisis in May 1999 (it later came back on its resignation), and second, following the conclusions of a report into the Srebrenica massacre on 16 April 2002. It continued in a demissionary capacity until it was replaced following the election of 2002.

Formation
The new cabinet was the successor of the First Kok cabinet (First Purple cabinet) and was formed from the same coalition of Labour Party, People's Party for Freedom and Democracy and Democrats 66. It was also known as the 'tweede paarse kabinet' ('second purple cabinet') called such because it contained both the social-democratic Labour Party (red) and the liberal People's Party for Freedom and Democracy (blue).

Term
The aim of the cabinet was to continue the policy of cabinet Kok I, which was concerned with economizing, tax reduction and making an end to unemployment. Wim Kok was the Prime Minister, Annemarie Jorritsma as the Deputy Prime Minister for the People's Party for Freedom and Democracy, and Els Borst for Democrats 66. The cabinet was considered boring, because both left-wing and right-wing political parties were a part of it. There was no strong opposition in the parliament. The cabinet completed processes of liberalisation which were started by the previous cabinet: the legalisation of prostitution in 2000, same-sex marriage in 2001 and Euthanasia in 2002. This cabinet was notable for resigning twice. The first time was in May 1999, when Democrats 66 stepped out of the coalition when proposed legislation entered by this party was blocked; through negotiations the crisis was solved and the cabinet stayed together. The second and final time was on 16 April 2002, just one month before the next election, when Prime Minister Kok wished to resign over the NIOD report into the genocide of Srebrenica in 1995 and the other ministers had no choice but to follow him. The Second Kok cabinet remained in place as a Demissionary cabinet until 22 July 2002, when it was replaced by the First Balkenende cabinet.

Changes
On 7 June 1999 Minister of Agriculture, Nature and Fisheries Haijo Apotheker (D66) resigned citing that as a former mayor he could not adjust to national politics. Minister of Social Affairs and Employment Klaas de Vries (PvdA) served as acting Minister of Agriculture, Nature and Fisheries until 9 June 1999 when Member of the European Parliament Laurens Jan Brinkhorst (D66), a former State Secretary for Foreign Affairs was appointed as his successor.

On 13 March 2000 Minister of the Interior and Kingdom Relations Bram Peper (PvdA) resigned after a report was released about inappropriate declarations he had made when he served as Mayor of Rotterdam. Minister for Integration and Urban Planning Roger van Boxtel (D66) served as acting Minister of the Interior and Kingdom Relations until 24 March 2000 when Minister of Social Affairs and Employment Klaas de Vries (PvdA) was installed as his successor. That same day State Secretary for Finance Willem Vermeend (PvdA) was appointed as Minister of Social Affairs and Employment and Member of the House of Representatives Wouter Bos (PvdA) was installed as State Secretary for Finance.

On 1 January 2001 State Secretary for Justice Job Cohen (PvdA) resigned after he was appointed as Mayor of Amsterdam. That same day Member of the House of Representatives Ella Kalsbeek (PvdA) was installed as his successor.

Trivia

 * Seven cabinet members had previous experience as scholars and professors: Els Borst (Medical Ethics), Bram Peper (Social Economics), Gerrit Zalm (Political Economics), Willem Vermeend (Fiscal Law), Laurens Jan Brinkhorst (International Law and Relations), Job Cohen (Jurisprudence) and Rick van der Ploeg (Macroeconomics).
 * Seven cabinet members (later) served as Party Leaders: Wim Kok (1986–2001), Wouter Bos (2002–2010), Job Cohen (2010–2012) of the Labour Party, Gerrit Zalm (2002–2004) and Jozias van Aartsen (2004–2006) of the People's Party for Freedom and Democracy, Els Borst (1998) and Laurens Jan Brinkhorst (1982) of the Democrats 66.
 * Eleven cabinet members (later) served as Mayor: Annemarie Jorritsma (Delfzijl and Almere), Bram Peper (Rotterdam), Jozias van Aartsen (The Hague and Amsterdam), Loek Hermans (Zwolle), Tineke Netelenbos (Ede, Haarlemmermeer and Oud-Beijerland), Haijo Apotheker (Muntendam, Veendam, Leeuwarden, Steenwijk, Steenwijkerland, Sneek, Súdwest-Fryslân, Waadhoeke and Noardeast-Fryslân), Job Cohen (Amsterdam), Henk van Hoof (Delfzijl), Annelies Verstand (Heteren and Zutphen), Geke Faber (Zeewolde, Wageningen, Den Helder and Zaanstad) and Johan Remkes (The Hague).

title: Second Kok cabinet