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Analytical hierarchy

In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy. The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers, $$\mathbb{N}$$, and over functions from $$\mathbb{N}$$ to $$\mathbb{N}$$. The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy.

The analytical hierarchy of formulas
The notation $$\Sigma^1_0 = \Pi^1_0 = \Delta^1_0$$ indicates the class of formulas in the language of second-order arithmetic with number quantifiers but no set quantifiers. This language does not contain set parameters. The Greek letters here are lightface symbols, indicating the language choice. Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarchy for details.

A formula in the language of second-order arithmetic is defined to be $$\Sigma^1_{n+1}$$ if it is logically equivalent to a formula of the form $$\exists X_1\cdots \exists X_k \psi$$ where $$\psi$$ is $$\Pi^1_{n}$$. A formula is defined to be $$\Pi^1_{n+1}$$ if it is logically equivalent to a formula of the form $$\forall X_1\cdots \forall X_k \psi$$ where $$\psi$$ is $$\Sigma^1_{n}$$. This inductive definition defines the classes $$\Sigma^1_n$$ and $$\Pi^1_n$$ for every natural number $$n$$.

Kuratowski and Tarski showed in 1931 that every formula in the language of second-order arithmetic has a prenex normal form, and therefore is $$\Sigma^1_n$$ or $$\Pi^1_n$$ for some $$n$$. Because meaningless quantifiers can be added to any formula, once a formula is given the classification $$\Sigma^1_n$$ or $$\Pi^1_n$$ for some $$n$$ it will be given the classifications $$\Sigma^1_m$$ and $$\Pi^1_m$$ for all $$m$$ greater than $$n$$.

The analytical hierarchy of sets of natural numbers
A set of natural numbers is assigned the classification $$\Sigma^1_n$$ if it is definable by a $$\Sigma^1_n$$ formula (with one free number variable and no free set variables). The set is assigned the classification $$\Pi^1_n$$ if it is definable by a $$\Pi^1_n$$ formula. If the set is both $$\Sigma^1_n$$ and $$\Pi^1_n$$ then it is given the additional classification $$\Delta^1_n$$.

The $$\Delta^1_1$$ sets are called hyperarithmetical. An alternate classification of these sets by way of iterated computable functionals is provided by the hyperarithmetical theory.

The analytical hierarchy on subsets of Cantor and Baire space
The analytical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor and Baire space because they fit with the language of ordinary second-order arithmetic. Cantor space is the set of all infinite sequences of 0s and 1s; Baire space is the set of all infinite sequences of natural numbers. These are both Polish spaces.

The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers can naturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classification $$\Sigma^1_n$$ if it is definable by a $$\Sigma^1_n$$ formula (with one free set variable and no free number variables). The set is assigned the classification $$\Pi^1_n$$ if it is definable by a $$\Pi^1_n$$ formula. If the set is both $$\Sigma^1_n$$ and $$\Pi^1_n$$ then it is given the additional classification $$\Delta^1_n$$.

A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from $$\omega$$ to $$\omega$$ to the characteristic function of its graph. A subset of Baire space is given the classification $$\Sigma^1_n$$, $$\Pi^1_n$$, or $$\Delta^1_n$$ if and only if the corresponding subset of Cantor space has the same classification. An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space. This alternate definition gives exactly the same classifications as the first definition.

Because Cantor space is homeomorphic to any finite Cartesian power of itself, and Baire space is homeomorphic to any finite Cartesian power of itself, the analytical hierarchy applies equally well to finite Cartesian powers of one of these spaces. A similar extension is possible for countable powers and to products of powers of Cantor space and powers of Baire space.

Extensions
As is the case with the arithmetical hierarchy, a relativized version of the analytical hierarchy can be defined. The language is extended to add a constant set symbol A. A formula in the extended language is inductively defined to be $$\Sigma^{1,A}_n$$ or $$\Pi^{1,A}_n$$ using the same inductive definition as above. Given a set $$Y$$, a set is defined to be $$\Sigma^{1,Y}_n$$ if it is definable by a $$\Sigma^{1,A}_n$$ formula in which the symbol $$A$$ is interpreted as $$Y$$; similar definitions for $$\Pi^{1,Y}_n$$ and $$\Delta^{1,Y}_n$$ apply. The sets that are $$\Sigma^{1,Y}_n$$ or $$\Pi^{1,Y}_n$$, for any parameter Y, are classified in the projective hierarchy, and often denoted by boldface Greek letters to indicate the use of parameters.

Examples

 * For a relation $$\prec$$ on $$\mathbb N^2$$, the statement "$$\prec$$ is a well-order on $$\mathbb N$$" is $$\Pi_1^1$$. (Not to be confused with the general case for well-founded relations on sets, see Lévy hierarchy)
 * The set of all natural numbers that are indices of computable ordinals is a $$\Pi^1_1$$ set that is not $$\Sigma^1_1$$.
 * These sets are exactly the $\omega_1^{CK}$-recursively-enumerable subsets of $$\omega$$. [ Bar75, p. 168]
 * A function $$f:\mathbb N\to\mathbb N$$ is definable by Herbrand's 1931 formalism of systems of equations if and only if $$f$$ is hyperarithmetical.
 * The set of continuous functions $$f:[0,1]\to\mathbb [0,1]$$ that have the mean value property is no lower than $$\Delta_2^1$$ on the hierarchy.
 * The set of elements of Cantor space that are the characteristic functions of well orderings of $$\omega$$ is a $$\Pi^1_1$$ set that is not $$\Sigma^1_1$$. In fact, this set is not $$\Sigma^{1,Y}_1$$ for any element $$Y$$ of Baire space.
 * If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself that is $$\Delta^1_2$$ and is the graph of a well ordering of Baire space. If the axiom holds then there is also a $$\Delta^1_2$$ well ordering of Cantor space.

Properties
For each $$n$$ we have the following strict containments:

 * $$\Pi^1_n \subset \Sigma^1_{n+1}$$,
 * $$\Pi^1_n \subset \Pi^1_{n+1}$$,
 * $$\Sigma^1_n \subset \Pi^1_{n+1}$$,
 * $$\Sigma^1_n \subset \Sigma^1_{n+1}$$.

A set that is in $$\Sigma^1_n$$ for some n is said to be analytical. Care is required to distinguish this usage from the term analytic set, which has a different meaning, namely $$\boldsymbol\Sigma_1^1$$.

title: Analytical hierarchy


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British Columbia Highway 17

Highway 17 is a provincial highway in British Columbia, Canada. It comprises two separate sections connected by a ferry link. The Vancouver Island section is known as the Patricia Bay Highway and connects Victoria to the Swartz Bay ferry terminal in North Saanich. The Lower Mainland section is known as the South Fraser Perimeter Road and connects the Tsawwassen ferry terminal to Delta and Surrey, terminating at an interchange with Highway 1 in the Fraser Valley.

Vancouver Island section
The Island section of Highway 17 is known as the Patricia Bay Highway (locally abbreviated as the Pat Bay Highway) after nearby Patricia Bay, and is the main artery through the Saanich Peninsula, mostly travelling along its eastern coast. The highway is four lanes all the way from Victoria to Swartz Bay. The total length of the highway on the Island is 32 km. Highway 17 has had its present course through the area since 1978 when the Blanshard extension was completed.

In the south, Highway 17 begins at the intersection of Belleville and Oswego streets, at the entrance to the Victoria Harbour ferry terminal, which provides a ferry connection to Port Angeles, Washington. It travels east for 600 m, past the grounds of the British Columbia Parliament Buildings, to Douglas Street where it intersects the Trans-Canada Highway (Highway 1). It travels another 180 m east, where the highway turns north as Blanshard Street (a six lane divided city street) for 3 km through the city of Victoria before leaving the city at Tolmie Avenue and for another 2 km north through the district of Saanich to the Uptown Shopping Centre; Highway 17 then becomes the Pat Bay Highway and turns into a 6 km-long freeway, with three interchanges. After the third interchange at Royal Oak Drive, Highway 17 turns into a 14 km mix of divided four-lane arterial and expressway including an interchange at McTavish Road, until it reaches the town of Sidney. After exiting Sidney 3 km later, the Pat Bay once again becomes a freeway, with two more interchanges along its length, toward its northern terminus at the Swartz Bay ferry terminal another 3 km north.

Ferry route
At Swartz Bay, Highway 17 leaves Vancouver Island, and starts on a 44 km (24 nmi)-long ferry route through the Southern Gulf Islands and the Strait of Georgia. The ferry route between Swartz Bay and the Mainland is the oldest and most heavily used route in the B.C. Ferries system. After winding through the Gulf Islands, the route enters a small passage between Galiano and Mayne Islands, known as Active Pass. Active Pass is the midway point on the Highway 17 ferry route, but it is also the most hazardous part, as it has strong tidal currents, and has historically been the site of two maritime collisions involving BC Ferries vessels, as well as one incident of a ferry running aground. Consequently, ferries going through Active Pass have to sound their whistles upon entering and leaving the passage, and must adhere to a lower speed limit while transiting through it.

After Active Pass, the Highway 17 ferry heads due northeast across the Strait of Georgia. Halfway across the Strait, the route begins transiting United States waters for just under 9 km (5 nmi). It then crosses the 49th parallel back into Canadian waters just before landing at the Tsawwassen Ferry Terminal.

Mainland section
On the Mainland, Highway 17 is known as the South Fraser Perimeter Road (SFPR), a component of the British Columbia Ministry of Transportation's Gateway Program. It is a four-lane highway with a mix of freeway and expressway sections. The highway has a speed limit of 80 km/h over most of its length, connecting the Tsawwassen Ferry Terminal and Highway 99 in Delta to Highway 1 at 176th Street in Surrey, and providing access to all five of the major Fraser River crossings in Metro Vancouver. The mainland section is notable for its extensive use of noise walls, split level construction through Delta and use in some urban sections of so-called Quiet Pavement to reduce traffic noise.

The South Fraser Perimeter Road is 44 km long. Beginning at the Tsawwassen Ferry Terminal, it heads northeast on a 1.8 km long causeway to land on the Tsawwassen Peninsula, then continues northeast for 5 km to an interchange with Highway 17A. It then follows the CN Railway's Deltaport chord and 72nd Street northeast for 6.2 km to an interchange with Highway 99. The road then runs through the west side of Burns Bog, but avoiding the conservation area, and past the south side of the Tilbury and Sunbury industrial areas for 9.7 km before it reaches the Fraser River at an interchange connecting to Highway 91 via Highway 91 Connector. It then proceeds east along the south bank of the Fraser River through North Delta and Surrey, passing under the Alex Fraser, Pattullo and Port Mann bridges, before turning southeast at Surrey Bend Regional Park to terminate at its junction with Highway 1 and Highway 15, 21 km from Highway 91 Connector.

Vancouver Island Section
Prior to 1978, the southern terminus of Highway 17 was at the present-day intersection of Highway 1 (Douglas Street) and Carey Road. In 1978, the highway was extended and aligned along Blanshard Street through Downtown Victoria to the Inner Harbour in an effort to improve traffic flow.

South Fraser Perimeter Road
Route 17 once followed what is now Route 17A and terminated at the Highway 99 interchange before the George Massey Tunnel. The provincial government proposed to build a new expressway, the South Fraser Perimeter Road (SFPR), linking the Tsawwassen Ferry Terminal with the Trans-Canada Highway. According to the Ministry of Transportation, the South Fraser Perimeter Road will improve the movement of goods and people across the region while also alleviating truck traffic on municipal roads. West–east travel times across the region will also be significantly reduced.

The South Fraser Perimeter Road project was opposed by a number of groups, citing a variety of concerns. The SFPR alignment does not go through the Burns Bog Ecological Conservancy Area. However, the Burns Bog Conservation Society has stated they are concerned that it will affect surrounding hydrology, and have an adverse effect on the Conservancy Area.

The Burns Bog Conservation Society expressed concerns that the South Fraser Perimeter Road would endanger Burns Bog. In response to the mitigation measures proposed the Environmental Stewardship Branch of Environment Canada wrote that "... the changes are not sufficient to alleviate its concerns related to the impacts of the Project on Pacific Water Shrew (PWS), hydrology, aerial deposition, and ecological integrity of Burns Bog." On November 24, 2010 the Burns Bog Conservation Society launched a lawsuit intend to force the re-routing or cancellation of the SFPR project. Burns Bog has been listed as threatened by the International Mire Conservation Group because of the impacts of the SFPR.

The Wilderness Committee and other groups have criticized the SFPR, and the Gateway Program in general, for increasing greenhouse gas emissions. In early 2011, a protest camp organized by StopThePave.org and the Council of Canadians occupied a SFPR construction site for almost two weeks.

The SFPR Project worked with the Burns Bog Scientific Advisory Panel to develop systems, which helped improve the existing drainage / hydrology of the bog.

A 10 km portion from 136 Street in Surrey to a junction with Highway 15 and Highway 1 was the first portion of the SFPR to be completed, opening on December 1, 2012. The remaining western segment of the road was completed on December 21, 2013. The former Mainland portion of Highway 17 north of the 28th Avenue overpass has been renamed to Highway 17A through Ladner. The total cost of building the SFPR was $1.26 billion.

On March 10, 2017, an announcement was made to address congestion at Highway 17 (Delta): an extension of westbound slip lane from Tilbury Connector (80th Avenue) intersection, replacement of the traffic light-controlled Highway 91 Connector with an interchange, and construction of a new interchange at River Road. This would make the road a freeway all the way to Tannery Road. Funding will come from the Canadian government, the BC Government, the Vancouver Fraser Port Authority, and the Tsawwassen First Nation; the total cost is expected to be just over $245 million. Construction started in spring 2020 and the new River Road interchange opened on December 3, 2021. The new Highway 91 Connector interchange reached completion in November 2022.

Future
On Vancouver Island, there have been numerous studies to upgrade the Island portion of Highway 17 to freeway standard. This seems to be edging closer with the BC Government's issuing of the Highway 17 Corridor Planning Strategy. This study envisions interchanges at Haliburton, Sayward, Keating Cross Road and/or Island View, Mt Newton Cross Road, McTavish Road (Complete April 9, 2011) and Beacon Avenue in Sidney. Construction of a flyover ramp from northbound Highway 17 to Keating Cross Road commenced in June 2023.

As part of the Pattullo Bridge replacement project, the initial plan included a new interchange to provide direct connections to and from Highway 17 to the bridge as well as an extension of Scott Road north of King George Boulevard to directly connect with Highway 17 to divert traffic from local streets in Surrey. However, the project was down-scaled as the study showed Highway 17 cannot handle increased traffic from induced demand and the significant traffic delay would be expected on the provincial road network. Under the final proposal tendered in 2020, the only direct connection from the Pattullo Bridge would be from eastbound Pattullo Bridge to westbound, while Highway 17 and Old Yale Road intersection will be closed and grade-separated.

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

title: British Columbia Highway 17


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Talk:Eicosanoid

Untitled
HI moderator, what the hell is up with the picture? It's completely messed up. A good diagram would succinctly illustrate the synthesis of different eicosanoids, but this really detracts from the page. Thankyou —Preceding unsigned comment added by 86.13.74.231 (talk) 03:16, 25 July 2009 (UTC)

Accuracy
This article is a very good piece of work and shows a lot of research. But let me offer a correction. It repeatedly states that all eicosanoids are derived from arachadonic acid. This is incorrect. I'm not sure about the prostacyclins and the leukotrienes. Also, arachadonic acid has other uses in the body besides eicosanoids, e.g. as a substrate for the endocannabinoids. David Throop 25 Oct 2005
 * The series-2 prostaglandins (PG) and thromboxanes (TX) are derived from arachadonic acid.
 * The series-1 PG and TX are derived from Gamma-linolenic acid through the DGLA intermediary.
 * The series-3 PG and TX are derived from EPA.

Also, the article states that the eicosanoids are a class of cytokines. But the cytokine article says that cytokines are proteins. And eicosanoids aren't proteins; they don't even have any nitrogen in their structure.

David Throop 13 Nov 2005

EETs
I think that there is a third way of arachidonic acid metabolism, called epoxygenase.
 * Yes, there is, the EETs, via the Cytochrome P450 path. I'd noted it but haven't read enough about it to say anything interesting. I've added a link for it, but there's nothing there yet. I also added a link to a future article on Neuroprotectin D, which is AA derived. Does anybody know more about the EETs? Particularly, does anybody know if they only come from AA, or is there an EPA derived subfamily for them too?David.Throop 15:02, 12 March 2006 (UTC)

General discussion
This page looks good. I've never studied this topic, so there's not much that I can say about it. It might be nice to add some diagrams of molecular structures. I know that Canvas is a good program for this, but I won't be able to draw them any time soon. Do you know anything about the receptors for these hormones? I'm going on a Wiki-vacation, so I won't be available for more input for a while. Keep up the good work! AdamRetchless 08:15, 4 May 2004 (UTC)

 * My source has quite a bit on the receptors, but they are quite specific for leukotrienes, prostacyclin etc. I thought it would be better to include this in the related articles, rather than on eicosanoid proper.  JFW |  T@lk  09:28, 4 May 2004 (UTC)

I just wanted to say, completely out of place really, that I am totally in awe of whoever made that brilliant diagram with all the links in it. People like you are exactly what the world needs more of. Wikipedia just blows me away.

Grammar
Is "that largely function as a autocrine and" supposed to refer to a chemical known as "a autocrine"? Perhaps it should be in italics -- it looks like a grammar error, but I am not sure if it is one. Cleduc 1 July 2005 19:46 (UTC)

Cleanup
Someone asked for cleanup, because:

''This article is a very good piece of work and shows a lot of research. But let me offer a correction. It repeatedly states that all eicosanoids are derived from arachadonic acid. This is incorrect. The series-2 prostaglandins (PG) and thromboxanes (TX) are derived from arachadonic acid. The series-1 PG and TX are derived from Gamma-linoleic acid through the DGLA intermediary. The series-3 PG and TX are derived from EPA. I'm not sure about the prostacyclins and the leukotrienes. Also, arachadonic acid has other uses in the body besides eicosanoids, e.g. as a substrate for the endocannabinoids.''

 * This is not good use of cleanup. Cleanup is when the fact are correct but presented clumsily. If you think the facts are incorrect, fix it! JFW | T@lk  20:51, 26 October 2005 (UTC)

References?
This article seems to be more or less accurate, but it needs to be properly referenced.--Virulent 78 13:49, 4 February 2006 (UTC)

 * I'll find a few, but with M.W. King's site linked I guess we don't need the unreferenced template anymore. JFW | T@lk  05:16, 5 February 2006 (UTC)

<SNIP!> removed the text that just repeated what I'd said in the Comments page...

Request for Help drawing some Pictures
There are two diagrams (figures 1 and 2) in that I'd really like to have for the Eicosanoid page. I've read a lot of eicosanoid papers and these are by fat the best diagrams I've seen. They need to be updated a little – some more research has come in since they were published. Is anybody listening who is a good graphic artist and would someone like to work with me on this? – David.Throop 23:22, 10 January 2007 (UTC)
 * (.pdf format)

Major Updates
I've brought material from Essential fatty acid interactions; it's a better fit here, I think. I've put Ciar's pictures into a single figure and added captioning. I've put the list of receptors in a table, and added another table of eicosanoid medicines. Introduced a section on eicosanoids in inflammation.

I used the wikitable template, which is mostly good. But the body text runs right up against the table (in my browser, Firefox.) If anybody can jiggle the handle on the tables so they don't do that I'd be obliged.David.Throop 15:00, 13 January 2007 (UTC)

Hi there
Excellent work. As a note for the future, using the Bookmarklet as detailed on this page makes adding references as easy as cut and paste. It adds into your browser as a button you can click and it pops up a window summarising the PubMed entry you are looking at in the correct format to paste into Wikipedia. TimVickers 18:08, 13 January 2007 (UTC)

protein cofactor?
Help! Near the end of the Nomenclature section, I disambiguated "cofactor" to "cofactor (biochemistry)", but we now have the phrase "protein cofactor" linking to a definition which specifically states "non-protein"! It seems to me that either the Eicosanoid page or the cofactor (biochemistry) page needs to be edited by someone with knowledge of whether cofactors can be protein or not. (perhaps the cofactor page needs to say "usually non-protein"?) --Coppertwig 17:24, 27 January 2007 (UTC)
 * A quick trip to scholar.google.com showed that most scholarly works were calling FLAP an enzyme or a co-enzyme. So I made that change. David.Throop 17:55, 27 January 2007 (UTC)
 * Well done. --Coppertwig 01:17, 31 January 2007 (UTC)

Biosynthesis
I find the table a little confusing. It says there are two pathways for free fatty acids, and then it says there are also additional pathways. Could it be made clear what it is about these additional pathways that precludes them from being included in the original count of two? Is it because they don't begin with free fatty acids? Is it because they involve oxygenation, while the original two perhaps don't? Is it because the first two are commonly called eicosanoids, while the others are only "technically" eicosanoids? Is it because all of the "other" pathways produce products with a number of carbon atoms not equal to 20? Any of these could be inferred from the table. Whatever is actually meant should be more clearly stated.
 * OK, someone has fixed it. It's fine now.  It contrasts "classical" eicosanoids with "other ...20-carbon..." and has a few other little fixes such as putting eicosanoids in italics in one place which just make the table quite clear to me now.  Thanks.  --Coppertwig 01:34, 2 February 2007 (UTC)

Also, this article should decide exactly which molecules it is or is not calling "eicosanoids". For example, in the introduction, it could say something like "x, y and z are also technically eicosanoids but, following the more common convention, will not be referred to as eicosanoids in this article." I don't much care which convention is followed in the article as long as it's clear to the reader which naming convention is being followed, and all very common conventions are mentioned in the introduction. When the "other" molecules are mentioned it could say "not classified here as eicosanoids" or "not called eicosonoids here" or "not universally referred to as eicosanoids". (sorry I can't think of a shorter phrase for that.) --Coppertwig 14:47, 28 January 2007 (UTC)

 * Coppertwig, thanks for some good, pointed observations.  The problem is that the field doesn't seem to have reache a consensus.  Not only that, they don't even seem to be arguing about it -- there are just a bunch of different authors using the terms in similar but distinct ways.  The review articles just state that the eicosanoids comprise the prostanoids and the leukotrienes.  Then I read the articles on the EETs, lipoxins etc and they just state that these compounds are eicosanoids.  Nobody AFAIK has proposed an alternative nomenclature.  I'm thinking that I'll need to email a few of the prominent authors; maybe they can point me to someplace where this has already been hashed out.   I will get this straighter before I put it up for Good/Featured Article review, tho.

 * A related issue is that about half the articles claim that all eicosanoids derive from arachidonic acid and completely ignore the series from EPA and DGLA. But I've just decided to omit that.  David.Throop 21:28, 29 January 2007 (UTC)

 * In my opinion, you can more-or-less arbitrarily choose one definition, state that it's the definition being used in this article, and then use it consistently throughout the article. For example, one table states that "all eicosanoids" have corresponding receptors.  If you haven't clarified what "eicosanoids" means in this article, such a statement is confusing, meaningless and likely misleading.

 * One way to choose a definition is to take the approximate median of the definitions used by the sources:  that is, to include a molecule as an eicosanoid in your definition if approximately half or more of the authors would include it.  The definition should however also be simple, logical and identical to the definition used by at least one author.

 * Another approach is to look at the etymology of the word, i.e. to use the definition used by whoever invented the word in the first place (and it would be good to name this person in the article, anyway). However, if that definition is not now a common usage that might not be a good approach.

 * The definition of "eicosanoid" is crucial, among other reasons because it defines what subject matter this article is supposed to be covering. --Coppertwig 13:49, 1 February 2007 (UTC)

 * I suggest editing part of the Nomenclature section to read:
 * Current usage limits this to the leukotrienes (LT) and three types of prostanoids—prostaglandins (PG) prostacyclins (PGI), and thromboxanes (TX), and this is the definition used in this article, although several other classes could technically be termed eicosanoid, including the hepoxilins, resolvins, isofurans, isoprostanes, lipoxins, epoxyeicosatrienoic acids (EETs) and some endocannabinoids.   - good.  accepted. -drt
 * (I inserted "and this is the definition used in this article", and changed "but" to "although", and "are technically" to "could technically be termed"). I think that would clear up the current ambiguity (provided the rest of the article does actually conform to that definition). --Coppertwig 14:39, 1 February 2007 (UTC)

The first two sentences of the "biosynthesis" section, and perhaps most or all of the first paragraph of the biosynthesis section, is merely a repeat of information already given in the introduction and needs to be deleted. If there is any important information in this paragraph it would be better moved elsewhere, e.g. fit it in later in the biosynthesis section or in another section; the beginning of what is now the second paragraph of the biosynthesis section would be a better lead-in to this topic. The first paragraph seems to be more about usage and breakdown than synthesis. --Coppertwig 14:55, 28 January 2007 (UTC)

Here is an edit of the paragraph beginning "The first step...". I've attempted to improve the style (flow of language), for example trying to eliminate parentheses if possible. However, this paragraph still has this problem: animals do not have "cell walls"!!

''The first step of eicosanoid biosynthesis occurs when a cell is activated by mechanical trauma, cytokines, growth factors, an eicosanoid from a neighboring cell or some other stimulus, which triggers the release of a phospholipase at the cell wall. This phospholipase (either A2 or C) travels to the nuclear membrane, where it catalyzes ester hydrolysis of a phospholipid (by A2) or of diacylglycerol (by phospholipase C). This hydrolysis reaction frees a 20-carbon essential fatty acid and appears to be the rate-determining step for eicosanoid formation. ''

I'm editing for clarity and flowing language here; please check that I haven't mangled the facts. --Coppertwig 15:23, 28 January 2007 (UTC)

Woops: the second-last paragraph of the biosynthesis section (both the current version and my version above) implies that there are only two possible types of phospholipase involved, while the following paragraph says there are "several" possible types; to me "several" suggests at least three. It doesn't actually list more than two examples, though, and it also seems to say that one particular type, cPLA2, is absolutely needed, ("... devoid ...") so maybe there's really only one? Clarification needed. --Coppertwig 16:13, 28 January 2007 (UTC)

Peroxydation section
The first sentence of the last paragraph sounds anticlimactic to me on first reading:
 * The generation of lipid hydroperoxides within or near the cell nucleus must confer a benefit.

It seems to be saying that something which has already been positively stated earlier may be true, i.e. that "The generation of lipid hydroperoxides ... must confer a benefit". This sentence can be made stronger by increasing the emphasis on the site at which the reaction occurs. For example, by changing it to:
 * The site of the generation of lipid hydroperoxides, within or near the cell nucleus, must confer a benefit. or
 * ''The location of generation of lipid hydroperoxides within or near the cell nucleus must confer a benefit."

or "siting" or "positioning" instead of "location", etc. --Coppertwig 16:33, 28 January 2007 (UTC)
 * OK, changed it to
 * The cell must realize some benefit from generating lipid hydroperoxides close-by its nucleus. PGs and LTs may  signal or regulate DNA-transcription there; LTB4 is ligand for PPARα.
 * Better? David.Throop 21:13, 29 January 2007 (UTC)
 * Yes, I think that's fine. The slightly different rhythm of the sentence carries it, I think.  Thanks.  --Coppertwig 01:14, 30 January 2007 (UTC)

I changed "oxidation proceeds with high stereospecificity" to "oxidation reaction introduces stereospecificity". First of all, it seems to me that a molecule is either stereospecific or it's not; the word "high" seems unnecessary. Secondly, I believe the reaction begins with two stereononspecific molecules, therefore it seems to me that the reaction ends with stereospecificity but it is not necessarily accurate to say that it "proceeds" with stereospecificity. --Coppertwig 01:44, 2 February 2007 (UTC)

I think you've misunderstood the term 'stereospecific', it's a feature of the reaction, not the molecule. 'The reaction proceeds with great stereospecificity' means that it produces only one of the possible chiral products. If it has low stereospecificity, you get nearly a racemic mixture. Follow the link to stereospecificity for more. David.Throop 01:56, 2 February 2007 (UTC)

Image
I drew the diagram of eicosanoid synthesis. If there are adjustments to be made, please let me know. I can also release the sxd file (OpenOffice.Org) for others to fiddle with. JFW | T@lk  20:43, 29 January 2007 (UTC)

biosynthesis of prostanoids
The terminology needs to be made more clear and consistent between the paragraph and the diagram. It says "can be seen in the diagram," but I don't see in the diagram the things that are mentioned in the paragraph. The paragraph says "PGH" (what is that?); I don't see the label "PGH" in the diagram. The paragraph describes a number of molecules with rings, but two of the molecules in the diagram do not seem to have rings. Which of the molecules in the diagram are "PGH compounds"? Which are "the rest"? Which are "derived prostaglandins"? Which are "thromboxanes"? I really can't match a single molecule in the diagram to a single reference in the paragraph. The reader should be able to understand the paragraph and related diagram without having to go back and reread and study the earlier section of the article. I hope this doesn't sound too critical; I'm trying to be helpful here. --Coppertwig 02:42, 30 January 2007 (UTC)

OK, I think I can mostly fix this, but someone will have to check that I haven't mangled any of the science. It said:
 * All three classes of prostanoids originate from PGH. All have distinctive rings in the center of the molecule. They differ in their structures. As the Structures of Selected Eicosanoids figure  shows, the PGH compounds (parents to all the rest) have a 5-carbon ring, bridged by two oxygens (a peroxide.) The derived prostaglandins contain a single, unsaturated 5-carbon ring. In prostacyclins, this ring is conjoined to another oxygen-containing rings. In thromboxanes the ring becomes a 6-member ring with one oxygen. (See more detail, including the enzymes involved, in diagrams at Prostanoid.)

I'm changing it to:
 * All three classes of prostanoids originate from PGH. All have distinctive rings in the center of the molecule. They differ in their structures. The PGH compounds (parents to all the rest) have a 5-carbon ring, bridged by two oxygens (a peroxide.)  As the example in Structures of Selected Eicosanoids figure  shows, the derived prostaglandins contain a single, unsaturated 5-carbon ring. In prostacyclins, this ring is conjoined to another oxygen-containing ring. In thromboxanes the ring becomes a 6-member ring with one oxygen.  The leukotrienes do not have rings. (See more detail, including the enzymes involved, in diagrams at Prostanoid.)

My purpose here is to make the paragraph seem to match what is shown in the figure. If I understand this paragraph correctly, PGH compounds (not shown in the figure) have an oxygen bridge in the ring, while prostaglandins do not. Please check that the paragraph as I've edited it has not become incorrect. --Coppertwig 02:08, 2 February 2007 (UTC)

Only arachidonic acid?
Someone mentioned above that arachidonic acid is not the only source of eicosanoids. The first diagram at the top of the article shows only arachidonic acid, so maybe it needs to be changed (or labelled as being an example). --Coppertwig 02:45, 30 January 2007 (UTC)

References format
Changing the reference style has greatly improved the page. I have added the following:

To convert from single-column to two-column format (it looks much nicer) but alas this will not display as such in every browser. (I believe it works in Firefox). If someone knows how to make it display in two columns in all browsers, please tell me how! Thanks in advance István 17:48, 30 January 2007 (UTC)

"The ω-3 and ω-6 series" subsection
The reduction in AA-derived eicosanoids and the diminished activity of the alternative products generated from ω-3 fatty acids serve as the foundation for explaining some of the beneficial effects of greater ω-3 intake.

Is this right? It sounds right that "the reduction in AA-derived eicosanoids" is an explanation for the beneficial effects of omega-3 intake. But it sounds wrong to me that "the diminished activity of the alternative products generated from ω-3 fatty acids" would serve as an explanation. I would expect increased omega-3 intake to produce an increased, not diminished, activity of "the alternative products generated from ω-3 fatty acids". --Coppertwig 01:44, 31 January 2007 (UTC)

 * I think its correct, but perhaps not as clear as it could be - here is an alternate version:
 * The reduced concentration of ω-6 derived eicosanoids due to the increased concentration of ω-3 derived eicosanoids serve as one explanation of some of the beneficial effects of greater dietary ω-3 intake.
 * I believe is less ambiguous as the concentration of n6 eicosanoids has been altered, but their activity per se has not. István 17:55, 1 February 2007 (UTC)
 * Sorry I didn't reply earlier. Yes, this sentence is much better.

Looking at the changes for the last day in this section, I'm concerned that it's become much more tentative.
 * yesterday: Increasing dietary ω-3 fat restricts AA-derived eicosanoids. The alternative products—generated from EPA—have comparatively lower activity. These changes serve as the foundation for explaining the beneficial effects of greater ω-3 intake.
 * today: Increasing dietary intake of ω-3 fatty acids restricts production of AA-derived eicosanoids; the two metabolic pathways utilise, i.e. compete for, the same enzymes. The reduced concentration of AA-derived eicosanoids due to the increased concentration of ω-3 derived eicosanoids serve as one explanation of some of the beneficial effects of greater dietary ω-3 intake.

This is wordier and weaker in places. Common usage is 'omega-3 fats', not 'omega-3 fatty acids' and in the diet, people aren't eating fatty acids, they're eating fat. The eicosanoids aren't just one explanation of some of the beneficial effects; they're the foundation
 * I used 'EPA-derived' rather than 'ω-3' in this case. I've mentioned in the article that DGLA-derived ω-6 eicosanoids are also anti-inflamatory.

I think 'serve as the foundation' is stronger, and justified by the references. David.Throop 19:58, 1 February 2007 (UTC)

Perhaps there are three points of discussion:

 ω-3 v EPA  fats v fatty acids "one explanation for..." v "the foundation of..."  1) ω-3 is more popularly recognised than is EPA which is another TLA (and there is a minor pathway of DHA>EPA back-conversion). 2) I agree with you as per your above statement, "fats" is better than "fatty acids" (and technically more correct in a dietary sense) and 3) I think its not really known for sure yet. We think its so, and I agree that current literature indicates this, but so much in the techincal literature has changed so quickly (and we still dont fully understand the mechanisms involved) that a qualified statement may be more accurate than a certain one - your call. István 20:32, 1 February 2007 (UTC)

 * 1)ω-3 will be easier for people to recognize; that doesn't mean other things can't also be mentioned.  The other things can be referred to by phrases like "longer-chain fatty acids" or "EPA (from fish in the diet, for example)" to make them more understandable to non-experts.  2) I suppose it's probably better to say "omega-3 fats" rather than "fatty acids" when talking about the diet, though I think of the fatty acids, not the fats as being the essential nutrients;  3) I agree with David that "serve as the foundation of" sounds good.  I don't think it's overstated given that there is both theoretical understanding of the mechanisms and observational evidence of effects.  The next paragraph with its A-level evidence etc. makes the level of evidence clear.

Intro on pro-inflammatory eicosanoids
In the introduction it says The ω-6 eicosanoids are generally pro-inflammatory; ω-3's are much less so. This introduction could give people the erroneous impression that if they want to reduce inflammation, they should stay away from both omega-3 and omega-6 fats in their diet. I think this needs to be changed, since some people will read only the introduction. It would be better to say:

''The ω-6 eicosanoids are generally pro-inflammatory; ω-3's tend to counter them. '' Or it can say "compete with" or "can soften their effects" or something rather than "counter". --Coppertwig 16:28, 5 February 2007 (UTC)

 * You make a good point, but this is tricky. This is an area where it's hard to be both correct and not misleading.  We have to distinguish between the activities of the ω-3 fats and the eicosanoids derived from them.
 * The prostacyclins in general are anti-inflammatory.

 * The thromboxanes are pro-inflammatory. TXA3 (from EPA) by itself induces inflammation, but weakly.  It may be anti-inflammatory in the presence of TXA2.  That is, TXA3's activity signalling inflammation may be less than its activity (competitive inhibition) blocking TXA2.  But I'm not sure about that and I don't have a cite.
 * LTB5 actually does counteract LTB4; and I've got a good cite for that. I believe the cysteinyl LTs from EPA are still pro-inflammatory, though.

 * See, much of the anti-inflammatory effect of EPA happens before any eicosanoids get generated at all—by inhibiting the formation of AA-derived eicosanoids. There are some cases where the EPA-derived eicosanoids are anti-inflammatory, but not enough to make a generalized statement.  And while EPA is anti-inflammatory, it's not a statement that goes in the introduction, because this is an article about eicosanoids, not about fish oil.David.Throop 17:47, 5 February 2007 (UTC)

Quoting on eicosanoids and omega-3
Look at how I reformatted your numbered list. :-) More detail here.

When I wrote this whole paragraph, it was more or less a direct quote from Fritsche. Maybe I should just quote him directly. There's a quotation template that I'm looking around for... 01:01, 2 February 2007 (UTC)

reader level
Some readers' eyes will glaze over when they come to the "Biosynthesis of leukotrienes", if not earlier. I don't suggest deleting any material. However, it might be helpful to put more accessible information (such as that omega-3 helps reduce inflammation -- the function and pharmacology section) earlier in the article than this challenging paragraph. Another alternative is to possibly convert the "Biosynthesis of leukotrienes" into a figure or table rather than a symbol-filled paragraph. Maybe it's OK to just leave it the way it is and hope some readers will skip the paragraph rather than throwing in the towel. --Coppertwig 02:18, 2 February 2007 (UTC)
 * Istvan also suggested moving the Biosynthesis material. The problem is that some of the biosynthesis ideas are prerequisite for understanding the health effects and why the pharmacology works.  You can't understand how aspirin works until you at least partially understand the COX pathway.  I can't explain how and why the omega-3's work—competing for the desaturases, elongases, phospholipases and COX or 5-LO—until I've introduced some of the biosynthesis.  Istvan's suggestion of having a brief overview of Biosynthesis, and moving the more difficult material into a subarticle, seems right.  But I think it still has to logically proceed the function, pharmacology and effects.David.Throop 04:12, 2 February 2007 (UTC)
 * To me, the section "The ω-3 and ω-6 series" is readable and does not require "Biosynthesis of leukotrienes" as a prerequisite. Readers can understand that different pathways compete for enzymes even if they haven't previously been introduced to the specific reactions.  Here are some additional editing alternatives to consider:
 * Have a simpler version of "Biosynthesis of leukotrienes", then the function and pharmacology section, then a "Detailed biosynthesis of leukotrienes" with all the technical information. Mentioning "see detailed ... below" can help guide the readers who are interested in more detailed information to refer to that part before reading the rest.
 * Keep "Biosynthesis of leukotrienes" where it is but add more plain English words to it to give the reader a rest. For example, the sentence beginning "The cell must realize some benefit..." could perhaps be moved into the interior of this paragraph.  Other sentences such as "A phospholipases is an enzyme which frees a fatty acid off from a fat molecule" or "It only takes a ten-thousandth of a second for a substance to diffuse across the distance from the cell membrane to the cell nucleus" (if that's accurate) could possibly be added, or just some additional words such as inserting "the enzyme" or "the leukotriene" etc. before some symbols, and inserting more words like "although", "and then" etc.  I could do some (more) stuff like this but will wait to see what you think.
 * Combine the biosynthesis section with the following sections for which you feel it's a prerequisite. I.e. whenever the reader needs to know a certain fact, state the fact at that time.
 * As I suggested above, make it into a table or figure. --Coppertwig 20:32, 3 February 2007 (UTC)

Connect the dots in "function and pharmacology"
In the first paragraph:
 * Eicosanoids have a short half-life, ranging from seconds to minutes. Dietary antioxidants inhibit the generation of some inflammatory eicosanoids, e.g. trans-resveratrol against thromboxane and some leukotrienes. Most eicosanoid receptors are members of the G protein-coupled receptor superfamily; see the Receptors table.

I suggest adding at least one or two sentences between the first and second sentence and at least another one or two between the second and third, if appropriate information can be found, splitting into more than one paragraph if necessary. This would serve two purposes: first, to make this part longer than the corresponding part of the introduction which should logically be a mere summary of it;  secondly, to guide the reader in the flow of ideas from one of these sentences to the next. Each of these sentences seems to me to be a sufficiently different topic from each other than perhaps each should be in a separate paragraph, with some other information added to fill out the paragraphs. --Coppertwig 13:41, 2 February 2007 (UTC)
 * yeah, I know. Right now, the paragraph is just a catch-all for snippets of text that I didn't know what to do with.   For right now I'm going to leave it alone.  Once we've reworked the Biosynthesis material, it will be clearer what needs to be covered here.  But feel free to take a swipe at it.  You (or anybody else tuning in) might try reading Simopoulos and see if you could summarize what he says.  Alternatively, edit Talk:Eicosanoid/to_do. Make a list of what points should be made in the sub-section and I'll try to come up with the text.  David.Throop 16:07, 2 February 2007 (UTC)

Nomenclature
After Its two letter abbreviation I suggest putting "as given above". Also, "PGI" is three letters, so maybe it should say "Its two (or sometimes three) letter abbreviation, as given above."

I just figured out that "An ABC sequence-letter" means a single letter. I had misread it as meaning a sequence of three letters. I think "A single letter" or "one letter" or "one uppercase letter" would be more clear instead. It's not clear what functional groups are referred to. For example, there is more than one functional group beginning with the letter H. When giving examples of eicosanoids at the end of the nomenclature section, I think it would be helpful to spell out the full name of at least one or perhaps all of them:

 * The EPA-derived prostanoids have three double bonds, (e.g. PGG3, PGH3, PGI3, TXA3) while its leukotrienes have five, (LTB5). It would be good to explain here that PGG3 means "a prostaglandin with (oh-oh, I don't see any functional group beginning with the letter G) and with 3 double bonds" and also spell out at least the H, I and A in PGH3, PGI3 and TXA3.

Alternatively, when the ABC sequence-letter idea is mentioned, examples of some functional groups can be given: "e.g. H for hydroxyl, A for aldehyde, ..." or whatever they actually are. --Coppertwig 23:36, 5 February 2007 (UTC)
 * Thanks for pointing out what is unclear; I've been so close to this stuff for so long that I take a lot for granted.
 * The A-B-C letters don't stand for anything; they just indicate (roughly) what order the compounds were named.  I spelled this out more clearly in the text, but what had been a simple bullet point was getting very long indeed.  So I shoved it down into a footnote (you are reading the footnotes, aren't you?)  See if this wording is more accessible.   Oh, and do follow the link to the Cyberlipid page; it's good. David.Throop 04:55, 6 February 2007 (UTC)

Assessment comment
Substituted at 14:14, 29 April 2016 (UTC)

"Preferred IUPAC name: icosanoid" not true
Preferred IUPAC names are preferred systematic names of specific organic compounds when the systematic IUPAC nomenclature gives several possibilities. The name eicosanoid is a trivial name, not a name constructed according to IUPAC nomenclature (-noid is not part of IUPAC nomenclature), so the IUPAC preference of icosa- over eicosa- as a prefix meaning "twenty" is not relevant here. Therefore, I deleted the information Preferred IUPAC name: icosanoid froim the article.--Biologos (talk) 15:38, 27 June 2016 (UTC)

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cPLA2's acts specifically on AA, EPA and GPLA??
What is GPLA? Should it be DGLA (dihomo gamma linoleic acid)? Wisapi (talk) 21:50, 18 January 2018 (UTC)

Introduction
There are statements in the introduction which require references. Around 20, by my count.

Classification as Hormones
I've seen in multiple sources that Eicosanoids are classified as hormones but couldn't find any mention of that fact in this page, worth looking into

Sources: https://med.libretexts.org/Bookshelves/Anatomy_and_Physiology/Anatomy_and_Physiology_(Boundless)/15%3A_Endocrine_System/15.2%3A_Hormones/15.2C%3A_Chemistry_of_Hormones

https://www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/eicosanoid

I'm sure there are more sources but I think those should suffice Fyukfy5 (talk) 12:22, 29 March 2024 (UTC)

 * I also just realized that the lead on the Hormone article mentions eicosanoids as a type of hormone so this shouldn't be an issue Fyukfy5 (talk) 15:56, 9 April 2024 (UTC)
 * Under the Nomenclature section, some eicosanoids are hormones, but not all eicosanoids have mechanisms and/or receptors identified to enable the classification as hormones. For the statement you wish to make, a WP:BMI-WP:MEDRS source would be needed for the all-inclusive classification. Please don't edit war, WP:WAR, but allow the input of other editors, WP:CON. Zefr (talk) 16:06, 9 April 2024 (UTC)
 * Not trying to edit war but it was my only way to talk to you, why are the sources I cited not good enough? And why is the fact that they dare classified as hormones on the Hormone article not enough?
 * I'm new to this so I don't know all the rules . Fyukfy5 (talk) 16:12, 9 April 2024 (UTC)
 * Also the word hormone doesn't appear in the nomenclature section from what I can see. Fyukfy5 (talk) 16:16, 9 April 2024 (UTC)

title: Talk:Eicosanoid


text:
Shove ha'penny

Shove ha'penny (or shove halfpenny), also known in ancestral form as shoffe-grote ['shove-groat' in Modern English], slype groat ['slip groat'], and slide-thrift, is a pub game in the shuffleboard family, played predominantly in the United Kingdom. Two players or teams compete against one another using coins or discs on a tabletop board.

Board

Shove ha'penny is played on a small, rectangular, smooth board usually made of wood or stone. A number of parallel lines or grooves run horizontally across this board, separated by about one-and-a-half coin diameters. The spaces between the lines (usually nine) are called the "beds". Five British halfpenny coins "ha'pennies" (now obsolete pre-decimalisation coinage, diameter 1 inch; 25mm) or similarly-sized coins or metal discs are placed one-by-one at one end of the board slightly protruding over the edge and are shoved forward toward scoring lines, with a blow from the palm of the hand.

In the opinionated (and rare) book The Shove Ha'penny Board Displayed, author Trelawney Dayrell-Reed asserted that the best boards are made of unvarnished walnut or mahogany. In parts of Southern England, primarily Somerset, Dorset and Hampshire, the board is made of slate and lubricated with arrowroot powder or French chalk, which makes the polished ha'pennies glide with a very light touch.

The five-coin turns are alternate, and the coins are cleared between turns, so there is no nudging of opponents' coins. Indeed, both players use the same coins, and it is a serious mistake to move the coins back to the bottom of the board before one's opponent has had a chance to check the scoring, as they may be distracted by drinking and so suspect one of cheating. Sometimes teams compete, playing in sequence, but scoring is as for single players. Experienced players (including Dayrell-Reed) conclude that merely placing the coins between the lines is too easy so that to score they must be placed almost exactly in the centre of the bed. An exact placement by a single shove, rather than by nudging into position by subsequent shots, is called a "flopper" and will command applause, especially if it is a winning shot. In competition games, a scoring placement is judged by a referee. In times past, considerable sums could be wagered on games of shove ha'penny, and influencing the referee – whether actual or suspected – could result in sudden and violent confrontations. Because of this, some public houses have a strong antipathy to games of shove ha'penny and will only allow trusted locals to play, sometimes keeping the board in a back room and denying its existence to strangers. This is particularly common in the case of the Dorset long board, making it difficult even for a shove ha'penny enthusiast to ascertain how many pubs still have this archaic board.

The Dorset long board is a somewhat different game, employing a larger board marked with a four-square and semicircles rather than horizontal lines. This board can be viewed in a British Pathé newsreel. The short-board game was regularly featured in Yorkshire TV's Indoor League programme during the 1970s.

Gameplay
A player shoves five coins or metal discs (ha'pennies) up the board in each turn. To prepare each coin to be pushed, the player positions the coin at the front of the board, with the rear of the coin just sticking over the edge of the board. Any part of the hand can then be used to strike the coin, shoving it up the board. If the coin does not reach the first line on the board, that coin does not count as having been played and can be shoved again.

At the end of the turn, each coin that is completely within a "bed" (between two horizontal lines and within the bordering vertical lines) scores a point for that player in that bed. The points are scored with chalk marks in the squares at either end of the bed on the edge of the board, one player owning the right side, the other, the left. The aim is to get three chalk marks in each of the squares - three scores in each of the nine beds. However, once three scores have been made in a bed, any further scores in that bed will be given to the opponent instead, unless the opponent already has three scores in the bed. The one exception to this is the winning point, which must be scored properly by the winning player, not given away.

Players will attempt to cause a coin to knock onto one or more previously pushed ha'pennies in an effort to improve their position, as well as trying to make a score with the ha'penny being played. A little thought is also required – it is not usually a good move to score the third coin in a bed until towards the end of a hand, so as to set up coins in positions that will increase the chances that later coins will score by coming up from below and stopping behind the earlier coins. This is particularly true in the "progressive" variant of the game, where, after a player's hand of five coins is complete, any scoring coins are recorded and then pulled back to be replayed in the same turn. A player may try to place early coins not only to promote scoring using later coins, but also in such a way as to be pushed by the later coins into positions where further scoring opportunities may be created. Thus, in this version of the game, a very gifted (and/or lucky) player could theoretically win the game in a single turn.

The game has its own body of specialised jargon, providing a host of colourful names for particular scoring opportunities, shots and board features, as well as special rules of etiquette. Trelawney Dayrell-Reed asserts (tongue in cheek) that the appropriate penalty for someone unfortunately wetting the board with a drink ring (and thus compromising its playing surface) is to be "painfully destroyed without benefit of clergy."

Another important etiquette is to wipe the scoring area, after completing a game, away from the playing area as the marking chalk is coarse and interferes with the lubrication of the surface. Indeed, the condition of the surface is considered so critical that a shortage of arrowroot from St. Vincent due to crop failures proved problematic, as supplies from alternative sources did not perform well.

title: Shove ha'penny


text:
Wikipedia:Articles for deletion/Gourmand

DicDef. Delete unless expanded significantly -- Chris 73 | Talk 08:15, 4 May 2004 (UTC)
 * Delete. Ditto Alcarillo 14:36, 4 May 2004 (UTC)
 * Delete - not even a good dicdef -- Cyrius|&#9998 15:30, May 4, 2004 (UTC)
 * Delete unless expanded significantly and promptly. Deleting a one-sentence article should be no big deal; if someone has three sentences' worth to say they can write the three sentences and re-create the article. with three sentences' worth. (Should at least have a discussion of gourmand vs. gourmet.). See The_perfect_stub_article Dpbsmith 19:33, 4 May 2004 (UTC)
 * Keep --A dictionary is an encylopedia of words.. You guys have an encylopdeia of words but hate dictionaries.. you're goofy.. Memes guys memes. This is a nice meme..  It may need work, but trafic what it is and the war and all.. it may take some time for society to make a proper showing for Gourmand..  what's with the "kill it before it grows" thing?Quickwik
 * Delete. Agree with Dpbsmith. The author really should start by writing the article on gourmet, then discuss the difference. --Michael Snow 20:17, 4 May 2004 (UTC)
 * Delete. john 03:57, 5 May 2004 (UTC)
 * Delete. Gwimpey 21:52, 5 May 2004 (UTC)
 * Delete Jiang 04:33, 9 May 2004 (UTC)
 * Delete or redirect to Binge eating disorder JFW | t@lk 20:20, 9 May 2004 (UTC)


text:
Wheel theory

A wheel is a type of algebra (in the sense of universal algebra) where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring.

The term wheel is inspired by the topological picture $$\odot$$ of the real projective line together with an extra point ⊥ (bottom element) such that $$\bot = 0/0$$.

A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.

Definition
A wheel is an algebraic structure $$(W, 0, 1, +, \cdot, /)$$, in which and satisfying the following properties:
 * $$W$$ is a set,
 * $${}0$$ and $$1$$ are elements of that set,
 * $$+$$ and $$\cdot$$ are binary operations,
 * $$/$$ is a unary operation,
 * $$+$$ and $$\cdot$$ are each commutative and associative, and have $$\,0$$ and $$1$$ as their respective identities.
 * $$/$$ is an involution, for example $$//x = x$$
 * $$/$$ is multiplicative, for example $$/(xy) = /x/y$$
 * $$(x + y)z + 0z = xz + yz$$
 * $$(x + yz)/y = x/y + z + 0y$$
 * $$0\cdot 0 = 0$$
 * $$(x+0y)z = xz + 0y$$
 * $$/(x+0y) = /x + 0y$$
 * $$0/0 + x = 0/0$$

Algebra of wheels
Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument $$/x$$ similar (but not identical) to the multiplicative inverse $$x^{-1}$$, such that $$a/b$$ becomes shorthand for $$a \cdot /b = /b \cdot a$$, but neither $$a \cdot b^{-1}$$ nor $$b^{-1} \cdot a$$ in general, and modifies the rules of algebra such that
 * $$0x \neq 0$$ in the general case
 * $$x/x \neq 1$$ in the general case, as $$/x$$ is not the same as the multiplicative inverse of $$x$$.

Other identities that may be derived are where the negation $$-x$$ is defined by $$ -x = ax $$ and $$x - y = x + (-y)$$ if there is an element $$a$$ such that $$1 + a = 0$$ (thus in the general case $$x - x \neq 0$$).
 * $$0x + 0y = 0xy$$
 * $$x/x = 1 + 0x/x$$
 * $$x-x = 0x^2$$

However, for values of $$x$$ satisfying $$0x = 0$$ and $$0/x = 0$$, we get the usual
 * $$x/x = 1$$
 * $$x-x = 0$$

If negation can be defined as below then the subset $$\{x\mid 0x=0\}$$ is a commutative ring, and every commutative ring is such a subset of a wheel. If $$x$$ is an invertible element of the commutative ring then $$x^{-1} = /x$$. Thus, whenever $$x^{-1}$$ makes sense, it is equal to $$/x$$, but the latter is always defined, even when $$x=0$$.

Wheel of fractions
Let $$A$$ be a commutative ring, and let $$S$$ be a multiplicative submonoid of $$A$$. Define the congruence relation $$\sim_S$$ on $$A \times A$$ via
 * $$(x_1,x_2)\sim_S(y_1,y_2)$$ means that there exist $$s_x,s_y \in S$$ such that $$(s_x x_1,s_x x_2) = (s_y y_1,s_y y_2)$$.

Define the wheel of fractions of $$A$$ with respect to $$S$$ as the quotient $$A \times A~/{\sim_S}$$ (and denoting the equivalence class containing $$(x_1,x_2)$$ as $$[x_1,x_2]$$) with the operations
 * $$0 = [0_A,1_A]$$     (additive identity)
 * $$1 = [1_A,1_A]$$     (multiplicative identity)
 * $$/[x_1,x_2] = [x_2,x_1]$$     (reciprocal operation)
 * $$[x_1,x_2] + [y_1,y_2] = [x_1y_2 + x_2 y_1,x_2 y_2]$$     (addition operation)
 * $$[x_1,x_2] \cdot [y_1,y_2] = [x_1 y_1,x_2 y_2]$$     (multiplication operation)

Projective line and Riemann sphere
The special case of the above starting with a field produces a projective line extended to a wheel by adjoining a bottom element noted ⊥, where $$0/0=\bot$$. The projective line is itself an extension of the original field by an element $$\infty$$, where $$z/0=\infty$$ for any element $$z\neq 0$$ in the field. However, $$0/0$$ is still undefined on the projective line, but is defined in its extension to a wheel.

Starting with the real numbers, the corresponding projective "line" is geometrically a circle, and then the extra point $$0/0$$ gives the shape that is the source of the term "wheel". Or starting with the complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere), and then the extra point gives a 3-dimensional version of a wheel.

title: Wheel theory


text:
Pieter Bleeker

Pieter Bleeker (10 July 1819 – 24 January 1878) was a Dutch medical doctor, ichthyologist, and herpetologist. He was famous for the Atlas Ichthyologique des Indes Orientales Néêrlandaises, his monumental work on the fishes of East Asia published between 1862 and 1877.

Life and work
Bleeker was born on 10 July 1819 in Zaandam. He was employed as a medical officer in the Royal Netherlands East Indies Army from 1842 to 1860, stationed in the Dutch East Indies (now Indonesia). During that time, he did most of his ichthyology work, besides his duties in the army. He acquired many of his specimens from local fishermen, but he also built up an extended network of contacts who would send him specimens from various government outposts throughout the islands. During his time in Indonesia, he collected well over 12,000 specimens, many of which currently reside at the Naturalis Biodiversity Center in Leiden. Bleeker corresponded with Auguste Duméril of Paris. His work in ichthyology and tropical medicine was recognised by two doctorates honoris causa (Leyden University, 1846; Utrecht University, 1849).

After his return to the Netherlands in 1860, he started publishing the Atlas Ichthyologique des Indes Orientales Néêrlandaises, a comprehensive account of his studies done in Indonesia, featuring over 1,500 illustrations. It was published in 36 volumes between 1862 and his death in 1878. Between 1977 and 1983, the Smithsonian republished the work in 10 volumes.

Bleeker published more than 500 papers on ichthyology, describing 511 new genera and 1,925 new species.

He also worked in herpetology, describing at least 14 species of reptiles, most of them described in Reptilien van Agam.

In 1855, he became correspondent of the Royal Netherlands Academy of Arts and Sciences, department Natuurkunde (then Natural Sciences), and in 1862 a member. In 1856, he was elected correspondent for the Muséum national d'histoire naturelle in Paris. In January 1864 he received a French knighthood of the Légion d'honneur. He was president of the Royal Netherlands Institute of Southeast Asian and Caribbean Studies.

Bleeker died on 24 January 1878 in The Hague.

Taxa named in his honor
Fish named after him include:
 * Osteochilus bleekeri is a species of cyprinid fish endemic to Borneo and Sumatra.
 * The Stone Loach Triplophysa bleekeri (Sauvage & Dabry de Thiersant, 1874)
 * Argyrops bleekeri Oshima, 1927

 * Chlorurus bleekeri, known commonly as Bleeker's parrotfish, is a species of marine fish in the family Scaridae.

Taxa described by him

 * See Category:Taxa named by Pieter Bleeker

title: Pieter Bleeker


text:
Talk:Serbian Hound

Country of origin?
Country of origin should probably be changed from Yugoslavia to nowadays existing country. The name of the dog suggests that this should probably be Serbia and Montenegro, but I'm not sure. --Romanm 08:27, 4 May 2004 (UTC)

 * Yes, the FCI lists the country of origin as Serbia and Montenegro, and that's apparently where most examples of the breed are, so I've changed it in the article (there may be a few similar problems with other breeds actually, although we've fixed a few recently) -- sannse (talk) 09:29, 4 May 2004 (UTC)

Hello
Where can I find more info about "Frank Laska" who described the breed in detail, along with other scenthounds of that time. --Partizanul (talk) 07:44, 15 March 2011 (UTC)

title: Talk:Serbian Hound


text:
Hank Thompson (musician)

Henry William Thompson (September 3, 1925 – November 6, 2007) was an American country music singer-songwriter and musician whose career spanned seven decades.

Thompson's musical style, characterized as honky-tonk Western swing, was a mixture of fiddles, electric guitar, and steel guitar that featured his distinctive, smooth baritone vocals.

His backing band, The Brazos Valley Boys, was voted the top Country Western Band for 14 years in a row by Billboard. Thompson pursued a "light" version of the Western swing sound that Bob Wills and others played; the primary difference between his music and that of Bob Wills was that Thompson, who used the swing beat and instrumentation to enhance his vocals, discouraged the intense instrumental soloing from his musicians that Wills encouraged; however, the "Hank Thompson sound" exceeded Bob Wills in top-40 country hits.

Although not as prominent on the top country charts in later decades, Thompson remained a recording artist and concert draw well into his 80s.

The 2013 game Grand Theft Auto V featured his version of the song "I Don't Hurt Anymore" in the fictional radio show, Rebel Radio.

The 1987 novel Crazy Heart by Thomas Cobb was inspired by Thompson's life, specifically by his practice of picking up a local band to back him when he toured. In 2009, Cobb's novel was turned into a successful film directed by Scott Cooper and starring Jeff Bridges in an Academy Award-winning performance.

Biography
Born in Waco, Texas, United States, Thompson was interested in music from an early age, and won several amateur harmonica contests. He decided to pursue his musical talent after serving in the United States Navy in World War II as a radioman and studying electrical engineering at Princeton University before his discharge. He had intended to continue those studies on the GI Bill following his 1946 discharge, and return to Waco. Later that year, after having regional hits with his first single "Whoa Sailor" for Globe Records, Dallas (Globe 124) and almost simultaneously "California Women" for another Dallas label (Blue Bonnet 123), he chose to pursue a full-time musical career.

1952 brought his first number-one single, "The Wild Side of Life", which contained the memorable line, "I didn't know God made honky-tonk angels". (This line inspired songwriter J. D. "Jay" Miller to write the 1952 answer song "It Wasn't God Who Made Honky Tonk Angels", which became the first hit single for pioneer female country vocalist Kitty Wells.) Other hits for Thompson followed in quick succession in the 1950s and 1960s.

Thompson began singing in a plaintive honky-tonk style similar to that of Ernest Tubb, but, desiring to secure more engagements in the dance halls of the Southwest, he reconfigured his band, the Brazos Valley Boys, to play a "light" version of the Western swing sound that Bob Wills and others made famous, emphasizing the dance beat and meticulous arrangements.

From 1947 to 1964, he recorded for Capitol Records, then joined Warner Bros. Records, where he remained from 1966 through 1967. From 1968 through 1980, he recorded for Dot Records and its successors, ABC Dot and MCA Records. In 1997, Thompson released Hank Thompson and Friends, a collection of solo tracks and duets with some of country music's most popular performers. In 2000, he released a new album, Seven Decades, on the Hightone label. The title reflected his recording history from the 1940s to 2000s.

Thompson was elected to the Country Music Hall of Fame in 1989, and was inducted into the Nashville Songwriters Hall of Fame in 1997. He continued touring throughout the U.S. until shortly before he became ill. Often, he worked with a reconstituted version of the Brazos Valley Boys that included a few original members.

Retirement and death
Thompson's last public performance was on October 8, 2007, in Waco, Texas, his birthplace. A smoker for most of his adult life, Thompson was admitted into a Texas hospital in mid-October for shortness of breath. He was diagnosed with a particularly aggressive form of lung cancer. On November 1, 2007, two days after being released, Thompson cancelled the rest of his 2007 "Sunset Tour" and retired from singing. He went into hospice care at his home in Keller, Texas, and died five days later, on November 6, 2007, aged 82.

According to Thompson's spokesman, Tracy Pitcox, president of Heart of Texas Records, Thompson requested that no funeral be held. On November 14, a "celebration of life", open to both fans and friends, took place at Billy Bob's Texas, a Ft. Worth country and western nightclub that bills itself as the World's Largest Honky-Tonk.

title: Hank Thompson (musician)


text:
User:Sanjeeth

WTHAI? (Who The Hell Am I?)
A Post Grad from the Indian Institute of Information Technology -Bangalore aka iiit-b.

INTERESTS
Computer Algorithms, Linux, Open source, Tamil Language, Calvin and Hobbes, Dramatics, Cricket

MY FAVOURITE TIME KILLERS
Googling, Wikipediaing, Coding (Primarily in C and Perl ), Reading, Quizzing, Writing and Thinking ;)

title: User:Sanjeeth


text: File:Homemap.jpg

title: File:Homemap.jpg


text:
User:AmarChandra

Not me

 * Himalaya - Nanda Devi National Park - Joe Brown - George Band - Noel Odell - George Mallory - Vilhjalmur Stefansson - Richard Francis Burton - Chris Bonington - David Breashears - Maynard Owen Williams - Roald Amundsen - Ernest Shackleton - Robert Falcon Scott - Thor Heyerdahl - Eric Shipton - Bill Tilman - Frank Smythe - Walter Bonatti - Tom Longstaff - Charles Evans

 * Yojimbo - Samurai Jack - A Fistful of Dollars - Once Upon a Time in the West - Kill Bill - Apocalypse Now - The Bourne Identity

 * The End (song) - Grateful Dead - Led Zeppelin
 * Snow leopard - Lammergeier - Bharal - Leopard - Bear - Firefly - Wasp - Ant - Beetle

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Lagrangian, Lagrangian mechanics, Gauge theory, Gauge boson, Isospin, Euler-Lagrange equations, Fundamental interaction, Particle in a ring, Quantum electrodynamics, Vladimir Igorevich Arnol'd, Bhagirathi, Gangotri, Garhwal, Alaknanda, Kanpur, Lucknow, Grand Trunk Road, IIT Kanpur, BS Chandrasekhar, Brihadisvara Temple, Kaveri

Creates
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title: User:AmarChandra


text:
The Glee Club

The Glee Club is a chain of independent live stand-up comedy and live music venues in the UK. The first Glee Club was opened by Mark Tughan in Birmingham's Chinese Quarter in 1994, the first dedicated comedy club to open in the United Kingdom outside London.

A second branch opened in Cardiff in 2001 with a third club opening in Oxford in April 2010 and a fourth, two room club debuted in Nottingham in September 2010.

The latest club to come online is The Glee Club Glasgow which opened in February 2019.

A number of major names cut their teeth playing at The Glee, including Jack Dee, Peter Kay, Lee Mack and Ross Noble.

The brand has been directly involved with numerous urban comedy festivals in their host cities, and also presented stand-up stages at major outdoor events, including V Festival from 2012-2017. The Glee Club Comedy Tent at V welcomed appearances from Sean Lock (V2012), Eddie Izzard and Josh Widdicombe (V2013), Alan Carr (V2014), Frankie Boyle and Russell Kane (V2016), and Joe Lycett (V2017).

Comedy show
The Glee's regular comedy show consists of three acts from the UK and international comedy circuits introduced by a compère. The Saturday shows are often followed by a disco in some venues. All clubs also host tour shows by more established acts. Unlike many other comedy clubs, the seating for all comedy gigs is arranged in a theatre style and the clubs' bars close during the performances.

Professional comedians enjoy appearing at the clubs; as Tom Allen reported: "The Glee Club people are excellent at creating an environment where people can enjoy themselves. There's a science to getting the venue and night out right and they are great at attention to detail, everything from the welcome at the door to getting the lighting right. In the past comedy could be quite a macho environment and the Glee Clubs have been great at tackling that and making sure their bills feature diverse performers."

Many professional acts also opt to play secret or low-key tour warm-up shows at The Glee. Among those who've previewed largescale theatre and arena shows, and test out new material, are John Bishop, Lee Evans, Michael McIntyre, Sarah Millican and Jack Whitehall.

Live music
The Glee Clubs have presented live music since April 2001, and presently host gigs in Birmingham, Nottingham, Cardiff and Glasgow. Among the acts who have appeared include Adele and Mumford and Sons. Music was initially brought to the Glee Club by promoter Markus Sargeant.

The Glee Birmingham
Situated in Southside, Birmingham, The Glee's original home opened in leisure development The Arcadian on 10 September 1994, and was the first dedicated comedy club to open in the UK outside London.

The 20th anniversary show, in 2014, was compered by Ross Noble.

The venue is home to several unique nights including Comedy Carousel and Rough Works. Hosted and created by comedian Andy Robinson, Comedy Carousel has won several awards including multiple What's On Readers' Awards for Best Comedy Club Night and a Chortle Award for Best Comedy Club Show 2015, while actress and comedian Jo Enight's Rough Works provides a platform for new and established acts try out new material. Also taking place at the club is The Birmingham Comedy Festival Breaking Talent Award, which recognises new and emerging talent from the West Midlands.

Many local comics have graced the Glee's stage, including Jasper Carrott, Joe Lycett, Lenny Henry and Guz Khan.

The Glee Cardiff
A second branch opened in Cardiff in 2001 and is located in the Mermaid Quay development in Cardiff Bay. The opening night line-up included Mark Lamarr and was "an instant hit" with audiences.

The venue has a seated capacity of 440 and a standing capacity of 600. ITV Wales broadcast a series of eight stand up shows from the venue in 2006.

It won the Chortle Award for Best Venue (Wales and West) 10 times between 2002-2017.

The Glee Oxford
The third Glee Club opened in Hythe Bridge Street, Oxford in April 2010 with a one-off appearance from Shappi Khorsandi and Angelos Epithemiou, and went on to host appearances from Sarah Millican, Russell Kane, Ross Noble and Jack Whitehall.

Due to redevelopment of the area, the club relocated to The Bullingdon in Cowley Road, Oxford, on 14 January 2017. Appearing on opening night were Tom Allen and Ellie Taylor. Tom described appearing as "a big honour."

The Glee Nottingham
A fourth, two room club debuted in Nottingham on 15 September 2010.

Situated in the British Waterways Building, on The Waterfront development, the venue programmes comedy, music and special events. Hollywood actor and Academy Award winner Tim Robbins and his Rogues Gallery Band was the venue's first music show, while a pre-fame Jake Bugg had a residency here in 2011,.

The club also hosts regular cabaret and burlesque nights hosted by The Gilded Merkin.

The Glee Glasgow
The Glee Club Glasgow opened on 1 February 2019 in Renfrew Street, and featured a rare club appearance by Sean Lock, plus Gary Little, Geoff Norcott and MC Jay Lafferty.

The club was planned for several years, and is open for regular mixed bill comedy nights on Fridays and Saturdays, with touring shows at other times.

A mural in the entrance/ bar area depicts Scottish comedians Kevin Bridges, Billy Connolly and Frankie Boyle by local artist Michael Scott.

Awards

 * Chortle Awards 2002: Best Venue - Midlands Runner-Up (Birmingham); West/Wales Winner (Cardiff)
 * Chortle Awards 2003: Best Venue West/Wales (Cardiff)
 * Chortle Awards 2004: Best Venue East/West Midlands (Birmingham)
 * Chortle Awards 2005: Best Venue West/Wales (Cardiff)
 * Chortle Awards 2006: Best Venue West/Wales (Cardiff)
 * Chortle Awards 2007: Best Venue West/Wales (Cardiff)
 * Chortle Awards 2008: Best Venue West/Wales (Cardiff)
 * Chortle Awards 2009: Best Venue Midlands/East (Birmingham); West/Wales (Cardiff)
 * Chortle Awards 2010: Best Venue Midlands/East (Birmingham)
 * Chortle Awards 2011: Best Venue Midlands/East (Birmingham); West/Wales (Cardiff)
 * Chortle Awards 2012: Best Venue Midlands/East (Birmingham)
 * Chortle Awards 2013: Best Venue Midlands/East (Birmingham)
 * Midlands Comedy Awards 2014: Best Comedy Club - Runner-up
 * Chortle Awards 2014: Best Venue Midlands/East (Birmingham); West/Wales (Cardiff)
 * Chortle Awards 2015: Best Club Night Midlands/East - Comedy Carousel (Birmingham); Best Venue Midlands/East (Birmingham)
 * Midlands Comedy Awards 2016: Best Comedy Club - Runner-up
 * Chortle Awards 2016: Best Venue Midlands/East (Birmingham)
 * Midlands Comedy Awards 2017: Best Comedy Club
 * Chortle Awards 2017: Best Venue Midlands/East (Birmingham); West/Wales Winner (Cardiff)
 * What's On Readers' Awards 2017: Best Midlands Comedy Night - Comedy Carousel (Birmingham)
 * Midlands Comedy Awards 2018: Best Comedy Club
 * What's On Readers' Awards 2018: Best Midlands Comedy Night - Comedy Carousel (Birmingham)
 * Chortle Awards 2019: Best Venue Midlands (Birmingham); Wales and The West Winner (Cardiff)
 * What's On Readers' Awards 2019: Best Midlands Comedy Night - Comedy Carousel (Birmingham)
 * Midlands Comedy Awards 2019: Best Comedy Club
 * Chortle Awards 2020: Best Venue Midlands & East Winner (Birmingham); Best Venue Wales Winner (Cardiff)
 * What's On Readers' Awards 2020: Best Midlands Comedy Night - Comedy Carousel (Birmingham)
 * Midlands Comedy Awards 2022: Best Comedy Club (Birmingham)
 * Chortle Awards 2023: Best Venue Midlands, West and Wales (Birmingham)

title: The Glee Club


text:
Talk:Wheel theory

References and definitions
Any references, definitions?

Charles Matthews 08:37, 4 May 2004 (UTC)

The link appears to give a reference to a forthcoming article by the contributor. While it appears to be perfectly legit, it seems like too much a case of original research at the moment...maybe wait a couple years until after publication, or once the definition (which isn't given here) has been used by a few others. Revolver 09:19, 15 Oct 2004 (UTC)

I've looked at an online paper; this area does seem to be of possible interest, though as you say this formulation is yet young. Charles Matthews 10:47, 15 Oct 2004 (UTC)

I've been able to find several other names associated with this. It seems to go back to the mid 1990s at least. I don't see any problem, but given it's pretty new, a list of references would be good. Revolver 09:40, 19 Oct 2004 (UTC)

Does anyone speak sweedish?

Quite a few Sweedes, I understand - but they're mostly in Sweeden, the country next to Noorway.

I started this article once. It is true that I am the author of the abovementioned paper, published in Mathematical Structures in Computer Science 14(1):143–184, Cambridge University Press, 2004. I have now learned that it is against the rules to write wiki articles about ones own inventions. I did it because people contacted me, saying that it was stated in Wikipedia that you cannot divide by zero, and they frequently thought that my results had disproved this claim. I had to explain to them that in the ordinary sense of division, you cannot divide by zero. I edited the article about division by zero, explaining roughly the same thing. I inserted a link there to this article, and I wrote it. Now, different people have expanded on the topic. A few minutes ago, I added a couple of sentences in the end. As long as noone objects, I will let the article remain here.

Jesper Carlström

 * No objections here. The reason so-called vanity articles are not allowed on Wikipedia is to ensure that the article does not contain original research, that the content is verifiable and notable, and that the article is written from a neutral point of view. Since you're citing a doctoral thesis and don't include weasel words, these requirements are definitely met.
 * — Kwi | Talk 19:57, 11 October 2006 (UTC)

Constants "0" and "1"
Are the constants "0" and "1" in wheels additive and multiplicative identities, respectively, as they are in fields? If so, shouldn't this be mentioned as an axiom? mike4ty4 00:12, 17 May 2007 (UTC)
 * Its in there
 * Addition and multiplication are commutative and associative, with 0 and 1 as units respectively
 * --Salix alba (talk) 00:29, 17 May 2007 (UTC)
 * "Unit" means something has a multiplicative inverse, but this does not specify what the multiplicative identity or additive identity is. "Unity" refers to the multiplicative identity, although I think "unit" can also be used to refer to identity, however it is confusing nonetheless. This is at least how it is in ring theory. Even then, in the article ring (mathematics) it is mentioned as a distinct axiom in defining a ring. Is the situation different for a wheel? Regardless, it is still ambiguous and definitions need to be crisp. mike4ty4 07:24, 17 May 2007 (UTC)
 * I think it was me who put it in there, so I've fixed it now. --Ørjan 00:49, 18 May 2007 (UTC)
 * Thanks! mike4ty4 08:31, 1 June 2007 (UTC)

0 = 1
Is the algebra still a wheel in the case where zero equals one? This seems rather degenerate. 76.190.157.141 (talk) 00:32, 26 February 2009 (UTC)
 * Well the axioms are variety axioms, so none of them require anything to be distinct, and a degenerate wheel is thus still a wheel.
 * A different question is whether 0=1 means everything else is also equal. Hm.  We get

x = 1x = 0x x + y = 0x + 0y = 0xy = xy (first derived identity in article) /0 + x = 0/0 + x = 0/0 = /0 /x = /(0x) = /0/x = /0 + /x = /0 x = //x = //0 = 0
 * So 0=1 makes everything else equal. --Ørjan (talk) 01:31, 26 February 2009 (UTC)

x/0
Consider the wheel made by extending the real numbers into a wheel. Are 1/0 and 2/0 distinct? Or are they the same? I can't seem to derive either result from the rules. Only that /0 + /0 = 0/0.

Is it that /0 + finite = /0 ? How do you prove that?

Or are these things not provable, but rather just one possible wheel that you can make out of the real numbers? --AndreRD (talk) 21:51, 1 December 2014 (UTC)

 * Let a be any value (e.g. any nonzero original real number) such that 0/a = 0, then

a/0 = //a /0 = /0 //a = /(0/a) = /0

 * Also, if 0z = 0 then

/0 + z = 1/0 + z + 0 0 = (1 + 0z)/0 = /0

 * --Ørjan (talk) 07:09, 3 December 2014 (UTC)

History and applications
It would be nice if this article had sections describing the history and applications.—Anita5192 (talk) 01:24, 17 November 2018 (UTC)

Has this concept ever appeared outside the articles that introduced it?
Also, in what way (if at all) does it clarify the question of division by 0. 73.89.25.252 (talk) 23:22, 15 June 2020 (UTC)

I added two more sources by the same guy, but, to be honest, I suspect this article does not meet Wikipedia's notability standards. I will revisit it later, to at least achieve understanding of the mathematics for myself ;) Notrium (talk) 23:43, 15 June 2020 (UTC)
 * Thanks. The abstract for the second article says that the 1/0 can be interpreted as "ERR", which does not sound like a formalism that accomplishes anything.  Abstract for first article says that it solves (with 1/0 defined as 0) some problem of equational presentation "without hidden functions".  I don't know what hidden functions means here, but this sounds potentially more interesting. 73.89.25.252 (talk) 23:54, 15 June 2020 (UTC)

/ used as binary operator
The article gives "0/0 + x = 0/0" as an example, but / is a unary operator, so this is incorrect, so I assume this is a typo somehow. 46.223.43.217 (talk) 20:34, 24 October 2020 (UTC)
 * I think what happening is that multiplication is represented as juxtaposition "x y". So using "*" for multiplication "x/y" is really "x*(/y)" i.e. the multiplication of x with the inverse of y. You can think of normal division in this way, the multiplication of x with the reciprocal of y. --Salix alba (talk): 21:26, 24 October 2020 (UTC)

title: Talk:Wheel theory


text:
Kaʻahumanu

Kaʻahumanu (March 17, 1768 – June 5, 1832) ("the feathered mantle") was queen consort and acted as regent of the Kingdom of Hawaiʻi as Kuhina Nui. She was the favorite wife of King Kamehameha I and also the most politically powerful, and continued to wield considerable power as co-ruler in the kingdom during reigns of his first two successors.

Early life
Kaʻahumanu was born in a cave called Puu Kauiki in Hāna on the Hawaiian island of Maui. She was born on 17 March 1768. The present Kaahumanu Society celebrates the birthday of its namesake on March 17. Her father was Keʻeaumoku Papaʻiahiahi, a fugitive aliʻi (noble) from the island of Hawaiʻi, and her mother was Nāmāhānaikaleleokalani, daughter of King Kekaulike Kalani-nui-Kui-Hono-i-Kamoku and the wife of her half-brother the late king of Maui, Kamehameha Nui. Through her mother she was related to many kings of Maui. Through her father, she was the third cousin of Kamehameha I, both sharing the common ancestor, Princess Kalanikauleleiaiwi of the island of Hawaiʻi. She was named after her father's rival, Kahekilinuiʻahumanu because it was from him that her father was fleeing at the time.

Her siblings include Governor John Adams Kuakini of Hawaii island, Queen Kalākua Kaheiheimālie, and Governor George Keʻeaumoku II of Maui. Her father became an advisor and friend to Kamehameha I, eventually becoming royal governor of Maui. He arranged for Kaʻahumanu to marry him when she was thirteen. Kamehameha had numerous wives but Kaʻahumanu would become a favorite and encouraged his war to unify the islands.

Queen Regent
Kaʻahumanu was one of Kamehameha I's favorite wives and his most powerful. Upon Kamehameha's death on May 8, 1819, Kaʻahumanu announced that late king had wished that she share governance over the Kingdom of Hawaiʻi with his 22-year-old son Liholiho, who took the name of Kamehameha II. The council of advisors agreed and created the post of kuhina nui for her, which was similar to co-regent or modern-day prime minister. Her power base grew and she ruled as Queen Regent during the reigns of both Kamehameha II and Kauikeaouli, who assumed the throne as Kamehameha III.

In some ways Kaʻahumanu was ahead of her time and championed the rights of native Hawaiian women, although this was to her own advantage. In what became known as the  'Ai Noa (free eating), Kaʻahumanu conspired with Keōpūolani, another of her late husband's wives who was also a Queen Regent during the reign of Kamehameha II, to eat at the same table with the young king. Notably, she also convinced the Kahuna-nui (translatable to High Priest) of the kingdom, Hewahewa, to support her efforts to abolish the kapu. Breaking a major kapu should have resulted in her death, but her son refused to kill her; this event effectively broke the monarchy's support of the kapu, and resulted in the system being outlawed.

Kaumualiʻi of Kauaʻi
The island of Kauaʻi and its subject island Niʻihau had never been forcibly conquered by Kamehameha. After years of resistance they negotiated a bloodless surrender in the face of Kamehameha's armada. In 1810 the island's King, Kaumualiʻi, became a vassal to Kamehameha. When Kamehameha I died, Kamehameha II and Kaʻahumanu feared Kauaʻi would break away from the kingdom. To preserve the union they kidnapped Kaumualiʻi on October 9, 1821, and Kaʻahumanu married him by force. After Kaumualiʻi died in 1824, and a rebellion by Kaumualiʻi's son Humehume was put down, she married his other son Kealiʻiahonui.

Conversion to Christianity
In April 1824, Kaʻahumanu publicly acknowledged her conversion to Protestant Christianity and encouraged her subjects to be baptized into the faith. That same year, she presented Hawaiʻi with its first codified body of laws modeled after Christian ethics and values and the Ten Commandments. Kaʻahumanu was baptized on December 5, 1825, at the site where Kawaiahaʻo Church stands today. She took the name "Elizabeth".

Missionaries persuaded Kaʻahumanu that the Roman Catholic Church, which had established the Cathedral of Our Lady of Peace in Honolulu, should be removed from the island nation. On July 7, 1827, she ordered the first Catholic missionaries to leave. In 1830, Kaʻahumanu signed legislation that forbade Catholic teachings and threatened to deport whoever broke the law.

In 1832, Kaʻahumanu visited Maui, and came to the site of what is now Kaʻahumanu Church, witnessing services being presided by Jonathan Smith Green. Upon seeing this, Queen Ka'ahumanu asked the Congregationalist mission to name the permanent church structure after her. However, this request was not honored until 1876 when Edward Bailey constructed the fourth and current structure on the site, naming it after the Queen.

Banning Hula
As regent of Hawai'i after the death of her husband, King Kamehameha I, Ka'ahumanu took it upon herself to enforce Christian policies with her power, banning of the Hawaiian Dance hula in 1830. After her death in 1832, some chiefs ignored this ban, including King Kamehameha III. However it was not until King Kalakaua's reign in 1886 that hula was celebrated openly once again: "Hula is the language of the heart and therefore the heartbeat of the Hawaiian people." Ka'ahumanu's policies on hula have had a ripple effect on the acceptability of the art form ever since.

Establishing American relations
Kaʻahumanu and King Kamehameha III negotiated the first treaty between the Kingdom of Hawaiʻi and the United States in 1826, under the administration of President John Quincy Adams. The treaty assumed responsibility on behalf of native Hawaiians with debts to American traders and paid the bill with $150,000 worth of sandalwood; this won her the support of chiefs who owed money to the traders. The same document was also a free trade treaty, ensuring Americans had the right to enter all ports of Hawaiʻi to do business. Americans were also afforded the right to sue in Hawaiian courts and be protected by Hawaiian laws.

In 1827, after Kaʻahumanu returned from a tour of the windward islands, her health steadily declined. During her illness missionaries printed the first copy, bound in red leather with her name engraved in gold letters, of the New Testament in the Hawaiian language. She kept it with her until her death of intestinal illness, June 5, 1832, in the Mānoa Valley near Honolulu. Her funeral was held at Kawaiahaʻo Church, often referred to as the Westminster Abbey of Hawaiʻi. Services were presided by Hiram Bingham. She was laid to rest on ʻIolani Palace grounds but was later moved to the Royal Mausoleum. The monument of Kaumualiʻi in Waiola Church cemetery includes the inscription, "Kaahumanu was his wife, Year 1822," leading some to mistakenly conclude that she is buried there.

title: Kaʻahumanu


text:
Wikipedia:Articles for deletion/Bene Tleixiau

Redundant, misspelt title name. Correct article at Bene Tleilax. - Zhen Lin 08:44, 4 May 2004 (UTC)

 * Now its a redirect to Bene Tleilax. No need to delete -- Chris 73 | Talk 08:57, 4 May 2004 (UTC)
 * Ah, but notice that there are absolutely no other articles that link to it. It should be cleared, should a real Bene Tleixiau need an article one day. It is a misspelling, not an alternative spelling. - Zhen Lin 09:00, 4 May 2004 (UTC)
 * There are quite a number of misspelling redirects. I think it does not hurt, and if there really is a Bene Tleixiau popping up one day, then we can un-redirect, and give him his own page. -- Chris 73 | Talk 12:15, 4 May 2004 (UTC)
 * Delete. This isn't a correct spelling, period, and as such, has no place on wikipedia. It's Bene Tleilaxu and it already has a redirect. Alcarillo 14:40, 4 May 2004 (UTC)
 * Keep misspelling redirect. -- Cyrius|&#9998 15:34, May 4, 2004 (UTC)
 * I agree with Cyrius. Rhymeless 21:13, 4 May 2004 (UTC)
 * Keep, unless we delete Tony Bliar. john 03:56, 5 May 2004 (UTC)
 * Tony B. Liar -- I like that! Do we have George W. Liar, too? ;-) -- Chris 73 | Talk 23:29, 5 May 2004 (UTC)
 * Delete. Unless an alternative spelling or a very frequent misspelling, I think there's no point to it. Suitov 10:40, 7 May 2004 (UTC)