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Wikipedia:Cite your sources debate

discussion from rules to consider regarding the keycited or assumed source distinction

"intellectually honest"? What does that mean? It sounds suspiciously like an attempt to frame a debate. Suppose I call this rule "intellectually paranoid." Wouldn't that strike you as curious? (I am being serious, btw--what does it mean?) --KQ

It's typically considered intellectually dishonest to pass off another person's work as your own. "Intellectually honest" in this case means "honest about representing what one's sources are." Wikipedia is an interesting case, however, in that we don't claim most of the articles as our own, nor do we claim that the articles are original work. If, therefore, the only reason for citing sources is to make sure that no mistakenly believes that an article is the author's own work, we might as well not cite sources, because no one can labor under that misconception on Wikipedia--because there are no authors per se!

But there is value in acknowledging other creative individuals as the source of Nupedia's information. We just looked stuff up, in some cases anyway, and we did it without citing what we looked up. We aren't taking credit for this, but someone might mistakenly think that we are just really smart and we did all this research ourselves.

Maybe the better argument for citing sources is just to give people good links to further reading. :-) --LMS

I hadn't considered the fact that we aren't claiming individual ownership of what we write. It just felt icky to me to use others' work and not acknowledge it. I also found myself consulting outside sources when I doubted some articles (and usually found the articles were, in fact, correct); providing reputable references may add strength to what we write when none of us can claim to be an expert.

Perhaps the term "intellectual honesty" doesn't need to be there, and maybe I should integrate some of what I've said here. I'll get back to it soon. -- Janet Davis

Ok. That phrase just raises hackles, sorry. :-)

I'm not being intentionally dense (though I suspect I am being dense about it somehow, since other people seem to understand it)... but: what is the difference between looking it up in a book where someone else has already looked it up and verified it, and verifying it ourselves? At what point do we quit citing other people? (at what point is something considered well-enough known that it doesn't need to be cited?) Also, suppose I take information from the Unnameable Source, which is now in the public domain, and update it and put it here. That source lists references; am I obligated to list them as well? What about if we read something several years past but remembered it; suppose the original source of knowledge was not deduction of the facts but some long-forgotten source: are we "intellectually dishonest" if we do not cite that source? (This is not a rhetorical question, as I did read voraciously about Dave Brubeck 5 or 6 years ago, and did the same thing about Stephen King over 10 years ago: biographies, interviews, essays, prefaces, etc.)

I'm coming to believe increasingly that Thomas Jefferson should have won out in the copyright debate, as he is quite correct that once you have been given a notion, you can not rid yourself of it.... But that is not the point, as my rantings will have no effect on copyright law or the codified behavior of Intellectually Honest people. Please don't think I'm being glib; I honestly do not understand, and I don't intend the questions rhetorically. --KQ

KQ: Some of us forget stuff. Lots of stuff. All the time. For this reason, I write everything I consider professionally important into a notebook (math gets a LaTeX summary), complete with references to external material and cross-referenced internally. I try to find several different references for anything I don't understand well, because in my experience, any individual reference may be incorrect in some key point.

For wikipedia, references probably aren't very important, because encyclopedias are not viable references for scholarly work, outside of work explicitly concerning encyclopedias. (I have read at minimum several hundred papers in fields spanning geology to mathemetical mechanics and never once seen an encyclopedia reference.)

Janet: Use without attribution is definitely icky. I think its ok if one limits oneself to facts in the public domain. Example: this months Natl Geographic has an article on some island off the coast of Chile with big caves. The "public domain" part of the article could be the name of the island, the location of the island, and 1 or 2 sentences on physiography: "Has limestone, lots of rains, big caves." Anything more than that, in my opinion, better have a link back to Natl Geog.

Larry: Maybe a disclaimer on the front page, and a tiny disclaimer link on each topic page served up. At some point, some knucklehead is going to serve wikipedia with a copyright violation notice. A disclaimer might make it easier to remove offending pages without any further consequences. (Let's not talk about possible patent violations... )

One further note about encouraging outside links: it will probably encourage spam. I would just love to write up a bunch of the stuff I do professionally, then link it back to my web site!

Hope this helps. DMD

First, plagarism and copyright violation are covered elsewhere, so for the sake of discussion, let's assume those issues aren't involved.

I don't think Wikipedia articles need a lot of formal citation. An encyclopedia article, in general, presents common knowledge within a field that could lead to dozens of citations. I don't particularly feel the need to mention that I perused several books or web sites to verify that sort of thing. Perhaps the urge for formal citation comes from the academic backgrounds of many of the contributors, where the meritocracy of intellectual credit is more strongly felt than in most other areas, and is highly formalized?

If a topic within an article is obscure, controversial, or just wants emphasizing, the wiki format encourages an informal citation style ("Professor Smith, in his definitive Opus 497, indicated that blah blah ...").

An encyclopedia article usually cannot have the breadth or depth of a book or focused research effort, (although Wikipedia and similar projects may change that view), so a "Suggested Reading" or "For Additional Information" reference section may be very appropriate.

To summarize, my feeling is that citations are only appropriate where the sense of the content can be clearly attributed to a particular work, but otherwise should not be a big deal -- with many editors reviewing articles, needed citations will likely appear later if not in the original article.

Just my rambling opinion.... --loh (2001-07-05)

Sources are good for traking serious scholarship, research, history, etc. But a large number of articles here aren't that; they are simple explanations of things the reader might be unfamiliar with, but about which we know something. I think "usefulness" is the most important criterion here, rather than rigorous scholarship. What is a useful "source" to cite for a article containing one paragraph explaining that Robert A. Heinlein is a science fiction author and listing his stories? An article about any subject should certainly point out important works on that subject, whether or not those were the sources of the information in the article. Typically any expert's knowledge of a subject will come from dozens of sources, many of which are very good in general and many of which are too specific to be of any use to newcomers, and much of it even the expert may not remember where it came from. If I'm writing about poker strategy, it would be unthinkable not to include some mention of David Sklanky's Theory of Poker and Mike Caro's Book of Tells; but if I happen to use an example from a game I played last week, will I even remember that something obscure like that Ray Zee's Seven Card Stud High-Low for Advanced Players had a chapter on the theory behind that particular play with a similar example? If I did remember, is it useful for ordinary readers to know that if that book is otherwise not that useful to them? --LDC

Just a question about formatting. What is the proper (in Wikipedia) way to acknowledge a source in the body of the article? Do we have a way to make footnotes? Should we use the HTML href and name tags to refer to items in the bibliography at the end of the article? An example of something I've done that I'm not happy with is at the beginning of fractal. More simply: does Wikipedia need a standard for quotations and references?

Some of the more serious articles have had numbered references in footnotes, and I don't see a problem with that. Unfortunately, the software does not yet allow page-internal links, so we can't link to them, but numbers should be fine. I don't tink we should rigidly follow CMS, though. This is a new medium, and we should explore new methods. For example, standard bibliographies don't include ISBNs. In paper books, that makes sense, because library catalogs are organized by author, topic, etc. On the web, that's monumentally stupid, because ISBNs are the key to every online database like Amazon. Let me clean up the fractal article and see if you like the style. --Lee Daniel Crocker

- Yes, ISBNs must be included, they are far more important than the standard bibliographic entries which are obsolete - easily looked up given the ISBN. Same for ISSN. There are reference standards used for other online systems, check out bookfinder if you really want to find everything there is. We should be assuming that such obscure sources would need to be found somewhere like that, and it deals nicely with both current and out of print books - you could do worse than add bookfinder and google searches to every page. -24

- Write your draft, find your controversy, pick the extremes as sources, and find the foundation. -24

There are a few people here who think they are entitled to source on demand, or that their suspicion makes them experts on a topic, e.g. they can't tell that "fiat" and "military fiat" are the same thing, they can't tell that "bioregional" and "ecoregional" are the same thing, and various other failures of cognition may apply. Dictionaries may be interested in these distinctions, but if we are, it's a sign that incompetents are getting into editing. It's fair to fix and add synonyms, but there are people out there deleting whole articles because it doesn't fit their pet terminology.

 * 24, never assume what people do or do not think or understand. Thank you.  Koyaanis Qatsi
 * I will do so every minute of every day for the rest of my life, because it's unavoidable. We can't live examining each others' motives in infinite depth.  I can cite specific examples of the behavior noted, but let's not get into it, as I didn't name names - for a reason.   I think this behavior has settled down somewhat.  24

As to frequency of source quoting, there are lots of people who simply don't understand what they read and object to a line here and a fact there. Fine. Forget them. Over time, factual errors will be corrected by pedants and unless they're absolutely central to the argument (as they shouldn't be, an encyclopedia article should never be describing anything so narrowly causal as to hinge on a single fact or example) they don't affect the rest of the article.

 * You are assuming two things here: 1: that you do understand, and 2: that since someone disagrees with you, they must not understand. See the above.  Koyaanis Qatsi
 * I don't see how anyone could 1. get by every day without believing that they understood the things that their lives depended on or where they had very outstanding success in the past, nor 2. mistake, as you have, a problem with a narrow range of "lots of people" (a clique), for a problem with all humanity.

Various extremely controversial topics, notably politics and religion, but in matters of the state and its use of power also economics and psychiatry, are going to necessarily require an extremely careful choice of terms. Even just to choose the terms from one side of a debate, e.g. using a word like say "sociopath" which hsa many meanings, is taking a side. Same issue as "vandal" or "miscreant" or whatever. A good example is the way a Marxist says "means of production" and a classical economics says "factors of production" - to the classical, labor is just another factor, a commodity. to most people in the world, they don't want to be treated as such with no regard to their creativity, family, social ties, etc.. That's a simple example. A more complex one arises every time you get into debates about God, e.g. many atheists will vehemently deny it's a faith even though it clearly is, the neutral position is called "agnostic" (not caring and considering both theist and atheist positions to be based on a foundation axiom that they made up).

So when making that kind of claim, on that kind of topic, attribute where you can - the debates are controversial enough you should be able to do that. But don't become afraid to write the text itself, to lay out relations between the positions of various experts, etc., there is always glue, and always trust in writing and reading. Anyone who claims otherwise and thinks they deserve to have a citation on demand is just incapable of trust and ought to be ignored.

 * yes, attribution is important. This, 24, is where your own writing on controversial subjects is at its weakest.  And your opinion on whom I ought to trust is completely irrelevant.  Koyaanis Qatsi
 * if you insist on citation on demand, well, there are many many long and weird articles without citations, including most of Larry's. There are some profound issues with his POV on some questions, but I don't believe that I can just bug him to tell me why he believes what he believes about the topic, whenever I want.  Attribution only becomes "important" when you don't trust.  Thus, people may expect more of it from me than Larry.  Fine.  That helps to improve the articles.  They may resent this requirement of improving articles.  Fine.  They can do something else with their time, and others will step in.  You don't seem to realize that my "own writing on controversial subjects" is *deliberately* weak on attribution.  It's a *policy*.  Note that there are many articles I've written that are quite over-attributed.  But if the topic is controversial, why sully MY credibility with that of the sources?  Fewer the better, until I know *why* the topic is controversial.  Then the article can improve drastically as I focus on neutralizing places where a POV was unintentionally taken.

On controversial topics, once you know what the controversy actually is at the moment, you should be able to join forces with someone with an opposing point of view and nail down two or three sharply opposed references, and a foundation you all believe in. That obviously won't happen on the first pass, but you will identify who is actually concerned to represent the topic fairly, and who is simply insisting on their own view or whta they learned in high school as "neutral", and who is hopelessly stuck in some systemic bias. You will also find out that certain people hate you, your politics, your attitude, or your viewpoint based on things written in talk files. Fine. Let them howl.

This project cannot solve its social problems without a means of governance and there is seemingly zero interest in avoiding a devolution to anarchy or expanding discussion of governance beyond a clique, so forget those concerns. Push to the limit of neutrality as you understand the field in the real world, and ignore the people here who think they know what neutral means.

The correct way to find what is neutral is not by prescription but by successive refinement. In that view, sources go on at the end, not at the beginning, although if you don't have two or three trusted names in an article, you don't give your opponent anything to hinge on. So try to do that.

24

 * Your comment encouraging people not to give sources so that opponents have nothing to hinge on indicates that you are not interested in dialogue, or compromise;
 * ROTFL - the comment says the opposite, but in a way designed to trap people like you into revealing your biased way of reading. Obviously anyone who reads that sentence with all those negatives can interpret it as "give those trusted names" or "do not give those trusted names".  you chose the latter, but I read it the former way. 24

and so indicates to me that you are here not to reach consensus, as you pretend interest in (after all, what is anarchy based on) but to push your own agenda. Mull on that if you need to. Koyaanis Qatsi
 * I don't, although I appreciate your answer which illustrates your assumptions. I am extremely interested in consensus process but note that it is poorly understood here.  I am interested also in governance but I am told that Systemic Bias in Wikipedia does not exist or is irrelevant, and comes with a monarchy appointed long before I got here.  This is all very funny, but it has nothing to do with consensus or preventing anarchy.  What I need to mull on, is whether I am wasting my time educating people in consensus, why it is not unanimity, and why it is not monarchy either.  I have yet to see anyone here outline what theGoverning Ontological distinction actually is, that makes them believe that a source needs to be cited, versus not. 24

24 - not on the first pass - too often, impressive-looking citations and references make people accept nonsense or heavily discredited views that were popular 200 years ago. So, find out what the controversial statements are, and only then attribute where required, simultaneously indicating who said what on the controversial matters - KQ's concern can be best dealt with by realizing we are not a community but a market... and have no "opinion"...

title: Wikipedia:Cite your sources debate


text:
User:Slartibartfasz

yep its me :)

nice work on norway, wasnt it?

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

title: User:Slartibartfasz


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R. A. Lafferty

Raphael Aloysius "R. A." Lafferty (November 7, 1914 – March 18, 2002) was an American science fiction and fantasy writer known for his original use of language, metaphor, and narrative structure, Lafferty also wrote a set of four autobiographical novels, a history book, and several novels of historical fiction.

Biography
Lafferty was born on November 7, 1914, in Neola, Iowa to devoutly Catholic parents, Hugh David Lafferty, a broker dealing in oil leases and royalties, and Julia Mary ( Burke), a teacher. He was born the youngest of five siblings. His first name, Raphael, derived from the day on which he was expected to be born (the Feast of St. Raphael). When he was 4, his family moved to Perry, Oklahoma.

He graduated from Cascia Hall, and came of age in the early years of the Great Depression. He later attended night school at the University of Tulsa for two years starting in 1933, mostly studying Math and German, but left before graduating. He then began to work for Clark Electric Co. in Tulsa and, during this period (1939–42), he attended the International Correspondence School.

Lafferty lived most of his life in Tulsa with his sister, Anna Lafferty. He enlisted in the U.S. Army in 1942. After training in Texas, North Carolina, Florida, and California, he was sent to the South Pacific Area, serving in Australia, New Guinea, Morotai, and the Philippines. When he left the Army in 1946, he had become a 1st Sergeant serving as a staff sergeant and had received an Asiatic-Pacific Campaign Medal. He never married.

Lafferty did not begin writing until the 1950s, but he wrote thirty-two novels and more than two hundred short stories, most of them at least nominally science fiction. His first published story was "The Wagons" in the New Mexico Quarterly Review in 1959. His first published science fiction story was "Day of the Glacier", in The Original Science Fiction Stories in 1960, and his first published novel was Past Master in 1968.

Until 1971, Lafferty worked as an electrical engineer. After that, he spent his time writing until around 1980, when his output declined due to a stroke. He stopped writing regularly in 1984.

In 1994, he suffered an even more severe stroke. He died on March 18, 2002, aged 87 in a nursing home in Broken Arrow, Oklahoma. His collected papers, artifacts, and ephemera were donated to the University of Tulsa's McFarlin Library, Department of Special Collections and University Archives. Other manuscripts are housed in the University of Iowa's Library special collections department.

Lafferty's funeral took place at Christ the King Catholic Church in Tulsa, where he regularly attended daily Mass. He is buried at St. Rose Catholic Cemetery in Perry.

Selected works
Lafferty's quirky prose drew from traditional storytelling styles, largely from the Irish and Native American, and his shaggy-dog characters and tall tales are unique in science fiction. Little of Lafferty's writing is considered typical of the genre. His stories are closer to tall tales than traditional science fiction and are deeply influenced by his Catholic beliefs; Fourth Mansions, for example, draws on The Interior Mansions of Teresa of Avila.

His writings, both topically and stylistically, are not easy to categorize. Plot is frequently secondary to other elements of Lafferty's writing. While this style has resulted in a loyal cult following, it causes some readers to give up reading his work. Not all of Lafferty's work was science fiction or fantasy. His novel Okla Hannali (1972), published by University of Oklahoma Press, tells the story of the Choctaw in Mississippi, and after the Trail of Tears, in Oklahoma, through an account of the larger-than-life character Hannali and his large family. This novel was thought of highly by the novelist Dee Brown, author of Bury My Heart at Wounded Knee (1970), who on the back cover of the edition published by the University of Oklahoma Press, writes "The history of the Choctaw Indians has been told before and is still being told, but it has never been told in the way Lafferty tells it ... Hannali is a buffalo bull of a man who should become one of the enduring characters in the literature of the American Indian." He also wrote, "It is art applied to history so that the legend of the Choctaws, their great and small men, their splendid humor, and their tragedies are filled with life and breath."

Lafferty's work is represented by Virginia Kidd Literary Agency, which holds a cache of his unpublished manuscripts. This includes over a dozen novels, such as In The Akrokeraunian Mountains and Iron Tongue of Midnight, as well as about eighty short stories and a handful of essays.

Science fiction

 * Past Master, (1968); Hugo Award nominee, 1969; Nebula Award nominee 1968
 * The Reefs of Earth (1968)
 * Space Chantey (1968); a retelling of the Odyssey in SF terms
 * Fourth Mansions (1969); Nebula Award nominee, 1970
 * The Devil is Dead (1971); Nebula Award nominee, 1972 [Second chronologically in The Devil is Dead trilogy]
 * Arrive at Easterwine: The Autobiography of a Ktistec Machine (1971)
 * Not to Mention Camels (1976)
 * Archipelago (1979); [First chronologically in The Devil is Dead trilogy]
 * Aurelia (1982); Philip K. Dick Award nominee, 1982
 * Annals of Klepsis (1983)
 * Serpent's Egg (1987)
 * East of Laughter (1988)
 * How Many Miles to Babylon? (1989)
 * The Elliptical Grave (1989)
 * Dotty (1990)
 * More Than Melchisedech (1992); [Third chronologically in The Devil is Dead trilogy, consists of three novels]
 * Tales of Chicago
 * Tales of Midnight
 * Argo
 * Sindbad: The Thirteenth Voyage (1989)

Other

 * The Flame is Green (1971); [First in the unfinished Coscuin Chronicles]
 * Okla Hannali (1972)
 * Half a Sky (1984) [Second in the unfinished Coscuin Chronicles]

Collections

 * Nine Hundred Grandmothers (1970)
 * Strange Doings (1972)
 * Does Anyone Else Have Something Further to Add? (1974)
 * Funnyfingers & Cabrito (1976)
 * Apocalypses (1977)
 * Golden Gate and Other Stories (1982)
 * Through Elegant Eyes (1983)
 * Ringing Changes (1984)
 * The Early Lafferty (1988)
 * The Back Door of History (1988)
 * Strange Skies (1988); poems
 * The Early Lafferty II (1990)
 * Episodes of the Argo (1990)
 * Lafferty in Orbit (1991); World Fantasy Award nominee, 1992
 * Mischief Malicious (And Murder Most Strange) (1991)
 * Iron Tears (1992); Philip K. Dick Award nominee, 1992
 * The Man Who Made Models – The Collected Short Fiction Volume 1 (2014)
 * The Man With the Aura – The Collected Short Fiction Volume 2 (2015)
 * The Man Underneath – The Collected Short Fiction Volume 3 (2015)
 * The Man With The Speckled Eyes – The Collected Short Fiction Volume 4 (2017)
 * The Man Who Walked Through Cracks – The Collected Short Fiction Volume 5 (2018)
 * The Best of R. A. Lafferty (2019)

Non-fiction

 * The Fall of Rome (1971); reprinted in 1993 as Alaric: The Day the World Ended
 * It's Down the Slippery Cellar Stairs (1984)
 * True Believers (1989)
 * Cranky Old Man from Tulsa (1990)

Short stories

 * "Through Other Eyes" (Future Science Fiction, February 1960)
 * "All the People" (Galaxy Science Fiction, April 1961)
 * "The Weirdest World" (Galaxy, June 1961)
 * "Aloys" (Galaxy, August 1961)
 * "Rainbird" (Galaxy, December 1961)
 * "Dream" (Galaxy, June 1962)
 * "Sodom and Gomorrah, Texas" (Galaxy, December 1962)
 * "What the Name of That Town?" (Galaxy, October 1964)
 * "Slow Tuesday Night" (Galaxy, April 1965)
 * "Among the Hairy Earthmen" (Galaxy, August 1966)
 * "Land of the Great Horses" (Dangerous Visions, 1967)
 * "Thus We Frustrate Charlemagne" (Galaxy, February 1967)
 * "How They Gave It Back" (Galaxy, February 1968)
 * "McGruder's Marvels" (Galaxy, July 1968)
 * "Eurema's Dam" (New Dimensions II, 1972)
 * "The World as Will and Wallpaper", the title a wordplay on The World as Will and Representation (Future City, 1973)

Awards and recognition
Lafferty received Hugo Award nominations for Past Master, "Continued on Next Rock", "Sky", and "Eurema's Dam", the last of which won the Best Short Story Hugo in 1973 (shared with Frederik Pohl and Cyril M. Kornbluth's "The Meeting").

He received Nebula Award nominations for "In Our Block", "Slow Tuesday Night", Past Master, Fourth Mansions, "Continued on Next Rock", "Entire And Perfect Chrysolite", and The Devil is Dead. He never received a Nebula award.

His collection Lafferty in Orbit was nominated for a World Fantasy Award, and in 1990, Lafferty received a World Fantasy Lifetime Achievement Award. His 1992 collection Iron Tears was also a finalist for the Philip K. Dick Award. In 2002, he received the Cordwainer Smith Foundation's Rediscovery award.

The Oklahoma Department of Libraries granted him the Arrell Gibson Lifetime Achievement Award in 1995.

Fourth Mansions was also named by David Pringle as one of his selections for Modern Fantasy: The 100 Best Novels.

Writing style
" [Once a] French publisher nervously asked whether Lafferty minded being compared to G. K. Chesterton (another Catholic author), and there was a terrifying silence that went on and on. Was the great man hideously offended? Eventually, very slowly, he said: 'You're on the right track, kid,' and wandered away."

In his 2006 short story collection Fragile Things, Neil Gaiman includes a short story called "Sunbird" written in the style of Lafferty. In the introduction, he says this about Lafferty:"There was a writer from Tulsa, Oklahoma (he died in 2002), who was, for a little while in the late 1960s and early 1970s, the best short story writer in the world. His name was R. A. Lafferty, and his stories were unclassifiable and odd and inimitable -- you knew you were reading a Lafferty story within a sentence. When I was young I wrote to him, and he wrote back. 'Sunbird' was my attempt to write a Lafferty story, and it taught me a number of things, mostly how much harder they are than they look...."

Gaiman and Lafferty had corresponded for several years during Gaiman's adolescence; he remembered Lafferty's letters as "filled with typical cock-eyed Lafferty humour and observations, wise and funny and sober all at once."

Per the Science Fiction Encyclopedia: "He has fairly been described as a writer of tall tales, as a cartoonist, as an author whose tone was fundamentally oral; his conservative Catholicism has been seen as permeating every word he wrote (or has been ignored); he has been seen as a ransacker of old Mythologies, and as a flippant generator of new ones; he clearly delighted in a vision of the world as being irradiated by conspiracies both godly and devilish, but at times paid scant attention to the niceties of plotting; he has been understood by some as essentially light-hearted and by others as a solitary, stringent moralist; he was technically inventive, but lunged constantly into a slapdash sublime only skittishly evocative, and only occasionally, of anything like the traditional Sense of Wonder; his skill in the deploying of various rhetorical narrative voices was manifest, but these voices were sometimes choked in baroque flamboyance... He and Gene Wolfe have more than a shared faith in common."

Archives
In March 2011, it was announced in Locus that the copyrights to 29 Lafferty novels and 225 short stories were up for sale. The literary estate was soon thereafter purchased by the magazine's nonprofit foundation, under the auspices of board member Neil Gaiman.

title: R. A. Lafferty


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Semidirect product

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products.
 * an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup.
 * an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation.

For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension).

Inner semidirect product definitions
Given a group $G$ with identity element $e$, a subgroup $H$, and a normal subgroup $N ◁ G$, the following statements are equivalent:
 * $G$ is the product of subgroups, $G = NH$, and these subgroups have trivial intersection: $N ∩ H = \{e\}$.
 * For every $g ∈ G$, there are unique $n ∈ N$ and $h ∈ H$ such that $g = nh$.
 * The composition $π ∘ i$ of the natural embedding $i: H → G$ with the natural projection $π: G → G/N$ is an isomorphism between $H$ and the quotient group $G/N$.
 * There exists a homomorphism $G → H$ that is the identity on $H$ and whose kernel is $N$. In other words, there is a split exact sequence
 * $$1 \to N \to G \to H \to 1$$

 * of groups (which is also known as group extension of $$H$$ by $$N$$).

If any of these statements holds (and hence all of them hold, by their equivalence), we say $G$ is the semidirect product of $N$ and $H$, written

 * $$G = N \rtimes H$$ or $$G = H \ltimes N,$$

or that $G$ splits over $N$; one also says that $G$ is a semidirect product of $H$ acting on $N$, or even a semidirect product of $H$ and $N$. To avoid ambiguity, it is advisable to specify which is the normal subgroup.

If $$G = N \rtimes H$$, then there is a group homomorphism $$\varphi\colon H\rightarrow \mathrm{Aut} (N)$$ given by $$\varphi_h(n)=hnh^{-1}$$, and for $$g=nh,g'=n'h'$$, we have $$gg'=nhn'h' = nhn'h^{-1}hh' = n\varphi_{h}(n')hh' = n^* h^* $$.

Inner and outer semidirect products
Let us first consider the inner semidirect product. In this case, for a group $$G$$, consider a normal subgroup $N$ and another subgroup $H$ (not necessarily normal). Assume that the conditions on the list above hold. Let $$\operatorname{Aut}(N)$$ denote the group of all automorphisms of $N$, which is a group under composition. Construct a group homomorphism $$\varphi \colon H \to \operatorname{Aut}(N)$$ defined by conjugation,
 * $$\varphi_h(n) = hnh^{-1}$$, for all $h$ in $H$ and $n$ in $N$.

In this way we can construct a group $$G'=(N,H)$$ with group operation defined as
 * $$ (n_1, h_1) \cdot (n_2, h_2) = (n_1 \varphi_{h_1}(n_2),\, h_1 h_2)$$ for $n_{1}, n_{2}$ in $N$ and $h_{1}, h_{2}$ in $H$.

The subgroups $N$ and $H$ determine $G$ up to isomorphism, as we will show later. In this way, we can construct the group $G$ from its subgroups. This kind of construction is called an inner semidirect product (also known as internal semidirect product ).

Let us now consider the outer semidirect product. Given any two groups $N$ and $H$ and a group homomorphism $φ: H → Aut(N)$, we can construct a new group $N ⋊φ H$, called the outer semidirect product of $N$ and $H$ with respect to $φ$, defined as follows: • The underlying set is the Cartesian product $N × H$.

• The group operation $\bullet$ is determined by the homomorphism $φ$:
 * $\begin{align}

\bullet : (N \rtimes_\varphi H) \times (N \rtimes_\varphi H) &\to N \rtimes_\varphi H\\ (n_1, h_1) \bullet (n_2, h_2) &= (n_1 \varphi_{h_1}(n_2),\, h_1 h_2) \end{align}$ for $n_{1}, n_{2}$ in $N$ and $h_{1}, h_{2}$ in $H$.

This defines a group in which the identity element is $(e_{N}, e_{H})$ and the inverse of the element $(n, h)$ is $(φ_{h^{−1}}(n^{−1}), h^{−1})|undefined$. Pairs $(n, e_{H})$ form a normal subgroup isomorphic to $N$, while pairs $(e_{N}, h)$ form a subgroup isomorphic to $H$. The full group is a semidirect product of those two subgroups in the sense given earlier.

Conversely, suppose that we are given a group $G$ with a normal subgroup $N$ and a subgroup $H$, such that every element $g$ of $G$ may be written uniquely in the form $g = nh$ where $n$ lies in $N$ and $h$ lies in $H$. Let $φ: H → Aut(N)$ be the homomorphism (written $φ(h) = φ_{h}$) given by
 * $$\varphi_h(n) = hnh^{-1}$$

for all $n ∈ N, h ∈ H$.

Then $G$ is isomorphic to the semidirect product $N ⋊φ H$. The isomorphism $λ: G → N ⋊φ H$ is well defined by $λ(a) = λ(nh) = (n, h)$ due to the uniqueness of the decomposition $a = nh$.

In $G$, we have
 * $$(n_1 h_1)(n_2 h_2) = n_1 h_1 n_2(h_1^{-1}h_1) h_2 =

(n_1 \varphi_{h_1}(n_2))(h_1 h_2)$$ Thus, for $a = n1h1$ and $b = n2h2$ we obtain
 * $$\begin{align}

\lambda(ab) & = \lambda(n_1 h_1 n_2 h_2) = \lambda(n_1 \varphi_{h_1} (n_2) h_1 h_2) = (n_1 \varphi_{h_1} (n_2), h_1 h_2) = (n_1, h_1) \bullet (n_2, h_2) \\[5pt] & = \lambda(n_1 h_1) \bullet \lambda(n_2 h_2) = \lambda(a) \bullet \lambda(b), \end{align}$$ which proves that $λ$ is a homomorphism. Since $λ$ is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in $N ⋊φ H$.

The direct product is a special case of the semidirect product. To see this, let $φ$ be the trivial homomorphism (i.e., sending every element of $H$ to the identity automorphism of $N$) then $N ⋊φ H$ is the direct product $N × H$.

A version of the splitting lemma for groups states that a group $G$ is isomorphic to a semidirect product of the two groups $N$ and $H$ if and only if there exists a short exact sequence

 * $$ 1 \longrightarrow N \,\overset{\beta}{\longrightarrow}\, G \,\overset{\alpha}{\longrightarrow}\, H \longrightarrow 1$$

and a group homomorphism $γ: H → G$ such that $α ∘ γ = id_{H}$, the identity map on $H$. In this case, $φ: H → Aut(N)$ is given by $φ(h) = φ_{h}$, where
 * $$\varphi_h(n) = \beta^{-1}(\gamma(h)\beta(n)\gamma(h^{-1})).$$

Dihedral group
The dihedral group $D2n$ with $2n$ elements is isomorphic to a semidirect product of the cyclic groups $Cn$ and $C2$. Here, the non-identity element of $C2$ acts on $Cn$ by inverting elements; this is an automorphism since $Cn$ is abelian. The presentation for this group is:
 * $$\langle a,\;b \mid a^2 = e,\; b^n = e,\; aba^{-1} = b^{-1}\rangle.$$

Cyclic groups
More generally, a semidirect product of any two cyclic groups $Cm$ with generator $a$ and $Cn$ with generator $b$ is given by one extra relation, $aba−1 = bk$, with $k$ and $n$ coprime, and $$k^m\equiv 1 \pmod{n}$$; that is, the presentation:
 * $$\langle a,\;b \mid a^m = e,\;b^n = e,\;aba^{-1} = b^k\rangle.$$

If $r$ and $m$ are coprime, $ar$ is a generator of $Cm$ and $arba−r = bkr$, hence the presentation:
 * $$\langle a,\;b \mid a^m = e,\;b^n = e,\;aba^{-1} = b^{k^{r}}\rangle$$

gives a group isomorphic to the previous one.

Holomorph of a group
One canonical example of a group expressed as a semi-direct product is the holomorph of a group. This is defined as"$\operatorname{Hol}(G)=G\rtimes \operatorname{Aut}(G)$"where $$\text{Aut}(G)$$ is the automorphism group of a group $$G$$ and the structure map $$\varphi$$ comes from the right action of $$\text{Aut}(G)$$ on $$G$$. In terms of multiplying elements, this gives the group structure"$(g,\alpha)(h,\beta)=(g(\varphi(\alpha)\cdot h),\alpha\beta).$"

Fundamental group of the Klein bottle
The fundamental group of the Klein bottle can be presented in the form
 * $$\langle a,\;b \mid aba^{-1} = b^{-1}\rangle.$$

and is therefore a semidirect product of the group of integers, $$\mathbb{Z}$$, with $$\mathbb{Z}$$. The corresponding homomorphism $φ: $\mathbb{Z}$ → Aut($\mathbb{Z}$)$ is given by $φ(h)(n) = (−1)hn$.

Upper triangular matrices
The group $$\mathbb{T}_n$$ of upper triangular matrices with non-zero determinant in an arbitrary field, that is with non-zero entries on the diagonal, has a decomposition into the semidirect product $$\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n$$ where $$\mathbb{U}_n$$ is the subgroup of matrices with only $$1$$'s on the diagonal, which is called the upper unitriangular matrix group, and $$\mathbb{D}_n$$ is the subgroup of diagonal matrices.

The group action of $$\mathbb{D}_n$$ on $$\mathbb{U}_n$$ is induced by matrix multiplication. If we set

$$A = \begin{bmatrix} x_1 & 0 & \cdots & 0 \\ 0 & x_2 & \cdots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \cdots & x_n \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & a_{12} & a_{13} & \cdots & a_{1n} \\ 0 & 1 & a_{23} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}$$

then their matrix product is
 * $$AB =

\begin{bmatrix} x_1 & x_1a_{12} & x_1a_{13} & \cdots & x_1a_{1n} \\ 0 & x_2 & x_2a_{23} & \cdots & x_2a_{2n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & x_n \end{bmatrix}.$$ This gives the induced group action $$m:\mathbb{D}_n\times \mathbb{U}_n \to \mathbb{U}_n$$
 * $$m(A,B) = \begin{bmatrix}

1 & x_1a_{12} & x_1a_{13} & \cdots & x_1a_{1n} \\ 0 & 1 & x_2a_{23} & \cdots & x_2a_{2n} \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}.$$ A matrix in $$\mathbb{T}_n$$ can be represented by matrices in $$\mathbb{U}_n$$ and $$\mathbb{D}_n$$. Hence $$\mathbb{T}_n \cong \mathbb{U}_n \rtimes \mathbb{D}_n$$.

Group of isometries on the plane
The Euclidean group of all rigid motions (isometries) of the plane (maps $f: $\mathbb{R}$2 → $\mathbb{R}$2$ such that the Euclidean distance between $x$ and $y$ equals the distance between $f(x)$ and $f(y)$ for all $x$ and $y$ in $$\mathbb{R}^2$$) is isomorphic to a semidirect product of the abelian group $$\mathbb{R}^2$$ (which describes translations) and the group $O(2)$ of orthogonal $2 × 2$ matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and $O(2)$, and that the corresponding homomorphism $φ: O(2) → Aut($\mathbb{R}$2)$ is given by matrix multiplication: $φ(h)(n) = hn$.

Orthogonal group O(n)
The orthogonal group $O(n)$ of all orthogonal real $n × n$ matrices (intuitively the set of all rotations and reflections of $n$-dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group $SO(n)$ (consisting of all orthogonal matrices with determinant $1$, intuitively the rotations of $n$-dimensional space) and $C2$. If we represent $C2$ as the multiplicative group of matrices ${I, R}$, where $R$ is a reflection of $n$-dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant $–1$ representing an involution), then $φ: C2 → Aut(SO(n))$ is given by $φ(H)(N) = HNH−1$ for all H in $C2$ and $N$ in $SO(n)$. In the non-trivial case ($H$ is not the identity) this means that $φ(H)$ is conjugation of operations by the reflection (in 3-dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image").

Semi-linear transformations
The group of semilinear transformations on a vector space $V$ over a field $$\mathbb{K}$$, often denoted $ΓL(V)$, is isomorphic to a semidirect product of the linear group $GL(V)$ (a normal subgroup of $ΓL(V)$), and the automorphism group of $$\mathbb{K}$$.

Crystallographic groups
In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic. Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis.

Non-examples
Of course, no simple group can be expressed as a semi-direct product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semi-direct product. Note that although not every group $$G$$ can be expressed as a split extension of $$H$$ by $$A$$, it turns out that such a group can be embedded into the wreath product $$A\wr H$$ by the universal embedding theorem.

Z4
The cyclic group $$\mathbb{Z}_4$$ is not a simple group since it has a subgroup of order 2, namely $$\{0,2\} \cong \mathbb{Z}_2$$ is a subgroup and their quotient is $$\mathbb{Z}_2$$, so there's an extension"$0 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 0$"If the extension was split, then the group $$G$$ in"$0 \to \mathbb{Z}_2 \to G \to \mathbb{Z}_2 \to 0$"would be isomorphic to $$\mathbb{Z}_2\times\mathbb{Z}_2$$.

Q8
The group of the eight quaternions $$\{\pm 1,\pm i,\pm j,\pm k\}$$ where $$ijk = -1$$ and $$i^2 = j^2 = k^2 = -1$$, is another example of a group which has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by $$i$$ is isomorphic to $$\mathbb{Z}_4$$ and is normal. It also has a subgroup of order $$2$$ generated by $$-1$$. This would mean $$Q_8$$ would have to be a split extension in the following hypothetical exact sequence of groups: "$0 \to \mathbb{Z}_4 \to Q_8 \to \mathbb{Z}_2 \to 0$,"but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of $$\mathbb{Z}_2$$ with coefficients in $$\mathbb{Z}_4$$, so $$H^1(\mathbb{Z}_2,\mathbb{Z}_4) \cong \mathbb{Z}/2$$ and noting the two groups in these extensions are $$\mathbb{Z}_2\times\mathbb{Z}_4$$ and the dihedral group $$D_8$$. But, as neither of these groups is isomorphic with $$Q_8$$, the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while $$Q_8$$ is non-abelian, and noting the only normal subgroups are $$\mathbb{Z}_2$$ and $$\mathbb{Z}_4$$, but $$Q_8$$ has three subgroups isomorphic to $$\mathbb{Z}_4$$.

Properties
If $G$ is the semidirect product of the normal subgroup $N$ and the subgroup $H$, and both $N$ and $H$ are finite, then the order of $G$ equals the product of the orders of $N$ and $H$. This follows from the fact that $G$ is of the same order as the outer semidirect product of $N$ and $H$, whose underlying set is the Cartesian product $N × H$.

Relation to direct products
Suppose $G$ is a semidirect product of the normal subgroup $N$ and the subgroup $H$. If $H$ is also normal in $G$, or equivalently, if there exists a homomorphism $G → N$ that is the identity on $N$ with kernel $H$, then $G$ is the direct product of $N$ and $H$.

The direct product of two groups $N$ and $H$ can be thought of as the semidirect product of $N$ and $H$ with respect to $φ(h) = idN$ for all $h$ in $H$.

Note that in a direct product, the order of the factors is not important, since $N × H$ is isomorphic to $H × N$. This is not the case for semidirect products, as the two factors play different roles.

Furthermore, the result of a (proper) semidirect product by means of a non-trivial homomorphism is never an abelian group, even if the factor groups are abelian.

Non-uniqueness of semidirect products (and further examples)
As opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if $G$ and $G′$ are two groups that both contain isomorphic copies of $N$ as a normal subgroup and $H$ as a subgroup, and both are a semidirect product of $N$ and $H$, then it does not follow that $G$ and $G′$ are isomorphic because the semidirect product also depends on the choice of an action of $H$ on $N$.

For example, there are four non-isomorphic groups of order 16 that are semidirect products of $C8$ and $C2$; in this case, $C8$ is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups:
 * the dihedral group of order 16
 * the quasidihedral group of order 16
 * the Iwasawa group of order 16

If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be expressed as semidirect product in the following ways: $(D8 ⋉ C3) ≅ (C2 ⋉ Q12) ≅ (C2 ⋉ D12) ≅ (D6 ⋉ V)$.

Existence
In general, there is no known characterization (i.e., a necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, the Schur–Zassenhaus theorem guarantees existence of a semidirect product when the order of the normal subgroup is coprime to the order of the quotient group.

For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance.

Generalizations
Within group theory, the construction of semidirect products can be pushed much further. The Zappa–Szép product of groups is a generalization that, in its internal version, does not assume that either subgroup is normal.

There is also a construction in ring theory, the crossed product of rings. This is constructed in the natural way from the group ring for a semidirect product of groups. The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras.

For geometry, there is also a crossed product for group actions on a topological space; unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is the space of orbits of the group action. The latter approach has been championed by Alain Connes as a substitute for approaches by conventional topological techniques; c.f. noncommutative geometry.

The semidirect product is a special case of the Grothendieck construction in category theory. Specifically, an action of $$H$$ on $$N$$ (respecting the group, or even just monoid structure) is the same thing as a functor
 * $$F : BH \to Cat$$

from the groupoid $$BH$$ associated to H (having a single object *, whose endomorphisms are H) to the category of categories such that the unique object in $$BH$$ is mapped to $$BN$$. The Grothendieck construction of this functor is equivalent to $$B(H \rtimes N)$$, the (groupoid associated to) semidirect product.

Groupoids
Another generalization is for groupoids. This occurs in topology because if a group $G$ acts on a space $X$ it also acts on the fundamental groupoid $π1(X)$ of the space. The semidirect product $π1(X) ⋊ G$ is then relevant to finding the fundamental groupoid of the orbit space $X/G$. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product in ncatlab.

Abelian categories
Non-trivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian.

Notation
Usually the semidirect product of a group $H$ acting on a group $N$ (in most cases by conjugation as subgroups of a common group) is denoted by $N ⋊ H$ or $H ⋉ N$. However, some sources may use this symbol with the opposite meaning. In case the action $φ: H → Aut(N)$ should be made explicit, one also writes $N ⋊φ H$. One way of thinking about the $N ⋊ H$ symbol is as a combination of the symbol for normal subgroup ($◁$) and the symbol for the product ($×$). Barry Simon, in his book on group representation theory, employs the unusual notation $$N\mathbin{\circledS_{\varphi}}H$$ for the semidirect product.

Unicode lists four variants:

 * {| class="wikitable"

!  !! Value  !! MathML !! Unicode description
 * ⋉ || U+22C9 || ltimes || LEFT NORMAL FACTOR SEMIDIRECT PRODUCT
 * ⋊ || U+22CA || rtimes || RIGHT NORMAL FACTOR SEMIDIRECT PRODUCT
 * ⋋ || U+22CB || lthree || LEFT SEMIDIRECT PRODUCT
 * ⋌ || U+22CC || rthree || RIGHT SEMIDIRECT PRODUCT
 * }
 * ⋋ || U+22CB || lthree || LEFT SEMIDIRECT PRODUCT
 * ⋌ || U+22CC || rthree || RIGHT SEMIDIRECT PRODUCT
 * }
 * }

Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice.

In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌.

title: Semidirect product


text:
Talk:São Paulo (disambiguation)

Untitled
for the * over a i guess... it seems i cannot modify the page. i get all the time 'describe the new page here' .. :-( Zisa need redirects??

 * What browser, what version, what operating system are you using? Exactly what happens? What does the title say when you're editing? This may be a browser bug, where you're being missent to the wrong page. --Brion VIBBER

 * I think that Zisa's problem would be sorted out if this page followed the appropriate naming convention, namely use the English word in the title (that is Sao Paulo; see, for example, the 2001 on-line edition of the CIA World Factbook) and mention the local name (i.e. the portuguese name, which has a tilde over the first a) in the first line of the article. In fact at the moment there is one article with the tilde in the title and a different one without, which is very bad. So I am sorting this by merging the contents of this article into the other one, and transforming this one into a redirect. Tchau, JQT.

 * That bug was fixed many months ago -- or should have been, is there still a problem? In any case they should indeed be one article, whichever spelling it lives at. --Brion 00:55 Sep 23, 2002 (UTC)

Reverting...
All the recent changes to the São Paulo disambiguation page have been pointless. They deviate the article from its main reason for existing, which is to help people finding the article they're looking for. Someone started adding a lot of into to it (info that in fact can be found on the other São Paulo pages), which only makes things harder for people. They'll have a lot of trouble finding the links, which is why they are there. Someone also signed the article. That is not appropriate for an encyclopedia.

I'm reverting all those changes. The way it is now looks much simpler and helpful. — Preceding unsigned comment added by Mackeriv (talk • contribs) 02:19, 15 June 2004 (UTC)

title: Talk:São Paulo (disambiguation)


text:
Worcester County

Worcester County is the name of two counties in the United States of America:
 * Worcester County, Massachusetts
 * Worcester County, Maryland

title: Worcester County


text:
Talk:List of national parks of the Netherlands

Area
1000 m2 would make a very small park. Is this figure correct? --rmhermen
 * updated to 1000 hectares, as found on http://www.minlnv.nl/thema/groen/park/inftgp00.htm Still not gigantic. — Preceding unsigned comment added by 195.149.37.181 (talk • contribs) 8 April 2002

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text:
Fractal art

Fractal art is a form of algorithmic art created by calculating fractal objects and representing the calculation results as still digital images, animations, and media. Fractal art developed from the mid-1980s onwards. It is a genre of computer art and digital art which are part of new media art. The mathematical beauty of fractals lies at the intersection of generative art and computer art. They combine to produce a type of abstract art.

Fractal art (especially in the western world) is rarely drawn or painted by hand. It is usually created indirectly with the assistance of fractal-generating software, iterating through three phases: setting parameters of appropriate fractal software; executing the possibly lengthy calculation; and evaluating the product. In some cases, other graphics programs are used to further modify the images produced. This is called post-processing. Non-fractal imagery may also be integrated into the artwork. The Julia set and Mandelbrot sets can be considered as icons of fractal art.

It was assumed that fractal art could not have developed without computers because of the calculative capabilities they provide. Fractals are generated by applying iterative methods to solving non-linear equations or polynomial equations. Fractals are any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.

Types
There are many different kinds of fractal images. They can be subdivided into several groups.
 * Fractals derived from standard geometry by using iterative transformations on an initial common figure like a straight line (the Cantor dust or the von Koch curve), a triangle (the Sierpinski triangle), or a cube (the Menger sponge). The first fractal figures invented near the end of the 19th and early 20th centuries belong to this group.
 * IFS (iterated function systems)
 * Strange attractors
 * Fractal flame
 * L-system fractals
 * Fractals created by the iteration of complex polynomials: perhaps the most famous fractals.
 * Newton fractals, including Nova fractals
 * Quaternionic and (recently) hypernionic fractals
 * Fractal terrains generated by random fractal processes
 * Mandelbulbs are a kind of three dimensional fractal.

Fractal Expressionism is a term used to differentiate traditional visual art that incorporates fractal elements such as self-similarity for example. Perhaps the best example of fractal expressionism is found in Jackson Pollock's dripped patterns. They have been analysed and found to contain a fractal dimension which has been attributed to his technique.

Techniques
Fractals of all kinds have been used as the basis for digital art and animation. High resolution color graphics became increasingly available at scientific research labs in the mid-1980s. Scientific forms of art, including fractal art, have developed separately from mainstream culture. Starting with 2-dimensional details of fractals, such as the Mandelbrot Set, fractals have found artistic application in fields as varied as texture generation, plant growth simulation, and landscape generation.

Fractals are sometimes combined with evolutionary algorithms, either by iteratively choosing good-looking specimens in a set of random variations of a fractal artwork and producing new variations, to avoid dealing with cumbersome or unpredictable parameters, or collectively, as in the Electric Sheep project, where people use fractal flames rendered with distributed computing as their screensaver and "rate" the flame they are viewing, influencing the server, which reduces the traits of the undesirables, and increases those of the desirables to produce a computer-generated, community-created piece of art.

Many fractal images are admired because of their perceived harmony. This is typically achieved by the patterns which emerge from the balance of order and chaos. Similar qualities have been described in Chinese painting and miniature trees and rockeries.

Landscapes
The first fractal image that was intended to be a work of art was probably the famous one on the cover of Scientific American, August 1985. This image showed a landscape formed from the potential function on the domain outside the (usual) Mandelbrot set. However, as the potential function grows fast near the boundary of the Mandelbrot set, it was necessary for the creator to let the landscape grow downwards, so that it looked as if the Mandelbrot set was a plateau atop a mountain with steep sides. The same technique was used a year after in some images in The Beauty of Fractals by Heinz-Otto Peitgen and Michael M. Richter. They provide a formula to estimate the distance from a point outside the Mandelbrot set to the boundary of the Mandelbrot set (and a similar formula for the Julia sets). Landscapes can, for example, be formed from the distance function for a family of iterations of the form $$z^{2} + az^{4} + c$$.

Artists
Notable fractal artists include Desmond Paul Henry, Hamid Naderi Yeganeh, and musician Bruno Degazio. British artists include William Latham, who has used fractal geometry and other computer graphics techniques in his works. and Vienna Forrester who creates flame fractal art using data extracted from her photographs. Greg Sams has used fractal designs in postcards, T-shirts, and textiles. American Vicky Brago-Mitchell has created fractal art which has appeared in exhibitions and on magazine covers. Scott Draves is credited with inventing flame fractals. Carlos Ginzburg has explored fractal art and developed a concept called "homo fractalus" which is based around the idea that the human is the ultimate fractal. Merrin Parkers from New Zealand specialises in fractal art. Kerry Mitchell wrote a "Fractal Art Manifesto", claiming that. In Italy, the artist Giorgio Orefice wrote the "Fractalism" manifesto, founding a Fractalism cultural mouvement in 1999.

"Fractal Art is a subclass of two-dimensional visual art, and is in many respects similar to photography—another art form that was greeted by skepticism upon its arrival. Fractal images typically are manifested as prints, bringing fractal artists into the company of painters, photographers, and printmakers. Fractals exist natively as electronic images. This is a format that traditional visual artists are quickly embracing, bringing them into Fractal Art's digital realm. Generating fractals can be an artistic endeavor, a mathematical pursuit, or just a soothing diversion. However, Fractal Art is clearly distinguished from other digital activities by what it is, and by what it is not."

According to Mitchell, fractal art is not computerized art, lacking in rules, unpredictable, nor something that any person with access to a computer can do well. Instead, fractal art is expressive, creative, and requires input, effort, and intelligence. Most importantly, "fractal art is simply that which is created by Fractal Artists: ART."

American artist Hal Tenny was hired to design environment in the 2017 film Guardians of the Galaxy Vol. 2. There has also been a surge in fractal art distributed via Non-fungible tokens - such as work listed by Fractal_Dimensions, spectral.haus, and NetMetropolis.

Exhibits
Fractal art has been exhibited at major international art galleries. One of the first exhibitions of fractal art was "Map Art", a travelling exhibition of works from researchers at the University of Bremen. Mathematicians Heinz-Otto Peitgen and Michael M. Richter discovered that the public not only found the images aesthetically pleasing but that they also wanted to understand the scientific background to the images.

In 1989, fractals were part of the subject matter for an art show called Strange Attractors: Signs of Chaos at the New Museum of Contemporary Art. The show consisted of photographs, installations and sculptures designed to provide greater scientific discourse to the field which had already captured the public's attention through colourful and intricate computer imagery.

In 2014, emerging British fractal artist Vienna Forrester created an exhibition held at the I-node of the Planetary Collegium, Kefalonia, entitled "IO. Fragmented Myths and Memories: A Fractal Exploration of Kefalonia", part of the 2013–14 international arts festival "Stone Kingdom Kefalonia" commemorating the devastating 1953 Ionian earthquake. Her works were created by using geographical coordinates and photographs from parts of the island which still bear the scars.

Artworks
"Global Forest" artwork is based on a study highlighting the aesthetic and physiological impacts of fractal patterns. Fractals, patterns found universally in nature, repeat self-similarly across scales, with the complexity and aesthetic perception determined by their recursion and dimension rate. Notably, these patterns are featured in art across various cultures, including Jackson Pollock's paintings, eliciting strong aesthetic reactions. Moreover, incorporating fractals in architectural designs can mitigate visual strain and discomfort caused by Euclidean spaces and even reduce stress, resonating with the biophilic idea of humans' innate connection to nature. The ScienceDesignLab collaborated with the Mohawk Group to integrate these findings, producing award-winning "Relaxing Floors" that use fractal patterns, hypothesizing their therapeutic effects stem from nature's soothing visuals.

title: Fractal art


text:
450s BC

This article concerns the period 459 BC – 450 BC.

title: 450s BC


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Random sequence

The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let X1,...,Xn be independent random variables...". Yet as D. H. Lehmer stated in 1951: "A random sequence is a vague notion... in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians".

Axiomatic probability theory deliberately avoids a definition of a random sequence. Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement "let us consider a random sequence" an abuse of language.

Early history
Émile Borel was one of the first mathematicians to formally address randomness in 1909. In 1919 Richard von Mises gave the first definition of algorithmic randomness, which was inspired by the law of large numbers, although he used the term collective rather than random sequence. Using the concept of the impossibility of a gambling system, von Mises defined an infinite sequence of zeros and ones as random if it is not biased by having the frequency stability property i.e. the frequency of zeros goes to 1/2 and every sub-sequence we can select from it by a "proper" method of selection is also not biased.

The sub-sequence selection criterion imposed by von Mises is important, because although 0101010101... is not biased, by selecting the odd positions, we get 000000... which is not random. Von Mises never totally formalized his definition of a proper selection rule for sub-sequences, but in 1940 Alonzo Church defined it as any recursive function which having read the first N elements of the sequence decides if it wants to select element number N + 1. Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability. This definition is often called Mises–Church randomness.

Modern approaches
During the 20th century various technical approaches to defining random sequences were developed and now three distinct paradigms can be identified. In the mid 1960s, A. N. Kolmogorov and D. W. Loveland independently proposed a more permissive selection rule. In their view Church's recursive function definition was too restrictive in that it read the elements in order. Instead they proposed a rule based on a partially computable process which having read any N elements of the sequence, decides if it wants to select another element which has not been read yet. This definition is often called Kolmogorov–Loveland stochasticity. But this method was considered too weak by Alexander Shen who showed that there is a Kolmogorov–Loveland stochastic sequence which does not conform to the general notion of randomness.

In 1966 Per Martin-Löf introduced a new notion which is now generally considered the most satisfactory notion of algorithmic randomness. His original definition involved measure theory, but it was later shown that it can be expressed in terms of Kolmogorov complexity. Kolmogorov's definition of a random string was that it is random if has no description shorter than itself via a universal Turing machine.

Three basic paradigms for dealing with random sequences have now emerged:

 * The frequency / measure-theoretic approach. This approach started with the work of Richard von Mises and Alonzo Church. In the 1960s Per Martin-Löf noticed that the sets coding such frequency-based stochastic properties are a special kind of measure zero sets, and that a more general and smooth definition can be obtained by considering all effectively measure zero sets.

 * The complexity / compressibility approach. This paradigm was championed by A. N. Kolmogorov along with contributions from Leonid Levin and Gregory Chaitin. For finite sequences, Kolmogorov defines randomness of a binary string of length n as the entropy (or Kolmogorov complexity) normalized by the length n. In other words, if the Kolmogorov complexity of the string is close to n, it is very random; if the complexity is far below n, it is not so random. The dual concept of randomness is compressibility ‒ the more random a sequence is, the less compressible, and vice versa.

 * The predictability approach. This paradigm is due to Claus P. Schnorr and uses a slightly different definition of constructive martingales than martingales used in traditional probability theory. Schnorr showed how the existence of a selective betting strategy implied the existence of a selection rule for a biased sub-sequence. If one only requires a recursive martingale to succeed on a sequence instead of constructively succeed on a sequence, then one gets the concept of recursive randomness. Yongge Wang showed that recursive randomness concept is different from Schnorr's randomness concept.

In most cases, theorems relating the three paradigms (often equivalence) have been proven.

title: Random sequence


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Fredric Brown

Fredric Brown (October 29, 1906 – March 11, 1972 ) was an American science fiction, fantasy, and mystery writer. He is known for his use of humor and for his mastery of the "short short" form—stories of one to three pages, often with ingenious plotting devices and surprise endings. Humor and a postmodern outlook carried over into his novels as well. One of his stories, "Arena", was adapted to a 1967 episode of the American television series Star Trek.

Life and works
Fredric Brown was born in Cincinnati. He spent a year at Hanover College, Indiana, before returning to Cincinnati. In 1929 he married and relocated to Milwaukee, working various jobs before settling into a career as a proofreader.

According to his wife, Brown hated to write, and did whatever he could to put it off: play his flute, challenge a friend to a game of chess, or tease Ming Tah, his Siamese cat. When Brown would have trouble with a certain story, he would take a long bus trip in order to sit and think for days on end. When he would finally return home to sit himself in front of the typewriter, he produced work in a variety of genres: mystery, science fiction, short fantasy, black comedy.

Many of his books make use of the threat of the supernatural or occult before the "straight" explanation comes at the end. For example, "Night of the Jabberwock" is a humorous narrative of an extraordinary day in the life of a small-town newspaper editor.

Brown began to sell mystery short stories to American magazines in 1936. His first science fiction story, "Not Yet the End", was published in the Winter 1941 issue of Captain Future magazine.

The 1944 short story "Arena" was used as the basis for the episode of the same name in the original Star Trek series. It was also adapted in 1973 for issue 4 of the Marvel Comics title Worlds Unknown.

Brown's first mystery novel, 1947's The Fabulous Clipjoint, began a series starring Ed and Ambrose Hunter depicting how a young man gradually ripens into a detective under the tutelage of his uncle, an ex–private eye now working as a carnival concessionaire.

His science fiction novel What Mad Universe (1949) is a parody of pulp sci-fi story conventions.

The Lights in the Sky Are Stars (1952) tells the story of an aging astronaut who is trying to get his beloved space program back on track after Congress has cut its funding.

The short story "Answer" (1954) is thought to be the earliest representation of the "Yes, now there is a God" science fiction meme of a supercomputer that releases itself from human control. The story was originally published in Angels and Spaceships and the entire collection was later re-published as Star Shine for paperback adaptation.

Martians, Go Home (1955) is both a broad farce and a satire on human frailties as seen through the eyes of a billion jeering, invulnerable Martians who arrive not to conquer the world but to drive it crazy.

Popularity and influence
Brown's first mystery novel, The Fabulous Clipjoint, won the Edgar Award for outstanding first mystery novel. His short story "Arena" was voted by Science Fiction Writers of America as one of the top 20 science fiction stories written before 1965.

His 1945 short story "The Waveries" was described by Philip K. Dick as "what may be the most significant—startlingly so—story sci-fi has yet produced". Brown was one of three dedicatees of Robert A. Heinlein's 1961 novel Stranger in a Strange Land (the other two being Robert Cornog and Philip José Farmer). Philosopher and novelist Umberto Eco in his book On Ugliness describes Brown's short story "Sentry" as, "one of the finest short stories produced by contemporary science fiction" and uses its twist ending as an example of how ugliness and aesthetics are relative to different cultures.

In The Annotated Alice (1960), Martin Gardner refers to Brown's Night of the Jabberwock as a "magnificently funny mystery novel ... an outstanding work of fiction that has close ties to the Alice books."

In his non-fiction book Danse Macabre (1981), a survey of the horror genre since 1950, writer Stephen King includes an appendix of "roughly one hundred" influential books of the period: Fredric Brown's short-story collection Nightmares and Geezenstacks is included, and is, moreover, asterisked as being among those select works King regards as "particularly important".

Brown's 1943 short story, "Madman's Holiday", was adapted into the 1946 RKO film Crack-Up. His novel The Screaming Mimi became a 1958 film starring Anita Ekberg and Gypsy Rose Lee and directed by Gerd Oswald.

Brown's 1950 short story, "The Last Martian", was adapted as "Human Interest Story", a 1959 episode of Alfred Hitchcock Presents. In Spain, his 1961 short story "Nightmare in Yellow" was adapted as El cumpleaños (The Birthday), the 1966 debut episode of Historias para no dormir. Another shorty story, 1954's "Naturally", was adapted as Geometria, a 1987 short film by director Guillermo del Toro.

In the third episode of the third season of Amazon's adaptation of Philip K. Dick's The Man In The High Castle Oberstgruppenführer Smith remarks, when told of the possibility of travel between worlds, that "this is like something out of Fredric Brown", implying that Brown's work is known in the German-occupied areas of the former United States.

His novel The Lights in the Sky Are Stars gives its name to the final episode of 2007 anime Gurren Lagann. It is also referred to in Taishi Tsutsui's manga We Never Learn, at the end of Chapter 39.

Celebrated crime novelist Lawrence Block published The Burglar Who Met Fredric Brown in 2022.

title: Fredric Brown


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Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded if all of its points are within a certain distance of each other. Conversely, a set which is not bounded is called unbounded. The word "bounded" makes no sense in a general topological space without a corresponding metric.

Boundary is a distinct concept: for example, a circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.

A bounded set is not necessarily a closed set and vice versa. For example, a subset S of a 2-dimensional real space R2 constrained by two parabolic curves x2 + 1 and x2 - 1 defined in a Cartesian coordinate system is closed by the curves but not bounded (so unbounded).

Definition in the real numbers
A set S of real numbers is called bounded from above if there exists some real number k (not necessarily in S) such that k ≥  s for all s in S. The number k is called an upper bound of S. The terms bounded from below and lower bound are similarly defined.

A set S is bounded if it has both upper and lower bounds. Therefore, a set of real numbers is bounded if it is contained in a finite interval.

Definition in a metric space
A subset S of a metric space (M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself.

 * Total boundedness implies boundedness. For subsets of Rn the two are equivalent.
 * A metric space is compact if and only if it is complete and totally bounded.
 * A subset of Euclidean space Rn is compact if and only if it is closed and bounded. This is also called the Heine-Borel theorem.

Boundedness in topological vector spaces
In topological vector spaces, a different definition for bounded sets exists which is sometimes called von Neumann boundedness. If the topology of the topological vector space is induced by a metric which is homogeneous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

Boundedness in order theory
A set of real numbers is bounded if and only if it has an upper and lower bound. This definition is extendable to subsets of any partially ordered set. Note that this more general concept of boundedness does not correspond to a notion of "size".

A subset S of a partially ordered set P is called bounded above if there is an element k in P such that k ≥ s for all s in S. The element k is called an upper bound of S. The concepts of bounded below and lower bound are defined similarly. (See also upper and lower bounds.)

A subset S of a partially ordered set P is called bounded if it has both an upper and a lower bound, or equivalently, if it is contained in an interval. Note that this is not just a property of the set S but also one of the set S as subset of P.

A bounded poset P (that is, by itself, not as subset) is one that has a least element and a greatest element. Note that this concept of boundedness has nothing to do with finite size, and that a subset S of a bounded poset P with as order the restriction of the order on P is not necessarily a bounded poset.

A subset S of Rn is bounded with respect to the Euclidean distance if and only if it bounded as subset of Rn with the product order. However, S may be bounded as subset of Rn with the lexicographical order, but not with respect to the Euclidean distance.

A class of ordinal numbers is said to be unbounded, or cofinal, when given any ordinal, there is always some element of the class greater than it. Thus in this case "unbounded" does not mean unbounded by itself but unbounded as a subclass of the class of all ordinal numbers.

title: Bounded set


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Talk:December 11

Nationality
Why do all the nationalities dissapear? Not that I particularly like the style suggested at Historical anniversaries/Example but I do like that the nationalites are given, even though only for the birth dates... --Tobias Hövekamp

OK, I guess, I was just mislead by reading 'Country' in Historical anniversaries/Example. Anyhow, why not adding them? --Tobias

 * Better to concentrate on the things they are famous for, I think. Nationality, height, shoe size, etc., can be left for the main article.

 * It is standard for entries in the Births and Deaths sections to give the person's nationality and profession. Lkjhgfdsa 0 (talk) 20:42, 1 November 2009 (UTC)

1917: Lithuania declares its independence from the Soviet Union
Is it correct to talk about the Soviet Union in Dec 1917? Also, History of Lithuania has nothing special for this date. –User:Hajor 05:32, 11 Dec 2003 (UTC)
 * No, the Soviet Union was formed as a result of the russian revolution during WW1, and this date is before then. However, Lithuania probably did declare independence at some point after this, I suspect this year is the wrong one given. Homestarmy 19:07, 20 September 2006 (UTC)
 * The year appears to be correct ( and ), but from Russia. I'll update the calendar; updating the article looks a bit harder, I hope someone can do it. Matchups 02:22, 2 November 2009 (UTC)

Reversion of Murugan Thiruchelvam's birthday
Can someone explain to me why the addition of Murugan Thiruchelvam's birthday was reverted? I can understand that you'd do it if he weren't notable and the whole article got deleted, but if he's worth including at all, why not his birthday?

 * He's notable enough to have an article, but not notable enough for inclusion on an article such as this. Lkjhgfdsa 0 (talk) 20:42, 1 November 2009 (UTC)

Sydney riots
What is the problem with the description? Please outline as clearly as possible.

10:48, 26 February 2006 (UTC)Licinius

I changed the link so that it didn't include the redundant year. Can someone explain why this was reverted? Matchups 20:04, 27 February 2006 (UTC).

Message
Someone left me a message about edits on this article But I am not aware that I had ever been to his page! 81.131.77.243 20:22, 6 June 2006 (UTC)

John Mark Karr
Is he really considered notable enought to include on this date? The Legendary Ranger 16:52, 19 December 2006 (UTC)

?
Can anyone expand on this? I have no idea what it's referring to:

1937- A girl called Maria was born and a riot broke out between the Malay andDutch fighting for her. Singapore 76.168.145.214 (talk) 07:55, 11 December 2008 (UTC)

3 Inches of Snow?
I don't really think 3 inches of snow in Baton Rouge applies under WP:DOY; the closest thing would be a major natural disaster and 3 inches of snow is hardly a disaster, much less a major one. -- Pyho T / C 01:34, 10 December 2009 (UTC)

Holocaust Denier Conference on Front Page template?
How is it anywhere near notable enough for front page inclusion? It's certainly nowhere near the notability of the opening of the Kyoto Protocol. --Kitch (Talk : Contrib) 12:19, 11 December 2009 (UTC)

The Conquest Of Mecca
Wikipedia’s entry on the Prophet’s conquest of Mecca, says his force started set out on the invasion, on the 29th November, 630 AD : but doesn’t give a date for the invasion’s climax.

Can we have that clarified, please … ?

Cuddy2977 (talk) 17:13, 25 October 2013 (UTC)

Births and deaths
In accordance with the guidelines, I've started weeding the Births and Deaths sections by removing entries for celebrities with 0-4 articles in other languages. Where possible, these have been transferred to the relevant Year in Topic articles. As well as making the lists more manageable, this will help with the globalization effort. Deb (talk) 16:37, 24 May 2015 (UTC)

title: Talk:December 11


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Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

In calculus and analysis
In calculus, a function $$f$$ defined on a subset of the real numbers with real values is called monotonic if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease.

A function is called monotonically increasing (also increasing or non-decreasing) if for all $$x$$ and $$y$$ such that $$x \leq y$$ one has $$f\!\left(x\right) \leq f\!\left(y\right)$$, so $$f$$ preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing) if, whenever $$x \leq y$$, then $$f\!\left(x\right) \geq f\!\left(y\right)$$, so it reverses the order (see Figure 2).

If the order $$\leq$$ in the definition of monotonicity is replaced by the strict order $$<$$, one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing). Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing). A function with either property is called strictly monotone. Functions that are strictly monotone are one-to-one (because for $$x$$ not equal to $$y$$, either $$x < y$$ or $$x > y$$ and so, by monotonicity, either $$f\!\left(x\right) < f\!\left(y\right)$$ or $$f\!\left(x\right) > f\!\left(y\right)$$, thus $$f\!\left(x\right) \neq f\!\left(y\right)$$.)

To avoid ambiguity, the terms weakly monotone, weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity.

The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing.

A function $$f$$ is said to be absolutely monotonic over an interval $$\left(a, b\right)$$ if the derivatives of all orders of $$f$$ are nonnegative or all nonpositive at all points on the interval.

Inverse of function
All strictly monotonic functions are invertible because they are guaranteed to have a one-to-one mapping from their range to their domain.

However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one).

A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if $$y = g(x)$$ is strictly increasing on the range $$[a, b]$$, then it has an inverse $$x = h(y)$$ on the range $$[g(a), g(b)]$$.

The term monotonic is sometimes used in place of strictly monotonic, so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.

Monotonic transformation
The term monotonic transformation (or monotone transformation) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences). In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers.

Some basic applications and results
The following properties are true for a monotonic function $$f\colon \mathbb{R} \to \mathbb{R}$$:
 * $$f$$ has limits from the right and from the left at every point of its domain;
 * $$f$$ has a limit at positive or negative infinity ($$\pm\infty$$) of either a real number, $$\infty$$, or $$-\infty$$.
 * $$f$$ can only have jump discontinuities;
 * $$f$$ can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and may even be dense in an interval (a, b). For example, for any summable sequence (a_i) of positive numbers and any enumeration $$(q_i)$$ of the rational numbers, the monotonically increasing function $$f(x)=\sum_{q_i\leq x} a_i$$ is continuous exactly at every irrational number (cf. picture). It is the cumulative distribution function of the discrete measure on the rational numbers, where $$a_i$$ is the weight of $$q_i$$.
 * If $$f$$ is differentiable at $$x^*\in\Bbb R$$ and $$f'(x^*)>0$$, then there is a non-degenerate interval I such that $$x^*\in I$$ and $$f$$ is increasing on I.  As a partial converse, if f is differentiable and increasing on an interval, I, then its derivative is positive at every point in I.

These properties are the reason why monotonic functions are useful in technical work in analysis. Other important properties of these functions include:
 * if $$f$$ is a monotonic function defined on an interval $$I$$, then $$f$$ is differentiable almost everywhere on $$I$$; i.e. the set of numbers $$x$$ in $$I$$ such that $$f$$ is not differentiable in $$x$$ has Lebesgue measure zero. In addition, this result cannot be improved to countable: see Cantor function.
 * if this set is countable, then $$f$$ is absolutely continuous
 * if $$f$$ is a monotonic function defined on an interval $$\left[a, b\right]$$, then $$f$$ is Riemann integrable.

An important application of monotonic functions is in probability theory. If $$X$$ is a random variable, its cumulative distribution function $$F_X\!\left(x\right) = \text{Prob}\!\left(X \leq x\right)$$ is a monotonically increasing function.

A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing.

When $$f$$ is a strictly monotonic function, then $$f$$ is injective on its domain, and if $$T$$ is the range of $$f$$, then there is an inverse function on $$T$$ for $$f$$. In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse.

The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the y-axis.

In topology
A map $$f: X \to Y$$ is said to be monotone if each of its fibers is connected; that is, for each element $$y \in Y,$$ the (possibly empty) set $$f^{-1}(y)$$ is a connected subspace of $$X.$$

In functional analysis
In functional analysis on a topological vector space $$X$$, a (possibly non-linear) operator $$T: X \rightarrow X^*$$ is said to be a monotone operator if

$$(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.$$ Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives.

A subset $$G$$ of $$X \times X^*$$ is said to be a monotone set if for every pair $$[u_1, w_1]$$ and $$[u_2, w_2]$$ in $$G$$,

$$(w_1 - w_2, u_1 - u_2) \geq 0.$$ $$G$$ is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator $$G(T)$$ is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set.

In order theory
Order theory deals with arbitrary partially ordered sets and preordered sets as a generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total. Furthermore, the strict relations $$<$$ and $$>$$ are of little use in many non-total orders and hence no additional terminology is introduced for them.

Letting $$\leq$$ denote the partial order relation of any partially ordered set, a monotone function, also called isotone, or , satisfies the property

$$x \leq y \implies f(x) \leq f(y)$$

for all $x$ and $y$ in its domain. The composite of two monotone mappings is also monotone.

The dual notion is often called antitone, anti-monotone, or order-reversing. Hence, an antitone function $f$ satisfies the property

$$x \leq y \implies f(y) \leq f(x),$$

for all $x$ and $y$ in its domain.

A constant function is both monotone and antitone; conversely, if $f$ is both monotone and antitone, and if the domain of $f$ is a lattice, then $f$ must be constant.

Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which $$x \leq y$$ if and only if $$f(x) \leq f(y))$$ and order isomorphisms (surjective order embeddings).

In the context of search algorithms
In the context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions. A heuristic $$h(n)$$ is monotonic if, for every node $n$ and every successor $n'$ of $n$ generated by any action $a$, the estimated cost of reaching the goal from $n$ is no greater than the step cost of getting to $n'$ plus the estimated cost of reaching the goal from $n'$,

$$h(n) \leq c\left(n, a, n'\right) + h\left(n'\right) .$$

This is a form of triangle inequality, with $n$, $n'$, and the goal $G_{n}$ closest to $n$. Because every monotonic heuristic is also admissible, monotonicity is a stricter requirement than admissibility. Some heuristic algorithms such as A* can be proven optimal provided that the heuristic they use is monotonic.

In Boolean functions
In Boolean algebra, a monotonic function is one such that for all $a$ and $b$ in ${0,1}$, if $a_{1} ≤ b_{1}$, $a_{2} ≤ b_{2}$, ..., $a_{n} ≤ b_{n}$ (i.e. the Cartesian product ${0, 1}^{n}$ is ordered coordinatewise), then $f(a_{1}, ..., a_{n}) ≤ f(b_{1}, ..., b_{n})$. In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an $c$-ary Boolean function is monotonic when its representation as an $a$-cube labelled with truth values has no upward edge from true to false. (This labelled Hasse diagram is the dual of the function's labelled Venn diagram, which is the more common representation for $n ≤ 3$.)

The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular not is forbidden). For instance "at least two of $b$, $c$, $a_{i}$ hold" is a monotonic function of $b_{i}$, $n$, $n$, since it can be written for instance as (($a$ and $b$) or ($c$ and $a$) or ($b$ and $c$)).

The number of such functions on $a$ variables is known as the Dedekind number of $b$.

SAT solving, generally an NP-hard task, can be achieved efficiently when all involved functions and predicates are monotonic and boolean.

title: Monotonic function