text:
Talk:Klondike (solitaire)

Different ways of playing
How many ways are there to make Klondike as easy or difficult as you like?? Here they are. You are free to add more if you can:

Redeals:

You can set not to allow redealing, or to allow 1, 2, 3, or 4 redeals, or to redeal as often as you like. Setting this option to a low number means a harder game.

Draw:

You can set this to 1, 2, or 3 cards at a time. Setting this option to a high number means a harder game.

Filling Empty Columns:

You can select this to any card, only a king, only a card from the stock as long as the stock isn't empty, and no card at all. The last of these would mean a really difficult game.

Suit discrimination:

Equal suit, equal color, unequal color, and no suit discrimination. This gives 16 variants because there are 2 suit discriminations that are not necessarily the same, how the foundations are built up and how the columns are built down. The easiest of these is obviously no suit discrimination at all.

Pulling from foundations:

Microsoft Solitaire allows pulling cards (even aces) back down from the foundations to the piles. This increases the odds of winning considerably, and might make the game more enjoyable (personal opinion), as some interesting strategies are made possible. I haven't seen this mentioned in any rule set yet, not even in Microsoft's help text.

Klondike (solitaire)
Moved from Klondike solitaire to be in line with other solitaire game article names. -- Netoholic 04:47, 15 Jul 2004 (UTC)

The "(5.0)" on the picture is unnecessary - the version number in the about box refers to the version of windows [5.0 is windows 2000] - solitaire itself doesn't have a version number [and, as far as i know, the only changes that have ever been made to it are to make it 32-bit and to load card images from an external library]

 * Yep. Just look at the status bar--still Windows 3.x style! --Jason McHuff 02:19, 1 October 2005 (UTC)

Canfield, Demon and Klondike
The Penguin Book of Patience has this division differently. According to them, "Klondike" is the name for either Demon (see Canfield (solitaire)), or Canfield (see Klondike (solitaire)). Should something be done about that? Mysha (nl)

Odds of Winning
Any deck that can be solved in Thoughtful Solitare can be solved in regular solitaire as well: simply make all the right choices.
 * Perhaps the question should be rephrased as, "given perfect use of limited knowledge, what percentage of decks will be solved"
 * Then it seems that the percentage of decks that can be solved in thoughtful solitaire would be an upper bound on the percentage of decks that would be solved using perfect strategy in standard solitaire.
 * Lower limit would be set by the best computer player devised so far, or the long run average for the best human player.

 * I don't understand why it's so difficult to compute the possible number of winning games of solitate. It would take time, yes, but I'm just shocked no one has done it. RobertM525 05:09, 6 June 2006 (UTC)

 * There's about 2 billion possible deals; if it ever is found out, I'm sure it won't just be one person doing all the work. 130.184.237.233 17:47, 22 June 2006 (UTC)
 * Not one person, of course. But a computer should be able to run through all of them given enough time... RobertM525 11:20, 23 June 2006 (UTC)

 * I remember reading about someone who kept records of his own games of Klondike and came up with an average of one game in 30. (This is for the one-card-at-a-time-with-no-redeal version.)  This seems about right to me.  Kostaki mou 01:18, 9 February 2007 (UTC)

At the moment the article states that: "A modified version of the game called "Thoughtful Solitaire" has a known solution strategy that works 70% of the time but requires significant computing power. This solution strategy does not solve every deck that is solvable, but is about twice as good as the best human players." Does it refer to games where redeals are allowed or not? Because with redeals it's easy to reach 70% of wins anyway. Here is a screenshots of solitaire statistics after 1000 games with 88% wins (redeal and undo allowed)

Actually, the basic premise here is false - not all solvable Thoughtful games are solvable in Klondike. In Thoughtful it is possible that the initial deal will include stacks topped by a downward sequence of cards in alternating colors. That sequence could then be moved off as a unit in Thoughtful, but not in Klondike, because in Klondike all but the top card would be face down. What people are trying to get at is the solvability when you know what the face down cards are, which is not the same as the solvability when you deal the cards face up. JanDWolter (talk) 21:52, 4 January 2014 (UTC)

 * I realise your comment is almost a decade old, but you've misunderstood what Thoughtful Klondike is. Thoughtful Klondike is not simply dealing all the cards face-up. It's exactly the same as normal Klondike, except that you know what each face-down card is before you turn it over. In your example, the alternating sequence from the initial deal cannot be moved as a unit in Thoughtful Klondike, because all but one of the cards are face down. Knowing the value of the face-down cards lets you strategize, but it doesn't let you move them, since only face-up cards can be moved. I think the article now makes that quite clear; probably it didn't, back in 2014. Dricherby (talk) 10:47, 12 September 2023 (UTC)

Merge from Wes Cherry
I claim that the Wes Cherry article should redirect to Klondike (solitaire), with any content unique to his article moved over, for what I think are fairly obvious reasons. There's no evidence that he's notable for anything other than developing Windows Solitaire, and his article says little or nothing about him except that he did so. Stellmach 21:56, 2 September 2006 (UTC)
 * Okay, this merge seems to be entirely uncontroversial. Going ahead. -Stellmach 13:23, 8 September 2006 (UTC)

Demerge from Wes Cherry
The information on Wes Cherry has been removed except for a link. Either this information should be restored, or the Wes Cherry article should be demerged. I would favour demerger because it is the more accident-resistent way. 86.140.97.102 10:14, 31 May 2007 (UTC)

"Most Played" assertion
Since computer games in general do not keep any kind of tracking information about how much time has been spent playing them, it seems as though the assertion that Klonidke has been the most commonly played one, while an appealing notion, will always be impossible to verify. -Stellmach 14:01, 6 September 2006 (UTC)
 * We can't do much about the data, but it seems like it should be easy to find one or more sources that make that claim. Seems more like a citation issue than an accuracy one, though it would be a good idea to add a comment similar to yours afterwards. -FunnyMan 05:03, 13 September 2006 (UTC)

I removed the assertion since it just can't be proved. — Hex    (❝ ?!  ❞)   23:55, 5 October 2006 (UTC)
 * I've added three sources that assert it is "most popular". While it is debatable whether or not those assertions can be proved as correct, it's not our job to decide what is proven and what isn't. It's a common assertion that the Solitaire shipped with windows is the most popular game on a personal computer/handheld - one that is reflected in popular culture and the media. Hence, finding sources is not an issue.  Icemotoboy (talk) 22:28, 25 February 2009 (UTC)

Maxcards?
What's "Maxcards", as referred to in the rules section? If it's the game referred to at maxcards.co.uk, I think we can do without this reference. Rojomoke 11:22, 8 February 2007 (UTC)

Windows version
Shouldn't we mention the Windows version a bit more? It's surely the most widely played PC as well as Windows game of all time and some would say is a significant drain on human labour in offices Nil Einne 23:31, 23 August 2007 (UTC)

scoring
I would like to see a scoring section that explains how the score is computed. in the windows version there are two different types of scoring and I think it would be interesting to know the difference.--Movesandpepper (talk) 17:41, 27 March 2008 (UTC)

 * On Windows Solitaire, the scoring is as follows:

 * Moving a new card to a table pile 5 points
 * Turning over a card 5 points
 * Moving a card to a base pile 10 points
 * Removing a card from a base pile -15 points
 * Turning the deck over -100 points for one card draw
 * Turning the deck over -20 points for three card draw after the third rotation
 * Undo command undoes the score on Vista, -3 points previous versions
 * In timed games, for each 5 seconds -2 points
 * In timed games there is also a time-dependent bonus of around 4000 to 6000 points on completion.

My Vista Home Premium version has an error that gives 10 points for moving the aces between bases. I noted it in the article and had it removed when someone else found their version did not have the error. My version also does not grant 5 points for turning a card over when moving an ace to the base on the same move. Could anyone verify the differences on their versions -- 129.101.34.186

The second mode is Vegas Scoring (which I believe is actually used in some casinos). You pay $52 for each game and receive $5 for each card on the base piles when you have exhausted the deck once – so 11 cards home is a win. --217.134.77.95 (talk) 19:54, 30 March 2008 (UTC)

Klondike/Thoughtful difference
I admit that this constitutes original research, but as far as I can tell, there is a difference between Klondike and Thoughtful which causes the odds of winning to be different. The difference is when the reversed card at the top of a pile of reversed cards is the number higher than and the opposite colour to the card at the bottom of the pile on top of it, those playing thoughtful will be able to see that the two cards can be used as a run of 2 (or more) cards, whereas those playing Klondike will have to flip over the reversed card before they can be treated as a run. Is this a genuine difference? If so, will this actually cause a different in the odds? If these questions can't be adequately answered, I don't think that we can actually say that Thoughtful and Klondike have the same chances of being able to be won.Labrat256 02:14, 13 January 2010 (UTC) —Preceding unsigned comment added by Labrat256 (talk • contribs)

 * You've misunderstood how Thoughtful works. Thoughtful just means that you know the values of the face-down cards. They're still face-down, and they can't be moved until you turn them face-up as in normal Klondike. Knowing their values means that you can strategize, because you know the effect of a move before you make it. But it doesn't allow you to make any moves that you couldn't make in ordinary Klondike. The article now makes this quite clear IMO, but probably it was less clear when you wrote your comment. Dricherby (talk) 10:51, 12 September 2023 (UTC)

A scholarly source...
check out "The Complexity of Solitaire" http://www.springerlink.com/content/b40703p37h74j220/ —Preceding unsigned comment added by 96.238.247.121 (talk) 03:24, 18 July 2010 (UTC)

Strategy
There should be a section on Klondike strategy. — Preceding unsigned comment added by 82.139.81.0 (talk) 09:40, 27 January 2013 (UTC)

History?
When was Klondike created, how long has people played this particular solitaire? It predates the computer solitaire, doesn't it? Swimsuit Zipper (talk) 11:48, 29 November 2013 (UTC)

Where does the name come from?
Where does the name Klondike come from for this form of solitaire? — Preceding unsigned comment added by 80.193.189.151 (talk) 16:47, 27 April 2015 (UTC)
 * I also came here to this article to try and find the answer to this question, as I just installed Solitaire on Windows 10 and it was the first time I've come across the name Klondike for the game. Irrelevant background information: I was first introduced to the game as being called Patience, long before I first played a computerised version of the game (which incidentally was also called Patience) on an Acorn Archimedes in the early 1990s. I first came across the name Solitaire years later on a Windows 95 machine. Obviously, I'm not disputing the name; I'm just curious. --Beeurd (talk) 18:39, 5 September 2015 (UTC)

Odds of winning definitions
In the Odds of winning section of the article, four outcomes are claimed, but only three are defined: Theoretically winnable, playable and unplayable. Is the fourth outcome "Games won?"

VastError (talk) 04:51, 15 February 2016 (UTC)

Number of possible hands
206.248.63.215 recently wrote that there are 8 x 10^67 or 52! possible number of hands instead of "7,000 trillion". I got rid of his rambling about the Big Bang but kept the number for now, because it seems to make sense (although there is no source and it is technically original research). Highly Ridiculous (talk) 16:46, 1 July 2016 (UTC)

 * In actuality, the number of meaningfully unique names would be less, as there would be functionally no difference if you swapped all spades for clubs or all hearts for diamonds, or all reds for blacks. I haven't done the math to see what that would drop it down to though. Tristan (talk) 17:59, 10 July 2018 (UTC)

Error in image caption?
This image and caption is in the "Probability of winning" section. Can't the seven of spades also be moved, onto the eight of hearts (carrying the six of diamonds with it)? Herostratus (talk) 02:21, 19 December 2018 (UTC)
 * Yes it certainly can be moved. IMHO it even seems playable, for one can move the 5♣ in the stock (it's the 13th card from right) onto the 6♦ and then move the 4♥ with 3♠ to it, in this way a downturned card will be freed. But I don't know if one can win this game afterwards. --WhitePhosphorus (talk) 14:41, 7 February 2019 (UTC)
 * Thoughtful Solitaire is defined to be draw three, in which case none of the available stock cards (Qd, 9c, Kc, Kd, 9h, 6s) are movable. I've added a note to that effect. Arcorann (talk) 01:09, 29 February 2020 (UTC)

History and name
I've tweaked the history and name information in the light of better, albeit no doubt still imperfect, information. Arnold is incorrect to say that Canfield is a misnomer for Klondike; it is the traditional English name for the game, although that tradition is waning in the face of computer game competition. It is true though that in America, Canfield might be perceived as a misnomer, since it is the name of another game. Parlett is likely to be right about this as he is a card game historian rather than a card game compiler like Arnold.

There is more work to be done though. I am suspicious about the game appearing at the time of the gold rush because I can find no record of "Klondike" before 1849. However, games called Demon, Triangle, Fascination and Canfield, names sometimes given as synonyms for Klondike, are recorded in the early 20th century and may be its antecedents. However, I have yet to establish whether they were played to the same rules. The trouble is that card game compilers often don't check their facts and frequently describe the origin of games with apparent certainty when in fact they are talking utter drivel. Among the more laughable myths is the century-old one that Bierspiel is a German card game of the Rams family. There is absolutely no evidence for that in German sources whatsoever, unsurprising really because Bierspiel just means "drinking game". Bermicourt (talk) 19:24, 12 October 2020 (UTC)

Editorializing about the inexplicability of the popularity of the game in the lead?
Is it really necessary to provide Parlett's musing on this fact here? Jyamine (talk) 17:34, 24 April 2023 (UTC)

Recent edits
Two recent edits by an anonymous editor removed David Parlett's assessment of Klondike, one with the comment "Parlett's remark is highly opinionated and should not be presented as fact. Maybe people like a challenge?" Leaving aside that the second sentence implies Parlett is right anyway, he is one of the world's leading gaming experts and a consultant on the subject to film producers wanting authenticity. He has also penned one of the most comprehensive books on patience and solitaire ever written. So his views carry weight. And they are hardly extreme: Klondike "offers one of the lowest success rates of any patience". This is not only evident to anyone who plays solitaire card games, but is corroborated by American card game experts Morehead and Mott-Smith who assess its success rate at 1 in 30. So I have reverted the edits concerned. Bermicourt (talk) 08:37, 8 October 2023 (UTC)

 * My primary objection is to quoting Parlett's sensational language which presumes that there is some obvious contradiction between a game's difficulty and popularity. As you will see above, I'm not the only one to raise the concern. 2601:642:4600:3F80:D905:A44F:AB76:D886 (talk) 14:39, 8 October 2023 (UTC)

 * The way Wikipedia works is that you are welcome to make a 'bold' edit, but if it is reverted, you should discuss it on the talk page (here) until a consensus is reached. Repeatedly reinstating your bold edit could be seen as WP:EDITWARRING which is likely to lead to a ban, the original text being restored and then you will be unable to participate in the debate. So please restore the original for now and then we can move forward. Bermicourt (talk) 14:45, 8 October 2023 (UTC)

 * Okay, I see the issue here. Parlett, Morehead, Mott-Smith, Arnold, various American Hoyles and other card game authors who describe the game as difficult and the odds as 1 in 30, are talking about the classic version in which cards are properly shuffled (randomised), there is only one pass through the stock and no worrying back (albeit some authors acknowledge easier options and give different odds for them).
 * Meanwhile the article describes 5 versions of Klondike based on how the stock is dealt without pointing out, crucially, that this results in different levels of challenge ranging from easy (deal 1 card at a time, unlimited passes) to difficult (deal 3 at a time, 1 pass only). Computer versions such as MS Klondike make life even easier by also allowing worrying back from foundations to columns and offering varying levels of difficulty in which, at the lower levels, the cards are not randomised and so the game will succeed more often than not. No wonder Parlett's remarks seem at odds with our experience of the game!
 * This can be resolved. The article should be amended, using appropriate sources, to point out the range of difficulty/level of challenge resulting from different ways of dealing the stock and the comments of the experts should suitably qualified by making it clear which version of the game they are actually referring to. Bermicourt (talk) 17:11, 9 October 2023 (UTC)

 * The impact of varying rules on the probabilities is confirmed by vegassolitaire.com thus: "The odds of winning with Klondike depend on what version of Solitaire (Klondike) you play. There are many variations in rules which all have their influence on the outcome and odds. The following rules/factors have a significant impact on the odds of winning Solitaire:
 * How many times you are allowed to go through the deck
 * How many cards you are allowed to turn over at a time from the pile (draw one or draw 3)
 * The game strategy (to finish all cards or to play a minimum of 11 cards to the Foundation)
 * If rules allow, “Worry Back,” where you can play cards back from the Foundation to the Tableau..." See Vegas Solitaire Odds.
 * Vegas Solitaire is of course simply Klondike with Richard Canfield's alleged system of paying $52 for the pack and winning $5 for every card on the foundations, albeit often attributed to the different game of Demon/Canfield. Bermicourt (talk) 08:27, 10 October 2023 (UTC)

Future of section on "Probability of winning"
The section entitled "Probability of winning" is highly problematic. For a start it is not comparing like with like. It opens by quoting the "1 in 30" probability that we now know refers to the classic form of Klondike where the stock is dealt singly and there is one pass with no other privileges. It then describes "Thoughtful Klondike" which does not equate to any real version of the game because the computer knows the location of all the cards from the start. It wrongly claims this is the only difference, but it also allows limitless and repeated undo moves to solve the game, something that is not normally part of the game, but which ave been introduced into computer simulations geared to making it easier to win (and hence presumably more marketable). No wonder the computer can win "Thoughtful Klondike" 82% of the time, but that is of no relevance to real Klondike where peeking at all the cards would be tantamount to cheating. Thus the section is highly misleading and either needs to be rewritten to clarify this and cut down or deleted: it takes up far too much space for a theoretical analysis of a variant with different rules that is tangential to the article topic and not playable in practice. Bermicourt (talk) 18:12, 9 October 2023 (UTC)

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

title: Talk:Klondike (solitaire)


text:
Mudra

A mudra (मुद्रा,, "seal", "mark", or "gesture"; ) is a symbolic or ritual gesture or pose in Hinduism, Jainism and Buddhism. While some mudras involve the entire body, most are performed with the hands and fingers.

As well as being spiritual gestures employed in the iconography and spiritual practice of Indian religions, mudras have meaning in many forms of Indian dance, and yoga. The range of mudras used in each field (and religion) differs, but with some overlap. In addition, many of the Buddhist mudras are used outside South Asia, and have developed different local forms elsewhere.

In hatha yoga, mudras are used in conjunction with pranayama (yogic breathing exercises), generally while in a seated posture, to stimulate different parts of the body involved with breathing and to affect the flow of prana. It is also associated with bindu, bodhicitta, amrita, or consciousness in the body. Unlike older tantric mudras, hatha yogic mudras are generally internal actions, involving the pelvic floor, diaphragm, throat, eyes, tongue, anus, genitals, abdomen, and other parts of the body. Examples of this diversity of mudras are Mula Bandha, Mahamudra, Viparita Karani, Khecarī mudrā, and Vajroli mudra. These expanded in number from 3 in the Amritasiddhi, to 25 in the Gheranda Samhita, with a classical set of ten arising in the Hatha Yoga Pradipika.

Mudra is used in the iconography of Hindu and Buddhist art of the Indian subcontinent and described in the scriptures, such as Nātyaśāstra, which lists 24 ' ("separated", meaning "one-hand") and 13 ' ("joined", meaning "two-hand") mudras. Mudra positions are usually formed by both the hand and the fingers. Along with āsanas ("seated postures"), they are employed statically in the meditation and dynamically in the practice of Hinduism.

Hindu and Buddhist iconography share some mudras. In some regions, for example in Laos and Thailand, these are distinct but share related iconographic conventions.

According to Jamgön Kongtrül in his commentary on the Hevajra Tantra, the ornaments of wrathful deities and witches made of human bones (Skt: ; ) are also known as mudra "seals".

Etymology and nomenclature
The word mudrā has Sanskrit roots. According to scholar Sir Monier Monier-Williams it means "seal" or "any other instrument used for sealing".

Buddhist Iconography
A Buddha image can have one of several common mudras, combined with different asanas. The main mudras used represent specific moments in the life of the Buddha, and are shorthand depictions of these.

Abhaya Mudrā
The Abhayamudra "gesture of fearlessness" represents protection, peace, benevolence and the dispelling of fear. In Theravada Buddhism it is usually made while standing with the right arm bent and raised to shoulder height, the palm facing forward, the fingers closed, pointing upright and the left hand resting by the side. In Thailand and Laos, this mudra is associated with the Walking Buddha, sometime also shown having both hands making a double abhaya mudra that is uniform.

This mudra was probably used before the onset of Buddhism as a symbol of good intentions proposing friendship when approaching strangers. In Gandharan art, it is seen when showing the action of preaching. It was also used in China during the Wei and Sui eras of the 4th and 7th centuries.

This gesture was used by the Buddha when attacked by an elephant, subduing it as shown in several frescoes and scripts.

In Mahayana Buddhism, the deities are often portrayed as pairing the Abhaya Mudrā with another Mudrā using the other hand.

Bhūmisparśa Mudrā
The bhūmisparśa or "earth witness" mudra of Gautama Buddha is one of the most common iconic images of Buddhism. Other names include "Buddha calling the earth to witness", and "earth-touching". It depicts the story from Buddhist legend of the moment when Buddha attained complete enlightenment, with Buddha sitting in meditation with his left hand, palm upright, in his lap, and his right hand touching the earth. In the legend, Buddha was challenged by the evil one, Mara, who argue for a witness to attest his right to achieve it. In response to Mara, Buddha touched the ground, and Phra Mae Thorani, the earth goddess, appeared to be the witnesses for the Buddha's enlightenment.

In East Asia, this mudra (also called the Maravijaya attitude) may show Buddha's fingers not reaching as far as the ground, as is usual in Burmese or Indian depictions.

Bodhyangi Mudrā
The Bodhyangi mudrā, the "mudrā of the six elements," or the "fist of wisdom," is a gesture entailing the left-hand index finger being grasped with the right hand. It is commonly seen on statues of the Vairocana Buddha.

Dharmachakra Pravartana Mudrā
The Buddha preached his first sermon after his Enlightenment in Deer Park in Sarnath. The dharmachakra Pravartana or "turning of the wheel" mudrā represents that moment. In general, only Gautama Buddha is shown making this mudrā except Maitreya as the dispenser of the Law. Dharmachakra mudrā is two hands close together in front of the chest in vitarka with the right palm forward and the left palm upwards, sometimes facing the chest. There are several variants such as in the Ajanta Caves frescoes, where the two hands are separated and the fingers do not touch. In the Indo-Greek style of Gandhara, the clenched fist of the right hand seemingly overlies the fingers joined to the thumb on the left hand. In pictorials of Hōryū-ji in Japan the right hand is superimposed on the left. Certain figures of Amitābha are seen using this mudra before the 9th century in Japan.

Dhyāna Mudrā
The dhyāna mudrā ("meditation mudra") is the gesture of meditation, of the concentration of the Good Law and the sangha. The two hands are placed on the lap, right hand on left with fingers fully stretched (four fingers resting on each other and the thumbs facing upwards towards one another diagonally), palms facing upwards; in this manner, the hands and fingers form the shape of a triangle, which is symbolic of the spiritual fire or the Three Jewels. This mudra is used in representations of Gautama Buddha and Amitābha. Sometimes the dhyāna mudrā is used in certain representations of as the "Medicine Buddha", with a medicine bowl placed on the hands. It originated in India most likely in Gandhāra and in China during the Northern Wei.

It is heavily used in Southeast Asia in Theravada Buddhism; however, the thumbs are placed against the palms. Dhyāna mudrā is also known as "samādhi mudrā" or "yoga mudrā",.

The mida no jōin (弥陀定印) is the Japanese name of a variation of the dhyāna mudra, where the index fingers are brought together with the thumbs. This was predominantly used in Japan in an effort to distinguish Amitābha (hence "mida" from Amida) from the Vairocana Buddha, and was rarely used elsewhere.

Varada Mudrā
The Varadamudrā "generosity gesture" signifies offering, welcome, charity, giving, compassion and sincerity. It is nearly always shown made with the left hand by a revered figure devoted to human salvation from greed, anger and delusion. It can be made with the arm crooked and the palm offered slightly turned up or in the case of the arm facing down the palm presented with the fingers upright or slightly bent. The Varada mudrā is rarely seen without another mudra used by the right hand, typically abhaya mudrā. It is often confused with vitarka mudrā, which it closely resembles. In China and Japan during the Northern Wei and Asuka periods, respectively, the fingers are stiff and then gradually begin to loosen as it developed over time, eventually leading to the Tang dynasty standard where the fingers are naturally curved.

In India, varada mudra is used by both seated and standing figures, of Buddha and boddhisattvas and other figures, and in Hindu art is especially associated with Vishnu. It was used in images of Avalokiteśvara from Gupta art (4th and 5th centuries) onwards. Varada mudrā is extensively used in the statues of Southeast Asia.

Vajra Mudrā
The Vajra mudrā "thunder gesture" is the gesture of knowledge.

Vitarka Mudrā
The Vitarka mudrā "mudra of discussion" is the gesture of discussion and transmission of Buddhist teaching. It is done by joining the tips of the thumb and the index together, and keeping the other fingers straight very much like the abhaya and varada mudrās but with the thumbs touching the index fingers. This mudra has a great number of variants in Mahayana Buddhism. In Tibetan Buddhism, it is the mystic gesture of Tārās and bodhisattvas with some differences by the deities in Yab-Yum. Vitarka mudrā is also known as Vyākhyāna mudrā ("mudra of explanation"). This is also called as chin-mudra.

Jñāna Mudrā
The Jñāna mudrā ("mudra of wisdom") is done by touching the tips of the thumb and the index together, forming a circle, and the hand is held with the palm inward towards the heart. The mudra represents spiritual enlightenment in the indian-origin religions. Sometimes sadhus chose to be buried alive in this samadhi position. A 2700 old skeleton arranged like this was found at Balathal in Rajasthan, suggesting that something like yoga may have existed at that time.

Karana Mudrā
The karana mudrā is the mudra which expels demons and removes obstacles such as sickness or negative thoughts. It is made by raising the index and the little finger, and folding the other fingers. It is nearly the same as the Western "sign of the horns", the difference is that in the Karana mudra the thumb does not hold down the middle and ring finger. This mudra is also known as tarjanī mudrā.

Indian classical dance
In Indian classical dance and derived dances (such as Khmer, Thai or Balinese), the term "Hasta Mudra" is used. The Natya Shastra describes 24 mudras, while the Abhinaya Darpana of Nandikeshvara gives 28. In all their forms of Indian classical dance, the mudras are similar, though the names and uses vary. There are 28 (or 32) root mudras in Bharatanatyam, 24 in Kathakali and 20 in Odissi. These root mudras are combined in different ways, like one hand, two hands, arm movements, body and facial expressions. In Kathakali, which has the greatest number of combinations, the vocabulary adds up to c. 900. Sanyukta mudras use both hands and asanyukta mudras use one hand. In Thai dances, there are 9 mudras.

Yoga

The classical sources for the yogic seals are the Gheranda Samhita and the Hatha Yoga Pradipika. The Hatha Yoga Pradipika states the importance of mudras in yoga practice: "Therefore the [ Kundalini ] goddess sleeping at the entrance of Brahma's door [at the base of the spine] should be constantly aroused with all effort, by performing mudra thoroughly." In the 20th and 21st centuries, the yoga teacher Satyananda Saraswati, founder of the Bihar School of Yoga, continued to emphasize the importance of mudras in his instructional text Asana, Pranayama, Mudrā, Bandha.

Hand Gestures
There are numerous hand gesture mudras in yoga. Each of the hand gestures is based on the concept of the five elements as they relate to one's fingers.

Hatha Yoga
The yoga mudras are diverse in the parts of the body involved and in the procedures required, as in Mula Bandha, Mahamudra, Viparita Karani, Khecarī mudrā, and Vajroli mudra.

Mula Bandha

Mula Bandha, the Root Lock, consists of pressing one heel into the anus, generally in a cross-legged seated asana, and contracting the perineum, forcing the prana to enter the central sushumna channel.

Mahamudra
Mahamudra, the Great Seal, similarly has one heel pressed into the perineum; the chin is pressed down to the chest in Jalandhara Bandha, the Throat Lock, and the breath is held with the body's upper and lower openings both sealed, again to force the prana into the sushumna channel.

Viparita Karani
Viparita Karani, the Inverter, is a posture with the head down and the feet up, using gravity to retain the prana. Gradually the time spent in the posture is increased until it can be held for "three hours". The practice is claimed by the Dattatreyayogashastra to destroy all diseases and to banish grey hair and wrinkles.

Khechari mudra
Khecarī mudrā, the Khechari Seal, consists of turning back the tongue "into the hollow of the skull", sealing in the bindu fluid so that it stops dripping down from the head and being lost, even when the yogi "embraces a passionate woman". To make the tongue long and flexible enough to be folded back in this way, the Khecharividya exhorts the yogi to make a cut a hair's breadth deep in the frenulum of the tongue once a week. Six months of this treatment destroys the frenulum, leaving the tongue able to fold back; then the yogi is advised to practise stretching the tongue out, holding it with a cloth, to lengthen it, and to learn to touch each ear in turn, and the base of the chin. After six years of practice, which cannot be hurried, the tongue is said to become able to close the top end of the sushumna channel.

Vajroli mudra
Vajroli mudra, the Vajroli Seal, requires the yogi to preserve the semen, either by learning not to release it, or if released by drawing it up through the urethra from the vagina of "a woman devoted to the practice of yoga".

Martial arts
Some Asian martial arts forms contain positions (Japanese: in) identical to these mudras. Tendai and Shingon Buddhism derived the supposedly powerful gestures from Mikkyo Buddhism, still to be found in many Ko-ryū ("old") martial arts Ryū (schools) founded before the 17th century. For example the "knife hand" or shuto gesture is subtly concealed in some Koryu kata, and in Buddhist statues, representing the sword of enlightenment.

title: Mudra


text:
Mac OS X 10.1

Mac OS X 10.1 (code named Puma) is the second major release of macOS, Apple's desktop and server operating system. It superseded Mac OS X 10.0 and preceded Mac OS X Jaguar. Mac OS X 10.1 was released on September 25, 2001, as a free update for Mac OS X 10.0 users. The operating system was handed out for free by Apple employees after Steve Jobs' keynote speech at the Seybold publishing conference in San Francisco. It was subsequently distributed to Mac users on October 25, 2001, at Apple Stores and other retail stores that carried Apple products.

Mac OS X 10.1 was codenamed "Puma" because the internal team thought it was "one fast cat."

System requirements
Supported computers: RAM: Hard Drive Space:
 * Power Mac G3
 * Power Mac G4
 * Power Mac G4 Cube
 * iMac G3
 * iMac G4,
 * eMac,
 * PowerBook G3, except for the original PowerBook G3
 * PowerBook G4
 * iBook
 * 128 megabytes (MB) (unofficially 64 MB minimum)
 * 1.5 gigabytes (GB)

Features
Apple introduced many features that were missing from the previous version, as well as improving overall system performance.

This system release brought some major new features to the Mac OS X platform:
 * Performance enhancements — Mac OS X 10.1 introduced large performance increases throughout the system.
 * Easier CD and DVD burning — better support in Finder as well as in iTunes
 * DVD playback support — DVDs can be played in Apple DVD Player
 * More printer support (200 printers supported out of the box) — One of the main complaints of version 10.0 users was the lack of printer drivers, and Apple attempted to remedy the situation by including more drivers, although many critics complained that there were still not enough.
 * Faster 3D (OpenGL performs 20% faster) — The OpenGL drivers and handling were vastly improved in this version of Mac OS X, which created a large performance gap for 3D elements in the interface, and 3D applications.
 * Improved AppleScript — The scripting interface now allows scripting access to many more system components, such as the Printer Center, and Terminal, thus improving the customizability of the interface. As well, Apple introduced AppleScript Studio, which allows a user to create full AppleScript applications in a simple graphical interface.
 * Improved filehandling - The Finder was enhanced to optionally hide file extensions on a per-file basis. The Cocoa API was enhanced to allow developers to set traditional Mac type and creator information directly without relying on Carbon to do it.
 * ColorSync 4.0, the color management system and API.
 * Image Capture, for acquiring images from digital cameras and scanners.
 * Menu Extras, a set of items the user can add to the system menu, replacing the supplied Dock Extras from Mac OS X 10.0 Cheetah.

Apple switched to using Mac OS X as the default on all then-new Macs with the 10.1.2 release.

Applications found on Mac OS X 10.1 Puma

 * Address Book
 * AppleScript
 * Calculator
 * Chess
 * Clock
 * CPU Monitor
 * DVD Player
 * Image Capture
 * iMovie
 * Internet Connect
 * Internet Explorer for Mac
 * iTunes
 * Mail
 * Preview
 * Process Viewer (now Activity Monitor)
 * QuickTime Player
 * Sherlock
 * Stickies
 * System Preferences
 * StuffIt Expander
 * TextEdit
 * Terminal

title: Mac OS X 10.1


text:
Adeline Records

Adeline Records was an American record label that was formed in Oakland, California in late 1997 and closed in August 2017.

History
In 1997, Adeline Records was founded by Billie Joe Armstrong, lead singer of Green Day, Screw 32's Doug Sangalang, Green Day guitarist Jason White, and professional skateboarder Jim Thiebaud. It was named after a street (Adeline Street) that runs from The Port of Oakland through West Oakland and Emeryville, terminating at Ward St and Shattuck Ave in Berkeley.

Green Day's management company, Pat Magnarella Management, took over ownership of the label and ran it from the company's office in San Diego for many years.

The label was closed in 2017. In July 2017, Pat Magnarella split from Green Day having managed them for over 21 years. Shortly thereafter Adeline Records abruptly closed down.

Merchandise
Adeline Records also sold clothing for men and women as well as other clothing, providing advertising for the label. The range was called Adeline Street and included clothes (skirts, tops, underwear, dresses, hats, hoodies, jackets), accessories and jewellery for men, women, and children. The Adeline street logo is a skull and crossbones with a heart in the center of the crossed bones in pink for women and khaki for men. The clothing line ceased trading in 2008. The company posted the following statement at the time: "Adeline Street has decided not to move forward with future designs while Billie Joe and the guys are working hard on their follow up to 'American Idiot.'"

Bands

 * The Frustrators
 * Green Day (vinyl releases only)
 * Stickup Kid
 * Jesse Malin
 * Living with Lions
 * Look Mexico
 * One Time Angels
 * One Man Army
 * Pinhead Gunpowder
 * Timmy Curran
 * White Wives
 * AFI
 * Agent 51
 * Broadway Calls
 * Fetish
 * Fleshies
 * The Frisk
 * The Hours
 * The Influents
 * The Living End
 * The Network
 * The Soviettes
 * The Thumbs
 * Emily's Army

title: Adeline Records


text:
Estate in land

An estate in land is, in the law of England and Wales, an interest in real property that is or may become possessory. It is a type of personal property and encompasses land ownership, rental and other arrangements that give people the right to use land. This is distinct from sovereignty over the land, which includes the right to government and taxation.

This should be distinguished from an "estate" as used in reference to an area of land, and "estate" as used to refer to property in general.

In property law, the rights and interests associated with an estate in land may be conceptually understood as a "bundle of rights" because of the potential for different parties having different interests in the same real property.

Categories of estates
Estates in land can be divided into four basic categories:
 * 1) Freehold estates: rights of conveyable exclusive possession and use, having immobility and indeterminate duration
 * 2) * fee simple
 * 3) ** fee simple absolute—most rights, least limitations, indefeasible
 * 4) * defeasible estate—voidable possession and use
 * 5) ** fee simple determinable
 * 6) ** fee simple subject to a condition subsequent
 * 7) ** fee simple subject to executory limitation
 * 8) * finite estate—limited to lifetimes
 * 9) ** life estate—fragmented possession and use for duration of someone's life
 * 10) ** fee tail—inalienable rights of inheritance for duration of family line
 * 11) Leasehold estates: rights of possession and use but not ownership.  The lessor (owner/landlord) gives this right to the lessee (tenant).  There are four categories of leasehold estates:
 * 12) estate for years (a term of year absolute or tenancy for years)—lease of any length with specific begin and end date
 * 13) periodic estate (periodic tenancy)—automatically renewing lease (month to month, week to week)
 * 14) estate at will (tenancy at will)—leasehold for no fixed time or period.  It lasts as long as both parties desire.  Termination is bilateral (either party may terminate at any time) or by operation of law.
 * 15) tenancy at sufferance—created when tenant remains after lease expires and becomes a holdover tenant, converts to holdover tenancy upon landlord acceptance.
 * 16) *Types of leases:
 * 17) ** gross lease
 * 18) ** net lease
 * 19) ** percentage lease
 * 20) Concurrent estates: owned or possessed by two or more individuals simultaneously.
 * 21) tenancy by the entirety
 * 22) joint tenancy
 * 23) tenancy in common
 * 24) statutory estates: created by statute:
 * 25) * community property
 * 26) * homestead — protection from claim by creditors
 * 27) * dower—interest a wife has in the property of her husband
 * 28) * curtesy—interest a husband has in the property of his wife
 * 29) Equitable estates: neither ownership nor possession
 * 30) * Future interests — interests in real or personal property, a gift or trust, or other things in which the privilege of possession or of enjoyment is in the future and not the present
 * 31) ** reversions
 * 32) ** possibilities of reverter
 * 33) ** powers of termination, also known as rights of reentry for condition broken
 * 34) ** remainders
 * 35) ** executory interests
 * 36) * Incorporeal interests — those that cannot be possessed physically, since they consist of rights of a particular user, or the right to enforce an agreement concerning use
 * 37) ** easement
 * 38) *** easement in gross
 * 39) *** easement appurtenant
 * 40) **** ingress
 * 41) **** egress
 * 42) ** profits
 * 43) ** real covenants
 * 44) *** covenant appurtenant
 * 45) *** covenant in gross
 * 46) ** equitable servitudes
 * 47) ** licenses
 * 48) * Lien
 * 49) ** general
 * 50) ** specific

title: Estate in land


text:
Mac OS X 10.0

Mac OS X 10.0 (code named Cheetah) is the first major release of macOS, Apple's desktop and server operating system. It was released on March 24, 2001, for a price of $129 after a public beta.

Mac OS X was Apple's successor to the classic Mac OS. It was derived from NeXTSTEP and FreeBSD, and featured a new user interface called Aqua, as well as improved stability and security due to its new Unix foundations. It introduced the Quartz graphics rendering engine for hardware-accelerated animations. Many technologies were ported from the classic Mac OS, including Sherlock and the QuickTime framework. The core components of Mac OS X were open sourced as Darwin.

Boxed releases of Mac OS X 10.0 also included a copy of Mac OS 9.1, which can be installed alongside Mac OS X 10.0, through the means of dual booting (which meant that reboots are required for switching between the two OSes). This was important for compatibility reasons; while many Mac OS 9 applications could be run under Mac OS X in the Classic environment, some, such as applications that directly accessed hardware, could only run under Mac OS 9.

Six months after its release, Mac OS X 10.0 was succeeded by Mac OS X 10.1, code named Puma.

Development
The development of Mac OS X 10.0 began in 1998, after Apple acquired NeXT Computer, which was founded by Steve Jobs after he was forcibly removed from Apple in the mid-1980s.

The initial development of Mac OS X was led by Avie Tevanian, who had previously worked at NeXT and had played a key role in the development of NeXTSTEP. The development team faced significant challenges in merging the classic Mac OS with the new Unix-based architecture, as well as in creating a modern user interface that would be familiar to Mac users.

Mac OS X 10.0 was released to the public on March 24, 2001, after several months of beta testing. The release was met with mixed reviews, with some users praising the new features and stability, while others criticized the lack of compatibility with older Mac applications.

Some of the key features of Mac OS X 10.0 included the Aqua user interface, which introduced translucent menus, drop shadows, and other visual effects; the Dock, a new way of launching and switching between applications; and a new file system called HFS+. The operating system also included built-in support for networking protocols such as TCP/IP and PPP, as well as for USB and FireWire devices.

In the years following the release of Mac OS X 10.0, Apple continued to refine and improve the operating system, releasing updates and new versions that added new features and improved performance and compatibility. Mac OS X was renamed OS X in 2011, and to macOS in 2016;, the latest version is macOS Sonoma, which was released in September 2023.

New and updated features

 * The features of the release include the Dock which was a new way of organizing one's Mac OS X applications on a user interface, and a change from the classic method of Application launching in previous Mac OS systems.
 * It included Terminal, a terminal emulator that provides access to Mac OS X's Unix command-line interface; the classic Mac OS had previously had the distinction of being one of the few operating systems with no command line interface.
 * The new Mail email client included the ability to configure the software to receive all of a user's email accounts in one list, the ability to file emails into folders, the ability to search for emails, and the ability to automatically append signatures to outgoing emails.
 * The Address Book was a new application which had features including exporting and importing cards to and from vCard format, API to interface with other applications, change of address notifications, contact groups, auto-merge when importing vCards, customizable fields and categories, the automatic formatting of phone numbers.
 * TextEdit replaced the SimpleText application with new features.
 * PDF support was added; it allows the user to create PDFs from any application.
 * The OS introduced the new Aqua UI.
 * Several features of Mac OS 9 were ported to Mac OS X, including the Sherlock desktop and web search engine.

Removed features

 * File-sharing client — The system can only use TCP/IP, not AppleTalk, to connect to servers sharing the Apple Filing Protocol. It cannot use SMB to connect to Windows or Samba servers.
 * File-sharing server — As a server, the system can share files using only the Apple Filing Protocol (over TCP/IP), HTTP, SSH, and FTP.
 * Optical media — Neither DVD playback nor burning CDs or DVDs is supported. However, audio CD burning was added in the Mac OS X 10.0.2 update, roughly 2 months after initial release.

Architecture
Mac OS X is built on Darwin, a Unix-like operating system derived from FreeBSD. Darwin includes a new kernel, XNU, derived from Mach and BSD, as a replacement for the Mac OS nanokernel used in classic Mac OS.

Unlike Mac OS 9, Mac OS X has protected memory and preemptive multitasking. This means that if an application's memory becomes corrupted due to a bug, the application will crash without the entire system crashing and needing to be rebooted.

Mac OS X also had support for OpenGL, AppleScript, and the Carbon and Cocoa APIs.

Language support
Mac OS X 10.0 began a short era (that ended with Mac OS X 10.2 Jaguar's release) where Apple offered two types of installation CDs: 1Z and 2Z CDs. The difference in the two lay in the extent of multilingual support.

Input method editors of Simplified Chinese, Traditional Chinese, and Korean were only included with the 2Z CDs. They also came with more languages (the full set of 15 languages), whereas the 1Z CDs came only with about eight languages and could not actually display simplified Chinese, traditional Chinese and/or Korean (except for the Chinese characters present in Japanese Kanji). A variant of 2Z CDs were introduced when Mac OS X v10.0.3 was released to the Asian market (this variant could not be upgraded to version 10.0.4). The brief period of multilingual confusion ended with the release of v10.2. Currently, all Mac OS X installer CDs and preinstallations include the full set of 15 languages and full multilingual compatibility.

Marketing
Mac OS X 10.0 was not externally marketed with its codename, a practice which began with Mac OS X Jaguar.

On March 23, 2001, the night before the launch day, Apple hosted a launch party in a Micro Anvika store located at Tottenham Court Road, London. Attendees were provided with live music, along with food and alcoholic beverages. The first 50 Mac OS X 10.0 customers of the participating stores in the UK, including the store hosting the launch party, would receive a free Apple Pro Mouse, while the next 100 customers would be given a commemorative Mac OS X T-shirt. These participating stores also provided demonstrations and offers for Mac products. Similarly, an Apple Authorized Service Provider (known back then as Apple Specialist), in Cupertino, California, held a launch party at midnight. The first 100 Mac OS X 10.0 customers would receive a free commemorative Mac OS X T-shirt. On the day, the store was completely packed with customers and fans of Apple products. Steve Wozniak, one of the co-founders of Apple, also attended the launch party.

In the US, multiple Apple authorized resellers also held events on the 24 and 25 of March 2001, to coincide with Mac OS X's launch.

A store located in Minneapolis reported that over 60% of their available Mac OS X stock were sold on launch day.

On April 3, 2001, Apple launched an bi-weekly email newsletter service named Mac OS X Product News that showcases the latest software for Mac OS X.

Reception
The release of Mac OS X 10.0 saw mixed reviews. ZDNet called the new OS "underdone" due to poor performance, kernel panics, with main applications such as Finder causing system freezes. CNET rated Mac OS X 10.0, a 6 out of 10, calling it "more stable than previous Mac OSs", along with compliments on its UI, memory management and speed, but isn't "ready for the masses", due to issues such as the lack of native third-party applications for the platform, missing DVD playback and hard to use user interfaces. David Pogue stated in a New York Times tech column, that while he felt Mac OS X was better looking and easier to use than Mac OS 9 with superior features and menus, it is not yet ready for the average user, as it's not as polished, with missing features such as CD burning (at launch), automated shutdown scheduling, Labels menu.

System requirements

 * Supported Computers: Power Macintosh G3 Beige, G3 B&W, G4, G4 Cube, iMac G3, PowerBook G3, PowerBook G4, iBook
 * RAM:
 * 128 MB (unofficially 64 MB minimum)
 * Hard Drive Space:
 * 1,500 MB (800 MB for the minimal install)

title: Mac OS X 10.0


text:
National Railroad Museum

The National Railroad Museum is a railroad museum located in Ashwaubenon, Wisconsin, US.

Founded in 1956 by community volunteers, the National Railroad Museum is one of the oldest and largest U.S. institutions dedicated to preserving and interpreting the nation's railroad history. Two years later, a joint resolution of Congress recognized the Museum as the National Railroad Museum. The museum has been a Nonprofit 501(c)(3) organization since 1958.

Its collection of locomotives and rolling stock spans more than a century of railroading. Notable items include an Aerotrain; Union Pacific Big Boy No. 4017, one of the world's largest steam locomotives; and British Railways Class A4 No. 60008 Dwight D Eisenhower (ex-London & North Eastern Railway No. 4496 Golden Shuttle) and train used by the Supreme Allied Commander and his staff in the United Kingdom and continental Europe during World War II.

A museum building houses a wide variety of railroad artifacts, an archive, and photography gallery. A standard gauge track rings the grounds. An 80 ft wooden observation tower has views of the Fox River and Green Bay.

The museum hosts an annual Day Out with Thomas event, where Thomas the Tank Engine pulls young friends past the exhibited rolling stock; and in October, "Terror on the Fox": Green Bay Preble Optimist Club's haunted attraction that includes "haunted" train rides after dark. The Frederick J. Lenfesty Center, an enclosed and climate-controlled structure was recently added that houses several of the unique and rarer locomotives and cars.

Expansion
A $15 million expansion project plan has been released that includes plaza along the Fox River and an additional display building. The state of Wisconsin, through the use of American Rescue Plan Act funds is contributing $7 million to this project.

Other collections
The museum's archives hold corporate records and documents, annual reports, maps, mechanical and engineering drawings, oral histories, and ephemera. The holdings represent various railroad companies, labor unions, and fraternal organizations.

Its library holds works on the social, economic, political, and cultural aspects of U.S. railroading history.

The National Railroad Museum holds over 5,000 artifacts, including textiles, uniforms, tools and personal items.

Its photograph collection includes 15,000 photographic prints, slides, and film negatives of U.S. railroading since 1890.

Capital campaign
As of 2019, the museum was working to raise money to build a roundhouse to surround its current buildings (except the train station) to shelter from the weather the locomotives and cars displayed in the open pavilion.

title: National Railroad Museum


text:
EKU

EKU or Eku may refer to:

 * Eku, Delta, a town in Nigeria
 * Eku Edewor (born 1986), Nigerian actress
 * Earliest known use, in philately
 * Eastern Kentucky University, a public, regional comprehensive, post-secondary institution in Richmond, Kentucky, United States
 * European Karate Union, now the European Karate Federation
 * Evangelical Church of the Union (Evangelische Kirche der Union), a defunct German Christian denomination
 * EKU, a former German brewery, now a brand of Kulmbacher Brewery
 * Extended Key Usage, an extension of Public Key Infrastructure to a Digital signature

title: EKU


text:
Invagination

Invagination is the process of a surface folding in on itself to form a cavity, pouch or tube. In developmental biology, invagination is a mechanism that takes place during gastrulation. This mechanism or cell movement happens mostly in the vegetal pole. Invagination consists of the folding of an area of the exterior sheet of cells towards the inside of the blastula. In each organism, the complexity will be different depending on the number of cells. Invagination can be referenced as one of the steps of the establishment of the body plan. The term, originally used in embryology, has been adopted in other disciplines as well.

There is more than one type of movement for invagination. Two common types are axial and orthogonal. The difference between the production of the tube formed in the cytoskeleton and extracellular matrix. Axial invagination can be formed at a single point along the axis of a surface. Orthogonal invagination is linear and trough.

Biology

 * Invagination is the morphogenetic processes by which an embryo takes form, and is the initial step of gastrulation, the massive reorganization of the embryo from a simple spherical ball of cells, the blastula, into a multi-layered organism, with differentiated germ layers: endoderm, mesoderm, and ectoderm. More localized invaginations also occur later in embryonic development,
 * The inner membrane of a mitochondrion invaginates to form cristae, thus providing a much greater surface area to accommodate the protein complexes and other participants that produce adenosine triphosphate (ATP).
 * Invagination occurs during endocytosis and exocytosis when a vesicle forms within the cell and the membrane closes around it.
 * Invagination of a part of the intestine into another part is called intussusception.

Amphioxus
The invagination in amphioxus is the first cell movement of gastrulation. This process was first described by Conklin. During gastrulation, the blastula will be transformed by the invagination. The endoderm will fold towards the inner part and that way the blastocoel will be gone transforming into a cup-shaped structure with a double wall. The inner wall will be called the archenteron; the primitive gut. The archenteron will open to the exterior through the blastopore. The outer wall will become the ectoderm. Later forming the epidermis and neural crest.

Tunicates
In tunicates, invagination is the first mechanism that takes place during gastrulation. The four largest endoderm cells induce the invagination process in the tunicates. Invagination consists of the internal movements of a sheet of cells (the endoderm) based on changes in their shape. The blastula of the tunicates is a little flattened in the vegetal pole making a change of shape from a columnar to a wedge shape. Once the endoderm cells were invaginated, the cells will keep moving beneath the ectoderm. Later, the blastopore will be formed and with this, the invagination process is complete. The blastopore will be surrounded by the mesoderm by all sides.

Geology
In geology, invagination is used to describe a deep depression of strata. Used by Donald L. Baars in "The Colorado Plateau".

title: Invagination


text:
Acid hydrolase

An acid hydrolase is an enzyme that works best at acidic pHs. It is commonly located in lysosomes, which are acidic on the inside. Acid hydrolases may be nucleases, proteases, glycosidases, lipases, phosphatases, sulfatases and phospholipases and make up the approximately 50 degradative enzymes of the lysosome that break apart biological matter.

types of Acid Hydrolase:

-Nucleases (P1 from Penicillium citrinum, used in the food industry for taste enhancement or present in Gouda cheese)

-Lipase: for example lysosomal acid lipase.

-Proteases

-Glycosidases

title: Acid hydrolase


text:
Little Albert experiment

The Little Albert experiment was a controversial study that mid-20th century psychologists interpret as evidence of classical conditioning in humans. The study is also claimed to be an example of stimulus generalization although reading the research report demonstrates that fear did not generalize by color or tactile qualities. It was carried out by John B. Watson and his graduate student, Rosalie Rayner, at Johns Hopkins University. The results were first published in the February 1920 issue of the Journal of Experimental Psychology.

After observing children in the field, Watson hypothesized that the fearful response of children to loud noises is an innate unconditioned response. He wanted to test the notion that by following the principles of the procedure now known as "classical conditioning", he could use this unconditioned response to condition a child to fear a distinctive stimulus that normally would not be feared by a child (in this case, furry objects). However, he admitted in his research article that the fear he generated was neither strong nor lasting.

Method
The aim of Watson and Rayner was to condition a phobia in an emotionally stable child. For this study, they chose a nine-month-old infant from a hospital. The child was referred to as "Albert" for the experiment. Watson followed the procedures which Ivan Pavlov had used in his experiments with dogs.

Before the experiment, Albert was given a battery of baseline emotional tests: the infant was exposed, briefly and for the first time, to a white rat, a rabbit, a dog, a monkey, masks (with and without hair), cotton, wool, burning newspapers, and other stimuli. Albert showed no fear of any of these items during the baseline tests.

For the experiment proper, by which point Albert was 11 months old, he was put on a mattress on a table in the middle of a room. A white laboratory rat was placed near Albert and he was allowed to play with it. At this point, Watson and Rayner made a loud sound behind Albert's back by striking a suspended steel bar with a hammer each time the baby touched the rat. Albert responded to the noise by crying and showing fear. After several such pairings of the two stimuli, Albert was presented with only the rat. Upon seeing the rat, Albert became very distressed, crying and crawling away. Apparently, the infant associated the white rat with the noise. The rat, originally a neutral stimulus, had become a conditioned stimulus and was eliciting an emotional (conditional) response similar to the distress (unconditioned response) originally given to the noise (unconditioned stimulus).

In further experiments, Little Albert seemed to generalize his response to the white rat. He became distressed at the sight of several other furry objects, such as a rabbit, a furry dog, a seal-skin coat, and even a Santa Claus mask with white cotton balls in the beard. However, this stimulus generalization did not extend to everything with hair.

Watson's experiment had many failings by modern standards. For example, it had only a single subject and no control subjects. Furthermore, such an experiment could be hard to conduct in compliance with current laws and regulations, given the expected risks to the subject.

Subsequent events
Albert was about one year old at the end of the experiment, and he reportedly left the hospital shortly thereafter. Though Watson had discussed what might be done to remove Albert's conditioned fears, he chose not to attempt such desensitization with Albert, and it is thought likely that the infant's fear of furry things continued post-experimentally.

Watson later gave a series of weekend lectures describing the Little Albert study. One of these lectures was attended by Mary Cover Jones, which sparked her interest in pursuing graduate work in psychology. Jones conducted an experiment to figure out how to eliminate fear responses in children and studied a boy named Peter, who was two years old. Peter shared similar fears of white rabbits and furry objects as Little Albert. Jones was able to increase Peter's tolerance of white rabbits by exposing him to the animal at gradually shorter distances and having Peter interact with children who were not afraid of the rabbit. Watson is listed in the publication of the study as advisor and editor. Mary Cover Jones was the first psychologist to desensitize, or decondition, a fear response and become known as the "Mother of Behavior Therapy".

Identifying Little Albert
According to some textbooks, Albert's mother worked in the same building as Watson and did not know the tests were being conducted. When she found out, she took Albert and moved away, letting no one know where they were going. A 2009 report, however, disputes that. The original report had stated that the baby's mother was a wet nurse at the hospital, who may have felt coerced and unable to turn down a request for her baby to be used in Watson's experiment. The claim of coercion was successfully challenged in an article published in the same journal as the coercion claim.

Douglas Merritte
In 2009, psychologists Hall P. Beck and Sharman Levinson published an article in which they claimed to have discovered the true identity of "Albert B." After reviewing Watson's correspondence and publications, as well as research in public documents (such as the 1920 United States Census and state birth and death records), Beck argued that "Albert B." was a pseudonym for Douglas Merritte, the son of Arvilla Merritte, then a woman who appears to have been a wet nurse at the Harriet Lane Home.

It was later found that Douglas Merritte had hydrocephalus, from which he died at the age of 6. With this condition, which is an accumulation of cerebrospinal fluid on the brain, Merritte may have had severe trouble seeing at the time of the experiment, and this disputes the claim that the child in question was average and healthy. According to researchers who looked at this case years later, if Douglas Merritte was, indeed, Little Albert, his actions during the conditioning sessions align with signs of neurological impairment. This includes Little Albert's use of hand-scooping, rather than grasping gestures typical of this age, as well as poor eye-scanning abilities and his lack of facial expressions. Other research has argued, however, that Douglas Merritte may not have been "Little Albert", who may, in fact, have been young Albert Barger (known later as William Albert Martin).

Albert Barger
The identity claimed by Beck et al. has been contested by psychology researchers Russ Powell, Nancy Digdon and Ben Harris, who offer an alternative identity based on available data. Albert Barger had been born within a day of Merritte, and his mother had also worked at the hospital where the experiment was conducted. In addition, his body weight and developmental condition closely matched the experiment's documentation of the subject baby's condition. Moreover, according to Watson and Rayner (1920), Little Albert was 12 months and 21 days the last day of the experiment when "he was taken from the hospital" (p. 12), the exact age Albert Barger had when he left the hospital. Finally, when Powell et al. were allowed to independently verify Douglas Merritte clinical file, it was revealed that he was "completely blind", which is at odds with the experiment's films where Little Albert engages in probable instances of object-directed action and social referencing.

Through the use of a professional genealogist, the researchers learned Barger had died in 2007 at age 87 and identified one close living relative, a niece. In an interview, Barger's niece stated that she and her uncle had been quite close throughout his life, and acknowledged Barger's antipathy toward dogs as a well-known fact that family members, particularly his wife, would tease him about (the researchers noted there was no way to determine whether or not this behavior was linked to Watson's experiment). She also informed researchers of her uncle's aversion to animals in general, not just dogs. Though it was not a particularly strong aversion, family members would often have to keep their dogs in a separate room when he visited. Outside of this, Barger's niece stated that she did not recall any other phobias he may have had. The researchers concluded that Barger would have been unaware of his role as an infant test subject.

Ethical considerations
The experiment today would be considered unethical according to the American Psychological Association's ethic code, and legislation has been passed to prevent such potentially harmful experiments. In the early 1970s, following widely publicized cases of research abuse, the National Commission for the Protection of Human Subjects of Biomedical and Behavioral Research (NCPHS) was created to study issues surrounding the protection of humans in research. In 1979, the Commission issued a report entitled Ethical Principles and Guidelines for the Protection of Human Subjects of Research (commonly called the Belmont Report), which provided the ethical framework on which current federal regulations for the protection of human participants in research are based. Under the NCPHS standards set in the late 1970s, an experiment such as Watson's would not have been allowed. Research with participants is also regulated by the Health Insurance Portability and Accountability Act of 1996 and the Public Health Service Act. In addition, training of researchers in the use of human participants has been required by the National Institutes of Health since 2000.

Criticisms
A detailed review of the original study and its subsequent interpretations by Jared Harris (1979) stated: "Critical reading of Watson and Rayner's (1920) report reveals little evidence either that Albert developed a rat phobia or even that animals consistently evoked his fear (or anxiety) during Watson and Rayner's experiment. It may be useful for modern learning theorists to see how the Albert study prompted subsequent research ... but it seems time, finally, to place the Watson and Rayner data in the category of 'interesting but uninterpretable' results."

It is difficult to be certain exactly what happened during the Little Albert experiment since concrete evidence and scientific records are lacking. Though a film was shot during the experiment, textbooks interpret the movie differently. Various sources give contradicting accounts of events that took place, and they raise questions about exactly what stimuli were used, which stimuli the baby came to fear, and what happened to the child after the experiment. It was said that most textbooks "suffer from inaccuracies of various degrees" while referring to Watson and Rayner's study. Texts often misrepresent, exaggerate, or minimize the range of Albert's post-conditioning fears.

Other criticisms stem from the health of the child (cited as Douglas Merritte) who was not a "healthy", "normal" infant as claimed in the study, but one who was very ill and had exhibited symptoms of hydrocephalus since birth—according to relatives he never learned to walk or talk later in life. The child would die five years after the experiment due to complications from the congenital disease. It is stated that the study's authors were aware of the child's severe cognitive deficit, abnormal behavior, and unusually frequent crying, but continued to terrify the sick infant and generalize their findings to healthy infants, an act criticized as academic fraud. These accusations were successfully challenged in the same journals that published the initial claims. The articles refuting the "sick baby" claim explain that the child was actually William (called Albert by his family) Barger, and that the child was, in fact, healthy.

title: Little Albert experiment


text:
Talk:Binomial coefficient

Multiplicative formula
Made an edit from (n-(k-1)) to (n-(k+1)) which should be the correct falling factorial. If I'm wrong revert me please. 2601:640:4100:4C70:81FA:2C7B:8F13:A0D4 (talk) 22:17, 5 September 2019 (UTC)
 * You're wrong so I have reverted you. Maybe you missed that n-(k-1) = n-k+1. PrimeHunter (talk) 23:29, 5 September 2019 (UTC)
 * Doh, completely spaced. It just felt weird at first glance. Thanks. 2601:640:4100:4C70:81FA:2C7B:8F13:A0D4 (talk) 00:42, 6 September 2019 (UTC)

Binomial Coefficient definition should precede discussion of Binomial Theorem and Pascal's Triangle
The definition of the binomial coefficient in terms of what it means numerically should precede discussion of binomial theorem. Reader quite possibly may only want to know what it means and have no interest in binomial theorem or Pascal Triangle.RHB100 (talk) 21:31, 11 August 2012 (UTC)

 * This is not a difficult article to navigate: a person interested in formulas for binomial coefficients need only look at the table of contents for the well-labeled sections with the word "formula" in their titles. Please see Manual of Style/Lead section to read about what a lead section is supposed to accomplish; notably, "The lead should be able to stand alone as a concise overview. It should define the topic, establish context, explain why the topic is notable, and summarize the most important points."  Your sentence is detrimental to almost all of these goals, and distracts emphasis from the important definitions to less-important formulas. --JBL (talk) 02:24, 11 August 2012 (UTC)

The article at present is either deliberately designed to confuse readers or it is so poorly written that it confuses the reader as much as if it had been deliberate. The statement above implying that the definition of the binomial coefficient, $$ \binom nk = \frac{n!}{k!\,(n-k)!} \quad \mbox{for }\ 0\leq k\leq n$$, where the notation, $$n!$$, called n factorial is defined by $$n! = n(n-1)(n-2) ... 1$$ with $$0! = 1$$, is of less importance than Pascal's triangle and such is a very ignorant and stupid statement. RHB100 (talk) 21:31, 11 August 2012 (UTC)

 * Go have a read of WP:Civility and perhaps we can talk. It seems to me possible that a formula (or a pointer to the section on formulas) could be worked into the introduction somewhere, but you're very much going about it the wrong way. --JBL (talk) 18:11, 13 August 2012 (UTC)

 * I fully agree with that —specially about the civility issue. The factorial expression could i.m.o. be easily incorporated in the lead — not the way RHB100 has in mind though, as that would indeed be 100% orthogonal to Manual of Style/Lead section. I have made an attempt at doing so. I think this can work. - DVdm (talk) 18:43, 13 August 2012 (UTC)

OK, I think it is somewhat better now. However, I think it should be kept in mind that some people may not be familiar with the factorial operator. Therefore I think a definition of the factorial operator for the simplest case of positive integers and zero before getting into the more advanced part of the article would be desirable. RHB100 (talk) 19:15, 13 August 2012 (UTC)

 * Mentioning factorials would i.m.o. clutter up the lead. The reader is explicitly pointed to factorials in all their glory in the section Binomial coefficient. Remember the guidelines about the lead: the keywords to go with are concise overview and summarize. - DVdm (talk) 20:00, 13 August 2012 (UTC)

 * I think the current phrasing (while obviously better than the original) is somewhat cluttered -- that sentence is now doing an awful lot of work. Better would be to make it its own sentence (a little later in the paragraph).  In fact, though, I think it would be even better to replace the formula with a sentence like the following: "There are several simple formulas (both recursive and explicit) for computing binomial coefficients -- see the section below for details."  This serves the purpose of mentioning and pointing to the formulas without cluttering the lead or detracting from the more important definitions (namely, that these are the coefficients that appear in the binomial theorem, and their combinatorial interpretation).  --JBL (talk) 20:09, 13 August 2012 (UTC)

 * Hm, I wouldn't opt for a see-below-like-kind-of-easteregg like you suggested. But yes, I see your point. Indeed, the sentence is doing too much work now, so I have moved the calculation into its own sentence — let's have the work done by two sentences, right? There are indeed several simple formulas, but I dare assume, so to speak, that we can all agree that the factorial expression is the most common and most notable... so how about this? - DVdm (talk) 21:29, 13 August 2012 (UTC)

 * Would you object to replacing "Under suitable circumstances" with "When n and k are nonnegative integers"? Should there be a remark about the fact that they can be defined more generally?  Or is this too much for the lead?  Edit: ok, the remark about generalizing comes later, is fine there, and shouldn't be repeated.  The first question stands.  --JBL (talk) 21:43, 13 August 2012 (UTC)

 * Re the first question: I wouldn't mind —provided n-k is required to be a nonnegint as well, which would add slightly more clutter again— but I don't think we really need that in this lead. - DVdm (talk) 22:05, 13 August 2012 (UTC)

 * Good point. --JBL (talk) 22:06, 13 August 2012 (UTC)

(In english please. It's hard to know what is just jibber jabber and what is solid useful math language!)

Pronunciation?
How is $$ \binom nk $$ usually pronounced? Shouldn't this be in the article? 84.29.139.151 (talk) 09:43, 9 June 2014 (UTC)
 * Try reading the second paragraph of the lead section ;). JBL (talk) 13:19, 9 June 2014 (UTC)
 * Since this part of the text has changed since the above comment was written 7 years ago, I'll quote the relevant sentence from that point in time: $$\tbinom nk$$ is often read as "n choose k". 194.39.218.10 (talk) 13:45, 2 July 2021 (UTC)
 * I've added this sentence back to the main article. 194.39.218.10 (talk) 13:58, 2 July 2021 (UTC)
 * You are mistaken that it has changed -- I mean, perhaps the location is different now than it was, but it's still in the lead, following "The binomial coefficients occur in many areas of mathematics, and especially in combinatorics." So I have reverted your redundant addition. --JBL (talk) 14:10, 2 July 2021 (UTC)

Another identity
You have (under 'Identities involving binomial coefficients') everything but the basis for the recursive formula that appears in 'Computing the value of binomial coefficients':

nCk = (n-1)C(k-1) + (n-1)Ck

when I copy/paste I get: \binom nk = \binom{n-1}{k-1} + \binom{n-1}k \quad \text{for all integers }n,k : 1\le k\le n-1

The second identity down is similar looking, but the recursive formula seems more direct.

It just seems like the 'recursive identity' should be included. 71.139.161.9 (talk) 05:12, 28 August 2014 (UTC)

Equation numbering
Although I approve of equation numbering (as found in this article) and certainly use it in my own publications, I am finding it a hinderance to editing this article. I am considering numbering an equation early in the article which doesn't have a number now. This will require a wholesale renumbering of all the equations below it and their reference tags. Unwillingness to do this may explain why a blatant but minor error has not been corrected for years. When I do this I will remove all equation numbers that are not referred to (I know that many authors put them in, "just in case", they need to refer to the equations, or for the possibility that they might want to do this in the future ... but it creates more work for editors) and I will move to a numbering within section scheme (equations numbered 2.1, 2.2, ..., 2.k in section 2 and 3.1, 3.2 ... in section 3, and this can be extended to subsections as well). This will mean that fewer changes are needed when additions and deletions are made to numbered equations. Comments? Bill Cherowitzo (talk) 06:55, 4 December 2014 (UTC)
 * I certainly approve of not numbering equations that we don't refer to. And in some cases, like (3), there is an actual name we can refer to ("Pascal's relation") that might make the text clearer as well as avoiding the need for numbers. As for the section-based numbering, I wonder if it really will make for fewer changes? Then we'd have to change all the numbers every time we change the section heading organization rather than every time we add or remove a numbered equation; both seem likely to be infrequent, but still often enough to be annoying. I just added my own first WP "numbered" equation a couple of days ago (in Euclid–Euler theorem) but in that case there was only one so I just used (*). —David Eppstein (talk) 07:07, 4 December 2014 (UTC)

I realized the problem with section reorganization shortly after I wrote the above. Perhaps using a letter scheme (A.1, A.2, etc. for section 3; B.1, B.2, etc. for section 5 (with no numbered equations in section 4)) and not going any deeper than main section levels would be about the best one could hope for given the fluid nature of our articles. I think that stars and names are to be preferred whenever they make sense. Bill Cherowitzo (talk) 18:03, 4 December 2014 (UTC)

Approximation
Hi, I explicitely checked the formula in the subsection APPROXIMATION, with Mathematica 10.0 and it is clearly wrong since the ratio of the two members goes to zero instead of 1. Could please someone tell me where to find the derivation of such a formula? — Preceding unsigned comment added by 163.1.241.224 (talk) 18:43, 11 May 2015 (UTC)

 * I think it's implicit in the section that $$ k = \Theta(n)$$, though of course this isn't stated explicitly. (And why is this section not part of the preceding section, anyhow?  Sigh.)  --JBL (talk) 21:10, 11 May 2015 (UTC)

 * I'd rather say that implicitly $$k-(n/2) = o(n)$$; worth to mention this. 192.54.190.20 (talk) 16:37, 3 July 2015 (UTC)

Omar Khayyam triangle and Yang Hui triangle
User:Joel B. Lewis, you removed information about Yang Hui's triangle, with the argument this is a redirection, this is not the case, this was as now a link to the author of the triangle, I also added a link to Omar Khayyam, as it also created this triangle in Iran. It is interesting to have history of this triangle. I don't understand why you want to remove it.Popolon (talk) 01:36, 18 October 2016 (UTC)
 * , the purpose of the lead section of an article is to summarize the body of the article. If you would like to include historical information, you should put it in the section titled "History," along with an appropriate source.  --JBL (talk) 02:11, 18 October 2016 (UTC)

The way I see it, this article is about binomial coefficients and there is lots to say about binomial coefficients. One part of what is important is the relationship of binomial coefficients to the Hui/Khayyam/Pascal triangle. That relationship deserves a mention, which is already present in the article using the most commonly accepted name Pascal's triangle. That the triangle itself has a long history is useful information for the article about the triangle, but, I am thinking that it is not sufficiently relevant to this article about binomial coefficients. 𝕃eegrc (talk) 13:32, 18 October 2016 (UTC)

Result of Singmaster
I removed the following:

The apparent reference (which I also removed, not given inline, but probably should have been) is here

The statement (as I'm interpreting it) appears to be false. For example, for d = 2, take N = 2M, and then f(N) is exactly 3M. So the expression inside the limit can be made arbitrarily small, certainly not something approaching 1.

Unfortunately, I can only view the first page of the Singmaster article, so if anyone else has access to it and can correct the statement, that would be great. Or maybe I'm just being dense, in which case, please revert me with extreme prejudice. --Deacon Vorbis (talk) 22:10, 2 May 2017 (UTC)

 * The number of even binomial coefficients up to 2M is certainly not 3M. It is the number of odd binomial coefficients that equals 3M. —David Eppstein (talk) 22:17, 2 May 2017 (UTC)
 * Yeah, sorry; I had a bad feeling it had to be something stupid on my part. Thanks. --Deacon Vorbis (talk) 00:16, 3 May 2017 (UTC)

WP:OR?
concludes with an interesting result (namely, that primality is equivalent to dividing all the intermediate binomial coefficients), but those statements look suspiciously like WP:OR. (Note, for example, that the paragraph is unsourced and includes a proof.) I couldn't find a source with a cursory Google search; does anyone else recognize the result as being published somewhere? 128.135.98.14 (talk) 18:03, 15 February 2019 (UTC)
 * There's a somewhat similar (but different) primality test based on looking at non-horizontal lines through Pascal's triangle in . This test itself is called "well-known" and is the starting point for the main results in a more recent paper, —David Eppstein (talk) 22:32, 15 February 2019 (UTC)

Preamble

 * This section is about

Let's continue here.
 * > giving a reason in an edit summary is better, but far better still, there's a reason= parameter that you're leaving blank It is not mandatory to specify reason in cases of obvious uncertainty. The word indexing is out of place and has very vague meaning. It probably means that indexing is expected to be done by using pascal's triangle but it is never meant anywhere and appropriate formula is given to calculate necessary number instead so it must be clarified for those clueless.
 * > still grammar problems, but more importantly, "In mathematics..." or "In biology..." or whatever are standard article openings in Wikipedia. Here, "term" is redundant, as a coefficient means the number sitting in front of a term. Again, if you still disagree, best to take to the talk page and discuss) The article is obviously mathematical so words "in mathematics" are redundant here. The "term" cannot be redundant cause binomial coefficient is inseparable from the term. It is like to say that referring to binomial expansion theorem is redundant here. Completely wrong. Those who are unfamiliar with theorem won't understand anything unless hinted by definitions so it must be left.
 * To reinforce my position I would like you to see what MOS:INTRO says about article accessibility.  DAVRONOV A.A. ✉ ⚑ 20:02, 17 February 2019 (UTC)
 * I find this "grousing" to be totally unconvincing. The concern about indexing can be simply handled with a link to Index notation. We use expressions like "In mathematics ..." to set the context of an article. Redundancy is not an issue, this is only redundant if you already know what the article is about from its title, an assumption that flies in the face of accessibility. The reason that "binomial theorem" is so popular an expression is because "binomial expansion theorem" is so overly pedantic, a charge that can be leveled against most of these comments.--Bill Cherowitzo (talk) 21:47, 17 February 2019 (UTC)
 * I find this "grousing" to be totally unconvincing.[...] Well, I'm not surprised. After reading the several times I found it difficult to understand anything until I referred to more friendly resources. Currently it seems to me  that it is dedicated for hard-core mathematicians and not ordinary readers. Wikipedia was never meant to be that hard-core scientific reference of anything. Checkout WP:NOTJOURNAL #7 For more info. It is advised that the introduction is understandable without any advanced knowledge. It's pretty clear and that's why I request to clarify terms and give details by myself if I can.
 * The concern about indexing can be simply handled with a link to Index notation.[...] Yeah, thanks for your explanation, it's much better now but it doesn't solve the main problem yet: there is still nothing that the indexing is related to.
 * We use expressions like "In mathematics ..." to set the context of an article.[...] There is no way to infer context from a title unless you know the subject in a advance or it is explicitly mentioned. The context of this article is more algebraic rather than of merely broad math. Why not to add this instead of "In mathematics ..."? I'm not insisting on deleting the expression though. I'm mostly concerned about primary definition currently.
 * because "binomial expansion theorem" is so overly pedantic[...] I hardly can disagree with this but it is unrelated to this topic I believe.  DAVRONOV A.A. ✉ ⚑ 10:15, 18 February 2019 (UTC)
 * Just a side note, but if you want to quote other editors, please use (for inline quotes), or / (for set-apart).  This avoids a bunch of span tags all over the place, and it uses a standard color that's less jarring and more accessible.  In any case, I'm just not really sure what the issues here are.  As the article even points out, binomial coefficients crop up in very widely different branches of math, so "In mathematics..." seems to be reasonably appropriate.  As far as the bit about indexing, you said "It is not mandatory to specify reason in cases of obvious uncertainty".  Except it's not obvious here (and it's almost never obvious anywhere) because it makes perfect sense to me and I don't know why someone would find it confusing.  I mean, we could say something like "parameterized by" instead, but that's a bit more MOS:JARGONy.  "Indexed by" means the same thing and is less technical, so I'm not sure what we could do here. — Preceding unsigned comment added by Deacon Vorbis (talk • contribs) 15:53, 18 February 2019 (UTC)

edit to "generating functions"
I edited the first equation under "generating functions" => "Ordinary generating functions" Previously read

 * $$\sum_{k=0}^\infty {n\choose k} x^k = (1+x)^n.$$

now reads
 * $$\sum_{k=0}^n{n\choose k} x^k = (1+x)^n.$$

Summing k above n is mildly confusing at best. — Preceding unsigned comment added by Affine.function (talk • contribs)

 * It would not be confusing to you if you read the article, where this is discussed at least twice: Binomial_coefficient and Binomial_coefficient. --JBL (talk) 12:31, 15 December 2020 (UTC)

Largest currently known positive integer n such that binomial(n,k) has exactly k prime factors, counted with multiplicity
There are some results.

My results are:

k=1, n=2^82589933–1=148894445742...325217902591 (24862048 digits; elipsis omits 24862024 digits) (see largest known prime);

k=2, n=2618163402417 * 2^1290001 - 1 (388342 digits) (see Cunningham chain).

Can you extend my results for k>2? — Preceding unsigned comment added by 109.106.227.160 (talk • contribs) 18:01, 25 August 2021 (UTC)
 * Unless you are basing this on reliable publications rather than your own original research, it is off-topic here. This page is for discussions on improvements to the article, which can only be based on sources. —David Eppstein (talk) 18:27, 25 August 2021 (UTC)

Can you find the rsults for k from 3 to 32, please? I can only find results for n=1 or n=2. I know that, e.g., binomial(799,15) has exactly 15 prime factors, counting multiplicity, but I don't know that 799 is the currently largest known n such that binomial(n,15) has exactly 15 prime factors, counting multiplicity. Joakjovan21 (talk) 15:19, 27 August 2021 (UTC)
 * If you want people to assist with calculations, this is not the place. Try the WP:HELPDESK. —David Eppstein (talk) 16:50, 27 August 2021 (UTC)

Some proposed changes in the article on Binomial Coefficients - section Congruences
If $$n$$ is a positive integer and $$p$$ is a prime number, then  where $$\nu_p(n)$$ is the $$-adic order or $p$-adic valuation of an integer $p$, being the exponent of the highest power of the prime number $n$ that divides $p$. Equation ($n$) follows from a more elementary fact. If $$j$$ is a positive integer, $$ \binom {n-1}{p^j - 1} \equiv \begin{cases} 1 \mod p & \text{if }\ p^j | n \\ 0 \mod p & \text{if }\ p^j \nmid n. \end{cases}$$ Dario T. de Castro (talk) 10:32, 1 September 2022 (UTC) Dario T. de Castro (talk) 10:32, 1 September 2022 (UTC)
 * Specific text to be added:
 * Reason for the change: I humbly think this result is a valid addition to the body of knowledge on this topic. Additionally, I believe that there is no review article in which the result I demonstrated is stated.
 * References supporting change: http://math.colgate.edu/~integers/w61/w61.pdf

 * Hey there, which article do you want to add this to? Thanks, --Ferien (talk) 19:05, 1 September 2022 (UTC)
 * Dear Ferien, the addition was intended for the article on 'Binomial Coefficients', in the 'Congruences' section.Dario T. de Castro (talk) 23:59, 1 September 2022 (UTC)
 * See my comment below for the other requested addition. My reaction here is the same. —David Eppstein (talk) 13:52, 26 September 2022 (UTC)

Proposed changes in the article on Binomial Coefficients - section Divisibility properties
Let $$\nu_p(n)$$ be the $p$-adic valuation of a positive integer $$, being the exponent of the highest power of the prime number $n$ that divides $p$. Then
 * Specific text to be added:
 * $$ \nu_p(n) = p \sum_{j=1}^{\lfloor \log_p{n}\rfloor}\left\{\binom n {p^j} \frac{p^{j-1}}{n} \right\}, $$

where $$\{ x \}$$ denotes the fractional part of $$x$$. Dario T. de Castro (talk) 06:40, 8 September 2022 (UTC) Dario T. de Castro (talk) 06:40, 8 September 2022 (UTC)
 * Reason for the change: I humbly think this result is a valid addition to the body of knowledge on this topic and in this section. Additionally, I believe that there is no review article in which the result I demonstrated is stated.
 * References supporting change: http://math.colgate.edu/~integers/w61/w61.pdf

 * Moved this and the preceding section from Dario's talk page. Rusalkii  (talk) 04:37, 24 September 2022 (UTC)

This article is already a mass of way too many big intimidating equations and way too little explanatory text. Both the previous change and this one would add to that problem, not improve it. —David Eppstein (talk) 04:54, 24 September 2022 (UTC)

 * agreed 24.52.219.172 (talk) 06:25, 26 September 2022 (UTC)
 * I don't think that this has demonstrated the impact that suggests it should go in the article. The other identities in the article are generally well-covered by textbooks in the field. Russ Woodroofe (talk) 13:13, 27 September 2022 (UTC)
 * Dear David, Russ and other contributors
 * I am reading the comments made in this article on Binomial Coefficient (sec. Congruences and sec. Divisibility Properties). I agree with Dr. Eppstein in that the article as a whole needs improvement. I am also aware that the editions I propose involve conflict of interest. So, it seems to me that I should choose not to express myself in this debate and hope that you and your collaborators decide on the best way to edit this material. Perhaps I should say that I proposed these changes with the intention of disseminating knowledge about a somewhat new connection between binomial coefficients and the prime factorization of integers. With kind regards,Dario T. de Castro (talk) 20:00, 27 September 2022 (UTC)

Name for an identity
Hi, in french the formula $$k{n \choose k} =n{n-1 \choose {k-1}} $$ is named "pawn formula" : https://fr.wikipedia.org/wiki/Formule_du_pion. Is there a name in English ? Robert FERREOL (talk) 11:39, 25 November 2022 (UTC)
 * FWIW, Concrete Mathematics calls the identity (or a slightly rearranged form of it) "absorption/extraction". A casual search finds some other uses of the term absorption, although I'm not certain it is widespread; I didn't find the term extraction. Russ Woodroofe (talk) 11:50, 25 November 2022 (UTC)
 * Thank you. I added this reference on https://fr.wikipedia.org/wiki/Formule_du_pion. Robert FERREOL (talk) 09:31, 27 November 2022 (UTC)
 * I have never felt that this identity had a standard English name. --JBL (talk) 17:38, 27 November 2022 (UTC)

Blank section on programming languages?
Can the section on programming languages be blanked? It adds nothing at all to the article; it reads like a homework assignment for high-school seniors. There are interesting things to say about algorithmic identities for binomials, but those algorithms customarily appear first in basic hypergeometric series (and date back to the 19th century, so pre-computer algorithms. I think Kummer gave some of the first ones.) So, yes, there could be a section on those algorithms, but this is not it. 67.198.37.16 (talk) 22:45, 5 January 2024 (UTC)

 * I was bold and just blanked. Looking at the page history, this section was an accretion of original research that was added over the years by anon authors making a case for their favorite programming languages. 67.198.37.16 (talk) 22:55, 5 January 2024 (UTC)

"NCk" listed at Redirects for discussion
The redirect [//en.wikipedia.org/w/index.php?title=NCk&redirect=no NCk] has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at  until a consensus is reached. Shhhnotsoloud (talk) 18:47, 6 February 2024 (UTC)

title: Talk:Binomial coefficient