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Talk:Karnaugh map

Image in top right is utter garbage
"This image actually shows two Karnaugh maps..." Oh, does it? Well I wish it didn't! It's quite unreadable. How is the person who has come here expected to learn if you throw them in at the deep end? How about having three images, with this being the third that shows how you have combined the first two? — Preceding unsigned comment added by 203.241.147.20 (talk) 05:56, 8 December 2023 (UTC)

Marquand x Karnaugh
Is really the same? Need a little topic here. See the redirectioning. — Preceding unsigned comment added by 201.68.255.23 (talk) 08:17, 11 June 2012 (UTC)

Current K map coloration impossible to read
The current example illustrations of K maps on this page are atrociously har to read due to the coloration system used. Such simple examples are made so awfully difficult to make out. I propose replacing these with normal, standard K map illustrations that use colored outlines and no shading.

152.3.68.83 (talk) 19:26, 5 March 2013 (UTC)

Veitch Diagram
Although there is now a reference to the Veitch diagram, this is a little misleading. The Karnaugh Map is a special case of a Veitch Diagram. Although the orig;linal form of the Veitch diagram doesn't enjoy any sort of uses now.

This also conflicts with the idea that the Karnaugh map was 'invented' in 1950 - it is more properly an extension of existing work. Anyone have any thoughts on how to resolve this? NVeitch

Additionally, The link to Veitch_chart is circular because it redirects back to this section. I'm not familiar with the preferred way to resolve. spartlow —Preceding undated comment added 20:24, 7 January 2019 (UTC)

Hm the C,D-row is upside-down. It should go as (0,0)-(0,1)-(1,1)-(1,0). --BL

Gray code
Perhaps we should mention that the rows and columns are ordered according to a gray code, or otherwise explain the importance of having precisely one input value flip between adjacent boxes. Bovlb 00:37, 2004 Mar 11 (UTC)

 * Done. Fresheneesz

I do understand that if you don't use Gray code, Karnaugh maps no longer work the way they do. What's the significance, reason and advantage behind using Gray code in Karnaugh maps?

ICE77 (talk) 22:22, 28 December 2010 (UTC)
 * Yeah the 'why' would be nice, though, the K-maps still work no matter which order you choose. But, you will get more min/max-terms if you don't go with the gray code which then can be reduced. If you use the K-map with the gray code scheme you probably don't have to reduce your terms at all. The final result, however, will still be the same.

Note: Might say it in general. Just tried it on a few examples. 178.9.117.214 (talk) 14:34, 30 January 2012 (UTC)

wrap around
Should there be mentionted that a K-diagram is wrapped around?

So the top left and top right connect and the same goes for bottom left and bottom right.

Kind regards

 * I second this. Fresheneesz 08:50, 8 March 2006 (UTC)

There is a line that reads "just as m0, m8, m2, and m10 can be combined into a four-by-four group."(accessed 21/jun/2010). I think that this should read 'a two-by-two group' because there are only two squares on each side making a total of four squares in the group when wrapping the four corners together. Note, I could be really wrong, because I don't understand any of this and there is likely to be more to it than I get. ElTimbalino (talk) 12:16, 21 June 2010 (UTC)

rectangular circling
Should there be a mention that the cells need to be cirlced in a rectangular shape?
 * Done (not by me tho) Fresheneesz 08:52, 8 March 2006 (UTC)
 * Not wishing to be pedantic, but is this strictly true? When doing them by hand any rough shape that groups the right things will do - mine usually come out ellipsoid. The shape of the "circling" is not relevant to the use of the K-map, it's the grouping that matters.Graham 11:32, 8 March 2006 (UTC)

 * Well, not only would that be confusing, but what we mean when we say that is that the circled numbers must trace a rectangular shape. Fresheneesz 19:39, 23 March 2006 (UTC)

Pronunciation
What is the correct pronunciation of Karnaugh's name? Stern 00:00, 1 Dec 2004 (UTC)

 * Same as Carnot, or CAR-NO. Graham 00:08, 1 Dec 2004 (UTC)

Second diagram
What's the purpose of the second diagram? I don't understand it, it appears to duplicate the first with a less clear labelling, and the text doesn't refer to it. I think it should go. Graham 00:29, 21 Dec 2004 (UTC)

 * It is an alternative syntax, the one we were taught at the university. I actually find it much more clear than the first one: in this one you can see directly where A, B, C and D are ON and where they are OFF just by looking at the line/row. It should actually be rotated 90 degrees so that it could be drawn more directly from the truth table of the function (and so should the already existing map). I'll create an improved version and add a description of it to the article soon. --ZeroOne 17:57, 21 Dec 2004 (UTC)

 * OK, you can see the improved version here. Refresh the image if it doesn't differ from the one you saw - it is updated:

 * The little numbers at the corner of each square are the row numbers of the truth table of the function. It is otherwise from 0 to 15, left to right, up to down but the two last columns and two last rows are flipped. There's nothing strange in that, the existing map has the same feature, too. The area "A{", two last rows, is where A is 1. The area "}B", two middle rows, is where B is 1. The area "C{", two last columns, is where C is 1 and the area "D{", two middle columns, is where D is 1. What do you think now? --ZeroOne 18:41, 21 Dec 2004 (UTC)

 * The updated version does not seem to be showing in the article, even after refresh. It is just an identical copy of the first image, as is the third. Arlo Barnes (talk) 18:43, 23 September 2010 (UTC)

 * Weeellll.... to be honest I've been using Karnaugh maps for years and have never seen this form. I'm trying to figure out exactly what it is portraying just by looking at it, and I'm afraid I still don't get it. Maybe I'm just too used to the "old" form. Seems to me the need for the row numbers to be put in each square to help make sense of it mean it might not be as effective as the "old" type - with that one, I can see at a glance what the terms need to be and can simply write them down without any interpretation - but again I'm willing to put that down to familiarity. The flipping of the columns you mention is indeed a feature of the normal map - that's because the bit ordering needs to follow a gray code - something that the article fails to mention, but probably should. Maybe the explanatory text you plan to write will help the lightbulb go on, but the diagram on its own doesn't work for me - but maybe others here will have more to say about it? Graham 22:39, 21 Dec 2004 (UTC)

 * You don't need those row numbers, you learn them quickly and can always deduce them again should you forget them. Actually the map I'm showing is no different from your map: it just emphasizes the ones better. That is, the rows and columns where a certain variable is 1 are "grouped" with the {'s. The squares which do not belong to a given group belong to the negation (where the variable is 0) of that group. I found a nice picture from the teaching material of my university: . It combines both of the maps in the same picture (because they actually are the same map). Maybe *that* will explain it to you too? :) To add, to me the current map looks almost like if it had 8 rows and 8 columns. The amount of variables shown is too big to handle effectively. --ZeroOne 23:22, 21 Dec 2004 (UTC)

 * Ah, OK, I get it now. The image on your uni site adds the crucial "lightbulb" detail for me - the AB/CD labelling in the top corner. That said I'm not really convinced that this method is really any better than the normal one - perhaps I was expecting the advantage to be much greater than what is really just a small change to the way the columns and rows are labelled. I'd like to hear some other opinions on this though. By the way, the appearance of 8 columns rather than 4 is not something I'd really found before, since it's clear to me that it is PAIRS of variables that are being considered by each column. Graham 23:47, 21 Dec 2004 (UTC)p

Minterms
Image:Karnaugh_map_showing_minterms.png

I made the above picture to specify how to find minterms from a karnaugh map. I didn't put this on the main page, because I'm not sure if this is univerasal. Comments please. Fresheneesz 04:57, 6 March 2006 (UTC)

 * I have made an SVG image Image:K-map minterms.svg. Cburnett 06:06, 22 December 2006 (UTC)

circling xor or xnor terms
This article should mention the possibility and how-to of finding xor terms when minimizing. Fresheneesz 01:56, 19 October 2006 (UTC)
 * A karnaugh map doesn't attempt to get the minimum form, it attempts to get the minimum form for a two stage implementation using a particular gate in each stage. Yes there may be tricks that allow one to occasionally spot that there is another way that might save a bit when working in discrete logic but they aren't the main purpose of the maps. Plugwash 16:04, 20 October 2006 (UTC)

''Are you talking about Reed-Muller logic ? --68.0.120.35 14:58, 7 December 2006 (UTC)''

Image changes
I created a few SVG images for this article and, in the process, discovered the listed minterms were not reflected correctly in the images. Please, give the example a thorough review if you have a chance. I did confirm what I could with. Cburnett 08:08, 22 December 2006 (UTC)

 * Under Race hazards, first bullet point - shouldn't that read when C is 1, D is 0 and not "when C and D are both 0"? I could be wrong - it's only been 30 years since I have looked at these. Great graphic by the way.--Larry In Cincinnati 03:31, 7 January 2007 (UTC)

 * Correct and I've fixed it. Thanks for the complements on the graphics. :) Cburnett 04:01, 7 January 2007 (UTC)

 * I feel it appropriate to raise the concern here that illustrations created "de novo" run the risk of being factually incorrect, and fail the encyclopedic requirement of a sound reference source. You can avoid this by using a trusted reference as the source of the illustration's details, even if you create a new rendering to meet copyright requirements. Does anyone else have the same concern? Cburnett seems to have edited the illustration several times to correct errors. This could be avoided by use of an authoritative source material/example. JoGusto (talk) 14:15, 10 August 2013 (UTC)

Map Entered Variables
Should these be mentioned?

Definition of Karnaugh Map
What is a Karnaugh map? Neither in your article, nor in the general literature on the subject, will you find a definition (with one exception). Given a function $$f:\{0,1\}^n \mapsto \{0,1\}$$, we call its $$2^n$$ 'arguments' $$(x_1,\dots,x_n)$$ — each an element of $$\{0,1\}^n$$ — an input combination or input event. We can now define:

A Karnaugh map for n input variables $$x_1,\dots,x_n$$ is a Venn diagram whose universe consists of all $$2^n$$ input events $$(x_1,\dots,x_n)$$. The sets defined and pictured on the universe are n non-disjoint sets $$\mathcal{X}_1,\dots,\mathcal{X}_n$$, any such set $$\mathcal{X}_i$$ containing all input events whose i-th input variable $$x_i$$ is 1, i.e., $$(x_1,\dots,x_i,\dots,x_n)\in \mathcal{X}_i\Leftrightarrow x_i=1$$.

Karnaugh maps and their usage are covered quite extensively in chapters 6, 7 and 21 (Reduced K-maps) of

Shimon P. Vingron:'Switching Theory. Insight through Predicate Logic' Springer-Verlag, 2004, ISBN 3-540-40343-4.

I have not put this, and other aspects touched on in the above reference, in the main article so as not to intrude on your work: I leave it to you whether you want to make use of the above comment.

S.P.Vingron, 81.217.16.172 14:16, 28 August 2007 (UTC)

Don't cares
Why must the blue inverse term be restricted by $$\overline{A}$$? If we really don't care about the m15 case, it can be treated as 1 for the non-inverse case and 0 for the inverse case; while this makes the non-inverse and inverse cases non-equivalent for all inputs, they are still equivalent for all inputs that we care about.

However, it doesn't really matter, since treating the don't-care as 1 allows the inverse function to be simplified to $$F = \left(A + B\right)\left(A + C\right)\left(A + \overline{D}\right)$$. I can't take credit for this second bit though, an applet gave it to me when I was checking the first paragraph. Anomie 17:07, 14 October 2007 (UTC)

Race hazards
Shouldn't the $$\left(A + \overline{D}\right)$$ term be added to the inverse case to remove a hazard when moving from m7 to either m3 or m5? Anomie 17:07, 14 October 2007 (UTC)

The section title "Elimination" seems unhelpful and obtuse for a section which deals with race hazards and how to eliminate them. The origin of a race hazard as the result of the reduction, how to fix the race hazard, when is the race hazard important (purely combinatoric logic) and when might it be ignored (registered, clocked "sequential" circuits). JoGusto (talk) 14:28, 10 August 2013 (UTC)

map
For the green encircling, besides A and B, C also maintains the same state of 1. So, shouldn't the second term be $$A\overline{B}C$$ and not $$A\overline{B}$$?

Sepia tone 04:40, 25 October 2007 (UTC)

 * I don't know which image you're looking at, but I can't find one where a green encircling has C remaining unchanged. In Image:K-map 6,8,9,10,11,12,13,14.svg, for example, the green circles all four minterms in the rightmost ($$A\overline{B}$$) column, overlapping with the right half of the red group. Anomie 11:20, 25 October 2007 (UTC)

You are right. My bad. Sepia tone (talk) 02:53, 30 November 2007 (UTC)

More than 4 variables
Can we have examples for 5 and 6 variables?

For 5 variables A,B,C,D,E: We can make a 4 variable frame A,B,C,D, then fill in each entry with 1,0,x(don't care),E,e(Ebar). Then put in the circles, noting that E is not 0 and is not 1.

Truth table ABCDE Out 00000 0 00001 1 00010 0 00011 1 00100 0 00101 1 00110 0 00111 1 01000 0 01001 1 01010 0 01011 1 01100 0 01101 1 01110 0 01111 1 10000 0 10001 0 10010 0 10011 0 10100 1 10101 1 10110 1 10111 1 11000 1 11001 1 11010 0 11011 0 11100 0 11101 0 11110 1 11111 1

Map |0110|C |0011|D --|| 00|EEEE| 10|0110| 11|1010| 01|EEEE| --|| AB|

Out=aE+AbC+ACD+ABcd

(lower case is not)

 * No, usually you would use Boolean Algebra when simplifying five, six, or more variable circuits. It's certainly possible, but do you really want to have thirty-two squares to fill out? ChyranandChloe (talk) 02:29, 14 June 2008 (UTC)
 * It is possible to work with 5 variables any beyond. For an odd number like 5 you would create two 2x2 maps. One for E = 0 and one for E = 1. However, like ChyranandChloe said it isn't wise to work with a huge K-map unless you have to but even then I'd rather look for a software solution. 178.9.117.214 (talk) 14:44, 30 January 2012 (UTC)

 * I agree... but you would probably use an automated tool based on the Quine-McCluskey minimization algorithm. Perhaps it should be mentioned. JoGusto (talk) 14:31, 10 August 2013 (UTC)

If you think that the above suggestion is about 32 entries or an additional dimension, then you have miss-read it. It is describing how to add an extra variable without increasing the number of dimensions. We learnt this at university circa 1992-1995. It may have another name. Look at the example it has 5 variable, but only 2 dimensions. From what I remember it allows 6 variable with 2 dimensions. Ctrl-alt-delor (talk) 12:21, 01 October 2019 (UTC)

You can use 4×4×2 cuboid for 5 variables. 220.228.194.112 (talk) 03:34, 11 December 2020 (UTC)

Link-to's
Shouldn't entries such as K-Map link to this page? Thanks

129.7.108.128 (talk) 18:28, 6 February 2008 (UTC)

 * K-map does, K-Map doesn't because "map" isn't capitalized. But if you put "K-Map" in the search box then you hit this article just fine.  Cburnett (talk) 19:03, 6 February 2008 (UTC)

Circuit Design Image
I think it would helpful if we could include a unsimplified digital electronic circuit design and a simplified circuit design in addition to the boolean algebra - it makes it difficult to apply if you don't have the circuit to refer to. ChyranandChloe (talk) 00:01, 30 May 2008 (UTC)

 * Good point! Can you do it?  yoyo (talk) 05:40, 9 June 2010 (UTC)

axion law?
What the heck is the "axion law"? This term is used at least twice in this article, and links to the boolean logic page, in which the word "axion" never appears. linas (talk) 21:34, 4 June 2008 (UTC)

 * It is "axiom laws" (an axion is a hypothetical elementary partial - in physics). An Axiom is a postulate, a assumption that is accepted and known to be true.  The way that is is used in the article (and in boolean algebra) is that we automatically accept that if you have A and NOT A they cancel each other out; it's like adding $$-1$$ and $$1$$, you get $$0$$.  They go to a lot of trouble explaining it in the Boolean algebra page:

"...Two Boolean laws having no numeric counterpart are the laws characterizing logical negation, namely x∧¬x = 0..."
 * ∧ = AND; ¬ = NOT by the way. The edit fixes are made, thanks. ChyranandChloe (talk) 01:55, 6 June 2008 (UTC)

Karnaugh map example
In the example diagram it is not obvious that the brown section is the overlapping of the red and green sections. This got me confused for quite a while! I suggest that using eg striped red and green for the overlapping section, or make the outlines easier to see.--Czar Kirk (talk) 04:30, 13 June 2008 (UTC)

 * I suggest we stop coloring our K-maps or at least the inverses. I'll make an effort to redo them, but I'm still learning how to use Inkscape. ChyranandChloe (talk) 02:29, 14 June 2008 (UTC)

 * Yeah, the current colouring scheme is illegible. I'd suggest using multiple maps to avoid the tangle. Dcoetzee 02:34, 14 June 2008 (UTC)

 * I am trying to learn about Karnaugh maps, I saw something a lot time ago during the first semester in school.
 * Diagrams in the style suggested by ZeroOne seems more clear to me.
 * As far as I remember, instead of using Gray codes, the arrangement was done by drawing the variables in the way that is shown in ZeroOne's diagram. If I am not wrong it is a way to achieve Gray codes always, if one follows that mnemonic rule.

 * It was not clear to me, why the lower 2 cells at the bottom right corner together with the 2 cells at the top right were discarded. It is a 2^2=4 set of cells filled with 1s.

 * I think this article may be substantially improved if this examples are worked, drawing a figure for each step, using diagrams like the one suggested by ZeroOne, and also adding a more formal definition of what a Karnaugh map is as suggested above, complemented by a formal proof that the method works.
 * The part about glitches detection is important if it can be explained in more detail would be great.
 * Thank you! —Preceding unsigned comment added by Elias (talk • contribs) 06:20, 3 January 2009 (UTC)

 * Different schools teach in different ways, when I first learned it, rather than gray code it was A'B' A'B AB A'B' with "skip the third". But you're right, if we could get step by step examples would it would substantially simplify the the learning process. This article has been essentially neglected, it was actually first written in Wikiversity and transplanted to Wikipedia, so improving it would not be hard. ChyranandChloe (talk) 06:32, 3 January 2009 (UTC)

The yellow square and teal square are both redundant groupings. The actual minimal POS and SOP equations should only have 3 terms each, not 4. 75.151.246.133 (talk) 22:11, 2 May 2010 (UTC)

 * Agree about the redundant groups in the top diagram. The equations should be F = AC' + AB' + BCD' and F = (A'+C)(A'+B)(B'+C'+D). I see they're written that way  when the diagram is repeated further down the page.  I don't want to edit it, however, since the diagram shows the redundant groups (AD' is shown on the diagram as yellow).  The diagram should be updated along with the equations.
 * Another point: I think it's confusing to label both equations F = ... when they represent the complements of each other. Perhaps F = ... and F' = ... would be better.  Alternatively, F = ... and G = ..., to at least indicate the two equations don't represent the same cells in the map. TeWaitere (talk) 06:37, 16 April 2024 (UTC)

compare with Punnett square?
I just discovered that geneticists describe Mendelian inheritance using a diagram called a Punnett square which strikes me as being both structurally and functionally very similar to a Karnaugh map. Is it worth mentioning this correspondence anywhere? (Perhaps just a See Also link?) —Steve Summit (talk) 03:33, 30 June 2009 (UTC)

 * I can see the "structurally similar" here (inputs along the borders) but not the "functionally similar" --- the analogy actually seems quite superficial, as any Cartesian product of mappings can be represented as somewhat along these lines. So some further explanation about how the Punnett square is functionally similar is required. The P square is not used (afaik) to find minimal representations or prime implicants, whatever the analogs in genetics might be. (I am both a geneticist and a logician, so I think I would have spotted any such analogy, but then I do not wish to claim outright that it isn't there just because I failed to observe it.) 2A01:CB0C:CD:D800:9C38:3BDB:AF80:1542 (talk) 15:56, 11 May 2021 (UTC)

Lead sentence is wrong

A Karnaugh map is not the same thing as a Veitch diagram. Veitch's diagram is used by virtually no one. I have the originals of both papers. The 2nd drawing on the right is a virtual reproduction of drawings from both papers. Notice that the Veitch diagram does not use the "Gray (en)coding" or the encoding from the "vertices of a hypercube".

I'll have change this paragraph eventually unless someone persuades me to do otherwise. Bill Wvbailey (talk) 18:20, 22 December 2009 (UTC)

I just went ahead and changed the lead paragraph. If anyone doubts the truth of the above I'll just photograph or pdf the relevant drawings from both papers to make the point. BTW: I was able to get the Karnaugh paper off the web, but I had to buy the damn Veitch paper; someone with access to an academic library can probably get it from the ACM. It is not a particularly satisfactory paper when compared to the Karnaugh paper. Bill Wvbailey (talk) 18:38, 22 December 2009 (UTC)

Is the second function in the example right?
I am by no means a subject matter expert, but the second function in the example does not look right to me. Can someone confirm that it is indeed correct? Specifically, I am referring to this:

 * $$f(A, B, C, D,E) = (\overline{A}BCD\overline{E}) + (A\overline{B}\,\overline{C}\,\overline{D}\overline{E}) + (A\overline{B}\,\overline{C}{D}E) + (A\overline{B}CD\overline{E}) + (A\overline{B}CDE) + (AB\overline{C}\,\overline{D}, \overline{E}) + (AB\overline{C}DE) + (ABCD\overline{E})$$

It doesn't seem to correspond with the previous example or what comes after.

I searched through the edit history and found a prior revision of that example that makes a lot more sense to me. Here is the revision where it changed from something I understand to something I don't.

207.182.200.34 (talk) 15:18, 24 August 2011 (UTC)

 * The example is thoroughly corrupted. It has 5 variables: A, B, C, D, E. This requires it to have 25 = 32 rows, 5 "input" columns labelled as A, B, C, D, E with all 0 & 1 patterns below them and one output column. The example has none of the above. A four-variable example would have 16 "input" rows and 4 input columns labelled A, B, C, D; one output column. I rolled the version back to the uncorrupted truth table, but the formulas underneath the drawings look corrupted, plus unlike a real K-map, the minterms (the 16 squares) don't have the numbers that correspond to the rows inside them. From upper left to lower right the minterm boxes are usually numbered as follows (base 10 with the binary equivalents below this matrix). Thus with the boxes numbered it's easy to go back and forth from truth-table to map:
 * 0,   1,    3,    2
 * 4,   5,    7,    6
 * 12, 13, 15, 14
 * 8,   9,    11,   10

 * 0000, 0001, 0011, 0010
 * 0100, 0101, 0111, 0110
 * 1100, 1101, 1111, 1110
 * 1000, 1001, 1011, 1010

If you look closely at the binary versions you'll see that as you go left to right or right to left, or up or down one square at a time (but not diagonally), only one "1" or "0" changes at a time. This includes the left-right edge wraparound and the top-bottom edge wrap-around. Other number-schemes are allowed but the "only-one-variable change" cannot be violated. That's the key to the Karnaugh map.

See the proper box-numbering scheme above (circa 2004 -- but this is a different example, but the numbering scheme is correct), and the 8-square version directly above.

Hope this helps. BillWvbailey (talk) 20:58, 24 August 2011 (UTC)

 * It definitely helps. Thanks. 207.182.200.34 (talk) 18:13, 25 August 2011 (UTC)

I went back to my sources to see their cell-numbering schemes. To a man/woman they numbered them in a different way than what I showed above. This includes M. Karnaugh himself in his very-most original paper. Here is how they all do it:
 * 0, 4, 12, 8
 * 1, 5, 13, 9
 * 3, 7, 15, 11
 * 2, 6, 14, 10

 * 0000, 0100, 1100, 1000
 * 0001, 0101, 1101, 1001
 * 0011, 0111, 1111, 1011
 * 0010, 0110, 1110, 1010

In words: the "classical" numbering scheme shown above is "my" numbering scheme mirror-imaged and rotated counter-clockwise. This is just to set the record straight. BillWvbailey (talk) 00:48, 28 August 2011 (UTC)

Picture in section "Solution" difficult to read / justification of revert
I reverted the recent edits made in good faith by 108.41.142.178; here is why: In order to avoid repetitions of 108.41.142.178's misunderstanding of the picture, the extensions of the red, green, and blue field should be made more clear in an improved version of the picture. There could be more emphasis on the borders and less on the areas; and the border lines of the red and green field should not coincide exactly, such that both are visible. If the partitioning of the 0 area is not discussed in the article (I didn't check that), it should be removed from the picture. - Jochen Burghardt (talk) 16:39, 30 September 2013 (UTC)
 * According to the Karnaugh map methodology, the "green group" in the picture in section "Karnaugh map" should be as large as possible in order to obtain the simplest possible expression realizing the function f. Since the whole rightmost column contains 1s, the green group should be at least that large (it can't be larger, on the other hand). That is, it should be AB' rather than AB'C only ("'" denoting complement here).
 * The inner caption of the picture says "F=AC'+AB'+BCD'" rather than "F=AC'+AB'C+BCD'". The text should be consistent with that.
 * Looking very closely at the square ABCD=1001, some green can be seen at its bottom, probably intended to indicate that the green area is not restricted to the two fields below it.

Solution Section Picture Error
From the article: "For the red grouping: [...] C does not change. It is always 0, so its complement, NOT-C, should be included. Thus, \overline{C} should be included."

This is true in the bottom picture, but not the top picture (File:Karnaugh.svg). That picture shows "CD" as "10" or "11" in the red group, in which case C is always 1. Can anyone fix either the image or description? 108.202.199.178 (talk) 15:38, 18 February 2015 (UTC)

Include Original Function for Example
Just before the sub-section labeled "Karnaugh map", it may be helpful to actually write out the complete (16-term/16-factor) function for both sum-of-products and product-of-sums. I figured this out on my own after carefully reading the math and the definition of minterm/maxterm (from the linked Canonical_normal_form), but the SoP/PoS notion is a lot simpler than either article makes it appear.

I think having the complete function would also really help to illustrate how the solution below (of 3 terms) is vastly simpler than the trivial (canonical?) version. However, if showing the entire function is thought out of scope, is there perhaps a better/more-straightforward description that could be linked at this point in the article?--preferably one that clearly explains how one might construct the full form, but not as overly complicated as the Canonical_normal_form article. 108.202.199.178 (talk) 16:14, 18 February 2015 (UTC)

Is this an example of a heuristic?
Are Karnaugh Map solutions to truth tables an example of a heuristic ? They will usually produce a good solution, even if they don't always find the best solution? Paul Murray (talk) 03:56, 18 June 2015 (UTC)

title: Talk:Karnaugh map


text:
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 * "Papa" (Prince song), a 1994 song from Come by Prince
 * "Pa-Pa", a song from Hell of a Tester by The Rasmus

Acronym

 * British Amateur Press Association (comics fandom), an amateur press association which first published under the name PAPA
 * Professional and Amateur Pinball Association
 * Parallax Aircraft Parking Aid, a device that indicates where pilots should stop in a stand

Other uses

 * PAPA syndrome, a genetic disorder in humans
 * Papa, the letter "P" in the NATO and ICAO phonetic alphabets
 * Papa-class Soviet submarines, the sole member being Soviet submarine K-222
 * Papa, the Spanish word for potato, used in the names of numerous Latin American potato-based dishes
 * Station P, an oceanographic measure station often called Station Papa

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

title: Papa


text:
Raki

Raki may refer to:

Liquors

 * Raki / Rakia (with a dotted "i"), fruit-based spirits in several countries in southeast Europe
 * Rakı (with a dotless "ı"), an anise-flavored spirit popular in Turkey, similar to ouzo

People and characters

 * Laya Raki (1927–2018), German dancer and actress
 * Raki Aouani (born 2004), Tunisian soccer player
 * Raquel "Raki" Martin, a competitor on 2013 Gran Hermano (Spanish season 14)
 * Raki (Claymore), a fictional character, from the manga/anime series Claymore
 * Livy Raki (イェスパー・リヴェ・ラキ, Yesupā Rive Raki), a fictional character, from the Japanese light novel series Dances with the Dragons
 * Raki, a mythological figure, an alternative name for Rangi, the Sky Father in the South Island dialect of Māori
 * Raki, the Dog King, a mythological figure from Norse mythology; see List of named animals and plants in Germanic heroic legend

Places

 * Raki, Iran; a village
 * Raki, Masovian Voivodeship, Poland; a village

title: Raki


text:
Rangi and Papa

In Māori mythology the primal couple Rangi and Papa (or Ranginui and Papatūānuku) appear in a creation myth explaining the origin of the world and the Māori people (though there are many different versions). In some South Island dialects, Rangi is called Raki or Rakinui.

Union and separation
Ranginui first married Poharua Te Po where they bore 3 offspring including Aorangi (or Aoraki as given in South Island). He later married Papatūānuku together becoming the primordial sky father and earth mother bearing over 70 children including Tāwhirimātea, Tāne and Tangaroa, all of whom are male. Both Ranginui and Papatūānuku lie locked together in a tight embrace, and their sons forced to live in the cramped darkness between them.

These children grow and discuss among themselves what it would be like to live in the light. Tūmatauenga, the fiercest of the children, proposes that the best solution to their predicament is to kill their parents. But his brother Tāne disagrees, suggesting that it is better to push them apart, to let Ranginui be as a stranger to them in the sky above while Papatūānuku will remain below to nurture them. The others put their plans into action—Rongo, the god of cultivated food, tries to push his parents apart, then Tangaroa, the god of the sea, and his sibling Haumia-tiketike, the god of wild food, join him. In spite of their joint efforts Rangi and Papa remain close together in their loving embrace.

After many attempts Tāne, god of forests and birds, forces his parents apart. Instead of standing upright and pushing with his hands as his brothers have done, he lies on his back and pushes with his strong legs. Stretching every sinew Tāne pushes and pushes until, with cries of grief and surprise, Ranginui and Papatūānuku were pried apart. Traditions of the Taranaki region, however, assign this separating role to Tangaroa, god of the sea.

War in heaven and earth
And so the children of Ranginui and Papatūanuku see light and have space to move for the first time. While the other children have agreed to the separation, Tāwhirimātea, the god of storms and winds, is angered that the parents have been torn apart. He cannot bear to hear the cries of his parents nor see the tears of Ranginui as they are parted, he promises his siblings that from henceforth they will have to deal with his anger. He flies off to join Rangi and there carefully fosters his own many offspring who include the winds, one of whom is sent to each quarter of the compass. To fight his brothers, Tāwhirimātea gathers an army of his children —winds and clouds of different kinds, including fierce squalls, whirlwinds, gloomy thick clouds, fiery clouds, hurricane clouds and thunderstorm clouds, and rain, mists and fog. As these winds show their might the dust flies and the great forest trees of Tāne are smashed under the attack and fall to the ground, food for decay and for insects. Then Tāwhirimātea attacks the oceans and huge waves rise, whirlpools form, and Tangaroa, the god of the sea, flees in panic. Punga, a son of Tangaroa, has two children, Ikatere father of fish, and Tū-te-wehiwehi (or Tu-te-wanawana) the ancestor of reptiles. Terrified by Tāwhirimātea's onslaught the fish seek shelter in the sea and the reptiles in the forests. Ever since Tangaroa has been angry with Tāne for giving refuge to his runaway children. So it is that Tāne supplies the descendants of Tūmatauenga with canoes, fishhooks and nets to catch the descendants of Tangaroa. Tangaroa retaliates by swamping canoes and sweeping away houses, land and trees that are washed out to sea in floods.

Tāwhirimātea next attacks his brothers Rongo and Haumia-tiketike, the gods of cultivated and uncultivated foods. Rongo and Haumia are in great fear of Tāwhirimātea but, as he attacks them, Papatūānuku determines to keep these for her other children and hides them so well that Tāwhirimātea cannot find them. So Tāwhirimātea turns on his brother Tūmatauenga. He uses all his strength but Tūmatauenga stands fast and Tāwhirimatea cannot prevail against him. Tū (or human kind) stands fast and, at last, the anger of the gods subsided and peace prevailed.

Tū thought about the actions of Tāne in separating their parents and made snares to catch the birds, the children of Tāne who could no longer fly free. He then made nets from forest plants and casts them in the sea so that the children of Tangaroa soon lie in heaps on the shore. He made hoes to dig the ground, capturing his brothers Rongo and Haumia-tiketike where they have hidden from Tāwhirimātea in the bosom of the earth mother and, recognising them by their long hair that remains above the surface of the earth, he drags them forth and heaps them into baskets to be eaten. So Tūmatauenga eats all of his brothers to repay them for their cowardice; the only brother that Tūmatauenga does not subdue is Tāwhirimātea, whose storms and hurricanes attack humankind to this day.

There was one more child of Ranginui and Papatūānuku who was never born and still lives inside Papatūanuku. Whenever this child is kicking the earth shakes and it causes an earthquake. Rūaumoko is his name and he is the god of earthquakes and volcanoes.

Yearning
Tāne searched for heavenly bodies as lights so that his father would be appropriately dressed. He obtained the stars and threw them up, along with the moon and the sun. At last Ranginui looked handsome. Ranginui and Papatūanuku continue to grieve for each other to this day. Ranginui's tears fall towards Papatūanuku to show how much he loves her. Sometimes Papatūanuku heaves and strains and almost breaks herself apart to reach her beloved partner again but it is to no avail. When mist rises from the forests, these are Papatūānuku's sighs as the warmth of her body yearns for Ranginui and continues to nurture mankind.

Names and epithets
Ranginui
 * Rangi ("Sky")
 * Raki ("Sky") in the South Island (see Māori language)
 * Ranginui ("Great Sky")
 * Rangi-pōtiki ("Rangi the Lastborn"): possibly another name of Rangi, or a closely allied deity

Papatuanuku
 * Papa ("world")
 * Papatūānuku ("world separated"), (Earth), (Mother Earth)

title: Rangi and Papa


text:
Tāne

In Māori mythology, Tāne (also called Tāne-mahuta, Tāne-nui-a-Rangi, Tāne-te-waiora and several other names) is the god of forests and of birds, and the son of Ranginui and Papatūānuku, the sky father and the earth mother, who used to lie in a tight embrace where their many children lived in the darkness between them (Grey 1956:2).

On Tahiti, Tane was the god of peace and beauty.

Separates his parents
The children of Rangi and Papa grew frustrated at their confinement in the cramped space between their parents. Tū, future god of war, proposes that they should kill their parents. But Tāne (or Tāne-mahuta) disagrees, suggesting that it is better to separate them, sending Rangi into the sky and leaving Papa below to care for them. Tāne's brothers Rongo, then Tangaroa, Haumia-tiketike and Tū all try in vain to separate the parents. After many tries, Tāne lies on his back and pushes with his strong legs, and finally forces his parents apart, and Rangi rises high into the heavens (Grey 1956:2-3). Then Tāne searched for heavenly bodies as lights so that his father would be appropriately dressed. He obtained the stars and threw them up, along with the Moon and the Sun. At last Rangi looked handsome (Orbell 1998:145).

Tāwhirimātea, the god of storms and winds, is angry that the parents have been torn apart. He joins his father in the sky and punishes the earth and sea with violent storms (Grey 1956:3-6, Tregear 1891:54). Tāwhirimātea attacks the forests of Tāne, snapping the trunks of the trees, shattering the trees to the ground, leaving them as food for decay and insects. Then he attacks the oceans, and Tangaroa, the god of the sea, flees. Two of Tangaroa's descendants, Ikatere, father of fish and Tu-te-wehiwehi (or Tu-te-wanawana), the ancestor of reptiles, are terrified by Tāwhirimātea's fury. The fish flee into the sea, and the reptiles into the forests. Ever since, Tangaroa has resented Tāne for hiding his runaway children. So it is that Tāne supplies the descendants of Tūmatauenga with canoes, fishhooks, and nets to catch the descendants of Tangaroa. Tangaroa retaliates by overturning canoes and sending floods that sweep away houses, land and trees (Grey 1971:5-6).

Progenitor of humankind
Some legends say that Tāne made the first man, named Tiki. More widely known is a tradition that Tāne was trying to find himself a wife, but at first he found only non-human females and fathered insects, birds, and plants. One such was Rangahore, who gave birth to a stone and was abandoned by Tāne. Then he made a woman by moulding her from the soil (Orbell 1998:145).

In some stories, Tāne marries his daughter Hine-tītama without her knowing who he is. Upon discovering that she has married her father, she flees to the underworld, and becomes the goddess of death, Hine-nui-te-pō. Tāne follows her and begs her to return. She tells him to return to the world and raise their children, while she will wait below to receive them when they die (Orbell 1998:38).

Other traditions tell of the three baskets of knowledge that Tāne brought down from the heavens (Orbell 1998:145).

title: Tāne


text:
Rehua

In Māori mythology, Rehua is a very sacred personage, who lives in Te Putahi-nui-o-Rehua in Rangi-tuarea, the tenth and highest of the heavens in some versions of Māori lore. Rehua is identified with certain stars. To the Tūhoe people of the North Island he is Antares. Others say he is Betelgeuse, or Sirius. Because he lives in the highest of the skies, Rehua is untouched by death, and has the power to cure blindness, revive the dead, and heal any disease (Orbell 1998:119-120). He is a son of Rangi and Papa, and the father of Kaitangata, as well as the ancestor of Māui (Tregear 1891:381). A Ngāi Tahu legend from the South Island speaks of Rehua as the eldest son of Rangi and Papa, who first manifested as lightning, but assumed a human shape when he travelled into the skies. Later his brother Tāne went to pay him a visit, Rehua had birds in his hair, feeding on his lice. Rehua had his servants cook and prepare the birds as a meal for Tāne, who was shocked and declined to eat them because the birds had eaten the lice from Rehua's head, which was extremely tapu (sacred). However Rehua gave him birds to bring down to this world, and showed him how to snare them. Tāne also brought with him the fruit trees that the birds fed on, and so it is that there are forests and birds on the earth (Orbell 1998:119).

In Tahiti, Rehua was a star-god, the star of the New Year, who produced the Twins as well as the Pleiades, and was considered the lord of the year.

title: Rehua


text:
Kaitangata

Kaitangata can mean the following:
 * Kaitangata, New Zealand, a small town near the coast of South Otago in New Zealand
 * Kaitangata Mine disaster, a coal mining disaster at Kaitangata in 1879
 * Kaitangata Line, a branch line from Stirling on the Main South Line to Kaitangata
 * Kaitangata (mythology), a character in Māori mythology

title: Kaitangata


text:
Centrosome

In cell biology, the centrosome (Latin centrum 'center' + Greek sōma 'body') (archaically cytocentre ) is an organelle that serves as the main microtubule organizing center (MTOC) of the animal cell, as well as a regulator of cell-cycle progression. The centrosome provides structure for the cell. The centrosome is thought to have evolved only in the metazoan lineage of eukaryotic cells. Fungi and plants lack centrosomes and therefore use other structures to organize their microtubules. Although the centrosome has a key role in efficient mitosis in animal cells, it is not essential in certain fly and flatworm species.

Centrosomes are composed of two centrioles arranged at right angles to each other, and surrounded by a dense, highly structured mass of protein termed the pericentriolar material (PCM). The PCM contains proteins responsible for microtubule nucleation and anchoring — including γ-tubulin, pericentrin and ninein. In general, each centriole of the centrosome is based on a nine-triplet microtubule assembled in a cartwheel structure, and contains centrin, cenexin and tektin. In many cell types, the centrosome is replaced by a cilium during cellular differentiation. However, once the cell starts to divide, the cilium is replaced again by the centrosome.

History
The centrosome was discovered jointly by Walther Flemming in 1875 and Edouard Van Beneden in 1876, and later described and named in 1888 by Theodor Boveri.

Functions
Centrosomes are associated with the nuclear membrane during the prophase stage of the cell cycle. During mitosis, the nuclear membrane breaks down, and the centrosome-nucleated microtubules can interact with the chromosomes to build the mitotic spindle.

The mother centriole, the older of the two in the centriole pair, also has a central role in making cilia and flagella.

The centrosome is copied only once per cell cycle, so that each daughter cell inherits one centrosome, containing two structures called centrioles. The centrosome replicates during the S phase of the cell cycle. During the prophase in the process of cell division called mitosis, the centrosomes migrate to opposite poles of the cell. The mitotic spindle then forms between the two centrosomes. Upon division, each daughter cell receives one centrosome. Aberrant numbers of centrosomes in a cell have been associated with cancer. Doubling of a centrosome is similar to DNA replication in two respects: the semiconservative nature of the process and the action of CDK2 as a regulator of the process. But the processes are essentially different in that centrosome doubling does not occur by template reading and assembly. The mother centriole just aids in the accumulation of materials required for the assembly of the daughter centriole.

Centrioles, however, are not required for the progression of mitosis. When the centrioles are irradiated by a laser, mitosis proceeds normally with a morphologically normal spindle. Moreover, development of the fruit fly Drosophila is largely normal when centrioles are absent due to a mutation in a gene required for their duplication. In the absence of the centrioles, the microtubules of the spindle are focused by motors, allowing the formation of a bipolar spindle. Many cells can completely undergo interphase without centrioles.

Unlike centrioles, centrosomes are required for survival of the organism. Cells without centrosomes lack radial arrays of astral microtubules. They are also defective in spindle positioning and in the ability to establish a central localization site in cytokinesis. The function of centrosomes in this context is hypothesized to ensure the fidelity of cell division, because it greatly increases the efficacy. Some cell types arrest in the following cell cycle when centrosomes are absent. This is not a universal phenomenon.

When the nematode C. elegans egg is fertilized, the sperm delivers a pair of centrioles. These centrioles will form the centrosomes, which will direct the first cell division of the zygote, and this will determine its polarity. It's not yet clear whether the role of the centrosome in polarity determination is microtubule-dependent or independent.

In human reproduction, the sperm supplies the centriole that creates the centrosome and microtubule system of the zygote.

Centrosome alterations in cancer cells
Theodor Boveri, in 1914, described centrosome aberrations in cancer cells. This initial observation was subsequently extended to many types of human tumors. Centrosome alterations in cancer can be divided in two subgroups — i.e., structural or numeric aberrations — yet both can be found simultaneously in a tumor.

Structural aberrations
Usually, structural aberrations appear due to uncontrolled expression of centrosome components, or due to post-translational modifications (such as phosphorylations) that are not adequate for the components. These modifications may produce variations in centrosome size (usually too large, due to an excess of pericentriolar material). In addition, because centrosomal proteins have a tendency to form aggregates, centrosome-related bodies (CRBs) are often observed in ectopic places. Both enlarged centrosomes and CRBs are similar to the centrosomal structures observed in tumors. Even more, these structures can be induced in culture cells by overexpression of specific centrosomal proteins, such as CNap-1 or Nlp. These structures may look very similar, yet detailed studies reveal that they may present very different properties, depending on their proteic composition. For instance, their capacity to incorporate γ-TuRC complexes (see also: γ-tubulin) can be very variable, and so their capacity to nucleate microtubules therefore affects the shape, polarity and motility of implicated tumor cells in different ways.

Numeric aberrations
The presence of an inadequate number of centrosomes is very often linked to the appearance of genome instability and the loss of tissue differentiation. However, the method to count the centrosome number (with two centrioles to each centrosome) is often not very precise, because it is frequently assessed using fluorescence microscopy, which does not have high enough optical resolution to resolve centrioles that are very close to each other. Nevertheless, it is clear that the presence of an excess of centrosomes is a common event in human tumors. It has been observed that loss of the tumor-suppressor p53 produces superfluous centrosomes, as well as deregulating other proteins implicated in cancer formation in humans, such as BRCA1 and BRCA2. (For references, see .) An excess of centrosomes can be generated by very different mechanisms: specific reduplication of the centrosome, cytokinesis failure during cell division (generating an increase in chromosome number), cell fusion (such as in cases of infection by specific viruses) or de novo generation of centrosomes. At this point, there is insufficient information to know how prevalent these mechanisms are in vivo, but it is possible that the increase in centrosome numbers due to a failure during cell division might be more frequent than appreciated, because many "primary" defects in one cell (deregulation of the cell cycle, defective DNA or chromatin metabolism, failure in the spindle checkpoint, etc.) would generate a failure in cell division, an increase in ploidy and an increase in centrosome numbers as a "secondary" effect.

Evolution
The evolutionary history of the centrosome and the centriole has been traced for some of the signature genes — e.g., the centrins. Centrins participate in calcium signaling and are required for centriole duplication. There exist two main subfamilies of centrins, both of which are present in the early-branching eukaryote Giardia intestinalis. Centrins have therefore been present in the common ancestor of eukaryotes. Conversely, they have no recognizable homologs in archea and bacteria and are thus part of the "eukaryotic signature genes". Although there are studies on the evolution of the centrins and centrioles, no studies have been published on the evolution of the pericentriolar material.

It is evident that some parts of the centrosome are highly diverged in the model species Drosophila melanogaster and Caenorhabditis elegans. For example, both species have lost one of the centrin subfamilies that are usually associated with centriole duplication. Drosophila melanogaster mutants that lack centrosomes can even develop to morphologically normal adult flies, which then die shortly after birth because their sensory neurons lack cilia. Thus, these flies have evolved functionally redundant machinery, which is independent of the centrosomes.

Associated nucleotides
Research in 2006 indicated that centrosomes from Atlantic surf clam eggs contain RNA sequences. The sequences identified were found in "few to no" other places in the cell, and do not appear in existing genome databases. One identified RNA sequence contains a putative RNA polymerase, leading to the hypothesis of an RNA-based genome within the centrosome. However, subsequent research has shown that centrosome do not contain their own DNA-based genomes. While it was confirmed that RNA molecules associate with centrosomes, the sequences have still been found within the nucleus. Furthermore, centrosomes can form de novo after having been removed (e.g., by laser irradiation) from normal cells.

title: Centrosome


text:
Stirling number

In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730). They were rediscovered and given a combinatorial meaning by Masanobu Saka in 1782.

Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind. Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them.

A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non-empty subsets, where each subset is endowed with a certain kind of order (no order, cyclical, or linear).

Notation
Several different notations for Stirling numbers are in use. Ordinary (signed) Stirling numbers of the first kind are commonly denoted:

 * $$ s(n,k)\,.$$

Unsigned Stirling numbers of the first kind, which count the number of permutations of n elements with k disjoint cycles, are denoted:

 * $$ \biggl[{n \atop k}\biggr] =c(n,k)=|s(n,k)|=(-1)^{n-k} s(n,k)\,$$

Stirling numbers of the second kind, which count the number of ways to partition a set of n elements into k nonempty subsets:

 * $$ \biggl\{{\!n\! \atop \!k\!}\biggr\} = S(n,k) = S_n^{(k)} \,$$

Abramowitz and Stegun use an uppercase $$S$$ and a blackletter $$\mathfrak S$$, respectively, for the first and second kinds of Stirling number. The notation of brackets and braces, in analogy to binomial coefficients, was introduced in 1935 by Jovan Karamata and promoted later by Donald Knuth. (The bracket notation conflicts with a common notation for Gaussian coefficients. ) The mathematical motivation for this type of notation, as well as additional Stirling number formulae, may be found on the page for Stirling numbers and exponential generating functions.

Another infrequent notation is $$s_1(n,k)$$ and $$s_2(n,k)$$.

Expansions of falling and rising factorials
Stirling numbers express coefficients in expansions of falling and rising factorials (also known as the Pochhammer symbol) as polynomials.

That is, the falling factorial, defined as $$\ (x)_{n} = x(x-1)\ \cdots(x-n+1)\ ,$$ is a polynomial in $x$ of degree $n$ whose expansion is

 * $$(x)_{n}\ =\ \sum_{k=0}^n\ s(n,k)\ x^k\ $$

with (signed) Stirling numbers of the first kind as coefficients.

Note that $$\ (x)_0 \equiv 1\ ,$$ by convention, because it is an empty product. The notations for the falling factorial and  for the rising factorial are also often used. (Confusingly, the Pochhammer symbol that many use for falling factorials is used in special functions for rising factorials.)

Similarly, the rising factorial, defined as $$\ x^{(n)}\ =\ x(x+1)\ \cdots(x+n-1)\ ,$$ is a polynomial in $x$ of degree $n$ whose expansion is

 * $$ x^{(n)}\ =\ \sum_{k=0}^n\ \biggl[{n \atop k}\biggr]\ x^k\ =\ \sum_{k=0}^n\ (-1)^{n-k}\ s(n,k)\ x^k\ ,$$

with unsigned Stirling numbers of the first kind as coefficients. One of these expansions can be derived from the other by observing that $$\ x^{(n)} = (-1)^n (-x)_{n} ~.$$

Stirling numbers of the second kind express the reverse relations:

 * $$\ x^n\ =\ \sum_{k=0}^n\ S(n,k)\ (x)_k\ $$

and

 * $$\ x^n\ =\ \sum_{k=0}^n\ (-1)^{n-k}\ S(n,k)\ x^{(k)} ~.$$

As change of basis coefficients
Considering the set of polynomials in the (indeterminate) variable x as a vector space, each of the three sequences
 * $$x^0,x^1,x^2,x^3,\dots \quad (x)_0,(x)_1,(x)_2,\dots \quad x^{(0)},x^{(1)},x^{(2)},\dots$$

is a basis. That is, every polynomial in x can be written as a sum $$a_0 x^{(0)} + a_1 x^{(1)} + \dots + a_n x^{(n)}$$ for some unique coefficients $$a_i$$ (similarly for the other two bases). The above relations then express the change of basis between them, as summarized in the following commutative diagram:
 * Stirling numbers as polynomial basis changes.png give coefficients for changing one basis of polynomials to another ]]

The coefficients for the two bottom changes are described by the Lah numbers below. Since coefficients in any basis are unique, one can define Stirling numbers this way, as the coefficients expressing polynomials of one basis in terms of another, that is, the unique numbers relating $$x^n$$ with falling and rising factorials as above.

Falling factorials define, up to scaling, the same polynomials as binomial coefficients: $\binom{x}{k} = (x)_k/k!$. The changes between the standard basis $$\textstyle x^0, x^1, x^2, \dots$$ and the basis $\binom{x}{0}, \binom{x}{1}, \binom{x}{2}, \dots$ are thus described by similar formulas:
 * $$x^n=\sum_{k=0}^n \biggl\{{\!n\! \atop \!k\!}\biggr\} k! \binom{x}{k} \quad \text{and} \quad \binom{x}{n}=\sum_{k=0}^n \frac{s(n,k)}{n!} x^k$$.

Example
Expressing a polynomial in the basis of falling factorials is useful for calculating sums of the polynomial evaluated at consecutive integers. Indeed, the sum of falling factorials with fixed k can expressed as another falling factorial (for $$k\ne-1$$)

 * $$\sum_{0\leq i < n} (i)_k = \frac{(n)_{k+1}}{k+1} $$

This can be proved by induction.

For example, the sum of fourth powers of integers up to n (this time with n included), is:

 * $$\begin{align}

\sum_{i=0}^{n} i^4 & = \sum_{i=0}^n \sum_{k=0}^4 \biggl\{{\!4\! \atop \!k\!}\biggr\} (i)_k = \sum_{k=0}^4 \biggl\{{\!4\! \atop \!k\!}\biggr\} \sum_{i=0}^n (i)_k = \sum_{k=0}^4 \biggl\{{\!4\! \atop \!k\!}\biggr\} \frac{(n{+}1)_{k+1}}{k{+}1} \\[10mu] & = \biggl\{{\!4\! \atop \!1\!}\biggr\} \frac{(n{+}1)_{2}}2 + \biggl\{{\!4\! \atop \!2\!}\biggr\} \frac{(n{+}1)_{3}}3 + \biggl\{{\!4\! \atop \!3\!}\biggr\} \frac{(n{+}1)_{4}}4 + \biggl\{{\!4\! \atop \!4\!}\biggr\} \frac{(n{+}1)_{5}}5 \\[8mu] & = \frac12 (n{+}1)_{2} + \frac73 (n{+}1)_{3} + \frac64 (n{+}1)_{4} + \frac15 (n{+}1)_{5}\,. \end{align}$$

Here the Stirling numbers can be computed from their definition as the number of partitions of 4 elements into k non-empty unlabeled subsets.

In contrast, the sum $\sum_{i=0}^n i^k$ in the standard basis is given by Faulhaber's formula, which in general is more complicated.

As inverse matrices
The Stirling numbers of the first and second kinds can be considered inverses of one another:

 * $$\sum_{j=k}^n s(n,j) S(j,k) = \sum_{j=k}^n (-1)^{n-j} \biggl[{n \atop j}\biggr] \biggl\{{\!j\! \atop \!k\!}\biggr\} = \delta_{n,k}$$

and

 * $$\sum_{j=k}^n S(n,j) s(j,k) = \sum_{j=k}^n (-1)^{j-k} \biggl\{{\!n\! \atop \!j\!}\biggr\} \biggl[{j \atop k}\biggr]= \delta_{n,k},$$

where $$\delta_{nk}$$ is the Kronecker delta. These two relationships may be understood to be matrix inverse relationships. That is, let s be the lower triangular matrix of Stirling numbers of the first kind, whose matrix elements $$s_{nk}=s(n,k).\,$$ The inverse of this matrix is S, the lower triangular matrix of Stirling numbers of the second kind, whose entries are $$S_{nk}=S(n,k).$$ Symbolically, this is written

 * $$s^{-1} = S\,$$

Although s and S are infinite, so calculating a product entry involves an infinite sum, the matrix multiplications work because these matrices are lower triangular, so only a finite number of terms in the sum are nonzero.

Lah numbers
The Lah numbers $$L(n,k) = {n-1 \choose k-1} \frac{n!}{k!}$$ are sometimes called Stirling numbers of the third kind. By convention, $$L(0,0)=1$$ and $$L(n,k)=0$$ if $$n<k$$ or $$k = 0 < n$$.

These numbers are coefficients expressing falling factorials in terms of rising factorials and vice versa:
 * $$x^{(n)} = \sum_{k=0}^n L(n,k) (x)_k\quad$$ and $$\quad(x)_n = \sum_{k=0}^n (-1)^{n-k} L(n,k)x^{(k)}.$$

As above, this means they express the change of basis between the bases $$(x)_0,(x)_1,(x)_2,\cdots$$ and $$x^{(0)},x^{(1)},x^{(2)},\cdots$$, completing the diagram. In particular, one formula is the inverse of the other, thus:

 * $$\sum_{j=k}^n (-1)^{j-k} L(n,j) L(j,k) = \delta_{n,k}.$$

Similarly, composing the change of basis from $$x^{(n)}$$ to $$x^n$$ with the change of basis from $$x^n$$ to $$(x)_{n}$$ gives the change of basis directly from $$x^{(n)}$$ to $$(x)_{n}$$:

 * $$ L(n,k) = \sum_{j=k}^n \biggl[{n \atop j}\biggr] \biggl\{{\!j\! \atop \!k\!}\biggr\} ,$$

and similarly for other compositions. In terms of matrices, if $$L$$ denotes the matrix with entries $$L_{nk}=L(n,k)$$ and $$L^{-}$$ denotes the matrix with entries $$L^{-}_{nk}=(-1)^{n-k}L(n,k)$$, then one is the inverse of the other: $$ L^{-} = L^{-1}$$. Composing the matrix of unsigned Stirling numbers of the first kind with the matrix of Stirling numbers of the second kind gives the Lah numbers: $$L = |s| \cdot S$$.

Enumeratively, $\left\{{\!n\! \atop \!k\!}\right\}, \left[{n \atop k}\right], L(n,k)$ can be defined as the number of partitions of n elements into k non-empty unlabeled subsets, where each subset is endowed with no order, a cyclic order, or a linear order, respectively. In particular, this implies the inequalities:
 * $$\biggl\{{\!n\! \atop \!k\!}\biggr\} \leq \biggl[{n \atop k}\biggr] \leq L(n,k).$$

Inversion relations and the Stirling transform
For any pair of sequences, $$\{f_n\}$$ and $$\{g_n\}$$, related by a finite sum Stirling number formula given by

 * $$g_n = \sum_{k=0}^{n} \left\{\begin{matrix} n \\ k \end{matrix} \right\} f_k, $$

for all integers $$n \geq 0$$, we have a corresponding inversion formula for $$f_n$$ given by

 * $$f_n = \sum_{k=0}^{n} \left[\begin{matrix} n \\ k \end{matrix} \right] (-1)^{n-k} g_k. $$

The lower indices could be any integer between $0$ and $n$.

These inversion relations between the two sequences translate into functional equations between the sequence exponential generating functions given by the Stirling (generating function) transform as

 * $$\widehat{G}(z) = \widehat{F}\left(e^z-1\right)$$

and

 * $$\widehat{F}(z) = \widehat{G}\left(\log(1+z)\right). $$

For $$D = d/dx$$, the differential operators $$x^nD^n$$ and $$(xD)^n$$ are related by the following formulas for all integers $$n \geq 0$$:

\begin{align} (xD)^n &= \sum_{k=0}^n S(n, k) x^k D^k \\ x^n D^n &= \sum_{k=0}^n s(n, k) (xD)^k = (xD)_n = xD(xD - 1)\ldots (xD - n + 1) \end{align} $$

Another pair of "inversion" relations involving the Stirling numbers relate the forward differences and the ordinary $$n^{th}$$ derivatives of a function, $$f(x)$$, which is analytic for all $$x$$ by the formulas

 * $$\frac{1}{k!} \frac{d^k}{dx^k} f(x) = \sum_{n=k}^{\infty} \frac{s(n, k)}{n!} \Delta^n f(x)$$

 * $$\frac{1}{k!} \Delta^k f(x) = \sum_{n=k}^{\infty} \frac{S(n, k)}{n!} \frac{d^n}{dx^n} f(x). $$

Similar properties
See the specific articles for details.

Symmetric formulae
Abramowitz and Stegun give the following symmetric formulae that relate the Stirling numbers of the first and second kind.

 * $$ \left[{ n \atop k } \right] = \sum_{j=n}^{2n-k} (-1)^{j-k} \binom{j-1}{k-1} \binom{2n-k}{j} \left\{{ j-k \atop j-n} \right\} $$

and

\left\{{n \atop k}\right\} = \sum_{j=n}^{2n-k} (-1)^{j-k} \binom{j-1}{k-1} \binom{2n-k}{j} \left[{j-k \atop j-n } \right] $$

Stirling numbers with negative integral values
The Stirling numbers can be extended to negative integral values, but not all authors do so in the same way. Regardless of the approach taken, it is worth noting that Stirling numbers of first and second kind are connected by the relations:

 * $$\biggl[{n \atop k}\biggr] = \biggl\{{\!-k\! \atop \!-n\!}\biggr\} \quad \text{and} \quad

\biggl\{{\!n\! \atop \!k\!}\biggr\} = \biggl[{-k \atop -n}\biggr]$$

when n and k are nonnegative integers. So we have the following table for $$\left[{-n \atop -k}\right]$$:

Donald Knuth defined the more general Stirling numbers by extending a recurrence relation to all integers. In this approach, $ \left[{n \atop k}\right]$ and $\left\{{\!n\! \atop \!k\!}\right\}$ are zero if n is negative and k is nonnegative, or if n is nonnegative and k is negative, and so we have, for any integers n and k,
 * $$\biggl[{n \atop k}\biggr] = \biggl\{{\!-k\! \atop \!-n\!}\biggr\} \quad \text{and} \quad

\biggl\{{\!n\! \atop \!k\!}\biggr\} = \biggl[{-k \atop -n}\biggr].$$

On the other hand, for positive integers n and k, David Branson defined $ \left[{-n \atop -k}\right]\!,$ $\left\{{\!-n\! \atop \!-k\!}\right\}\!,$ $ \left[{-n \atop k}\right]\!,$  and $\left\{{\!-n\! \atop \!k\!}\right\}$ (but not $ \left[{n \atop -k}\right]$  or $\left\{{\!n\! \atop \!-k\!}\right\}$ ). In this approach, one has the following extension of the recurrence relation of the Stirling numbers of the first kind:

 * $$ \biggl[{-n \atop k}\biggr]

= \frac{(-1)^{n+1}}{n!}\sum_{i=1}^{n}\frac{(-1)^{i+1}}{i^k} \binom ni $$,

For example, $\left[{-5 \atop k}\right] = \frac1{120}\Bigl(5-\frac{10}{2^k}+\frac{10}{3^k}-\frac 5{4^k}+\frac 1{5^k}\Bigr).$  This leads to the following table of values of $\left[{n \atop k}\right]$  for negative integral n.

In this case $\sum_{n=1}^{\infty}\left[{-n \atop -k}\right]=B_{k} $ where $$B_{k}$$ is a Bell number, and so one may define the negative Bell numbers by $\sum_{n=1}^{\infty}\left[{-n \atop k}\right]=:B_{-k}$.

For example, this produces $\sum_{n=1}^{\infty}\left[{-n \atop 1}\right]=B_{-1}=\frac 1e\sum_{j=1}^\infty\frac1{j\cdot j!}=\frac 1e\int_0^1\frac{e^t-1}{t}dt=0.4848291\dots$, generally $B_{-k}=\frac 1e\sum_{j=1}^\infty\frac1{j^kj!} $.

title: Stirling number


text:
Tangaroa

Tangaroa (Takaroa in the South Island) is the great atua of the sea, lakes, rivers, and creatures that live within them, especially fish, in Māori mythology. As Tangaroa-whakamau-tai he exercises control over the tides. He is sometimes depicted as a whale.

In some of the Cook Islands he has similar roles, though in Manihiki he is the fire deity that Māui steals from, which in Māori mythology is instead Mahuika, a goddess of fire.

Māori traditions
Tangaroa is son of Ranginui and Papatūānuku, Sky and Earth. After joining his brothers Rongo, Tū, Haumia, and Tāne in the forcible separation of their parents, he is attacked by his brother Tāwhirimātea, the atua of storms, and forced to hide in the sea.

Tangaroa is the father of many sea creatures. Tangaroa's son, Punga, has two children, Ikatere, the ancestor of fish, and Tū-te-wehiwehi (or Tū-te-wanawana), the ancestor of reptiles. Terrified by Tāwhirimātea's onslaught, the fish seek shelter in the sea, and the reptiles in the forests. Ever since, Tangaroa has held a grudge with Tāne Mahuta, the atua of forests, because he offers refuge to his runaway children.

The contention between Tangaroa and Tāne Mahuta, the father of birds, trees, and humans, is an indication that the Māori thought of the ocean and the land as opposed realms. When people go out to sea to fish or to travel, they are in effect representatives of Tāne Mahuta, entering the realm of Tāne Mahuta's enemy. For this reason, it was important that offerings were made to Tangaroa before any such expedition.

The Kāi Tahu version of the origin of Takaroa maintains that he is the son of Temoretu, and that Papatūānuku is his wife. Papatūānuku commits adultery with Rakinui while Takaroa is away, and in the resulting battle on the beach Takaroa's spear pierces Rakinui through both his thighs. Papatūānuku then marries Rakinui. In another legend, Tangaroa marries Te Anu-matao (chilling cold). They are the parents of the atua ‘of the fish class’, including Te Whata-uira-a-Tangawa, Te Whatukura, Poutini, and Te Pounamu. In some versions, Tangaroa has a son, Tinirau, and nine daughters.

Cook Islands

 * In Rarotonga, Tangaroa is the god of the sea and fertility. He is the most important of all the departmental gods. Carved figures made from wood carvings are very popular on the island today.
 * In Mangaia, Tangaroa is a child of Vatea (daylight) and Papa (foundation) and the younger twin brother of Rongo. Rongo and Tangaroa share food and fish: Tangaroa's share is everything that is red (the red taro, red fish and so on). Tangaroa is said to have yellow hair and when Mangaians first saw Europeans they thought they must be Tangaroa's children.
 * In Manihiki, Tangaroa is the origin of fire. Māui goes to him to obtain fire for humankind. Advised to reach Tangaroa's abode by taking the common path, he takes the forbidden path of death infuriating Tangaroa who tries to kick him to death. Māui manages to prevent that and insists that Tangaroa give him fire. Māui kills Tangaroa. When his parents are horrified, Māui uses incantations to bring him back to life.

Elsewhere
Tangaloa is one of the oldest Polynesian deities and in western Polynesia (for example, Samoa and Tonga) traditions has the status of supreme creator god. In eastern Polynesian cultures Tangaroa is usually considered of equal status to Tāne and thus not supreme.

 * In Rapa Nui tradition Tangaroa was killed at Hotu-iti bay and was buried in the surrounding area.
 * In Ra'iātea a legend reported by Professor Friedrich Ratzel in 1896 gave a picture of his all-pervading power.
 * In the Marquesas Islands, the equivalent deities are Tana'oa or Taka'oa.
 * In Rennell and Bellona Islands (Polynesian cultures in the southern Solomon Islands) Tangagoa is a sea god who stayed on the coastal cliff of east Rennell known as Toho, and flew in the night with a flame in the sky. Tangagoa was believed to take spirits of the dead, so when someone was near death, the sparkling fire would be seen at night. Some can still recall the time when this god appeared in the night as a flame in the sky, and have many tales of it. Tangagoa started to disappear in the 1970s and early 1980s when Christian missionaries visited the cliff and reportedly 'cast' him out.

A legendary figure named Tagaro also features in the Melanesian cultures of north-eastern Vanuatu. In the beliefs of North Pentecost island, Tagaro appears as a destructive trickster, while in other areas, he is an eternal creator figure, and names cognate with Tagaro (such as Apma Takaa) are applied nowadays to the Christian God.

Moriori
In the mythology of the Moriori of the Chatham Islands, Tangaroa is a fish deity alongside Pou.

title: Tangaroa


text:
Nikephoros Kallistos Xanthopoulos

Nikephoros Kallistos Xanthopoulos (Νικηφόρος Κάλλιστος Ξανθόπουλος; Latinized as Nicephorus Callistus Xanthopulus; c. 1256 – 1335) was a Greek ecclesiastical historian and litterateur of the late Byzantine Empire. His most popular work, the voluminous Ecclesiastica historia, constitutes a significant documentary source on primitive Christianity and its doctrinal controversies, as well as for hagiographical, liturgical, and legendary texts from Byzantine culture.

Life
Most of our knowledge of Xanthopoulos' biography comes from the opening of his ecclesiastical history. The preface says that he began working on his history at age thirty-six. Nicephorus was trained in the florid, rhetorical style of Renaissance Byzantine historiography. He became a priest at Hagia Sophia and thus gained access to the patriarchal library. He also gave lessons in rhetoric and theology for which he prepared many new progymnasmata (exercises). Xanthopoulos was a friend of Theodoros Metochites, who dedicated his Poem 12 to him. During his later years it is possible that he became a monk.

Ecclesiastica Historia
Xanthopoulos' 23-volume Ecclesiastica Historia (Εκκλησιαστική Ιστορία; "Church History"), of which only the first eighteen volumes survive, starts the historical narrative from the time of Christ and continues until the execution of the emperor Phokas in 610. The work includes descriptions of secular events, such as the accession of emperors and military campaigns, but emphasizes ecumenical councils, doctrinal disputes, and the four eastern patriarchates.

Xanthopoulos begun his project around 1310 using the basilica's manuscript library and he completed it sometime after 1317. For the first four centuries, the author is largely dependent on his predecessors, Eusebius, Socrates Scholasticus, Sozomen, Theodoret and Evagrius, his additions showing very little critical faculty. Also relied heavily on a collection of church histories in the Baroccianus Graecus 142, which contains the histories of Sozomen, Evagrios, and a variety of other excerpts on church history. Among the comments and notes that were added by Xanthopoulos, the book contains the phrase Κύριε βοήθει τῶ δούλω Νικοφόρω Καλλίστω ("Lord help your slave Nikephoros Kallistos"), which served as both a prayer and a signature. The work was dedicated to the emperor Andronicus II Palaeologus and it contributed to the monarch’s nationalist movement exalting Greek culture and Orthodoxy above Latin Christianity. His later work, which is based upon documents now no longer extant, is much more valuable.

A table of contents of another five books, continuing the history to the death of Leo VI the Wise in 911, also exists, but whether the books were ever actually written is doubtful. Some modern scholars are of opinion that Nicephorus appropriated and passed off as his own the work of an unknown author of the 10th century. The plan of the work is good and, in spite of its fables and superstitious absurdities, contains important facts which would otherwise have been unknown.

Only one manuscript of the history is known. It was stolen by a Turkish soldier from the library at Buda during the reign of Matthias Corvinus of Hungary and taken to Constantinople, where it was bought by a Christian and eventually reached the imperial library at Vienna.

Other
Among Xanthopoulos' other works are commentaries on the writings of the patristic Greek theologian Gregory of Nazianzus and of the Byzantine monk John Climacus. He was also the author of lists of the emperors and patriarchs of Constantinople, of a poem on the capture of Jerusalem, and of a synopsis of the Scriptures, all in iambics; and of commentaries on liturgical poems. As a hagiographer, his writings include a history of miracles that occurred at the shrine of Zoödochos Pege, as well as the lives of Saint Nicholas of Myra and Euphrosyne the Younger. He also wrote many of the synaxaria in use in the Eastern Orthodox Church.

title: Nikephoros Kallistos Xanthopoulos


text:
Ta'aroa

Ta'aroa is the supreme creator god in the mythology of the Society Islands of French Polynesia. While the use of the ʻeta is appropriate given the pronunciation of his name, it is often omitted in practice, as is typically the case with Tahitian words.

The Myth
In the beginning, there was only Ta'aroa, creator of all, including himself. He waited alone in his shell, which appeared as an egg spinning in the empty endless void of the time before the sky, before the Earth, before the Moon, before the Sun, before the stars. He was bored, alone in his shell, and so he cracked it with a shake of his body and slid out of its confines, finding everything somber and silent outside, finding himself alone in the nothingness.

So he broke the shell into pieces and from them formed the rocks and the sand, and the foundation of all the world, Tumu-Nui. With his backbone he created the mountains; with his tears he filled the oceans, the lakes, the rivers; with his fingernails and toenails he made the scales that cover the fish and the turtles; with his feathers he created the trees and the bushes; with his blood he colored the rainbow.

Ta'aroa then called forth artists who came with their baskets filled with To'i, so that they might sculpt Tane, the first god. Then came Ru, Hina, Maui, and hundreds of others. Tane decorated the sky with stars and hung the sun in the sky to illuminate the day and the moon to illuminate the night. Ta'aroa decided then to complete his work by creating man.

He divided the world into 7 levels. On the bottommost level lived man, and he multiplied quickly, which delighted Ta'aroa. Sharing the space as he did with creatures and plants of all sorts, it was not long before man felt crowded in his space and so decided to expand his domain by opening a hole into the level above his. Man continued in this fashion, filling one level and then climbing to the next, one level at a time, until all levels were occupied.

And so man filled the Earth, but still all belonged to Ta'aroa, who was master of all.

title: Ta'aroa


text:
Buda

Buda was the historic capital of the Kingdom of Hungary and, since 1873, has been the western part of the Hungarian capital Budapest, on the west bank of the Danube. Buda comprises a third of Budapest's total territory and is mostly wooded. Landmarks include Buda Castle, the Citadella, and the president of Hungary's residence, Sándor Palace.

Etymology
According to a legend recorded in chronicles from the Middle Ages, the name "Buda" comes from the name of Bleda (Buda), brother of Hunnic ruler Attila."Attila went in the city of Sicambria in Pannonia, where he killed Buda, his brother, and he threw his corpse into the Danube. For while Attila was in the west, his brother crossed the boundaries in his reign, because he named Sicambria after his own name Buda's Castle. And though King Attila forbade the Huns and the other peoples to call that city Buda's Castle, but he called it Attila's Capital, the Germans who were terrified by the prohibition named the city as Eccylburg, which means Attila Castle, however, the Hungarians did not care about the ban and call it Óbuda [Old Buda] and call it to this day.""The Scythians are certainly an ancient people and the strength of Scythia lies in the east, as we said above. And the first king of Scythia was Magog, son of Japhet, and his people were called Magyars [Hungarians] after their King Magog, from whose royal line the most renowned and mighty King Attila descended, who, in the 451st year of Our Lord’s birth, coming down from Scythia, entered Pannonia with a mighty force and, putting the Romans to flight, took the realm and made a royal residence for himself beside the Danube above the hot springs, and he ordered all the old buildings that he found there to be restored and he built them in a circular and very strong wall that in the Hungarian language is now called Budavár [Buda Castle] and by the Germans Etzelburg [Attila Castle]"

Demographics

The Buda fortress and palace were built by King Béla IV of Hungary in 1247, and were the nucleus around which the town of Buda was built, which soon gained great importance, and became in 1361 the capital of Hungary.

While Pest was mostly Hungarian in the 15th century, Buda had a German majority; however according to the Hungarian Royal Treasury, it had a Hungarian majority with a sizeable German minority in 1495. In 1432, Bertrandon de la Broquière wrote that Buda "is governed by Germans, as well in respect to police as commerce, and what regards the different professions". He noted a significant Jewish population in the city, proficient in French, many of whom were descendants of Jews previously expelled from France.

Buda became part of Ottoman-ruled central Hungary from 1541 to 1686. It was the capital of the province of Budin during the Ottoman era. By the middle of the seventeenth century Buda had become majority Muslim, largely resulting from an influx of Balkan Muslims.

In 1686, two years after the unsuccessful siege of Buda, a renewed European campaign was started to enter Buda, which was formerly the capital of medieval Hungary. This time, the Holy League's army was twice as large, containing over 74,000 men, including German, Dutch, Hungarian, English, Spanish, Czech, French, Croat, Burgundian, Danish and Swedish soldiers, along with other Europeans as volunteers, artillerymen, and officers, the Christian forces reconquered Buda (see Siege of Buda).

After the reconquest of Buda, bourgeoisie from different parts of southern Germany moved into the almost deserted city. Germans — also clinging to their language — partly crowded out, partly assimilated the Hungarians and Serbians they had found here. As the rural population moved into Buda, in the 19th century Hungarians slowly became the majority there.

Notable residents

 * Andrew III of Hungary, (ca.1265–1301) King of Hungary and Croatia, 1290 to 1301, buried in the Greyfriars' Church, a Franciscan church in Buda
 * Jadwiga of Poland, (ca.1373–1399), born in Buda, first female monarch of the Kingdom of Poland
 * John Corvinus (1473–1504) illegitimate son of Matthias Corvinus, King of Hungary, and his mistress, Barbara Edelpöck.
 * Louis II of Hungary (1506–1526) King of Hungary, Croatia and Bohemia from 1516 to 1526.
 * Aaron ben Joseph of Buda (ca. 1686), poet
 * Mihail G. Boiagi, (1780 – ca.1842) an Aromanian grammarian and professor
 * László Szalay (1813–1864) a Hungarian statesman and historian.
 * József Eötvös (1813–1871) a Hungarian writer and statesman.
 * Ignaz Philipp Semmelweis (1818–1865), a Hungarian physician and scientist. An early pioneer of antiseptic procedures, he proposed doctors start the practice of disinfecting their hands.
 * Kornelije Stanković, (1831–1865) notable Serbian composer, born and died in Buda
 * Edmund Hauler (1859–1941), classicist and philologist

Twin cities

 * 🇮🇹 Capestrano, Italy

title: Buda


text:
Tāwhirimātea

In Māori mythology, Tāwhirimātea (or Tāwhiri) is the god of weather, including thunder and lightning, wind, clouds and storms. He is a son of Papatūānuku (earth mother) and Ranginui (sky father). Tawhirimatea is the second oldest of 7 children, all of whom are boys. In his anger at his brothers for separating their parents, Tāwhirimātea destroyed the forests of Tāne (god of forests), drove Tangaroa (god of the sea) and his progeny into the sea, pursued Rongo and Haumia-tiketike till they had to take refuge in the bosom of their mother Papa, and only found in Tūmatauenga a worthy opponent and eternal enemy (Tregear 1891:499). To fight his brothers, Tāwhirimātea gathered an army of his children, winds and clouds of different kinds - including Apū-hau ("fierce squall"), Apū-matangi, Ao-nui, Ao-roa, Ao-pōuri, Ao-pōtango, Ao-whētuma, Ao-whekere, Ao-kāhiwahiwa, Ao-kānapanapa, Ao-pākinakina, Ao-pakarea, and Ao-tākawe (Grey 1971). Grey translates these as 'fierce squalls, whirlwinds, dense clouds, massy clouds, dark clouds, gloomy thick clouds, fiery clouds, clouds which preceded hurricanes, clouds of fiery black, clouds reflecting glowing red light, clouds wildly drifting from all quarters and wildly bursting, clouds of thunder storms, and clouds hurriedly flying on' (Grey 1956:5).

Other children of Tāwhirimātea are the various kinds of rain, mists and fog. Tāwhirimātea's attacks on his brothers led to the flooding of large areas of the land. The names of the beings involved in this flooding include Ua-nui (terrible rain), Ua-roa (long-continued rain), Ua-whatu (fierce hailstorms), and Ua-nganga (sleet); after these, their children in turn took up the fight: Hau-maringi (mist), Hau-marotoroto (heavy dew), and Tōmairangi (light mist) (Grey 1956:10-11, Grey 1971:5). Tregear mentions Hau-maringiringi as a personification of mists (Tregear 1891:54).

Tāwhirimātea lives on the sky with his father Rangi and brother, the star Rehua.

The Divine War and Tāwhirimātea's conquest over his brothers
Eons ago, Rangi, the Sky Father, and Papa, the Earth Mother, were in an eternal embrace because of their love for each other. Their union gave rise to many powerful sons, who lived in between their parents.

As their sons grew up, they soon began to grow tired of living in a cramped up space, forever in darkness. One brother, Tūmatūenga, the God of War and Humans, suggested that they slay their parents. However, his brother, Tāne, the God of Forests, suggested that they separate their parents forever.

Except for Tāwhirimātea, all other brothers accepted the proposal. The brothers individually tried to separate their parents, but Tāne put his head on the earth and feet in the sky and pushed them apart.

Tāwhirimātea was enraged, as he saw it as a move to usurp his kingdom. So the god flew to the sky and communed with his father. Rangi reluctantly agreed to help his son wage a brutal war on his siblings. They were the spirits of winds, storms, and rain. Tāwhirimātea assembled his huge army and set out to conquer his brothers.

Tāwhirimātea first attacked Tāne, and razed his forests, causing Tāne to flee. Next, Tāwhirimātea attacked his brother, Tangaroa, the Sea God. He caused huge storms and waves, spreading panic in Tangaroa. Tangaroa was himself helpless before Tāwhirimātea, as the sea was in such a chaotic rage, harming all living beings. Having never seen such chaos at sea, many of Tangaroa's children deserted their father and took shelter with Tāne. Since then Tangaroa is at war with Tāne.

Tāwhirimātea pursued his brother, Rongo, and Haumea, the gods of cultivated and uncultivated food, but they were cleverly hidden by their mother, Papa, who still loved her children. Finally, Tāwhirimātea began to fight Tumatuenga.

This time, however, Tumatuenga firmly embedded his feet in the earth, saving him from Tāwhirimātea's storms. He cast spells, which turned the storms into gentle weather, suitable for humans. But neither brother could prevail against each other. Tāwhirimātea slowly withdrew.

To punish his brothers for cowardice, Tumatuenga invented the arts of hunting, woodcutting, agriculture, cooking, and fishing, to subjugate their respective denizens as food for humans. However, Tumatuenga and Tawhirimatea still fight each other to this day. Another result of the war was that most of the land was submerged into the ocean, because of Tāwhirimātea causing heavy rains and thunderstorms. He was maybe the only brother out of all of them to want their parents together.

title: Tāwhirimātea


text:
TU

Tu or TU may refer to:

Language

 * Tu language
 * Tu (cuneiform), a cuneiform sign
 * tu or tú the 2nd-person singular subject pronoun in many languages; see personal pronoun
 * T–V distinction (from the Latin pronouns tu and vos), the use in some languages, of a different personal pronoun for formality or social distance
 * Tsu (kana), also romanized as tu

People and names

 * Tū (Tūmatauenga), a supernatural being in Māori mythology
 * Tu people, the Monguor people of the People's Republic of China
 * Tu language
 * Tu Holloway (born 1989), basketball player for Maccabi Rishon LeZion in the Israeli Basketball Premier League
 * Tu (surname) 屠, a rare Chinese family name
 * Du (surname) 杜 or Tu, a common Chinese family name

Music

 * Tú (Canadian band), a Canadian pop music duo in the late-1980s
 * Tu (American band), an American duo, formed by member of King Crimson

Songs

 * "Tu" (Umberto Tozzi song), 1978
 * "Tu", a song by Umberto Bindi, 1959
 * "Tu", a song by Ewa Farna, 2015

Other

 * Tu (film) Here (2003 film) or Tu, a Croatian film
 * Tu, a brand of clothing from Sainsbury's
 * Tu (cake), a type of Tibetan cake

Companies and organizations

 * TU (union), the international union for T-Mobile workers
 * TU Media, a Digital Multimedia Broadcasting mobile companies in Korea
 * Toimihenkilöunioni (Union of Salaried Employees), a Finnish trade union
 * Tunisair (IATA airline code TU)
 * Transunion, a consumer credit reporting agency
 * Tupolev, a Russian aerospace and defence company
 * Teknisk Ukeblad, a Norwegian engineering magazine
 * The Times Union, an Albany newspaper
 * New York City Teachers Union (1916–1964), commonly known as the "TU"
 * Trout Unlimited, conservation group

Units of measurement

 * TU (time unit), a unit of time equal to 1024 microseconds
 * Transmission unit, a historical unit of loss in long distance telephony
 * Tritium unit, a measure of tritium concentration in water
 * Tuberculin Units, a measure of strength of tuberculin

In the United States

 * Taylor University, Indiana
 * Temple University, Pennsylvania
 * Thomas University, Georgia
 * Touro University California, California
 * Touro University Nevada, Nevada
 * Towson University, Maryland
 * Trinity University, Texas
 * Troy University, Alabama
 * Tufts University, Massachusetts
 * Tulane University, Louisiana
 * Tuskegee University, Alabama
 * University of Tulsa, Oklahoma
 * t.u., a derisive reference to the University of Texas at Austin by students and supporters of in-state rival Texas A&M University

In other countries

 * University of Tartu (Tartu Ülikool), Estonia
 * Any Technische Universität, i.e. university of technology in German-speaking countries
 * Technische Universiteit Delft, Netherlands
 * Technische Universiteit Eindhoven, Netherlands
 * Tezpur University, Assam, India
 * Thapar University, Patiala, India
 * Thammasat University, Thailand
 * Tianjin University, Tianjin, China
 * Tibet University, Lhasa, Tibet Autonomous Region, China
 * Tooling University, an American non-profit educational technology company
 * Tribhuvan University, Kathmandu, Nepal
 * Technological Universities of Myanmar
 * Triam Udom Suksa School

Other uses

 * Tren Urbano, the metro system in San Juan, Puerto Rico
 * Tu Hundred, a district of Vccästmanland in Sweden
 * Universal Time, "Tempus Universalis"
 * Thulium, a chemical element with obsolete symbol Tu
 * Translation unit, a single cognitive unit of text
 * Translation unit (programming), the ultimate input to the compiler

title: TU