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## RPG Games For Girls!!!

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Posted 04 December 2001 - 11:18 AM

I suppose I''m obligated to reply to this, seeing as how Mooglez has pointed a big neon arrow at me. *chuckle*

Hmm, how to attack this topic...

1) Girls are not that different from guys, really! We are certainly not incomprehensible - If I can understand both girls and guys, you should be able to to, with a little study of gendered fiction and sociology of gender roles and stuff.

2) Girls are a little different from guys, and the differences are fairly predictable. Girls are raised to think they are socially obligated to be pretty/stylish/fashionable, while all humans are naturally creative, this tends to be discouraged in boys. Female creativity is what all you poor guys would have had you not grown up in a gender-stereotyping society that treats women as more ornamental than men. To answer the question of how many hairstyles/clothes/etc would satisfy a girl, I think if you had a thousand possibilities (for all the categories together) that would be sufficient. Mush more practical to design into an MM game than a singleplayer, but any game can use a color selector to let the player customize things.

3) Girls tend to be more extroverted (people oriented) then boys, I''m not sure whether this is nature or nurture. So girls like games where NPCs have strong and dramatic personalities, and the player is asked to make choices that have social value, like dialogue choices. Japanese ren''ai games are the place to look for examples of this.

4) Girls are taught that some of the worst thing you can be are callous, indiscriminately violent, unfeeling, etc. This is one reason girls tend to react poorly to games where the player must slaughter monsters or aliens, or games where there are scream sfx and blood splatters. There''s also the fact that monster battles are really boring because they have no social or artistic element at all, and very little strategic element.

5) Girls are taught to be mothers. What do mothers do? They fix problems and injuries, give gifts, provide comfort and advice, create various things (meals, clothes), arrange things (laying out a garden, interior decorating). Look to Harvest Moon and Microsoft Bob, as well as most adventure and pattern/puzzle games for ideas about things like this.

6) The one demonstrable way that girls are naturally different from guys is their taste in people''s appearances. Get a female artist to design some of your characters. Look at one of the characters so popular with women that many have written fanfics about him: Sephiroth from FF7. Women generally like bishounens (pretty boys), sparkly things, fluffy things, slinky things (e.g. a siamese cat), and dramatic color displays (e.g. a peacock). Women also seem to be more sensitive to fractal-type things than men - examples are feathers, flowers or just petals, lightning bugs, clouds, ripples in water, spiral patterns like in a seashell or a DNA, the rainbows an oil spill makes. Shoujo anime is the area to study to learn about this.

7) It is not true that girls are completely non-violent. Girls can find violence appealing if it is individualized, romanticized, poetic justice, or creative revenge with an element of humor involved.

Did I miss anything?

### #22GalaxyQuest  Members

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Posted 04 December 2001 - 11:40 AM

First, i gotta say some of you sound like 13 year old BOYS with the pink and fluffy nonsense.

The STAR TREK series often had worlds where only women ruled, also, Buck Rogers had many roles where women where the leaders, such as that HOT N SEXY brown haired queen in the huge space ship (queen amadala??), along with planets where the women where in control and the men were the sissy's.

MMRPG's include female roles to play, such as in everquest and others. They make kick ass wizards or whatever. Although, I havnt seen much attempt to bring them together. There are a lot of female characters in everquest, at least in character!

Maybe to bring the women together, introduce women leaders(such as a queen for the empire as NPCs), where to be a royal guard, you must first be a women, then secondly an expert in protection(spells, melee, etc). Men also protect the queen. Just take a look at history.

Another idea, as I mentioned, is to define a planet, a guild, or a region of territory, where the women can begin playin there characters TOGETHER. Hopefully, this would spawn more women, in game, and out, to play.

But yes, it hasnt been done correctly, probably because most, if not all, developers and project managers and corporate  is run by men who havnt thought about the other half and what to do with them.

Edited by - GalaxyQuest on December 4, 2001 6:42:16 PM

Edited by - GalaxyQuest on December 4, 2001 6:43:55 PM

### #23MSW  Members

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Posted 04 December 2001 - 01:16 PM

Question: From a female perspective, what role would female players like to play in an singleplayer RPG?...I don''t meen a typical D&D type midevil fantasy basied RPG...but a RPG that takes place in a more current setting...

here is a idea for a RPG:

Imagine a singleplayer RPG that takes place in Chicago around 1923-24...Players control a young female school teacher...in a time when women had recently been granted the right to vote...in a time when Jazz was king and silent movies only cost a penny to watch...lots of interesting story, charactor and setting potential.

The young students in the players class would..in a way...represent the players ''stats''...the better the students are doing the higher the related stat...thins would allow for some ''side quest'' elements...like "why is Timmy doing so poorly?" and the such...allowing players to express some of thier mothering insticts...

Then add in the larger setting where America is just starting to come to grips with how women fit into society as the industrial age is in full swing...again lots of potential

### #24Drizzt DoUrden  Members

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Posted 04 December 2001 - 01:43 PM

quote:
I suppose I''m obligated to reply to this, seeing as how Mooglez has pointed a big neon arrow at me. *chuckle*

Hmm, how to attack this topic...

1) Girls are not that different from guys, really! We are certainly not incomprehensible - If I can understand both girls and guys, you should be able to to, with a little study of gendered fiction and sociology of gender roles and stuff.

I agree that understanding the other gender should not be hard, but I guess for some people it just is. Those who believe it is too hard to understand usually are those who are unwilling to understand it.

quote:

2) Girls are a little different from guys, and the differences are fairly predictable. Girls are raised to think they are socially obligated to be pretty/stylish/fashionable, while all humans are naturally creative, this tends to be discouraged in boys. Female creativity is what all you poor guys would have had you not grown up in a gender-stereotyping society that treats women as more ornamental than men. To answer the question of how many hairstyles/clothes/etc would satisfy a girl, I think if you had a thousand possibilities (for all the categories together) that would be sufficient. Mush more practical to design into an MM game than a singleplayer, but any game can use a color selector to let the player customize things.

Is that hinting that you like Baldurs Gate''s character creation ability Are you giving away the secrets of what makes games that females enjoy!? lol.

quote:

3) Girls tend to be more extroverted (people oriented) then boys, I''m not sure whether this is nature or nurture. So girls like games where NPCs have strong and dramatic personalities, and the player is asked to make choices that have social value, like dialogue choices. Japanese ren''ai games are the place to look for examples of this.

I like games where NPC''s have "strong and dramatic" personalities, and I am sure other guys do, too. I don''t think this can be limited to females.

quote:

4) Girls are taught that some of the worst thing you can be are callous, indiscriminately violent, unfeeling, etc. This is one reason girls tend to react poorly to games where the player must slaughter monsters or aliens, or games where there are scream sfx and blood splatters. There''s also the fact that monster battles are really boring because they have no social or artistic element at all, and very little strategic element.

This also can not be limited to females. I hate games like Rainbow Six and Counter Strike, or other games the promote terrorism, etc. I do, however, like to kill Orc and Ogre''s and Ogrillons. So I don''t know

quote:

5) Girls are taught to be mothers. What do mothers do? They fix problems and injuries, give gifts, provide comfort and advice, create various things (meals, clothes), arrange things (laying out a garden, interior decorating). Look to Harvest Moon and Microsoft Bob, as well as most adventure and pattern/puzzle games for ideas about things like this.

Guys are taught to be fathers. Watch football, work, drink beer, order their wife and kids to do things for them, etc. Every once in a while a boy will break out of that fatherly teachings and become something more.

quote:

6) The one demonstrable way that girls are naturally different from guys is their taste in people''s appearances. Get a female artist to design some of your characters. Look at one of the characters so popular with women that many have written fanfics about him: Sephiroth from FF7. Women generally like bishounens (pretty boys), sparkly things, fluffy things, slinky things (e.g. a siamese cat), and dramatic color displays (e.g. a peacock). Women also seem to be more sensitive to fractal-type things than men - examples are feathers, flowers or just petals, lightning bugs, clouds, ripples in water, spiral patterns like in a seashell or a DNA, the rainbows an oil spill makes. Shoujo anime is the area to study to learn about this.

??? Are you saying that I am a girl because I like clouds, and Sephiroth from Final Fantasy 7 I mean, these things are just more stereotypes of what these guys think girls are. I guess it is just because all the girl-friends (like friends that are girls, don''t get the two confused ) I have like everything I like, and do most of the things I do. (The ones who I am not friends with wake up at 3 A.M. to put on the makeup needed for school at 7).

quote:

7) It is not true that girls are completely non-violent. Girls can find violence appealing if it is individualized, romanticized, poetic justice, or creative revenge with an element of humor involved.

Did I miss anything?

See thats more like it. What you said in the beginning: "Girls are not that much different than boys, really!" was enough. There are too many different types of personalities, and so many different types of people, noone can really say what is different.

Some people do, some people don''t.

That statement is as general as it gets.

Everyone should note that that statement doesn''t say:

"Some girls do, some girls don''t." or "Some guys do, some guys don''t."

It says people. People are general.

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Posted 04 December 2001 - 02:37 PM

quote:
Original post by MSW

Question: From a female perspective, what role would female players like to play in an singleplayer RPG?...I don''t meen a typical D&D type midevil fantasy basied RPG...but a RPG that takes place in a more current setting...

Hmm... A school teacher could work. If it''s the ''20s... how about the costume designer for a club with showgirls. The costume designer acts as a mom to the girls, and defends them from the boss and customers, and manages gangsters so they don''t destroy the club...

Drizzt - I was trying to convey the idea of a statistical difference between males and females, you know, when there''s a curve representing males and a curve representing females and the 2 overlap by 70% but you can still tell that the mean, median, and mode are all further to the right for one gender than the other.

### #26Eight  Members

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Posted 04 December 2001 - 08:53 PM

quote:
Original post by digitaldirt
Not at all. But that is the attitude of a "guy" developer. They are ignorant of the famale audience in games, as they are in relationships i suppose. who knows...

I think there is a lot you can do, without resulting to "Pink" and "My Little Pony".

Sorry, was being sarcastic to highlight that the age old sexist views are still hanging around (Mary Kate & Ashley games for a start).

I don''t think the female''s lack of interest in games has anything to do with the content. Rather, society has told them (and all of us) that computer games are for little boys. Time is changing that for everyone as they become more mass-market anyway.

E

### #27Hase  Members

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Posted 04 December 2001 - 09:26 PM

I think a lot of you are still going about this from the wrong angle. You´re trying to guess at what girls want, and then create that.

1) It´s not about the main character, or the representation of that character. Male players have no problem playing female characters and vice versa.

2) It´s not only about catering to the "female skills" (at least this is what it sounds like when you describe them), the things female players go for are usally things which male players can enjoy just as much.
To create a good game you don´t have to go out of your way to include the mothering instinct or whatevery you consider to be a typically female trait.
Similarly, attacking the "female pride", as eng3d describes it won´t work. If you print on your box "GUYS ONLY", the potential female customers will think something along the lines of "what do i care" and pass it by. As chance has it, most products in our society which are labelled "guys only", are usually interesting to or primarily bought by guys alone.

3) I think the games girls are traditionally bad at have been stated pretty correctly above, but that distinction is NOT due to some inherent difference between men and women (someone suggested motor control, i think that´s bs, no offense intended), but because these games (2D platformers, FPSs, RPGs and martial arts games / beat ´em ups) are games which are the least attractive to the casual gamer, which women in nine out of ten cases are.
In order to be good at Quake Deathmatch you have to have a lot of 3d experience, quick reflexes and lots of practise. In order to be able to play a beat ´em up well you have to know a lot of moves.
Its not that women CANT be good at those games, I think they don´t want to, because they usually lack the background in those genres which equals the fact that they lack the training, which in turn makes those games too hard. The same thing goes for games with a high degree of realism (especially management sims), these (as the others mentioned) are hard to get into, and require a lot of time to be learned properly.
On a personal note: I´ve had a lot of fun playing games like SoulCalibur and Tekken with girls, but it only works when you pick up the game together (=start at more or less the same point of the learning curve), or find some way to bridge the skill gap.

For those of you who are really interested: THere is already a lot of literature in the field of media studies / gender studies, so it might be a good idea to pick some of those up at the local university library.

### #28digitaldirt  Members

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Posted 04 December 2001 - 10:40 PM

The whole naked women ruling the planet while the men are sissies, show girls, buck rogers and so on. are guy ideas or possibly fantasies. i don't know. but yeah, someone said it right when they said "you guys sound like a bunch of 13 year old boys".

I brought this subject up, and I thank the ones who have replied with thoughtful responses, instead of ignorant "i don't undersand them." remarks. Its very interesting to see how the online game community see's the opposite side of things. I've been holding study groups with women of different ages, with presentations and content brainstorms, and so far i've held 32 groups of 16 women. From my studies, Women don't want to "see what little timmy is doing".. they want more worldly plots, not just "womens rights/voting". They really enjoy the fantasy genre more than the futuristic genre, but they have nothing against that genre. its been a 80/20 split. I've been using all of this in my game design for a new title.

It is my hope that we are making more strides to grow the female audience in interactive entertainment.

Edited by - digitaldirt on December 5, 2001 5:42:35 AM

### #29MSW  Members

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Posted 05 December 2001 - 12:10 AM

Haste - Ever heard of "chick flicks" and romance novels? Arn''t those aimed more at females then males?

I''m not saying a female isn''t going to be interested in guy type forms of entertainment (or vice versa)... or that certain examples in entertainment cannot "cross over"...but there are some very general stories, ideas, concepts, characters, and themes that attract females more then men (and vice versa).

Games being a form of entertainment are no different...nor should they be...

Digitaldirt - I''m just throwing ideas up to see what sticks...It''s good that you''ve included females in your brainstorming sessions, it really is...But, from your postings, you seem to be giving them two different game settings to chose from...fantasy and futuristic...Has the potential for a more modern game setting been brought up? something that doesn''t fit into the common cliched RPG game settings?

I''m also curios about the "different ages" of females you have included in your groups...what are the age ranges? do any have kids, families? in school? single, married, divorced? grandparents? teens?

### #30Sandman  Members

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Posted 05 December 2001 - 01:13 AM

Yay, another games for girls thread

I agree with Hase here. You don't have to target women particularly. Funnily enough, just like men, women like different things. Some girls might like playing My Little Pony's Adventure In Happy Fun Land, others enjoy Quake III.

You could argue that there are two types of women gamer:

1: Regular Gamers. These play the same games as men through choice, and are therefore unlikely to even consider My Little Pony's Adventure In Happy Fun Land unless they get to frag some guys in it. I believe there are women only quake clans dedicated to fragging blokes.

2: Occasional Gamers. Most women I know fall into this category. These people generally don't play games, since they often have 'better' things to do, but that doesn't mean they are completely opposed to playing games. They certainly don't spend much money on games, preferring to play on their boyfriends/brothers/fathers computer. Thus they are very unlikely to play My Little Pony's Adventure In Happy Fun Land since it is very unlikely that any guy will be seen dead with it installed on his computer. They tend to prefer games which don't take a long time to play and relatively simple controls. In otherwords, stuff which you can pick up and play very quickly. Violence isn't really an issue, neither is big boobs. For example, my girlfriend likes Kill 'em All (a missile command style game where you shoot cute bunnies with parachutes) Lemmings, Tetris, Tekken 3, stuff like that. Half of the games she likes could be written in an afternoon.

In short, by writing a game like My Little Pony's Adventure In Happy Fun Land, you are not only alienating the male gaming population, but you also alienate most of the female gaming population too.

In other words, just write the game how you like. Some women will like it, some women will hate it. Some men will like it, some men will hate it.

Edited by - Sandman on December 5, 2001 8:18:22 AM

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Posted 05 December 2001 - 01:21 AM

quote:
Original post by MSW
Hase - Ever heard of "chick flicks" and romance novels? Arn''t those aimed more at females then males?

But I can enjoy a good "chick flick" now and then too! SPECIALLY with my girlfriend.

I''ve been reading this thread, and what comes to mind most clearly in the whole guys vs. girls, nature vs. nurture thing is that perhaps, on the whole, girls are simply discouraged from playing computer games AT ALL.
A good game will appeal to female gamers. But it is VERY difficult to design a computer game that will actually bring more women to the PC to play games. I take my sister as an example: she''s nearly computer illiterate - knows just enough to get on the Internet and use the word processor to type out her reports and things. She plays "Spider" (a silly card game) and "patience" while she''s working sometimes, but I don''t see her ever getting into a game that requires full attention. I don''t think I could convince her to play a full game.

People might not remember what you said, or what you did, but they will always remember how you made them feel.

### #32 Anonymous Poster_Anonymous Poster_*   Guests

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Posted 05 December 2001 - 01:21 AM

quote:
Original post by Drizzt DoUrden
[...]and the girls around here spend more than 14 hours a day on making themselves look like the girls on the Jay Rule video or whatever the hell that dumb-shits name is.

God, And I thought that was only my town. I now fear for all of humanity. I know many girls who dont act this way, and respect them for that. Everyone else... Hey, if you want to act stupid, be my guest.

Z.

### #33 Anonymous Poster_Anonymous Poster_*   Guests

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Posted 05 December 2001 - 01:32 AM

quote:

Original post by Hase

but because these games (2D platformers, FPSs, RPGs and martial arts games / beat ´em ups) are games which are the least attractive to the casual gamer, which women in nine out of ten cases are.
In order to be good at Quake Deathmatch you have to have a lot of 3d experience, quick reflexes and lots of practise. In order to be able to play a beat ´em up well you have to know a lot of moves.

I would be interested to know why the 2D platform genre does not get acceptance among women. Do these not count as a subset of the problem solving genre which traditionally appeal to women?

### #34Fox Mc Cloud  Members

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Posted 05 December 2001 - 01:34 AM

quote:
Original post by Hase
1) It´s not about the main character, or the representation of that character. Male players have no problem playing female characters and vice versa.

Well, maybe for single-player games, but when it comes to an online game you don''t want to fool the others by looking like a girl / a girl doesn''t want the other players to think she''s a guy.
I know I wouldn''t take a female character in an online game, and my girlfriend always takes female characters too (in Darkstone, Soldier of Fortune, Quake3...). She wanted to play CounterStrike and was getting quite good at it but dropped it because she was annoyed that there''s no female character...

But again, in a single player game, I don''t mind and I don''t think girls do either.

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### #35Hase  Members

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Posted 05 December 2001 - 02:06 AM

@MSW:

I´m familiar with the term "chick flicks", and I think that there might come a time when computer games along those lines could work. One of the problems is that this genre of novel or film relies heavily on characters, not actions. I think what we´re talking about here is a soap game (the sims?), which would create big problems in the way of interactive storytelling. Also the genre of the classical adventure games is as good as dead, so I wouldn´t consider doing a game like that. I also think that creating games along the lines of "chick flicks" are not what the market needs right now. It is a very specialised form of entertainment geared towards the female audience, and while it may work reasonably well with films and books, I don´t think it´s wise to make games like that right now, because usually female gamers DONT want to be classified as the rosamunde-pilcher reading stereotype.
Maybe you can integrate some of the themes into a game, but you have to be very, very careful there as the line between good narrative and cliché is extremely thin with games. And before you go for the stereotypes *again*, you´d better not do it at all.
Another consideration is the current market situation. Nobody can afford making a triple A title which is geared for an audience which is as unpredictable and unreliable as female casual gamers (even if they´d like it there´s no guarantee they´d buy it - there are only very few marktet studies... etc.)

So, while I think that there may come a time when games like that are possible, I doubt that it will be within the next few years. I´m talking about full retail titles, not some cheap budget games - There would probably be an option there, but personally I wouldn´t want games to go that way. If we make games for girls, we should do it to the best of our abilites, not some crap which sells because it´s pink and has ponies in it.

@Sandman

agreed on all points, especially alienating the audience.

Sure you can enjoy a chick flick with your girl, but the situation is a bit different. If you have a game then you want someone to actively buy and play it, not consume it passively.

So maybe all we can do is make the games we would make anyways, and just make them a little bit more attractive for girls?

@AP (2)

The 2d-platform genre is as good as dead anyways (at least on the PC), so I don´t think it would be very fruitful to pursue that path. I don´t think that these games are a subset of the puzzle solving genre, as you don´t have to really solve anything, if at all, the puzzles are more about finding the right location, with a lot of running around in between. The puzzles women tend to go for are more immediate and direct, there is no context or environment you have to put it in.

@Fox Mc Cloud

I don´t think that it´s much of a problem in online games, as its all about playing a role. If someone wants to be a barbarian, you can do that and nobody will be upset if your gender doesn´t correspond with that of your character. It´s true, the deeper the immersion goes (MMORPGS), the more the players tend to choose characters which share traits with them. But including the options of female player characters isn´t really that much of a bother, and putting some gender-related gameplay issues in won´t make that much of an extra effort.

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Posted 05 December 2001 - 02:13 AM

quote:
Original post by Hase
So maybe all we can do is make the games we would make anyways, and just make them a little bit more attractive for girls?

I think that''s indeed all we can do. Perhaps get some female programmers on our teams, some female testers, artists, etc.
And ALWAYS keep in the back of your mind "what would my sister/girlfriend/wife/mother think about this feature?". Doing it that way, you will simply create a good, enjoyable game that is not blatantly sexist and therefore doesn''t scare off the female players.

People might not remember what you said, or what you did, but they will always remember how you made them feel.

### #37Fox Mc Cloud  Members

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Posted 05 December 2001 - 02:33 AM

quote:
Original post by Hase
Also the genre of the classical adventure games is as good as dead, so I wouldn´t consider doing a game like that.

That nearly no new adventure game is being developed doesn''t mean it wouldn''t be a good idea... I miss those games so much ! why doesn''t anyone make games like these any longer? Sam And Max and Day Of The Tentacle must be my favorite games ever
The latest (major) adventure game must be Grim Fandango from LucasArts (THE masters of the genre)... Any others apart from that?
I think it''s a genre that suits female well too.

(by the way, my girlfriend''s favorite games are real-time strategy... Total Annihilation, Total Annihilation Kingdoms etc... I guess girls rely more on reflexion than us and our "me-see-me-kill" quake-likes )

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### #38digitaldirt  Members

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Posted 05 December 2001 - 03:05 AM

btw: "Chick-Flick" is just another ignorant guy term. Say a new movie is a love story, like "Serendipity"... oh no... another chick flick. Its not marketed as a chick flick. thats kinda just a slang term that has appeared by beer drinkin, football watching, guys... (I must point, those 2 examples are also stereotypes..)

The reason Fantasy and Fiction are so pleasing, is becuase they are "unknown" and that appeals to a lot of people, to discover a new land... If you did an RPG game based in a normal everyday society and the year is 2002... well... first off, try to create that world will ya... too big. In fantasy realms, the artists can really do whatever they want to do, making for a more intersting story, landscape and characters. as apposed to a suburban neighborhood and your main character is a skateboarder who has to steal homework from jimi''s house and work up the urge to ask a girl across the street to a dance. not too fun for guys or girls. just silly, like that powerranger pop-star idea. (not saying its bad, just not too fun.)

Space Quest and Kings Quest were even fun and they were both on the opposite ends of the RPG genre. excellent games though.

Btw: A genre of game that is equally enjoyed by both male and females is side-scroller games like super mario brothers, commander keen, and so on.

### #39Sturm  Members

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Posted 05 December 2001 - 04:28 AM

Okay I just want to add a little personal note on this one.

(I''m jusing man/woman here)

Okay I''ve tried to write something here, but came up short since it all sounds like it came from a beer drinkin, football watching, guy, and if these one thing I ain''t then its that (I don''t drink and sport is something you do not watch on TV, if I had one).

You say that Commander Keen is fun or at least enjoyable, but who want to make/play those any more. I mean they were great for their time but that''s more than a decade ago.

If I should create a game that was made with women as the target group, I would ask them what they want to play. After reading through the responces I would properly come to this conclution:

A multi path adaptive story, with a dynamically changing environment that focused on character interaction and nonviolence (Or at least without all the blood and body parts)conflicct solving, using realistic situations. (Actually sounds interesting)

Where as íf I asked a male group:

Monsters, Great FX, fast.

I could do the first thing, but that would take me like a decade to write that, or the second, which could be done in less than a year.

The problem isn''t that it can''t be done, it''s just that it''s too complex.

Well that was just my thoughts on that.

-- Sturm

P.S.
What is a good game for women, in your oppinion?

---------------------------------------------------
Life after death? No thanks, I want to live NOW
--- Sturm 2001

### #40digitaldirt  Members

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Posted 05 December 2001 - 05:35 AM

Yes I agree with you very much. the football/beer thing is a classic american depiction... thats all it is.

Commander Keen and Super Mario Brothers will always be fun games. The problem is, The big companies who make video games don''t see a point in making games like that, becuase even if you did, they wouldn''t sell for very much or even make the company money. You ask any new game company what they are making and no matter what genre, they all want "cut scenes"... those magical movie-like character building "cut scenes". And just by adding this little movie-isque feature into your game, you can sell it for atleast 39.99 upon initial release. Blizzard is perhaps one of the most successful companies at the whole cut-scene cinematic escapes.

But if you look at your options, everything has been done. I rack my brain for ideas of a new genre of game, and fall short. Ever since 3D came out, we''ve boxed ourselves in this reality that says... "ok... it has to be 3D, becuase its new and ''state of the art''". The whole 2D-sprites thing has also been done a million times over.

If you look at it in the whole picture of things. All industries are like this. They reach a peak in their development, where they cannot go any higher, just faster, more effects. It depends on the platform. We have yet to see a game that is fully rendered 3d action. I''m not talking about the texture, light, smoke and blood effects of the Quake III engine or Unreal2. I''m talking about serious movie quality 3D Rendering. Close to life, such as what ''Square Pictures'', did for that Final Fantasy flick.
Even that, could use a lot of improvement. But it depends on the hardware. If we all had Cray Systems or SGI setups, then we might stand a chance. But we''re stuck with our puny 2Ghz systems. hehe.

Anyway, I just wanted to further thank everyone who has been involved in this topic/discussion. It has been an interesting ride, and appreciate all the input. This posting gamedev.net community is awesome. Despite a few posters who post the outright dumbest things, this has been a wonderful community to be apart of.

Old topic!

Guest, the last post of this topic is over 60 days old and at this point you may not reply in this topic. If you wish to continue this conversation start a new topic.


text:
New member warned about not using the homework template
I want to solve this contour integral
$$J(\omega)= \frac{1}{2\pi}\frac{\gamma_i\lambda^2}{(\lambda^2+(\omega_i-\Delta-\omega)^2)}$$
$$N(\omega)=\frac{1}{e^{\frac{-\omega t}{T}}-1}$$

$$\int_0^\infty J(\omega)N(\omega)$$

there are three poles I don't know how I get rid of pole on zero (pole in N(w))
Thanks

## Answers and Replies

mfb
Mentor
Doesn't look like a convergent integral to me.

For small ω,
$$N(\omega)=\frac{1}{e^{\frac{-\omega t}{T}}-1} \approx \frac{-T}{\omega t}$$

While J(ω) quickly approaches some constant.

format: plain


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# Dark Sectors of Astroparticle Physics (AstroDark-2021): Axions, Neutrinos, Black Holes and Gravitational Waves

7-10 December 2021
Online
Asia/Tokyo timezone

## Dark Sector Freeze-out due to a Non-Boltzmann Suppression

7 Dec 2021, 08:20
30m
Online

#### Online

Details are notified to registrants.
Poster

### Speaker

Sougata Ganguly (Indian Association for the Cultivation of Science)

### Description

Commonly known as Boltzmann suppression is the key ingredient to create chemical imbalance for thermal dark matter. In a degenerate/quasi degenerate dark sector chemical imbalance can also be generated from a different mechanism which is analogous to the radioactive decay law, known as co-decaying dark matter. In this work, we have studied the dynamics of a multicomponent thermally decoupled degenerate dark sector in a hidden $U(1)_X$ extension of the Standard Model. We compute the relic density and the temperature ($T^\prime$) evolution of the hidden sector by considering all possible $2\rightarrow2$ and $3\rightarrow2$ processes. We find that the production of energetic particles from $3\rightarrow2$ processes increase the temperature of the dark sector whereas the rate of growth of temperature is decelerated due to the presence of $2\rightarrow2$ processes and expansion of the Universe. We also study the prospect of detecting neutrino and $gamma$-ray signals from DM annihilation via one step cascade processes. We find that in the present scenario, all the existing indirect detection constraints arising from measured fluxes of atmospheric neutrinos by Super-Kamiokande and diffuse $\gamma$-rays by EGRET, Fermi-LAT, and INTEGRAL respectively can easily be evaded for the degenerate dark sector. However for the quasi degenerate scenario the constraints are significant.

### Primary authors

Sougata Ganguly (Indian Association for the Cultivation of Science)


text:
## Computational complexity of evolutionary stable strategies

John Maynard Smith and John Forbes Nash, Jr.

Yesterday, I shared a video of John Maynard Smith introducing evolutionary game theory (EGT) to the London Mathematical Society. I suggested that at its foundation, EGT was just like classical game theory, and based on equilibrium analysis — the evolutionary stable strategy (Maynard Smith & Price, 1973). Given a utility function $U(\sigma,\sigma')$ that gives the expected payoff for mixed strategy $\sigma$ interacting with mixed strategy $\sigma$, we say that $\sigma^*$ is an evolutionary stable strategy (ESS) if:

1. For all $\sigma' \neq \sigma^*$ we have $\quad U(\sigma^*,\sigma^*) \geq U(\sigma',\sigma^*)$, and
2. If $U(\sigma^*,\sigma^*) = U(\sigma',\sigma^*)$ then $U(\sigma^*,\sigma') > U(\sigma',\sigma')$.

Or we look at the contrapositive: a strategy $\sigma'$ can invade a host strategy $\sigma$ if (1) it has a better payoff in a population of $\sigma$s than $\sigma$s have against themselves, or (2) if they are neutral then a small population of $\sigma'$ benefits themselves more than they benefit the host population. If some strategy $\sigma^*$ has no invading strategy $\sigma'$ then it is an evolutionary stable strategy.

If you are an observant reader and familiar with classical game theory then you probably noticed that the first condition of the direct definition is equivalent to the definition of a Nash equilibrium (NE). The second clause joined with an “and” means that every NE is an ESS, but not every ESS is an NE. However, that second condition seems innocuous enough, surely it can’t change the the qualitative nature of the equilibrium concept?
Read more of this post


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### Electronic Research Archive

2020, Issue 2: 795-805. doi: 10.3934/era.2020040

# On Seshadri constants and point-curve configurations

• Received: 01 February 2020 Revised: 01 April 2020
• 14C20, 14N10, 14N20

• In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.

Citation: Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations[J]. Electronic Research Archive, 2020, 28(2): 795-805. doi: 10.3934/era.2020040

### Related Papers:

• In the note we study the multipoint Seshadri constants of $\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ centered at singular loci of certain curve arrangements in the complex projective plane. Our first aim is to show that the values of Seshadri constants can be approximated with use of a combinatorial invariant which we call the configurational Seshadri constant. We study specific examples of point-curve configurations for which we provide actual values of the associated Seshadri constants. In particular, we provide an example based on Hesse point-conic configuration for which the associated Seshadri constant is computed by a line. This shows that multipoint Seshadri constants are not purely combinatorial.

 [1] Th. Bauer, Ł. Farnik, K. Hanumanthu and J. Huizenga, Mini-Workshop: Seshadri Constants, Oberwolfach Report No. $53/2019$, 2020. [2] Simplicial arrangements with up to 27 lines. Discrete Comput. Geom. (2012) 48: 682-701. [3] I. Dolgachev, A. Laface, U. Persson and G. Urzúa, Chilean configuration of conics, lines and points, Preprint. [4] F. Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and Geometry, Vol. II, Progr. Math., Birkhäuser, Boston, Mass., 36 (1983), 113–140. [5] D. Kohel, X. Roulleau and A. Sarti, A special configuration of 12 conics and generalized Kummer surfaces, Preprint. [6] Seshadri constants in a family of surfaces. Math. Ann. (2002) 323: 625-631. [7] Seshadri constants and special point configurations in the projective plane. Rocky Mountain J. Math. (2019) 49: 963-978. [8] Bounded negativity, Harbourne constants and transversal arrangements of curves. Ann. Inst. Fourier (Grenoble) (2017) 67: 2719-2735. [9] P. Pokora and T. Szemberg, Conic-line arrangements in the complex projective plane, arXiv: 2002.01760. [10] Remarks on the Nagata conjecture. Serdica Math. J. (2004) 30: 405-430.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.833 0.8


text:
# 12. Appendix: Markov Chain Monte Carlo¶

## 12.1. Monte Carlo Methods in Bayesian Analysis¶

Bayesian analysis often requires integration over multiple dimensions that is intractable both via analytic methods or standard methods of numerical integration. However, it is often possible to compute these integrals by simulating (drawing samples) from posterior distributions. For example, consider the expected value of a random variable $$\mathbf{x}$$:

$E[{\bf x}] = \int {\bf x} f({\bf x}) d{\bf x}, \qquad {\bf x} = \{x_1,...,x_k\}$

where $$k$$ (the dimension of vector $$x$$) is perhaps very large. If we can produce a reasonable number of random vectors $$\{{\bf x_i}\}$$, we can use these values to approximate the unknown integral. This process is known as Monte Carlo integration. In general, MC integration allows integrals against probability density functions:

$I = \int h(\mathbf{x}) f(\mathbf{x}) \mathbf{dx}$

to be estimated by finite sums:

$\hat{I} = \frac{1}{n}\sum_{i=1}^n h(\mathbf{x}_i),$

where $$\mathbf{x}_i$$ is a sample from $$f$$. This estimate is valid and useful because:

• By the strong law of large numbers:
$\hat{I} \rightarrow I \mbox{ with probability 1}$
• Simulation error can be measured and controlled:
$$Var(\hat{I}) = \frac{1}{n(n-1)}\sum_{i=1}^n (h(\mathbf{x}_i)-\hat{I})^2$$

Why is this relevant to Bayesian analysis? If we replace $$f(\mathbf{x})$$ with a posterior, $$f(\theta|d)$$ and make $$h(\theta)$$ an interesting function of the unknown parameter, the resulting expectation is that of the posterior of $$h(\theta)$$:

$$E[h(\theta)|d] = \int f(\theta|d) h(\theta) d\theta \approx \frac{1}{n}\sum_{i=1}^n h(\theta)$$

### 12.1.1. Rejection Sampling¶

Though Monte Carlo integration allows us to estimate integrals that are unassailable by analysis and standard numerical methods, it relies on the ability to draw samples from the posterior distribution. For known parametric forms, this is not a problem; probability integral transforms or bivariate techniques (e.g Box-Muller method) may be used to obtain samples from uniform pseudo-random variates generated from a computer. Often, however, we cannot readily generate random values from non-standard posteriors. In such instances, we can use rejection sampling to generate samples.

Posit a function, $$f(x)$$ which can be evaluated for any value on the support of $$x:S_x = [A,B]$$, but may not be integrable or easily sampled from. If we can calculate the maximum value of $$f(x)$$, we can then define a rectangle that is guaranteed to contain all possible values $$(x,f(x))$$. It is then trivial to generate points over the box and enumerate the values that fall under the curve (Figure Rejection sampling of a bounded form. Area is estimated by the ratio of accepted (open squares) to total points, multiplied by the rectangle area.).

Rejection sampling of a bounded form. Area is estimated by the ratio of accepted (open squares) to total points, multiplied by the rectangle area.

$\frac{\mbox{Points under curve}}{\mbox{Points generated}} \times \mbox{box area} = \lim_{n \to \infty} \int_A^B f(x) dx$

This approach is useful, for example, in estimating the normalizing constant for posterior distributions.

Rejection sampling of an unbounded form using an enveloping distribution.

If $$f(x)$$ has unbounded support (i.e. infinite tails), such as a Gaussian distribution, a bounding box is no longer appropriate. We must specify a majorizing (or, enveloping) function, $$g(x)$$, which implies:

$g(x) \ge f(x) \qquad\forall x \in (-\infty,\infty)$

Having done this, we can now sample $${x_i}$$ from $$g(x)$$ and accept or reject each of these values based upon $$f(x_i)$$. Specifically, for each draw $$x_i$$, we also draw a uniform random variate $$u_i$$ and accept $$x_i$$ if $$u_i < f(x_i)/cg(x_i)$$, where $$c$$ is a constant (Figure Rejection sampling of an unbounded form using an enveloping distribution.). This approach is made more efficient by choosing an enveloping distribution that is “close” to the target distribution, thus maximizing the number of accepted points. Further improvement is gained by using optimized algorithms such as importance sampling which, as the name implies, samples more frequently from important areas of the distribution.

Rejection sampling is usually subject to declining performance as the dimension of the parameter space increases, so it is used less frequently than MCMC for evaluation of posterior distributions [Gamerman1997].

## 12.2. Markov Chains¶

A Markov chain is a special type of stochastic process. The standard definition of a stochastic process is an ordered collection of random variables:

$\{X_t:t \in T\}$

where $$t$$ is frequently (but not necessarily) a time index. If we think of $$X_t$$ as a state $$X$$ at time $$t$$, and invoke the following dependence condition on each state:

$Pr(X_{t+1}=x_{t+1} | X_t=x_t, X_{t-1}=x_{t-1},\ldots,X_0=x_0) = Pr(X_{t+1}=x_{t+1} | X_t=x_t)$

then the stochastic process is known as a Markov chain. This conditioning specifies that the future depends on the current state, but not past states. Thus, the Markov chain wanders about the state space, remembering only where it has just been in the last time step. The collection of transition probabilities is sometimes called a transition matrix when dealing with discrete states, or more generally, a transition kernel.

In the context of Markov chain Monte Carlo, it is useful to think of the Markovian property as “mild non-independence”. MCMC allows us to indirectly generate independent samples from a particular posterior distribution.

### 12.2.1. Jargon-busting¶

Before we move on, it is important to define some general properties of Markov chains. They are frequently encountered in the MCMC literature, and some will help us decide whether MCMC is producing a useful sample from the posterior.

• Homogeneity:

A Markov chain is homogeneous at step $$t$$ if the transition probabilities are independent of time $$t$$.

• Irreducibility:

A Markov chain is irreducible if every state is accessible in one or more steps from any other state. That is, the chain contains no absorbing states. This implies that there is a non-zero probability of eventually reaching state $$k$$ from any other state in the chain.

• Recurrence:

States which are visited repeatedly are recurrent. If the expected time to return to a particular state is bounded, this is known as positive recurrence, otherwise the recurrent state is null recurrent. Further, a chain is Harris recurrent when it visits all states $$X \in S$$ infinitely often in the limit as $$t \to \infty$$; this is an important characteristic when dealing with unbounded, continuous state spaces. Whenever a chain ends up in a closed, irreducible set of Harris recurrent states, it stays there forever and visits every state with probability one.

• Stationarity:

A stationary Markov chain produces the same marginal distribution when multiplied by the transition kernel. Thus, if $$P$$ is some $$n \times n$$ transition matrix:

${\bf \pi P} = {\bf \pi}$

for Markov chain $$\pi$$. Thus, $$\pi$$ is no longer subscripted, and is referred to as the limiting distribution of the chain. In MCMC, the chain explores the state space according to its limiting marginal distribution.

• Ergodicity:

Ergodicity is an emergent property of Markov chains which are irreducible, positive Harris recurrent and aperiodic. Ergodicity is defined as:

$\lim_{n \to \infty} Pr^{(n)}(\theta_i \rightarrow \theta_j) = \pi(\theta) \quad \forall \theta_i, \theta_j \in \Theta$

or in words, after many steps the marginal distribution of the chain is the same at one step as at all other steps. This implies that our Markov chain, which we recall is dependent, can generate samples that are independent if we wait long enough between samples. If it means anything to you, ergodicity is the analogue of the strong law of large numbers for Markov chains. For example, take values $$\theta_{i+1},\ldots,\theta_{i+n}$$ from a chain that has reached an ergodic state. A statistic of interest can then be estimated by:

$\frac{1}{n}\sum_{j=i+1}^{i+n} h(\theta_j) \approx \int f(\theta) h(\theta) d\theta$

## 12.3. Why MCMC Works: Reversible Markov Chains¶

Markov chain Monte Carlo simulates a Markov chain for which some function of interest (e.g. the joint distribution of the parameters of some model) is the unique, invariant limiting distribution. An invariant distribution with respect to some Markov chain with transition kernel $$Pr(y \mid x)$$ implies that:

$\int_x Pr(y \mid x) \pi(x) dx = \pi(y).$

Invariance is guaranteed for any reversible Markov chain. Consider a Markov chain in reverse sequence: $$\{\theta^{(n)},\theta^{(n-1)},...,\theta^{(0)}\}$$. This sequence is still Markovian, because:

$Pr(\theta^{(k)}=y \mid \theta^{(k+1)}=x,\theta^{(k+2)}=x_1,\ldots ) = Pr(\theta^{(k)}=y \mid \theta^{(k+1)}=x)$

Forward and reverse transition probabilities may be related through Bayes theorem:

$\frac{Pr(\theta^{(k+1)}=x \mid \theta^{(k)}=y) \pi^{(k)}(y)}{\pi^{(k+1)}(x)}$

Though not homogeneous in general, $$\pi$$ becomes homogeneous if Do you ever call the stationary distribution itself homogeneous?:

• $$n \rightarrow \infty$$
• $$\pi^{(i)}=\pi$$ for some $$i < k$$

If this chain is homogeneous it is called reversible, because it satisfies the detailed balance equation:

$\pi(x)Pr(y \mid x) = \pi(y) Pr(x \mid y)$

Reversibility is important because it has the effect of balancing movement through the entire state space. When a Markov chain is reversible, $$\pi$$ is the unique, invariant, stationary distribution of that chain. Hence, if $$\pi$$ is of interest, we need only find the reversible Markov chain for which $$\pi$$ is the limiting distribution. This is what MCMC does!

## 12.4. Gibbs Sampling¶

The Gibbs sampler is the simplest and most prevalent MCMC algorithm. If a posterior has $$k$$ parameters to be estimated, we may condition each parameter on current values of the other $$k-1$$ parameters, and sample from the resultant distributional form (usually easier), and repeat this operation on the other parameters in turn. This procedure generates samples from the posterior distribution. Note that we have now combined Markov chains (conditional independence) and Monte Carlo techniques (estimation by simulation) to yield Markov chain Monte Carlo.

Here is a stereotypical Gibbs sampling algorithm:

As we can see from the algorithm, each distribution is conditioned on the last iteration of its chain values, constituting a Markov chain as advertised. The Gibbs sampler has all of the important properties outlined in the previous section: it is aperiodic, homogeneous and ergodic. Once the sampler converges, all subsequent samples are from the target distribution. This convergence occurs at a geometric rate.

1. Choose starting values for states (parameters): $${\bf \theta} = [\theta_1^{(0)},\theta_2^{(0)},\ldots,\theta_k^{(0)}]$$

2. Initialize counter $$j=1$$

3. Draw the following values from each of the $$k$$ conditional distributions:

\begin{eqnarray*} \theta_1^{(j)} &\sim& \pi(\theta_1 | \theta_2^{(j-1)},\theta_3^{(j-1)},\ldots,\theta_{k-1}^{(j-1)},\theta_k^{(j-1)}) \\ \theta_2^{(j)} &\sim& \pi(\theta_2 | \theta_1^{(j)},\theta_3^{(j-1)},\ldots,\theta_{k-1}^{(j-1)},\theta_k^{(j-1)}) \\ \theta_3^{(j)} &\sim& \pi(\theta_3 | \theta_1^{(j)},\theta_2^{(j)},\ldots,\theta_{k-1}^{(j-1)},\theta_k^{(j-1)}) \\ \vdots \\ \theta_{k-1}^{(j)} &\sim& \pi(\theta_{k-1} | \theta_1^{(j)},\theta_2^{(j)},\ldots,\theta_{k-2}^{(j)},\theta_k^{(j-1)}) \\ \theta_k^{(j)} &\sim& \pi(\theta_k | \theta_1^{(j)},\theta_2^{(j)},\theta_4^{(j)},\ldots,\theta_{k-2}^{(j)},\theta_{k-1}^{(j)}) \end{eqnarray*}
4. Increment $$j$$ and repeat until convergence occurs.

## 12.5. The Metropolis-Hastings Algorithm¶

The key to success in applying the Gibbs sampler to the estimation of Bayesian posteriors is being able to specify the form of the complete conditionals of $${\bf \theta}$$. In fact, the algorithm cannot be implemented without them. Of course, the posterior conditionals cannot always be neatly specified. In contrast to the Gibbs algorithm, the Metropolis-Hastings algorithm generates candidate state transitions from an alternate distribution, and accepts or rejects each candidate probabilistically.

Let us first consider a simple Metropolis-Hastings algorithm for a single parameter, $$\theta$$. We will use a standard sampling distribution, referred to as the proposal distribution, to produce candidate variables $$q_t(\theta^{\prime} | \theta)$$. That is, the generated value, $$\theta^{\prime}$$, is a possible next value for $$\theta$$ at step $$t+1$$. We also need to be able to calculate the probability of moving back to the original value from the candidate, or $$q_t(\theta | \theta^{\prime})$$. These probabilistic ingredients are used to define an acceptance ratio:

$a(\theta^{\prime},\theta) = \frac{q_t(\theta^{\prime} | \theta) \pi(\theta^{\prime})}{q_t(\theta | \theta^{\prime}) \pi(\theta)}$

The value of $$\theta^{(t+1)}$$ is then determined by:

$\begin{split}\theta^{(t+1)} = \left\{\begin{array}{l@{\quad \mbox{with prob.} \quad}l}\theta^{\prime} & \min(a(\theta^{\prime},\theta),1) \\ \theta^{(t)} & 1 - \min(a(\theta^{\prime},\theta),1) \end{array}\right.\end{split}$

This transition kernel implies that movement is not guaranteed at every step. It only occurs if the suggested transition is likely based on the acceptance ratio.

A single iteration of the Metropolis-Hastings algorithm proceeds as follows:

The original form of the algorithm specified by Metropolis required that $$q_t(\theta^{\prime} | \theta) = q_t(\theta | \theta^{\prime})$$, which reduces $$a(\theta^{\prime},\theta)$$ to $$\pi(\theta^{\prime})/\pi(\theta)$$, but this is not necessary. In either case, the state moves to high-density points in the distribution with high probability, and to low-density points with low probability. After convergence, the Metropolis-Hastings algorithm describes the full target posterior density, so all points are recurrent.

1. Sample $$\theta^{\prime}$$ from $$q(\theta^{\prime} | \theta^{(t)})$$.
2. Generate a Uniform[0,1] random variate $$u$$.
3. If $$a(\theta^{\prime},\theta) > u$$ then $$\theta^{(t+1)} = \theta^{\prime}$$, otherwise $$\theta^{(t+1)} = \theta^{(t)}$$.

### 12.5.1. Random-walk Metropolis-Hastings¶

A practical implementation of the Metropolis-Hastings algorithm makes use of a random-walk proposal. Recall that a random walk is a Markov chain that evolves according to:

\begin{eqnarray*} \theta^{(t+1)} &=& \theta^{(t)} + \epsilon_t \\ \epsilon_t &\sim& f(\phi) \end{eqnarray*}

As applied to the MCMC sampling, the random walk is used as a proposal distribution, whereby dependent proposals are generated according to:

$q(\theta^{\prime} | \theta^{(t)}) = f(\theta^{\prime} - \theta^{(t)}) = \theta^{(t)} + \epsilon_t$

Generally, the density generating $$\epsilon_t$$ is symmetric about zero, resulting in a symmetric chain. Chain symmetry implies that $$q(\theta^{\prime} | \theta^{(t)}) = q(\theta^{(t)} | \theta^{\prime})$$, which reduces the Metropolis-Hastings acceptance ratio to:

$a(\theta^{\prime},\theta) = \frac{\pi(\theta^{\prime})}{\pi(\theta)}$

The choice of the random walk distribution for $$\epsilon_t$$ is frequently a normal or Student’s $$t$$ density, but it may be any distribution that generates an irreducible proposal chain.

An important consideration is the specification of the scale parameter for the random walk error distribution. Large values produce random walk steps that are highly exploratory, but tend to produce proposal values in the tails of the target distribution, potentially resulting in very small acceptance rates. Conversely, small values tend to be accepted more frequently, since they tend to produce proposals close to the current parameter value, but may result in chains that mix very slowly. Some simulation studies suggest optimal acceptance rates in the range of 20-50%. It is often worthwhile to optimize the proposal variance by iteratively adjusting its value, according to observed acceptance rates early in the MCMC simulation [Gamerman1997].


text:
A Combined NNLO Lattice-Continuum Determination of $L_{10}^r$

Research output: Contribution to journalArticle

Open

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Original language English Physical Review D 89 094510 10.1103/PhysRevD.89.094510 Published - 29 May 2014

Abstract

The renormalized next-to-leading-order (NLO) chiral low-energy constant, $L_{10}^r$, is determined in a complete next-to-next-to-leading-order (NNLO) analysis, using a combination of lattice and continuum data for the flavor $ud$ $V-A$ correlator and results from a recent chiral sum-rule analysis of the flavor-breaking combination of $ud$ and $us$ $V-A$ correlator differences. The analysis also fixes two combinations of NNLO low-energy constants, the determination of which is crucial to the precision achieved for $L_{10}^r$. Using the results of the flavor-breaking chiral $V-A$ sum rule obtained with current versions of the strange hadronic $\tau$ branching fractions as input, we find $L_{10}^r(m_\rho )\, =\, -0.00346(32)$. This result represents the first NNLO determination of $L_{10}^r$ having all inputs under full theoretical and/or experimental control, and the best current precision for this quantity.

Research areas

• hep-ph, hep-ex, hep-lat

No data available

ID: 17197402


text:
## Sunday, December 23, 2012

### Nimbers, the numbers for Nim game

If you are in Hong Kong and you need help for university mathematics courses, please visit www.all-r-math.com.

In previous posts, we have introduced some simple games. One of them is $G'(n)$, which consists a board with n pieces, two players in turn can take any number of pieces from the block, the winner is the one who takes the last piece.

Looks like a stupid game, right? The first player can take all pieces at his first step and win.

Not so easy now. It is the famous Nim game. Some says that Nim originated from China, but no one can sure.

 ** ****** *****
The Nim game G'(2)+G'(6)+G'(5) consists of three heaps, each time a player can take away any number of pieces from a heap, the objective is to take the last piece.

To solve Nim, mathematicians have introduced the concept of nimber, which is basically ordinal numbers represented in binary numeral system under the position-wise boolean addition (i.e. 0+0=1+1=0, 0+1=1+0=1).

For example 3+7=(011)+(111)=(100)=4, 5+11=(0101)+(1011)=(1010)=10.

There is also nimber multiplication, which is omitted here.

Before we get into the relation between Nim and nimber, lets investigate the properties of nimber addition:
1. n+m=0 if and only if n=m
2. If n has a 1 at the 2k-th position and m is a k-digit nimber, then n+m<n. (The proof is easy: (1...1...)+ (1...)=(1...0...)<(1...1...). )
3. If the nim sum of several nimbers is a k-digit nimber, then at least one nimber has a 1 at the 2k-th position.
4. If the nim sum of several nimbers is 0, then replace any one of them by a different nonzero nimber, the new nim sum is nonzero. (Proof: Given n1+n2+...+nr=0. Suppose we replace the first nimber by m, then m+n2+...+nr=m+n1+n1+n2+...+nr=m+n1≠0.)
Now we are ready to study Nim.

For each heap, there are a nimber of pieces. Take the nim sum of the nimbers.
 ** ****** *****
The Nim sum of G'(2)+G'(6)+G'(5)is 2+6+5=(010)+(110)+(101)=(001).

The objective of the game is to take away all the pieces. At that last moment, the number of pieces is 0. From Property (4), if the nim sum is already 0, take away any number of pieces from a heap cannot keep the nim sum be 0, and therefore cannot win the game.

What if the nim sum, say m is a k-digit nimber?
From Property (3), there is a heap of which the nimber, say n, has a 1 at the 2k-th position. From Property (2), n+m<n. Take away some pieces fromt that heap so that there are n+m pieces left. The nim sum becomes
nim sum of all other heaps+(n+m)=(m-n)+(n+m)=m+m=0.

Therefore, the winning strategy is to maintain the nim sum to be 0. If the nim sum at the beginning is 0, the second player wins; if the nim sum at the beginning is nonzero, the first player wins.

Consider the game G'(2)+G'(6)+G'(5).
2=(10), 6=(110), 5=(101).
2+6+4=(10)+(110)+(100)=(001).
Only 5 have a 1 at the 1-th position. (101)+(001)=(100)=4, so the player has to left 4 in the last heap.

The game becomes G'(2)+G'(6)+G'(4).
The nim sum is 0. No matter the opponent does, he will disturb the balance. Suppose he takes away 3 pieces from the second heap.

The game becomes G'(2)+G'(3)+G'(4).
The nim sum is (101). The player now leaves (100)+(101)=(001)=1 pieces in the last heap.

The game becomes G'(2)+G'(3)+G'(1).

I suppose the readers can figure out the rest.


text: - The Difference Between 5x5 Doubly Nonnegative and Completely Positive Matrices Samuel Burer (samuel-bureruiowa.edu) Kurt M. Anstreicher (kurt-anstreicheruiowa.edu) Mirjam Duer (M.E.Durrug.nl) Abstract: The convex cone of $n \times n$ completely positive (CPP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CPP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for $n \le 4$ only, every DNN matrix is CPP. In this paper, we investigate the difference between $5\times 5$ DNN and CPP matrices. Defining a {\em bad\/} matrix to be one which is DNN but not CPP, we: (i) characterize all $5 \times 5$ extreme DNN matrices, in particular bad ones; (ii) design a finite procedure to decompose any $n \times n$ DNN matrix into the sum of a CPP matrix and a bad matrix, which itself cannot be further decomposed; (iii) show that every bad $5 \times 5$ DNN matrix is the sum of a CPP matrix and a single bad extreme matrix; and (iv) demonstrate how to separate bad extreme matrices from the cone of $5 \times 5$ CPP matrices. Keywords: completely positive matrices, doubly nonnegative matrices, copositive matrices Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Linear, Cone and Semidefinite Programming (Semi-definite Programming ) Category 3: Global Optimization (Theory ) Citation: Linear Algebra and its Applications 431 (2009), 1539-1552. Download: [PDF]Entry Submitted: 05/27/2008Entry Accepted: 05/27/2008Entry Last Modified: 08/17/2009Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.


text:
## How do you change font settings of labels in the "Group X" section in the graph builder?

Community Trekker

Joined:

May 8, 2013

I have several categories in the group X (and group Y) sections, in the graph builder. And I'd like to decrease the size of the font for these so that the text fits better. Is there any way to do this? I've checked the documentation and it indicates how to change the font of axis labels, but not group labels. Thanks in advance.

1 ACCEPTED SOLUTION

Accepted Solutions

Super User

Joined:

Jun 23, 2011

Solution

I have not found a specific way to adjust the font size of the group labels. Label size seems to be controlled by the the text font preferences, i.e. they can be decreased by choosing a smaller size in preferences/fonts/text. But that will affect the label size in other reports as well as some other elements, legend text size for example. Although more general than desired, this preference has more restricted effects than the global shortcut (cmd+- or Make Text Smaller in the View Menu). For a particular graph it still can be useful.

Preferences can also be changed with jsl. Here's an example how to decreases size while retaining font type.

// Run first two lines once

p_expr = Expr( Get Preferences( text font ) );

p_start = p_expr;

// Repeat following two lines until font size of group labels are OK

p = p_expr;

Preferences( text font( Arg( Arg( p, 1 ), 1 ), Arg( Arg( p, 1 ), 2 ) - 1 ) );

// Run this to reset

Eval( p_start );

2 REPLIES

Super User

Joined:

Jun 23, 2011

Solution

I have not found a specific way to adjust the font size of the group labels. Label size seems to be controlled by the the text font preferences, i.e. they can be decreased by choosing a smaller size in preferences/fonts/text. But that will affect the label size in other reports as well as some other elements, legend text size for example. Although more general than desired, this preference has more restricted effects than the global shortcut (cmd+- or Make Text Smaller in the View Menu). For a particular graph it still can be useful.

Preferences can also be changed with jsl. Here's an example how to decreases size while retaining font type.

// Run first two lines once

p_expr = Expr( Get Preferences( text font ) );

p_start = p_expr;

// Repeat following two lines until font size of group labels are OK

p = p_expr;

Preferences( text font( Arg( Arg( p, 1 ), 1 ), Arg( Arg( p, 1 ), 2 ) - 1 ) );

// Run this to reset

Eval( p_start );

Community Trekker

Joined:

May 8, 2013

Thanks very much. That did the trick in this case. Fortunately, I also needed to decrease the size of the text in the Group Y. But too bad the two can't be specified independently.


text:
# Triangle TBC

TBC is isosceles triangle with base TB with base angle 63° and legs length |TC| = |BC| = 25. How long is the base TB?

Result

|TB| =  22.7

#### Solution:

$\cos 63 ^\circ = \dfrac{|TB|/2}{ |TC|} \ \\ \ \\ |TB| = 2 \cdot 25 \cdot \cos(63 ^\circ ) = 22.7$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

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Tips to related online calculators

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text:
# Opening Portals

## 题目描述

Pavel plays a famous computer game. A player is responsible for a whole country and he can travel there freely, complete quests and earn experience.
This country has $n$ cities connected by $m$ bidirectional roads of different lengths so that it is possible to get from any city to any other one. There are portals in $k$ of these cities. At the beginning of the game all portals are closed. When a player visits a portal city, the portal opens. Strange as it is, one can teleport from an open portal to an open one. The teleportation takes no time and that enables the player to travel quickly between rather remote regions of the country.
At the beginning of the game Pavel is in city number $1$. He wants to open all portals as quickly as possible. How much time will he need for that?

## 代码

#include <iostream>
#include <algorithm>
#include <cstring>
#include <queue>
using namespace std;
long long n, m, k, nume, tot, ans, mi = 1e18;
long long h[100001], p[100001], d[100001], fa[100001], pre[100001];
bool in[100001];
struct edge { long long v, w, nxt; } e[200001];
struct line {
long long u, v, w;
bool operator < (const line &a) const {
return d[u] + w + d[v] < d[a.u] + a.w + d[a.v];
}
} l[200001];
void add_edge(long long u, long long v, long long w) {
e[++nume].v = v, e[nume].w = w, e[nume].nxt = h[u], h[u] = nume;
e[++nume].v = u, e[nume].w = w, e[nume].nxt = h[v], h[v] = nume;
}
void spfa() {
queue<long long> que;
while (!que.empty()) que.pop();
for (int i = 1; i <= n; ++i)
if (!d[i]) in[i] = true, que.push(i);
while (!que.empty()) {
int u = que.front(); in[u] = false, que.pop();
for (int i = h[u]; i; i = e[i].nxt)
if (d[e[i].v] > d[u] + e[i].w) {
d[e[i].v] = d[u] + e[i].w, pre[e[i].v] = pre[u];
if (!in[e[i].v]) in[e[i].v] = true, que.push(e[i].v);
}
}
}
long long get_fa(long long t) {
return t == fa[t] ? t : fa[t] = get_fa(fa[t]);
}
int main()
{
cin >> n >> m;
for (int i = 1; i <= m; ++i) {
long long u, v, w; cin >> u >> v >> w; add_edge(u, v, w);
}
memset(d, 0x1f, sizeof d), d[1] = 0, spfa(); cin >> k;
for (int i = 1; i <= k; ++i) {
cin >> p[i]; if (d[p[i]] < mi) mi = d[p[i]];
}
ans = mi, memset(d, 0x1f, sizeof d);
for (int i = 1; i <= n; ++i) fa[i] = i, pre[i] = i;
for (int i = 1; i <= k; ++i) d[p[i]] = 0; spfa();
for (int i = 1; i <= n; ++i)
for (int j = h[i]; j; j = e[j].nxt)
l[++tot].u = i, l[tot].v = e[j].v, l[tot].w = e[j].w;
sort(l + 1, l + tot + 1);
for (int i = 1; i <= tot; ++i) {
long long u = get_fa(pre[l[i].u]), v = get_fa(pre[l[i].v]);
if (u != v) fa[u] = v, ans += d[l[i].u] + l[i].w + d[l[i].v];
}
cout << ans << endl;
return 0;
}

418 I'm a teapot


text:
# question about mixed spectrum of a linear operator $\mathcal{L}$

Suppose $\mathcal{L}$ is a bounded linear operator and I have the solution to Eigenvalue problem

$\mathcal{L} \phi + \lambda \phi = 0$

wish to solve the following PDE

$\left(-\partial_t + \mathcal{L}\right)u = 0$.

If the spectrum of $\mathcal{L}$ is continuous or discrete, then a general solution to the PDE is

$\int C_q e^{- \lambda_q t} \phi_q dq$

or

$\sum_q C_q e^{- \lambda_q t} \phi_q$,

where the $C$'s are constants.

But, what if the spectrum of $\mathcal{L}$ is mixed and has a continuous part, a discrete part, and a singular part? Is there a general way to write the solution to the above PDE if I do not know the spectrum of $\mathcal{L}$?

This has come up in my research because I need to work with the $e^{- \lambda_q t}$ but I do not know what the spectrum of $\mathcal{L}$?

-

Well, I have to confess that I am not sure about your goals. In which space are you working? But there is always a way to represent the solution as a generazied exponential function, called operator semigroup. If your operator $\mathcal L$ is indeed bounded, then you can represent the solution via Dunford-Riesz calculus. A good reference for this and much more general cases is Engel-Nagel: A short course on operator semigroups, Springer.
I'll check out the "short course on operator semigroups" as you suggested. In the mean-time, I think I have found an answer to my quandary (on wikipedia embarrassingly). I can just write my solution as $\int e^{-\lambda t} \phi_\lambda \mu( d\lambda )$ where $\mu$ is the spectral measure. At least, I think this works (I'm a physicist ... we aren't normally taught things like this in our coursework--at least I never was--so we need to learn this as we go in research). In any case, thank you very much for your reply to my question. –  psyduck Feb 15 '11 at 1:42


text:
# Question #2000450: Other Calculus Problems

Question: A study indicates that $$x$$ months from now the population of a certain town will be increasing at the rate of $$5+3{{x}^{2/3}}$$ people per month. By how much will the population of the town increase over the next 8 months?

Solution: The solution consists of 83 words (1 page)
Deliverables: Word Document

0


text:
# Non-linear, non-convex optimization problem

I would like to solve the following optimization problem:

$$\max\{\min_{k=1,2,...,m}\{R_d^k\}+\sum_{k=1}^m R_e^k\} \\\text{s.t.} \\ \mathrm{trace}(W_d^kW_d^{kT})=1 \\ \mathrm{trace}(W_e^kW_e^{kT})=1 \\0<P_d^k<P_d^{Max} \\0<P_e^k<P_e^{Max}$$

where

• $R_d^k$ and $R_e^k$ are both non-linear, non-convex functions of variables $W_d^k$, $W_e^k$, $P_d^k$, and $P_e^k$,
• $W_d^k$ and $W_e^k$ are 2-by-2 matrices, and
• $P_d^k$ and $P_e^k$ are scalar values.

What method can I use to solve this problem?

-


text:
# 10th Class Mathematics Pair of Linear Equations in Two Variables Pair of Linear Equations In Two Variables

Pair of Linear Equations In Two Variables

Category : 10th Class

Pair of Linear Equations in Two Variables

• Linear equation in two variables: An equation of the form$\text{ax}+\text{by}=\text{c}$, where $a\ne 0,b\ne 0$ and a, b and c are real numbers is known as a linear equation in two variables x and y.
• A pair of linear equations in two variables: Two linear equations in the same two variables are called a pair of linear equations in two variables.
• General form of a pair of linear equations in two variables: The general form of a pair of linear equations in two variables is ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$, where ${{a}_{1}},{{a}_{2}},{{b}_{1}},{{b}_{2}},{{c}_{1}}$and${{c}_{2}}$are real numbers such that $a_{1}^{2}+b_{1}^{2}\ne 0$and$a_{2}^{2}+b_{2}^{2}\ne 0$.

Note: A pair of linear equations in two variables is called a system of simultaneous linear equations.

• Solution of a pair of linear equations in two variables: A pair of values (x, y) that satisfies both the linear equations is called a solution of the system of simultaneous equations.

Note: (i) A pair of values (x, y) that satisfies an equation is called its solution. (ii)  Every linear equation in two variables has an infinite number of solutions.

(iii) Every solution of a linear equation is a point on the line representing it.

• Methods of solving a pair of linear equations in two variables: A pair of linear equations in two variables can be solved by (i) Graphical method (ii) Algebraic method.

(i) Graphical method: The graph of a linear equation is a straight line. The graph of a pair of linear equations in two variables is represented by two lines.

(a) If the two lines intersect at a point, then the pair of linear equations has a unique solution (the point) and is said to be consistent.

(b) If the two lines coincide, then the pair of linear equations has infinitely many solutions (each point on the line being a solution), and is said to be dependent or consistent.

(c) If the two lines are parallel, then the pair of linear equations has no solution (no common point) and is said to be inconsistent.

In other words, there are three types of solutions of a pair of linear equations in two variables:

(a) Unique solution

(b) Infinitely many solutions

(c) No solution.

Note: Graphical method does not give an accurate answer as error is likely to occur in reading the coordinates of a point.

(ii) Algebraic methods: To obtain accurate result of solution of simultaneous linear equations, algebraic methods are used.

There are three algebraic methods to learn for this year.

(a) Method of elimination by substitution.

(b) Method of elimination by equating the coefficients.

(c) Method of cross multiplication.

• Conditions for solvability of a pair of linear equations in two variables:

Consider the pair of linear equations ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0;\,{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$

 Conditions Types of lines No. of Solutions Type of equation $\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}$ Intersecting lines 1 (Unique solution) Consistent $\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}$ Coincident lines Infinitely many Dependent (or) consistent $\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}$ Parallel lines No Solution Inconsistent

• Homogeneous equations: The system of equations ${{a}_{1}}x+{{b}_{1}}y=0$ and$\,{{a}_{2}}x+{{b}_{2}}y=0$is called the system of homogeneous equations, and has only one solution x = 0 and y = 0. When $\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}}$, the system of homogeneous equations has only one solution and the system is consistent.

When$\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}$the system of homogeneous equations has infinitely many solutions and the system is consistent.

• Some equations are not linear. But they can be reduced to linear form by making some suitable substitutions and solving them using any convenient method.

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text:
# Terminology for drill component?

I built/assembled a small drill press recently and used a part from a Junk Drill because I couldn't figure out what a particular component I needed was called.

What I was looking for was a way to step up in chuck size so I could accommodate a bigger bit on a rod that was a smaller size (see: 3.17mm motor shaft driving 1/4"+ bits).

I now know the function of what I was looking for, it's a chamber of gears driven by a central D shaft gear the motor shaft can be inserted into. It's the internal component used to drive keyless chucks available in most retail drills.

Does anyone know what this would be called so I could look for it properly next time? I resorted to stripping an existing finished product for a part just because I couldn't figure it out, I was originally wanting to mount a keyed chuck

• I don't know if this question qualifies for EE.SE. You might be better off in DIY. However, I think it is not the chuck, but you've taken the planetery gearbox*(full of planetary gears, may be called orbital gears instead) and the *torque limiter from a battery operated drill, so it seems likely one of those words will help you. If I may ask, is there a reason you've gone this route instead of buying a cheap drill press? Mine isn't industrial grade, but it's more robust than the one in your picture and only cost $40 CAD on sale. – K H Mar 13, 2019 at 4:28 • @KH THAT'S EXACTLY IT. I recently got into electrical engineering as a hobby and realized I need to be able to make cases, so to better my understanding I wanted to build a drill rather than buying one. It had a few small purposes but I just wanted to learn and be flexible. I very well might end up swapping my current one out with one of the more preferred gearboxes to get more height to work with on the press. Mar 13, 2019 at 4:32 • K I realised that qualified as an answer so I made it one. – K H Mar 13, 2019 at 4:33 ## 2 Answers I think it is not the chuck you want, but you've taken the planetery gearbox(full of planetary gears, may be called orbital gears instead) and the torque limiter from a battery operated drill, so it seems likely one of those words will help you. If I may ask, is there a reason you've gone this route instead of buying a cheap drill press? Mine isn't industrial grade, but it's more robust than the one in your picture and only cost$40 CAD on sale.

Also be advised that if you want the torque limiting function, broken battery operated hand tools can often be found cheaply on second hand sources, so you could take the gearbox from a higher quality hand drill and put yours back in service. If you go shopping for broken drills I'd go for Milwaukee(Extremely well documented, high quality) if you can find it.

Chuck

Some filler text because SE won't allow me to post fewer than 30 characters!

• generally just looking for a chuck will restrict me to options that only accommodate bits as large as the shaft they're mounted to. Looking for 3.17mm chucks return results for chucks that support up to 3/16, looking for larger chucks do not return chucks that can be mounted on a smaller motor shaft. I believe the narrower term I'm looking for is what's inside the chuck - though I could be wrong and it's one of those things that just is the way it is
– Sani Evans
Mar 13, 2019 at 4:21


text:
<meta http-equiv="refresh" content="1; url=/nojavascript/"> The Kruskal-Wallis Test and the Runs Test | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: CK-12 Probability and Statistics - Advanced (Second Edition) Go to the latest version.

# 12.3: The Kruskal-Wallis Test and the Runs Test

Created by: CK-12

## Learning Objectives

• Evaluate a hypothesis for several populations that are not normally distributed using multiple randomly selected independent samples with the Kruskal-Wallis Test.
• Determine the randomness of a sample using the runs test to access the number of data sequences and compute a test statistic using the appropriate formula.

## Introduction

In the previous sections, we learned how to conduct nonparametric tests, including the sign test, the sign rank test, the rank sum test, and the rank correlation test. These tests allowed us to test hypotheses using data that did not meet the assumptions of being normally distributed or having homogeneity with respect to variance. In addition, each of these non-parametric tests had parametric counterparts.

In this last section, we will examine another nonparametric test$-$the Kruskal-Wallis one-way analysis of variance (also known simply as the Kruskal-Wallis test). This test is similar to the ANOVA test, and the calculation of the test statistic is similar to that of the rank sum test. In addition, we will also explore something known as the runs test, which can be used to help decide if sequences observed within a data set are random.

### Evaluating Hypotheses Using the Kruskal-Wallis Test

The Kruskal-Wallis test is the analog of the one-way ANOVA and is used when our data set does not meet the assumptions of normality or homogeneity of variance. However, this test has its own requirements: it is essential that the data set has identically shaped and scaled distributions for each group.

As we learned in Chapter 11, when performing the one-way ANOVA test, we establish the null hypothesis that there is no difference between the means of the populations from which our samples were selected. However, we express the null hypothesis in more general terms when using the Kruskal-Wallis test. In this test, we state that there is no difference in the distributions of scores of the populations. Another way of stating this null hypothesis is that the average of the ranks of the random samples is expected to be the same.

The test statistic for this test is the non-parametric alternative to the $F$-statistic. This test statistic is defined by the following formula:

$H=\frac{12}{N(N+1)} \sum^m_{k=1} \frac{R^2_k}{n_k}-3(N+1)$

where:

$N=\sum n_k$.

$n_k$ is number of observations in the $k^{\text{th}}$ sample.

$R_k$ is the sum of the ranks in the $k^{\text{th}}$ sample.

$m$ is the number of samples.

Like most nonparametric tests, the Kruskal-Wallis test relies on the use of ranked data to calculate a test statistic. In this test, the measurement observations from all the samples are converted to their ranks in the overall data set. The smallest observation is assigned a rank of 1, the next smallest is assigned a rank of 2, and so on. Similar to this procedure in the rank sum test, if two observations have the same value, we assign both of them the same rank.

Once the observations in all of the samples are converted to ranks, we calculate the test statistic, $H$, using the ranks and not the observations themselves. Similar to the other parametric and non-parametric tests, we use the test statistic to evaluate our hypothesis. For this test, the sampling distribution for $H$ is the chi-square distribution with $m-1$ degrees of freedom, where $m$ is the number of samples.

It is easy to use Microsoft Excel or a statistical programming package, such as SAS or SPSS, to calculate this test statistic and evaluate our hypothesis. However, for the purposes of this example, we will perform this test by hand.

Example: Suppose that a principal is interested in the differences among final exam scores from Mr. Red, Ms. White, and Mrs. Blue’s algebra classes. The principal takes random samples of students from each of these classes and records their final exam scores as shown:

Mr. Red Ms. White Mrs. Blue
52 66 63
46 49 65
62 64 58
48 53 70
57 68 71
54 73

Determine if there is a difference between the final exam scores of the three teachers.

Our hypothesis for the Kruskal-Wallis test is that there is no difference in the distributions of the scores of these three populations. Our alternative hypothesis is that at least two of the three populations differ. For this example, we will set our level of significance at $\alpha=0.05$.

To test this hypothesis, we need to calculate our test statistic. To calculate this statistic, it is necessary to assign and sum the ranks for each of the scores in the table above as follows:

Mr. Red Overall Rank Ms. White Overall Rank Mrs. Blue Overall Rank
52 4 66 13 63 10
46 1 49 3 65 12
62 9 64 11 58 8
48 2 53 5 70 15
57 7 68 14 71 16
54 6 73 17
Rank Sum 29 46 78

Using this information, we can calculate our test statistic as shown:

$H=\frac{12}{N(N+1)} \sum^m_{k=1} \frac{R^2_k}{n_k}-3(N+1)=\frac{12}{(17)(18)} \left(\frac{29^2}{6}+\frac{46^2}{5}+\frac{78^2}{6}\right)-(3)(17+1)=7.86$

Using the chi-square distribution, we determine that with $3-1=2$ degrees of freedom, our critical value at $\alpha=0.05$ is 5.991. Since our test statistic of 7.86 exceeds the critical value, we can reject the null hypothesis that stated there is no difference in the final exam scores among students from the three different classes.

### Determining the Randomness of a Sample Using the Runs Test

The runs test (also known as the Wald-Wolfowitz test) is another nonparametric test that is used to test the hypothesis that the samples taken from a population are independent of one another. We also say that the runs test checks the randomness of data when we are working with two variables. A run is essentially a grouping or a pattern of observations. For example, the sequence $++--++--++--$ has six runs. Three of these runs are designated by two positive signs, and three of the runs are designated by two negative signs.

We often use the runs test in studies where measurements are made according to a ranking in either time or space. In these types of scenarios, one of the questions we are trying to answer is whether or not the average value of the measurement is different at different points in the sequence. For example, suppose that we are conducting a longitudinal study on the number of referrals that different teachers give throughout the year. After several months, we notice that the number of referrals appears to increase around the time that standardized tests are given. We could formally test this observation using the runs test.

Using the laws of probability, it is possible to estimate the number of runs that one would expect by chance, given the proportion of the population in each of the categories and the sample size. Since we are dealing with proportions and probabilities between discrete variables, we consider the binomial distribution as the foundation of this test. When conducting a runs test, we establish the null hypothesis that the data samples are independent of one another and are random. On the contrary, our alternative hypothesis states that the data samples are not random and/or not independent of one another.

The runs test can be used with either nominal or categorical data. When working with nominal data, the first step in conducting the test is to compute the mean of the data and then designate each observation as being either above the mean (i.e., $+$) or below the mean (i.e., $-$). Next, regardless of whether or not we are working with nominal or categorical data, we compute the number of runs within the data set. As mentioned, a run is a grouping of the variables. For example, in the following sequence, we would have 5 runs. We could also say that the sequence of the data switched five times.

$++ - - - - + + + - +$

After determining the number of runs, we also need to record each time a certain variable occurs and the total number of observations. In the example above, we have 11 observations in total, with 6 positives $(n_1=6)$ and 5 negatives $(n_2=5)$. With this information, we are able to calculate our test statistic using the following formulas:

$z &= \frac{\text{number of observed runs}-\mu}{\sigma}\\\mu &= \text{expected number of runs}=1+\frac{2n_1n_2}{n_1+n_2}\\\sigma^2 &= \text{variance of the number of runs}=\frac{2n_1n_2(2n_1n_2-n_1-n_2)}{(n_1+n_2)^2(n_1+n_2-1)}$

When conducting the runs test, we calculate the standard $z$-score and evaluate our hypotheses, just like we do with other parametric and non-parametric tests.

Example: A teacher is interested in assessing if the seating arrangement of males and females in his classroom is random. He observes the seating pattern of his students and records the following sequence:

MFMMFFFFMMMFMFMMMMFFMFFMFFFF

Is the seating arrangement random? Use $\alpha=0.05$.

To answer this question, we first generate the null hypothesis that the seating arrangement is random and independent. Our alternative hypothesis states that the seating arrangement is not random or independent. With $\alpha=0.05$, we set our critical values at 1.96 standard scores above and below the mean.

To calculate the test statistic, we first record the number of runs and the number of each type of observation as shown:

$R=14 \quad M:n_1=13 \quad F:n_2=15$

With these data, we can easily compute the test statistic as follows:

$\mu &= \text{expected number of runs}=1+\frac{(2)(13)(15)}{13+15}=1+\frac{390}{28}=14.9\\\sigma^2 &= \text{variance of the number of runs}=\frac{(2)(13)(15)[(2)(13)(15)-13-15]}{(13+15)^2(13+15-1)}=\frac{(390)(362)}{(784)(27)}=6.67\\\sigma &= 2.58\\z &= \frac{\text{number of observed runs}-\mu}{\sigma}=\frac{14-14.9}{2.58}=-0.35$

Since the calculated test statistic is not less than $z=-1.96$, our critical value, we fail to reject the null hypothesis and conclude that the seating arrangement of males and females is random.

## Lesson Summary

The Kruskal-Wallis test is used when we are assessing the one-way variance of a specific variable in non-normal distributions.

The test statistic for the Kruskal-Wallis test is the non-parametric alternative to the $F$-statistic. This test statistic is defined by the following formula:

$H=\frac{12}{N(N+1)} \sum^m_{k=1} \frac{R^2_k}{n_k}-3(N+1)$

The runs test (also known as the Wald-Wolfowitz test) is another non-parametric test that is used to test the hypothesis that the samples taken from a population are independent of one another. We use the $z$-statistic to evaluate this hypothesis.

On the Web

http://tinyurl.com/334e5to Good explanations of and examples of different nonparametric tests.

http://tinyurl.com/33s4h3o Allows you to enter data and then performs the Wilcoxon sign rank test.

http://tinyurl.com/33s4h3o Allows you to enter data and performs the Mann Whitney Test.

Keywords

Kruskal-Wallis test
The Kruskal-Wallis test is the analog of the one-way ANOVA and is used when our data set does not meet the assumptions of normality or homogeneity of variance.
Mann-Whitney $v-$test
This test is sensitive to both the median and the distribution of the sample and population.
Non-parametric tests
we use the test statistic to evaluate our hypothesis
Rank correlation coefficient
to measure the strength, magnitude, and direction of the relationship between two variables.
Rank correlation test
to measure the strength, magnitude, and direction of the relationship between two variables.
Rank sum test
We use the rank sum test (also known as the Mann-Whitney $v-$test) to assess whether two samples come from the same distribution. This test is sensitive to both the median and the distribution of the samples.
Run
A run is essentially a grouping or a pattern of observations.
Runs test
The runs test (also known as the Wald-Wolfowitz test) is another nonparametric test that is used to test the hypothesis that the samples taken from a population are independent of one another.
Sign rank test
A more useful test that assesses the difference in size between the observations in a matched pair is the sign rank test.
Sign test
The sign test examines the difference in the medians of matched data sets.
Spearman rank correlation coefficient
Spearman rank correlation coefficient (also known simply as the rank correlation coefficient, $p$, or ‘rho’) to measure the strength, magnitude, and direction of the relationship between two variables.
$U-$distribution
the $U-$distribution approaches the normal distribution as the sizes of both samples grow.
$U-$statistic
we use the $U-$statistic to calculate the standard $z-$score.
Wald-Wolfowitz test
The runs test (also known as the Wald-Wolfowitz test) is another nonparametric test that is used to test the hypothesis that the samples taken from a population are independent of one another.
Wilcoxon sign rank test
The sign rank test (also known as the Wilcoxon sign rank test) resembles the sign test, but it is much more sensitive. Similar to the sign test, the sign rank test is also a nonparametric alternative to the paired Student’s $t-$test.

Feb 23, 2012

Dec 15, 2014