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Hi all - my question is pretty much as in the title: why is it that, assuming we take the shorter of the 2 arc lengths between any 2 points on S^2, the perimeter of any given spherical triangle is strictly bounded above by ? I tried using Gauss-Bonnet (overkill?) but to no avail. I've managed to prove the triangle inequality which is fairly trivial, if that's any use!
Thanks very much, Delaypoems101 ( talk) 00:51, 29 January 2010 (UTC)
I guess my point is that I'm wondering how to actually show that that construes the largest possible triangle - for example, a triangle with one point 'A' at the north pole and 2 antipodal points on the equator would also have perimeter length pi+pi/2+pi/2=2pi, though I would consider these degenerate cases since 'A' isn't really a vertex and I'm not sure 3 collinear points is considered a triangle on the sphere - how would I go about showing that for any side lengths a, b, c, of a non-degenerate spherical triangle, we have a+b+c < 2pi? (Rather than saying 'this case looks like it should be a maximum, and has perimeter 2pi, hence any other triangle probably has a smaller perimeter than that'?) Thanks for all responses, Delaypoems101 ( talk) 04:29, 29 January 2010 (UTC)
Is there a (in some sense) mathematically optimal strategy for unique bid auctions, eg. a Nash equilibrium strategy? Has any work been done on optimal strategies in real-life unique bid auctions? -- The Anome ( talk) 01:02, 29 January 2010 (UTC)
Update:An earlier version of the article suggests that there can't be any deterministic optimal strategy, as if more than one player uses it, they will all select the same bid, and thus all lose, therefore contradicting the premise. However, I can't see any reason for a probabilistic strategy not to exist. -- The Anome ( talk) 02:16, 29 January 2010 (UTC)
I'm taking a computer science course in Discrete Structures. One of the questions is asking about
cardinality, which is defined on wikipedia and my text book as the number of elements in a set. Therefore:
{x} = 1
{{x}} = 1
{x, {x}} = 2
{x, {x}, {x, {x}}} = 3 but why is this true? shouldn't the answer be 4??
and this one:
{2, {3, 4, 10, {4, 0}, 6, 3}, 6, 12} the answer is listed as 4...but why, that makes no since.
-- penubag ( talk) 08:51, 29 January 2010 (UTC)
At: http://en.wikipedia.org/wiki/Demographics_of_Australia#Population_growth_rate There appears to be a numerical error in the following section:
As of the end of June 2009 the population growth rate was 2.1%.[7] This rate was based on estimates of:[8]
* one birth every 1 minute and 45 seconds, * one death every 3 minutes and 40 seconds, * a net gain of one international migrant every 1 minutes and 51 seconds leading to * an overall total population increase of one person every 1 minutes and 11 seconds.
In 2009 the estimated rates were:
* Birth rate - 12.47 births/1,000 population (Rank 164) * Mortality rate - 6.68 deaths/1,000 population (Rank 146) * Net migration rate - 6.23 migrant(s)/1,000 population. (Rank 15)
The ratio between: one birth every 1 minute and 45 seconds and a net gain of one international migrant every 1 minutes and 51 seconds is approximately 1:1.06
whereas the ratio between: Birth rate - 12.47 births/1,000 and Net migration rate - 6.23 migrant(s)/1,000 is approximately 1:0.5
Should these ratios not be equal?
I am unable to discern which figures are correct and which is wrong, so am in no position to edit the article, but would like an answer to satisfy my interpretation of the figures or show why I am wrong to expect actual or near equality between the two ratios. —Preceding unsigned comment added by Briandcjones ( talk • contribs) 10:55, 29 January 2010 (UTC)
The page Talk:Demographics_of_Australia is the proper place for your question, I think. Bo Jacoby ( talk) 16:41, 29 January 2010 (UTC).
Basically, say I had a list of characters in a production, and knew which scenes each character appeared in. Then say I needed to assign actors to characters, with each character only needing one actor, and each actor being able to play any number of characters as long as none of their characters share scenes with each other. Assuming none of the actors need any specific requirements to play each character, how would I find the minimum number of actors required? (Note: I only know discrete maths at A-Level standard - I tried looking at stuff about P and NP and didn't understand, so if you could make your explanation as simple as possible, I'd appreciate it.) Thanks! Anthrcer (click to talk to me) 12:17, 29 January 2010 (UTC)
I see that this method would be what I need, but I really don't understand the articles, and the Simple English Wikipedia doesn't have an article specifically about graph colouring. Can the method be explained in words I can understand? Or can an appropriate website be linked? Anthrcer (click to talk to me) 20:27, 29 January 2010 (UTC) (edited by Anthrcer (click to talk to me) 22:08, 29 January 2010 (UTC))
I see. Thanks for helping! Anthrcer (click to talk to me) 13:08, 30 January 2010 (UTC)
If you were asked to prove that the square root of 2 is an irrational number with proofs, how would it be done? 198.188.150.134 ( talk) 13:02, 29 January 2010 (UTC)
Hi, I'm having a little trouble finishing this one differentiation problem I'm trying to do. I have right now:
y^2 = sqrt(b^2- (b^2x/a^2))
for positive y values only. Thanks, -- Fbv 65 e del — t — c // 20:00, 29 January 2010 (UTC)
-- JohnBlackburne words deeds 20:12, 29 January 2010 (UTC)
The function y is implicitly defined anyway, so why not get rid of the square root and the fraction: , and differentiate: Bo Jacoby ( talk) 20:47, 29 January 2010 (UTC).
Solving equations by algebra alone is often hard and sometimes impossible. Differentiation of polynomials is easy. So if you are not explicitly requested to provide an explicit expression, implicit differentiation is sufficient. For example, the function y = f(x), defined implicitly by the equation y5+y = x, cannot be expressed explicitly, but the equation can easily be differentiated. Bo Jacoby ( talk) 16:36, 30 January 2010 (UTC)
We know the probability is 1/3 if you dont switch your guess and 50% if you do, so whats wrong with this reasoning: lets look at what happens from the perspective of what was under your original guess. In 1/3 of the cases there was a car under it; if you switch your guess when Monty gas removed one goat from the possibilities, you will be guaranteed to switch to the other goat. But in 2/3 of the cases there is a goat under your original door: in these cases, Monty removed the other one, leaving you guaranteed to switch to a car. Therefore, from this perspective, 2/3 of the time you will get a car with the switchin strategy. (but this is false; in fact it is onlz 50 % of the time see monty hall problem. So, whats wrong with the line of reasoning I outlined above? Where is the logical error? Thank you. 80.187.105.225 ( talk) 22:54, 29 January 2010 (UTC)
Mathematics desk | ||
---|---|---|
< January 28 | << Dec | January | Feb >> | January 30 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Hi all - my question is pretty much as in the title: why is it that, assuming we take the shorter of the 2 arc lengths between any 2 points on S^2, the perimeter of any given spherical triangle is strictly bounded above by ? I tried using Gauss-Bonnet (overkill?) but to no avail. I've managed to prove the triangle inequality which is fairly trivial, if that's any use!
Thanks very much, Delaypoems101 ( talk) 00:51, 29 January 2010 (UTC)
I guess my point is that I'm wondering how to actually show that that construes the largest possible triangle - for example, a triangle with one point 'A' at the north pole and 2 antipodal points on the equator would also have perimeter length pi+pi/2+pi/2=2pi, though I would consider these degenerate cases since 'A' isn't really a vertex and I'm not sure 3 collinear points is considered a triangle on the sphere - how would I go about showing that for any side lengths a, b, c, of a non-degenerate spherical triangle, we have a+b+c < 2pi? (Rather than saying 'this case looks like it should be a maximum, and has perimeter 2pi, hence any other triangle probably has a smaller perimeter than that'?) Thanks for all responses, Delaypoems101 ( talk) 04:29, 29 January 2010 (UTC)
Is there a (in some sense) mathematically optimal strategy for unique bid auctions, eg. a Nash equilibrium strategy? Has any work been done on optimal strategies in real-life unique bid auctions? -- The Anome ( talk) 01:02, 29 January 2010 (UTC)
Update:An earlier version of the article suggests that there can't be any deterministic optimal strategy, as if more than one player uses it, they will all select the same bid, and thus all lose, therefore contradicting the premise. However, I can't see any reason for a probabilistic strategy not to exist. -- The Anome ( talk) 02:16, 29 January 2010 (UTC)
I'm taking a computer science course in Discrete Structures. One of the questions is asking about
cardinality, which is defined on wikipedia and my text book as the number of elements in a set. Therefore:
{x} = 1
{{x}} = 1
{x, {x}} = 2
{x, {x}, {x, {x}}} = 3 but why is this true? shouldn't the answer be 4??
and this one:
{2, {3, 4, 10, {4, 0}, 6, 3}, 6, 12} the answer is listed as 4...but why, that makes no since.
-- penubag ( talk) 08:51, 29 January 2010 (UTC)
At: http://en.wikipedia.org/wiki/Demographics_of_Australia#Population_growth_rate There appears to be a numerical error in the following section:
As of the end of June 2009 the population growth rate was 2.1%.[7] This rate was based on estimates of:[8]
* one birth every 1 minute and 45 seconds, * one death every 3 minutes and 40 seconds, * a net gain of one international migrant every 1 minutes and 51 seconds leading to * an overall total population increase of one person every 1 minutes and 11 seconds.
In 2009 the estimated rates were:
* Birth rate - 12.47 births/1,000 population (Rank 164) * Mortality rate - 6.68 deaths/1,000 population (Rank 146) * Net migration rate - 6.23 migrant(s)/1,000 population. (Rank 15)
The ratio between: one birth every 1 minute and 45 seconds and a net gain of one international migrant every 1 minutes and 51 seconds is approximately 1:1.06
whereas the ratio between: Birth rate - 12.47 births/1,000 and Net migration rate - 6.23 migrant(s)/1,000 is approximately 1:0.5
Should these ratios not be equal?
I am unable to discern which figures are correct and which is wrong, so am in no position to edit the article, but would like an answer to satisfy my interpretation of the figures or show why I am wrong to expect actual or near equality between the two ratios. —Preceding unsigned comment added by Briandcjones ( talk • contribs) 10:55, 29 January 2010 (UTC)
The page Talk:Demographics_of_Australia is the proper place for your question, I think. Bo Jacoby ( talk) 16:41, 29 January 2010 (UTC).
Basically, say I had a list of characters in a production, and knew which scenes each character appeared in. Then say I needed to assign actors to characters, with each character only needing one actor, and each actor being able to play any number of characters as long as none of their characters share scenes with each other. Assuming none of the actors need any specific requirements to play each character, how would I find the minimum number of actors required? (Note: I only know discrete maths at A-Level standard - I tried looking at stuff about P and NP and didn't understand, so if you could make your explanation as simple as possible, I'd appreciate it.) Thanks! Anthrcer (click to talk to me) 12:17, 29 January 2010 (UTC)
I see that this method would be what I need, but I really don't understand the articles, and the Simple English Wikipedia doesn't have an article specifically about graph colouring. Can the method be explained in words I can understand? Or can an appropriate website be linked? Anthrcer (click to talk to me) 20:27, 29 January 2010 (UTC) (edited by Anthrcer (click to talk to me) 22:08, 29 January 2010 (UTC))
I see. Thanks for helping! Anthrcer (click to talk to me) 13:08, 30 January 2010 (UTC)
If you were asked to prove that the square root of 2 is an irrational number with proofs, how would it be done? 198.188.150.134 ( talk) 13:02, 29 January 2010 (UTC)
Hi, I'm having a little trouble finishing this one differentiation problem I'm trying to do. I have right now:
y^2 = sqrt(b^2- (b^2x/a^2))
for positive y values only. Thanks, -- Fbv 65 e del — t — c // 20:00, 29 January 2010 (UTC)
-- JohnBlackburne words deeds 20:12, 29 January 2010 (UTC)
The function y is implicitly defined anyway, so why not get rid of the square root and the fraction: , and differentiate: Bo Jacoby ( talk) 20:47, 29 January 2010 (UTC).
Solving equations by algebra alone is often hard and sometimes impossible. Differentiation of polynomials is easy. So if you are not explicitly requested to provide an explicit expression, implicit differentiation is sufficient. For example, the function y = f(x), defined implicitly by the equation y5+y = x, cannot be expressed explicitly, but the equation can easily be differentiated. Bo Jacoby ( talk) 16:36, 30 January 2010 (UTC)
We know the probability is 1/3 if you dont switch your guess and 50% if you do, so whats wrong with this reasoning: lets look at what happens from the perspective of what was under your original guess. In 1/3 of the cases there was a car under it; if you switch your guess when Monty gas removed one goat from the possibilities, you will be guaranteed to switch to the other goat. But in 2/3 of the cases there is a goat under your original door: in these cases, Monty removed the other one, leaving you guaranteed to switch to a car. Therefore, from this perspective, 2/3 of the time you will get a car with the switchin strategy. (but this is false; in fact it is onlz 50 % of the time see monty hall problem. So, whats wrong with the line of reasoning I outlined above? Where is the logical error? Thank you. 80.187.105.225 ( talk) 22:54, 29 January 2010 (UTC)