Math: Fleeting Thoughts
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Re: Math: Fleeting Thoughts
Maybe it doesn't hold both ways. It's clear that you can go from right to left, but left to right loses information.
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Re: Math: Fleeting Thoughts
Patashu wrote:I suspect something like this holds: a + bw + cw^2 === (a + 0.5b  0.5c) + (sqrt(3)/2)bcj
Having thought a little more, I think that the way to nudge you is to ask you to find elements of the split complex numbers, the complex numbers or the dual numbers such that their cube is 1. Hint:
Spoiler:
Patashu wrote:Maybe it doesn't hold both ways. It's clear that you can go from right to left, but left to right loses information.
Well, commutativity doesn't really have a direction. Either wz = zw for all z and w, or not. But talking about losing information is on the right track it seems.
Anyway, new set of assumptions. We are interested in [imath]K = \{a + bk : a,b\in \mathbf{R} \}[/imath] under the operations + and *. We assume that it is a commutative additive group, with 0 + 0k the identity element, that multiplication distributes over addition and also that multiplication is associative ie (ab)c = a(bc). I actually needed that before as well, and it was rather foolish of me not to see it.
Another technicality is that I implicitly assume that [imath]rk = r\cdot k[/imath]. This seems obvious, but it does seem to be an underlying assumption so I should state it.
With those assumptions its possible to identify at least one other algebra. There may well be others too, but it comes down to a question whose answer I don't know.
Spoiler:
So, apart from the case where [imath]k\cdot 1 = 0[/imath] and things isomorphic to that, to resolve the issue you need to know what functions satisfy A(r+s) = A(r)+A(s) and A(rs) = rA(s)+sA(r).
Edit: So, A is a derivation apparently. More research...
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Re: Math: Fleeting Thoughts
Two things that are interesting:
The geometric mean of the natural numbers up to [imath]n[/imath], i.e. [imath]\sqrt[n]{n!}[/imath]
And then this beast, which is defined for [imath]x\; \epsilon \; C[/imath]...
[math]\lambda (x) = \sum_{k=1}^{\infty }\frac{x^{k}}{k^{k}}[/math]
The geometric mean of the natural numbers up to [imath]n[/imath], i.e. [imath]\sqrt[n]{n!}[/imath]
And then this beast, which is defined for [imath]x\; \epsilon \; C[/imath]...
[math]\lambda (x) = \sum_{k=1}^{\infty }\frac{x^{k}}{k^{k}}[/math]
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Re: Math: Fleeting Thoughts
Eastwinn wrote:Two things that are interesting:
The geometric mean of the natural numbers up to [imath]n[/imath], i.e. [imath]\sqrt[n]{n!}[/imath]
And then this beast, which is defined for [imath]x\; \epsilon \; C[/imath]...
[math]\lambda (x) = \sum_{k=1}^{\infty }\frac{x^{k}}{k^{k}}[/math]
Try taking the ratio of n and the geometric mean of the numbers up to n for large n... it's identities like that make me love math (and please, nobody explain exactly why to me  It'll just lose some of the magic that way. EDIT: Nvm, I learned why... ): )
I don't understand what the second function is supposed to do, should it just grow very fast?
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Re: Math: Fleeting Thoughts
Hmmm, I'm finding it hard to prove what the cube roots of 1 in complex numbers are, outside of converting to modulusargument form and using the obvious geometric interpretation  except I don't know what the geometric interpretation of a splitcomplex number is. Probably something hyperbolic, given what I remember of conics.
Dual numbers only have 1 as a cube root of 1, of course.
*looks it up*
Aah: exp(j*theta) = cosh(theta) + j * sinh(theta), according to wikipedia.
Dual numbers only have 1 as a cube root of 1, of course.
*looks it up*
Aah: exp(j*theta) = cosh(theta) + j * sinh(theta), according to wikipedia.
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Re: Math: Fleeting Thoughts
try equating real and imaginary parts in
1 = (a + bi)^{3} = a^{3}  3ab^{2} + (b^{3} + 3a^{2}b)i
and similarly for the split complex'. You should be able to work out b^{2} in terms of a^{2} using the imaginary part (and this has to be the same for both algebras). Sub that into the real part and see how simple (or not) what you get is.
1 = (a + bi)^{3} = a^{3}  3ab^{2} + (b^{3} + 3a^{2}b)i
and similarly for the split complex'. You should be able to work out b^{2} in terms of a^{2} using the imaginary part (and this has to be the same for both algebras). Sub that into the real part and see how simple (or not) what you get is.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: Math: Fleeting Thoughts
If you're trying to prove rather than find, then just go through them, cube them all  and quote FTA to say that there aren't any more.
Re: Math: Fleeting Thoughts
squareroot wrote:I don't understand what the second function is supposed to do, should it just grow very fast?
It does have an interesting curve on the positive side, but all the action is on the negative side where it takes a big dive, come up a bit, and then slowly levels out. I don't know whether that leveling out is asymptotic or if it's more like a log, where it grows incredibly slowly.
Edit: Also Mathematica doesn't appear to know how it works.
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Re: Math: Fleeting Thoughts
The Boy or Girl paradox revisited
Please don't hurt me!
When I first saw this problem a day or so ago, I thought "Ah, the old Boy or Girl paradox that I read about in Martin Gardner's books when I was in high school. Easy!"
So I reasoned that the answer must be 2/3, i.e. the "Smith" case from the Wikipedia page, but now I'm not so sure. Firstly, there's the issue that both kids are named John. I'd like to reject this possibility, partly because of the parent's phrasing, but also on the grounds that it's a bad idea to import red herrings into simple probability / combinatorics puzzles. Secondly, there's the possibility that John is a girl, which I'd also like to reject, for the much the same reason.
But then there's the "gone to camp" v "left at home" distinction. I'm beginning to think that this is equivalent to the older v younger distinction, which would make the scenario equivalent to the "Jones" case from the Wikipedia, making the answer 1/2. But I'm not quite convinced.
So, is it really the "Jones" case in disguise, or is this yet another red herring? And am I justified in rejecting the other red herrings?
Help!
I didn't want to start a thread on this, considering the other thread I found on the classic version of the paradox was locked (for good reason, IMHO), but I'd be happy for it to move into its own thread if people think it merits one.
Please don't hurt me!
You overhear a parent: "Well, I've got two kids, but John's gone to camp this week". What's the probability that the other kid left at home is a girl?
When I first saw this problem a day or so ago, I thought "Ah, the old Boy or Girl paradox that I read about in Martin Gardner's books when I was in high school. Easy!"
So I reasoned that the answer must be 2/3, i.e. the "Smith" case from the Wikipedia page, but now I'm not so sure. Firstly, there's the issue that both kids are named John. I'd like to reject this possibility, partly because of the parent's phrasing, but also on the grounds that it's a bad idea to import red herrings into simple probability / combinatorics puzzles. Secondly, there's the possibility that John is a girl, which I'd also like to reject, for the much the same reason.
But then there's the "gone to camp" v "left at home" distinction. I'm beginning to think that this is equivalent to the older v younger distinction, which would make the scenario equivalent to the "Jones" case from the Wikipedia, making the answer 1/2. But I'm not quite convinced.
So, is it really the "Jones" case in disguise, or is this yet another red herring? And am I justified in rejecting the other red herrings?
Help!
I didn't want to start a thread on this, considering the other thread I found on the classic version of the paradox was locked (for good reason, IMHO), but I'd be happy for it to move into its own thread if people think it merits one.
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Re: Math: Fleeting Thoughts
With the assumption that the other child is *not* at camp, then it falls into the Jones case, as one child holds a distinguished position. You just need to impose an ordering on the children, such that you know the ordinal of the kid who's gender was provided, to make it a Jones case.
If both children could plausibly be at camp, but the parent is just for some reason only mentioning the campness of one of them, then you lose that ordering and return to the Smith case.
If both children could plausibly be at camp, but the parent is just for some reason only mentioning the campness of one of them, then you lose that ordering and return to the Smith case.
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Re: Math: Fleeting Thoughts
Thanks, Xanthir! Much appreciated.
Edit:
In case you're wondering, this puzzle arose on another forum I frequent. The debate still rages, with some people totally not getting the Smith scenario, although most of the discussion now centres on the wording of the original question (which I quoted verbatim above). Some say it's ambiguous, or that the at camp / at home distinction isn't the same as the older / younger distinction.
I'm still (mostly) convinced that it is equivalent to the Jones scenario, and that comparing it to the case of tossing a gold coin & a sliver coin seems to be valid.
But I just had a troubling thought while thinking about the at camp / at home thing.
How do we tackle the following question?
Q: I have two children, Chris and Pat. One of them is at camp, one is at home.
What's the probability that Chris is the one at camp?
A_{1}: There's not enough information to give a valid probability.
A_{2}: We don't know how the decision was made to send one of the children to camp, so the best probability we can assign is 1/2.
As you might guess, I'm not particularly happy with either of these answers, or this state of affairs. But is my new question even relevant to the original puzzle?
Edit:
In case you're wondering, this puzzle arose on another forum I frequent. The debate still rages, with some people totally not getting the Smith scenario, although most of the discussion now centres on the wording of the original question (which I quoted verbatim above). Some say it's ambiguous, or that the at camp / at home distinction isn't the same as the older / younger distinction.
I'm still (mostly) convinced that it is equivalent to the Jones scenario, and that comparing it to the case of tossing a gold coin & a sliver coin seems to be valid.
But I just had a troubling thought while thinking about the at camp / at home thing.
How do we tackle the following question?
Q: I have two children, Chris and Pat. One of them is at camp, one is at home.
What's the probability that Chris is the one at camp?
A_{1}: There's not enough information to give a valid probability.
A_{2}: We don't know how the decision was made to send one of the children to camp, so the best probability we can assign is 1/2.
As you might guess, I'm not particularly happy with either of these answers, or this state of affairs. But is my new question even relevant to the original puzzle?
Re: Math: Fleeting Thoughts
To me, the trouble with these questions is that when people volunteer information, you have to analyze why they offered that piece of information and not a different true statement.
Let me put it this way. Let's say that Susan took a math test and an English test today. Based on past data, you expect a 50% probability that she did well on either of them, and those are independent events. So you see Susan and have the following discussion:
You: How did you do on the tests?
Susan: I did well on the math test.
What is the probability that she did well on the English test? As a matter of psychology, I think you'd be naive to assume that it was still 50%, or else she'd have said that she did well on both. You have no formal evidence that the result of the English test is dependent on her statement, but the circumstantial evidence is heavy. But we'd be arguing that until the cows came home if we were of a mind to.
Let me put it this way. Let's say that Susan took a math test and an English test today. Based on past data, you expect a 50% probability that she did well on either of them, and those are independent events. So you see Susan and have the following discussion:
You: How did you do on the tests?
Susan: I did well on the math test.
What is the probability that she did well on the English test? As a matter of psychology, I think you'd be naive to assume that it was still 50%, or else she'd have said that she did well on both. You have no formal evidence that the result of the English test is dependent on her statement, but the circumstantial evidence is heavy. But we'd be arguing that until the cows came home if we were of a mind to.
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Re: Math: Fleeting Thoughts
PM 2ring, the problem you are having is that you have not defined "probability." Presumably here you are asking which is more likely to be the case: That a person named A is going to camp, or that a person named B is going to camp. In this case, it is obvious that the two are equal. Assuming exactly one of the two scenarios is true, then clearly the probability of each must be 0.5.
However, if you are asking which of those two specific people is more likely to go to camp, then you do not know, because you do not know anything about them. However, given no more information than their name, the best guess you can make is assigning an equal probability to each outcome. This is the equivalent of the question, "Which is more likely, event A or event B?" Since you know nothing about the event, the best answer is, "I don't know, but supposing one is more likely than the other, it could just as easily be the one labeled A as the one labeled B, so they are equally likely given the information at hand."
In either case, given the information you have, the probabilities are 50/50. Probabilities are always calculated given a limited knowledge of the situation.
However, if you are asking which of those two specific people is more likely to go to camp, then you do not know, because you do not know anything about them. However, given no more information than their name, the best guess you can make is assigning an equal probability to each outcome. This is the equivalent of the question, "Which is more likely, event A or event B?" Since you know nothing about the event, the best answer is, "I don't know, but supposing one is more likely than the other, it could just as easily be the one labeled A as the one labeled B, so they are equally likely given the information at hand."
In either case, given the information you have, the probabilities are 50/50. Probabilities are always calculated given a limited knowledge of the situation.
Re: Math: Fleeting Thoughts
Tirian wrote:To me, the trouble with these questions is that when people volunteer information, you have to analyze why they offered that piece of information and not a different true statement.
Good point, Tirian, and thanks for you comments & example. This issue has been raised on the other forum in reference to the original question. My stance is that it's unknown why they offered that particular piece of information, so we can't let it bias our probability calculation.
Eebster the Great wrote:PM 2ring, the problem you are having is that you have not defined "probability."
Gosh! I didn't realize I was allowed to do that. * ponders the implications *
Eebster the Great wrote:Presumably here you are asking which is more likely to be the case: That a person named A is going to camp, or that a person named B is going to camp. In this case, it is obvious that the two are equal. Assuming exactly one of the two scenarios is true, then clearly the probability of each must be 0.5.
Ok, that sounds reasonable.
Eebster the Great wrote:However, if you are asking which of those two specific people is more likely to go to camp, then you do not know, because you do not know anything about them. However, given no more information than their name, the best guess you can make is assigning an equal probability to each outcome. This is the equivalent of the question, "Which is more likely, event A or event B?" Since you know nothing about the event, the best answer is, "I don't know, but supposing one is more likely than the other, it could just as easily be the one labeled A as the one labeled B, so they are equally likely given the information at hand."
In either case, given the information you have, the probabilities are 50/50.
I think I've covered those two viewpoints in my A_{1} and A_{2}, haven't I?
Eebster the Great wrote:Probabilities are always calculated given a limited knowledge of the situation.
Thanks for reminding me of that, Eebster. I guess it's no different to a standard coin toss: with sufficiently accurate knowledge of the forces involved in the toss, we could use the laws of physics to predict the outcome before the coin comes to rest, but without that information we can't do better than assigning the usual 50:50 probability.
Of course, if we were looking at a fundamental quantum situation, eg a single radioactive decay event, then we can't even do the prediction using the laws of physics, but I don't want to open that can of worms at this stage.
Please keep the comments coming, folks. I feel that thinking about this stuff is giving me a deeper appreciation for how probability really works, and what it means. Or maybe I'm just confusing myself by overthinking. And undersleeping.
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Re: Math: Fleeting Thoughts
Well quantum mechanics do not use the same statistics as as classical probability, so it isn't exactly equivalent. But you are correct that quantum events exhibit true randomness.
Re: Math: Fleeting Thoughts
afarnen wrote:Bassoon wrote:It hasn't been said yet, but I hate it when people say "timesing" or "minusing" instead of "multiplying" or "subtracting". But it's overly ridiculous when people say "plussing" instead of "adding". It makes me want to scream.
Although I have heard people use "minus" as a verb, I don't think I've heard it used in gerund form. I also don't believe I've heard "times" or "plus" as verbs at all. Interesting...
Here the *ing is being used as a verb which makes it the present participle. A gerund would be the noun form of the verb with *ing.
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Re: Math: Fleeting Thoughts
vookaloop wrote:afarnen wrote:Bassoon wrote:It hasn't been said yet, but I hate it when people say "timesing" or "minusing" instead of "multiplying" or "subtracting". But it's overly ridiculous when people say "plussing" instead of "adding". It makes me want to scream.
Although I have heard people use "minus" as a verb, I don't think I've heard it used in gerund form. I also don't believe I've heard "times" or "plus" as verbs at all. Interesting...
Here the ing form is being used as an adjective which makes it the present participle. A gerund would be the noun form of the verb with ing.
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Re: Math: Fleeting Thoughts
Eebster the Great wrote: an adjective
If you are saying "I am timesing these two numbers together" it would be a present participle verb. If used in the sense of an adjective, i.e. "I have to do these timesing problems" then it would be a gerund (noun) acting in an adjective function as a premodifier of a noun. Considering the analog was "multiplying" I would assume verb but that is just my interpretation. Anyway I'll stop because this is math thoughts, just had to respond cause of the altered text.
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Re: Math: Fleeting Thoughts
vookaloop wrote:Eebster the Great wrote: an adjective
If you are saying "I am timesing these two numbers together" it would be a present participle verb. If used in the sense of an adjective, i.e. "I have to do these timesing problems" then it would be a gerund (noun) acting in an adjective function as a premodifier of a noun. Considering the analog was "multiplying" I would assume verb but that is just my interpretation. Anyway I'll stop because this is math thoughts, just had to respond cause of the altered text.
False. "I am verbing" is the present continuous, which does use the present participle in its construction, but as a whole it is a finite verb (specifically, one in the indicative mood, active voice). Notice that this construction is not inconsistent with the participle being an adjective, since the subject is being described as "verbing." That is to say, if I say "I am working," then I am describing myself as "working," which can alternatively be seen as a predicate nominative.
If on the other hand I say, "I have to do these timesing problems," you are correct that it is a gerund, since it is describing the problems as "involving multiplication," but if instead I say, "The running man was next to the timesing woman," I mean that a man is running next to a woman who is performing multiplication. Both are adjectives, but the former is a gerund and the latter is a participle. An example of the gerund used as a noun might be, "I hate timesing," indicating my dislike of multiplication itself. This is similar to the infinitive, "I hate to times."
At least, that's the way I see it. It gets more complicated if we want to consider gerundives and periphrasis.
Re: Math: Fleeting Thoughts
Make it stop... please never say that word again! It makes my ears bleed.
Re: Math: Fleeting Thoughts
Eebster the Great wrote:False. "I am verbing" is the present continuous, which does use the present participle in its construction, but as a whole it is a finite verb (specifically, one in the indicative mood, active voice). Notice that this construction is not inconsistent with the participle being an adjective, since the subject is being described as "verbing." That is to say, if I say "I am working," then I am describing myself as "working," which can alternatively be seen as a predicate nominative.
Interesting. Thanks for the explanation.
Re: Math: Fleeting Thoughts
MFT: In science, "model" has exactly the opposite meaning to "model" in mathematics.
One of them is a set of axioms based on a phenomenon they describe, the other is a construct obeying a set of axioms.
One of them is a set of axioms based on a phenomenon they describe, the other is a construct obeying a set of axioms.
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Re: Math: Fleeting Thoughts
Why is the adjective form of "arithmetic" spelled "arithmetic" but pronounced "airithmaatic"?
Metafleetingthought: is this a mathematics fleeting thought, or a linguistics fleeting thought?
That's true; I never thought of it that way. Funny.
Metafleetingthought: is this a mathematics fleeting thought, or a linguistics fleeting thought?
Macbi wrote:MFT: In science, "model" has exactly the opposite meaning to "model" in mathematics.
One of them is a set of axioms based on a phenomenon they describe, the other is a construct obeying a set of axioms.
That's true; I never thought of it that way. Funny.
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Re: Math: Fleeting Thoughts
skeptical scientist wrote:Why is the adjective form of "arithmetic" spelled "arithmetic" but pronounced "airithmaatic"?
Clearly your spelling is broken. Spell it with an A. Problem solved.
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Re: Math: Fleeting Thoughts
Xanthir wrote:skeptical scientist wrote:Why is the adjective form of "arithmetic" spelled "arithmetic" but pronounced "airithmaatic"?
Clearly your spelling is broken. Spell it with an A. Problem solved.
Well spelling is far more standardized than pronunciation, so that seems to be pretty backward.
Re: Math: Fleeting Thoughts
skeptical scientist wrote:Why is the adjective form of "arithmetic" spelled "arithmetic" but pronounced "airithmaatic"?
Wah? In my dialect, the noun is əRITHmətic, the adjective is ArithMETic.
Re: Math: Fleeting Thoughts
What's the smallest molecule which is topologically nonplanar (where it's atoms are nodes and its bonds are edges)?
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Re: Math: Fleeting Thoughts
Ammonia? (Shouldn't this be in Science rather than Mathematics?)Macbi wrote:What's the smallest molecule which is topologically nonplanar (where it's atoms are nodes and its bonds are edges)?
Re: Math: Fleeting Thoughts
Because there's no Science: Fleeting Thoughts thread?
Also, Ammonia is nonplanar when you look at it in its physical configuration, but he was asking about nonplanarity as a graph property. And, after thinking for several minutes, I don't actually know any nonplanar molecules. Even something like a fullerene is planar when you blow up one of the faces to be the exterior face, and arrange everything else inside it. But then, I know very little about chemistry.
Also, Ammonia is nonplanar when you look at it in its physical configuration, but he was asking about nonplanarity as a graph property. And, after thinking for several minutes, I don't actually know any nonplanar molecules. Even something like a fullerene is planar when you blow up one of the faces to be the exterior face, and arrange everything else inside it. But then, I know very little about chemistry.
Re: Math: Fleeting Thoughts
Ah. Good point.but he was asking about nonplanarity as a graph property.
Re: Math: Fleeting Thoughts
Indeed. Even your avatar is planar (if indeed it even is a molecule). If anyone had any examples, I would really like to see them. A molecule with K_{5} or K_{3,3} as a minor would be really interesting.
Re: Math: Fleeting Thoughts
I found lots in this paper (pdf), but the smallest one (the SimmonsPaquette molecule, right at the bottom) still uses 35 atoms.
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Re: Math: Fleeting Thoughts
Macbi wrote:I found lots in this paper (pdf), but the smallest one (the SimmonsPaquette molecule, right at the bottom) still uses 35 atoms.
There is also a threerung molecular Mobius ladder (at least 60 atoms, but not all such molecules are intrinsically chiral according to this link) and a molecular trefoil knot (much larger). Some even occur naturally, such as certain proteins or circular DNA strands (one book lists the result of the action of enzyme TN3 Resolvase on circular DNA).
Here is a book on the subject.
This article suggests that Au_{34} nanoclusters have intrinsic chirality, one less atom than the SimmonsPaquette molecule (if you can call nanoclusters "molecules").
Unfortunately, I can't find any source on the smallest synthesized intrinsically chiral molecule.
Re: Math: Fleeting Thoughts
skeptical scientist wrote:Why is the adjective form of "arithmetic" spelled "arithmetic" but pronounced "airithmaatic"?
This has always bothered me.
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Re: Math: Fleeting Thoughts
Same as "address", noun and verb, I guess. (ADdress vs. adDRESS)
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Re: Math: Fleeting Thoughts
The Scyphozoa wrote:Same as "address", noun and verb, I guess. (ADdress vs. adDRESS)
I think he's objecting to /ɛ/ being realized as [ɛɪ], but I don't think that's the only time that happens in English. I can't think of another example off the top of my head.
I also think the pronunciation varies with the region.
Re: Math: Fleeting Thoughts
I was trying to find a way to expand (2n)! without using [imath]\pi[/imath] but was unsuccessful .
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Re: Math: Fleeting Thoughts
Eastwinn wrote:I was trying to find a way to expand (2n)! without using [imath]\pi[/imath] but was unsuccessful .
Funny, I've been toying around with (2n choose n) as (n goes to infinity) for a bit... at one point, I thought it went to c*4^n, and I thought I had c, but then I noticed a flaw... I'm thinking it's something on the order of (4^n)/n now, but again I'm not sure.
EDIT: WolframAlpha, with a bit of guess and check, told me it's sqrt(pi)*(4^n)/sqrt(n). Hmm.
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Re: Math: Fleeting Thoughts
Just need to get these thoughts written down cos it's bugging me:
Suppose we have an m x n rectangle where m and n are integers and the rectangle is partitioned itself into integer edgelength rectangles (can be any size and position as long as they partition the large rectangle into at least two smaller rectangles). Call this partition of the rectangle [imath]R_0[/imath] and call [imath]R_{i+1}[/imath] a refinement of [imath]R_i[/imath] if it can be formed by replacing a proper subset of the 'tiles' with their union in the obvious way (this union must itself be a rectangle).
Intuitively we're starting off with a set of rectangles and then rubbing out lines on the partition so that we still have a partition (and never rub out every line).
Call [imath]R_k[/imath] a total refinement if [imath]R_k[/imath] is the partition of the m x n rectangle in to exactly two other rectangles (and so cannot be further refined). Call [imath]R_k[/imath] a nontrivial refinement if [imath]R_k[/imath] is a partition of the m x n rectangle in to more than two rectangles and no further refinement exists (this is why the proper subset property was needed for refinements).
Conjecture; Given [imath]R_0[/imath] and it is known that [imath]R_k[/imath] is a nontrivial refinement, no other distinct nontrivial refinement exists. That is, [imath]R_{k'}=R_k[/imath].
Ok, I knew writing it down would help. A counter example exists: a 2x2 square with a 'border' of 1x1 squares has at least 2 distinct nontrivial refinements (mirror images of each other) and a few more that aren't mirror images.
Ok, new conjecture; Any [imath]R_0[/imath] for which more than one nontrivial refinement exists can also be totally refined.
I'll have a think about this one as a counterexample seems a little bit more out of reach for this one. Yay for fleeting thoughts. Feel free to contribute guys, I thought it was a pretty nice construction of a problem.
Suppose we have an m x n rectangle where m and n are integers and the rectangle is partitioned itself into integer edgelength rectangles (can be any size and position as long as they partition the large rectangle into at least two smaller rectangles). Call this partition of the rectangle [imath]R_0[/imath] and call [imath]R_{i+1}[/imath] a refinement of [imath]R_i[/imath] if it can be formed by replacing a proper subset of the 'tiles' with their union in the obvious way (this union must itself be a rectangle).
Intuitively we're starting off with a set of rectangles and then rubbing out lines on the partition so that we still have a partition (and never rub out every line).
Call [imath]R_k[/imath] a total refinement if [imath]R_k[/imath] is the partition of the m x n rectangle in to exactly two other rectangles (and so cannot be further refined). Call [imath]R_k[/imath] a nontrivial refinement if [imath]R_k[/imath] is a partition of the m x n rectangle in to more than two rectangles and no further refinement exists (this is why the proper subset property was needed for refinements).
Conjecture; Given [imath]R_0[/imath] and it is known that [imath]R_k[/imath] is a nontrivial refinement, no other distinct nontrivial refinement exists. That is, [imath]R_{k'}=R_k[/imath].
Ok, I knew writing it down would help. A counter example exists: a 2x2 square with a 'border' of 1x1 squares has at least 2 distinct nontrivial refinements (mirror images of each other) and a few more that aren't mirror images.
Ok, new conjecture; Any [imath]R_0[/imath] for which more than one nontrivial refinement exists can also be totally refined.
I'll have a think about this one as a counterexample seems a little bit more out of reach for this one. Yay for fleeting thoughts. Feel free to contribute guys, I thought it was a pretty nice construction of a problem.
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