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going to take her to lunch on day N and you don't do so. Then she can't know that you'll take her to lunch on any day, even Friday, since there's no law of physics compelling you to do so; thus you can fulfill the bargain by taking her out on any day, even Friday. Alternatively we could define "know" such that there are no worlds where all three of the following are true: she knows you'll take her to lunch on day N, you don't do so, and you ultimately fulfill the bargain. In that case she can know on Friday but not on any earlier day, since on
Thursday she's forced to acknowledge the existence of a possible world where (for some reason) she doesn't know on Friday and you take her out then. Or we could define "know" even more broadly by allowing her to eliminate possible worlds based on plausible assumptions about her own future behavior. In that case she can (choose in advance to) "know" you'll take her to lunch every day, and so you can't fulfill the bargain. Once you define your terms precisely enough the problem can be solved, and the solution depends on the definition. If you don't define your terms and try to reason intuitively about your knowledge/beliefs/predictions then you're doomed, because they'll just keep flip-flopping. --
668:
not at the day she said would be the best for her. It was nice of you that you asked her because obviously it would be bad if she had some other occupation at the time you scheduled the dinner, but that she can choose some almost impossible criterion for the day wasn't really part of the penalty. On the other hand, you could also try to show up on any day and surprise her in some way other than by the choice of the day. –
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it's a problem of psychology rather than mathematics. You can replace "expect" with "logically deduce" and turn it into a problem of formal logic. The article is misleading when it says that "no consensus on its correct resolution has yet been established", since one doesn't expect a consensus on the meaning of an utterance in the
English language. If you ever find yourself in the condemned prisoner's position, keep in mind
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features, specifically my graph is a weighted (but undirected) graph so I would like to be able to visualise these weights either by colour or by thickness of the edge. In addition, it would be really great if the program would allow me to have a colour/size scale for the nodes based on how connected they were or some other number that I could assign to each node.
465:"Dear, dear sister. Please keep in mind that there is always the smallest probability that even the best made and most carefully thought-out promises cannot be kept (due to unexpected hospital stays for example), and that despite best efforts, it is always possible that I might have to take you to lunch after you get back from your trip."
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The naive method is of course to optimize each coordinate separately in every iteration. However, this can fail to converge, and at best converges linearly. I tried a basic google search, but could only find discussions of Nash equilibria for mixed strategies from a discrete space (then again, I have
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Take her out on Friday. Monday morning, she will think that you have thought through the puzzle and will take her out on Monday because it is the "most" surprising. Tuesday morning, she will think that you are being clever and waiting until
Tuesday. On Wednesday she will be yet more expectant, and
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That's one solution, anyway; it's not the only one because there's more than one way of interpreting
English words like "expect". You can turn it into a problem about the slippery nature of belief (can your sister really make herself believe in Thursday after having reasoned as I just did?), but then
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But there's a problem: If I were to happen to roll a 5 (Friday - the last possible day) then on Friday morning my sister will think to herself: "He didn't take me to lunch yet - and he promised to do it this week - so today MUST be the day"...so it's not a surprise when I show up on Friday lunchtime
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That's true, but I'm more looking to use the graph as just a broad visual representation of the data. The idea is not so much to visualise the existance of the connections, but more the weight of the connections either through colour or line thickness. Unfortunately all the programs seem to require
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I have looked at graphviz and the GUI is basically non-existant and it looks like I would have to reenter all the data and I looked at an excel frontend for it but that wasn't very promising either. I also found a program called yEd but that doesn't seem to allow me to enter in an adjacency matrix.
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it every day, and I'm happy to allow her to, but the problem becomes trivial if we do. It's all about the vagueness of the terms. Replace "expect" with "know". Define "know" such that one can't know something that doesn't come to pass, i.e. there are no possible worlds where your sister knows you're
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I had a bet with my sister (which I lost) my penalty was that I had to take her to lunch. I ask her when we should meet up. Being a fun-loving person, she says "Well, today is Sunday - I'm going on vacation on
Saturday morning so it has to be before then. Surprise me! Just show up at work around
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Was the criterion that she can choose the day for the dinner also in the bet? If you promised in the bet that you'd take her to a dinner, but then later you just casually asked what day would be the best for her, then I think it might not be a break of your promise if you took her to a dinner but
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Thanks for the link! The parable of the dagger has some surprising applications... For example, when faced with the Monty Hall problem, no longer do you have to rely on mere probabilities - once you have two doors remaining, you simply put two post-it notes on the doors, one of them saying "either
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you can't do it on Friday). But lunchtime comes and goes and you don't show up. Now what? If we allow her to expect to be taken out on Friday also, then your argument is valid, but all it shows is the trivial fact that if she expects to be taken out every day then she won't be surprised on the day
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I have an adjacency matrix (in Excel 07) from which I would like to generate a graph in order to be able to visualise the data nicely. Is there a program that will generate a graph from the adjacency matrix specified in any format Excel can output (e.g. xls, csv, etc.). I also need a couple of
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I reworded the paragraph to try to clarify it. It would probably have made more sense if I'd used "predict" instead of "expect", since that implies more certainty. It doesn't sound fair to predict that the lunch will happen every day and then claim to be vindicated when it finally does. It sounds
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The way I would look at it is like this: Since every day has been ruled out, there is the same probability (albeit zero) that it will occur on any day of that week. Since it must occur (because you always keep your promises), and it has an equal probability of occuring any day, then there is no
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for root-finding (in the sense of using nothing more then algebraic operations on the values of the function). If there are only good algorithms which require derivative evaluation, I could try to find an expression for the derivatives, but that would be a challenge on its own, and evaluation of
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OK - so I have to choose between 1=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday and re-roll the dice if I get a 5 or a 6. But now we have the same problem with
Thursday. She knows (because we've stipulated that she thinks exactly like me) that Friday wouldn't be a surprise - so it's impossible.
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If the original quote is exact, there's a whole world of opportunity. Don't think so small. For instance, you could hide above a ceiling tile the night before, and drop down behind her when she isn't looking. That'd be a surprise. You might have to make a deal with security ahead of time.
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certainty as to when it will occur or will not occur. Thus, any day (except Friday) it would be a suprise. If you take them any day but Friday, they don't know if you'll wait until the next day or not. (This is the first time I have heard of this paradox, so that may be flawed logic.)
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both post-it notes are true or both are false" and the other saying "there is a prize behind this door". Once you have done this, it is logically impossible that there is no prize behind the latter door and you can safely choose it. I never knew logic could be so practically useful.
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The problem of mixed strategies over a discrete strategy space is obviously a special case of pure strategies over a continuous (possibly multidimensional) strategy space. If you insist on each player's strategy being a single real number, you need an injection from
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But his/her sister is also intelligent (they think alike) so when she gets up on
Thursday, she'll know that it has to be the day (because Friday is out) and therefore it's not a surprise. And the same reasoning can be applied to Wednesday, Tuesday and Monday.
1344:, then you will (eventually) converge to a NE, though which NE you'll find may depend in the order of coordinates in which you have iteratively discharged strategies. This idea may not be useful in general, depending on the structure of the utility functions
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that she is. If we don't allow her to expect to be taken out every day, then your argument breaks down immediately; you can surprise her on any day, including Friday, because there's a good chance she'll have blown her one chance at unsurprise before then.
1830:. This is indeed the kind of algorithm I had in mind, and it does seem to solve the problem of a repelling equilibrium. If implemented property, its performance will probably be good enough, but I'll be happy to hear any other suggestions. --
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Thanks. Unfortunately, I am not sure about how to go about eliminating dominated strategies in practice. My functions are quite complicated and I see no way to analyze them symbolically; I treat them as an oracle to which I can provide an
205:, but I'm having a hard time proving it or finding a counterexample (the only examples I know of a bounded but not totally bounded metric spaces are disconnected). Is this intuition true or false? And how might I go about proving it? --
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that this is at all possible. In this particular case it is of course a non-issue as providing an explicit choice function is trivial, but this goes to show you that the description is not perfect. A possible fix is to replace
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Sadly, now that
Thursday is out of the question - so is Wednesday...and that means that Tuesday is impossible...and that only leaves Monday - and that won't be a surprise because it's the only possible day remaining.
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It depends. For some of my applications the functions will be analytic, for some they will have manifolds of nondifferentiability (Just to demonstrate what I mean, they will be, smoothness-wise, similar to
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So I could take her on Monday, Tuesday, Wednesday, Thursday or Friday. Let's stipulate that I always keep my promise, we're both intelligent, we think very much alike. Is it possible to surprise her?
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More directly, you can make any metric space bounded without making it totally bounded or changing its topology in the slightest. Just define a new distance to be the minimum of the old distance and 1.
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itself cannot have nice properties. In particular, my functions are continuous and almost everywhere differentiable, and I'll be happy with an algorithm that assumes the functions are smooth. --
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Where is your sister going? If it's very close, then you could contrive to take her to lunch on
Saturday. (And if to you the week starts on Monday, Sunday is also an option.) Eric.
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In this way, you are dismissing sections of the entire space (or a hypercube if the domain is somehow restricted) that will never host a NE. If you replace the ": -->
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On the face of it, it seems easy - I just need a random number. So maybe I should roll a dice, 1=Monday, 2=Tuesday...5=Friday, and on a 6, I re-roll. Easy - right?
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OK, let's take a stab at finding a counterexample. Start with a countably infinite metric space in which the distance between any two points is 1. Clearly bounded,
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are the best responses of each player (function) to the remaining coordinates of the vector? In case it's really costly to find, you may pick a representative φ
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This seems like a bogus argument - but I can't find a logical hole in it. Is it true that it's impossible to truly surprise someone under these circumstances?
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on
Thursday she will be 100% sure that that is the day. On Friday she will think you have forgotten. This way, you surprise her 5 times instead of just once.
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Hmmm. Even if you find appropriate graph generation software, will it be of much use to you ? A highly connected graph with ~40 nodes is very unlikely to be
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Ah, yep, I got it now. I thought about it one way and it was logical, then the other way, and it was illogical. Personally, I like Predtigitator's idea.
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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
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323:? Maybe if you can figure out reasonable answers to those questions, you'll get a space that is bounded (no distance greater than 1), connected, and
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That's not fair. It is unreasonable to expect someone to assume the OP knew about the article after they'd typed out that scenario in such detail.
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2747:", because there are only a finite number of choices and it doesn't matter which one you take. However, if you want this to apply to a variable
2200:, but this may leave you with somewhat ugly payoff functions. Still, it at least suggests that your problem may be hard in the general case. —
140:. Unless the graph has some special symmetries, I would think a diagram is going to be very messy, and visually almost indistinguishable from a
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Cool - it's good to know the mathematicians are earning their keep answering everyday problems! Many thanks - I'm off to read it carefully.
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Hence: on Thursday morning, she knows that I can't leave it until Friday to take her to lunch because that wouldn't be a surprise - so it
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at the other, and let the distance be the ordinary metric on . Now it's connected, but what is the distance between a point between
2876:{\displaystyle R:\mathbb {C} \to {\mathcal {P}}(\mathbb {C} ),\ t\mapsto \left\{\zeta _{r}^{n}{\sqrt{f(t)}}:0\leq n\leq r-1\right\}}
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of my choice and which, at a significant computational cost, outputs the values. I was hoping for an algorithm reminiscent of the
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Nice to know, but it's not so relevant, as the paper seems to discuss the case of mixed strategies over a discrete space. --
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Reduce the problem to two days (Thursday and Friday). Say your sister expects you to take her out on Thursday (since
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is, the faster the algorithm will arrive at a NE, but the chances are greater that the algorithm fails to converge.
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I know you don't want to talk about conditions in derivatives, so I assume that the fact that for an interior NE
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It looks as not an easy task. Speaking about the secant method, how about the following: Fix an initial vector
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I don't want to do it manually because the graph is highly connected and has ~40 nodes. Any ideas?--
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be happening today...so again, it won't be a surprise. That means that Thursday is impossible too.
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I have a logic problem that's been bothering me for years - it comes in the form of a true story:
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All that said, if all you want is a concise notation for the set of roots, why not just write
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I’d like to check my understanding here. I’m trying to write the r roots of a function of
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408:. That article explains some approaches that have been attempted to resolve the problem.
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BenRG's argument is very persuasive... also, that "parable of the dagger link" is great.
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In response to your last point: That's a set of complex numbers which depends on
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in a concise manner. Specifically say I want to write the set of all r roots of
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manual entry or I would need to undergo a steep learning curve to use them.
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I can think of no monotonous function to apply to simplify the problem. --
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2385:{\displaystyle \left\{\zeta _{r}^{n}{\sqrt{f(t)}}:0\leq n\leq r-1\right\}}
3150:, the OP seems to be looking for a set of functions. If you have a fixed
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which is continuous everywhere but has a circle of nondifferentiability).
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which at least represents the correct sign of the variation. The greater
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If you want to be even more precise, you can specify explicitly that
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Not sure if it's at all relevant to your question, but it has been
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totally bounded. Now let's connect it: between any pair of points
1504:, then you cannot assume continuity, and even the simple equation
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Welcome to the Knowledge (XXG) Mathematics Reference Desk Archives
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Try looking at infinite-dimensional examples. Have you had any
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isn't of much help in your case. Hope anything of this helps.
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is a parameter of speed of adjustment of the procedure, and
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879:{\displaystyle x=(x_{1},\ldots ,x_{p})\in \mathbb {R} ^{p}}
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is through an oracle. That doesn't mean that the function
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369:- which means that Friday cannot be an acceptable result.
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I see. So the naive method will be a special case with
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No, I have not, but I will take a look. Thank you. --
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The first idea that comes to my mind is to think of
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585:if he/she doesn't take her out on Thursday, she'll
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578:the fact that she can't expect to be taken out
3058:th root" (which we take to be known to exist).
1312:{\displaystyle f_{i}(\mathbf {x} _{-}i,x): -->
594:In either case, I think that we have to allow
384:Hence no day is truly (logically) a surprise.
1784:If it converges at all, it clearly does to a
553:. Not that it'll save you from your fate. --
356:- don't tell me you're coming, just show up."
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1108:iterated elimination of dominated strategies
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352:noon someday this week and we'll do lunch...
104:Drawing Graphs (Graph Theory, not Functions)
189:Connected and Totally Bounded metric spaces
2133:yesterday where he mentioned this result.
1714:{\displaystyle \varphi (\mathbf {x} ^{n})}
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1110:: You may discard for each coordinate
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1319:f_{i}(\mathbf {x} _{-}i,y)}" /: -->
813:of the functions, that is, a point
2883:- it gets a little trickier. Now "
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404:This problem is well known as the
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1216:{\displaystyle x\in \mathbb {R} }
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809:. I am interested in finding the
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1091:So, any suggestions? Thanks. --
2129:-complete. I was at a talk by
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2990:{\displaystyle {\sqrt{f(t)}}}
2916:{\displaystyle {\sqrt{f(t)}}}
2751:- that is, define a function
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2098:1\end{array}}\right.}" /: -->
911:{\displaystyle 1\leq i\leq p}
802:{\displaystyle 1\leq i\leq p}
571:I don't follow. When you say
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3158:at all, and not just give
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2691:{\displaystyle \zeta _{r}}
2529:{\displaystyle \zeta _{r}}
2412:{\displaystyle \zeta _{r}}
1888:)) can be used instead of
589:to be taken out on Friday?
467:Then take her Tuesday. --
406:unexpected hanging paradox
354:but I want to be surprised
3047:{\displaystyle {\sqrt{}}}
2303:. Is it correct to write
551:the parable of the dagger
3176:22:57, 24 May 2008 (UTC)
3142:22:16, 24 May 2008 (UTC)
2500:21:51, 24 May 2008 (UTC)
2212:13:45, 28 May 2008 (UTC)
2154:12:06, 28 May 2008 (UTC)
2113:17:45, 27 May 2008 (UTC)
1911:09:27, 27 May 2008 (UTC)
1840:17:45, 27 May 2008 (UTC)
1800:04:51, 27 May 2008 (UTC)
1545:17:18, 26 May 2008 (UTC)
1522:16:54, 26 May 2008 (UTC)
1482:16:19, 26 May 2008 (UTC)
1427:15:17, 26 May 2008 (UTC)
1101:20:06, 24 May 2008 (UTC)
715:05:58, 30 May 2008 (UTC)
696:17:53, 25 May 2008 (UTC)
673:11:35, 25 May 2008 (UTC)
659:19:51, 26 May 2008 (UTC)
636:00:10, 25 May 2008 (UTC)
608:22:23, 24 May 2008 (UTC)
575:, are you referring to:
563:22:08, 24 May 2008 (UTC)
528:20:19, 24 May 2008 (UTC)
511:19:36, 24 May 2008 (UTC)
492:19:19, 24 May 2008 (UTC)
477:19:11, 24 May 2008 (UTC)
453:17:41, 24 May 2008 (UTC)
431:17:27, 24 May 2008 (UTC)
413:16:13, 24 May 2008 (UTC)
399:15:29, 24 May 2008 (UTC)
337:14:34, 26 May 2008 (UTC)
258:11:06, 24 May 2008 (UTC)
248:08:36, 24 May 2008 (UTC)
234:08:25, 24 May 2008 (UTC)
215:07:51, 24 May 2008 (UTC)
181:10:03, 25 May 2008 (UTC)
154:14:30, 24 May 2008 (UTC)
131:04:42, 24 May 2008 (UTC)
2954:not immediately obvious
1578:, and iterate following
1084:not yet fully mastered
723:Finding Nash equlibrium
3126:
3054:is some branch of the
3048:
3020:
2991:
2946:
2917:
2877:
2741:
2721:
2692:
2665:
2579:
2578:{\displaystyle f(t)=a}
2530:
2485:
2453:
2413:
2386:
2297:
2268:
2245:
2190:
2131:Christos Papadimitriou
2093:
1824:
1768:
1715:
1671:
1572:
1461:
1405:
1338:
1314:
1217:
1189:
1136:
1074:
940:
912:
880:
803:
771:
87:current reference desk
3127:
3049:
3021:
2992:
2947:
2918:
2878:
2742:
2722:
2693:
2666:
2580:
2531:
2486:
2484:{\displaystyle f(t)?}
2454:
2414:
2387:
2298:
2269:
2246:
2191:
2094:
2092:1\end{array}}\right.}
1825:
1769:
1716:
1672:
1573:
1462:
1406:
1339:
1337:{\displaystyle \geq }
1315:
1218:
1190:
1137:
1075:
941:
913:
881:
804:
772:
727:Hi. I have functions
3166:) its own name?). --
3074:
3030:
3019:{\displaystyle f(t)}
3001:
2961:
2945:{\displaystyle f(t)}
2927:
2887:
2755:
2731:
2702:
2675:
2597:
2554:
2513:
2505:Mostly yes. However:
2463:
2423:
2396:
2307:
2296:{\displaystyle f(t)}
2278:
2258:
2244:{\displaystyle f(t)}
2226:
2163:
1922:
1808:
1725:
1687:
1583:
1553:
1436:
1358:
1328:
1229:
1199:
1146:
1118:
950:
922:
890:
886:such that for every
817:
781:
731:
315:and a point between
299:and a point between
3065:must be an integer.
2820:
2619:
2329:
2198:space-filling curve
1823:{\displaystyle k=1}
1760:
621:somewhat fairer to
3122:
3044:
3016:
2997:is some r root of
2987:
2942:
2923:is some r root of
2913:
2873:
2806:
2737:
2727:is some r root of
2717:
2688:
2661:
2605:
2575:
2526:
2481:
2459:is some r root of
2449:
2409:
2382:
2315:
2293:
2264:
2241:
2186:
2088:
2083:
1865:analytic functions
1820:
1764:
1741:
1711:
1667:
1568:
1457:
1401:
1334:
1309:
1213:
1185:
1142:such that for all
1132:
1070:
936:
908:
876:
799:
767:
3042:
2985:
2911:
2842:
2794:
2740:{\displaystyle a}
2715:
2630:
2447:
2351:
2267:{\displaystyle t}
2210:
1393:
327:totally bounded.
178:
174:
170:
128:
124:
120:
93:
92:
73:
72:
3185:
3131:
3129:
3128:
3123:
3103:
3102:
3090:
3053:
3051:
3050:
3045:
3043:
3041:
3036:
3034:
3025:
3023:
3022:
3017:
2996:
2994:
2993:
2988:
2986:
2984:
2979:
2965:
2951:
2949:
2948:
2943:
2922:
2920:
2919:
2914:
2912:
2910:
2905:
2891:
2882:
2880:
2879:
2874:
2872:
2868:
2843:
2841:
2836:
2822:
2819:
2814:
2793:
2786:
2778:
2777:
2768:
2746:
2744:
2743:
2738:
2726:
2724:
2723:
2718:
2716:
2714:
2706:
2697:
2695:
2694:
2689:
2687:
2686:
2670:
2668:
2667:
2662:
2660:
2656:
2631:
2629:
2621:
2618:
2613:
2584:
2582:
2581:
2576:
2535:
2533:
2532:
2527:
2525:
2524:
2490:
2488:
2487:
2482:
2458:
2456:
2455:
2450:
2448:
2446:
2441:
2427:
2418:
2416:
2415:
2410:
2408:
2407:
2391:
2389:
2388:
2383:
2381:
2377:
2352:
2350:
2345:
2331:
2328:
2323:
2302:
2300:
2299:
2294:
2273:
2271:
2270:
2265:
2250:
2248:
2247:
2242:
2204:
2195:
2193:
2192:
2187:
2185:
2177:
2176:
2171:
2099:
2096:
2095:
2089:
2087:
2084:
2074:
2073:
2061:
2060:
2040:
2039:
2027:
2026:
2001:
2000:
1988:
1987:
1976:
1975:
1963:
1962:
1829:
1827:
1826:
1821:
1797:
1792:
1786:Nash equilibrium
1773:
1771:
1770:
1765:
1759:
1754:
1746:
1737:
1736:
1720:
1718:
1717:
1712:
1707:
1706:
1701:
1676:
1674:
1673:
1668:
1663:
1662:
1657:
1645:
1644:
1639:
1618:
1617:
1612:
1603:
1602:
1591:
1577:
1575:
1574:
1569:
1567:
1566:
1561:
1466:
1464:
1463:
1458:
1456:
1455:
1450:
1424:
1419:
1410:
1408:
1407:
1402:
1394:
1392:
1391:
1390:
1377:
1376:
1375:
1362:
1343:
1341:
1340:
1335:
1320:
1317:
1316:
1310:
1296:
1295:
1290:
1281:
1280:
1256:
1255:
1250:
1241:
1240:
1222:
1220:
1219:
1214:
1212:
1194:
1192:
1191:
1186:
1184:
1183:
1172:
1160:
1159:
1154:
1141:
1139:
1138:
1133:
1131:
1079:
1077:
1076:
1071:
1060:
1059:
1044:
1043:
1025:
1024:
1000:
999:
975:
974:
962:
961:
945:
943:
942:
937:
935:
917:
915:
914:
909:
885:
883:
882:
877:
875:
874:
869:
857:
856:
838:
837:
811:Nash equilibrium
808:
806:
805:
800:
776:
774:
773:
768:
766:
758:
757:
752:
743:
742:
582:be surprised, or
176:
172:
168:
126:
122:
118:
75:
38:Mathematics desk
34:
3193:
3192:
3188:
3187:
3186:
3184:
3183:
3182:
3094:
3072:
3071:
3035:
3028:
3027:
2999:
2998:
2966:
2959:
2958:
2925:
2924:
2892:
2885:
2884:
2823:
2805:
2801:
2753:
2752:
2729:
2728:
2700:
2699:
2678:
2673:
2672:
2604:
2600:
2595:
2594:
2552:
2551:
2516:
2511:
2510:
2461:
2460:
2428:
2421:
2420:
2399:
2394:
2393:
2332:
2314:
2310:
2305:
2304:
2276:
2275:
2256:
2255:
2252:
2224:
2223:
2166:
2161:
2160:
2082:
2081:
2065:
2052:
2050:
2031:
2018:
2009:
2008:
1992:
1979:
1977:
1967:
1954:
1946:
1919:
1918:
1896:
1883:
1867:? Note that if
1858:
1850:Do you know if
1806:
1805:
1795:
1790:
1777:
1728:
1723:
1722:
1721:'s coordinates
1696:
1685:
1684:
1652:
1634:
1607:
1586:
1581:
1580:
1556:
1551:
1550:
1445:
1434:
1433:
1422:
1417:
1382:
1378:
1367:
1363:
1356:
1355:
1349:
1326:
1325:
1285:
1272:
1245:
1232:
1226:
1225:
1197:
1196:
1167:
1149:
1144:
1143:
1116:
1115:
1051:
1035:
1010:
985:
966:
953:
948:
947:
920:
919:
888:
887:
864:
848:
829:
815:
814:
779:
778:
747:
734:
729:
728:
725:
469:Prestidigitator
344:
287:at one end and
271:connected, and
203:totally bounded
199:connected space
191:
106:
101:
30:
29:
28:
12:
11:
5:
3191:
3189:
3181:
3180:
3179:
3178:
3134:Meni Rosenfeld
3121:
3118:
3115:
3112:
3109:
3106:
3101:
3097:
3093:
3089:
3085:
3082:
3079:
3067:
3066:
3059:
3040:
3015:
3012:
3009:
3006:
2983:
2978:
2975:
2972:
2969:
2941:
2938:
2935:
2932:
2909:
2904:
2901:
2898:
2895:
2871:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2840:
2835:
2832:
2829:
2826:
2818:
2813:
2809:
2804:
2800:
2797:
2792:
2789:
2785:
2781:
2776:
2771:
2767:
2763:
2760:
2736:
2713:
2709:
2685:
2681:
2659:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2628:
2624:
2617:
2612:
2608:
2603:
2574:
2571:
2568:
2565:
2562:
2559:
2544:
2523:
2519:
2507:
2506:
2480:
2477:
2474:
2471:
2468:
2445:
2440:
2437:
2434:
2431:
2406:
2402:
2380:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2349:
2344:
2341:
2338:
2335:
2327:
2322:
2318:
2313:
2292:
2289:
2286:
2283:
2263:
2251:
2240:
2237:
2234:
2231:
2220:
2219:
2218:
2217:
2216:
2215:
2214:
2202:Ilmari Karonen
2184:
2180:
2175:
2170:
2146:Meni Rosenfeld
2118:
2117:
2116:
2115:
2105:Meni Rosenfeld
2101:
2086:
2080:
2077:
2072:
2068:
2064:
2059:
2055:
2051:
2049:
2046:
2043:
2038:
2034:
2030:
2025:
2021:
2017:
2014:
2011:
2010:
2007:
2004:
1999:
1995:
1991:
1986:
1982:
1978:
1974:
1970:
1966:
1961:
1957:
1953:
1952:
1949:
1945:
1942:
1939:
1936:
1933:
1930:
1927:
1892:
1879:
1854:
1848:
1847:
1846:
1845:
1844:
1843:
1842:
1832:Meni Rosenfeld
1819:
1816:
1813:
1783:
1775:
1763:
1758:
1753:
1750:
1745:
1740:
1735:
1731:
1710:
1705:
1700:
1695:
1692:
1677:
1666:
1661:
1656:
1651:
1648:
1643:
1638:
1633:
1630:
1627:
1624:
1621:
1616:
1611:
1606:
1601:
1598:
1595:
1590:
1579:
1565:
1560:
1537:Meni Rosenfeld
1488:If a function
1486:
1485:
1484:
1474:Meni Rosenfeld
1454:
1449:
1444:
1441:
1400:
1397:
1389:
1385:
1381:
1374:
1370:
1366:
1352:
1347:
1333:
1322:
1308:
1305:
1302:
1299:
1294:
1289:
1284:
1279:
1275:
1271:
1268:
1265:
1262:
1259:
1254:
1249:
1244:
1239:
1235:
1224:
1211:
1207:
1204:
1182:
1179:
1176:
1171:
1166:
1163:
1158:
1153:
1130:
1126:
1123:
1093:Meni Rosenfeld
1069:
1066:
1063:
1058:
1054:
1050:
1047:
1042:
1038:
1034:
1031:
1028:
1023:
1020:
1017:
1013:
1009:
1006:
1003:
998:
995:
992:
988:
984:
981:
978:
973:
969:
965:
960:
956:
934:
930:
927:
907:
904:
901:
898:
895:
873:
868:
863:
860:
855:
851:
847:
844:
841:
836:
832:
828:
825:
822:
798:
795:
792:
789:
786:
765:
761:
756:
751:
746:
741:
737:
724:
721:
720:
719:
718:
717:
699:
698:
684:
681:
676:
675:
664:
663:
662:
661:
643:
642:
641:
640:
639:
638:
613:
612:
611:
610:
592:
591:
590:
583:
566:
565:
545:
544:
535:
534:
533:
532:
531:
530:
516:
515:
514:
513:
495:
494:
480:
479:
460:
459:
458:
457:
456:
455:
436:
435:
434:
433:
416:
415:
359:
358:
343:
340:
265:
264:
263:
262:
261:
260:
240:196.210.152.31
207:196.210.152.31
190:
187:
186:
185:
184:
183:
157:
156:
142:complete graph
105:
102:
100:
97:
95:
91:
90:
82:
81:
71:
70:
64:
48:
41:
40:
31:
15:
14:
13:
10:
9:
6:
4:
3:
2:
3190:
3177:
3173:
3169:
3165:
3161:
3157:
3153:
3149:
3145:
3144:
3143:
3139:
3135:
3113:
3107:
3104:
3099:
3095:
3091:
3083:
3080:
3069:
3068:
3064:
3060:
3057:
3038:
3010:
3004:
2981:
2973:
2967:
2955:
2936:
2930:
2907:
2899:
2893:
2869:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2838:
2830:
2824:
2816:
2811:
2807:
2802:
2795:
2790:
2761:
2758:
2750:
2734:
2711:
2707:
2683:
2679:
2657:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2626:
2622:
2615:
2610:
2606:
2601:
2592:
2588:
2572:
2569:
2563:
2557:
2549:
2545:
2542:
2539:
2521:
2517:
2509:
2508:
2504:
2503:
2502:
2501:
2497:
2493:
2478:
2472:
2466:
2443:
2435:
2429:
2404:
2400:
2378:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2347:
2339:
2333:
2325:
2320:
2316:
2311:
2287:
2281:
2261:
2235:
2229:
2213:
2208:
2203:
2199:
2173:
2157:
2156:
2155:
2151:
2147:
2143:
2142:
2140:
2136:
2132:
2128:
2124:
2120:
2119:
2114:
2110:
2106:
2102:
2078:
2075:
2070:
2066:
2062:
2057:
2053:
2047:
2044:
2036:
2032:
2028:
2023:
2019:
2012:
2005:
2002:
1997:
1993:
1989:
1984:
1980:
1972:
1968:
1964:
1959:
1955:
1947:
1943:
1937:
1934:
1931:
1925:
1915:
1914:
1912:
1908:
1904:
1900:
1895:
1891:
1887:
1882:
1878:
1874:
1870:
1866:
1862:
1857:
1853:
1849:
1841:
1837:
1833:
1817:
1814:
1811:
1803:
1802:
1801:
1798:
1793:
1787:
1781:
1756:
1751:
1748:
1733:
1729:
1703:
1690:
1682:
1678:
1659:
1649:
1641:
1628:
1622:
1619:
1614:
1604:
1599:
1596:
1593:
1563:
1548:
1547:
1546:
1542:
1538:
1534:
1530:
1527:My access to
1526:
1525:
1523:
1519:
1515:
1511:
1507:
1503:
1500:) when asked
1499:
1495:
1491:
1487:
1483:
1479:
1475:
1470:
1469:secant method
1452:
1442:
1439:
1430:
1429:
1428:
1425:
1420:
1414:
1398:
1395:
1387:
1383:
1372:
1368:
1353:
1350:
1331:
1303:
1300:
1297:
1292:
1277:
1273:
1269:
1263:
1260:
1257:
1252:
1237:
1233:
1205:
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1195:there exists
1180:
1177:
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629:
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388:
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329:Michael Hardy
326:
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195:metric spaces
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23:
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3163:
3159:
3155:
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3055:
2748:
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2589:th roots of
2586:
2547:
2540:
2537:
2253:
1898:
1893:
1889:
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1876:
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1505:
1501:
1497:
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1489:
1412:
1345:
1111:
1090:
1082:
726:
707:Black Carrot
622:
600:Zain Ebrahim
595:
586:
579:
572:
539:
503:Zain Ebrahim
464:
445:Zain Ebrahim
389:
386:
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222:Banach space
192:
111:
107:
94:
78:
688:81.159.33.9
26:Mathematics
2593:would be "
2536:must be a
918:and every
423:SteveBaker
410:Algebraist
255:Algebraist
224:theory? --
177:(contribs)
127:(contribs)
2538:primitive
2492:GromXXVII
2222:Roots of
2135:Oliphaunt
1903:Bo Jacoby
1679:...where
1514:Bo Jacoby
1223:for which
540:obviously
489:¡hábleme!
342:Surprise!
226:Trovatore
146:Gandalf61
50:<<
3026:" with "
1791:Pallida
1418:Pallida
1411:for all
1114:the set
946:we have
525:Ζρς ι'β'
486:Ζρς ι'β'
24: |
22:Archives
20: |
670:b_jonas
165:AMorris
115:AMorris
89:pages.
2671:where
2392:where
1863:) are
1324:" for
777:, for
623:expect
587:expect
169:(talk)
138:planar
119:(talk)
99:May 24
67:May 25
46:May 23
3168:Tango
3132:? --
2123:shown
2076:: -->
1270:: -->
628:BenRG
555:BenRG
69:: -->
63:: -->
62:: -->
44:<
16:<
3172:talk
3138:talk
2496:talk
2207:talk
2150:talk
2139:talk
2127:PPAD
2109:talk
1907:talk
1836:talk
1796:Mors
1541:talk
1518:talk
1478:talk
1423:Mors
1097:talk
711:talk
692:talk
655:talk
632:talk
604:talk
596:that
573:that
559:talk
507:talk
473:talk
449:talk
427:talk
395:talk
375:must
333:talk
319:and
311:and
303:and
295:and
279:and
244:talk
230:talk
211:talk
150:talk
2546:If
1088:).
580:and
325:not
273:not
269:not
201:is
60:Jun
56:May
52:Apr
3174:)
3140:)
3084:∈
2863:−
2857:≤
2851:≤
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2799:↦
2770:→
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2639:≤
2607:ζ
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2401:ζ
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2003:≤
1913:.
1909:)
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1691:φ
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1543:)
1524:.
1520:)
1480:)
1443:∈
1380:∂
1365:∂
1332:≥
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1253:−
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1178:−
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1157:−
1125:∈
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144:.
58:|
54:|
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3120:}
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3100:r
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3088:C
3081:z
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2042:)
2037:2
2033:y
2029:+
2024:2
2020:x
2016:(
2013:2
2006:1
1998:2
1994:y
1990:+
1985:2
1981:x
1973:2
1969:y
1965:+
1960:2
1956:x
1948:{
1944:=
1941:)
1938:y
1935:,
1932:x
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173:●
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