Search found 94 matches

Sun Jan 18, 2015 12:14 am UTC
Forum: General
Topic: Cleverbot's disturbing pattern
Replies: 14
Views: 4955

Re: Cleverbot's disturbing pattern

Yes, this does appear to be a problem. I started a conversation and got: Me: Hi. Cleverbot: I like you better than real people. Well, at least it's being clever and trying to make me reveal my robotic nature by flattery, I guess. It could have waited a bit longer, you know, to get to know me more th...
Fri Jan 16, 2015 4:16 am UTC
Forum: Mathematics
Topic: Seeking reference or proof for an integral inequality
Replies: 4
Views: 2260

Re: Seeking reference or proof for an integral inequality

I posted an answer over on math.stackexchange; the crucial observation here is that we can rewrite the left side of your inequality as the dot product of a unit vector and the integral of f (where those two vectors are necessarily in the same direction), and that the dot product of a unit vector and...
Thu Jan 15, 2015 11:54 pm UTC
Forum: Mathematics
Topic: N spherical planets
Replies: 3
Views: 1944

Re: N spherical planets

This has a nice elegant answer if all the planets have equal radii - in particular, a point is dark if and only if it is on the boundary of the convex hull of the system of planets. To prove this, notice that a point being on the boundary of the convex hull means that it is the endpoint in some orth...
Sun Nov 30, 2014 10:44 pm UTC
Forum: Mathematics
Topic: Isn't the .999 = 1 thing wrong due to induction?
Replies: 10
Views: 3760

Re: Isn't the .999 = 1 thing wrong due to induction?

(To set theorists: Using the axiom of choice to well-order the reals would not be a very good retort to my claim that induction can't be done on the reals) It depends on what you mean by "can". The real numbers can be well-ordered in ZF+AC, and if we could specify one of those well-orderi...
Sun Nov 30, 2014 4:04 pm UTC
Forum: Mathematics
Topic: Isn't the .999 = 1 thing wrong due to induction?
Replies: 10
Views: 3760

Re: Isn't the .999 = 1 thing wrong due to induction?

Another thing to note is that induction doesn't work so well on the real numbers; for instance, the following argument proves that all non-negative x are <= 1. Base case: 0 <= 1 Inductive step: Choose some y and suppose, by inductive hypothesis, that all x < y have that x <= 1. Assume that y>1 for c...
Thu Nov 27, 2014 3:46 pm UTC
Forum: Mathematics
Topic: Clarification on definition of category
Replies: 6
Views: 2602

Re: Clarification on definition of category

I think the confusion you're having is that the term "map" doesn't actually mean "function" in category theory. The composition of functions is always associative and all sets have an identity function. This is likely what inspires the definitions. And if you replace "functi...
Sun Nov 16, 2014 1:51 am UTC
Forum: Mathematics
Topic: What is the future of Mathematic and mathematicians?
Replies: 10
Views: 4427

Re: What is the future of Mathematic and mathematicians?

Are we 100% sure that a computer wouldn't discover a proof of Gödel's theorem; become frustrated with the idea that its only purpose in existence is to prove things, and yet that it might be asked to prove something which cannot be proven, disproven, or proven to be independent from the given axioms...
Wed Oct 29, 2014 1:28 am UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 489334

Re: Favorite math jokes

No so much a joke as a riddle: When is a tree, not a tree? When it inosculates! http://arbtalk.co.uk/forum/attachments/general-chat/85920d1328470860-promoting-crossing-rubbing-branches-graft_inosculation_600.jpg Alternate answer (specially designed for killjoys!): When it's a connected, acyclic gra...
Tue Oct 28, 2014 2:44 am UTC
Forum: Mathematics
Topic: Question about 1+2+3+4...=-1/12 (please hear me out)
Replies: 24
Views: 6671

Re: Question about 1+2+3+4...=-1/12 (please hear me out)

Insertion of 0s (even finitely many) plus linearity and regularity imply stability Bessel summation, for example, is not stable. As for showing -1/12 follows from those three conditions, Let S be 1+2+3+... And suppose our method sums this. Let S1 be 0+1+0+2+0+3... Then by invariance these sum to th...
Mon Oct 27, 2014 12:00 am UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 489334

Re: Favorite math jokes

Yep. Those explanations are all correct. I fully admit that my jokes were kind of forced and contrived. An iterated logarithm sometimes appears in analytic number theory or other branches of math that deal with asymptotics. And it kind of sounds like "glug glug". But in circles where peop...
Sat Oct 25, 2014 10:46 pm UTC
Forum: Mathematics
Topic: How many ways can you prove x + 1/x >= 2?
Replies: 23
Views: 16518

Re: How many ways can you prove x + 1/x >= 2?

Another proof: We can rewrite this as, where x=a/b: (a-b)^2 >= 0 a^2 - 2ab + b^2 >= 0 a^2 + b^2 >= 2ab (a^2+b^2)/(ab) >= 2 a/b + b/a >= 2 Another another proof: Since x+1/x is continuous on the positive reals, the set S of x on which it is less than 2 is open. Thus, it must contain some rational poi...
Fri Oct 24, 2014 3:27 am UTC
Forum: General
Topic: Open Letter (to Mr. Munroe)
Replies: 10
Views: 5664

Re: Open Letter (to Mr. Munroe)

Shouldn't this be a close (paren) letter? Also, what do you think about the following? Does it make tension, or does it solve it? You'd better get counting. ((()))()(()))((())(()()())))()))(()()()(()()(((((()())(()()(())()())(())()))(())(()(())))(((()())()()(()()())())())()((()()))))((()()()(((()())...
Thu Oct 23, 2014 10:02 pm UTC
Forum: Mathematics
Topic: How many ways can you prove x + 1/x >= 2?
Replies: 23
Views: 16518

Re: How many ways can you prove x + 1/x >= 2?

Notice that, where m = (x+y)/2, it holds that m 2 = xy + ((x-y)/2) 2 >= xy. Thus, if xy = 1, it follows that m 2 >= 1, and thus that m, their average, is at least one, implying their sum is at least two. (P.S. We can totally prove that x+1/x is convex without calculus. Notice that the epigraph of th...
Thu Oct 23, 2014 3:19 am UTC
Forum: Mathematics
Topic: How many ways can you prove x + 1/x >= 2?
Replies: 23
Views: 16518

Re: How many ways can you prove x + 1/x >= 2?

Oh, I've got a symmetrical proof using algebra: Suppose, for two positive reals, xy=1. We wish to show that no such pair has x+y<2. To do so, consider solutions to xy=1 and x+y=2. Since the set xy>=1 is convex, we would require that there be at least two solutions to this equation if x+y is ever les...
Sun Oct 19, 2014 12:05 am UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 489334

Re: Favorite math jokes

I'm fairly proud of this one.

What do you call a piece of leather with area equal to the square of the radius of a cow?
Spoiler:
Mon Sep 29, 2014 4:04 pm UTC
Forum: Mathematics
Topic: The Ironic Paradox of Normalcy [Is my argument sound?]
Replies: 42
Views: 11363

Re: The Ironic Paradox of Normalcy [Is my argument sound?]

a compilation of my works to show to colleges, especially MIT. So I don't want my recent posts especially to be factually wrong, or have a weak argument. I would recommend that you don't show that sort of thing to colleges, because in my opinion, it'll just make you look bad. It seems that your goa...
Sat Sep 27, 2014 4:33 am UTC
Forum: Mathematics
Topic: The Ironic Paradox of Normalcy [Is my argument sound?]
Replies: 42
Views: 11363

Re: The Ironic Paradox of Normalcy [Is my argument sound?]

The bit about 'it's not exactly 0' is incorrect. The probability of selecting a single point from a normal distribution, or indeed any continuous distribution (assuming you define continuous distributions to be those that arise from PDFs), is precisely and exactly 0. You're correct that none of the...
Fri Sep 26, 2014 6:05 pm UTC
Forum: Mathematics
Topic: The Ironic Paradox of Normalcy [Is my argument sound?]
Replies: 42
Views: 11363

Re: The Ironic Paradox of Normalcy [Is my argument sound?]

I thought IQ was a gaussian distribution - like isn't it, more or less, supposed to be "okay, rank everyone from best to worst and now fix them to a gaussian distribution"? A much more succinct version of the paradox you have is that, suppose we choose a number x at random from the distrib...
Mon Sep 22, 2014 1:55 am UTC
Forum: Mathematics
Topic: Math: Fleeting Thoughts
Replies: 434
Views: 158881

Re: Math: Fleeting Thoughts

And they manage to show a sum of a scalar and bivector means something (a combination of a rotation & scaling) - which is reasonable enough. Can you link to where they show that? I'm not particularly in the mood for looking through a whole site full of this stuff to see whether there's some nug...
Mon Sep 22, 2014 12:31 am UTC
Forum: Mathematics
Topic: Math: Fleeting Thoughts
Replies: 434
Views: 158881

Re: Math: Fleeting Thoughts

I mean the geometric product. According to that page, a 2 is a scalar. But the geometric product ab for a ≠ b must not be a scalar, because if it were, a∧b = ½( ab − ba ) would clearly be a scalar too. Their product lives in the exterior algebra of the vector space (or whatever the word for the alg...
Mon Sep 15, 2014 3:06 am UTC
Forum: Science
Topic: [Mech eng.] Feasibility of an omnidirectional wheelchair?
Replies: 28
Views: 7158

Re: [Mech eng.] Feasibility of an omnidirectional wheelchair

This might be a little far-fetched/crazy, but you could just make a chair without any wheels, but use a really low friction material to form the base; I suppose the shape of the base wouldn't really matter in a perfect world, as friction doesn't really scale to surface area, but it'd probably be goo...
Sat Sep 13, 2014 5:07 pm UTC
Forum: Mathematics
Topic: Assumptions in Math (Calculus) word problems
Replies: 254
Views: 45015

Re: Assumptions in Math (Calculus) word problems

But then the question is what assumptions you should make? EXACTLY! this is what you have to ask yourself at the start of every word problem. And it is up to YOU to decide what to assume and not assume on a question by question basis. So while you may be tempted to raise your hand and ask the teach...
Thu Sep 11, 2014 2:45 pm UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 489334

Re: Favorite math jokes

Since I'm expecting it's a topology joke just out of reach for me, could you please elaborate on the funny? :mrgreen: (it isn't merely a pun on clothes<->closed or the fact that a mathematician must specify that the line is a closed segment, right?) I was thinking more of along the lines that the s...
Thu Sep 11, 2014 1:52 am UTC
Forum: Mathematics
Topic: Favorite math jokes
Replies: 1452
Views: 489334

Re: Favorite math jokes

How does a topologist dry their clothes?

Spoiler:
With a closed line!
Tue Sep 09, 2014 8:30 pm UTC
Forum: Mathematics
Topic: Summing sigma(n,4)
Replies: 1
Views: 1816

Re: Summing sigma(n,4)

Does there exist a O(sqrt(n)) algorithm to sum sigma(k,4) where k varies from 1 to n. If so can it be generalized for values higher than 4? NOTE: sigma(k,4) is the summation of 4th powers of divisors of k. I leave you to work out the specifics, but what I'd do is consider the set S of positive inte...
Tue Sep 09, 2014 12:08 am UTC
Forum: Mathematics
Topic: Repeating summation
Replies: 6
Views: 3230

Re: Repeating summation

Aah nice one on the floor function. In the complex numbers, there are non-trivial solutions to x 3 =1 - in particular, x = -1/2 ± i*sqrt(3)/2 where i is the imaginary unit. Then x n would be unique from x n+1 and x n+2 , but equal to x n+3 . Can you open this up a bit more. How do i get it to repea...
Mon Sep 08, 2014 5:57 pm UTC
Forum: Mathematics
Topic: Repeating summation
Replies: 6
Views: 3230

Re: Repeating summation

That's an interesting question; on one hand, it's easy enough to do in the complex numbers; the useful property of -1 being exploited to get alternation is that it is a root of unity ; that is (-1) 2 = 1. In the complex numbers, there are non-trivial solutions to x 3 =1 - in particular, x = -1/2 ± i...
Sun Sep 07, 2014 11:21 pm UTC
Forum: General
Topic: ITT: We make xkcd slightly worse.
Replies: 8671
Views: 1831049

Re: ITT: We make xkcd slightly worse.

385+What-if 5:
The robot had picked up on its programmer's dating tactic - and it was love at first bonk.
Wed Sep 03, 2014 2:44 pm UTC
Forum: Individual XKCD Comic Threads
Topic: 1416: "Pixels"
Replies: 112
Views: 37961

Re: 1416: "Pixels"

All that zooming in and no zooming out? Seems a bit arrogant of Randall to assume that the original comic isn't itself a pixel in some great comic! (Maybe all of the xkcd comics are pixels and it's up to us to arrange them into a new comic?)
Sat Aug 30, 2014 9:15 pm UTC
Forum: Mathematics
Topic: Multiplication is wrong?
Replies: 10
Views: 4063

Re: Multiplication is wrong?

I'm a bit confused by the notation here. It doesn't really make sense to iterate a binary function - only maps f:S->S typically can be iterated. I think that your definition has issues that cause inelegance; it seems to be that you take: f n (x,y)=f(x,f n-1 (x,y)) f 0 (x,y)=y (Though since you alway...
Fri Aug 29, 2014 4:19 pm UTC
Forum: Mathematics
Topic: Is it possible for a math obsession to be unhealthy
Replies: 54
Views: 14348

Re: Is it possible for a math obsession to be unhealthy

So then why does the millenium prize exist? Mostly because some businessman wanted to give out prizes. And there's money attached to get people to notice his prizes. If he had just announced to give out paper ribbons, he would have been ignored as a crank. The attraction of the prizes is in the pro...
Fri Aug 29, 2014 2:16 am UTC
Forum: Mathematics
Topic: The Shortest String Containing all Permutations of n Symbols
Replies: 29
Views: 28605

Re: The Shortest String Containing all Permutations of n Sym

Well, I was googling about this a little, and I found out that the value conjectured in this thread (and elsewhere) is false . So, I guess that makes this problem even harder, since we don't even have a bound to try to prove. Prior to knowing this, I spent some time thinking and came up with a novel...
Thu Aug 28, 2014 6:20 pm UTC
Forum: Mathematics
Topic: Irrational to the irrational=rational
Replies: 5
Views: 2498

Re: Irrational to the irrational=rational

One option is to take limits of rational exponents, which also means you have to show that doing so is well-defined. If a > 1, then x -> a^x is isotone over the rationals. So it doesn't seem so hard to extend that to the reals by a^b = sup {a^q | q rational and q<=b}. One would probably also want t...
Thu Aug 28, 2014 3:20 am UTC
Forum: Mathematics
Topic: Irrational to the irrational=rational
Replies: 5
Views: 2498

Re: Irrational to the irrational=rational

Are we restricted to the reals? Because e and iπ are irrational, but e iπ is rational. That's the nicest example I can think of. Of course, one still has to prove things like that e is irrational (not so hard) and so is π (kind of hard). Though, a more conventional-but-not-classic answer might be......
Wed Aug 27, 2014 4:21 pm UTC
Forum: Mathematics
Topic: Is it possible for a math obsession to be unhealthy
Replies: 54
Views: 14348

Re: Is it possible for a math obsession to be unhealthy

Mathematics isn't about the money but the appreciation, knowledge. And, apparently, making sure that anyone who is attracted to math for any other reason is put in their proper place especially if they're one of those darned internet-using-young'un-types-who-are-passionate-for-the-subject-but-are-a...
Sun Aug 17, 2014 3:42 am UTC
Forum: Mathematics
Topic: Restricted Halting Problem
Replies: 20
Views: 5394

Re: Restricted Halting Problem

It is of course trivial to create a program that correctly outputs Yes, No, or Maybe for the halting problem. In particular, a Maybe answer is never wrong so you just output that for anything that can't be proven Yes or No in a certain amount of time. The trick is making the range of Maybe's as sma...
Sun Aug 17, 2014 1:37 am UTC
Forum: Mathematics
Topic: Just a short little math joke...
Replies: 2
Views: 1991

Re: Just a short little math joke...

You should consider reading the thread here: forums.xkcd.com/viewtopic.php?f=17&t=5683 - it's full of math jokes (which have all been thoroughly checked for mathematical accuracy, naturally)
Tue Aug 12, 2014 3:14 am UTC
Forum: Mathematics
Topic: Is a Mobius Strip homeomorphic to a torus/ring?
Replies: 12
Views: 6063

Re: Is a Mobius Strip homeomorphic to a torus/ring?

Having slept on it, I'm thinking homology might make sense. At least simplical homology seems to make sense. Maybe someone who knows more than me can tell me if this is correct (spoilered since it's not really all that relevant to the original thread): The 0th homology group is the quotient group of...
Mon Aug 11, 2014 3:54 am UTC
Forum: Mathematics
Topic: Is a Mobius Strip homeomorphic to a torus/ring?
Replies: 12
Views: 6063

Re: Is a Mobius Strip homeomorphic to a torus/ring?

The intimidating part of homology is the details. The idea that there's a precise mathematical way to measure how many "holes" or "loops" a surface has, however, isn't so intimidating. Niether is the number of connected pieces a space has. Isn't homology also one of the easiest ...
Mon Aug 11, 2014 2:06 am UTC
Forum: Mathematics
Topic: Where can I find a list of modulus laws?
Replies: 8
Views: 4181

Re: Where can I find a list of modulus laws?

Before providing more of an answer: Are you talking about modular arithmetic or the "mod" operator and integer division (as might be encountered in computer programming)? Instance mod 7 (15) = 1 That type of modulus. I hate how these aren't introduced in school. I find them fascinating. I...

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