Search found 140 matches

by aleph_one
Wed Sep 14, 2011 6:32 pm UTC
Forum: Science
Topic: What do you feel in orbit around a massive object?
Replies: 18
Views: 2594

Re: What do you feel in orbit around a massive object?

Thanks for the answers! If I understand correctly, orbit, freefall, and uniform motion are perceived the same, because after a non-inertial change into coordinates where one is at rest, the situations become equivalent, as the virtual forces cancel out the actual ones. Riddle me this: I recall readi...
by aleph_one
Wed Sep 14, 2011 5:07 pm UTC
Forum: Science
Topic: What do you feel in orbit around a massive object?
Replies: 18
Views: 2594

What do you feel in orbit around a massive object?

Suppose you are in a circular orbit around a massive body, at a distance so that the gravitational acceleration is much greater than 1g (that on the surface of the Earth). My question is: do you perceive the crushing gravitational force? For instance, would there be adverse effects on your body? Als...
by aleph_one
Tue Aug 02, 2011 5:21 am UTC
Forum: Logic Puzzles
Topic: Neologisms
Replies: 20
Views: 5891

Re: Neologisms

Since multiple poster starting with smakibbfb have given the correct answer, I'll post the solution as I worded it: The formation of words in the first list required breaking up existing morphemes (units of meaning), whereas words on the second list were formed solely by combining existing m...
by aleph_one
Tue Jul 26, 2011 8:01 pm UTC
Forum: Logic Puzzles
Topic: Neologisms
Replies: 20
Views: 5891

Neologisms

What feature do words from the first list share that ones on the second list don't?

'Yes' instances:
    backronym
    cheeseburger
    chocoholic
    gaydar
    gerrymander
    mathlete
    monokini
    telethon
'No' instances:
    blamestorm
    genericide
    metrosexual
    paratrooper
    psychonaut
    technophobe
    wikipedia
    wordsmith
by aleph_one
Thu Jan 13, 2011 5:54 pm UTC
Forum: Mathematics
Topic: You make a measurement.
Replies: 9
Views: 1044

Re: You make a measurement.

Let me rephrase to avoid considering a prior distribution on the standard deviations: Given a single sample from a Gaussian of mean zero and unknown standard deviation, the standard deviation that gives the highest probability density of producing that sample equals the absolute value of the sample.
by aleph_one
Thu Jan 13, 2011 6:55 am UTC
Forum: Mathematics
Topic: You make a measurement.
Replies: 9
Views: 1044

Re: You make a measurement.

I think it might be this fact: Given a simple sample from a Gaussian of mean zero and unknown standard deviation, the most likely standard deviation equals the absolute value of the sample.
by aleph_one
Sun Jan 09, 2011 6:58 pm UTC
Forum: The Help Desk
Topic: Remap to single-key volume control
Replies: 0
Views: 620

Remap to single-key volume control

On my Asus K50IJ-X8 laptop, Mute, Volume Down, and Volume Up are activated by pressing Fn+F10, Fn+F11, and Fn+F12 respectively. This is very inconvenient because I need two hands to press these key combinations, and they are easy to mispress. Is there a way to assign some unused keys or combinations...
by aleph_one
Wed Jan 05, 2011 2:00 am UTC
Forum: Mathematics
Topic: Weird Tensor Identity
Replies: 7
Views: 2602

Re: Weird Tensor Identity

I can prove that the set of vectors of form v^{\odot n} for unit v spans the whole space, and the set of U^{\odot n} acts transitively on these, which I believe suffices. I realized I got this wrong. There's no reason that the representation acting transitively on a spanning set should imply irredu...
by aleph_one
Mon Jan 03, 2011 9:50 pm UTC
Forum: Logic Puzzles
Topic: Card guessing game
Replies: 18
Views: 11693

Re: Card guessing game

The winning probability is that at the start of the game. It is the maximum over all strategies of the probability of that strategy winning on a randomly shuffled deck. The probability of winning at any point in the game (over the distribution of decks conditioned on what you've) has been calculated...
by aleph_one
Mon Jan 03, 2011 5:14 am UTC
Forum: Mathematics
Topic: Weird Tensor Identity
Replies: 7
Views: 2602

Re: Weird Tensor Identity

Thanks for the pointers to the book; I'll take a look at it tomorrow when my school's library reopens. I've been meaning to learn about Lie groups, as I keep seeing unexpected references to them in quantum papers. I've looked up the Schur orthogonality relations, and I think I see how they are relev...
by aleph_one
Wed Dec 29, 2010 5:48 pm UTC
Forum: Mathematics
Topic: Weird Tensor Identity
Replies: 7
Views: 2602

Re: Weird Tensor Identity

Thanks for the response, Bob! I'm afraid I only have a passing familiarity with the Schur Orthogonality Relations, so I don't understand your proof. Please do enlighten me. However, I think I can reconstruct most of your proof without them. Thm: Let P = Exp_{U\in U(m)} \left[ U \otimes U^{-1...
by aleph_one
Thu Dec 23, 2010 1:21 am UTC
Forum: Logic Puzzles
Topic: Card guessing game
Replies: 18
Views: 11693

Re: Card guessing game

jbwraith, the strategy of guessing only guess after seeing 26 black will fail if the last card is black, since you'll have to guess before it. In fact, it only works half the time, exactly if the last card is red. You suggested simply guessing after you get a large number of black cards - can you re...
by aleph_one
Tue Dec 21, 2010 5:55 am UTC
Forum: Mathematics
Topic: Fractal Groups?
Replies: 5
Views: 1435

Re: Fractal Groups?

I'm not sure if this is the type of thing you're talking about, but the free group (say on two elements) is often depicted with a fractal Cayley graph. The self-similar nature of the group is used in Banach and Tarski's construction for the eponymous paradox.
by aleph_one
Wed Dec 15, 2010 8:16 pm UTC
Forum: Mathematics
Topic: Weird Tensor Identity
Replies: 7
Views: 2602

Weird Tensor Identity

While playing around with some quantum stuff, I found a weird identity: Exp_{U\in U(m)} \left[ (Uv) \otimes (U^{-1}w) \right] = \frac{1}{m} w \otimes v where the expectation is taken over the Haar-uniform distribution of unitary matrices. Now, I can prove the identity by chec...
by aleph_one
Mon Dec 13, 2010 8:40 pm UTC
Forum: Mathematics
Topic: Game Theory at a high school level
Replies: 11
Views: 2267

Re: Game Theory at a high school level

I second xiron's recommendation of the Straffin book.
by aleph_one
Fri Dec 03, 2010 7:57 pm UTC
Forum: Mathematics
Topic: What do conics in C^n look like?
Replies: 2
Views: 729

Re: What do conics in C^n look like?

I like your proof for the n=2 case. I had not seen before the trick of homogenizing to \mathbf{CP}^n . I used the less generalizable trick of doing a coordinate transform from z_1^2 + z_2^2 = 1 to z_1 z_2= 1 , which makes it easy. Could there be another non-degenerate conic for n variables that's ea...
by aleph_one
Thu Dec 02, 2010 1:12 am UTC
Forum: Individual XKCD Comic Threads
Topic: 0828: "Positive Attitude"
Replies: 92
Views: 19601

Re: 0828: "Positive Attitude"

Someone beat me to mentioning Barbara Ehrenreich's book, "Bright-Sided: How the Relentless Promotion of Positive Thinking Has Undermined America." She does a good job of debunking the idea that positive thinking helps cure health problems (or anything else). I highly recommend her talk Sm...
by aleph_one
Thu Dec 02, 2010 12:29 am UTC
Forum: Mathematics
Topic: What do conics in C^n look like?
Replies: 2
Views: 729

What do conics in C^n look like?

What does the subset of \mathbf{C}^n that satisfies z_1^2 + \dots + z_n^2 =1 look like topologically (interpreting \mathbf{C}^n as \mathbf{R}^{2n}) ? It's two points for n=1 and I think its \mathbf{R}^{2}-(0,0) = R \times S^1 for n=2 , but I can't find a systematic way to imagine this su...
by aleph_one
Thu Dec 02, 2010 12:04 am UTC
Forum: Logic Puzzles
Topic: Card guessing game
Replies: 18
Views: 11693

Card guessing game

(Stolen from this MathOverflow answer by Kevin Buzzard. Don't look at the comment after it; it has the key tio the solution.) A friend deals cards face-up one a time from a shuffled deck of 26 red cards and 26 black cards. At some point before all the cards are dealt, you must call "stop",...
by aleph_one
Wed Dec 01, 2010 11:34 pm UTC
Forum: Logic Puzzles
Topic: Private and Public
Replies: 6
Views: 2176

Re: Private and Public

What video did you find this problem from? I'd be curious to hear others, as this was quite a nice one. EDIT: I figured out how to make my proof rigorous: To proof the fact that the ties (point on boundaries of private regions) are measure zero, not that the directions in which there's a...
by aleph_one
Wed Dec 01, 2010 10:55 pm UTC
Forum: Logic Puzzles
Topic: Private and Public
Replies: 6
Views: 2176

Re: Private and Public

I got a simple solution that's similar to Token's. Define the tangent direction of a point on a planet's surface be the direction from it's planet's center to it. Such a point is visible only from points that lie further in space along its tangent direction than it. So, a point is privat...
by aleph_one
Wed Oct 13, 2010 1:03 am UTC
Forum: Mathematics
Topic: Bayesian Puzzle
Replies: 3
Views: 892

Re: Bayesian Puzzle

Since there haven't been replies, I'll post the answers (but not yet the solutions).

1) Uniform over {0,1,...,n} (so 1/(n+1) for each).
2) (k+1)/(n+2)
by aleph_one
Sun Oct 10, 2010 3:31 pm UTC
Forum: Mathematics
Topic: Giving slogans for mathematical disciplines
Replies: 13
Views: 2288

Re: Giving slogans for mathematical disciplines

- Combinatorics: Bijections or bust.
- Real analysis: Fix epsilon > 0.
- Multilinear algebra: It's canonical, really!
by aleph_one
Sun Oct 10, 2010 12:34 am UTC
Forum: Mathematics
Topic: Bayesian Puzzle
Replies: 3
Views: 892

Bayesian Puzzle

You have a weighted coin for which your prior distribution on its weighting, i.e. its probability of landing on heads, is uniform on [0,1]. In other words, you may imagine that all you know about the coin is that it was made in a factory that produces each coin by picking a random value in [0,1], an...
by aleph_one
Sat Oct 09, 2010 7:28 pm UTC
Forum: Mathematics
Topic: Seperate branch for indeterminate
Replies: 16
Views: 1436

Re: Seperate branch for indeterminate

You might like non-standard analysis. It is a way to do calculus without limits by using a variant of the real numbers that includes infinite and infinitesimal values. But, no, 0/0 is still not defined.
by aleph_one
Sat Oct 09, 2010 6:50 pm UTC
Forum: Mathematics
Topic: Set addition with the empty set?
Replies: 5
Views: 851

Re: Set addition with the empty set?

Here's another way to see it. Note that memberwise addition can be defined as first taking the Cartesian product, then taking the image of the result under the map that takes each tuple (x,y) to x+y. Since the Cartesian of a set and an empty set is the empty set, it follows that the pairwise sum is ...
by aleph_one
Thu Oct 07, 2010 1:34 am UTC
Forum: Logic Puzzles
Topic: Color Hanjie
Replies: 3
Views: 1823

Re: Color Hanjie

This puzzle looks rather familiar. Was it in the Microsoft College Puzzle Challenge a couple a couple of years back? Edit: It indeed is the puzzle Resistance is Futile from 2008. Unfortunately, it requires a login to see. The name should give a hint as to how the colors correspond to numbers. My tea...
by aleph_one
Mon Oct 04, 2010 6:49 pm UTC
Forum: Mathematics
Topic: point not equidistant from two lattice points
Replies: 5
Views: 1580

Re: point not equidistant from two lattice points

There's an easy nonconstructive proof that nearly all points work: For each pair of lattice points, the locus of points that is equidistant to them is a line. So, the prohibited points are the union of countably many lines, which can't be whole plane because each line has measure 0 and the measure o...
by aleph_one
Thu Sep 30, 2010 12:53 am UTC
Forum: Mathematics
Topic: A rectangle with rounded edges is not an oval
Replies: 22
Views: 17307

Re: A rectangle with rounded edges is not an oval

Well, it doesn't work to round off the corners as circles, as these have nonzero curvature everywhere, so the place where the line and the circle meet will have an undefined second derivative. There probably is an smooth way to join two perpendicular line segments, considering that there are smooth ...
by aleph_one
Wed Sep 29, 2010 3:51 am UTC
Forum: Mathematics
Topic: showing sets equinumerous
Replies: 52
Views: 4228

Re: showing sets equanumerous

You could construct a bijection (one-to-one function) between the sets.
by aleph_one
Wed Sep 22, 2010 7:31 pm UTC
Forum: Mathematics
Topic: Wanted: Simple, rational orthogonal matrix
Replies: 11
Views: 2344

Re: Wanted: Simple, rational orthogonal matrix

Thanks, Nitrodon! This is excellent. I'm curious how you found the second matrix.

Also, I found that multiplying the matrices in the opposite order gives a result with less repetition of values.
by aleph_one
Tue Sep 21, 2010 8:51 pm UTC
Forum: Mathematics
Topic: Can different groups have similar multiplication tables?
Replies: 5
Views: 1243

Re: Can different groups have similar multiplication tables?

After writing it more clearly, I realized that the identification I proposed doesn't work, because it gives only 1/2 + 1/(4n) overlap, rather than the 3/4 that I promised. So, let me write out one that does work. Let N be even - we only need this for the two groups we'll use to be non-isomorphic. We...
by aleph_one
Mon Sep 20, 2010 8:12 pm UTC
Forum: Mathematics
Topic: Polynomial Inequality
Replies: 15
Views: 2641

Re: Polynomial Inequality

Ah, nice. Do you know where I can find a proof of this? Edit: Wait, I think I can show it. Any positive quadratic is the sum of two squares, by completing the square. Now, by fully factorizing the polynomial over the reals, we express it as a product of positive quadratic terms. Finally, the product...
by aleph_one
Mon Sep 20, 2010 7:54 pm UTC
Forum: Mathematics
Topic: Polynomial Inequality
Replies: 15
Views: 2641

Re: Polynomial Inequality

Wikipedia wrote:Every real polynomial in one variable is non-negative on ℝ if and only if it is a sum of two squares of real polynomials in one variable.

How strange. Can every sum of squares really be written as a sum of two squares? What about 1+x^2+x^4?
by aleph_one
Mon Sep 20, 2010 6:27 am UTC
Forum: Mathematics
Topic: Why aren't any good cryptosystems NP-complete?
Replies: 12
Views: 1830

Re: Why aren't any good cryptosystems NP-complete?

letterX wrote:Blah. Reading comprehension. I will use it next time.

Nah, it was me being unclear.
by aleph_one
Mon Sep 20, 2010 5:17 am UTC
Forum: Mathematics
Topic: Topology Extra Credit Problem
Replies: 12
Views: 1380

Re: Topology Extra Credit Problem

Something that helped me appreciate the subtlety and necessity of measure theory, even just for lengths of unions of real intervals, is the example on the Tricki titled Measure Theory is not Trivial about proving that an interval of length 2 cannot be covered by intervals of length {1/2, 1/4, 1/8, ....
by aleph_one
Mon Sep 20, 2010 3:53 am UTC
Forum: Mathematics
Topic: Why aren't any good cryptosystems NP-complete?
Replies: 12
Views: 1830

Re: Why aren't any good cryptosystems NP-complete?

CRGreathouse wrote:
BlackSails wrote:Afaik, nobody has proven where factoring falls in the complexity zoo. For all we know, factoring is in NP


Of course it's in NP (or, rather, its decision-problem variant is in NP). It's just not believed to be NP-complete, which would require NP = co-NP.

And also NP [imath]\subset[/imath] BQP.
by aleph_one
Mon Sep 20, 2010 1:56 am UTC
Forum: Mathematics
Topic: A rectangle with rounded edges is not an oval
Replies: 22
Views: 17307

Re: A rectangle with rounded edges is not an oval

Squares with rounded corners are so inelegant. Their boundary paths are not even infinitely differentiable!
by aleph_one
Mon Sep 20, 2010 1:52 am UTC
Forum: Mathematics
Topic: Wanted: Simple, rational orthogonal matrix
Replies: 11
Views: 2344

Re: Wanted: Simple, rational orthogonal matrix

Thanks for answers. When I get a chance, I'll compose some rational rotations and see if I can get decent matrix.

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