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July 27

Atoms in measure theory

So, there is a qualifying exam problem which defines an atom to be

$A_{x}=\{\cap B:x\in B\in {\mathcal {M}}\}$

But, in the article in Wikipedia, Atom_(measure_theory), it says that Lebesgue measure has no atoms. Is this a situation where both are right, some people define it one way and some another? StatisticsMan (talk) 00:12, 27 July 2009 (UTC)[reply]

It's defining atoms differently, and individual points in space are the "atoms" of the sigma-algebra of Lebesgue-measurable sets. Notice that A_x is not required to have positive measure here.Ray^Talk 00:27, 27 July 2009 (UTC)[reply]

If I hear the word "atom" in measure theory, the first thing that comes to mind is a set of strictly positive measure that contains only one point. Or it may be a set of strictly positive measure that has no measurable subsets. "Atom" traditionally means something that cannot be divided into smaller pieces, so it should mean something that cannot be divided into smaller measurable parts.

In particular, in the theory of Boolean algebras, one sometimes considers measures on the algebra of all clopen subsets of a Stone space (not generally a sigma-algebra), and in that context, the standard usage is a one-element set whose measure is strictly positive. Michael Hardy (talk) 02:47, 27 July 2009 (UTC)[reply]

In the context of measure theory I think atom is universally defined (see e.g. Halmos) as a measurable set of positive measure that has no subsets of strictly less, yet positive measure. In other words, the measure restricted to it has only two values. A measure is pure atomic iff any sets of positive measure contains an atom (e.g. the counting measure); it is non-atomic or divisible iff it has no atoms (e.g the Lebesgue's). And I'd like to introduce "heavy atom" for an atom of infinite measure. --pma (talk) 18:10, 29 July 2009 (UTC)[reply]

Statistical terms.

Dear Wikipedians:

Within the context of statistics, what do the terms "degree of performance" and "degree of consistency" mean?

Thanks.

174.88.241.78 (talk) 00:01, 27 July 2009 (UTC)[reply]

I haven't encountered either term, but consistency (statistics) could have some relevant material. On the other hand, there are various terms having to do with application of statistics to particular fields, in which the term might have other meanings. Michael Hardy (talk) 02:41, 27 July 2009 (UTC)[reply]

Thanks Dr. Hardy for your response. 174.88.241.78 (talk) 21:02, 27 July 2009 (UTC)[reply]

Resolved

Chaos

I'm looking for an example of a 2D continuous chaotic map, but the only continuous maps I've found are in three dimensions. Does anybody know of a 2D example? If so, what are its defining equations? —Preceding unsigned comment added by 72.197.202.36 (talk) 05:21, 27 July 2009 (UTC)[reply]

Do you want a map from

\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{2}

? I can do

\mathbb {R} \rightarrow \mathbb {R} ^{2}

. --Tango (talk) 14:18, 27 July 2009 (UTC)[reply]

For the R2->R2 case, try the Henon map. risk (talk) 15:11, 27 July 2009 (UTC)[reply]

A 2d phase space for a continuous dynamical system can't be chaotic, see Poincaré–Bendixson theorem. Dmcq (talk) 17:40, 27 July 2009 (UTC)[reply]

Ah, but he was asking for a continuous map, which would imply a discrete dynamical system, where the function defining it is continuous. At least that's how I interpreted it. risk (talk) 18:34, 27 July 2009 (UTC)[reply]

I must admit that sounds more likely, I think I've got in the habit of wondering if people mean something else and just taking key words out of what they say :) Dmcq (talk) 19:17, 27 July 2009 (UTC)[reply]

Is the ~~Lorenz attractor~~ (oops, 3D) Smale horseshoe not the most famous one? 70.90.174.101 (talk) 09:46, 28 July 2009 (UTC)[reply]

Easter eggs in baskets

Hey all,

My math teacher gave me a problem today: given twenty baskets, put a number of easter eggs in each basket such that no two baskets have the same number of eggs and the total number of eggs is minimized.

After confirming that the number of eggs in each basket had to be a positive integer, I said $\sum _{i=1}^{20}i=210$ . He says that's wrong and that the answer is nontrivial, but refuses to tell me what he thinks it is.

Confused, I proved that the above sum is the optimal solution for a set of positive integers - he said my proof was valid, but my answer was not.

No idea what he's getting at, and beginning to think that it's semantics - any ideas? Thanks, Aseld ^talk 10:47, 27 July 2009 (UTC)[reply]

It's possible he's one of the people who considers zero to be positive. Algebraist 11:10, 27 July 2009 (UTC)[reply]

Can you put a basket in a basket? In which case you could get away with twenty eggs even without a basker of zero eggs. Dmcq (talk) 12:00, 27 July 2009 (UTC)[reply]

What was the question? You suppose that you are requested to provide the total number of eggs in the baskets, but why suppose so? Perhaps the question is in how may ways it can be done, and the answer is 2432902008176640000. Bo Jacoby (talk) 12:16, 27 July 2009 (UTC).[reply]

I think putting baskets inside each other is probably the answer wanted, however for a mathematician I think the correct answer is to put ever larger and larger IOU's for eggs in each basket, they can be paid back in coffee and biscuits and odd jobs ;-) Dmcq (talk) 12:36, 27 July 2009 (UTC)[reply]

The problem description requests a positive number of eggs to be put in each basket. — Emil J. 13:12, 27 July 2009 (UTC)[reply]

Oh I see, I thought that was just an assumption but it was confirmed. Very good you never know what kind of devilish tricky thing they might be up to. Twenty eggs in total it is then. Negative numbers can be used to solve problems as at Math Jokes and Archimedes in the one where the mathematician, biologist and physicist are sitting in a street cafe. 13:56, 27 July 2009 (UTC)

"Given twenty baskets" is the key, as the solution depends on the form of the baskets. If they are undeformable and the same size they can't be put inside each other, unless they are tapering and will fit together with an egg-sized space between. "Twenty shopping bags" would be better, as they may be presumed to be flexible.86.148.186.45 (talk) 14:12, 27 July 2009 (UTC)[reply]

Thanks for the responses everyone. I think the stacking baskets is the answer that's wanted - will find out tomorrow. Little disappointed in the problem, I was expecting some pretty mathematical solution. I would have never thought of that myself - thank you. ^_^ Aseld ^talk 14:28, 27 July 2009 (UTC)[reply]

Yes I'd like some quantum mechanical sort of solution where you had fewer than twenty eggs but whenever you looked at any two baskets they always had a different number of eggs. Can't see how to so something like that though. And they'd have to be very small eggs. Dmcq (talk) 14:37, 27 July 2009 (UTC)[reply]

If a basket wraps around an egg n times, shouldn't we say it contains it with multeplicity n? So my solution is: 1 egg. Hey! I see many simple eggs fling to me. --pma (talk) 21:17, 29 July 2009 (UTC)[reply]

Representing an affine transformation

I'm trying to find a good intuitive representation for an n-dimensional affine map, as a vector of real numbers. The simplest representation would be to represent the map as a transformation matrix and a translation vector (n^2 + n real values), but I'd like each value to have a single clearly defined function.

I'm thinking about splitting the transformation into a translation, a scaling, a rotation and various shears. My first question is, does this cover all affine operations? I've found a paper that says so, but I'm not sure if they're just referring to the 2D case.

The translation and scaling are each defined by n values with a straightforward interpretation. The rotation can be represented by (n^2 -n)/2 angles, but I can't find any good references for the shear. From shear matrix, I gather that there are at least (n^2 -n) ways to create an elementary skew matrix (one for each non-diagonal element). But then I started thinking, I can do a QR decomposition on a matrix, and end up with a rotation matrix, and an upper diagonal matrix, which means that the skew component can be encoded in just the upper non-diagonal elements of the matrix. I've done a lot of googling, but everything I've found so far is specific to the 2D case. It seems like this would be something that someone worked out a long time ago, but I can't find the right things to search for. 146.50.144.17 (talk) 15:01, 27 July 2009 (UTC)[reply]

The singular value decomposition expresses a matrix as a composition of a rotation, a scaling, and a rotation. JackSchmidt (talk) 15:10, 27 July 2009 (UTC)[reply]

Lebesgue Density

Qual prob as usual. I have a solution but I'm stuck on the last little bit. They define the Lebesgue density of E at a point $x\in \mathbb {R}$ by

  $\delta (x)=\lim _{h\to 0+}{\frac {m[(x-h,x+h)\cap E]}{2h}},$

provided this limit exists. Actually, in the problem they have $\lim _{h\to 0}$ which doesn't make sense. The question is: Let $E=\bigcup _{n=1}^{\infty }(1/(2n+1),1/(2n))$ . Calculate the density of E at 0.

Proof is straightforwrd. The basic idea is for a given natural number i, assume 1 / (2i) < h < 1 / (2i-1) and define

  $S_{i}=m(E\cap (0,h))=\sum _{n=i}^{\infty }({\frac {1}{2n}}-{\frac {1}{2n+1}})={\frac {1}{4}}\sum _{n=i}^{\infty }{\frac {1}{n(n+1/2)}}$ .

Then, this is clearly bounded by

  ${\frac {1}{4}}\sum _{n=i}^{\infty }{\frac {1}{(n+1)^{2}}}\leq S_{i}\leq {\frac {1}{4}}\sum _{n=i}^{\infty }{\frac {1}{n^{2}}}$ .

Using the idea of upper and lower Riemann sums and the integral $\int _{i}^{\infty }1/s^{2}$ , we get

  ${\frac {1}{4(n+1)}}\leq S_{i}\leq {\frac {1}{4(n-1)}}$ .

Very last step says: "Hence $S_{i}/2h\to 1/4$ as $i\to \infty$ ." And, this is where I get confused. The thing is, in doing this limit, we are skipping infinitely many points as h gets closer and closer to 0. The way we defined h, h is always in some interval 1 / (2i) < h < 1 / (2i-1). But, not every h close to 0 is in such an interval as these intervals comprise 1/2 < h < 1, 1/4 < h < 1/3, 1/6 < h < 1/5. So, how does this show the limit as h goes to 0 is 1/4 if we don't use large chunks of h in any neighborhood about 0? StatisticsMan (talk) 16:29, 27 July 2009 (UTC)[reply]

You got ~~all~~ many of the order signs backwards, it should be 1/(2i) < h < 1/(2i − 1), 1/2 < h < 1, 1/4 < h < 1/3, etc. Now, back to the point: the exact same calculation works assuming just 1/(2i) ≤ h ≤ 1/(2i − 2), then you do not leave out any points. — Emil J. 16:54, 27 July 2009 (UTC)[reply]

By the way, notice that you can estimate

\sum _{n=i}^{\infty }{\frac {1}{n(n+1/2)}}\geq \sum _{n=i}^{\infty }{\frac {1}{n(n+1)}}=\sum _{n=i}^{\infty }\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)={\frac {1}{i}}

\sum _{n=i}^{\infty }{\frac {1}{n(n+1/2)}}\leq \sum _{n=i}^{\infty }{\frac {1}{(n-1/2)(n+1/2)}}=\sum _{n=i}^{\infty }\left({\frac {1}{n-1/2}}-{\frac {1}{n+1/2}}\right)={\frac {1}{i-1/2}}

without doing any integrals. — Emil J. 17:06, 27 July 2009 (UTC)[reply]

Okay, I think I fixed at least most of the mistakes. Thanks for noticing that. And, more importantly, thanks for the help on the problem. StatisticsMan (talk) 21:25, 27 July 2009 (UTC)[reply]

One small caveat to what Emil J said. For 1/(2i) ≤ h ≤ 1/(2i − 2),

{\frac {m[(-h,h)\cap E]}{2h}}

isn't always equal to S_i but is between S_i and S_i-1, which still works just fine. Rckrone (talk) 00:59, 28 July 2009 (UTC)[reply]

Yea, I already noticed that, but thanks in case I didn't. I am considering 1/(2i) ≤ h ≤ 1/(2i − 2) and got rid of what I called S_i. I just used the summation stuff EmilJ mentioned to get the bound

1/4\cdot 1/(i+1)\leq m(E\cap (0,h))\leq 1/4\cdot 1/(i-1)

. Then, I divide through by 2h and use the first inequalities to get

i\leq 1/(2h)\leq i+1

so I get

\lim _{h\to 0+}{\frac {1}{4}}{\frac {i}{i+1}}\leq \delta (0)\leq \lim _{h\to 0}{\frac {1}{4}}{\frac {i+1}{i-1}}

.

Of course, the left side is slightly less than 1/4 and the right side is slightly more than 1/4 and both sides go to 1/4 as i goes to infinity, which happens as h goes to 0. So, limit exists and is 1/4. StatisticsMan (talk) 01:24, 28 July 2009 (UTC)[reply]

A beautiful solution?

Dear Wikipedians:

While trying to solve a question through the messy "cases" approach, I hit upon an alternative idea which led me to what I believe is a beautiful solution, but I need your expertise in validating my solution.

The question reads, "An elevator in a building starts with 6 people and stops at 7 floors. If each passenger is equally likely to get off at any floor and independent of other passengers, find the probability that exactly 3 people in total get off at the 3rd or 4th floor".

After trying, futilely, the messy aproach of P(3 on 3rd floor) + P(3 on 4th floor) + P(1 on 3rd & 2 on 4th) + P(2 on 3rd floor & 1 on 4th floor), I hit upon an elegant idea of combining floors 3 and 4 into a single floor with P(get off) = 2/7. Using the binomial distribution, I was able to solve the question using one short expression:

${6 \choose 3}\left({\frac {2}{7}}\right)^{3}\left({\frac {5}{7}}\right)^{3}=0.17$

Is it right?

Thank.

70.29.26.86 (talk) 21:16, 27 July 2009 (UTC)[reply]

Looks right to me. Rckrone (talk) 02:13, 28 July 2009 (UTC)[reply]

Thanks! 174.88.243.93 (talk) 15:22, 28 July 2009 (UTC)[reply]

Resolved