Wikipedia:Reference desk/Archives/Mathematics/2013 August 15

Welcome to the Wikipedia Mathematics Reference Desk Archives
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 current reference desk pages.

August 15

How does one define "symmetrized Kronecker product"

Hello,

I am somewhat familiar with representations and characters of groups, and I know that by taking the Kronecker product of two representations of the same group, one ends up with a character which is the product of the two original characters, but I bumped into "symmetrized Kronecker product". However, I found it surprisingly hard to find a clear definition of this on the web or in several books.

Can anyone help me with this? Many thanks! Evilbu (talk) 11:52, 15 August 2013 (UTC)

I believe it's the same except instead of using the tensor product you use the Symmetric tensor product. Note that this works only with the product of a representation with itself. Note also that there is an antisymmetric product as well. Start with a module V, then the subspace of VxV generated by expression of the form axb-bxa is a submodule. Call the quotient the symmetrized product and the submodule antisymmetrized product. It should be possible to find the characters of the two corresponding representations from the characteristic polynomials of the corresponding matrices: If the polynomial is l²-al+b then the character of the antisymmetric product is b (the determinant) and the symmetric product a²-b. Higher order tensors work similarly but are a bit more complex.--RDBury (talk) 12:56, 15 August 2013 (UTC)

If the arithmetic mean of seven different numbers is 5...

If the arithmetic mean of seven different numbers is 5, and Column A has a median of the seven numbers, and Column B is 5, then which column is greater/bigger/larger?

Answer Key: The relationship cannot be determined from the information given.

• How should you tackle this type of problem, practically speaking?
• What is the strategy behind this problem?
• What exactly are they asking?
• Why cannot the relationship be determined from the information given?

In my head, I have the sum of seven different numbers set to 35, because the arithmetic mean is stated. Now, using the same seven numbers, the median number is Column A. The only problem is, what if one or more numbers in the set is/are a very big positive/negative number, and the result just comes out to be 35? How would that compare with Column B, which is 5? Would Column A be greater/lesser than Column B? Could that be the reason why the relationship cannot be determined from the information given - that it is unclear whether Column A has a very big positive/negative number? Sneazy (talk) 13:26, 15 August 2013 (UTC)

Maybe the 7 numbers are 1, 2, 3, 4, 5, 6, 14 with a median of 4, which is < 5. Or maybe the numbers are 1, 2, 3, 6, 7, 8, 8 with a median of 6, which is > 5. But the problem says there are "seven different numbers" -- does different mean distinct, so that no two of them can be equal? Then replace the second example with 0, 1, 2, 6, 7, 8, 11, with median of 6 which is > 5. But by "numbers" do they mean positive integers? Then with distinct positive integers the highest you can get the median to be is with 1, 2, 3, 5, 7, 8, 9, with a median of 5 which is = column B. Duoduoduo (talk) 14:19, 15 August 2013 (UTC)

There is really no strategy involved here. The question is testing whether you understand that outliers affect the mean without affecting the median -- therefore knowing one does not tell you the other. In other words, the question is testing whether you have a basic intuitive understanding of means and medians. Looie496 (talk) 14:47, 15 August 2013 (UTC)

While it's true that the question is testing whether you understand the difference between mean and median, I disagree that there's no strategy involved in answering the question. Once the question has been made clear, perhaps from context, as to what numbers (negatives? zero?) are permitted, then the strategy is to try to find an example of 7 numbers that makes column A bigger, an example where both columns are equal, and an example where column B is bigger, or to prove that one or more of these is impossible to find an example for. Duoduoduo (talk) 15:23, 15 August 2013 (UTC)

Continued fraction types.

I was looking at the pages on Continued Fractions and they fall into a couple of categories:

• Finite continued fractions which are the rational numbers
• Repeating pattern infinite contined fractions which (plus the rationals) are quadratic irrationals (the solutions to quadratic equations) Q(sqrt(c))
• Repeating patterns with alteration that are finitely describable (e = [2;1,2,1,1,4,1,1,6,1,1,8,1,1,...]

"2;1 followed in blocks of three starting with 2,1,1 increasing the value of the first of the three by 2 each time)

• "Apparently Random" patterns. (Pi is [3;7,15,1,292,1,1,1,2,1,3,1,…] )

Is there a description of the numbers that fit the first three but not the last (so 4, (1+sqrt(8)/17), and e) are in the set but Pi is not?Naraht (talk) 16:27, 15 August 2013 (UTC)

This is not the answer, but Liouville numbers are somewhat related. They have unbounded coefficients in the continued fraction. But e is like that, but it is in the third category. Bubba73 ^{You talkin' to me?} 16:41, 15 August 2013 (UTC)

The issue is coming up with a good formalization of "finitely describable". Most natural formalizations will include Pi's continued fraction, since Pi is computable.--96.245.213.29 (talk) 21:18, 15 August 2013 (UTC)

(stupid VE!) pi is only describable optimally by computing pi itself, while OTOH the c.f. of e can be described by a very short algorithm:

x <- 1, y <- 2, z <-1

Output 2;

(*)

Output x,y,z,

Add 2 to y,

Repeat from (*)

That's 6 lines of code, or 5 if you don't count the label (*) - but it is not a finite state machine because y goes through all even positive integers.

Not sure if you can define the quality of a number that way, though. It lacks scientific rigor and one would have to specify what operations are allowed in the algorithm.

(IT's not an algorithm right now because it doesn't end, either.) - ¡Ouch! (^{hurt me} / _{more pain}) 11:59, 16 August 2013 (UTC)

Maybe a good formalization for "describable" patterns will be linear recurrence relations. For example, for e you have $a_{n}=2a_{n-3}-a_{n-6}$. -- Meni Rosenfeld (talk) 08:11, 18 August 2013 (UTC)

That works. You'd have to define the first 8 terms I think, but otherwise, that's a good definition.Naraht (talk) 14:26, 19 August 2013 (UTC)