Wikipedia:Reference desk/Archives/Mathematics/2014 March 25

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March 25

Abstract Algebra Integral Domains

Let A be a finite integral domain. Prove the following: If A has characteristic 3, and 5·a=0, then a=0.

I have For the sake of contradiction, assume a≠0. 5·a=0 a+a+a+a+a=0 char(A)=5 which is a contradiction to the given statement char(A)=3 Therefore, a=0.

Any suggestions? I am not sure how to get a contradiction to a≠0. — Preceding unsigned comment added by Abstractminter (talk • contribs) 00:02, 25 March 2014 (UTC)

This looks like homework to me. A suitable hint would be: what is the definition of an integral domain? RomanSpa (talk) 02:46, 25 March 2014 (UTC)

Is the characteristic 3 or is it 0? I don't see how a finite ring can have characteristic zero. —Kusma (t·c) 08:26, 25 March 2014 (UTC)

Tribbonacci numbers

Wikipedia gives only one definition to the Tribbonacci numbers, but OEIS gives 3 different definitions. For what these definitions are, go to OEIS and type Tribonacci and the first 3 definitions are the 3 definitions it gives. Georgia guy (talk) 17:15, 25 March 2014 (UTC)

As far as I can see, the 3 OEIS sequences simply differ in the values of the first three terms (0,0,1), (1,1,1) or (0,1,0). The recurrence relationship is the same for all 3 sequences. Gandalf61 (talk) 17:27, 25 March 2014 (UTC)

But Wikipedia thinks thtat (0,0,1) is by definition the Tribbonacci sequence. Any thoughts?? Georgia guy (talk) 17:30, 25 March 2014 (UTC)

Exactly the same convention applies to Tribonacci sequences as to Fibonacci sequences. "The" Tribonacci sequence begins 0, 0, 1 but you can define "a" sequence beginning with any numbers you choose. Dbfirs 17:41, 25 March 2014 (UTC)

Unlike the Fibonacci sequence, where starting with 1, 0 or 0, 1 or 1, 1 only results in different offsets, in the Tribonacci sequence you get different numbers for different starting values, even when the starting values seem to be reasonable extensions of the Fibonacci definition. So the definition of "the" tribonacci numbers is somewhat vague. The WP article being referenced is poorly sourced and what sources are given are not always reliable, so there may be a certain amount of OR there that shouldn't be taken too seriously. OEIS is somewhat inconsistent as well sometimes referring to A000073 and sometimes to A001590 as "the" tribonacci numbers. I'm guessing there isn't really a consensus in the literature on what a tribonacci number actually is. --RDBury (talk) 21:39, 25 March 2014 (UTC)

I agree with everything that RDBury writes above except that he doesn't mention the possibility of starting a Fibonacci sequence with numbers other than "1, 0 or 0, 1 or 1, 1". Starting with other different values gives a different sequence that is often called a Fibonacci sequence, though possibly it ought to be called a Fibonacci-type sequence See 1, 4, 5, 9, 14, 23, 37, 60, 97 for example. Dbfirs 08:09, 26 March 2014 (UTC)

Using 1-s2/s1 as a measure of one-dimensionality

Let $s_{i}$ be the singular values of a nonzero $m\times n$ matrix $M$ with $m>n$, sorted in descending order such that $s_{i}>s_{i+1}$.

In particular $s_{1}$ is the largest singular value.

Then, the quantity $q=1-s_{2}/s_{1}$ is 1 if the columns of $M$ are linearly dependent, i.e. if the data in $M$ is purely one-dimensional. In any case $0\leq q\leq 1$. High $q$ means that $M$ is nearly one-dimensional, low $q$ means a second dimension is also important (and maybe also a third, fourth ...).

Is this quantity known in the mathematical or statistical literature? Does it have a name?

82.69.98.189 (talk) 19:44, 25 March 2014 (UTC)