# leslie townes

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 May7 awarded Necromancer May7 awarded Good Answer Apr25 awarded Necromancer Mar29 awarded Nice Answer Dec25 awarded Good Answer Dec22 asked Constructions of the smallest nonabelian group of odd order Nov15 awarded Necromancer Oct23 awarded Yearling Oct1 comment Boundary between all convergent series on one side and divergent series on the other sideI have never liked this kind of phrasing in textbooks (exercises of this sort are common in analysis books). In my view, it is more appropriate to say that these theorems show that one way (or perhaps even a class of ways) of attempting to formalize a "boundary" between convergent and divergent series will not work. (The fact that one could likely come up with examples dooming any way of trying to do this is interesting, but almost "meta-mathematical," and not really the point.) Someone will surely give more detail about the relevance of this example to a "boundary" in an answer. Aug14 comment Is there a great book on eigenvalues?A study of abstract linear algebra (ie, vector spaces and linear operators, not matrices and $\mathbb{R}^n$) might provide the closest thing to an answer to this question. It can sometimes seem like the "eigenvalues" people mention in varied contexts are different from abstract linear algebra eigenvalues (e.g. because no linear operator or vector space is mentioned, just a physical system, or a differential equation). But with effort it is often possible to identify an operator, specific to each applied context, whose linear algebra eigenvalues are the "eigenvalues" under discussion. Jul1 answered Solving ODE and finding maxima Jun14 comment Finite groups of functions under function compositionFWIW if an $2 \times 2$ integer matrix has finite order, the order must be $1$, $2$, $3$, $4$, or $6$. (The minimal polynomial of such a matrix must divide some polynomial of the form $x^n - 1$, and hence be a product of cyclotomic polynomials. Degree considerations rule out all possibilities for the minimal polynomial except $\Phi_1$, $\Phi_2$, $\Phi_1 \Phi_2$, $\Phi_3$, $\Phi_4$, and $\Phi_6$. And each of these divides one of the polynomials $x^k - 1$, for $k = 2$, $3$, $4$, or $6$. Simple examples show that all of these orders do happen.) Jun14 comment Finite groups of functions under function compositionI think I understand what you want now. It would surprise me if an example were possible (outside of cheap shots, like partitioning a subset $\mathbb{R}$ or $\mathbb{C}$ into five congruent pieces and permuting the pieces). Jun14 comment Finite groups of functions under function composition... but, isn't any rotation of $\mathbb{C}$ (about the origin) given by an algebraic rule of the form $x \mapsto ax$, for some $|a|=1$? I guess I want a better idea of what "simple" means. Complex multiplication (as opposed to real) isn't simple? Could that be it? Jun14 comment Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.Reference: problem 2 in round 2 of the British Mathematical Olympiad, January 2012 (bmoc.maths.org/home/bmo2-2012.pdf). It may be useful to notice/prove that the sequence $d(n) = f(n) - f(n-1)$ satisfies $d(2) = 1$, $d(3k) = d(3k+1) = d(2k)$, and $d(3k+2) = 2 d(2k+1)$ for all $k \geq 1$. This allows $d(n)$ to be rapidly computed (and shows quite clearly that $d(n)$ is always a power of $2$). The sequence of $n$ for which $d(n) > n$ is in the OEIS: oeis.org/A205594 Jun14 revised Relation between defining polynomials and irreducible components of varietychanged V_n to (what I believe was intended) f_n Jun14 answered Density of the matrices with all eigenvalues of non-zero real part. Jun14 comment Infinite products - reference needed!I don't have any books with me, but the basics of infinite products are often covered in introductory complex analysis textbooks (because they are useful in any number of places in complex analysis, and arguably, rigorous development of the formal use of infinite products was one of the main historical motivations for complex analysis / analysis in general). Ahlfors's Complex analysis and Conway's Functions of one complex variable I both come to mind as possible sources for this. Jun14 comment Is there a sequence that contains every rational number once, but with the “simplest” fractions first?@endolith yeah, that'd be it (although, maybe begin with $0, 1$). Starting with that list, then inserting inverses after each element of that list would work. I hadn't thought of that, but it's an interesting way to do it. Jun14 comment Is there a sequence that contains every rational number once, but with the “simplest” fractions first?You may be interested in the so called "Farey sequences" (see en.wikipedia.org/wiki/Farey_sequence for details). Letting $F_n$ denote the Farey sequence of order $n$, if you restrict your attention to $[0,1]$, you can list all of the rationals in this interval in a way that I think you might like by first listing the elements of $F_1$, then the elements of $F_2$ that aren't in $F_1$, then the elements of $F_3$ that aren't in $F_2$, and so on. Not quite what you are asking about (because of the $[0,1]$ restriction), but very similar.