User steven spallone - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:19:12Z http://mathoverflow.net/feeds/user/15478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100618/a-reference-book-for-schurs-lemma/100738#100738 Answer by Steven Spallone for A reference book for Schur's lemma Steven Spallone 2012-06-27T03:23:01Z 2012-06-27T03:23:01Z <p>This is also in Bourbaki's Algebra 8 (most recent edition), Section 3, number 2, Example, page 43. This reference is online if you have access to SpringerLink.</p> http://mathoverflow.net/questions/95536/stable-conjugacy-for-integer-matrices Stable Conjugacy for Integer Matrices Steven Spallone 2012-04-30T00:15:01Z 2012-04-30T02:25:01Z <p>Let $F$ be a field, and $E$ an extension field. Then two matrices in $GL_n(F)$ are conjugate if and only if they are conjugate in $GL_n(E)$. I'm curious whether the analogous fact holds for rings of integers. </p> <p>Is the following true?</p> <p>Two matrices in $GL_n(\mathbb Z)$ are conjugate if and only if they are conjugate in $GL_n(\mathbb A)$, where $\mathbb A$ is the ring of algebraic integers.</p> http://mathoverflow.net/questions/79742/motivating-algebra-and-analysis-for-average-undergraduates/79836#79836 Answer by Steven Spallone for Motivating Algebra and Analysis for Average Undergraduates Steven Spallone 2011-11-02T14:17:09Z 2011-11-02T14:17:09Z <p>I taught abstract algebra to a class dominated by education majors for a few years. Much of the "applications" from the course were either algebraic facts which are familiar to calculus students, or modular arithmetic. For example, we spent a couple weeks on Peano arithmetic. The theory of polynomials illuminates the theory of partial fractions. Many "individual concepts" appear along the way. For example the concept of a ring is quite natural after you've been discussing numbers and polynomials in the same style. While more difficult, the notion of equivalence classes appear when you define rational numbers or rational functions precisely. I found that education students were sufficiently motivated by the clarity and authority that comes from proving such standard facts. And modular arithmetic is just fun for everyone. I have some notes on my OU webpage with a large "collection of questions" for this course.</p> http://mathoverflow.net/questions/50343/what-would-you-want-to-see-at-the-museum-of-mathematics/79089#79089 Answer by Steven Spallone for What would you want to see at the Museum of Mathematics? Steven Spallone 2011-10-25T15:40:35Z 2011-10-25T15:40:35Z <p>There are some cool little formulas you can "prove" by putting together a 3d block puzzle. For example, the formula for the sum of the first n squares can be seen by putting together 6 puzzle pieces to form a square prism with sides n, n+1, and 2n+1. Here, each piece is a "staggered square pyramid" of volume 1+ ...+ n^2. (Suggestion: Take n=5) There are other such puzzle-ready formulas like the sum of triangular numbers. I believe "The Book of Numbers" by Conway and Guy has some. You could build nice big soft ones that schoolchildren can play with and grownups can appreciate.</p> http://mathoverflow.net/questions/72229/quotients-of-number-rings Quotients of number rings Steven Spallone 2011-08-06T08:06:25Z 2011-08-06T21:24:18Z <p>Hi,</p> <p>Here's a question that comes up every now and then. Of course, the quotient of a number ring (ring of integers of a number field) by an ideal $I$ is a finite (Artin) ring. If we take $I$ to be the power of a prime, we obtain a finite local (Artinian) ring. Is there a characterization of finite local rings which arise in this way?</p> <p>In particular, can we obtain the quotient rings $k[x]/x^n$, where $k$ is a finite field?</p> <p>Much thanks!</p> http://mathoverflow.net/questions/66478/powers-of-maps-on-finite-sets Powers of maps on finite sets Steven Spallone 2011-05-30T18:28:25Z 2011-06-11T01:33:01Z <p>Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in $F(X_n,X_n)$ are $m$th powers of other maps? In other words, how big is the image of the function which takes each map to its $m$th power? I am curious if there is a nice formula.</p> <p>Note: This is entry A102709 at the Encyclopedia of Integer Sequences. However they don't seem to have a general formula.</p> <p>Thanks!</p> http://mathoverflow.net/questions/95536/stable-conjugacy-for-integer-matrices/95544#95544 Comment by Steven Spallone Steven Spallone 2012-05-01T01:29:38Z 2012-05-01T01:29:38Z Thank you very much this is good to know. http://mathoverflow.net/questions/95536/stable-conjugacy-for-integer-matrices/95539#95539 Comment by Steven Spallone Steven Spallone 2012-05-01T01:29:02Z 2012-05-01T01:29:02Z Thanks this is helpful. http://mathoverflow.net/questions/79742/motivating-algebra-and-analysis-for-average-undergraduates/79836#79836 Comment by Steven Spallone Steven Spallone 2011-11-03T14:59:36Z 2011-11-03T14:59:36Z By the way, Thierry, I notice you got your degree from Purdue. I'm referring to the M453 class which I taught there many times. http://mathoverflow.net/questions/79742/motivating-algebra-and-analysis-for-average-undergraduates/79836#79836 Comment by Steven Spallone Steven Spallone 2011-11-03T14:02:30Z 2011-11-03T14:02:30Z It's interesting that you think equivalence classes would be easy, because I assure you that this is the most difficult part of the course. Not simply the definition of an equivalence relation, but how to work with a quotient set in the proper way. For example why does multiplication work out for Z/nZ in the naive way, but not exponentiation? http://mathoverflow.net/questions/72229/quotients-of-number-rings/72234#72234 Comment by Steven Spallone Steven Spallone 2011-08-07T07:29:26Z 2011-08-07T07:29:26Z Thank you; this is very helpful. To answer your comment above, as a student I was curious about the classification of finite rings, and this method of production suggested itself. More recently, I am hoping to learn about methods to transfer characteristic zero results to characteristic p ones, and such isomorphisms are evidently used. Of course, one may also pose the analogous question for integers of division algebras... http://mathoverflow.net/questions/72229/quotients-of-number-rings/72231#72231 Comment by Steven Spallone Steven Spallone 2011-08-06T09:43:20Z 2011-08-06T09:43:20Z Actually, my question is whether given a ring $R=k[x]/(x^n)$, does there exist a number ring $A$, a prime ideal $P$, and a positive integer $i$ so that $A/P^i$ is isomorphic to $R$? Certainly the converse is not true. Thanks though. http://mathoverflow.net/questions/66478/powers-of-maps-on-finite-sets/67475#67475 Comment by Steven Spallone Steven Spallone 2011-06-22T20:21:04Z 2011-06-22T20:21:04Z Thank you this is helpful. To write out some details underlying your assertions above: Suppose $f$ is a map, and $f^b=f^{b+d}$. Then if $a,a' \geq b$ and $a \equiv a' \mod d$, we have $f^a=f^{a'}$. If we then take $p$ to be the least common multiple of the various $d$, and $N$ sufficiently large, we have $f^N=f^{N+p}$. Therefore the $a_k$ has period (at most) $p$. Anyway I have not yet found an answer to the question as posed, but this is a good start.