# Michael Joyce

New Orleans, LA

math.tulane.edu/~mjoyce3

Age: 37

I am interested in algebraic combinatorics, especially in the combinatorics of orbits of a Borel subgroup on spherical varieties.
 2d comment Covergence of the integral $\int_0^1 \ln(1+\frac{a^2}{x^2})dx$But, e.g., $\frac{\ln(1+a^2u^2)}{\sqrt{u}} \rightarrow 0$ as $u \rightarrow \infty$. 2d comment Covergence of the integral $\int_0^1 \ln(1+\frac{a^2}{x^2})dx$Hint: Argue that $\frac{\ln(1 + a^2 u^2)}{u^2} \leq \frac{C}{u^{\alpha}}$ for appropriate constants $C$ and $\alpha$ such that $\int_1^{\infty} \frac{C}{u^{\alpha}} \, du$ converges. Nov 24 comment Can students tell the difference between the "definition if" and the "theorem if"?Great question. Anecdotally, I had professors who made an attempt to write iff for the definitional if each time, though force of habit sometimes made such quests not fully successful. There's also the alternative when lecturing to use the $\Rightarrow$ and $\Leftrightarrow$ arrows in lieu of the 'if ... then' and 'iff' linguistic constructions. Nov 24 comment How long would it take to teach proper limit calculations?@JessicaB: Better late than never? I've given an answer somewhat related to the comments above. Nov 24 answered How long would it take to teach proper limit calculations? Nov 22 comment How to use false theorems or proofs?This is essentially the same flaw as in the "all horses are the same color" proof. It's definitely a good exercise in whatever variant you offer it, because it forces to really think critically about the structure of an argument and how one subtle flaw can lead to an absurdly false claim. Nov 20 awarded Nice Answer Nov 20 revised Is this just a mistake or a more serious misconception?added 73 characters in body Nov 20 answered Is this just a mistake or a more serious misconception? Nov 18 comment Seeking Your Recommendation on Problem-Solving Books (preparing for Putnam)Although the allure of competition in contest mathematics is often compelling for people your age, there is much more value in developing mathematical maturity in problem solving (even if that does not result in your highest possible score). Nov 18 comment What should I do if I have a student 'hiding' their working out?The kneeling (or squatting if that is more feasible) idea also has worked for me in my experience. If there is still resistance, I think a private conversation would be best. Explain that you looking to offer help and listen to their reasons for wanting to shield their work (if they will divulge). While you should insist that the student be willing to show you their work, you should also try to accommodate any needs or insecurities they have to help break down the barrier that is preventing communication between the two of you. Nov 17 comment How long would it take to teach proper limit calculations?On the other hand, if you take the limit laws as axioms (without any attempt to justify them), you can compute these examples from the single fact that $1/x \rightarrow 0$ as $x \rightarrow \infty$. I don't personally see this as a particularly compelling way to introduce students to limits, but you could then explain this solution in a single class meeting. It should be clear that the first solution does not fit the given axiomatic framework, but I don't see any reason why students would understand why the given axiomatic framework is preferable to one based on rules to manipulate $\infty$. Nov 17 comment How long would it take to teach proper limit calculations?In my university, this level of understanding usually does not occur unless a student takes a first course in analysis. The time it takes to go through a careful definition of limit, prove the basic facts about limits from the definition and then show how to apply the facts to solve problems such as these is considered by many (myself included) as a less than optimal use of the available time in the course. My estimate of the time taken to cover exactly what is necessary to properly understand the logic involved in this one particular problem and nothing more is about two weeks. Nov 17 comment How long would it take to teach proper limit calculations?It's not clear what properly grasp the second solution means. We all have an idea of what a truly proper grasp involves ... a definition of limit, understanding of how to prove the basic limit laws as well the fact that $1 / x \rightarrow 0$ as $x \rightarrow \infty$ using this definition, and then application of these limit laws in the particular problem at hand. Not meeting either of these three criteria seems, to me, to indicate a lack of a proper understanding of the given approach... Nov 17 comment When $-x$ is positive@JpMcCarthy: Yes, in $\mathbb{R}$, $\text{negative}(x) = \text{minus}(0,x)$ and $\text{minus}(x,y) = x + \text{negative}(y)$, so each can be recovered from the other in $\mathbb{R}$. As above, this can fail in other algebraic structures, such as $\mathbb{N}$. And Serge Lang would not let you forget it. :) Nov 17 comment When $-x$ is positive@JpMcCarthy: Alas, Serge Lang is not around to give you his rant. His point, as I recall it, is that minus can be defined as a partial function on pairs of natural numbers (e.g. minus(7,3) = 4, but minus (3,7) is undefined)) and does not require the existence of negative numbers. (Keep in mind that there is no negative number function if you limit your world to only natural numbers, i.e. to only positive integers.) Nov 13 comment What is mean of $\dot{\gamma}^j(t)$?It depends on the conventions of the source you are using, but I imagine that $\gamma$ is a curve in $\mathbb{R}^n$, in particular a function $\gamma : I \rightarrow \mathbb{R}^n$ for some interval $I \subseteq \mathbb{R}$. Then $\gamma^j$ likely refers to the $j$-th coordinate function of $\gamma$, i.e. $\gamma(t) = (\gamma^1(t), \gamma^2(t), \dots, \gamma^n(t))$. And then the derivative of $\gamma$ is given by $\dot{\gamma}(t) = (\dot{\gamma}^1(t), \dots, \dot{\gamma}^n(t))$. Nov 11 comment As a TA, how to reduce imprecise notations/statements in students' exams@Fryie: That's a sensible system. In the US, students have a range of experience in high school mathematics, from no calculus to a large base of calculus knowledge (almost never rigorous). At my university, we have all of those backgrounds in one course. In practice, that makes it impossible to teach the desired content in a fully rigorous manner in a way that is accessible to students with no or limited background in calculus. Or at least, that is my experience. I certainly would prefer your situation to the one I currently deal with. Nov 11 comment Intuition behind Re(z) not being analytical$\text{Re}(z)$ does not satisfy the Cauchy-Riemann equations. (This is essentially formalizing Umberto P.'s comment.) Nov 10 comment Are there any situations in which L'Hopital's Rule WILL NOT work?@Thomas: I would prefer to say that the rule can NEVER be used when the conditions for using it are not satisfied, even though the conclusion of the theorem may sometimes still hold (by 'luck' if you will).