User ts - MathOverflow

Answer by TS for Examples of common false beliefs in mathematics.

2010-10-09T20:00:50Z

Draw the graph of a continuous function $f$ (from $\mathbb{R}$ to $\mathbb{R}$). Now draw two dashed curves: one which everywhere a distance $\epsilon$ above the graph of $f$ and one which is everywhere a distance $\epsilon$ below the graph of $f$. Then the open $\epsilon$-ball around $f$ (with respect to the uniform norm) is all functions which fit strictly between the two dashed curves.

Answer by TS for Equality vs. isomorphism vs. specific isomorphism

2010-05-14T01:23:53Z

This specific example may not count, as I don't know if anyone really views all separable infinite dimensional Hilbert spaces as being the same. But I imagine that just making the identification $L^2(0, 2\pi)=l^2(\mathbb{Z})$ based on the fact that all separable infinite Hilbert spaces are isomorphic doesn't gain you as much insight as knowing that the Fourier transform from $L^2(0, 2\pi)$ to $l^2(\mathbb{Z})$ is an isomorphism.

Answer by TS for Math paper authors' order

2010-03-31T21:54:49Z

My limited experience agrees with Ryan Williams's answer. As an undergraduate, I wrote a paper with my advisor (last name Mills) and she insisted that my name appear first (my last name is Shelly) so that readers would know that I did most of the work. She was actually being quite generous, and I think really she just wanted the publication to benefit me as much as possible. As she said, if people saw my name second they would assume that I helped out with some trivial aspects of the paper.

Is there an "adjacency matrix" for weighted directed graphs that captures the weights?

2010-03-28T19:25:21Z

I am currently writing up some notes on the max-plus algebra $\mathbb{R}_{\max}$ (for those that have never seen the term "max-plus algebra", it is just $\mathbb{R}$ with addition replaced by $\max$ and multiplication replaced by addition. For some reason, authors whose main interest is control-theoretic applications never seem to use the term "tropical", and I have been reading from such authors). There is a nice result which says the following:

$\textbf{Theorem.}$ Let $G$ be a directed graph on $n$ vertices such that each arc $(i,j)$ in $G$ has a real weight $w(i,j)$. Define the $n \times n$ matrix $A$ by $(A)_{ij} = w(i,j)$ if $(i,j)$ is an arc, and $(A)_{ij} = -\infty$ otherwise. Then for each $k > 0$, the maximum weight of a path of length $k$ from vertex $i$ to vertex $j$ is given by $(A^{\otimes k})_{ij}$ (here, $A^{\otimes k}$ is just the $k$th power of $A$, computed using the $\mathbb{R}_{\max}$ operations).

This result is certainly analagous to the standard result that the $ij$-entry of the $k$th power of the adjacency matrix gives the number of walks of length $k$ from vertex $i$ to vertex $j$. When writing up my notes I found myself claiming that the above theorem provides some evidence that $\mathbb{R}_{\max}$ is in fact a natural setting in which to study weighted digraphs, since there is no natural definition of an ``adjacency matrix'' of a weighted digraph (in the usual setting of $\mathbb{R}^{n \times n}$) that gives useful information about the weights. This seemed like too strong of a claim, especially since I am no expert in networks or combinatorial optimization. This leads to the question:

$\textbf{Question.}$ Is there a standard matrix (in $\mathbb{R}^{n \times n}$) associated with a weighted digraph that is analogous to the adjacency matrix and captures in a useful way the weights of the arcs?

$\textbf{Clarification:}$ By "analogous to the adjacency matrix" I mean a matrix that is defined simply in terms of the graph (vertices, arcs, and weights). I imagine there are all sorts of matrices associated to weighted digraphs so that computers can be used to analyze networks. But I am not interested in, say, a matrix that requires a complicated algorithm to compute its entries.

Comment by TS

2010-11-13T02:25:12Z

All of the papers published in the journal Involve have a significant student contribution. You can read the abstracts at their website (involvemath.org). I am not saying this is the place to go to find a research topic (or that you should attempt to find a topic on your own via the internet). But I know from experience that it can be fun and inspirational to see that your peers are researching and publishing in a variety of areas.

Comment by TS

2010-10-24T23:14:27Z

(As I was enjoying your notes, I noticed a minor typo: on page 12 in the paragraph before "We now begin the proof..." I believe you want H_i \cap H_j = {1}.)

Comment by TS

2010-10-10T19:24:04Z

You are right, I should have specified open ball, thanks. I think it is just barely false for the open ball. Honestly, I held this false belief until a couple of days ago, and I haven't thought much about correcting my belief. Probably the real open epsilon ball is the union of all functions that fit between dashed curves a distance strictly less than epsilon away from f? At any rate, I think the above picture is the right way to think about it most of the time. But it gives results such as $tan^{-1}$ being in the open ball of radious pi/2 centered at 0 if you interpret it literally.

Comment by TS

2010-09-27T19:52:46Z

Sorry. Your previous version was asking how many did not coincide with a max operation. I am saying that all of them do appear as a max with respect to some ordering.

Comment by TS

2010-09-27T12:42:09Z

Well if you have a commutative idempotent operation + then the relation $\leq$ defined by $a\leq b$ iff $a+b=b$ is a partial order. This is why (if I am correctly interpreting what you are asking) I think the answer to your second question is zero.

Comment by TS

2010-09-27T02:59:28Z

Regarding your second question: Do you mean to ask how many have an addition which doesn't come from a max operation? I ask because I am really only familiar with the semiring one gets by using "max" as the addition and "plus" as the multiplication on the reals. However, if this is your intended question, then the answer may be zero. Also, have you looked at any of Golan's texts on semirings?

Comment by TS

2010-08-11T16:25:30Z

This is more or less how I go about things too. Somewhere along the line I started writing "the last display" when citing the previous displayed equation. I mostly reserve this for the case where there is some smaller in-line equation (or even other remarks with symbols) between the display and my citation of it.

Comment by TS

2010-08-08T19:24:50Z

mathforum.org/library/drmath/view/55979.html

Comment by TS

2010-08-04T02:37:26Z

Oops, please ignore the above...

Comment by TS

2010-08-04T02:36:12Z

Doesn't this mean b = b+(a+a) = (b+a)+a = b+a, so that a = 0?

Comment by TS

2010-05-11T20:35:37Z

I would like to add, to Adam J, that it seems as though you have a good intuition about the graph isomorphism concept. Before reading the FAQ and asking this question on another site, I would encourage you to make a serious effort to answer your own question rigorously.

Comment by TS

2010-04-14T03:55:00Z

Also related, it is difficult to give an example of a continuous real valued function on [0,1] that is not monotone on any interval, but it is not too difficult to prove using the Baire category theorem that almost all functions are like this.

Comment by TS

2010-04-03T21:14:54Z

I agree that this is, in some instances, a good way to incorporate author A's contribution and make sure that author A gets credit. But could it cause problems if author A thinks his/her contribution is too significant to be banished to an appendix? I mean, if author A was not a particularly agreeable person and thought the paper would be rubbish without his/her "crucial" proof, author A might fight with the other authors for a full co-authorship. I have no idea if this situation really happens. Perhaps this is why Hardy and Littlewood had their fourth axiom...

Comment by TS

2010-03-28T23:05:14Z

Thanks. It is nice to know that the matrix is actually commonly referred to as the weighted adjacency matrix, as that is what I have been calling it. The literature I am familiar with on the topic (which is all in the realm of modeling transportation systems) seems to define the graph from the matrix, never the other way around, so it gives no name to the matrix. To address your comment, I personally don't want anything else from the matrix, but I didn't want to exclude the possibility that there was another natural and useful matrix to represent a weighted digraph.

Comment by TS

2010-03-28T20:09:34Z

Thanks! I had actually dismissed the product of the weights as being information that was "useless", stupidly ignoring the obvious (and perfectly usual) case where the weights are probabilities. This is what I get for looking at one particular application of digraphs for too long. I attempted to vote up your answer, but alas, I was reputationally (not a word) denied.