Notation lost in Space

2009 March 17

One of the more commonly used terms in linear algebra, and other areas that make heavy use of linear algebra, is that of a subspace. This term, I’ve been teaching a course on linear algebra, and we’ve gotten to the chapter on subspaces. Frequently, I end up saying X is a subspace of Y or some such thing. It seems to me that much of mathematical notation comes about because certain phrases and words are repeated so often that it greatly simplifies [1] writing if there is notation for them.

I’ve said this so frequently in class, and had to write it, that I’m beginning to wish that there were notation for subspace, yet, to the best of my knowledge, there isn’t. “X is a subspace of Y” is analogous to both “x is less than y” or “A is contained in B”/”A is a subset of B”. These three relations all fit in the framework of partially ordered sets. There is notation for the latter two. Why haven’t we come up with a corresponding notation for subspace? Or have we, and I just didn’t get the memo? (If I ever teach an algebra course, I’ll be complaining about the lack of notation for subgroup as well.)

[1] Well, it makes for less time writing, anyway. On the other hand, students don’t much care for new notation, so maybe it’s better to do without it.


Asymmetric metric

2008 May 24

A metric on a space X is a function d from X \times X to R satisfying the following properties

  1. d(x,y)\geq 0
  2. d(x,y)=0 if and only if x=y
  3. d(x,y)\leq d(x,z) +d(z,y) for all x,y \in X
  4. d(x,y)=d(y,x) for all x,y\in X

The standard example of this is probably the usual Euclidean metric on Euclidean space. Now it is easy to find “real life” examples in which the fourth axiom doesn’t hold, but the other three do. For example, pick any city that has at least one one-way street. Then the distance from point A to point B along that one way street will be different from B to A, unless perhaps the one way street is a circle and A and B are on opposite points on the circle. For any other pair of points, the distances are not equal.

A discrete example is a strongly connected digraph (the graph must be strongly connected, for if not, then there is a pair of vertices a and b such that there is no path from a to b, and thus the distance between the two is infinite, which contradicts that fact that the distance must be a real number).

A continuous example is a circle in which it is only possible to travel clockwise.

A function which satisfies all but condition 4 in the definition of metric is called a quasimetric (The word quasimetric may have different meanings, depending on the context). Elsewhere it is simply referred to as an asymmetric metric, or asymmetric distance, (these meaning of these terms are also inconsistent). Whatever you call it, there seems to be little of substance on the matter. The term quasimetric seems to yield the best results on Google.

Is there a candidate for a “standard” example of a (non-trivial) quasimetric space? Is there a standard quasimetric on Euclidean space?

Is there a relevant source that covers the non-trivialities of quasimetrics?

Are there any other real-life examples of quasi-metrics besides cities with one way streets?

Checking In

2008 May 22

It’s been quite some time since I’ve written anything to this blog. In principle, the blog is still active, though in practice, I’ve been too busy with other things to post. I’ve also had a number of changes in computer hardware, the consequence of which is that the files (pre-blog) that are supposed to form the content of this blog are not within easy reach. I do intend on awakening this blog from its slumber at some point. For the time being, though, I will let it sleep.

New Graph Theory Book

2008 February 1

For years, one of the standard references in graph theory was “Graph Theory with Applications” by J.A. Bondy and U.S.R Murty, first published in 1976. More recently other texts have played that role. On November 30, 2007 a follow up to Bondy and Murty’s book, entitled simply “Graph Theory”, was published.

The blog for the book can be found here.

The Springer page for the book can be found here.

I haven’t been able to find a review of the book. If you know of one, please let me know.

Wading into Wiki Waters

2007 September 1

I just made my first contribution to Wikipedia, fixing up some notation and wording in the article on Schmidt Decomposition. There may still be some bugs to fix up there, but I corrected what I could based on what I know.

I’m always a bit reluctant to reference Wikipedia in public, though admittedly, I spend quite a lot of time perusing their articles. In particular, I’ve spent a fair bit of time looking for definitions of mathematical concepts that don’t appear in any of the math texts that I have access to. There have been a few articles that are not very good, but for the most part, I think the math articles are fairly well written. While I may have some quibbles about wording, the information in the articles on subjects that I do know a lot about is almost always correct.

On a more general note, Colby Cosh tells us why perhaps we shouldn’t be so hard on the online encyclopedia.

Geometry of Skewness and Kurtosis?

2007 August 5

Investigation into the standard deviation has given me more insight into why it is defined as it is. It seemed a sensible definition when I was introduced to it, but I didn’t question the motivation for defining it in that particular way, when there may have been other possibilities which, without any further explanation, seemed just as valid. For example, the absolute deviation. Recently, however, I learned of a geometric relationship between the standard deviation and the mean which justifies its usage as a measure of deviation from the mean (it has to do with how distance is measured). A similar relationship exists between the median and the absolute deviation. In either case, the choice of how to measure deviation is the natural choice.

Does this geometric approach lend itself to analysis of higher order moments, or parameters or statistics based on them, such as the skewness and kurtosis? Is there a geometrical relationship between the third moment, the mean, and the standard deviation that makes the definition of skewness natural? Likewise for the 4th moment and the skewness, or higher moments and parameters defined in terms of them.

Can one define an “absolute skewness” or an “absolute kurtosis” in a natural way that describes the asymmetry or the peakedness about the median?

More on Lp spaces

2007 August 5

L_1, L_2 and L_{\infty} norms on R^n are commonplace and each is appropriate in certain contexts. The definitions of these easily extend to general L_p norms for p\geq 1. Are there any contexts where an L_p norm for p other than 1,2 or \infty is appropriate?


2007 August 5

Suppose that, instead of using the usual Euclidean norm on R^2, we use an arbitrary p-norm. Then the distance between two points x and y is given by \sqrt[p]{|x_1-y_1|^p + |x_2-y_2|^p}.

What are the geodesics in R^2 with the p-norm? (or in R^n, though I suspect that answering the question in R^2 answers the question in R^n).

A maximization problem

2007 June 19

Suppose that \sum_{i=1}^m s_i=s and \sum_{i=1}^n t_i=t for some m\leq n. What is

\max_{\pi\in \Pi} \sum_{i=1}^m s_it_{\pi(i)},

where \Pi is the set of functions from [m] to [n] ([m]=\{1,\ldots,m\})?

That is, if two sets of numbers (possibly multisets) sum to numbers s and t, what is the maximum sum of products of pairs of numbers, where each pair contains exactly one element from each set?

Is the question different if we restrict to the case where s_i and t_j are integral for all i and j?

LaTeX to LaTeX

2007 May 1

It seems that this blog is deviating from its intended purpose before it has actually been used for its intended purpose. I do have a question. It’s just not about math.

Is there a script or program that will take a \LaTeX document consisting of many files (using the \include and \input commands) to produce one file consisting of all of the material from the original many?

I suspect it would not be hard to write one. Given enough time, I’m sure that I could do it myself. I’d rather not reinvent the wheel, however.

Any help would be appreciated.