## Common Metrics

Forum for the GRE subject test in mathematics.
zuluyankee
Posts: 16
Joined: Wed Jul 22, 2009 1:20 am

### Common Metrics

Hi, I was just thinking maybe we could help each other for the tests by listing the metrics and nonmetrics that we already know. That would be helpful as we already "recognize" them if ever we are asked to identify which is/is not a metric in a given list in a GRE (or any multiple choice) test.

Listed are what I can think of off hand:

1. $$d(x, y)=|x-y|$$

2. $$d(x, y)=\min\{|x-y|, 1\}$$ (is this true if 1 is replaced by any $$r\in\mathbb{R}$$?)

3. $$d(x, y)=0$$ if $$x=y$$ and $$1$$ otherwise. (can replace 1 by any $$r\in\mathbb{R}$$)

4. $$d(x, y)=d'(x, y)/(1+d'(x, y))$$ whenever $$d'(x, y)$$ is a known metric.

5. $$d(x, y)=|x-y|/3$$ (can probably replace $$|x-y|$$ by any known metric $$d'$$ or 3 by any real >0.

Here is an example of a nonmetric:

1. $$f(x, y)=(x-y)^2$$

Please feel free to add to the list. Motivation behind the metric would also be useful, so is the area of mathematics where it crops up.

Please feel free to correct if there are mistakes.

origin415
Posts: 61
Joined: Fri Oct 23, 2009 11:42 pm

### Re: Common Metrics

For 2 and 3, r > 0 works always.
For 5, you are correct, if d is a metric, kd is a metric any k > 0.

The common metrics in $$\mathbb{R}^n$$, where d any metric in $$\mathbb{R}$$

1. Euclidean $$\sqrt{\sum{(d(x_i, y_i))^2}}$$
2. Taxicab $$\sum{d(x_i, y_i)}$$
3. Sup $$\sup\{d(x_i, y_i)\}$$

mtey
Posts: 22
Joined: Tue Oct 27, 2009 12:30 pm

### Re: Common Metrics

This is also a metric in $$\mathbb{R}^n$$

$$S_p(\overline{x}, \overline{y}) = \displaystyle \sqrt[p]\sum_{i = 1}^{n}(x_i - y_i)^p$$ for $$p \in \mathbb{N}, p \geq 2$$

joey
Posts: 32
Joined: Fri Oct 16, 2009 3:53 pm

### Re: Common Metrics

Also, if d is a metric, then $$\sqrt{d}$$ is, too. However, $$d^2$$ might not be, as your first nonexample shows. I have a flashcard that asks me to differentiate between these two, as I often forget which is which. Good idea with this topic, by the way. I wish there was a lot more analysis and topology on the exam so I could justify studying those topics more frequently. Wait, let me qualify that: I wish there was a lot more non-complex analysis and topology on the exam.

zuluyankee
Posts: 16
Joined: Wed Jul 22, 2009 1:20 am

### Re: Common Metrics

Most likely, there is at most one question on identifying metric. Being able to answer by a quick glance would be very helpful!

If there are two then it is great if each is answered by quick glance through. If the whole test is about identifying metric, then we finish the test in under 165 sec., assuming each question takes at most 2.5 sec. to answer. (Just a wild imagination... Haha...)