Natural examples of metrics that are not Translation invariant.

In summary, two examples of metrics that are not translation invariant are given by d(x,y) = |log(x/y)| on the positive real line and d(x,y) = |x| + |y| in Euclidean space where the distance is measured via 0. These may not appear naturally in pure mathematics, but they provide simple and understandable examples for those studying general metric spaces. It is also important to note that not all metrics on spaces have a vector space structure, making the term "translation invariant" irrelevant in these cases.
  • #1
deluks917
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I am trying to explain metric spaces and finding it hard to come up with simple to understand, interesting examples of metrics that are not translation invariant. The audience is people who are just now studying general metric spaces.
 
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  • #2
I just looked up "metric spaces" on wikipedia and two examples stood out:
1) Give the positive real line the metric d(x,y) = |log(x/y)|
2) In Euclidean space suppose that instead of considering the direct distance from x to y, we want to travel via 0, then the distance is given by:
d(x,y) = |x| + |y|
unless x=y in which case d(x,x) = 0.

I couldn't say where these appear naturally in pure mathematics, but they seem easy enough to understand and it is certainly not unthinkable that such space may at the very least be used in applied mathematics and therefore that our abstract study of metric spaces should include them.

You should of course also keep in mind that lots of metrics are on spaces without a vector space structure so the term "translation invariant" does not even make sense for general metric spaces.
 

1. What is a metric that is not translation invariant?

A metric that is not translation invariant is a measure of distance that changes when the position or orientation of an object is shifted. This means that the value of the metric will depend on the reference point or frame of reference used.

2. Can you give an example of a natural phenomenon that exhibits non-translation invariant behavior?

An example of a natural phenomenon that exhibits non-translation invariant behavior is the measurement of seismic activity. The distance between two points on the earth's surface may change depending on the location and movement of the tectonic plates, making it a non-translation invariant metric.

3. How does non-translation invariance affect scientific studies?

Non-translation invariance can have a significant impact on scientific studies, as it can lead to inaccurate measurements and conclusions. For example, in studies of climate change, using a non-translation invariant metric to measure temperature may result in misleading data and incorrect predictions.

4. Are there any benefits to using non-translation invariant metrics?

Yes, there can be benefits to using non-translation invariant metrics in certain situations. For instance, in computer vision and image processing, using non-translation invariant metrics can help capture and analyze patterns and features that are not affected by translation, such as texture and shape.

5. How can scientists account for non-translation invariance in their research?

Scientists can account for non-translation invariance in their research by carefully choosing and understanding the metrics they use. If possible, they can also try to account for non-translation invariance through mathematical adjustments or by using multiple metrics to validate their results.

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