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What is time invariance?

In: Mathematics

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A time-variant system is any system were the inputs vary by time. But the “inputs” really depend on how you define the problem.

* So in the example of getting to work, the drive time depends on rush hour which depends on the time of day. Leaving 1 hour earlier might save 30 minutes off your commute.

* Crossing a river might depend on the water level, which might depend on the season.

* Transferring a file on the internet might depend on congestion, which might depend on the time of day (everyone watches NetFlix after work, but fewer people are watching early in the morning).

* Solving a Rubik’s Cube can be considered time-variant if you assume people will be slower if you wake them up in the middle of the night to do the task.

* Walking around a track can be considered time-variant if the track is crowded at certain times, forcing you to change your pace.

A time-invariant system has inputs that don’t depend on the time:

* Making a baby takes 9 months ([no matter how many women are assigned to the task](https://en.wikipedia.org/wiki/The_Mythical_Man-Month)). Sure, not every baby takes 9 months, but it’s not like you can change the gestation period by deciding to star earlier or later.

* Solving a Rubik’s Cube can be considered time-invariant if you only consider small delays (starting now vs 1 hour from now)

* Walking around a empty track can be considered time-invariant. (You could turn that into a time-variant task by saying “the lights are turned off at night, so it’s harder to navigate”)

A task is ‘time invariant’ if when you do it doesn’t affect how long it takes you to do it.

Suppose you live next to a tract of woods with a path that leads to your work. Alternatively, you can drive to work on the city streets.

If you walk to work, it’s always going to take you roughly the same amount of time. You simply walk through the empty woods along the path. That’s a time invariant task.

If you drive to work, the time of day is going to vary the length of your trip because traffic may or may not interfere. If it’s during rush hour, it’ll take you a lot longer than it would in the dead of night. In this case, you have a time dependent task.

Mathematically, time invariant systems do not use time as a factor in their calculations while non-invariant systems do.