“Recursive” Sequence Reaching Every Open Interval

In summary, the conversation discusses the movement of a particle on a number line, with the ability to move to either a-a2+ax or x-ax-a+a2 for a fixed value of 0<a<1. It is proven that any open interval within the interval (a-1,a) will contain a point that the particle can reach, as it can continuously move in one direction and "swap" to get closer to the origin. The conversation also mentions the possibility of using a binary approach for a=0.5, which can be generalized to other values of a.
  • #1
Don Jon
2
0
Thread moved from the technical forums, so no Homework Template is shown
Let x denote the position of a particle on the number line. From x, it can move to either the point a-a2+ax or to the point x-ax-a+a2 for some fixed 0<a<1. Suppose the particle starts at the origin. Prove that any open interval that is a subset of the interval (a-1,a) contains a point that the particle can reach.

It’s fairly clear the particle can get arbitrarily close to the origin (by moving in one direction continuously and then suddenly swapping). Thus you can get arbitrarily close to a point that is reached by moving in one direction continuously. But I don’t know how to fill in the “holes” in between these points
 
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  • #2
Is this homework?

Does the particle do many consecutive steps, following these rules?

For a=0.5 it looks like you can approach any point in a binary way (reducing the distance by 2 with each additional step). I guess this can be generalized to other values of a.
 
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  • #3
mfb said:
Is this homework?

Does the particle do many consecutive steps, following these rules?

For a=0.5 it looks like you can approach any point in a binary way (reducing the distance by 2 with each additional step). I guess this can be generalized to other values of a.

Yes. The particle does do many consecutive steps. Could you be more explicit about the binary approach for a=0.5; I’m not quite understanding it. Thanks!
 
  • #4
This is an iterated function system, altough a rather boring one in one dimension. Everything under "Properties" in the wikipedia article applies here. If I is the interval of interest (a-1,a) the main thing is to find out what f_1(I) ∪ f_2(I) is. (the union of the ranges of f_1 and f_2).
 
  • #5
This is an iterated function system, altough a rather boring one in one dimension. Everything under "Properties" in the wikipedia article applies here. If I is the interval of interest (a-1,a) the main thing is to find out what f_1(I) ∪ f_2(I) is. (the union of the ranges of f_1 and f_2).
 

Related to “Recursive” Sequence Reaching Every Open Interval

1. What is a “Recursive” Sequence Reaching Every Open Interval?

A “Recursive” Sequence Reaching Every Open Interval is a mathematical concept that involves creating a sequence of numbers using a specific rule or formula. This sequence is called a "recursive" sequence because each term in the sequence is defined in terms of the previous term. This sequence is designed to reach every possible decimal number within a given interval, or range of numbers.

2. How is a “Recursive” Sequence Reaching Every Open Interval different from a regular sequence?

A regular sequence is a list of numbers that follow a specific pattern or rule, but the terms are not necessarily dependent on each other. In contrast, a “Recursive” Sequence Reaching Every Open Interval uses the previous term in the sequence to define the next term, creating a more complex and interconnected pattern.

3. What is the significance of a “Recursive” Sequence Reaching Every Open Interval?

This sequence has practical applications in fields such as computer science, physics, and economics. It can be used to model real-life scenarios that involve continuous change, such as population growth, financial investments, and radioactive decay. It is also important in understanding the concept of infinity and the behavior of numbers on a continuous scale.

4. Can you give an example of a “Recursive” Sequence Reaching Every Open Interval?

One example of a “Recursive” Sequence Reaching Every Open Interval is the Fibonacci sequence, in which each term is defined as the sum of the two previous terms. This sequence is designed to reach every possible decimal number between 0 and 1. Another example is the logistic map, which is used to model population growth and can reach every possible decimal number between 0 and 1.

5. How is a “Recursive” Sequence Reaching Every Open Interval calculated?

The calculation of this sequence depends on the specific rule or formula used to define the terms. In general, the first term in the sequence is given, and then subsequent terms are calculated using the rule or formula, with each term depending on the previous one. The sequence continues until it reaches every possible decimal number within the given interval.

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