The random function's universality of multiplicative identity

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Discussion Overview

The discussion revolves around the concept of the multiplicative identity of a random function r(x) and its interaction with another function f(x). Participants explore theoretical implications, properties of randomness, and potential applications, including the search for zeros of f(x). The conversation includes technical reasoning and speculative ideas.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that the multiplicative identity of the random function r(x) can be any function f(x) except where f(x)=0.
  • Others argue that if f(x) is a non-random mapping, multiplying it by r(x) retains the randomness of r(x), except when f(x)=0.
  • A participant questions the implications of defining f(x) as k/r(x), suggesting it may not align with the initial premise about randomness.
  • Concerns are raised about whether multiplying r(x) by a constant preserves its properties, particularly if r(x) is defined to return values within a specific range.
  • Some participants discuss the nature of randomness and whether the definition of r(x) needs to be refined to maintain its properties under multiplication.
  • There is speculation about the probability distribution of r(x) and whether it can be normalized to retain its randomness.
  • One participant suggests that a random number multiplied by another random or non-random number results in a random number, prompting further inquiry into the nature of these functions.
  • Another participant introduces the idea of modular multiplication as a way to constrain the range of results from r(x).

Areas of Agreement / Disagreement

Participants express differing views on the properties of randomness and the implications of various definitions of f(x). There is no consensus on the nature of the multiplicative identity or the conditions under which randomness is preserved.

Contextual Notes

Participants note limitations regarding the definitions of randomness and the potential need for tighter constraints on the properties of r(x). There are unresolved questions about the behavior of these functions under multiplication and the implications for their ranges.

Who May Find This Useful

Readers interested in the theoretical aspects of random functions, mathematical properties of multiplicative identities, and the exploration of randomness in mathematical contexts may find this discussion relevant.

Loren Booda
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The multiplicative identity of the random function r(x) is any other function f(x) except where f(x)=0.
 
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In other words, a field in x of random values r(x) each multiplied by a corresponding value for a (non-)random mapping f(x), always retains a random mapping r(x), except where f(x)=0.

Can you imagine this situation?

Also, may this be a great way (in theory) to obtain 0's of f(x)?
 
What if f(x)=k/r(x) ? Huh ?
Maybe you wanted to say f(r(x))...or I just don't understand...:frown:
 
Originally posted by Loren Booda
In other words, a field in x of random values r(x) each multiplied by a corresponding value for a (non-)random mapping f(x), always retains a random mapping r(x), except where f(x)=0.

Can you imagine this situation?

Also, may this be a great way (in theory) to obtain 0's of f(x)?


Does it always maintain the properties of the random function r(x). For example you might have specified that if the domain is the real numbers, the function r(x) returns values in the interval [0, 1], multiplying this by 2 does not preserve this property.

Do you need to make your definition of randomness more tight, or am I missing something.

As to Bogdan's point, f(x)=k/r(x) will be a random function...and thus not allowed by the Loren Booda's initial statement.
 
Last edited:
In a set X equipped with a binary operation called a product, the multiplicative identity is an element e such that e * x = x * e = x, for all x in X...got it...

So e = f(x)...hmmm... r(x)*f(x)=r(x)...hmmm...still don't get it...
 
Originally posted by bogdan
What if f(x)=k/r(x) ? Huh ?
Maybe you wanted to say f(r(x))...or I just don't understand...:frown:

Certainly he does not mean f(r(x)) as you could just set f(x) = 1 for all x.

What he appears to be saying that there if r contains no information, i.e. it returns a value on the range with equal probability, then multiplying it by any conventional (not including random) and not equal to zero at x ,function returns a result which contains no information as defined above.

Perhaps his statement is more general?
 
I don't know for sure...maybe he'll explain better...
 
plus
Does it always maintain the properties of the random function r(x). For example you
might have specified that if the domain is the real numbers, the function r(x)
returns values in the interval [0, 1], multiplying this by 2 does not preserve this
property.

Do you need to make your definition of randomness more tight, or am I missing
something.
What is returned is the property of randomness, apparently not the function itself as first defined. Indeed I need to "loosen" my definition of random r(x). Perhaps I should require its returned interval to be [-[oo],[oo]]. Thanks for your feedback.

I hope this helps your understanding of the problem also, bogdan?
 
Yeap...if (non-)random means not random... ...those brackets...
 
  • #10
bogdan-

A random number times a random number, or times a non-random number, is a random number.
 
  • #11
But why ?
if f(x)=1/r(x), then both are random...
Or am I just stupid ? Or worse ?
 
  • #12
What is the probability distribution of your random function on the real line going to look like?
 
  • #13
Actually, plus, I don't know if r(x) can be "seen," as its average magnitude may be infinite. Can you think of a finite random function that returns the property of randomness to similar bounds (or better yet, as I initiated, the random function itself?) Perhaps I should require normalization of r(x) and r(x)f(x)?

bogdan, I appreciate your interest too. Think of a random function with magnitude r(x) returning values from 0 to [oo], multiply each by a nonzero arbitrary number f(x), and one returns the random function along x. The cardinality of "randomness" apparently is greater than that of the real number line.
 
  • #14
The only way I could think of to get this in a finite range would be to have modular multiplication. so if it was mod 10,

8.18* 10 = 0.9 (mod 10)
 
  • #15
Beautiful idea. The credit is yours. Do your see any application for the infinite range r(x) (like finding zeroes of a function)?
 

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