The random function's universality of multiplicative identity

• Loren Booda
In summary, the multiplicative identity of the random function r(x) is any other function f(x) except where f(x)=0. This means that 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. This situation can be imagined and in theory, it could be a great way to obtain 0's of f(x) as long as the definition of randomness is sufficiently tight. However, it may not always maintain the properties of the random function r(x) and a more general statement may be needed. It is also possible to have a finite random function by using modular multiplication, but for
Loren Booda
The multiplicative identity of the random function r(x) is any other function f(x) except where f(x)=0.

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...

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.

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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...

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...

bogdan-

A random number times a random number, or times a non-random number, is a random number.

But why ?
if f(x)=1/r(x), then both are random...
Or am I just stupid ? Or worse ?

What is the probability distribution of your random function on the real line going to look like?

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.

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)

Beautiful idea. The credit is yours. Do your see any application for the infinite range r(x) (like finding zeroes of a function)?

1. What is the random function's universality of multiplicative identity?

The random function's universality of multiplicative identity refers to the property of the random function where the output of the function remains unchanged when multiplied by the multiplicative identity, which is typically represented as 1. In other words, when the random function is multiplied by 1, the result is the same as the original input.

2. Why is the universality of multiplicative identity important?

The universality of multiplicative identity is important because it allows for the random function to be used in a wide range of mathematical equations and calculations. It acts as a neutral element that preserves the original input, making it a useful tool in various scientific and mathematical fields.

3. How is the universality of multiplicative identity proven in the random function?

The universality of multiplicative identity is proven in the random function through mathematical analysis and testing. This involves using a variety of inputs and multiplying them by 1 to observe if the output remains the same. If the output does not change, it is considered to have the property of universality of multiplicative identity.

4. Are there any limitations to the universality of multiplicative identity in the random function?

While the universality of multiplicative identity is a valuable property of the random function, it does have its limitations. For example, it only applies to the multiplication operation and does not necessarily hold true for other operations such as addition or division. Additionally, it may not hold true for certain types of random functions that have specific constraints or parameters.

5. How is the universality of multiplicative identity relevant in real-world applications?

The universality of multiplicative identity has many practical applications in various fields such as finance, engineering, and computer science. For example, it is used in financial modeling to determine the risk and return of investment portfolios. In computer science, it is utilized in algorithms for data encryption and compression. Overall, its universality makes it a versatile and powerful tool in solving complex problems in different industries.

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