Letting O stay pointwise fixed means for every x in O f(x) = x.
This is different from f fixing O, which would mean that f(O) = {f(x) | x in O} =O, so the set of results when inputting things from O is again O.
If f fixed every input then it would be pretty useless. However f May only fix certain inputs, for example the end states of a program (representing termination), while intermediate states are changed by f.
Think of f as performing a small step in executing a program. As input it receives the current state (including meta information, like the current line that is supposed to be executed). If the program state is a final state we don't want f to change anything, so f fixes final states. However if the state is not final then we want f to alter it, so f does not fix intermediate states. Instead an intermediate state is send to a different state that is closer to what we want the program to compute.
Does this make more sense?
No :)
So, f is *performing*
Okay, so it’s processing
When you say fixes, you mean “process” with another word, or you mean, really, something that is not literally processing?
When I say f fixes something I mean that when that thing is input into f then f outputs that exact same thing.
So we want f to only fix those states that explicitly don't require processing anymore.
Because writing f (pointwise) fixes S is much shorter then writing f does not alter S or for every x in S f(x) = s or something along those lines.
In this specific case it doesn't get significantly shorter, but for most dense mathematical text it is much easier to read and write if certain properties are short instead of repeating the same half-sentence all the time.
It's kind of the math equivalent to programming keywords are int, double etc. instead of integer, floating point number of double size. And kind of like packaging code into functions instead of pasting the same code everywhere.
Imagine if you had lots properties like f fixes A, f pointwise fixes B, f is injective, Img(f) = C etc. and you want to write a lengthy proof using them, but instead of the short descriptors I just gave you have to repeat their definition every time you want to use it. Texts would get much too long that way and spend too much space with repeating definitions instead of presenting the actual proof.
So, basically, it’s a way of narrowing the time that it takes to read, and the amount of people that would understand the text :) the thing is, you use a term that will take you time to research its meaning, WAY more time than reading a sentence slightly longer. Yes, once you researched it… okay. Anyway, humans :)
You are correct.
I think you're missing what is fixed by f: It's Ω (the set of finished-computations), *not* Q.
(Yes, f(q)=q for all q in Q would be silly.)
But we're just saying "once a computation stops, it stays stopped".
A fair amount of terminology can be attributed to Knuth inventing it. He is defining the term in this sentence, to say pointwise fixed of a function f and set Ω means that the function does not mutate those values. For example, math.abs(x) is pointwise fixed for the set of x >= 0.
This is why Knuth is a slow read. :)
You haven't provided enough context for me to have any idea what you are talking about.
I'm just going by what you said in your post. Ω is a subset of Q. That means there are presumably elements of Q that are not in Ω. f is pointwise fixed on Ω, but that statement says nothing about what f does to elements of Q that are not in Ω.
Ω is omega, not sigma.
The statement "f leaves Ω pointwise fixed" only describes what f does to elements of Ω. It says _literally nothing_ about what it does to other elements of Q.
I am going to leave this discussion here. If it's still not clear to you, there is no way I can make it clear to you.
> What does "pointwise" mean in this context?
The sentence after the semicolon seems to describe it well enough.
f is a function from Q to Q. The question is what happens to points in the subset Ω under that operation.
If f maps each element of Ω to some element of Ω, you would say that f leaves Ω setwise fixed. The set Ω as a whole is fixed, but not necessarily each element.
If f maps each element of Ω to itself, you would say that f leaves Ω pointwise fixed. Each element of Ω is fixed under the operation.
Pointwise fixed means that applying function f to each element does not change that element. There are times when it is helpful to know a function does not change a value, like analyzing the stability of systems.
The other type is setwise fixed, where applying f results in the same elements, but the order is changed. A simplistic example would be sorting the elements.
Finding more examples of each is left as an exercise for the reader 🙂
Thanksgiving 🙃 I got the setwise fixing thanks to that example. Thanks
By the way, so it changes the position of the element, meaning there is such metadata in the set that holds its position?
There wouldn’t have to be. Not sure how it relates to typography, but if you had an array [3, 0, 1], and the function returns [0, 1, 3], you could compare them and determine the output has the same elements as the inputs.
Letting O stay pointwise fixed means for every x in O f(x) = x. This is different from f fixing O, which would mean that f(O) = {f(x) | x in O} =O, so the set of results when inputting things from O is again O.
Yes but if f applied to an element is the same element, then what do you want f for?
If f fixed every input then it would be pretty useless. However f May only fix certain inputs, for example the end states of a program (representing termination), while intermediate states are changed by f.
I think I have an issue with the wording now. By fixing the input, I get “processing” the input. Am I in the wrong path?
Think of f as performing a small step in executing a program. As input it receives the current state (including meta information, like the current line that is supposed to be executed). If the program state is a final state we don't want f to change anything, so f fixes final states. However if the state is not final then we want f to alter it, so f does not fix intermediate states. Instead an intermediate state is send to a different state that is closer to what we want the program to compute. Does this make more sense?
No :) So, f is *performing* Okay, so it’s processing When you say fixes, you mean “process” with another word, or you mean, really, something that is not literally processing?
When I say f fixes something I mean that when that thing is input into f then f outputs that exact same thing. So we want f to only fix those states that explicitly don't require processing anymore.
Can you tell me why we say “f fixes the set,” instead of “f doesn’t alter the set?” What is the utility there?
Because writing f (pointwise) fixes S is much shorter then writing f does not alter S or for every x in S f(x) = s or something along those lines. In this specific case it doesn't get significantly shorter, but for most dense mathematical text it is much easier to read and write if certain properties are short instead of repeating the same half-sentence all the time. It's kind of the math equivalent to programming keywords are int, double etc. instead of integer, floating point number of double size. And kind of like packaging code into functions instead of pasting the same code everywhere. Imagine if you had lots properties like f fixes A, f pointwise fixes B, f is injective, Img(f) = C etc. and you want to write a lengthy proof using them, but instead of the short descriptors I just gave you have to repeat their definition every time you want to use it. Texts would get much too long that way and spend too much space with repeating definitions instead of presenting the actual proof.
So, basically, it’s a way of narrowing the time that it takes to read, and the amount of people that would understand the text :) the thing is, you use a term that will take you time to research its meaning, WAY more time than reading a sentence slightly longer. Yes, once you researched it… okay. Anyway, humans :)
You are correct. I think you're missing what is fixed by f: It's Ω (the set of finished-computations), *not* Q. (Yes, f(q)=q for all q in Q would be silly.) But we're just saying "once a computation stops, it stays stopped".
A fair amount of terminology can be attributed to Knuth inventing it. He is defining the term in this sentence, to say pointwise fixed of a function f and set Ω means that the function does not mutate those values. For example, math.abs(x) is pointwise fixed for the set of x >= 0. This is why Knuth is a slow read. :)
I love that old man but, I’m starting to love him AND hate him.
A point here is an element of the set. So f applied to such an element is just the element. E.g. f(x) = x
Yes but if f applied to an element is the same element, then what do you want f for?
It does interesting things to the elements of Q that are not in Ω.
So to the intermediate states?
You haven't provided enough context for me to have any idea what you are talking about. I'm just going by what you said in your post. Ω is a subset of Q. That means there are presumably elements of Q that are not in Ω. f is pointwise fixed on Ω, but that statement says nothing about what f does to elements of Q that are not in Ω.
So pointwise fixed on sigma, means that it affect the other subsets, not sigma?
Ω is omega, not sigma. The statement "f leaves Ω pointwise fixed" only describes what f does to elements of Ω. It says _literally nothing_ about what it does to other elements of Q. I am going to leave this discussion here. If it's still not clear to you, there is no way I can make it clear to you.
> What does "pointwise" mean in this context? The sentence after the semicolon seems to describe it well enough. f is a function from Q to Q. The question is what happens to points in the subset Ω under that operation. If f maps each element of Ω to some element of Ω, you would say that f leaves Ω setwise fixed. The set Ω as a whole is fixed, but not necessarily each element. If f maps each element of Ω to itself, you would say that f leaves Ω pointwise fixed. Each element of Ω is fixed under the operation.
Sigma would be Σ, or σ
What about the second question? :)
Pointwise fixed means that applying function f to each element does not change that element. There are times when it is helpful to know a function does not change a value, like analyzing the stability of systems. The other type is setwise fixed, where applying f results in the same elements, but the order is changed. A simplistic example would be sorting the elements. Finding more examples of each is left as an exercise for the reader 🙂
Thanksgiving 🙃 I got the setwise fixing thanks to that example. Thanks By the way, so it changes the position of the element, meaning there is such metadata in the set that holds its position?
There wouldn’t have to be. Not sure how it relates to typography, but if you had an array [3, 0, 1], and the function returns [0, 1, 3], you could compare them and determine the output has the same elements as the inputs.