you see, it seems like we can define a limit as a converging value or structure. for example, a dirac delta function takes a curve and shows that it can converge to a vertical line right?
What i want to know is, can a vertical line be mathematically unwrapped or deconstructed to show that the converse is true? that a vertical line diverges to a curve? Is there a duality for such a thing? Can a series be shown that it converges to a value, and that conversely, the value can be used with some initial value or bound of an equation so that it tapers or "expands" out into the function or series you originally were discussing as being convergent to a limit?
I don't know if this is dual to a limit, or just nonsense, but i would like some insight from someone here about this concept of unraveling or expanding a compressed form like a line.
Josiah Campbell
It doesn't really feel like the limiting process is "bijective" in that sense.
>the value can be used with some initial value or bound of an equation Maybe? It could be like ordinary differential equations where not specifying an initial value gives you a parametrized family of possible solutions.
You'd be looking at something like [eqn]\text{lim}^{-1}(f) = \left\{ \left\{f_n\right\}_{n=1}^{\infty} : \lim\limits_{n\to \infty} f_n = f \right\}[/eqn]
Lincoln Price
so could that be generalized into an operator that maps a function who has a limit with a function who may or may not have a limit, but initial value is that first functions limit? i don't know how to describe "initial value" here, but it's the condition that gives you the evolution of the function away from the limit, you know?
Jaxson James
Given an object x in a certain 'space', you can usually find sequence converging to it. Actually, there's one really easy to find: take the constant sequence x,x,x,...
As user before me said, the association sequence => limit is not bijective at all. For example, if your space contains at least two points x and y, you can find many different sequences converging to x: for example y,x,x,x,x,... x,y,x,x,x,x,... y,y,x,x,x,x,... and the like. This means that a given object has no canonical sequence associated to it, in general.
Joshua Thomas
>i don't know how to describe "initial value" here I'm guessing it would just be a constraint on [math]f_1[/math].
But I'm not sure how you would go about mandating "smoothness" on the sequence of functions, since you could just have [math]\left\{f_n\right\}_{n=1}^{\infty} = \left\{ f_1, f, f, f, \dots \right\}[/math] where [math]f_1[/math] is just some random function or your initial value if it's specified.
I don't know very much functional analysis, but that seems like where you'd want to look for your answers, OP.
Brody Murphy
Cauchy sequences form equivalence classes based on their limits (same limit => equivalent sequence). Because of this fact, there is obviously more than one sequence that can be the result of a limit, so you wouldn't really be finding useful information about the original function (ex. both the normal distribution as the standard deviation goes to 0 and the sequence (n/x^2) as n goes to 0 both approach the dirac delta function, but they are very different functions in nature other than the fact that they are both non-negative (but we could even change that)).
Hunter Bennett
>dual to a limit So literally colimits then.
Sebastian Anderson
no what i mean if something like the angle of a door that you open. There's only one value that you can give the angle to say the door is closed, which can be considered a convergence when talking about the sequence of angles leading to a closed door right? is there a duality between that and a limit stating a door is opened, even if there are an infinite amount of sequences that could lead to periodic motions of swinging the door closed and opened?
Ryder Bell
>I google things and pretend to understand what comes up Fuck off role-player
Mason Gray
thinly vailed bait for dirac delta user
Hunter Roberts
Depends on if the function converges uniformly or not. If not, you lose information and can't reverse the process.
Ian Turner
is there no way to extract that information?
Brody Bailey
just nonsense
no
all you have is a point and its neighborhoods. you have no idea which way you came through
Levi Howard
No an angle can't be considered as *a* convergence. There are many different motions of the door that lead to the status door=closed, meaning you can give the door different speeds and accelerations, etc. etc.
Of course you can identify a value with the set of all sequences converging to it, but that's fucking pointless.
Hunter Edwards
not even with...point set topology and functional analysis having a baby or something?
William Wilson
>What i want to know is, can a vertical line be mathematically unwrapped or deconstructed to show that the converse is true? that a vertical line diverges to a curve? Is there a duality for such a thing? Can a series be shown that it converges to a value, and that conversely, the value can be used with some initial value or bound of an equation so that it tapers or "expands" out into the function or series you originally were discussing as being convergent to a limit?
Yeah, see the heat equation with an impulse starting condition.
Carson Torres
are you just saying the names of courses you haven't taken yet, but will have to take in 2 years?
Connor Young
nah holmes i just think that they might help, i mean isn't toplogy aboute the correlation of points and their neighbourhoods, and trying to extrapolate an underlying structure with as little initial detail as possible?
Hunter Johnson
it can, but not uniquely. see convergence as a funnel. water goes in and is pushed to a point. If you send water through the funnel the other way around, its not pushed and will show some behaviour other than convergence.
Eli Campbell
i have a pre-university level of understanding of mathematics so i dont know how to attack this question mathematically
however, it seems to me that in the case trying to make defining the limit of a curve reversible you run into the problem of information loss.
there are an infinite number of curves that converge into a line. you cannot possibly know which curve a line you are looking at was derived (as it were) from.
someone who understands entropy and information theory could give you a more rigorous answer but to be honest it seems pretty fucking obvious that its not reversible if you are just given a line drawn on a piece of paper for gods sake
Nathan Richardson
maybe not on a piece of paper, but what about a specific manifold or function space
Logan Russell
not really sure what that means but the likely answer is that you could find the set of all possible curves in that specific manifold of function space which may or may not be infinite
if there was only one possible curve in said manifold or function space then it would be the line you were given in the first place
also, if you are given a situation with a finite amount of curves, how could there even be a limit?
Juan Wood
>dirac delta function
Wyatt Diaz
No, the space of tempered distributions is defined as the weak topological closure of generalized functions under the L^2 norm, and generalized functions are limits of regular smooth functions. If the Dirac delta function converges to some regular function then it should lie within that space, which is patently not true since we know that space of regular functions is a proper subset of tempered distributions. To see why the last statement is true, by the Painleve property the Fourier transform is a norm-preserving automorphism on the space of tempered distributions but not on smooth functions (can can't even define the Fourier transform of the constant function 1 in it). The Fourier transform in a sense "takes you out" of the space of smooth functions but it's "exactly the right size" for tempered distributions, meaning the space of smooth functions is strictly smaller than the space of tempered distributions.
Levi Russell
>Paul Dirac called it a function >shitposter objects gtfo
