Another math problem! (Not HW)

[Kaylia]

Wouldn't the individual delta impuse not exist anywhere else on the graph elsewhere? Since the impulse extends to infinity, but only has magnitude of 1, at least, from a signals/systems point of view.

It would be defined only on this point, but the path from (0,0) to (1,0) where your impulse is defined still has a length of 1 if it's continuous. You can't jump from (0,0) to (1,0) and keep the continuity if you're working in real space.

I am only talking about tiling horizontally, not vertically. It should look (something) like this: (each ___ has a length of .1, ten tiles in total)

Edit: my make shift picture didn't come out well. But yes if you only tile horizontally (and not vertically) you will get a continuous ( but not differentiable) path of length one.

You would end on (1,0) if you tile 10 horizontal line next to each other, not (1,1). You would need 10 horizontal tiles, and 10 vertical tiles to reach (1,1).


[edit]
nevermind, kinda misread what you said in your first post. Your function isn't continuous. It does have a length of 1, but losing continuity isn't allowed here.

[Korietsu]

I need sleep, time to stop posting.

[Trajan]

It would be defined only on this point, but the path from (0,0) to (1,0) where your impulse is defined still has a length of 1 if it's continuous. You can't jump from (0,0) to (1,0) and keep the continuity if you're working in real space.



You would end on (1,0) if you tile 10 horizontal line next to each other, not (1,1). You would need 10 horizontal tiles, and 10 vertical tiles to reach (1,1).

Edit: Just read your edit lol. It isn't differentiable, but I am fairly certain that it is continuous. I will go through the old calc1 book tomorrow to verify though.

[Kaylia]

I think he means as if there were jump discontinuities, as if a peacewise function.

I'm getting sleepy. I misunderstand posts after posts.


The integral between 0 and 1 would be 1 in this case, but you don't find a path using dirac delta like this. If we were looking for an integral, you could simply use the function y=1, and integrate between 0 and 1, you would get the same result you do with a delta.

Of course, you could say it's a v(t)=1 m/s graphic, and the integral between t=0 and t=1 is 1, which mean the distance travelled is 1, but the question was poorly worded if it's the case.

[Kaylia]

Edit: Just read your edit lol. It isn't differentiable, but I am fairly certain that it is continuous. I will go through the old calc1 book tomorrow to verify though.

I'm too lazy (and tired) to make the demonstration, but I will ask you to take my word on this. I'm absolutely certain it's not continuous in RxR space. You could redefine the space to make it continous (y axis = natural number), but that's cheating.


Topology demonstration involves balls, so it's too gay for me tonight, but I'm pretty certain you would get the same answer using any definitions.

[Woozie]

That's just impossible then. Arc length will always be longer than a straigth line, and the straigth line is obviously sqrt(2) with normal metric.

At least, as long you have triangle inequality, you won't be able to find such function, and sincerely, I don't know how you can break triangle inequality without changing the metric completely. That's why I proposed a new metric earlier, because I don't see how it's possible otherwise.

If you can find a solution, I would really like to hear it though. There is always weird mathematics that I don't know about.

That's what I was thinking when I first saw the problem. It's pretty easy to prove that any path between these points will have a length of at least sqrt(2). I figured there was some sort of loophole in the statement of the problem that I was overlooking. The way it was presented to me indicated that a solution existed, so I figured I was missing something.

[Julian]

sounds weird, I can't help but think it's going to be some stupid trick like using the 1-norm instead of the 2-norm

the only actual interesting possibility i can imagine would be something like
Cantor function - Wikipedia, the free encyclopedia

except that thing has arc length 2

yeah any path between (0,0) and (1,1) is, by definition, at least as long as the shortest path-- the geodesic with length sqrt2

I think the answer may be something similar to this.

I went up to him at the end of class and asked him, if there was going to be some "physics trick" or some "math trick" to solve it, since the guy is a physics grad student, thought it could be some weird physics thing we hadn't learned yet. He said that it was just math.

I pointed out that the "shortest distance" is sqrt 2, which is greater than 1, and he said something along the lines of "Well then, that's not the shortest distance then, is it?" and then went on to say something about how most of the math we know is stuff that was discovered 300 years ago, and that there's new stuff from the past ~150 years that we don't really learn/haven't learned yet. He also said how some guy came up with some (I guess function?) and all the other Mathematicians went "Oh", and then "proved that the function really was continuous", I'm guessing he was referring to what he wanted us to find.

Also @ the "dead fly" guy, this TA is a really cool and nice guy. When we ask him questions, he actually answers them so that we really understand it (even if it does give away the response on the lab report, lol) and he isn't a complete dick that just looks through the reports looking for any reason to take points off of it.

[Woozie]

Well if there's no trick or loophone, then, as Kaylia pointed out, in the metric space (R^2,d), where R^2 are points in the 2D plane and the distance function is the familiar distance we're used to, the shortest distance HAS to be sqrt(2) because the triangle inequality holds in all metric spaces. If we're using Euclidean geometry, I can use calculus of variations to show that the sqrt(2) is an extrema, then generate at least one longer path to show that it's a minima. So either Kaylia was right and we need to use a different metric space (was topology developed in the past 300 years?) or we need to use non-euclidean geometry (which was definitely developed within the past 300 years).

[Julian]

Honestly, idk This class is on friday morning, I'll try taking some of the "solutions" that we came up with here, see if he buys any of them, then I'll post the "real solution" here sometime on friday. (If I get out of that class, Friday morning, otherwise, afternoonish/evening)

[Kaylia]

I think the answer may be something similar to this.

I went up to him at the end of class and asked him, if there was going to be some "physics trick" or some "math trick" to solve it, since the guy is a physics grad student, thought it could be some weird physics thing we hadn't learned yet. He said that it was just math.

I pointed out that the "shortest distance" is sqrt 2, which is greater than 1, and he said something along the lines of "Well then, that's not the shortest distance then, is it?" and then went on to say something about how most of the math we know is stuff that was discovered 300 years ago, and that there's new stuff from the past ~150 years that we don't really learn/haven't learned yet. He also said how some guy came up with some (I guess function?) and all the other Mathematicians went "Oh", and then "proved that the function really was continuous", I'm guessing he was referring to what he wanted us to find.

Also @ the "dead fly" guy, this TA is a really cool and nice guy. When we ask him questions, he actually answers them so that we really understand it (even if it does give away the response on the lab report, lol) and he isn't a complete dick that just looks through the reports looking for any reason to take points off of it.

Typically, you don't see much topology (if not any) in physics, unless you go out of your way. I'm willing to bet he misquoted/misunderstandood something, because it goes against common sense (the way it's worded).


In topology, the space itself is defined first, then you add an inner product property, and if you want to push thing farther, you define a distance property using the operation you defined (addition, dot product). RxR space initially has nothing defined, but in normal environment, you will have the dot product, and from it, you define a distance function (vector norm).

When you use dot product and normal distance function, you will get a distance > sqrt(2), no matter how fucked up you path continuous path is, because triangle inequalities is a direct consequence of the initial axioms that defined your space. Many solutions proposed involved changing the space completely, or function to move something from one space to another...it could works, but you don't have to go this far to change a distance. Defining a new inner products or distance function also works.

Because distance is a "function", and you can define a new one using the same RxR space, it works well with the question, and seem to go around many issues here. Also, since topology is relatively recent in mathematics (started with euler, but wasnt fleshed out until last century), it could be the "weird" function he was talking about.

[Elric]

How about the greatest integer function?

You only move in the x coordinate 1 unit, then immediately jump to (1,1).

then find a proof that the lim x->1+ = lim x->1-

[Kaylia]

How about the greatest integer function?

You only move in the x coordinate 1 unit, then immediately jump to (1,1).

then find a proof that the lim x->1+ = lim x->1-

You can easily make a proof that it's not continuous using the delta/epsilon definition around f(1), or using the ball definition (both are more general than the derivative).

And well, the derivative at x=1 in this case wouldn't exist, since it's different in both direction (-infinity from right left, and infinity from left to right). This definition of continuity require the derivative to exist everywhere.

[edit]
Cantor function posted earlier is an exemple of a continuous function with no derivative. That's why the epsilon is a more general definition of continuity, and for similar reasons, balls definition is even more general.

[joft]

Kaylia:

Weierstrass function - Wikipedia, the free encyclopedia

it's continuous everywhere but differentiable NOWHERE

there's a heirarchy of different types of "continuous"
Lipschitz > Absolute > Uniform > topological contuinity (the usual one with epsilon/delta or balls)

if a function is absolutely continuous then it is differentiable almost everywhere (the set of points where it is not differentiable has measure zero)

there's a theorem that any continuous function is uniformly continuous on any compact subset of its domain. so the Weierstrass function above is even uniformly continuous (on compact subsets) but still differentiable nowhere

[Sylvrdragon]

Did he say the coordinate system had to be Orthogonal? If not, then tell him you used Scew Coordinates with the y axis rotated 90 degrees. >.>

[Kaylia]

Kaylia:

Weierstrass function - Wikipedia, the free encyclopedia

it's continuous everywhere but differentiable NOWHERE

there's a heirarchy of different types of "continuous"
Lipschitz > Absolute > Uniform > topological contuinity (the usual one with epsilon/delta or balls)

if a function is absolutely continuous then it is differentiable almost everywhere (the set of points where it is not differentiable has measure zero)

there's a theorem that any continuous function is uniformly continuous on any compact subset of its domain. so the Weierstrass function above is even uniformly continuous (on compact subsets) but still differentiable nowhere

That's what I was saying...I think? It's the case of most fractals, not just weierstrass. The function that was listed wasn't continuous with any definitions. I simply pointed out the more general definition to avoid falling in a trap where something isn't continuous with the strict derivative continuity, but is considered continuous in topology.


I don't understand why lipschitz is more general, it should be the strictest, (> should be <)? I might be wrong here, but the way I see it

Lipschitz: College definition with derivative (doesn't work on fractal, cantor and other shitty function invented by evil mathematician)
Absolute: delta/epsilon on a f(x) function
Uniform: balls in a space with metric
topology: use of open and convergence of a set on another.

Did he say the coordinate system had to be Orthogonal? If not, then tell him you used Scew Coordinates with the y axis rotated 90 degrees. >.>

It's the euclidean vs non euclidean space argument again. It's one of the many possible solutions, but the question was poorly worded if it's the case, because it's asking you to find a continuous function that draw a path between 0,0 and 1,1.

Changing the space, changing the operator, and changing the metrics are all potential solutions, but none involve a function defined in euclid space.

If such thing existed, I'm pretty sure Woozie would have heard about it because it's kinda groundbreaking imo.

[Sylvrdragon]

I was mostly just being a smartass. I never went to college (biggest regret ever) so I don't have any higher mathematics education. Everything I know beyond High School Algebra (didn't even finish pre-calc ; ; ), I learned from Wiki- and that isn't much.

It's something I would really love to get into, but without some sort of curriculum guide, I have no idea where to start.

[Woozie]

I was mostly just being a smartass. I never went to college (biggest regret ever) so I don't have any higher mathematics education. Everything I know beyond High School Algebra (didn't even finish pre-calc ; ; ), I learned from Wiki- and that isn't much.

It's something I would really love to get into, but without some sort of curriculum guide, I have no idea where to start.

Are you saying you want to start learning mathematics and you just need to know where to start/what direction to go? If that's what you're asking we could easily help you with that. But I don't want to post a bunch of stuff if that's not what you're saying.

[Kaylia]

I was mostly just being a smartass. I never went to college (biggest regret ever) so I don't have any higher mathematics education. Everything I know beyond High School Algebra (didn't even finish pre-calc ; ; ), I learned from Wiki- and that isn't much.

It's something I would really love to get into, but without some sort of curriculum guide, I have no idea where to start.

You werent being "smartass" here, it was a correct answer. It's just the question that need to be clarified

[Sylvrdragon]

Are you saying you want to start learning mathematics and you just need to know where to start/what direction to go? If that's what you're asking we could easily help you with that. But I don't want to post a bunch of stuff if that's not what you're saying.

That is what I'm saying. I wanna be able to know what I'm looking at when I see a bunch of math symbols in front of me (namely Physics equations). I did buy a book on Calculus not too terribly long ago. I got about 1/3 of the way through it and promptly misplaced it when I moved. I'm not sure how much of that book I managed to retain... High School math was so long ago (9 years >.<) that I'm probably gonna have to relearn a lot of the Algebra I once knew.

Feel free to take it to PMs so as not to clutter this thread any further.

Edit: Have to get ready for work now, so I can't respond to any further posts/messages till tonight.

[friedfunk]

Yea the easy answer is to make your co-ordinate system non orthogonal.

If your X-axis is horizontal, make your Y-axis tilted 30 degrees to the left. Then the distance between (0,0) and (1,1) is one side of an equilateral triangle which = 1.

Edit: tilted 30 degrees, not 45... the inside angles are 60 degrees