# Math, anyone?



## Nightriser

Anyone else here a math major or have any sort of interest in advanced math? I chose Galois theory as a senior thesis topic and am now wondering what I've gotten myself into. Some of it I think I understand, but I want to know if there's someone here I could ask any questions I may have. Thanks.


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## cryptonia

aha hahah... yeah, no. I'm just in the very base level math theory class, doing bijections and counting and the like, where we're only concerned with two levels of infinity, if it comes up at all... haha sorry.


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## Nightriser

cryptonia said:


> aha hahah... yeah, no. I'm just in the very base level math theory class, and I probably won't be continuing in it. Sorry.


Thanks anyway.


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## Happy

Geez you INTP's are so into math. hehe.:crazy:


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## cryptonia

haha if no one could help me with that little problem, there's sure no way she's gonna get much help with something like her's.


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## Happy

Maybe you can ask her to solve that problem you had earlier.:wink:


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## cryptonia

irrelevant now. The homework's been due for a week or so, and the solution's probably posted by now. I thought the same thing, though.


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## Happy

Well at least someone who possesses the same level of math intellect is in the forum.:happy:


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## cryptonia

way more... I wish I thought like a mathematician. 12 years of schooling has beaten the true math out of me, to the point where now all I can do is use what the mathematicians have come up with to do other stuff. I don't much like when people insist on the purity of the field as something important, as if learning the most pure field were something to brag about... but I don't much like it when people call the theory useless and hound on the formulaic applications, as if Maxwell's equations just invented themselves, either. lol... I generally remain jealous of people who can actually do math unless they start to turn snobbish about it.


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## Happy

Do you see equations when you look at an object? One of my teachers in accounting told me that if your advanced in mathematics you start seeing equations on things just like the guy from Beautiful Mind. Quite extraordinary.:happy:


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## Nightriser

cryptonia said:


> way more... I wish I thought like a mathematician. 12 years of schooling has beaten the true math out of me, to the point where now all I can do is use what the mathematicians have come up with to do other stuff. I don't much like when people insist on the purity of the field as something important, as if learning the most pure field were something to brag about... but I don't much like it when people call the theory useless and hound on the formulaic applications, as if Maxwell's equations just invented themselves, either. lol... I generally remain jealous of people who can actually do math unless they start to turn snobbish about it.


Ha, there are quite a few snobs, unfortunately. I went in sort of the opposite direction. I went from thinking in terms of number crunching ("plug and chug," as I've heard some say) to actually intuitively modeling the objects. The problem is that I have trouble translating from my intuitive models to the actual writing. I'm fairly confident that I have the right idea and my prof often says I do, but I like to omit writing what seems obvious and unnecessary to me. It often turns out that what's obvious is not so obvious. :frustrating: 



Lance said:


> Do you see equations when you look at an object? One of my teachers in accounting told me that if your advanced in mathematics you start seeing equations on things just like the guy from Beautiful Mind. Quite extraordinary.:happy:


Equations need not be the only thing you see. I don't really see equations, I just look for patterns and models (I really like Venn diagrams and using sets to make sense of relationships, which is a lot of what math is anyway). For a while, though, I did see vectors everywhere. Damn free-body diagrams....*shakes fist* 
But, of course, I can't speak for all mathematicians.


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## Happy

nightriser13 said:


> Ha, there are quite a few snobs, unfortunately. I went in sort of the opposite direction. I went from thinking in terms of number crunching ("plug and chug," as I've heard some say) to actually intuitively modeling the objects. The problem is that I have trouble translating from my intuitive models to the actual writing. I'm fairly confident that I have the right idea and my prof often says I do, but I like to omit writing what seems obvious and unnecessary to me. It often turns out that what's obvious is not so obvious. :frustrating:
> 
> 
> 
> Equations need not be the only thing you see. I don't really see equations, I just look for patterns and models (I really like Venn diagrams and using sets to make sense of relationships, which is a lot of what math is anyway). For a while, though, I did see vectors everywhere. Damn free-body diagrams....*shakes fist*
> But, of course, I can't speak for all mathematicians.


How I wish I can live your life and see what you see.:frustrating:


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## cryptonia

That's very funny... I actually work the exact reverse way. When I see the world... well... quite frankly, I don't see the world. I pay less attention and take less notice of my surroundings than anyone I've ever met. When someone shows me an equation that describes something, though, it's worth more in terms of explaining how something works than anything the person could have ever said.

On the other hand, math goes a loong way in describing philosophy... so I realized, while I don't fit patterns to things in the real world anymore, I definitely do it to ideas. When I was learning basic calculus, and even the general rules for what the graphs of certain equations looked like during pre-calc, I started fitting graphs to theological concepts pretty regularly. It's been 4 or 5 years since those days, though, and calc fell into the form of chugging through calculations years ago... so I'm not sure I even have that ability anymore.


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## Nightriser

cryptonia said:


> That's very funny... I actually work the exact reverse way. When I see the world... well... quite frankly, I don't see the world. I pay less attention and take less notice of my surroundings than anyone I've ever met. When someone shows me an equation that describes something, though, it's worth more in terms of explaining how something works than anything the person could have ever said.
> 
> On the other hand, math goes a loong way in describing philosophy... so I realized, while I don't fit patterns to things in the real world anymore, I definitely do it to ideas. When I was learning basic calculus, and even the general rules for what the graphs of certain equations looked like during pre-calc, I started fitting graphs to theological concepts pretty regularly. It's been 4 or 5 years since those days, though, and calc fell into the form of chugging through calculations years ago... so I'm not sure I even have that ability anymore.


Interesting. I alternate between attentiveness to my surroundings and complete obliviousness. I can go for weeks without noticing a change in what people look like or how the furniture's arranged, unless someone makes a comment (fortunately, no one's ever been offended, to my knowledge). On the other hand, I have times when I deliberately pay attention to certain things around me. 

I think it was Plato or Aristotle, speaking of their particular establishment, "Let no one ignorant of Euclid's Elements pass through this gate." Or something to that effect. Can you give an example? There was an interesting graph called Gabriel's Horn (or Torricelli's Trumpet) which had finite volume and infinite surface area. But then again, that's what a fractal would be like, if I'm not mistaken.


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## cryptonia

akh, if only I could. The only thing that comes to mind now is the idea of the three sets, {1, 4, 7, ... 3n + 1, ...}, {2, 5, 8, 3n + 2, ...}, and {0, 3, 6, 9, ..., 3n, ... }, as representations for the christian doctrine of a trinity. Three sets partitioning the natural numbers whose union contains no more elements than any individual one. Each is quite clearly different from the others, and yet their elements share the same properties, the properties of natural numbers.

I feel like that was shallow, and that I had a bunch of better or more interesting ones at one point... but they fell into that category of "not worth sharing because people just don't get my ideas," and got forgotten over time. I'm pretty sure if I were talking theology with someone and they didn't get some part of it, and I had a math idea to fit to it at some point, it would pop back into my head... but not without some external idea for the Ne to pick up on and trigger the relationship in memory. If I remember right, many of them had to do with graphs with an asymptote somewhere that I could fit different axes to to get a rough estimate of change over time, or indeterminate limits (I think that's the word... when they're infinity/infinity or something similar) that you'd have to do some work to find, the idea of L'Hopital's rule gave me some way to tell that when the world is pushing in two opposite ways, the ratio of the two still tend towards the same value if you look at the rates of change of each, rather than the specific (harder to find) absolute values instead.

er... lol, I hope that made sense.


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## Stickynotee

Hah. I'm in the same situation. 
I'm actually doing a guys homework to get paid for it, and, funny thing is, he;'s in grade 10, I'm in grade 11.....And I get everything, but I know there is a formula, and I can't seen to figure it out.

Eg.>> 2/3 (Two-thirds) TIMES x EQUALS 12.
Basically, FInd x.
I can do this, but I forgot the formula for it Lol.

Anyone care to help? xD


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## Nightriser

Stickynotee said:


> Hah. I'm in the same situation.
> I'm actually doing a guys homework to get paid for it, and, funny thing is, he;'s in grade 10, I'm in grade 11.....And I get everything, but I know there is a formula, and I can't seen to figure it out.
> 
> Eg.>> 2/3 (Two-thirds) TIMES x EQUALS 12.
> Basically, FInd x.
> I can do this, but I forgot the formula for it Lol.
> 
> Anyone care to help? xD


You mean 2/3(x) = 12 ?

Multiply each side by the reciprocal of 2/3, which would be 3/2. You would get 18. 

But I'm guessing it's more complicated than that? What's the specific subject, algebra I?


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## Happy

nightriser I think you solved it. lol.


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## Nightriser

cryptonia said:


> akh, if only I could. The only thing that comes to mind now is the idea of the three sets, {1, 4, 7, ... 3n + 1, ...}, {2, 5, 8, 3n + 2, ...}, and {0, 3, 6, 9, ..., 3n, ... }, as representations for the christian doctrine of a trinity. Three sets partitioning the natural numbers whose union contains no more elements than any individual one. Each is quite clearly different from the others, and yet their elements share the same properties, the properties of natural numbers.


Hm, interesting. We used that a similar example recently (sans the theological connotation) for a group theory thing. I think you mean integers? It can be extended to negatives, so the sets form an even more general set. Anyway, those sets could be represented as a sort of group called the integers mod three. 

On the subject of sets, have you heard of Cantor's continuum hypothesis and varying sizes of infinity? His proof of varying sizes of infinity was actually controversial for their theological implications. Perhaps taking away from the intrigue of differing sizes is that if one models different sets of numbers as groups, infinite groups, such as the integers (with respect to addition), all have the same structure anyway, despite differing cardinalities. 



> I feel like that was shallow, and that I had a bunch of better or more interesting ones at one point... but they fell into that category of "not worth sharing because people just don't get my ideas," and got forgotten over time. I'm pretty sure if I were talking theology with someone and they didn't get some part of it, and I had a math idea to fit to it at some point, it would pop back into my head... but not without some external idea for the Ne to pick up on and trigger the relationship in memory. If I remember right, many of them had to do with graphs with an asymptote somewhere that I could fit different axes to to get a rough estimate of change over time, or indeterminate limits (I think that's the word... when they're infinity/infinity or something similar) that you'd have to do some work to find, the idea of L'Hopital's rule gave me some way to tell that when the world is pushing in two opposite ways, the ratio of the two still tend towards the same value if you look at the rates of change of each, rather than the specific (harder to find) absolute values instead.
> 
> er... lol, I hope that made sense.


No, talk some more! I like this! :laughing: 
Come to think of it, there was a book I started reading a while ago about the relationship between math and Christianity. (I unfortunately had to return it to the library.) It talked about some of the mathematical concepts that seem to conflict with Christian theology and attempting to resolve them. I remember it addressing some theological problems posed by mathematical Platonism and attempting to resolve the conflict, but that's about it. I'll have to read it again.


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## Nightriser

So do you do the same thing for physics, cryptonia?


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## Schwarz

What I came up with was essentially the same as what Nightriser did. No matter how small you make the vertical and horizontal lines, there's still a shorter way between them, which is the diagonal line drawn between them. So the actual line would have a length of sqrt (2), even though the length of the zigzagging line segments would be two.
Fun conversation, though.
I have a math question I'm wondering about, but I may have to wait a while to ask it, because it's connected to a math project I'm doing and we're not supposed to get help from anyone outside of class. 

Anyway, here's an unrelated question. Does anyone know of a convenient generalization for the difference of two powers (like difference of squares, difference of cubes, etc.)? I figured out something that I think works for x^n-y^n so long as n is a prime number, but for non-prime powers things work out a little less cleanly.


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## cryptonia

gotcha, thanks both of you (haha that was detailed, night). I wasn't really wondering about a "why don't I" sum using the hypotenuses , because that solution obviously works... lol I was more wondering why this other solution _doesn't_ work. Math should be consistent, no? Whenever I hit something like this I usually find out that I misunderstood something.

but yeah... if I get what you're both saying, the problem is where I suspected the problem was in the first place--that the diagonal of a triangle is not actually the sum of infinitesimal steps in alternating different directions. Thanks!

I'll try to think about that other one, schwarz... but I wouldn't expect much. I'm pretty lousy at math still. What was the generalization you came up with for if n is prime?


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## Nightriser

Haha, no problem. I was busy scratching away. But yeah, the hypotenuse is not just the sum of the step path lengths, because the step path lengths are simply the original triangle's sides evenly divided then rearranged. 

Sorry I couldn't explain it better, I apparently have a way of explaining things that make sense to almost no one. It makes perfect sense to me, but when I try to explain to someone else, I like to say that it's like explaining Japanese grammar in Arabic to a fellow American. :crazy:
I need to learn how to communicate things to other people. :dry:


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## Schwarz

cryptonia: If the pattern holds, if n is prime it should be (x-y)[x^(n-1)+x^(n-2)*y+...+x*y^(n-2)+y^(n-1)].
I don't have a proof for it or anything, though, I just worked it out with specific examples up to the 12th power or so.
Not that most people have much reason for using the 12th power and beyond anyway, but these things are just fun to figure out.


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## strygwyr

nightriser13 said:


> Anyone else here a math major or have any sort of interest in advanced math? I chose Galois theory as a senior thesis topic and am now wondering what I've gotten myself into. Some of it I think I understand, but I want to know if there's someone here I could ask any questions I may have. Thanks.


on my school days on secondary level here in philippines i ddnt know about the galois theory.. i have study on math major are..geometry..hmm i forgot to mention the other  coz its been a long time i ddnt visit on school xD


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## Schwarz

I actually found a solution to the problem I had before, so if any of you wants to give it a try, here it is:
Given:
u=1+(x^3)/3!+(x^6)/6!+(x^9)/9!+....
v=x+(x^4)/4!+(x^7)/7!+(x^10)/10!+...
w=(x^2)/2+(x^5)/5!+(x^8)/8!+(x^11)/11!+...
Show that: 
u^3+v^3+w^3-3uvw=1.


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## mcgooglian

I've never really been interested in math despite the fact that I'm good at it. I find it a bit weird that I don't like it considering the fact that I'm studying Arts and Business in university.


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## ArenaHomme

No, but I enjoy how clear cut math can be. It's enjoyable on that level, at least to me.


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## Nightriser

It's a fun challenge. I hadn't even considered majoring in math until I was in college. I originally entered to study engineering, but at the college I go to, the path to an engineering major is to major in applied math in this school, graduate after three years, then go to a partner school to get a master's in the engineering of our choice. After discrete, I fell in love with math. Up until then, everything had been boring, mechanical memorization (which I was good at nonetheless). 

schwarz, that looks like fun. I'll give it a shot, if I have time. :happy:


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## cryptonia

I got another one. I'll just post the problem here first, in case anyone wants to work on it, but if you don't have time/don't feel like it I'll post the proof too... because it was pretty bad ass IMO.

Prove that for any sphere who's surface is 90% covered by water (with any distribution), there exists a cube that can be inscribed such that all of it's corners are submerged.


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## cryptonia

so... it's time to rekindle this thread (even if it's very likely sifr will be the only one to answer :laughing

I was working with a generating function for Legendre polynomials today, which indirectly required me to do a binomial expansion on 1/sqrt(1 + v^2). I wasn't even sure what the formula for a binomial expansion was, so I looked it up on some site, and it required me to take (-1/2) choose 1.

Now, I've always known "n choose k" as n!/[(n-k)!k!].... so what gives? How are factorials defined for fractions, or even negative numbers for that matter? At first I thought that I was just being dumb and misapplying the formula, but I punched it into google and got the answer: -1/2. I then did it for the next term in the series, (-1/2) choose 2, and got back 3/8 (also the right answer). Eventually I got confused, and just typed (-1/2)!... and it returned 1.77245385.

If there's a pattern there, I sure can't find it.... and looking around on google (halfheartedly) just returned a whole host of "fractional factorial designs" pages. So... what's going on here?


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## Duke

If n is not a positive integer, i.e. either a negative integer or a fraction, then we have

(1 + x)^n = 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + ... + [n(n-1)(n-2)...(n-r+1)/r!]x^r + ... --- (1)

If you're wondering what r is, it is the (term number - 1). This implies that the r value for the 4th term in the series is 3.

1/√(1 + v²) = (1 + v²) ^(-1/2)

Therefore x = v², n = -1/2. Substitute values into (1), solve!

Don't stone me if you find any error in there! :shocked: I haven't done math in two years, but I've kept my notes and shambles of my memory. And I'm minoring in Math in college, which starts in about six months? Hurhur. I hope this helped, or at least narrowed your search... somehow. :tongue:

PS: Gawd, I hate it that they don't have the superscript function here. :dry:


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## Jrquinlisk

cryptonia said:


> so... it's time to rekindle this thread (even if it's very likely sifr will be the only one to answer :laughing
> 
> I was working with a generating function for Legendre polynomials today, which indirectly required me to do a binomial expansion on 1/sqrt(1 + v^2). I wasn't even sure what the formula for a binomial expansion was, so I looked it up on some site, and it required me to take (-1/2) choose 1.
> 
> Now, I've always known "n choose k" as n!/[(n-k)!k!].... so what gives? How are factorials defined for fractions, or even negative numbers for that matter? At first I thought that I was just being dumb and misapplying the formula, but I punched it into google and got the answer: -1/2. I then did it for the next term in the series, (-1/2) choose 2, and got back 3/8 (also the right answer). Eventually I got confused, and just typed (-1/2)!... and it returned 1.77245385.
> 
> If there's a pattern there, I sure can't find it.... and looking around on google (halfheartedly) just returned a whole host of "fractional factorial designs" pages. So... what's going on here?


...Can you even do that? I've never used Legendre polynomials in any of my classes, but I don't think you can do a binomial expansion on that. I don't think it even makes sense.

EDIT:



Duke said:


> If n is not a positive integer, i.e. either a negative integer or a fraction, then we have
> 
> (1 + x)^n = 1 + nx + [n(n-1)/2!]x^2 + [n(n-1)(n-2)/3!]x^3 + ... + [n(n-1)(n-2)...(n-r+1)/r!]x^r + ... --- (1)
> 
> If you're wondering what r is, it is the (term number - 1). This implies that the r value for the 4th term in the series is 3.
> 
> 1/√(1 + v²) = (1 + v²) ^(-1/2)
> 
> Therefore x = v², n = -1/2. Substitute values into (1), solve!
> 
> Don't stone me if you find any error in there! :shocked: I haven't done math in two years, but I've kept my notes and shambles of my memory. And I'm minoring in Math in college, which starts in about six months? Hurhur. I hope this helped, or at least narrowed your search... somehow. :tongue:
> 
> PS: Gawd, I hate it that they don't have the superscript function here. :dry:


Well, _now_ I feel stupid. In my defense, my Combinatorics class where we dealt with this stuff was so boring that I taught myself Japanese kana instead.


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## Duke

Jrquinlisk said:


> ...Can you even do that? I've never used *Legendre* polynomials in any of my classes, but I don't think you can do a binomial expansion on that. I don't think it even makes sense.


I haven't heard of Legendre polynomials either... It was always about Binomial expansions, Maclaurin's Series...

Integration and Differentiation. -jumps off the ledge-


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## Jrquinlisk

Duke said:


> I haven't heard of Legendre polynomials either... It was always about Binomial expansions, Maclaurin's Series...
> 
> Integration and Differentiation. -jumps off the ledge-


Heh heh... You haven't even gotten started. Laurent series! Bring on the residue theory!


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## Duke

Jrquinlisk said:


> Laurent series! Bring on the residue theory!


Zomg, I'd rather chew on Francis Bacon any other day. :tongue:


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## Shai Gar

Really? Because I've got three original old english Francis Bacon books here. Not first editions, but reprints of first editions.


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## cryptonia

oh wow, some more of the new people do do math, then :laughing:... excellent.

right, Duke. What you described was a Maclaurin Series (Taylor expansion around the point x=0). That's one type of expansion, but the formula for the expansion that I was using was the Binomial Theorem -- from Wolfram MathWorld (I think doing a strait Taylor expansion would have worked just fine and given the same answer... but it's beside the point now).

If you try to plug the expression given (I was wrong, by the way... it was 1/sqrt(1+v)... but it makes no real difference) into the binomial theorem, you get "(-1/2) choose k" for k as some integers. Anyway... haha most of that I gave was just as background, anyway. The main thing I was wondering about is "how is something like (-1/2)! defined?"

Oh... Legendre polynomials popped up because we were solving the Legendre Differential Equation -- from Wolfram MathWorld. I don't know if math majors would bother, because it's a physics-y math course and these types of diff eq's apparently pop up a lot in physics problems.


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## Nightriser

I can see what I can pick from it. I vaguely remember something about Legendre polynomials, from numerical analysis. I'll warn you that my education in math _is_ limited, since I go to a very small school. Get back to you later on that. :happy: 

OTOH, more people who know math! Awesome! I invite any of you to proofread my paper on Galois theory. I will soon post a newer draft with more information and corrections, so keep in mind that what I have up is only a rough draft. 

Galois theory: solvability of polynomials - PersonalityCafe (link)


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## Duke

I think I just got roflstomped. It started at 



cryptonia said:


> Binomial Theorem -- from *Wolfram MathWorld*


Halp.

:wink:


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