lambda complex

@lambda-complex

MMMMM YES.

prepare for a lot of headcanons… sorry, lol.

when I started shipping it if I did: for as long as i’ve been in the fandom, which has been over a year ! golly, i’ve been in this hellfest for a while, haven’t i ?

my thoughts: IT’S A GREAT PAIRING THAT HOLDS SO MUCH POTENTIAL ! they were old friends back at black mesa, thus they have definite history together. with freemance, it’s a bit more iffy. relationships can blossom in stressful situations, such as the setting hl2 is in, but they also fizzle out faster. ( plus the two barely even know one another and alyx was a baby when gordon was chronologically 27 BUT YA KNOW, SOMETHING I’VE ALREADY EMPHASIZED IN THE PAST. ) but with freehoun, you have this 20 year gap of separation and quite possibly a lot of pain on barney’s end, if they were in a relationship prior to the black mesa incident. they could rebuild and heal together, and that is what makes it so much more wholesome and enjoyable. listen, i’m all for relationship development, but freemance would happen too quickly and die out too fast, and they both deserve better than that :“-)

What makes me happy about them: THE BLACK MESA ERA, HONESTLY. just the two goofs being grossly cute together in front of their co-workers, gordon causing havoc while barney tries and fails not to get roped into his bf’s shenanigans, PULLING PRANKS ON ONE ANOTHER IS A BIG ONE… and also, on my end, barney’s the very first person that gordon truly falls in love with. he was a fairly promiscuous person, but to finally feel something more than lust for somebody was honestly the best feeling in the world for gordon.

What makes me sad about them: aw MAN, the black mesa incident era and the hl2 era, especially. my gordon goes mad with power during hl2, since all the praise of being the savior of humanity fed into this wall of narcissism that pretty much kept him from crumbling into a paranoid mess. the latter does eventually happen after ep2, and barney’s pretty much the only one willing to help him fix himself. ( also adrián but w / e pFFFF )

things done in fanfic that annoys me: there’s like, literally 3 freehoun fanfics that i’ve found, and they were all good, so i can’t say there’s anything that annoys me. maybe the fact that there aren’t more freehoun fanfics.

things I look for in fanfic: UHHH, FREEHOUN ?? GIVE IT TO ME ANY WAY, I’LL PROBABLY LOVE IT.

Who I’d be comfortable them ending up with, if not each other: iunno. i’m not a big fan of freemance and shepman is Far from canon, so, ya know.

My happily ever after for them: MMMMMMMMM SETTLING DOWN IN A COZY HOUSE AND LIVING PEACEFULLY. PRETTY MUCH WHAT THE TWO HAVE ALWAYS WANTED. JUST. THEM BEING HAPPY.

who is the big spoon/little spoon: barmey… big spoon… gordang… LITTLE SPOO

what is their favorite non-sexual activity: mmmm for gordon, it’s just. Being with Barney. he loves his company so much. and for barney, he really likes playing with gordon’s hair, esp during black mesa, bc it was long and pretty.

Cauchy-Schwarz Inequality, Time, and Space

Say you’re sitting at home, and you thought to yourself, if I threw myself off the 40th floor balcony in my apartment building, what would my average speed be when I hit the ground? You don’t really intend to jump to find out, so you start doing some math. At this point, 9,999 out of 10,000 people would say that since the acceleration of gravity is 9.8 meters per second per second, and assuming each floor was about ten feet long (total of 122 meters), you would probably guess that your average speed would be about 24.5 m/s and leave it at that. BUT you are a mathematician (which is probably why you were contemplating jumping off a building in the first place), and you want to make sure you’re right. What you just calculated was a time average, that is:

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But, what if you wanted a different viewpoint? What if you wanted to look at the speed as a function of the distance fallen just to make sure the number you got was the right one? The equation is remarkably similar,

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but, gives you strange answers. Since L=122 and y=4.9t^2, we can see that:

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That’s obviously quite different from 24.5 m/s. You could argue (probably for hours) about the meaning of this result, but since you’re a mathematician and not a physicist, you’re more interested in whether

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regardless of the boundary conditions. You need a way to compare the two, so you think to yourself, what kind of math could I do with two integrated functions? 

Well, if we start with the realization that:

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If f(t) and g(t) are real valued functions, and U, and, L are real numbers (possibly infinite) and, lambda is real (for now). 

Expanding, we can see that:

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If we label each separate definite integral c,b,a respectively, we get:

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This has a simple geometric interpretation, the function of lambda can not cross the lambda-axis, the most it can do is touch it, as we are dealing with the area under functions. This must mean that lambda is complex for h(lambda)=0, or has a double zero at that point.

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So, we get the inequality:

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This finally gives the inequality

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So how are we going to use this to see if your velocity measured with time is less than or equal to your velocity measured using space?

Well, we can let g(t)=v(t)/T, and let f(t)=1. The Cauchy-Schwarz Inequality says that:

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and

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Taking advantage of the notation, 

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We’ve now effectively changed the variable to which we are integrating, so we have to change the limits on the integral, namely T=L and t=y

So, 

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We’re really close now! Since 

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We can change the left side of the inequality to be

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Which gives us

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and finally,

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So,

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Strange isn’t it?

I’ve deliberately left a bunch of questions unanswered, in the hopes that you’ll look into them. For one, what is the physical interpretation of the velocity in space that we derived in the first section? Also, would drag affect the answer in any way?

In any case, I hope this at least made you think. Pretty cool right? :)