lambda complex

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? :)