Here I tried to prove the Existence Theorem for Laplace Transforms. I don't know what the/a "conventional proof" looks like, but this is what I came up with.
What I can say is that the inequality comes from assumption that f is of exponential order. That is, that f is of exponential order ⇒ |f(t)| ≤ Me^{ct}. Since the definition looks at the the absolute value of f my assumption was that the sign of f didn't matter. Maybe that is wrong.
What would you suggest for making the proof more robust?
@johncarlosbaez I'm confused (I guess that much is clear 😞).
That said, if by exponential order we mean that |f(t)| ≤ Me^{ct} then it would seem that -e^{e^t} is not of exponential order because if we take logs of both sides of |-e^{e^t}| ≤ Me^{ct} we get e^{t} ≤ ln(M) + ct, with t ∈ [0,∞). e^{t} is going to grow faster than the right hand side for constants M and c.
So apparently f(t) = -e^{e^t} is not of exponential order and therefore doesn't meet the prerequisite of the theorem and hence f(t) = -e^{e^t} doesn't have a Laplace transform (this logic seems a bit shaky).
@johncarlosbaez Thanks. I came to that conclusion as well. I need to take a different approach to the proof to emphasize that showing that the integral
[M \int\limits_{0}^{\infty}e^{(c-s)t}dt ]
converges means that the Laplace transform exists..
The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].
Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].
Category theory friends: Is there a standard way to describe a functor?
I was using a two-case function to describe functor, where one case is what the functor does to objects and the other case is what the functor does to morphisms (see the image). However, I haven't been able to find a standard form in any of the literature I've been reading...
On May 17, 1902, Valerios Stais discovered the Antikythera Mechanism in a wooden box in the Antikythera shipwreck on the Greek island of Antikythera. The Mechanism is the oldest known mechanical computer and can accurately calculate various astronomical quantities.
As Tony Freeth says, "It is a work of stunning genius" [1].
My notes are mostly about Derek J. de Solla Price’s proposed Metonic Cycle gearing and how Micheal Wright actually figured out how that part of the mechanism worked (including the genius pin-and-slot device).
As Freeth said, "It is a work of stunning genius".
There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.
We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining
v⋅w= -½(vw + wv), v×w = ½(vw - wv)
We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.
The following stuff is pretty well-known: the group of linear transformations of ℝ³ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on ℝ³ because there's no linear subspace of ℝ³ that's mapped to itself by every transformation in SO(3).
Much to my surprise, it seems that SO(3) also has an irreducible representation on ℝ⁷ where every transformation preserves the dot product and cross product in 7 dimensions!
It's not news that SO(3) has an irreducible representation on ℝ⁷. In physics we call ℝ³ the spin-1 representation of SO(3), or at least a real form thereof, while ℝ⁷ is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on ℝ⁷ preserves the dot product. But I didn't know it also preserves the cross product on ℝ⁷, which is a much more exotic thing!
In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:
Just started writing up a few of my notes on introductory Category Theory. Not much here yet (it took me awhile to get Figure 1 to look right, and it's still not perfect).