Staff/Principal Engineer @ Apple. B.S. EECS Berkeley. M.S. CS GaTech. 🇺🇸 Malaysian-American. Childhood hometown: Boise, Idaho. Happily Married. Opinions own.
SF Bay Area, CA
Joined July 2012
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I remember back in my undergrad “Probability Theory and Stochastic Processes” class learning about this history and being so shocked how young of a field Probability Theory was.
Fun fact: Probability theory only became the formalized axiomatic mathematically rigorous system we know today in the 1930s. Before that it had tons of holes.
A brilliant Russian mathematician named Kolmogorov used
Measure Theory developed by Lebesgue which fixed issues with Reimann integral (basically Lebesgue integration allows Integrals at functions with finite discontinuities which is pretty important achieved roughly by ignoring “Measure 0 sets” as they don’t contribute to area under the curve pardon my ignorance something like that I haven’t taken measure theory)
and Distribution Theory by Strichartz which gave proper rigor to being able to take Fourier Transforms and Inverse Fourier Transform for things they aren’t classical devotion of a function like the “Dirac delta” or even the number “1” which Physicists like Heaviside and especially Dirac invented/used in their work especially Dirac in Quantum Physics but never made rigorous.
So Probability Theory and its 3 axioms are actually one of the most recent branches of mathematics, 1930s is extremely new from a rigorous mathematical model point of view.
Another cool thing about the Fourier Transform (and its inverse transform) is how it repeats itself mod 4 when being applied over and over.
I was reading this book and one section was going over Fourier Duality Properties so I drew some pictures by hand and then asked Claude to make these pictures.
I remember people trying to memorize duality properties back in undergrad but this cycle graph I made I feel makes it easier.
Sharing in case it’s useful to others.
Pretty cool huh? Any peridic function/signal that has a finite number of discontinuities can be represented as a (possibly infinite) sum of sines and consines (or equivalently complex exponentials).
Now notice the weird wiggling though. If the discontinuity is in the original function then you’ll see that wiggling no matter how many finite sums of harmonics you use it never converges to that exact value only the average of the values (which makes sense there is no “one value” because it’s discontinuous at that point it has two different values at that point depending on which direction you approach from in the limit).
But if you look at the triangle one, the approximation is nearly exact looking.
Why? Because the discontinuity is not in the original triangle function but in its derivative.
Will you always need an infinite sum? If there are discontinuities in the original function or any of its derivatives then yes you will need infinite sum.
But if the periodic function is infinitely differentiable (you can keep taking derivatives and each stays smooth and continuous) then finite number should work.
Intuitively this makes sense because sines and cosines are curvy so things that have edges (discontinuities) you need to sum infinite curvy things to create them something non-curvy.
Will always be my greatest game of all time (It still has the highest meta critic rating of any video game in the world at 99/100)
And now they are making a remake of it!
Adventure through Hyrule and become the Hero of Time when The Legend of #Zelda: Ocarina of Time releases Nov 5 on Nintendo Switch 2. #NintendoDirect
I don’t think I’ve ever done a “I like this product” type of tweet but I have been really impressed with my @maticrobots and the fact that it vacuums, sweeps, and mops is super cool. And the App UI/UX is fantastic too. The ability to select custom regions from your mapped out floors to custom clean was implemented very nicely for example.
Also just to be clear I’m not sponsored or anything but I really like how because it’s computer vision based like the Tesla you can just keep iteratively improving the CV algorithms in tandem with w/e reinforcement learning and SLAM it might also be doing.
I’ve seen how much the Tesla FSD has improved and the recent HW4 (even the distilled HW4lite which is what I have since I have an older Tesla) are phenomenal.
And the fact that they do final assembly in Mountain View, CA is also awesome.
Just so many positive things about it. I hope it does well.
I’m proud and grateful to put my heart and soul into this company for over a decade and counting!
Tim is amazing and I’m equally excited for the future with John as well.
If you have a pretty solid understanding and intuition on basic continuous function calculus and also of Linear Algebra from undergrad,
And wanted to know how those concepts and intuitions can be extended into real world infinite dimensional vector spaces like functional spaces and also how derivatives/differentiation and integration definitions can be changed/extended to work for real world but bad behaving “functions” that you see in Physics and Engineering like Electrical Engineering (Signal Processing)
Without all the hand waving that they do in your undergraduate physics or engineering classes where they ignore the fact the original definitions like Riemann integration doesn’t work for a lot of these real world signals and systems.
Then this short intuition PDF might interest you.
I generated this short 12 page PDF from these questions and answer chat sessions I was having with ChatGPT and Claude when I’m curious or bored.
I attached the PDF in a LinkedIn post I made so you can find it there.
It’s only been day 1 but so far impressed!
We got one and it’s pretty cool!
It’s actually neat seeing it seemingly do a lot of the algorithms from the AI in Robotics class I did almost a decade ago in my younger days like the SLAM (Simultaneous Localization and Mapping) algorithm for mapping out the rooms.
And I’m guessing when you add a new room by deep clicking to the closest point in existing map and it creates a line from Matic Origin to that point you clicked, it’s probably path planning that using some version of A* search to create the shortest valid path to that point.
The Computer Vision object detection and identification stuff seems cool too it correctly identified the stairs in our townhouse and didn’t move forward and fall down the stairs.
Also when one of the smaller carpets got caught it knew to turn off the vacuum and back away until the clog stopped.
I’m sure your algorithms are far more advanced than what I’m mentioning here, but it’s pretty neat seeing it all work!
Great work!
Yeah I’m deleting this app from my phone it’s too distracting.
Whenever I’ve made progress in my weekend labs or want to share something mathy I’ll post from my computer but having this app on my phone causes anxiety because of silliness of some of the prevailing views shared.
What I hadn’t realized before is that the dimension of a vector space is not just a property of the set that makes up the vector space BUT ALSO the field F chosen to be the scalars that can be used in scalar multiplication on that set.
I mean it seems obvious when you use C^N as your set and you use real scalars R versus complex scalars C on it and the dimension changes based on which field you choose R or C.
But it didn’t strike me until today lol.
I asked the LLM if there was a notion of extending this then to using Fields that can give you greater degrees of Freedom (Reals give you one degree of freedom, Complex gives you two degrees of freedom, can you go more?) and the short answer is for all practical purposes no.
You truly do have all of humanity’s knowledge and research at your fingertips because of these LLM thingamajigs.
There will be remote villages in Africa and India with gifted teenagers or young adults who will be able to learn harmonic or spectral analysis and maybe custom signal processing solutions so solve some problem in their village instead do relying on some expensive Western company with engineers to do it.
I know that sounds naive and idealistic but I think it’ll happen in the next 20 years.
The next time a mathematician asks if you want something like a door “open” or “closed” tell them you want it “clopen” and if they get annoyed tell them that’s how most normal people feel about some definitions in topology lol.
(Yes yes the definitions makes sense and they are good definitions with mostly good physical, geometric, or visual intuitions especially in terms of metric spaces, but I still find it funny a set can be both “open” and “closed” and thus “clopen”)