- madhadron parentI’m not sure what you mean by a change of basis making a nonlinear system linear. A linear system is one where solutions add as elements of a vector space. That’s true no matter what basis you express it in.
- All these transforms are switching to an eigenbasis of some differential operator (that usually corresponds to a differential equation of interest). Spherical harmonics, Bessel and Henkel functions, which are the radial versions of sines/cosines and complex exponential, respectively, and on and on.
The next big jumps were to collections of functions not parameterized by subsets of R^n. Wavelets use a tree shapes parameter space.
There’s a whole, interesting area of overcomplete basis sets that I have been meaning to look into where you give up your basis functions being orthogonal and all those nice properties in exchange for having multiple options for adapting better to different signal characteristics.
I don’t think these transforms are going to be relevant to understanding neural nets, though. They are, by their nature, doing something with nonlinear structures in high dimensions which are not smoothly extended across their domain, which is the opposite problem all our current approaches to functional analysis deal with.
- No, there's a senate to get the small states in the late 18th century to ratify the constitution. There's still a senate because various groups added states without enough population to pass the bar of statehood in order to get extra votes without having to actually have democratic support.
- > (I should add, for context, that my friend and I are talking about writing beautiful essays here. If you want to write the most precise thing possible, you need to edit mercilessly and accept that the writing ends up flat and disjointed.)
There speaks someone who has never spent time writing poetry, and probably isn't very good at editing.
- The BCG vaccine is not protective against tuberculosis in adults. It helps prevent miliary tuberculosis in children.
I did my graduate work on tuberculosis. Those of us who weren't vaccinated because of our country of origin refused to be because the vaccine wouldn't help us and it changes testing for TB from a quick skin test to a lung x-ray.
It's not barbaric or corrupt or anti-vaccine in this case. It's details of this particular vaccine.
- As usual, it depends. My time in public schools (K through 7th grade in various places in the southeast US) was a mixed bag. Newport News, VA with all the kids of engineers and naval officers? Awesome. I loved school. Most other places? Meh. Rural western Virginia? Terrible. Bullied until I finally snapped and left someone half conscious on the playground (and the football coaches watched as I handed out that beating). I was homeschooled from 8th grade until I left for college at the suggestion of the teachers in the school because they were running out of classes at the high school for me when I was in 7th grade.
I was fortunate to have parents that are extremely well educated and my homeschooling gave me an education that is simply unavailable in a school. Not many kids have sat on the back deck in the Appalachians with their father, learning how to read Virgil in Latin.
There were lots of other homeschoolers in our county who were all religious nuts. Fortunately Virginia requires you to come in and take standardized tests every couple years to see if you're at grade level if you're homeschooling, so the worst cases got corrected. The school district also proctored my AP tests for me, even though they weren't classes the school offered.
My kids are in public school. The public schools where I live are excellent and actually deal with bullying. My kids would rather not go to school, but they're not being traumatized and they're getting a good education and have lots of friends. There's a major emphasis on social-emotional learning, which turns out to be heavily correlated with later performance. Our biggest problem is in high school with parents pressure cooking their kids to try to go to places like Harvard or Yale. I do what I can to counsel the kids and get them off that path. My own kids are firmly convinced that they're going to guaranteed admission state schools, and don't have to try to build a ridiculous resume in high school.
Schools don't have to be horrible. They just have a history of being poorly run in many places.
- It's a tradeoff. You can have excluded middle in your logic or infinitestimals in your extended reals. For mathematicians dealing with all the wild stuff coming out of studying infinities in the calculus, getting rid of excluded middle was a non-starter, so the system based on limits was created. If non-constructible proofs via contradiction aren't useful to you, as in physics, then you can certainly use infinitesimals.
- > how do you guys keep up to date with tech?
I don't. There really isn't much that's new that shows up. Spend your time exploring stuff people have already done, then learn whatever version of it this employer is using as needed at work. For example, at one job I needed to learn Ruby for a project. But here's an object-based, single dispatch, dynamically typed, interpreted language with some warts. You figure out what they call the inevitable things that are there in such a language and what the warts are, read some code to get a sense for the idiom, and then get on with life. If it's your first such language it feels challenging. If it's your third or fourth, it's just filling in the blanks. SQL Server has lots of warts and differences from PostgreSQL, but the stuff you expect is all there under some name. Any distributed database has to solve the same problems as any other one.
So don't worry about the new things. Worry about the old things with staying power, and the underlying designs and principles and patterns of thought that get reused again and again.
- Biosphere II is worth a visit. They're still running it as a non-isolated research station. The original had several very specific problems (trees need wind to form strong roots, fresh concrete absorbs a lot of oxygen, their crops besides sweet potatos had serious soil nutrient problems, which meant they spent almost all their time desperately subsistence farming). Based on what they did we could run a Biosphere III today pretty well, which means I would call the Biosphere II experiment successful.
- I think the difference between this and Jacques Cousteau's ConShelf in the 1960's and the saturation diving habitats used today is the depth and duration? Cousteau's group spent weeks at 10m and the saturation diving habitats are meant for living in for a few days at a time until your shift is over.
They do not explain anywhere why you would want to live at 200m, where it's dark and boring and you sound squeaky all the time. Living at 10m I totally get, especially in somewhere like the Red Sea.
- > I wonder what enabled the Inca to imagine all of these things...
Probably the same way as the Chinese, Romans, Persians, Hittites, Assyrians, Olmec, Mexica, and all the other large scale political polities of the world.
> How do you even spread those ideas across such a wide area without literacy?
Most of the world has been illiterate for most of history. Even after the invention of writing, most practical knowledge was orally transmitted. Writing was for warehouse records so you could prove that the peasants really did owe you that ox.
- > Nobody wants to drive what feels like a Prius
I'm quite happy with my Prius. I think it's a skill issue. Roof racks and a tow hitch on a normal sized car handle the vast majority of cases where people imagine they need a truck or SUV. But I remember a young couple showing up to buy a mattress from my roommate, and being flummoxed when the mattress wouldn't fit inside their Subaru Forester with built-in roof rails. I had to show them how to tie it on the roof. They never did return my rope... If you can only imagine putting things inside the vehicle or in the bed of a pickup truck, then having a truck or a big SUV must seem essential. What makes it even weirder is that the Prius has more interior space than a lot of SUVs.
- > Penrose claims that human intelligence relies on quantum effects and demonstrates certain structures in neurons that makes it at least a plausible argument.
Except that Penrose's argument is biologically nonsense, which probably means that he has an incorrect definition of intelligence, since I'm pretty confident in his ability to reason from axioms.
- That's because they're very closely related languages. English is a mix of Anglo-Saxon and Norman French, after the Norman invasion of England in the 11th century. Norman French is how the Norse diaspora in Normandy mixed their Germanic tongue with the local dialects of Latin. And the Romance and Germanic languages are both pretty closely related branches of Indo-European, which is also a pretty narrow language family to start with. For example, 'tu' from French comes from mixing Norse 'du' with Latin 'tuus'...both of which come from deeper roots. Proto Indo-European reconstructs this as 'tuH'.
- Markets are at best a tiny piece of the question on housing. It has more to do with zoning, infrastructure planning, incentives among the various roles involved in building, the foibles of real estate sales, the voice of incumbent residents who moved there because of the state of affairs that now has insufficient possibility for housing...
At the far end of that you have a market for what is there. It's nearly impossible to buy a well designed, well built house in many places because there aren't any. But the invisible hand of the market largely just skyrockets prices for limited, poor buildings because it doesn't have much real effect on the rest.
- The fact that we call the curve fitting/optimization/compression that we do to fit machine learning models to input data "learning" is really unfortunate and leads to this kind of conflation.
If we trace the path of how we ended up here, it's similar to how people incorrectly refer to loci of DNA as genes. We have behavior analysis where we speak of learning as the conditioning via the antecedant-behavior-consequence loop. There was the Hebbian theory of how the ABC loop manifested physically in neurons. Early neural net papers took inspiration from that that mechanism and called it learning.
Meanwhile, actual learning is far, far richer than the Hebbian theory of synaptic strengthening, and has a lot more going on than just operant conditioning.
So, please, it's time for everyone to stop pretending that the fact that ML inherited the word "learn" as a term of art for curve fitting has any philosophical weight.
- > If you want to learn about Sparta, it's pretty easy to find
Oh god. Why Sparta? Why this particular historiographical can of worms, where the history of the history is far more convoluted than the history of Lacedaemon itself?
No. Someone without a pretty solid knowledge of Ancient Greek history and historiography and the skills to be able to track the changes in the research literature is going to have a damn hard time finding the best book on Sparta.