jacobolus

The basic problem is that to get to the objects you mention here is at least 3 or 4 years of full-time study away from the kind of math people learn for a typical college degree in science or engineering. If you really want to understand them, to make your "digging" efficient you should probably just get a pure math degree, but setting aside several years to satisfy occasional curiosity is not feasible for most people for various reasons.

One unfortunate feature of published pure math research is that often the ideas are quite accessible and straightforward and don't really require special abstractions or terminology, but those get used anyway because for someone who already has a math PhD it saves a bit of effort.

The big problem I have with Axler's presentation of the SVD is that it's backwards. It leads with a bunch of completely dry and technical minutiae written formally, loads students up with tedious and confusing technical exercises they aren't likely to appreciate, and defers the motivation, context, explanation, and pictures until a few dozen pages later (probably multiple weeks later for a course), and in my opinion doesn't do a great job with them even then. I think this does a big disservice to students, and I wouldn't recommend anyone learn about the SVD this way. (Admittedly, the SVD is treated more like a curious aside than a centrally important tool in Axler's book; he's not trying to train people to use numerical methods.)

The best place to learn about the SVD is probably in the context of some kind of concrete problem (the most illuminating would be a problem in statistics, image processing, geometry modeling, or whatever, but it could also be a more abstract pure math problem, something about quadratic forms or something) that demonstrates an actual need for it, and then introduce the idea with an intuitive explanation supported by pictures and spatial reasoning. The actual technical details are not really that complicated or hard to figure out once you understand the concept, but if you don't understand the concept then trying to prove a bunch of obscure technical statements just seems like pointless busywork.

That's all well and good, but you also shouldn't excuse authors for not providing context, motivation, or explanation under the theory that the students should really be figuring out the whole subject from scratch for themselves.

If you are really lucky the result of not explaining things might be an occasional exceptional student who works out a correct personal concept. But more commonly the result is just an unfilled gap in understanding and either a moderately motivated student who develops fluency with symbol twiddling but doesn't get the point of what they're doing or a less-motivated student who decides the topic sucks and gives up.

I've read a lot of linear algebra books, and I personally find Axler's to have average quality exposition and a not tremendously insightful point of view. I know other people who swear by the book though, so YMMV. It's more appropriate for a well prepared pure math student who wants to go to grad school and has a goal of internalizing a lot of jargon so they can read/write pure math papers than for a scientist or computer programmer.

You can represent any arbitrary colors using RGB values for whatever "R", "G", and "B" primaries you like, they just might not fit in the range [0, 1].

(At least, under the strong assumptions we make in color modeling using the CIE system of colorimetry; the basic keyword is "Grassmann's laws".)

Really? I find the part about the SVD in Axler's book extremely unhelpful, big blobs of opaque formulas and jargon with next to no explanation or context that basically require either knowing the topic fully beforehand or a huge amount of effort to parse.

e.g. Axler's definition of singular values is the extremely dry and technical:

Suppose T is in L(V, W). The singular values of T are the nonnegative square roots of the eigenvalues of T†T, listed in decreasing order, each included as any times as the dimension of the corresponding eigenspace of T†T.

(Using a dagger instead of an asterisk for the conjugate transpose since HN interprets and asterisk to mean italics.)

If you already just proved a lot of stuff about eigenvalues, this could be a serviceable definition; at any rate it saves space. But it doesn't really explain the point.

I'd recommend anyone interested in this or related topics read Trefethen & Bau (1997) Numerical Linear Algebra.

Housing has gotten much more expensive in many places, and jobs less stable.

Scott Perry didn't present any evidence for those claims. Ideologically he is opposed to the humanitarian work USAID does around the world, and he cheered as Elon Musk dismantled the agency.

Perry was also one of the key figures in the criminal conspiracy trying to overturn the result of the 2020 election and overthrow the US government to install Donald Trump as a dictator based on a series of outrageous lies. If Trump hadn't been elected again in 2024, interrupting the relevant investigations, there's a decent chance Perry would be either on trial or perhaps even already convicted for his crimes (conscious of his guilt, he directly asked Trump for a pardon).

I wouldn't consider any claims he makes credible without some corroborating evidence.

In practice what happens is that on average the tool-user's wages go up slightly but most of the jobs in the field are eliminated, and the resulting large profit mostly goes to managers and financiers.

Each congressperson's staff gathers all of the comments they get, counts them up, and reports back to the congressperson, passing along anything they think would be useful information; if you write a clear and careful letter about an obscure topic there's a decent chance the congressperson themself will see it. Congresspeople typically have local ties in their area and caring what their constituents think is their job.

It's a serious problem that there are some congresspeople who don't do any local events, send all comments straight to the trash, etc. You should vote those folks out.

In the longer term, we should push for significantly increasing the size of the US House of Representatives to 5–10x the current size and implement serious campaign finance reforms. In combination, these will help make congresspeople more responsive to constituents and less reliant on donors.

One is people's lived experience: "Hard-working families immigrate to a land of better opportunity and build a life for themselves, integrating as upstanding members of the community."

The other is nativist propaganda: "Hordes of scary 'aliens' are coming to take your jobs and destroy your way of life, bringing their drugs and crime and turning your neighborhood into a trash heap. They might even eat your pets!"

People have difficulty noticing that the second story is supposed to be a description of what they or their ancestors personally lived as the first story; people compartmentalize and sometimes believe the propaganda version even though it directly contradicts their lived experience.

Scholars aren't supported by sales of their published work, but by teaching/research salaries, much of the money for which comes from the public via government grants.

Musicians by and large aren't supported by record sales, especially in the streaming era, but by concert tickets, merch, etc., or often by other income sources like paid lessons, session work, one-off commissions for specific customers, etc.

Very few fiction authors make a living at it, and most of those who do are barely scraping by.

Journalism is in a very sorry state in the 2020s; its long-time essential income source – classified ads – collapsed a couple decades ago under pressure from free or cheap online substitutes and the industry still hasn't figured out a viable alternative at scale. There has been a 75% drop in local journalists since 2000, most important local news now goes unreported (in many places there is no local reporting whatsoever) and regional/national scale journalism has been increasingly co-opted by the super-wealthy and turned to propaganda. Independent industry leaders with integrity are, over time, replaced by shills and the ethics of industry culture is degenerating.

Big budget TV/movies is probably closest to matching your argument, since these require large-scale coordination by hundreds of people to produce, but here too there are significant complications.

In all of these industries, the people making most of the profit are businesspeople rather than creators, though a trivial number of celebrity creators make good money.

Much of the published culture you mention is done entirely as a hobby, and our current copyright regime actually stands in the way of creation as much as supports it.

One thing to keep in mind is that many (most?) of the books and papers in these archives are decades old, usually no longer in print, make zero or vanishingly small amounts of money for their original creators, are sometimes only physically available from distant libraries that are challenging to access, etc.

In doing scholarly research, it's extremely helpful to be able to quickly search and skim hundreds of vaguely relevant sources, but simply wouldn't be worth the trouble to pay for or track down a "legitimate" copy of every one, and in many cases would be physically impossible. These "pirate" archives make doing real library research, previously limited to scholars at top-tier universities, accessible to orders of magnitude more people.

There really isn't that much profit in most of these works, and whether a scholar reads one on their laptop screen vs. in a physical book in a university library somewhere doesn't have any material impact on the original authors, editor, illustrator, translator, printer, etc.

That sounds plausible. Most of the features are kind of gimmicky bolt-ons added piecemeal and not really integrated with each-other. They make for cool 10-second demos but then most users ignore them because they aren't part of a coherent system. The result is a menu after menu of gimmicks, like a cabinet of hyper-specialized kitchen tools bought from infomercials. There has been limited product vision about the core abstractions and their basic composability. If you give a skilled user a photoshop version from the early 2000s they'll largely be able to do what they need, because there hasn't really been much fundamental innovative improvement in the past ~25 years.

Plenty of students succeed just fine without owning a graphing calculator (they can spend a few minutes learning the handful of test-relevant features and borrow one for the exam). Thankfully as of this year there is also a Desmos option.

Ideally the tests would not require external tools at all. There's nothing that needs to be tested in the context of a high school course that can't done with pencil and paper.

NEET means "Not in Education, Employment, or Training". The stereotype is an unemployed young adult living with their parents and playing video games all day.

They had different models with different capabilities. As they made minor style changes, they bumped the numbers slightly. The 81–82–83–84 were basically the same concept, as were the 85–86. The 89 and 92 were higher-end models. The 80 and 73 are simpler models intended for middle school.

All of them are basically a multi-generational scam perpetrated against the hapless parents of American high school students who were told that they needed to buy overpriced anachronistic calculators for their kids to succeed in school. In my opinion the calculators have overall caused more pedagogical harm than benefit; the students would be better served by some combination of (a) problems that can be solved without the tedious but trivial numerical calculations these calculators support, or (b) are solved using a real programming language. If someone really wants to assign simple numerical problems, give the kids slide rules.

Calculators of this type used to make sense for an engineer doing work in the field somewhere, but make no sense in the context of a classroom.

You should try to avoid continuous static load on your muscles, especially the smaller ones. So you should find a typing position where that doesn't happen. You also want to use your muscles in the strong and comfortable part of their range of motion, which depends on the entire chain of joints, because tendons have to stretch past several joints to get to whatever bone they attach to – so for fluent finger motions, you want to keep wrists and hands in as neutral a position as you can.

If your wrists are not straight while typing a lot, that's really bad. I constantly see people typing with their wrists either significantly flexed or significantly extended; doing that a lot is a fast road to RSI, and even doing it a little is pretty unpleasant and inadvisable.

If you are going to type a whole lot at a stretch (say, as a programmer or writer), you want your arms to be mostly passively supported from the shoulder. Having your arm bent at the elbow doesn't cause much strain, as long as the upper arm is hanging loosely down with your shoulder relaxed – so bring the keyboard relatively close to your torso. Resting your wrists, palms, or forearms on some surface and then typing generally causes more strain than having your wrists and palms "floating" above the keyboard while actively typing. You can rest the fingertips lightly on the key tops if you want. You can rest your palms on a palmrest or arms on an armrest (or table, or lap, or whatever) while you are taking a break from typing. It's generally a good idea to take regular breaks.

the monitor should be at eye level vertically

This is slightly misleading advice. The ideal place for the display has the top of the display at roughly eye level, or for a very large display maybe slightly above, which puts most of the display below eye level. Humans actually have great ability to look slightly downward for long periods of time while doing stuff with their hands, even while keeping their head held up straight, and indeed our eyes can more comfortably focus on close objects in the lower part of our field of view than straight ahead. What you don't want to do is slouch or bend your neck too much.

A laptop display attached to the keyboard usually isn't an ideal placement, but it's generally not too bad.

Having the desk low, the chair high, or putting a laptop on your lap is okay. Having the desk or table "high" (i.e. at normal height for writing with a pen or eating a meal) is generally worse but not an insurmountable problem.

In either case, the most important thing is to keep your wrists in as straight and neutral position as possible, with your palms and wrists "floating" rather than resting on anything while actively typing. Having the wrists either flexed downward or extended upward is a really bad idea. Having the wrists turned out to the side isn't great either, but not as bad.

The keyboard should be positioned close enough to your body so that your shoulders can be relaxed with your upper arms hanging loosely. The laptop surface should be roughly parallel to your forearms, so if you have a high desk or table relative to your torso you will need to prop up the far side to tilt it up a bit.

save our wrists

If your wrist is in contact with the edge of the laptop while you are actively typing, then your typing style has a good chance of giving you RSI. You'd be better off trying to fix that than trying to make the fast path to RSI more convenient.

Magnus doesn't "dictate the rules of the game or the platforms by which it is played", so I'm not sure what your point is.

You think Magnus should be forced to participate in chess events he doesn't enjoy / doesn't like the format of? Or you think organizations like FIDE or chess.com should be blocked from trying to entice Magnus to participate in their events?

Please be more specific and concrete.

Magnus "drives meaningful outcomes" because he's really good at chess and members of the public enjoy watching him play, so various chess-related businesses will pay him money for sponsorship. How do you propose to "not allow him" that influence? Ban all use of people's names in marketing and products?

There are LED lamps which have a "warm dim" feature so that the appear oranger as you reduce the brightness.

Providing a day-like view of the scene should not be the primary goal of nighttime lighting.

The sodium lamps are in fact safer for driving, because they preserve drivers' night vision, which improves visibility into the shadows, and because they cause less glare.

What they aren't good for is LED manufacturers' bottom line, and the lighting industry spent a lot of lobbying money to entice friendly politicians to heavily subsidize them with public infrastructure budgets, with those subsidies then misleadingly sold to the public as "efficient" and "environmentally friendly".

They're also not very good for reading the newspaper or doing critical color analysis. Thankfully such tasks do not need to be done at night in the middle of the street.

It's not especially close to the peak sensitivity of the human eye (in either bright or dim conditions), but that's entirely okay. The goal should be to not affect people's level of dark adaptation.

If you use shorter ("bluer") wavelengths, as happens with white LEDs which consist of a blue LED + phosphor, it causes people's eyes to become bright adapted and effective night vision is ruined, causing people to have much worse vision in the shadows.

Also, if you use bluer light, the lights themselves cause dramatically more glare in peripheral vision, because the shorter-wavelength-sensitive "S" cone cells and rod cells are mostly absent from the fovea (center of the retina), and prevalent in the outer areas of the retina. This is why LED headlamps on cars are so obnoxious for drivers going the opposite direction.

Also, the LEDs clobber people's circadian rhythms and are extremely disruptive to wildlife.

Finally, the light pollution caused by the LEDs is much worse for seeing the stars, which is maybe not as important as the other harms, but still kind of sad.

Yes, that's the point. It will only break down in water when the water is hot. Here's the paper https://link.springer.com/article/10.1007/s11696-022-02286-x

Yeah, I imagine it's considerably slower at ambient ocean temperature. Don't throw your PLA bags in the ocean or a river. Here's a different paper:

For example, PLA is not biodegradable in freshwater and seawater at low temperatures [32,36–39]. There are two primary reasons for this: (i) The hydrophobic nature of PLA, which does not easily absorb water [40–42]. In aqueous environments, the lack of hydrophilicity diminishes the hydrolysis process, which is crucial for the initial breakdown of PLA into smaller, more degradable fragments. (ii) Resistance to enzymatic attack; the enzymes that degrade PLA are not prevalent or active under typical freshwater and seawater conditions [39,43,44]. The microbial communities in these environments may not produce the necessary enzymes in sufficient quantities or at the required activity levels to effectively breakdown PLA. Additionally, the relatively stable and crystalline domains of PLA can further resist enzymatic degradation.

Also:

It should be emphasized that neat PLA cannot be classified as a completely biodegradable polymer, as it generates microplastics (MPs) during biodegradation.

My understanding is that cellophane generally does biodegrade in most settings. Polylactic acid (those cornstarch-derived bags) mostly biodegrades in hot enough compost or (after several years) in ambient-temperature soil, but not very well in cooler water (One study: "The half-life period of degradation [of polylactic acid in artificial seawater] is 12 [days at 90° C] or 468 days [at 60° C]").