BRK salary is 100k; the corp pays for the private jet ("the Indefensible"). Aside from Berkshire, he also has a personal stock portfolio worth 9 digits or so [in essence due to investment work prior to Berkshire]. He is able to compound that at high rates and is tax savvy so almost all returns are long-term cap gains (or at dividend rates); this is what he's pointed out when stating how low his overall tax rate is vs say typical administrative staff.
Evidently neither you nor the person you're arguing with actually knows much about this topic. It seems like you already have an extremely strong opinion on the man and are trying to back-fill it with "facts". Oh well.
Yea. Most of the media reporting on Buffett is... sloppy to put it politely. The guy lives with Astrid (and some body guards) in a 5 bed house that he sentimentally likes. He also has a couple nice houses in CA (and possibly other states) and a private jet. If someone reads any biography on him they'd know this. (IIRC it also came up when he was advising Gov Arnold on re-doing real estate taxes and he wrote some OP-eds.)
What's more surprising to me is how well he's doing (physically and cognitively) in his 90's as someone who eats at McDonalds every single day and I think is largely sedentary. The other side of the coin is his dad barely made it to 60 though his mom died in her 90's.
if you read Johnson's work, it's typically both. Direct fructose consumption as well as the polyol pathway being on and your body converting 'extra' glucose into fructose.
excess fructose (in particular in ultra-processed forms) causes metabolic derangement but consuming too much of those other 3 esp from ultra processed foods, also activates the "switch" leading to endogenous fructose production and more derangement.
The role of sedentarism in all this is somewhat under-explored though.
It seems misleading to call this "symmetry" because what you wrote is not a group.
The only idempotent element in a group is the identity. And in abelian groups where the binary operation is given by "+" the identity is denoted by "0". Yet you have "1" being idempotent, so conclude that this is not a group (or that this is a sleight of hand because you've made the identification that 1 = 0).
and you can actually use up all the various desaturase and elongase enzymes in the process so that 10% upper bound drops markedly as you consume more ALA. Put differently long chain n-3's production definitely isn't linearly dependent on short chain n-3s like ALA and stearidonic acid.
Steele's The Cauchy-Schwarz Masterclass is actually quite good and seemingly designed for self study. (A lingering result of this book is I heavily use inequalities even outside of analysis.)
Artin's Algebra probably has had the most impact on my math thinking. The development of groups and rings while tightly linking them to linear algebra was rather brilliant.
It's interesting. My first instinct was to disagree with this post, but on reflection I think I mostly agree with it. A couple useful mental models are (i) deliberate practice and (ii) train-validation-test(/out of sample) sets from machine learning
Your point (2) about compiler/interpreter in programming giving you rapid objective feedback is spot on and a vital component for deliberate practice that most people don't think on. You can kind of get this in math, in particular when you have some familiarity with the subject matter so the machinery isn't "too abstract" for you to sort through. (I.e. you should be able to confirm whether your proof/answer is accurate the vast majority of the time.) This is much trickier for first exposure to a subject though and the checking effort is on you, not the compiler.
The biggest issue I've seen with people self studying or in small math groups is your final (non-aside) paragraph which is perhaps more a psychological problem than and aptitude problem. When things get tough there's an enormous temptation to delude yourself to think you understand something that you are clueless about. The typical, schoolroom, way of mitigating this is via a final exam and you can check your grade at the end of the class; this gets typically gets short circuited in self guided study. Exams, btw, are essentially validation data sets you compare your math knowledge/model against. (We can call them 'test' sets if you prefer). The most important step really is repeatedly seeing how your knowledge works out of sample i.e. on 'new' stuff that comes out of the wood works and math.stackexchange is a perfect place for this when dealing with undergrad to mid-grad level problems. I do this all the time to get a sense of my understanding of a new subject I've recently acquired. But most people refuse this final step. People will tell me its 'too hard' and 'takes too much time' (meanwhile they start a new math book) but I strongly suspect it's in large part due to cognitive dissonance. (Another kind of out of sample test comes up when working on a subject matter that uses something you just "learned" as a pre-req, though there's a recursive element here and at some point they basically need to interact with 3rd parties.)
I suppose my relatively minor quibble is how much effectiveness depends on being "very gifted" [in some sort of math specific sense] vs understanding the basics of self-learning and being psychologically aware (astute?) enough to not go into denial. Insert quote from Feynman or whomever about how easy it is to fool yourself.
My understanding of one guess around e.g. annual flu shots is not so much protection from infection as frequent stimulation of the immune system being beneficial in particular for Apoe4 carriers.
E.g. from p. 329 of Lieberman's Exercised:
> Although Westerners who carry the two copies of a gene called Apoe4 (a protein that transports fats in the bloodstream) are three to fifteen times more likely to get Alzheimer's in old age, elderly Tsimane with the same ApoE4 gene are less likely to show declines in cognitive performance if they suffer from many infections.
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I have heard something similar about ApoE4 carriers in Nigeria though don't have a source on hand. Basically a lot of variations of the hygiene hypothesis and possibly compatible with "angry immune cells". Definitely not a "mystery solved" though.
What is Mathematics by Courant and Robbins is surprisingly good. For something a little more specific Pinter's A Book of Abstract Algebra would fit the bill.
> Do round numbers to the nearest whole number [and do not use decimals unless it's money]
is very bad advice and may lead to all kinds of inappropriate rounding. Trying to find an additional representation (possibly visually) without using decimals or percentage would be much better guidance.
There's a lot of literature on various kinds of cardio (esp HIIT and 'zone 2') and lactate production -> BDNF which is loosely understood to be very important for your brain.
Some more recent stuff on myokine release (from strength or cardio training) and warding off cognitive decline. Similar with training your balance (a parameter in exercise that is often forgotten).
More speculatively: crosswords and sudoku seem fine, but what about something much more cognitively demanding like doing serious mathematics?
E.g. Goodfellow et al did even worse than this sin in the Deep Learning book when they claimed the condition number for a square (but not necessarily normal) matrix is defined in terms of eigenvalues. This is false, but nevertheless see 4.2 in https://www.deeplearningbook.org/contents/numerical.html . When I've raised this with people in real life, I typically get some reflexive response that it should be a useful approximation, but as this blog points out, that isn't true either.
I'd encourage people to actually read the above link. It is very early days and quite far from definitive, but an encouraging attempt that's of special interest for people who are APOE-4 positive and/or have a family history of Alzheimers.
Setting aside early onset Alzheimers, which is arguably a different disease, Alzheimers has major genetic and (ill specified) lifestyle components, tends to manifest in one's ~70s but the damage is believed to start accumulating several decades earlier, hence this study on people in their 40's is of interest.
I still think regular cardio work has the best evidence behind it but if you are at heightened risk of Alzheimers, be pragmatic and make the best out of fuzzy data.
You also excrete a lot of metals via vigorous sweating (as well as urine and stool). All the more reason to do sustained cardio (or sauna) plus drink lots of water and eat lots of fibrous food. I.e. having a well rounded 'generically healthy' lifestyle (though I am suggesting a lot more sweating than most Americans are used to).
I was at a talk featuring Richard Thaler a few years ago, and someone in the audience asked a question in a vein that runs through the comments here -- 'is it better to teach high school students stats or calc'?
Thaler's response: I don't think it matters as people will forget either. For example, by show of hands, who here remembers anything of substance from their highschool chemistry class?
I was mortified that only I and ~5 others raised their hands out of a very large group. (I picked up the ideal gas law and dimensional analysis from said chemistry class and have found the latter to be quite useful.)
a more persuasive theory imo is that yo-yo dieters that don't do serrious resistance training, have repeated cycles of sarcopenia and osteopenia -- i.e. lose 'good stuff' along with fat on the diet phase and gain back disproportionately fat after. The negative effects compound over time with each cycle.
Evidently neither you nor the person you're arguing with actually knows much about this topic. It seems like you already have an extremely strong opinion on the man and are trying to back-fill it with "facts". Oh well.