5 ms·

To those starting on a similar path (self-taught, but interested in the mathematical part), what would you say? What was easy to progress in, where did you flou

by spyrosg 11y ago

To those starting on a similar path (self-taught, but interested in the mathematical part), what would you say? What was easy to progress in, where did you flounder and why the difference? Is a Ph.D. requisite?

graycat 11y ago
Well, for nearly any topic in applied math, there are some really simple treatments; they might be fun to read, but they are like a bicycle -- don't want to use them to cross the Rockies. Instead, often want the real stuff. For that, usually need a good version of the math prerequisites or will struggle. Where I had good prerequisites, I did fine. Otherwise I struggled. The graduate work that did me the most good was some quite serious work in optimization, measure theory, functional analysis, probability theory, and stochastic processes. So, then, presto, bingo, look at Fourier theory, L^1, L^2, etc. and prefer it to the best ice cream and cake. Gorgeous. Powerful. Great fun -- and I understand it. Before the grad studies, I was never quite sure what the heck had a Fourier transform and what didn't, when I could interchange order of integration, etc. It was like driving a car but not knowing how many wheels it had. At one point I got pushed hard into the fast Fourier transform, and taken narrowly that was okay, but the narrow view is not the one really want, and I had to struggle with the broader view. E.g., if have the power spectrum of the noise and that of the signal, what filter do you want? Generally if do know what linear filter want, then can use the FFT to implement it. At one point it would have been good to have done such things, but I didn't know how. Now I'm sure I could read it, likely from N. Weiner and then from more recent sources and find it easy reading and/or just derive it myself. Just why is the power spectrum the Fourier transform of the auto-covariance or some such result? Then I didn't quite have the background for that; now I'm tempted just to derive it myself. Ergodic theory? It was also great fun. Before the grad school work, ergodic was just a mystery. Now I can browse and thoroughly understand and enjoy Luenberger, Optimization by Vector Space Techniques, e.g., Kalman filtering, deterministic optimal control, high end versions of Lagrange multipliers. A lot of it is based on the Hahn-Banach theorem, and I saw a rock solid version of that. Before grad school, Luenberger would have been a bit much -- I would have struggled, often not quite sure just what I was doing, limited to some painting by the numbers. Before grad school, at times I wanted to know Kalman filtering and deterministic optimal control but struggled. It would have helped for me to have known those topics. I wrote my dissertation in stochastic optimal control, but I wished I'd been able to have a good, high end course in such things, including with stochastic differential equations, etc., but there wasn't much in courses to pick from. The number of US departments that know and teach that stuff is tiny. Off and on I considered just some independent reading courses for that material, but various exogenous events caused me to run short on time and cash and rush to finish instead. Before the grad work, I found statistics to be a cookbook of bad tasting meals. Now the usual statistics books bore me; I don't know of a really good statistics book; and I usually end up just deriving what I need for myself. With a good background in probability, that's the way to go. I had a pretty good undergrad major in math, and when that was enough as prerequisites I did well. E.g., I touched on linear algebra as an ugrad but wanted more. So, I got a stack of books on LA and dug in. Of course, the best was Halmos, Finite Dimensional Vector Spaces, but that didn't cover everything closely related, e.g., applications to calculus of several variables, numerical linear algebra, connections with multi-variate statistics, etc., and I got books on those for more. My ugrad work didn't do really well with multi-variable calculus -- no wonder because too soon really need measure theory. So, I got partly caught up on such calculus on my own and got a lot more in grad school, but I still don't know differential geometry well enough to find general relativity easy and wish I did. At least now I have the prerequisites to learn differential geometry. E.g., as an ugrad, the courses never covered the inverse and implicit function theorems (just local nonlinear versions of what is obvious in the general case of solving linear equations), but I got some good treatments in my independent reading. E.g., in computing, sure, Bachus-Naur form was easy enough, really is basically just set theory, but I never got how to take such BNF and automatically write a parser. Once it might have been nice to have done that! Or just use Yacc, right? > Ph.D. requisite? A good Ph.D. is a good thing. It's the training in how to work with things that are new. So, learn to zip through a lot of stuff don't need to know well, where K-12 and college and even grad courses have an implicit norm that such zip work is not good. Good? Heck, it's crucial! Learn the pros/cons of stuff that is new. Do begin to lose patience with things that are old -- too often they are not as good as might assume in a course and should be improved on or even just set aside. In doing research, just have to look at the material in a more effective way than the usual way as a student trying to make good grades. No longer trying to make good grades and, instead, are trying to do something new, e.g., improve on the stuff in the text books or papers. Also are no longer hanging on every word of a prof and, instead, are trying to do own stuff. Some of the coursework for a Ph.D. is also usually darned good. At a good school, i.e., a top research university, the difference in quality is like that between a dinner at McDonald's and one at a Michelin 3 star -- no joke. Or, if have a course in ugrad school and think that the course was fine, if get a course from a good researcher, very bright, who knows the material from various approaches, can get a LOT more. The good researchers are smarter -- and not by just a little bit. My view is that math, and mostly the advanced stuff and often new stuff, will be the key to the future of computing, at least until the software is able to do math better than humans. Computing only what we can think of without math will become way too limiting.

by spyrosg 11y ago

graycat 11y ago

Well, for nearly any topic in applied math, there are some really simple treatments; they might be fun to read, but they are like a bicycle -- don't want to use them to cross the Rockies. Instead, often want the real stuff. For that, usually need a good version of the math prerequisites or will struggle. Where I had good prerequisites, I did fine. Otherwise I struggled. The graduate work that did me the most good was some quite serious work in optimization, measure theory, functional analysis, probability theory, and stochastic processes. So, then, presto, bingo, look at Fourier theory, L^1, L^2, etc. and prefer it to the best ice cream and cake. Gorgeous. Powerful. Great fun -- and I understand it. Before the grad studies, I was never quite sure what the heck had a Fourier transform and what didn't, when I could interchange order of integration, etc. It was like driving a car but not knowing how many wheels it had. At one point I got pushed hard into the fast Fourier transform, and taken narrowly that was okay, but the narrow view is not the one really want, and I had to struggle with the broader view. E.g., if have the power spectrum of the noise and that of the signal, what filter do you want? Generally if do know what linear filter want, then can use the FFT to implement it. At one point it would have been good to have done such things, but I didn't know how. Now I'm sure I could read it, likely from N. Weiner and then from more recent sources and find it easy reading and/or just derive it myself. Just why is the power spectrum the Fourier transform of the auto-covariance or some such result? Then I didn't quite have the background for that; now I'm tempted just to derive it myself. Ergodic theory? It was also great fun. Before the grad school work, ergodic was just a mystery. Now I can browse and thoroughly understand and enjoy Luenberger, Optimization by Vector Space Techniques, e.g., Kalman filtering, deterministic optimal control, high end versions of Lagrange multipliers. A lot of it is based on the Hahn-Banach theorem, and I saw a rock solid version of that. Before grad school, Luenberger would have been a bit much -- I would have struggled, often not quite sure just what I was doing, limited to some painting by the numbers. Before grad school, at times I wanted to know Kalman filtering and deterministic optimal control but struggled. It would have helped for me to have known those topics. I wrote my dissertation in stochastic optimal control, but I wished I'd been able to have a good, high end course in such things, including with stochastic differential equations, etc., but there wasn't much in courses to pick from. The number of US departments that know and teach that stuff is tiny. Off and on I considered just some independent reading courses for that material, but various exogenous events caused me to run short on time and cash and rush to finish instead. Before the grad work, I found statistics to be a cookbook of bad tasting meals. Now the usual statistics books bore me; I don't know of a really good statistics book; and I usually end up just deriving what I need for myself. With a good background in probability, that's the way to go. I had a pretty good undergrad major in math, and when that was enough as prerequisites I did well. E.g., I touched on linear algebra as an ugrad but wanted more. So, I got a stack of books on LA and dug in. Of course, the best was Halmos, Finite Dimensional Vector Spaces, but that didn't cover everything closely related, e.g., applications to calculus of several variables, numerical linear algebra, connections with multi-variate statistics, etc., and I got books on those for more. My ugrad work didn't do really well with multi-variable calculus -- no wonder because too soon really need measure theory. So, I got partly caught up on such calculus on my own and got a lot more in grad school, but I still don't know differential geometry well enough to find general relativity easy and wish I did. At least now I have the prerequisites to learn differential geometry. E.g., as an ugrad, the courses never covered the inverse and implicit function theorems (just local nonlinear versions of what is obvious in the general case of solving linear equations), but I got some good treatments in my independent reading. E.g., in computing, sure, Bachus-Naur form was easy enough, really is basically just set theory, but I never got how to take such BNF and automatically write a parser. Once it might have been nice to have done that! Or just use Yacc, right? > Ph.D. requisite? A good Ph.D. is a good thing. It's the training in how to work with things that are new. So, learn to zip through a lot of stuff don't need to know well, where K-12 and college and even grad courses have an implicit norm that such zip work is not good. Good? Heck, it's crucial! Learn the pros/cons of stuff that is new. Do begin to lose patience with things that are old -- too often they are not as good as might assume in a course and should be improved on or even just set aside. In doing research, just have to look at the material in a more effective way than the usual way as a student trying to make good grades. No longer trying to make good grades and, instead, are trying to do something new, e.g., improve on the stuff in the text books or papers. Also are no longer hanging on every word of a prof and, instead, are trying to do own stuff. Some of the coursework for a Ph.D. is also usually darned good. At a good school, i.e., a top research university, the difference in quality is like that between a dinner at McDonald's and one at a Michelin 3 star -- no joke. Or, if have a course in ugrad school and think that the course was fine, if get a course from a good researcher, very bright, who knows the material from various approaches, can get a LOT more. The good researchers are smarter -- and not by just a little bit. My view is that math, and mostly the advanced stuff and often new stuff, will be the key to the future of computing, at least until the software is able to do math better than humans. Computing only what we can think of without math will become way too limiting.