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When I was writing my undergraduate honors paper on group representations, I tried something like your dependency structure! Result: Mostly all I got was just

When I was writing my undergraduate honors paper on group representations, I tried something like your dependency structure! Result: Mostly all I got was just a simple sequence with nothing interesting or surprising. For that nearly 2000 page PDF, actually I know enough about the contents to suspect that again a dependency structure graph, especially for anyone with a background in Baby Rudin (Principles of Mathematical Analysis), Big Rudin (Real and Complex Analysis or maybe Functional Analysis), surprisingly, would not look very interesting. From just going through the table of contents, there are a lot of interesting and uncommon topics in those ~2000 pages, but the book looks more like a small encyclopedia or reference than an integrated whole. And for something that long and in places with some rare topics, there is, surprisingly, especially for the goal of ML (machine learning), comparatively little on probability, stochastic processes, and mathematical statistics. But in those many pages, there are many topics possibly valuable and not easy to find elsewhere, so keep a copy of the PDF and use it as a reference when appropriate. Here is a short remark for readers without much background: Don't be afraid to dig in nearly anywhere in the ~2000 pages; maybe you will have to backtrack a little but in general can start nearly anywhere and don't have to read sequentially chapter 1, 2, .... Warning: Some topics are covered relatively superficially while others have some really rare details. Broad, Curious Point: The ~2000 pages may be this and that, but it is relatively close to the applied mathematics of operations research and numerical analysis, closer to those fields than what is usually regarded as computer science until computer science wants to absorb the math of operations research and numerical analysis. For more, some of the numerical analysis was, as I recall, mostly of interest for numerical solutions of partial differential equations. That field is also to be subsumed by computer science?

wenc 7y ago
> is relatively close to the applied mathematics of operations research and numerical analysis I know these areas well, and I would be hesitant to agree with this statement.
- graycat 7y ago
  My Ph.D. from one of the world's best research universities is in essentially the math of operations research. My most important prof was a star student of E. Cinlar, long editor in chief of Mathematics of Operations Research. I've published peer-reviewed original research, taught graduate courses, and applied to US national security and business the math of operations research. I stand behind my statement.
  - wenc 7y ago
    Then I would be doubly surprised -- perhaps you did not notice the lack of coverage of certain important OR areas like (mixed-) integer programming and the many advances therein? Or in nonlinear optimization, interior point methods? Active-sets? SSOCs? Constraint qualifications? Also pretty thin on theory of statistics, queuing theory, combinatorics, etc. These are all fundamental OR topics but the book makes different choices on topics to cover. (to be fair, the book does not claim to target OR, so I'm not disagreeing with the book's premise, only yours). I do think though that the book could benefit from feedback from specialists in each individual area. In some areas, I find the selection of topics to be a little bit "unconventional".
    - graycat 7y ago
      You are not disagreeing with what I stated. I said that the book was close to the math of OR (operations research), that is, much of the book is a subset of the math of OR. You outlined some major OR omissions in the book -- I agree fully. So, you are saying that the book does not cover OR, that its OR material is a proper subset of OR. Fine. I agree. I thought his linear programming coverage was superficial. In particular, I didn't notice any coverage of linear programming for the problem of least cost flows on a network where each arc has a maximum capacity. The nice anti-cycling rule for the version of the simplex algorithm, with strongly feasible bases, is from W. Cunningham, long Chair at the Waterloo Department of Combinatorics and Optimization. It turns out that that algorithm makes a nice dent in the challenges of linear integer programming. Yes, I saw not much on constraint qualifications. I'm the guy who showed that the Zangwill and Kuhn-Tucker constraint qualifications are independent for problems with only functional constraints -- my work also solved a problem stated but not solved in the famous Arrow, Hurwicz, Uzawa paper on constraint qualifications in mathematical economics. Any really good book on linear programming is "close" to OR but can be attacked with your objection that it omits lots of OR.
siddboots 7y ago
> Mostly all I got was just a simple sequence with nothing interesting or surprising. It absolutely depends on the text of course. One one end of the spectrum you have something like Euclid's Elements, where a DAG representation seems to be very well suited to spirit of the text. Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. You might be right about the linked textbook. I suspect that there would be a core linear algebra subgraph that most of the rest of the book depends on, but otherwise little else in the way of complex structure. One thing that occurred to me was that the reductio ad absurdum of this ends up being something like mathematical logic. As you try to capture the dependency structure with more and more precision, you find yourself trying to build a proof assistant. Still, I think that there's something to be said for studying the structure of great textbooks purely as cultural artifacts.
- graycat 7y ago
  > Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. Naw! Take another pass through Baby Rudin! Here is some help: With appropriate mild assumptions, a continuous, real value function on a compact set is uniformly continuous. For the finite dimensional space of interest, compact is the same as closed and bounded. That is the main content of the first chapters. First Application: With uniform continuity, get to show that the Riemann and Riemann Stieltjes sums converge so that those integrals do exist. Second Application: The uniform limit of a sequence of continuous functions is continuous. This was a question on my Analysis Ph.D. qualifying exam -- I got it! Thus a certain normed space is complete, that is, is a Banach space -- I'm not sure this remark is in the book. The chapter(s) on sequences and series is used to define ln, sin, cosine so that can do Fourier series. The exterior algebra is essentially separate. That's the main stuff. Don't take the Riemann integral very seriously: The good stuff is Lebesgue's integral (partitions the range instead of the domain) as in the first half of Rudin's Real and Complex Analysis. There also get to see Banach space, Hilbert space, von Neumann's novel proof of the Radon-Nikodym theorem (grown up version of the fundamental theorem of calculus and with astounding power) and the Fourier integral. I won't take off points if you don't read the second half. Rudin's Functional Analysis -- get to see distributions.

wenc 7y ago

> is relatively close to the applied mathematics of operations research and numerical analysis I know these areas well, and I would be hesitant to agree with this statement.

graycat 7y ago
My Ph.D. from one of the world's best research universities is in essentially the math of operations research. My most important prof was a star student of E. Cinlar, long editor in chief of Mathematics of Operations Research. I've published peer-reviewed original research, taught graduate courses, and applied to US national security and business the math of operations research. I stand behind my statement.
- wenc 7y ago
  Then I would be doubly surprised -- perhaps you did not notice the lack of coverage of certain important OR areas like (mixed-) integer programming and the many advances therein? Or in nonlinear optimization, interior point methods? Active-sets? SSOCs? Constraint qualifications? Also pretty thin on theory of statistics, queuing theory, combinatorics, etc. These are all fundamental OR topics but the book makes different choices on topics to cover. (to be fair, the book does not claim to target OR, so I'm not disagreeing with the book's premise, only yours). I do think though that the book could benefit from feedback from specialists in each individual area. In some areas, I find the selection of topics to be a little bit "unconventional".
  - graycat 7y ago
    You are not disagreeing with what I stated. I said that the book was close to the math of OR (operations research), that is, much of the book is a subset of the math of OR. You outlined some major OR omissions in the book -- I agree fully. So, you are saying that the book does not cover OR, that its OR material is a proper subset of OR. Fine. I agree. I thought his linear programming coverage was superficial. In particular, I didn't notice any coverage of linear programming for the problem of least cost flows on a network where each arc has a maximum capacity. The nice anti-cycling rule for the version of the simplex algorithm, with strongly feasible bases, is from W. Cunningham, long Chair at the Waterloo Department of Combinatorics and Optimization. It turns out that that algorithm makes a nice dent in the challenges of linear integer programming. Yes, I saw not much on constraint qualifications. I'm the guy who showed that the Zangwill and Kuhn-Tucker constraint qualifications are independent for problems with only functional constraints -- my work also solved a problem stated but not solved in the famous Arrow, Hurwicz, Uzawa paper on constraint qualifications in mathematical economics. Any really good book on linear programming is "close" to OR but can be attacked with your objection that it omits lots of OR.

siddboots 7y ago

> Mostly all I got was just a simple sequence with nothing interesting or surprising. It absolutely depends on the text of course. One one end of the spectrum you have something like Euclid's Elements, where a DAG representation seems to be very well suited to spirit of the text. Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. You might be right about the linked textbook. I suspect that there would be a core linear algebra subgraph that most of the rest of the book depends on, but otherwise little else in the way of complex structure. One thing that occurred to me was that the reductio ad absurdum of this ends up being something like mathematical logic. As you try to capture the dependency structure with more and more precision, you find yourself trying to build a proof assistant. Still, I think that there's something to be said for studying the structure of great textbooks purely as cultural artifacts.

graycat 7y ago
> Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered. Naw! Take another pass through Baby Rudin! Here is some help: With appropriate mild assumptions, a continuous, real value function on a compact set is uniformly continuous. For the finite dimensional space of interest, compact is the same as closed and bounded. That is the main content of the first chapters. First Application: With uniform continuity, get to show that the Riemann and Riemann Stieltjes sums converge so that those integrals do exist. Second Application: The uniform limit of a sequence of continuous functions is continuous. This was a question on my Analysis Ph.D. qualifying exam -- I got it! Thus a certain normed space is complete, that is, is a Banach space -- I'm not sure this remark is in the book. The chapter(s) on sequences and series is used to define ln, sin, cosine so that can do Fourier series. The exterior algebra is essentially separate. That's the main stuff. Don't take the Riemann integral very seriously: The good stuff is Lebesgue's integral (partitions the range instead of the domain) as in the first half of Rudin's Real and Complex Analysis. There also get to see Banach space, Hilbert space, von Neumann's novel proof of the Radon-Nikodym theorem (grown up version of the fundamental theorem of calculus and with astounding power) and the Fourier integral. I won't take off points if you don't read the second half. Rudin's Functional Analysis -- get to see distributions.