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Just follow the usual path for education in relatively applied math. Here is a nutshell description: The standard high school level subjects are algebra, plan

Just follow the usual path for education in relatively applied math. Here is a nutshell description: The standard high school level subjects are algebra, plane geometry, second year algebra, trigonometry, and solid geometry. The standard college level subjects are calculus, abstract algebra, linear algebra, advanced calculus, ordinary differential equations. Might also take elementary courses in probability and statistics. Standard graduate school topics are point set topology, measure theory, functional analysis. For more in applications, can do (1) optimization -- linear, network linear, linear integer, quadratic, non-linear, and dynamic programming -- and (2) probability, statistics, and stochastic processes based on measure theory. Here is an overview: The course in abstract algebra will make you familiar with a long list of topics that come up in computing including sets, Boolean algebra, logic, relations, mappings, integers, prime numbers, modular arithmetic, rational numbers, the fundamental theorem of arithmetic, the fundamental theorem of algebra, elementary number theory, and the Euclidean greatest common divisor algorithm. You will also see finite field theory which has long been important in algebraic coding theory. You may touch on the question of P versus NP. You may see some of the work of Goedel, etc. and model theory. You will get good with math based on definitions, theorems, and proofs and develop 'mathematical maturity', i.e., ability to read, understand, and do abstract mathematics. Your ability to describe logical material in writing will improve. Famous authors in abstract algebra include Birkhoff, MacLane, Herstein, Lang. More recent authors likely touch on algebraic geometry. Linear algebra, mostly about 'vectors' and linear transformations, is the core of an ocean of applications in multivariate statistics (regression, factor analysis, analysis of variance, discriminate analysis, and more and, thus, applications in ad targeting, machine learning, data mining, anomaly detection, recommendation engines, etc.), optimization, group representation theory as in molecular spectroscopy, Shannon's information theory, and more. E.g., linear algebra is the place to learn about solving systems of linear equations, e.g., by Gauss elimination, and, thus, the place to get started with matrix inversion and the simplex algorithm of linear programming (optimization) and network linear programming. A good first text in linear algebra is by Ben Noble. The crown jewel is by Paul Halmos, 'Finite Dimensional Vector Spaces', written when Halmos was an assistant to von Neumann; the book approaches linear algebra much like it was functional analysis and, thus, is a baby version of Hilbert space theory and, thus, sometimes used to teach Hilbert space theory to physicists for quantum mechanics. Richard Bellman wrote tons. See also Roger Horn's books. Evar Nering is a good first book (although his treatment of linear programming is not good); so is Hoffman and Kunze (apparently now available on the Internet in PDF for free). For numerical linear algebra, see Forsythe and Moler. The standard advanced calculus text to teach the theorems and proofs of calculus is Walter Rudin's 'Principles of Mathematical Analysis'. After that book, notation such as O( n ln(n) ) as in Knuth's TACP will be child's play. Note: At one time the Halmos 'Finite Dimensional Vector Spaces' and Rudin's 'Principles' were used in Harvard's Math 55 with a colorful description in http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers A course in trigonometry, the use of trigonometry in calculus, and Rudin's treatment of Fourier series will help in data compression (e.g., JPG), time-invariant linear systems, digital filtering, and Shannon's information theory. Also good coverage of the binomial theorem in high school or abstract algebra and an elementary course in probability will be a help in following Knuth's TACP. Measure theory will take a huge load off your back: Calculus, that is, Riemann integration theory as taught through Rudin's 'Principles', has some rough edges, understood back to at least 1900, and Lebesgue and Borel and measure theory (heavily due to Lebesgue) provide a clean solution. As from Kolmogorov, the solution also makes a clean foundation for probability, stochastic processes, and statistics. Likely the nicest first text in measure theory is Royden's 'Real Analysis'. Also good is the first, real, half of Rudin's 'Real and Complex Analysis' where he also gives nice introductions to both Hilbert and Banach spaces and a nice treatment of the Fourier transform. My favorite authors in probability based on measure theory are J. Neveu (long in Paris) and L. Breiman (long at Berkeley). Or just read from their teacher, M. Loeve (long at Berkeley but with a writing style that looks like some cross between English and French). This background will also let you do much more in both pure and applied math in many directions and for many applications. For learning math, it is "not a spectator sport". Most of the work is between your ears as you think about the material. A good teacher in abstract algebra is likely necessary to get you started. Also without at least occasional contact with a solid university department it is too easy to get off track. Still, nearly all the work is to be done alone in a quiet room, and there a book is fine. In principle videos could help, but so far I've never seen even one that I would recommend for any utility at all in learning math.

dvse 14y ago
There is a number of really quite good applied maths courses on youtube (roughly in this order from each group): MIT 18.06, 18.085, 18.086, 6.262, 6.450 Stanford EE263, EE364A, EE261 Profs Strang, Boyd and Gallager are quite a bit better with maths than the typical engineering lecturer, even though their courses are not exactly at the level of Rudin, Breiman et al.
- lhh 14y ago
  My favorite math class was probably 18.310, intro to applied math, which is on ocw: http://ocw.mit.edu/courses/mathematics/18-310c-principles-of-applied-mathematics-fall-2007/index.htm http://ocw.mit.edu/courses/mathematics/18-310c-principles-of... It's not comprehensive by any means, and you probably need to know at least calculus to be ready for most of this, but it covers some pretty cool stuff, including RSA.
jasomill 14y ago
While I love Rudin's Real and Complex Analysis and Functional Analysis, I've always thought Loomis and Sternberg's Advanced Calculus could serve as an interesting alternative to more conventional texts like Baby Rudin for introductory analysis. It also seems particularly suited to self-study: surprisingly self-contained, good exercises, a nice selection of applications, and available free from Sternberg's Web site [1]. [1] http://www.math.harvard.edu/~shlomo/ http://www.math.harvard.edu/~shlomo/
- ten_fingers 14y ago
  'Advanced calculus' is a huge, ill-defined subject, partly a catch-all of introductions to several large topics. So I tried not to give a 'clarifying guide to advanced calculus' and, really, instead to concentrate on what would be of more interest to the OP. In particular, I listed only Rudin's 'Principles'. Generally in advanced calculus I avoided the discussion of, and much connection with, geometry, Stokes theorem, and exterior algebra. So, I avoided Buck, Fleming, Spivak, and of course also, now in English, Henri Cartan, 'Differential Forms'. I even avoided the classic applied advanced calculus text, long used at MIT, Francis B. Hildebrand, 'Advanced Calculus for Applications'. For Loomis and Sternberg, I agree with you, and have both the hard copy and the PDF. Since I've mentioned such advanced calculus, I will try to save many students: Students, there's a secret. The secret is that vector analysis, Stokes theorem, etc. are important in physics and engineering; they will also be important in computing when computing concentrates on such physics and engineering. But still mostly what you will find in physics and engineering is vector analysis much as it was done in the 19th century which the late 20th century math departments liked about as much as a skunk at a garden party. If you read the modern treatments, complete with differential forms, then you will be at the head of the class in an advanced class in general relativity (e.g., Misner, Thorne, and Wheeler) but will still be lost in much of old physics and engineering! So, what to do? Sure, go to Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957. The good thing about this book is the lie in the title -- it's mostly a 19th century treatment and not "modern"! So do whatever you have to do to get a copy. And get the 1957 edition and NOT a more recent edition where he omitted the 'good stuff'! Then, in about 20 pages of the sweetest dessert you ever tasted, with line integrals, conservative force fields, and potentials, volume and surface integrals, nice stuff like that, you will find a charmingly clear presentation of what you need. Right: The treatment is not up to the precision of Rudin and actually needs pictures. Still it's what you need for much of physics and engineering. It's, uh, 'intuitive' math; trying to make that material as precise as Rudin could take you, well, a long time. And it's EASY -- can take it with a couple of beers and have a really fun evening. Then don't tell anyone where you learned it! Besides, at its core, it's just nice uses of the fundamental theorem of calculus you saw in freshman calculus! Did I mention, it's easy? The key point about Rudin's 'Principles' is the care with which he covers the real numbers, compactness, continuity and uniform continuity, sequences and series, and the Riemann integral (yes, patched up with the Stieltjes extension which isn't much different). So, he concentrates hard on the foundations. For someone like the OP, getting those foundations solid is likely more important than rushing into many of the more famous topics in 'advanced calculus' -- Fourier series, the heat equation, Lagrange multipliers, vibrating strings (boundary value problems), the Navier-Stokes equations, series solutions to ordinary differential equations, etc. While I like Rudin, 'cut many of my math teeth' on Rudin, and really like some of his treatments of some topics, I omitted some notes on how to read Rudin; some such notes could be helpful. In particular, Rudin has some places where it's easy to get stuck, and students should be advised not to get stuck (don't assume that just because you can't see how to solve some one exercise must be missing something important) and if necessary just to look for other sources, ask for help, skip over and come back, or just f'get about it. Rudin was one of the best writers of his material, but he was not perfect, varied, got easier to read as he wrote more, but still is relatively severe. Due to the severity, there have been some people, e.g., at Courant, who didn't like Rudin! As much as I like the real half of his 'R&CA', he gets a bit severe and obscure in a few places (his novel and surprising but long and 'unstructured' construction of Lebesgue measure and his work on regular Borel measures); net, for most students it would be good to read Royden first or in parallel. Rudin has two exercises that can slow people down: (1) Every closed set is the union of a perfect set and a set that is at most countable and (2) there are no countably infinite sigma algebras. Both exercises require paying attention to what is countable versus uncountable. The first one I worked on about 14 hours a day for two weeks before someone mentioned 'uncountable' at which time I got it in about 90 seconds. The second one took me a long evening, but I was the only one in the class who got it. For the first one, eventually Rudin included the hint. Students: Don't get stuck on such exercises. For Rudin's 'Functional Analysis', I nearly went to Brown's Division of Applied Math but at the last moment went to Hopkins instead. Brown was using Rudin's FA, so I got a copy and at Hopkins asked for a reading course in it. Alas, the prof had never heard of that book and declined to participate! So, Rudin's FA, along with his 'Fourier Analysis on Groups' or some such are still sitting new on my shelf as I write software!

dvse 14y ago

There is a number of really quite good applied maths courses on youtube (roughly in this order from each group): MIT 18.06, 18.085, 18.086, 6.262, 6.450 Stanford EE263, EE364A, EE261 Profs Strang, Boyd and Gallager are quite a bit better with maths than the typical engineering lecturer, even though their courses are not exactly at the level of Rudin, Breiman et al.

lhh 14y ago
My favorite math class was probably 18.310, intro to applied math, which is on ocw: http://ocw.mit.edu/courses/mathematics/18-310c-principles-of-applied-mathematics-fall-2007/index.htm http://ocw.mit.edu/courses/mathematics/18-310c-principles-of... It's not comprehensive by any means, and you probably need to know at least calculus to be ready for most of this, but it covers some pretty cool stuff, including RSA.

jasomill 14y ago

While I love Rudin's Real and Complex Analysis and Functional Analysis, I've always thought Loomis and Sternberg's Advanced Calculus could serve as an interesting alternative to more conventional texts like Baby Rudin for introductory analysis. It also seems particularly suited to self-study: surprisingly self-contained, good exercises, a nice selection of applications, and available free from Sternberg's Web site [1]. [1] http://www.math.harvard.edu/~shlomo/ http://www.math.harvard.edu/~shlomo/

ten_fingers 14y ago
'Advanced calculus' is a huge, ill-defined subject, partly a catch-all of introductions to several large topics. So I tried not to give a 'clarifying guide to advanced calculus' and, really, instead to concentrate on what would be of more interest to the OP. In particular, I listed only Rudin's 'Principles'. Generally in advanced calculus I avoided the discussion of, and much connection with, geometry, Stokes theorem, and exterior algebra. So, I avoided Buck, Fleming, Spivak, and of course also, now in English, Henri Cartan, 'Differential Forms'. I even avoided the classic applied advanced calculus text, long used at MIT, Francis B. Hildebrand, 'Advanced Calculus for Applications'. For Loomis and Sternberg, I agree with you, and have both the hard copy and the PDF. Since I've mentioned such advanced calculus, I will try to save many students: Students, there's a secret. The secret is that vector analysis, Stokes theorem, etc. are important in physics and engineering; they will also be important in computing when computing concentrates on such physics and engineering. But still mostly what you will find in physics and engineering is vector analysis much as it was done in the 19th century which the late 20th century math departments liked about as much as a skunk at a garden party. If you read the modern treatments, complete with differential forms, then you will be at the head of the class in an advanced class in general relativity (e.g., Misner, Thorne, and Wheeler) but will still be lost in much of old physics and engineering! So, what to do? Sure, go to Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957. The good thing about this book is the lie in the title -- it's mostly a 19th century treatment and not "modern"! So do whatever you have to do to get a copy. And get the 1957 edition and NOT a more recent edition where he omitted the 'good stuff'! Then, in about 20 pages of the sweetest dessert you ever tasted, with line integrals, conservative force fields, and potentials, volume and surface integrals, nice stuff like that, you will find a charmingly clear presentation of what you need. Right: The treatment is not up to the precision of Rudin and actually needs pictures. Still it's what you need for much of physics and engineering. It's, uh, 'intuitive' math; trying to make that material as precise as Rudin could take you, well, a long time. And it's EASY -- can take it with a couple of beers and have a really fun evening. Then don't tell anyone where you learned it! Besides, at its core, it's just nice uses of the fundamental theorem of calculus you saw in freshman calculus! Did I mention, it's easy? The key point about Rudin's 'Principles' is the care with which he covers the real numbers, compactness, continuity and uniform continuity, sequences and series, and the Riemann integral (yes, patched up with the Stieltjes extension which isn't much different). So, he concentrates hard on the foundations. For someone like the OP, getting those foundations solid is likely more important than rushing into many of the more famous topics in 'advanced calculus' -- Fourier series, the heat equation, Lagrange multipliers, vibrating strings (boundary value problems), the Navier-Stokes equations, series solutions to ordinary differential equations, etc. While I like Rudin, 'cut many of my math teeth' on Rudin, and really like some of his treatments of some topics, I omitted some notes on how to read Rudin; some such notes could be helpful. In particular, Rudin has some places where it's easy to get stuck, and students should be advised not to get stuck (don't assume that just because you can't see how to solve some one exercise must be missing something important) and if necessary just to look for other sources, ask for help, skip over and come back, or just f'get about it. Rudin was one of the best writers of his material, but he was not perfect, varied, got easier to read as he wrote more, but still is relatively severe. Due to the severity, there have been some people, e.g., at Courant, who didn't like Rudin! As much as I like the real half of his 'R&CA', he gets a bit severe and obscure in a few places (his novel and surprising but long and 'unstructured' construction of Lebesgue measure and his work on regular Borel measures); net, for most students it would be good to read Royden first or in parallel. Rudin has two exercises that can slow people down: (1) Every closed set is the union of a perfect set and a set that is at most countable and (2) there are no countably infinite sigma algebras. Both exercises require paying attention to what is countable versus uncountable. The first one I worked on about 14 hours a day for two weeks before someone mentioned 'uncountable' at which time I got it in about 90 seconds. The second one took me a long evening, but I was the only one in the class who got it. For the first one, eventually Rudin included the hint. Students: Don't get stuck on such exercises. For Rudin's 'Functional Analysis', I nearly went to Brown's Division of Applied Math but at the last moment went to Hopkins instead. Brown was using Rudin's FA, so I got a copy and at Hopkins asked for a reading course in it. Alas, the prof had never heard of that book and declined to participate! So, Rudin's FA, along with his 'Fourier Analysis on Groups' or some such are still sitting new on my shelf as I write software!