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[Helgoland]

Pure mathematics - always go for pure mathematics. (I'll count Analysis as applied here, since that's how we do it at my university.)

1) You'll be learning abstract stuff, concepts, ideas - and mostly without all the technical machinery that the applied subfields need.
2) With the applied stuff you'll probably end up in a job where you actually apply it. With pure mathematics, you'll get a job where they need your finely-honed analytical skills in their pure form. At least to me the latter holds much more appeal.
3) With applied mathematics you're essentially figuring out how a tiny sliver of the world works. With pure mathematics, you dig deep towards the very fundamentals of the human mind/the inner workings of the universe, depending on what you hold mathematics to be.

[prefuzek]

At my university, C&O is separated from applied math, and analysis is a decently big part of the pure math program (though not applied analysis - there are separate courses for that). Generally I prefer theory over computation (integration and ODE's aren't my favourite), so I was a little worried about courses like differential geometry, but I guess optimization/linear programming would probably have a lot more computation. Looking more closely at course descriptions it seems like the pure math department does a good job of keeping the applied math out.

Regarding 2), are there many jobs out there for a pure math major? It seems like C&O would be a more employable field. I'm thinking of going into academia/teaching anyway, so it might not matter as much, but options are nice.

[Helgoland]

Well, no jobs where you'll work as a mathematician - but other than that, there's plenty. Turns out lots of places have use for a certified smart person. If you're just after money, consulting and finance is where you should go, and if you've got different interests, the world is your oyster.

Take the topology courses if you can, that field is pretty much the love baby of abstraction and vividness. You get to paint pictures as well as spend a whole week devising a proof that in the end takes no more than six lines! Also the logic and set theory courses are really worth it, if you can overcome the initial cultural shock.

[Arx]

As a counterargument in favour of applied mathematics in the spirit of

3) With applied mathematics you're essentially figuring out how a tiny sliver of the world works. With pure mathematics, you dig deep towards the very fundamentals of the human mind/the inner workings of the universe, depending on what you hold mathematics to be.

...

With pure mathematics, you're doing proofs that are sometimes entirely divorced from reality and you might never actually see a purpose for. Applied maths, on the other hand, offers a way to encapsulate the behaviours of entire, sometimes extremely complicated, systems in a few lines of maths. Want to understand how an ecosystem ticks? That's an applied maths problem. Want to know where the universe came from? Cosmology is a discipline of applied maths. Want to see the future or the past? Ask an applied mathematician!

But you're really choosing between branches of pure maths here, not applied.

I'd go for combinatorics and optimisation, myself, but that's because I don't enjoy

abstract stuff, concepts, ideas - and mostly without all the technical machinery

[prefuzek]

With pure mathematics, you're doing proofs that are sometimes entirely divorced from reality and you might never actually see a purpose for.

Heh, that's the best part! At that point it's basically art.

Take the topology courses if you can, that field is pretty much the love baby of abstraction and vividness. You get to paint pictures as well as spend a whole week devising a proof that in the end takes no more than six lines! Also the logic and set theory courses are really worth it, if you can overcome the initial cultural shock.

Awesome. I've done a little logic/set theory in my algebra courses, and I think one of my calculus courses involved some basic analysis (sequential characterization of limits, etc.) so I'm pretty excited.

One more question: What is optimization/linear programming like? From what I've seen it looks like more complicated systems of linear equations, but I'm sure there's more to it than that.

[RedWarrior0]

Given the current topic of conversation, I have a compelling urge to link this

[MagmaMcFry]

Linear programming is solving a set of linear inequalities so that the solution maximizes a linear function over all other possible solutions, that's about it. It's not hard, but also not very broad or deep.

Optimization is a much broader topic, it's basically the theory of numerically finding optimal or near-optimal solutions of all kinds of mathematical functions. For example suppose you have some real function in k real variables on your computer and want to find its maximum, but all you can do is evaluate it at single points. Optimization gives you techniques to find the maximum.

[da_nang]

One more question: What is optimization/linear programming like? From what I've seen it looks like more complicated systems of linear equations, but I'm sure there's more to it than that.

Fun is what it is.

LP, NLP, QP, QCP, MILP, MINLP... once you know the basics, there are so many areas of application. Not just industrial, I know I use it when optimizing designs in games e.g. ship and missile designs in Aurora 4X.

Spoiler: Single-level generic MINLP (click to show/hide)

min_x,y f(x,y)

s.t.

g(x,y) <= 0

h(x,y) = 0

y ∈ ℤ

Above is a generic example, but it's not the only form (or class!) of optimization problems.

You'll obviously need to learn to model the scenario (or even relax it to make it tractable), figure out if the problem is convex or not and apply appropriate methods/algorithms. Be prepared to learn GAMS, as it's a widely used industrial software. There's a free version of it (limited as always  ) but if you can make the problem small enough to run on the free version, you can scale it up and run on NEOS servers for free. Excel also has solvers though I don't recall if any of them are global solvers, so you can only expect local optima. OpenOffice Calc has a MILP solver. Python's scipy I believe also has solvers.

That is, if you plan on going a bit more into the engineering side of things.

On the theoretical side of things, I imagine you'd be looking into creating algorithms, finding ways to rewrite non-convex expressions to convex expressions, or conditions for optima such as the Karush-Kuhn-Tucker conditions. I haven't seen much use for algebraic solutions for anything non-trivial (as numeric solutions are far vaster).

[Reelya]

LP, NLP, QP, QCP, MILP, MINLP...

MILP: Optimization is Magic?

[Helgoland]

With pure mathematics, you're doing proofs that are sometimes entirely divorced from reality and you might never actually see a purpose for.

Heh, that's the best part! At that point it's basically art.

This guy gets it!

Quick question though, because I'm not familiar with how the American (I presume) system works: What sort of mathematics have you seen so far? Because if sequences and limits is all you've done in Analysis so far, chances are you've got lots of things left to encounter. In that case I'd recommend to you taking a wide range of different subjects - that's what they force on the freshmen here, and it does a very good job of getting everyone into a field they like. It'd be a shame if you overcommitted to just one branch and never found out that, I dunno, numerical statistics was what you were really good at.

[prefuzek]

Okay, looks like optimization might not be quite my cup of tea. I might do a minor in C&O anyway, since parts of it seem really useful and interesting.

Quick question though, because I'm not familiar with how the American (I presume) system works: What sort of mathematics have you seen so far? Because if sequences and limits is all you've done in Analysis so far, chances are you've got lots of things left to encounter. In that case I'd recommend to you taking a wide range of different subjects - that's what they force on the freshmen here, and it does a very good job of getting everyone into a field they like. It'd be a shame if you overcommitted to just one branch and never found out that, I dunno, numerical statistics was what you were really good at.

Canada, actually. I've done Calc 1 & 2, an algebra course that covered proof techniques and some basic number theory, and linear algebra. Calc 1 and linear algebra were more theoretical than the standard first year courses here - they were quite proof-heavy and covered things like Cauchy sequences, open and closed sets, cardinality/ordinality, and Zorn's Lemma. But yeah, I recognize that there's still a lot out there. I do have quite a bit of flexibility (it's not difficult to switch majors) and will still be taking a pretty good variety of courses next year: some statistics, multivariable calculus, combinatorics, more linear algebra, etc.

[Strife26]

I'm actually reading a textbook in preparation for a test. How far have I fallen with my poor courseload decisions?

[hops]

So, on Paul's online math notes: http://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx



Why is it that we are able to assume that tan theta will be positive?

[Strife26]

He's right to remove the absolute value, but his motivation is wrong. The left side of that equation can never be negative (for any x outside of the range (-2/5, 2/5); inside that range the function does not exist).

This would make more sense if they bothered to use the triangle at the beginning of the explanation of trig sub, rather than cram it in at the end as a way of back-substituting, but that's a problem with literally every explanation of trig sub.

Trig substitution is innately problematic.

[hops]

Is there a trick to solving this painlessly? Every time I use Gaussian elimination I somehow get something different each time.