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2d
comment Is the area of linear programming dead right now?
No, it's not dead. The development of algorithms for solving very large scale convex optimization problems (including linear programs) is an active area of research. For example, in recent years new algorithms for solving linear programs arising in compressed sensing have been developed. LPs are classically solved with interior point methods (or the simplex method), but for very large scale problems these classical methods are too expensive and so we might try using proximal algorithms, a current hot topic in convex optimization research.
2d
comment Prerequisites for Spivak's Calculus on Manifolds
You also need to know some linear algebra.
Feb
6
comment A good companion to Axler's “Linear Algebra Done Right”?
Yes, this book certainly doesn't go overboard with applications. It does have a few nice applications, such as sections on special relativity, solving homogeneous linear systems of ODEs, and Markov chains.
Feb
6
comment Why do we teach Calculus in High School instead of, say, programming?
If physics and engineering aren't sufficient applications, machine learning and statistics make heavy use of calculus. And data science is one of the hottest jobs right now. However, in defense of this question, there is an article called The Calculus Trap on artofproblemsolving.com which points out that moving into calculus asap may not always be the best idea.
Feb
5
comment Why are math textbooks that are considered good books so hard to read? Why do authors make their books difficult to read?
One thing that may help to explain the style of baby Rudin and similar math books is to note that, while they may not be written in a way that you think is easy to learn from, they are written in a way that is very mathematically elegant. And mathematicians tend to place high value on this elegance, for aesthetic reasons. Check out other authors like Strang or Axler or Scott Aaronson, etc, for some styles you might find easier to learn from (while still being elegant).
Feb
3
comment what should I read after elements?
Tangential comment: it always strikes me as a bit strange that people still read Euclid as a textbook when we use modern textbooks for everything else. As great a genius as Euclid was, surely in the past 2000 years we've managed to write a better geometry textbook (building on Euclid's work, of course).
Feb
2
comment Linear Algebra book for beginners
I think you might have an easier time with Introduction to Linear Algebra, which is sometimes used as a textbook for lower division undergrad linear algebra courses.
Jan
26
comment Discrete Linear Programming over Finite Fields?
How are finite fields relevant to this? (Maybe they are, I'd just like to understand how. I've never heard of doing "linear programming over a finite field".)
Jan
25
comment How did Von Neumann think of the formula for scalar product?
This is a well known identity for the 2-norm -- for example, this identity is used to prove that a linear transformation that preserves the 2-norm also preserves inner products. You can prove the identity easily (for the 2-norm) by expanding the right hand side.
Jan
23
comment Book on applied mathematics
I think this book is written for applied math students just as much as it's written for engineering and physics students.
Jan
23
comment Calculus chain rule as “expanded form”
I'm actually not a fan of this way of stating the chain rule, because two different functions are both given the name $f$. Different functions should have different names.
Jan
22
comment Non-linear systems convergence
In convex optimization, convergence for many important iterative methods such as the Douglas-Rachford method is shown by expressing the method as fixed-point iteration for a certain nonlinear operator, then invoking convergence results for fixed-point iteration. One important convergence result is that if an operator is firmly nonexpansive and has a fixed point then fixed-point iteration converges to a fixed point. I'd be interested in learning what results are available for Gauss-Seidel iteration.
Jan
21
comment Why is $|x · y| ≤ ||x||_1||y||_∞$?
You could start by using the triangle inequality on the left hand side.
Jan
18
comment University-level books focusing on intuition?
I think Gilbert Strang's books have great intuition. For example, Linear Algebra and its Applications, and also his book Introduction to Applied Mathematics.
Jan
17
comment Should I continue trying to solve Spivak or pick up a lighter book?
I'd say try a bunch of different books until you find one that really clicks with you.
Jan
12
comment Is there an efficient way to evaluate the proximal operator of $f(x) = \|x\|_2 + I_{\geq 0}(x)$?
Thank you, this is great!
Jan
12
comment Is $ \operatorname{rank}A =\operatorname{rank} A^T$?
@MartinSleziak I think this question is not quite the same as the more general question in your first link, because here we have the special assumption that $A$ has real-valued entries, which allows us to use the fact that the range of $A^T$ is the orthogonal complement of the null space of $A$. A similar proof can be given in the general case, but we must substitute annihilators for orthogonal complements (this is explained in Lax's linear algebra book, for example) and I think it's not quite as obvious.
Jan
11
comment How to find a potential of a differential form?
I think it wants you to find a differential form $\eta$ such that $d \eta = \omega$. The term "potential form" is defined here.
Jan
11
comment Is there an efficient way to evaluate the proximal operator of $f(x) = \|x\|_2 + I_{\geq 0}(x)$?
@whyyes Thank you, but while those notes give the prox-operator of the 2-norm (on slide 9-3), I don't think they show how to compute this prox-operator (where $ x $ is constrained to the nonnegative orthant).
Jan
10
comment Freshman calculus - Stokes's theorem proof
I would like to see this too, but I think even just defining a surface with boundary precisely is difficult at the freshman calculus level. And I think you'd have to introduce partitions of unity to define a surface integral precisely, which is another difficult step. But I could be wrong.