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A mathematician, a programmer, etc. etc.


Feb
20
comment Simple proof exercise recommendation, with full answers
"It's 2am, and I've spent more time than I really want to think about on an exercise which was clearly not meant to cause someone this much distress." You might be pushing too hard. Instead of forcing yourself to think about the problem, try taking a walk. I've noticed it help considerably, and many mathematicians recommend it (e.g., Alain Connes).
Feb
14
comment Symbolic math engines barf on this ostensibly tractable integral.
The trick is to expand this out using $2\cos(x)=\exp(ix)+\exp(-ix)$, consolidate the terms as $\int\exp(Ct)\,\mathrm{d}t$ then execute it.
Feb
13
comment integral of 1/(sqrt(e^x)) from 0 to infinity(Improper integral)
For 3, consider $\int (1+e^{ix})e^{-x}dx$ since $\exp(ix)=\cos(x)+i\sin(x)$ we take the imaginary part of the resulting integral...
Feb
12
comment Symbolic math engines barf on this ostensibly tractable integral.
What is the variable you're integrating? (There's no $dx$ factor in the integrand)
Feb
12
comment Vector Delta Function Identity
These are all simple roots, too, so none of them would have a multiplicity greater than 1...
Feb
12
comment Applications of Operator Algebras to modern physics
@Jeff Since you're at Caltech (I gather), go find Matilde Marcolli and speak with her. She's an expert on Noncommutative geometry, which basically amounts to applying operator algebras to...everything...
Feb
4
comment Is computer science a branch of mathematics?
@CarlMummert: true, but I figured everyone else would give rigorous books related to the formal field "the theory of computation", like the ones you noted. So, I wanted to just note there's a programming book that uses the axiomatic method :)
Feb
2
comment How must I understand concepts equations of physics?
It might be wise to study (a) differential geometry, and (b) partial differential equations. Differential geometry for the obvious reasons. But (b) because classical field theory, for the most part, boils down to PDEs [or, depending on your outlook, some "abstract nonsense" involving sections of fiber bundles ;)].
Jan
29
comment If 1 $\leq x$, then $\sqrt{x} \leq x $
Just a style thing, I would begin the proof writing "Proof. Assume for contradiction $x<\sqrt{x}$. Square both sides of $x\geq1$ gives us $x^{2}\geq x$. But squaring both sides of our assumption gives us $x^{2}<x$ which is a contradiction. We reject our assumption, and conclude $\sqrt{x}\leq x$." Just to make crystal clear which assumption we are trying to contradict. But I'm a windbag, so...
Jan
24
answered Is computer science a branch of mathematics?
Jan
24
comment Does $\frac12+\frac14+\frac18+\dots$ equal $1$?
Wait, there's a typo in your answer: $\sum^{n}_{k=1}2^{-k}=1-2^{-n-1}$ (you wrote the RHS as $1-2^{-k-1}$).
Jan
24
comment Metric space not a vector space
@SanathDevalapurkar: well, yes and no. Yes, because tangential thought is always helpful in speculative reasoning. Ostensibly, if you were interested in this direction, you would then ask about inverses and automorphisms, then about the diffeomorphism group's structure. That's a good approach, since the "distinct" Riemannian metrics would be $\mathbf{Riem}/\mathbf{Diff}$ (the space of all Riemann metrics modulo the diffeomorphism group). But it won't help coerce $\mathbf{Riem}$ into a vector space :(
Jan
24
comment Metric space not a vector space
@SanathDevalapurkar: You're talking about matrix multiplication? That's kind of tricky, there's a lot of nuances to that question. But remember the flat Riemannian manifold has its metric be (in suitable coordinates) the identity matrix...but that won't help considering the space of metrics as-a vector space (or as-a metric space --- it may be a tangential thought that made you ask, but I felt compelled to note it won't help).
Jan
24
comment Metric space not a vector space
@SanathDevalapurkar The space of all (Semi|Pseudo|vanilla) Riemannian metrics couldn't possibly be a vector space: what's the additive identity? They form a convex cone, though...see, e.g., Arthur Besse's Einstein Manifolds, Ch 4, section B for details and references.
Jan
24
comment Metric space not a vector space
It sounds like you're asking about the Moduli space of Riemannian metrics, which is a "manifold-like" space (not necessarily a vector space).
Jan
24
comment Metric space not a vector space
A relevant discussion: math.stackexchange.com/questions/98179/…
Jan
1
comment Closed form for this continued fraction
Note, though, for $x=i$, the convergents just fluctuates, cycling through $1$, $1+i$, $(1+i)/2$, without settling. So here's a puzzle: for which values of $x\in i(-2,2)$ does the partial fraction experience a divide-by-zero problem? (I mean, aside from the obvious $x=0$ solution!)
Dec
31
comment Evaluating the precision in the calculation of $\mathrm{e}$
@user21820 I think there is some ambiguity surrounding the term "arbitrary precision arithmetic". Since the OP was interested in having a rational approximation good to $n$ digits, I assumed the OP would be using something like MIT-Scheme which has "Bignum" arbitrary precision rationals. (Hence my concern about the number of division operations: they are the most costly for Bignum arithmetic.) Your use of the phrase "arbitrary precision" seems non-technical...but we have gone far from the ranch, and it seems completely irrelevant to the OP.
Dec
31
comment Evaluating the precision in the calculation of $\mathrm{e}$
@user21820 err, you seem to be forgetting the topic of discussion is computing $e$ to some desired precision. (I am not worried about $\exp(x)$ for general $x$, just $x=1$.) The 20th continuant of the continued fraction cited is good beyond 100 digits, requires 40 addition operations, 80 multiplication ops, and 1 division op. The Taylor series, OTOH, requires 38 terms, in Horner form that's 38 multiplication ops + 38 addition ops + the killer 38 division ops. That number of division operations makes it practically unacceptable...
Dec
30
comment Evaluating the precision in the calculation of $\mathrm{e}$
@user21820 A division operation is costlier than a multiplication operation. (About 5 times costlier, in fact, for x86 floating point...and rational arithmetic would be worse, computing the gcd and then performing 2 division operations...) The continued fraction expansion any sane person would have in mind could be googled in a second. Remember $e=\exp(1)$...