887 reputation
29
bio website code.google.com/p/notebk
location
age
visits member for 1 year, 11 months
seen 17 mins ago

A mathematician, a programmer, etc. etc.


Apr
15
comment Solve $\sqrt{1+\sqrt{1-4x^2}}=x\left( 1+\sqrt{1+\sqrt{1+2\sqrt{1-4x^2}}}\right).$
Perhaps set $u=\sqrt{1-4x^{2}}$, so $x = \sqrt{1-u^{2}}/2$ and the problem becomes $2=\sqrt{1-u}(1+\sqrt{1+\sqrt{1+2u}})$...?
Apr
11
comment Evaluate the definite integral from 0 to pi/4 (6 - 6sin^2x) / (2cos^2x) dx
So...what have you attempted so far?
Mar
31
comment Tensor fields and vector bundles
To be precise, on the level of sections, don't we have an isomorphism $\Gamma(E\otimes F)\cong\Gamma(E)\otimes\Gamma(F)$ and not a strict equality?
Mar
31
comment Divergent Alternating Series
But if $p=1/2$ then $b_{9} = -1/2 < 0$, contradicting one of the premises of the problem ($b_{n}>0$ for all $n$).
Mar
30
comment Tangent space of the tangent bundle
Well...is $dg_{(x,0)}$ surjective?
Mar
8
comment Topological Quantum Field theories
@SanathDevalapurkar, also, where I'm studying -- I studied at UC Davis as an undergraduate. My current situation is rather strange (not that I'm private about it, I just cannot describe it in 140 characters!). I still study quantum gravity, though :)
Mar
8
comment Topological Quantum Field theories
In, e.g., 1+1 dimensional TQFT, dynamics is done by specifying the number of loops you begin with at time $t=0$, and how many you have at $t=1$, as well as the topology of the world-sheet for $0<t<1$. BUT the partition function (controlling dynamics) then becomes a function of the topological invariants (which invariants depends on the TQFT). This is good for, e.g., BF-theory since computing topological invariants is simpler than, say, solving the Wheeler-DeWitt equation :)
Mar
8
comment Topological Quantum Field theories
@SanathDevalapurkar, time reparametrization invariance forces the Hamiltonian to be a constraint; a great review of this can be found in Henneaux and Teitelboim's Quantization of Gauge Systems, viz. chapter 4.
Mar
3
comment Superspace as the Hilbert Space for Quantum Gravity
Well, no, you never do that in -- say -- using geometric quantization for the canonical formalism for electromagnetism (or, a simpler example, a scalar field). You don't do it when geometrically quantizing a mechanical system either. The symplectic manifold $\mathcal{M}$ is a necessary ingredient for constructing the Hilbert space (or Fock space for field theories), but not sufficient to qualify as the Hilbert space (resp. Fock space) for the full quantum theory. NB: you can use phase space path integrals, but this is an irrelevant fun fact rather than pertinent information.
Mar
3
comment Superspace as the Hilbert Space for Quantum Gravity
You might want to read Woodhouse's Geometric Quantization. You don't turn $\mathcal{Riem}$ into a Hilbert space: you use it (well, technically $\mathcal{Riem}/\mathcal{Diff}$) as the configuration space, then construct an infinite-dimensional symplectic manifold $\mathcal{M}$ which you use as the underlying manifold for a complex line bundle. The space of sections on this line bundle then in (insert magical full quantization step here) and you've got a Fock space.
Feb
20
comment How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?
Hint: consider $g(x)=xF(x)$ and Differentiate under the Integral Sign
Feb
20
comment Simple proof exercise recommendation, with full answers
And play with the first equation. The $\sqrt{ab}$ strongly hints at squaring both sides of the equation, just to see what happens...
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 ;)].