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Jul
26
comment Matrix is conjugate to its own transpose
@MarcvanLeeuwen Sorry for the necromancy here, but I think I have a slightly more direct argument: Using rational canonical form as you suggested, it suffices to show that the companion matrix $C_p$ to $p(t)=t^n+a_{n-1}t^{n-1}+\ldots+a_0$ is similar to its transpose. I think we can prove this with just elementary linear algebra by noting: 1) The minimal polynomial of $C_p$ is $p(t)$ 2) In general, the minimal polynomials of $A$ and $A^t$ are equal 3) Any two matrices with cyclic bases (i.e. $v,Av,\ldots, A^{n-1}v$) and equal minimal polynomials are similar 4) $C_p^t$ has a cyclic basis
Jul
26
awarded  Critic
Jul
20
comment Reconstruction of a function from its moments
I'm not sure about this if you integrate on all of $(-\infty,\infty)$, but if you are integrating over some finite interval, $[a,b]$, and $f$ is continuous, then yes, $f$ is determined by its moments. This is a classic problem in undergraduate analysis: to use the Weierstrass approximation theorem to show that if $f$ is continuous on $[a,b]$ and $\int_a^b x^n f(x)\ dx=0$ for all natural numbers $n$, then $f=0$. (Can you see that this implies that the moments of a function are unique, by linearity of integration?)
Jul
18
comment Galois Group of the splitting field of the polynomial of $x^{11} - 7$ over $\mathbb{Q}$
Can you explicitly write some generators of $\mathbb L$ over $\mathbb Q$? Hint: there are two natural generators which have degrees $11$ and $10$ over $\mathbb Q$.
Jul
13
comment How to prove that the Schur algebra is isomorphic to a certain endomorphism ring?
You should reproduce the theorem for those of us who don't have the book :P
Jul
13
comment Exercise on a holomorphic $f$ on a strip satisfying $|f(z)|\leq A(1+|z|)^\eta$
A tighter bound would be to keep R a variable, include $(1+R)^\eta$ in your $A_n$, and then let $R\to 1$. Not that it matters for this problem in isolation
Jul
12
comment find local inverse functions of $f(x,y)=(x\sin(y),x\cos(y)), (x,y)\in (0,\infty)\times (0,3\pi)$
Have you not noticed that $f$ is basically the change-of-coordinates function from Cartesian to polar?
Jul
12
comment Questions about distributions on $l$-spaces.
Another piece of intuition: for certain distributions (those of order 0), the Riesz representation applies, and we can write them as $\langle T,f\rangle=\int_{\mathbb{R}^n} f\ d\mu$, where $\mu$ is a measure and the integral is a genuine measure-theoretic integral. This is in Friedlander-Joshi somewhere early.
Jul
12
comment Questions about distributions on $l$-spaces.
In the Euclidean case, the Schwartz functions are defined as smooth functions $f: \mathbb{R}^n\to\mathbb{C}$ with compact support. The intuition for the integral sign you asked about also is comes from noting that any (locally Lebesgue) integrable function $f$ defines a distribution, usually also denoted $f$, by the map $g\mapsto\int_{\mathbb{R}^n} fg dV$. There are some technical conditions to do with analysis that we put on functionals here, in addition to having them in $S^*(X)$. A standard reference is Friedlander and Joshi's "Intro to the Theory of Distributions."
Jul
12
comment Questions about distributions on $l$-spaces.
It comes from the definition of distributions over, say, Euclidean spaces $\mathbb{R}^n$. The name distribution (I think) comes from charge distribution - the intuition is that on Euclidean space, they represent things you can integrate (test) functions against. I.e. the Dirac delta distribution, $\delta(f)=f(0)$, which would, in E&M, represent something like a point charge at the origin.
Dec
23
awarded  Organizer
Dec
23
revised Banach Fixed Point Theorem problem contradiction
This is not about Banach spaces, nor really about functional analysis
Dec
23
suggested approved edit on Banach Fixed Point Theorem problem contradiction
Dec
22
answered Borel $\sigma$-algebra on non- second countable topological space
Dec
17
comment A formal justification for this “physicism”?
You're right. I think I need to ask my friend in what context he used this "trick" so that I can see what physical quantity the expression $\sum_{k=1}^\infty e^{ikx}$ represents, so I know how to think of your $\mathbb{C}[[e^{ix}]]$ as a collection of physical objects, i.e. interpret it physically.
Dec
14
comment A formal justification for this “physicism”?
OK, thanks a lot, that's helpful. I doubt I'll be able to understand his book, but I'll give it a shot and see if it contains an answer to what I'm looking for.
Dec
12
comment Generalized Cauchy's theorem (group theory)?
@GeoffRobinson yea, Sylow's theorems
Dec
12
comment A formal justification for this “physicism”?
Do you know of anywhere I could read about such kind of construct in more detail, by any chance? I haven't ever seen rings except which are semigroups under multiplication, so I can't see how this fits into any bigger mathematical theory.
Dec
12
comment A formal justification for this “physicism”?
But as a positive answer, what actually is it? What category does it belong to?
Dec
12
comment A formal justification for this “physicism”?
OK, fair. So what kind of object is $\mathbb{C}[[e^{ix},e^{-ix}]]$ supposed to be then?