User jon - MathOverflow

Answer by Jon for Theorems with unexpected conclusions

2010-03-14T09:11:56Z

Definition: Let $A$ and $B$ be self-adjoint matrices, with the partial order $A\ge B$ if $A-B$ is positive semidefinite. If $A$ is self-adjoint with spectrum in the interval $[a,b]$ and $f\colon [a,b] \to \mathbb{R}$ is a real-function, define $f(A)$ using the spectral theorem. The function $f$ is called matrix monotone if $A\ge B$ implies $f(A)\ge f(B)$ for all $A,B$ with spectra in the domain $[a,b]$ of $f$.

Loewner's theorem: A function $f\colon [a,b] \to \mathbb{R}$ is matrix monotone iff it has an analytic extension to the upper and lower half-planes so that the each of these half-planes is mapped into itself.

Answer by Jon for What is your favorite "strange" function?

2010-04-23T06:56:18Z

The Osgood curve ("A Jordan Curve of Positive Area") is an injective map from [0,1] into $\mathbb{R}^2$ which traces out an image of positive area. (This differs from standard space-filling curves, which are not injective.)

Answer by Jon for Demystifying complex numbers

2010-07-01T22:06:46Z

In answer to

"Why do we need to study numbers which do not belong to the real world?"

you might simply state that quantum mechanics tells us that complex numbers arise naturally in the correct description of probability theory as it occurs in our (quantum) universe.

I think a good explanation of this is in Chapter 3 of the third volume of the Feynman lectures of physics, although I don't have a copy handy to check. (In particular, similar to probability theory with real numbers, the complex amplitude of one of two independent events A or B occuring is just the sum of the amplitude of A and the amplitude of B. Furthermore, the complex amplitude of A followed by B is just the product of the amplitudes. After all intermediate calculations one just takes the magnitude of the complex number squared to get the usual (real number) probability.)

Answer by Jon for What is your favorite "strange" function?

2010-04-23T06:09:30Z

It is pretty obvious after you've seen it, but I like the crinkled curve from Halmos's Hilbert Space Problem book:

Let $f:\mathbb{R}\rightarrow(0,\infty)$ be an $L^2$ function, and define $t\mapsto g_t:\mathbb{R}\rightarrow L^2(\mathbb{R})$ by $$g_t(x)=\chi_{(-\infty,t)}(x) \times f(x).$$

Then $g_t$ has the property that for all $t_1 < t_2 < t_3$ the secants $g_{t_2}-g_{t_1}$ and $g_{t_3}-g_{t_2}$ are mutually orthogonal. (The curve turns a corner at every point.)

Answer by Jon for A non-commutative Radon-Nikodym derivative.

2010-05-01T04:12:33Z

Does this paper have any relevance?

http://arxiv.org/PS_cache/math-ph/pdf/0303/0303056v3.pdf

Answer by Jon for A question on a trace inequality

2010-04-11T04:28:03Z

Fedja's answer crossed this [deleted] post, which extended the earlier version of his argument.

Answer by Jon for A question on a trace inequality

2010-04-10T23:04:59Z

For what it is worth, a weaker conjecture is proved below.

Applying the formula for the derivative of the inverse $$d(M^{-1}) = -M^{-1}\ dM\ M^{-1},$$ to compute the t=0 derivative of the LHS of $$Tr(A^2+A(t^{1/2}B)^2A)^{-1}-Tr(A^2+(t^{1/2}B)A^2(t^{1/2}B))^{-1} \ge 0$$ gives $$Tr(A^{-2}BA^2BA^{-2})\ge Tr(A^{-1}B^2A^{-1})=Tr(BA^{-2}B).$$ Replacing $A^{-2}$ by $P$ gives the weaker conjecture that $$Tr(PBP^{-1}BP)\ge Tr(BPB)$$ for positive B and P.

PROOF OF WEAKER CONJECTURE: By the spectral theorem, we may take P=Diag($p_1,p_2,...$). Then $$Tr(BPB)=\Sigma p_j |B_{ij}|^2=\Sigma |B_{ij}|^2 (p_i+p_j)/2 $$ and $$Tr(PBP^{-1}BP)=\Sigma |B_{ij}|^2 p_i^2 p_j^{-1}=\Sigma |B_{ij}|^2 (p_i^2 p_j^{-1}+p_i^{-1}p_j^2)/2.$$ It remains to show that $$p_i^2 p_j^{-1}+p_i^{-1}p_j^2\ge p_i+p_j$$ for positive $p_{i,j}$. By homogeneity we may take $p_i=1$. Multiplying through by $p_j$, the inequality now follows from the identity $$1+p^3-p-p^2=(p-1)^2(1+p)\ge 0.$$ $\square$

Answer by Jon for What is the difference between matrix theory and linear algebra?

2010-04-01T21:08:12Z

Although some years ago I would have agreed with the above comments about the relationship between Linear Algebra and Matrix Theory, I DO NOT agree any more!

See, for example Bhatia's "Matrix Analysis" GTM book. For example, doubly-(sub)stochastic matrices arise naturally in the classification of unitarily-invariant norms. They also naturally appear in the study of quantum entanglement, which really has nothing to do with a basis. (In both instances, all sorts of NONarbitrary bases come into play, mainly after the spectral theorem gets applied.)

Doubly-stochastic matrices turn out to be useful to give concise proofs of basis-independent inequalities, such as the non-commutative Holder inequality:

tr |AB| $\le$ $||A||_p$ $||B||_q$

with 1/p+1/q=1, $|A|=(A^*A)^{1/2}$, and $||A||_p = (tr |A|^p)^{1/p}$

Answer by Jon for Oneupsmanship and Publishing Etiquette

2010-02-20T12:37:35Z

This suggestion may only work in certain situations, but is it possible to split your paper into two? One could be for your new results in the most elegant form, and the other could be primarily for comparing your results to the previous work. You can sent the former to a good journal and throw the latter in a lesser journal or just post on the arXiv.

Answer by Jon for When to split/merge papers?

2010-01-11T00:46:40Z

If it's not done already I'd say split it. It's hard for someone else to get motivated to read a long paper, including referees for conferences. Also I have a hard time forcing myself to carefully wordsmith anything longer than 15 pages, and I so get pretty inefficient.

Comment by Jon

2010-07-12T16:39:22Z

In quantum information theory it is common for other people to publicly and immediately comment on arXiv posts using the website scirate.com. The arXiv links back to the entry. Other fields don't seem to use this website, though.

Comment by Jon

2010-07-02T16:19:37Z

Yes, I think it was there. (The "strange theory of light and matter" book.)

Comment by Jon

2010-06-03T22:07:40Z

Unfortunately, most physicists don't know much analysis. (This statement is left ambiguous, so that everyone will agree.)

Comment by Jon

2010-05-10T05:49:11Z

Can you use this program to disable the "caps lock" key? I find it is much more trouble than it is worth.

Comment by Jon

2010-04-23T06:18:52Z

I heard that after Schwartz got the Fields medal someone quipped "Now they're giving the Fields medal for integration by parts."

Comment by Jon

2010-04-15T05:56:49Z

I see the "link|edit|flag" links, but there is no delete.

Comment by Jon

2010-04-11T03:21:17Z

minwalin: I clarified the weaker conjecture, it is as stated just above "PROOF."

Comment by Jon

2010-04-11T03:20:12Z

fedja: Sorry, there was a gap in the part where I reduced consideration to small B. I took it out, I don't currently know how to extend to prove the full conjecture.

Comment by Jon

2010-04-07T01:33:22Z

In the application to the Holder inequality, one uses the fact that if U is a unitary operator, then replacing the matrix elements of U by the squares of their absolute values yields a doubly-stochastic matrix.

Comment by Jon

2010-04-04T20:18:22Z

Yemon, do you think the Journal of Linear Algebra and its applcations should be renamed?

Comment by Jon

2010-04-04T18:06:42Z

It is ironic that a textbook on analysis would make such an outrageous claim on the trivially of another field: the analytic parts of linear algebra are truly deep and quite actively researched. See, for example, Loewner's classification of matrix-monotone functions, or most any paper in quantum Shannon theory. Additionally, the entire field of quantum information theory (QIT) is essentially the study of unitary and self-adjoint operators on tensor products of Hilbert spaces, and a large majority the interesting questions in QIT retain 99% of their interest in the finite-dimensional case.