User jon - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:41:08Z http://mathoverflow.net/feeds/user/3139 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18100/theorems-with-unexpected-conclusions/18156#18156 Answer by Jon for Theorems with unexpected conclusions Jon 2010-03-14T09:11:56Z 2010-12-13T20:50:16Z <p>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 <em>matrix monotone</em> if $A\ge B$ implies $f(A)\ge f(B)$ for all $A,B$ with spectra in the domain $[a,b]$ of $f$.</p> <p><strong>Loewner's theorem</strong>: 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.</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22308#22308 Answer by Jon for What is your favorite "strange" function? Jon 2010-04-23T06:56:18Z 2010-11-13T14:51:14Z <p>The Osgood curve ("<a href="http://www.jstor.org/stable/1986455" rel="nofollow">A Jordan Curve of Positive Area</a>") 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.)</p> http://mathoverflow.net/questions/30156/demystifying-complex-numbers/30240#30240 Answer by Jon for Demystifying complex numbers Jon 2010-07-01T22:06:46Z 2010-07-01T22:06:46Z <p>In answer to</p> <p>"Why do we need to study numbers which do not belong to the real world?"</p> <p>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. </p> <p>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.)</p> http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22305#22305 Answer by Jon for What is your favorite "strange" function? Jon 2010-04-23T06:09:30Z 2010-07-01T18:31:14Z <p>It is pretty obvious after you've seen it, but I like the crinkled curve from Halmos's Hilbert Space Problem book:</p> <p>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 $$g_t(x)=\chi_{(-\infty,t)}(x) \times f(x).$$</p> <p>Then $g_t$ has the property that for all $t_1 &lt; t_2 &lt; t_3$ 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.)</p> http://mathoverflow.net/questions/23154/a-non-commutative-radon-nikodym-derivative/23158#23158 Answer by Jon for A non-commutative Radon-Nikodym derivative. Jon 2010-05-01T04:12:33Z 2010-05-01T04:12:33Z <p>Does this paper have any relevance? </p> <p><a href="http://arxiv.org/PS_cache/math-ph/pdf/0303/0303056v3.pdf" rel="nofollow">http://arxiv.org/PS_cache/math-ph/pdf/0303/0303056v3.pdf</a></p> http://mathoverflow.net/questions/20924/a-question-on-a-trace-inequality/20981#20981 Answer by Jon for A question on a trace inequality Jon 2010-04-11T04:28:03Z 2010-04-11T04:40:16Z <p>Fedja's answer crossed this [deleted] post, which extended the earlier version of his argument.</p> http://mathoverflow.net/questions/20924/a-question-on-a-trace-inequality/20964#20964 Answer by Jon for A question on a trace inequality Jon 2010-04-10T23:04:59Z 2010-04-11T03:34:34Z <p>For what it is worth, a weaker conjecture is proved below.</p> <p>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.</p> <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 $$1+p^3-p-p^2=(p-1)^2(1+p)\ge 0.$$ $\square$</p> http://mathoverflow.net/questions/11669/what-is-the-difference-between-matrix-theory-and-linear-algebra/20108#20108 Answer by Jon for What is the difference between matrix theory and linear algebra? Jon 2010-04-01T21:08:12Z 2010-04-04T18:10:56Z <p>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! </p> <p>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.)</p> <p>Doubly-stochastic matrices turn out to be useful to give concise proofs of basis-independent inequalities, such as the non-commutative Holder inequality:</p> <p>tr |AB| $\le$ $||A||_p$ $||B||_q$</p> <p>with 1/p+1/q=1, $|A|=(A^*A)^{1/2}$, and $||A||_p = (tr |A|^p)^{1/p}$</p> http://mathoverflow.net/questions/15836/oneupsmanship-and-publishing-etiquette/15890#15890 Answer by Jon for Oneupsmanship and Publishing Etiquette Jon 2010-02-20T12:37:35Z 2010-02-20T12:37:35Z <p>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.</p> http://mathoverflow.net/questions/11366/when-to-split-merge-papers/11388#11388 Answer by Jon for When to split/merge papers? Jon 2010-01-11T00:46:40Z 2010-01-11T00:46:40Z <p>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.</p> http://mathoverflow.net/questions/31337/how-do-i-fix-someones-published-error/31428#31428 Comment by Jon Jon 2010-07-12T16:39:22Z 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. http://mathoverflow.net/questions/30156/demystifying-complex-numbers/30240#30240 Comment by Jon Jon 2010-07-02T16:19:37Z 2010-07-02T16:19:37Z Yes, I think it was there. (The &quot;strange theory of light and matter&quot; book.) http://mathoverflow.net/questions/26919/real-analysis-has-no-applications/26920#26920 Comment by Jon Jon 2010-06-03T22:07:40Z 2010-06-03T22:07:40Z Unfortunately, most physicists don't know much analysis. (This statement is left ambiguous, so that everyone will agree.) http://mathoverflow.net/questions/15595/math-keyboard-does-it-exist/15606#15606 Comment by Jon Jon 2010-05-10T05:49:11Z 2010-05-10T05:49:11Z Can you use this program to disable the &quot;caps lock&quot; key? I find it is much more trouble than it is worth. http://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function/22194#22194 Comment by Jon Jon 2010-04-23T06:18:52Z 2010-04-23T06:18:52Z I heard that after Schwartz got the Fields medal someone quipped &quot;Now they're giving the Fields medal for integration by parts.&quot; http://mathoverflow.net/questions/20924/a-question-on-a-trace-inequality/20981#20981 Comment by Jon Jon 2010-04-15T05:56:49Z 2010-04-15T05:56:49Z I see the &quot;link|edit|flag&quot; links, but there is no delete. http://mathoverflow.net/questions/20924/a-question-on-a-trace-inequality/20964#20964 Comment by Jon Jon 2010-04-11T03:21:17Z 2010-04-11T03:21:17Z minwalin: I clarified the weaker conjecture, it is as stated just above &quot;PROOF.&quot; http://mathoverflow.net/questions/20924/a-question-on-a-trace-inequality/20964#20964 Comment by Jon Jon 2010-04-11T03:20:12Z 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. http://mathoverflow.net/questions/11669/what-is-the-difference-between-matrix-theory-and-linear-algebra/20108#20108 Comment by Jon Jon 2010-04-07T01:33:22Z 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. http://mathoverflow.net/questions/11669/what-is-the-difference-between-matrix-theory-and-linear-algebra/19884#19884 Comment by Jon Jon 2010-04-04T20:18:22Z 2010-04-04T20:18:22Z Yemon, do you think the Journal of Linear Algebra and its applcations should be renamed? http://mathoverflow.net/questions/11669/what-is-the-difference-between-matrix-theory-and-linear-algebra/19884#19884 Comment by Jon Jon 2010-04-04T18:06:42Z 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.