Green's Functions and Correlations

I had the idea of writing this post a couple of weeks ago, but I didn't feel like I had enough stuff to write here at that time. Now I do, so here goes. (Also, here's hoping that inputting LaTeX into this post works once more.)

When I took 18.03 — Differential Equations in 2010 fall, one of the topics covered was linear time-invariant systems. The general system of interest was $Lu(t) = f(t)$ where $L$ is a linear time-invariant operator. The technique of course is to find a weight function $w(t)$ where $Lw(t) = \delta(t)$, and once that is done, the solution is $u(t) = \int_{-\infty}^{\infty} f(t') w(t - t') dt'$ which is a convolution of the input $f$ with the weight $w$. The professor mentioned that it is essentially akin to inverting the operator $L$, but while I could see the general utility in this method, I never quite understood why it might be considered inversion on any deeper level.

Last semester, I took 8.07 — Electromagnetism II, and there we discussed Green's functions a little more in the context of electromagnetism & electrodynamics. In a static situation, the Green's function comes up in solving the Poisson equation $\nabla^2 \phi = -\rho$. In this case, $\nabla^2 G(\vec{x}, \vec{x}') = -\delta(\vec{x} - \vec{x}')$ is solved by the familiar potential of a unit point charge $G(\vec{x}, \vec{x}') = \frac{1}{4\pi |\vec{x} - \vec{x}'|}$. I started to see a little more clearly why this worked, because if a general charge distribution was some superposition of point charges, then a general potential distribution should be the same superposition of point charge potentials. However, it still wasn't entirely clear to me how this was "inversion" per se. Follow the jump to see what changed.


Back Online After TOS Violation due to Malware Issue

For the last day or two, this blog was taken down again, this time due to a terms of service violation. I could not for the life of me think of what I might have written or done here to account for that. Today, I looked into it, and found that although this blog had been restored after the malware attack, the malware itself had not been removed. After removing a lot of third-party code, I found out that the issue lay in my Archives page, where the third-party code allowing the "Archives" widget to be displayed as a separate page had been infected with malware. I'm starting to be a little more wary of how Google handles things now, and while I do intend to stay with Blogger for the foreseeable future, I'm not counting a move entirely out.


Eighth Semester at College

I'm at the home stretch! This is my eighth and last semester as an undergraduate at MIT. Classes start tomorrow. I'll be taking 8.334 — Statistical Mechanics II (which is really statistical field theory), 8.962 — General Relativity, 14.15 — Networks, and 8.THU — Undergraduate Physics Thesis. The cool thing is that 8.334 — Statistical Mechanics II and 14.15 — Networks will have a bit of overlap in some places, as both discuss graph theory, collective phenomena, and phase transitions to varying degrees. More importantly, 8.THU — Undergraduate Physics Thesis is basically going to be my UROP, formalized into credits contingent on me producing a thesis at the end of it. That's also how I can start a new UROP project on nanoparticle absorption and scattering of infrared radiation. Even though I'm only taking 3 classes besides my UROP and [as far as I can tell] none of them have final exams, the semester will still keep me quite busy, but this will be the last semester where I can take more random classes that I want to take, as graduate school will likely only let me take classes related to my research interests. Here's hoping that my last semester of my undergraduate career turns out to be the best one yet, and good luck to everyone else for the new semester!


Featured Comments: Week of 2014 January 26

There were two posts this past week that got a comment each, so I'll repost both of those.

Revisited: Linux Mint 16 "Petra" KDE + Xfce

An anonymous reader had this vote of support: "welcome back prashanth ji.............." [following the takedown of this blog for a large portion of the month of 2014 January].

Review: Pinguy OS 13.10 Beta 3

Another anonymous commenter suggested, "Elementary OS, with all the stability you need. :)"

Thanks to both of those people for leaving those comments. This coming week, I will have a post about the semester ahead. After that, the semester will start and will certainly become busy, so I likely will drop the frequency of posts after that, as I have done in past semesters. Anyway, if you like what I write, please continue subscribing and commenting!


Reflection: 2014 IAP

This IAP was quite a bit more hectic near the end of it. I was starting to wrap up my current UROP project on photonic crystal enhancement of spontaneous emission and start learning about a new project on nanoparticle absorption & scattering of infrared radiation. Also, especially in the last week, I was doing a lot for making a video for the MIT-K12 project. Finally, there was organization to be done for the SPS Lightning Lectures on the last day of IAP. Overall, it was quite productive. At the moment, I am still awaiting graduate admission results (except for one positive one so far). And I await and anticipate one last semester of classes and research as an undergraduate at MIT!