Here are a couple of neat papers that I came across in the last week.  (Planning to write something about multiferroics as well, once I have a bit of time.) The idea of directly extracting useful energy from the rotation of the earth sounds like something out of an H. G. Wells novel.  At a rough estimate (and it's impressive to me that AI tools are now able to provide a convincing step-by-step calculation of this; I tried w/ gemini.google.com) the rotational kinetic energy of the earth is about \(2.6 \times 10^{29}\) J.  The tricky bit is, how do you get at it?  You might imagine constructing some kind of big space-based pick-up coil and getting some inductive voltage generation as the earth rotates its magnetic field past the coil.  Intuitively, though, it seems like while sitting on the (rotating) earth, you should in some sense be comoving with respect to the local magnetic field, so it shouldn't be possible to do anything clever that way.  It turns out, though, that Lorentz forces still apply when moving a wire through the axially symmetric parts of the earth's field.  This has some conceptual contact with Faraday's dc electric generator.   With the right choice of geometry and materials, it is possible to use such an approach to extract some (tiny at the moment) power.  For the theory proposal, see here.  For an experimental demonstration, using thermoelectric effects as a way to measure this (and confirm that the orientation of the cylindrical shell has the expected effect), see here.  I need to read this more closely to decide if I really understand the nuances of how it works. On a completely different note, this paper came out on Friday.  (Full disclosure:  The PI is my former postdoc and the second author was one of my students.)  It's an impressive technical achievement.  We are used to the fact that usually macroscopic objects don't show signatures of quantum interference.  Inelastic interactions of the object with its environment effectively suppress quantum interference effects on some time scale (and therefore some distance scale).  Small molecules are expected to still show electronic quantum effects at room temperature, since they are tiny and their electronic levels are widely spaced, and here is a review of what this could do in electronic measurements.  Quantum interference effects should also be possible in molecular vibrations at room temperature, and they could manifest themselves through the vibrational thermal conduction through single molecules, as considered theoretically here.  This experimental paper does a bridge measurement to compare the thermal transport between a single-molecule-containing junction between a tip and a surface, and an empty (farther spaced) twin tip-surface geometry.  They argue that they see differences between two kinds of molecules that originate from such quantum interference effects. As for more global issues about the US research climate, there will be more announcements soon about reductions in force and the forthcoming presidential budget request.  (Here is an online petition regarding the plan to shutter the NIST atomic spectroscopy group.)  Please pay attention to these issues, and if you're a US citizen, I urge you to contact your legislators and make your voice heard.

This will be a really quick one! Over the last two weeks I’ve been finishing up a big project to make DOIs for all the papers published in TAC, and my code takes a while to run. So while testing I would hit “go” and have like 10 minutes to kill… which means it’s time to start answering questions on mse again! I haven’t been very active recently because I’ve been spending a lot of time on research and music, but it’s been nice to get back into it. I’m especially proud of a few recent answers, so I think I might quickly turn them into blog posts like I did in the old days! In this post, we’ll try to understand the Capping Algorithm which turns a graph embedded in a surface into a particularly nice embedding where the graph cuts the surface into disks. I drew some pretty pictures to explain what’s going on, and I’m really pleased with how they turned out! So, to start, what’s this “capping algorithm” all about? Say you have a (finite) graph $G$ and you want to know what surfaces it embeds into. For instance planar graphs are those which embed in $\mathbb{R}^2$ (equivalently $S^2$), and owners of this novelty mug know that even the famously nonplanar $K_{3,3}$ embeds in a torus1: Obviously every graph embeds into some high genus surface – just add an extra handle for every edge of the graph, and the edges can’t possibly cross each other! Also once you can embed in some surface you can obviously embed in higher genus surfaces by just adding handles you don’t use. This leads to two obvious “extremal” questions: What is the smallest genus surface which $G$ embeds into? What is the largest genus surface which $G$ embeds into where all the handles are necessary? Note we can check if a handle is “necessary” or not by cutting our surface along the edges of our graph. If the handle doesn’t get cut apart then our graph $G$ must not have used it! This leads to the precise definition: Defn: A $2$-Cell Embedding of $G$ in a surface $S$ is an embedding so that all the conected components of $S \setminus G$ are 2-cells (read: homeomorphic to disks). Then the “largest genus surface where all the handles are necessary” amounts to looking for the largest genus surface where $G$ admits a 2-cell embedding! But in fact, we can restrict attention to 2-cell embeddings in the smallest genus case too, since if we randomly embed $G$ into a surface, there’s an algorithm which only ever decreases the genus and outputs a 2-cell embedding! So if $S$ is the minimal genus surface that $G$ embeds in, we can run this algorithm to get a 2-cell embedding of $G$ in $S$. And what is that algorithm? It’s called Capping, see for instance Minimal Embeddings and the Genus of a Graph by J.W.T. Youngs. The idea is to cut your surface along $G$, look for anything that isn’t a disk, and “cap it off” to make it a disk. Then you repeat until everything in a disk, and you stop. The other day somebody on mse asked about this algorithm, and I had a lot of fun drawing some pictures to show what’s going on2! This post basically exists because I was really proud of how these drawings turned out, and wanted to share them somewhere more permanent, haha. Anyways, on with the show! We’ll start with an embedding of a graph 𝐺 (shown in purple) in a genus 2 surface: we’ll cut it into pieces along $G$, and choose one of our non-disk pieces (call it $S$) to futz with: Now we choose3 a big submanifold $T \subseteq S$ which leaves behind cylinders when we remove it. Pay attention to the boundary components of $T$, called $J_1$ and $J_2$ below, since that’s where we’ll attach a disk to “cap off” where $T$ used to be We glue all our pieces back together, but remove the interior of $T$ and then, as promised “cap off” the boundary components $J_1$ and $J_2$ with disks. Note that the genus decreased when we did this! It used to be genus 2, and now we’re genus 1! Note also that $G$ still embeds into our new surface: Let’s squish it around to a homeomorphic picture, then do the same process a second time! But faster now that we know what’s going on: At this point, we can try to do it again, but we’ll find that removing $G$ cuts our surface into disks: This tells us the algorithm is done, since we’ve successfully produced a 2-cell embedding of $G$ ^_^. Wow that was a really quick one today! Start to finish in under an hour, but it makes sense since I’d already drawn the pictures and spent some time doing research for my answer the other day. Maybe I’ll go play flute for a bit. Thanks for hanging out, all! Stay safe, and see you soon ^_^ This photo of a solution was taken from games4life.co.uk ↩ You know it’s funny, even over the course of drawing just these pictures the other day I feel like I improved a lot… I have half a mind to redraw all these pictures even better, but that would defeat the point of a quick post, so I’ll stay strong! ↩ It’s possible there’s a unique “best” choice of $T$ and I’m just inexperienced with this algorithm. I hadn’t heard of it until I wrote this answer, so there’s a lot of details that I’m fuzzy on. If you happen to know a lot about this stuff, definitely let me know more! ↩

Randomness is essential to some research, but it’s always been prohibitively complicated. Now, we can use “pseudorandomness” instead. The post The High Cost of Quantum Randomness Is Dropping first appeared on Quanta Magazine

The fashion retailer, H&M, has announced that they will start using AI generated digital twins of models in some of their advertising. This has sparked another round of discussion about the use of AI to replace artists of various kinds. Regarding the H&M announcement specifically, they said they will use digital twins of models that […] The post H&M Will Use Digital Twins first appeared on NeuroLogica Blog.

London’s mid-rise architecture