I was recently looking for a paper on the font of all scientific knowledge these days, Google Scholar, and couldn’t find the original paper presenting the concept of manufactured solutions. I found the paper, but also found this image of most of the attendees that I thought should be offered up for the historical record.
The original paper is
Lingus, C. “Analytical test case’s for neutron and radiation transport codes.” Second conference on transport theory. 1971. PDF
This commercial showing an exciting play where Christian McCaffery scores a touchdown has been out for some time now. In it McCaffery discusses how it was improbable (14.2% of the time or about 1/7) that a touchdown would be scored with four defenders at a given distance away (3.0, 3.3, 7.5, and 8.3 yards away). Due to the magic of “Next Gen Stats”, this play becomes a statistical anomaly. Let’s analyze this claim and see if it tells us the outcome of this play is remarkable, from a statistical point of view.
We can stipulate that for a play starting at about the 5 yard line, when there are four unblocked defenders within 8.3 yards of the runner, it is unlikely that a touchdown will be scored. I would conjecture, however, that a large majority of the time those defenders would be in front of the runner. Indeed, it might be that only 15% of the time that none of the four closest defenders are in front of the runner. In that case, the touchdown probability for the way this play initially unfolded could be 94.7% (that is, 14.2/15=0.9466…), if all of the touchdowns occurred when no defenders were in front of the runner.
An experienced football fan would look at the stills above and would bet it is more likely than not for the runner to score (even if it was clear there was a defensive back right at the goal line). This is because there is still a lot of room to the sideline and the bad angle the defender is taking. It seems like the worst possible outcome is that a tackle is made at the 2 or 3 yard line.
The fact that there are four players at the given distances does not tell us very much about the play. Indeed, it seems like two of those defenders aren’t really even running at this point. What is important, and not quantified here (possibly because it is much harder), is that none of the defenders are in the path of the runner and that the nearest defender is on the goal line.
This is an example of using the data at hand to make a decision that doesn’t really have any insight. Looking at the available data, we can infer that yes, if there are that many unblocked players, that close to a runner, it is unlikely he will score. Of course, most of those plays will be cases where the runner is swarmed in the backfield by defenders that surround him from the start of the play. However, once we have a bit more information, like that none of those defenders stand between the runner and the end zone, it becomes much more likely to score a touchdown. The eye test would tell you that the probability is much better than the roughly 1 in 7 chance this play has a score a touchdown.
The Gram-Schmidt process takes a set of N vectors or functions and an inner product. It then builds a set of N orthogonal vectors/functions from the original ones. It is a process that can be easily automated using symbolic calculations.
This is done using Mathematica in this notebook. It begins with defining the inner product as a Mathematica function:
ip[u_, v_] := Integrate[u v, {x, 0, X}]
The special issue of Nuclear Science and Engineering featuring papers from the 2019 Mathematics and Computation Meeting held in Portland is now available. I was the guest editor for this issue.
You can read my introduction to the issue, and find all of the papers at this link: https://www.tandfonline.com/doi/full/10.1080/00295639.2020.1802915
Thanks to all of the contributors to the issue.
Reed’s problem [1] is a common test problem for transport codes. It is comprised of heterogeneous materials with strong absorber, vacuum, and scattering regions. These regions are valuable to testing different aspects of numerical discretizations. For instance, the vacuum region is troublesome for second-order forms of the transport equation.
Jim Warsa published solutions obtained from eigenfunctions expansions for Reed’s problem in 2002 [2]. The are in the form of exponentials and hyperbolic trig functions and have several terms that must be added together carefully to avoid numerical instability.
I have written a Mathematica notebook to evaluate the solutions that you can download here. Or, if you just want the solutions, you can download a CSV of them here. The solution is also plotted here.
[1] Reed, William H. “New difference schemes for the neutron transport equation.” Nuclear Science and Engineering 46.2 (1971): 309-314.
[2] Warsa, J. (2002). Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs. Annals of Nuclear Energy, 29(7), 851–874.
When material is moving, the physics of radiative transfer changes to account for Doppler shifts in the frequency, and the motion of the material causes the opacities to need to be evaluated at different energies. This complication has been studied for a long time, but accurate numerical methods (and their implementation is a bit of a challenge).
One common locus of argument is how important these corrections are when the material is moving slow relative to the speed of light. Many of the correction terms appear to be small when the material speed is much smaller than the speed of light.
In a recent extended summary for the 2021 M&C Meeting, Nick Gentile and I developed some benchmark solutions for the spectrum of light that would be measured in front of a moving slab. We show that, even at a speed of 5% the speed of light, the spectrum one measures is significantly different when one either ignores the material motion corrections or leaves of some seemingly unimportant terms. Moreover, the errors that result from ignoring these terms get worse as the spectrum resolution is increased. See the full document here.
The Mathematica notebooks used to produce the benchmarks can be found on Github.
I just posted the presentation I gave to the Notre Dame Radiation Laboratory (the radiation chemistry people at ND). The talk was this morning and it went well, despite my rather inchoate understanding of low energy electron physics in liquids.
The talk can be found here.
Recently, I realized, as one does, that I did know as much about the Thirty Years War as I thought I should. I sought to rectify this by reading about the conflict (or as it turns out, series of conflicts) by doing some reading. Several different books were recommended, but when I found out that a classic fictional account was considered to be a realistic firsthand account of the war, I decided to read it. The Adventures of Simplicius Simplicissimus by Hans Jakob Christoffel von Grimmelshausen is a mouthful of a title/author combination, but a relatively short read. In the introductory notes in the Penguin edition (translated by J.A. Underwood), it is noted that this is a well known book in the German world. Nevertheless, my German colleagues in academia had not heard of the book (or for that matter much about the Thirty Years War as, ahem, more recent wars dominate the history the curriculum in school).
The book was a good read for me for the mundane reason that the chapters are short (typically less than 4 pages), so it was a great pre-bedtime read because I could feel like I accomplished something by reading for a few minutes. In short my episodic reading matched the episodic telling of the narrative.
The story follows the Simplicius. As a boy he runs away when a the Swedish army raids his family’s modest homestead in the Spessart, a low mountain range in Hesse and Bavaria. He lives in the woods for a period of time, is a fixture in the court of a Governor, finds himself on both sides of the war, becomes famous as the Hunstman of Soest, finds riches and loses them repeatedly, and struggles with demons both inner and outer. He also discovers who his true father is in a scene the reminded me of Bill Pullman’s character in Spaceballs. My summary cannot do the book justice, but I do think that it was worth the read.
I will admit that I did not learn much about the ins and outs and what-have-yous of the Thirty Years War. One of the most remarkable things is that it is not clear at many points what side Simplicius finds himself on. I think this is one of the biggest takeaways I had from the book. The armies that were fighting were, in a large part, separate from the people of the land they were fighting in. They raided from peasants and villagers what they could, and the armies, from either “side”, were typically equal trouble for the citizenry. Indeed, in one episode Simplicius goes to Switzerland, a land not involved in the war, and he remarks that it feels entirely different to be in a place where raids, conscription, and attack are not the order of the day.
Another takeaway I had was that the violence in the book was not as lurid as I expected. The Thirty Years War has a reputation for its bloody savagery. For sure there is violence and death to beat the band in the this book, but it is really a sidelight to the story. Perhaps that should be the most shocking thing: the violence was not the remarkable aspect of the war. The violence was almost prosaic, even a pervasive in the ambient environment. That might be the most shocking finding for modern readers.
Here is a crossword puzzle that made today. I woke up and I thought I had some nice clues for a crossword puzzle, so I decided to try my hand at it.
Artistry?
If you want to solve this interactively, click here.
I made this Google map so that any where I travel, I can see if there is a Caravaggio painting close by. Enjoy.