text: Suppose $X$ and $Y$ are schemes of finite type over a field, and let $f: X\rightarrow Y$ be a morphism. Let $\Gamma_f$ be the closed subscheme of $X\times Y$, then the first example of the Fourier-Mukai transform says that $f_*()$ = $p_Y{_*}$ $(p_X^*()$ $\otimes_{\mathcal{O}{X\times Y}}$ $\mathcal{O}{\Gamma_f})$; $f_*()=p_Y{_*}(p_X^*()\bigotimes_{\mathcal{O}_{X\times Y}}\mathcal{O}_{\Gamma_f}),$$ similarly there is an expression for f^*. Has anyone checked this before? I am having some difficulties (at least on some commutative algebra) in verifying them. Does anyone know why those formulas are true? 3 added 251 characters in body Suppose X and Y are schemes of finite type over a field, and let f: X\rightarrow Y be a morphism. Let \Gamma_f be the closed subscheme of X\times Y, then the first example of the Fourier-Mukai transform says that f_*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}_{X\times f_*() = p_Y{_*} (p_X^*() \otimes_{\mathcal{O}{X\times Y}} \mathcal{O}{\Gamma_f}); similarly there is an expression for f^*. Has anyone checked this before? I am having some difficulties (at least on some commutative algebra) in verifying them. Does anyone know why those formulas are true? 2 deleted 340 characters in body; added 58 characters in body Suppose X and Y are schemes of finite type over a field, and let f: X\rightarrow Y be a morphism. Let \Gamma_f be the closed subscheme of X\times Y, then the first example of the Fourier-Mukai transform says that f_*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}{X\times f_*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}_{X\times Y}}$$f*()=p_Y{_*}(p_X^*()\otimes_{\mathcal{O}{X\times Y}}\mathcal{O}{\Gamma_f})$, similarly there is an expression for $f^*$. Has anyone checked this before? I am having some difficulties (on the commutative algebra at least) in verifying them. Does anyone know why those formulas are true?


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# All of gwern's Comments + Replies

What are the gears of gluten sensitivity?

If you are concerned about gluten sensitivity, why not directly test for the antibodies or celiac-related genetic variants (eg 23andMe)? You can do both at home via mail for like $200 total. That information sounds much more dispositive than reducing gluten and maybe observing some effect, and given the long-term harms of problems like celiac, this is not a problem one wants to cheap out on solving. 2G Gordon Worley III14dI'll look into it. I was unaware until this comment that such testing existed. My 23andMe results show I don't have the markers so celiac disease, and SNPedia findings don't show anything likely to be related to gluten. I'll see if it's possible to find testing that might indicate non-allergic food sensitivities that isn't also bogus. 1bvbvbvbvbvbvbvbvbvbvbv14dMy several cents : It's not just about celiac disease. There are several other disorders that seem to be worsened by gluten, the first one that comes to my mind is endometriosis. The way I see it, it seems that some people are susceptible to pro inflammatory effects of gluten, whereas most are not. So if you already have an inflammatory disorder or autoimmune condition AND have this inflammatory sensitivity then gluten will worsen it. I don't know if this susceptibility to gluten's inflammatory effect is genetic, or genetically identified / identifiable. Whereas you can indeed identify celiac related antibodies using various tests. 6ChristianKl14dMany people who speak about gluten sensitivity (and other allergic symptoms) think that often even when the antibody levels don't reach levels that are detectable by antibody tests there can be still gluten sensitivity and making the experiment of reducing gluten to test whether there's an effect is more sensitive then the antibody tests. While running tests is you can rationally deal with the results can be useful, actually doing the experiment of whether reducing gluten will have a noticeable effect is worthwhile. If you do run antigen tests, do blood tests. Skin tests don't really provide useful medical data use but are easy to bill to insurance companies. When it comes to running the experiment sometimes heartrate or HRV are sensitive enough to detect allergy reactions that you wouldn't notice otherwise. The real high cost action would be "Eat only rice for a week, see if symptoms disappear, then add additional foods one-by-one to see when symptoms reappear". "Decision Transformer" (Tool AIs are secret Agent AIs) Rewards need not be written in natural language as crudely as "REWARD: +10 UTILONS". Something to think about as you continue to write text online. And what of the dead? I own that I thought of myself, at times, almost as dead. Are they not locked below ground in chambers smaller than mine was, in their millions of millions? There is no category of human activity in which the dead do not outnumber the living many times over. Most beautiful children are dead. Most soldiers, most cowards. The fairest women and the most learned men – all are dead. Their bodi ... (read more) Curated conversations with brilliant rationalists IMO, that's shockingly cheap, and there's little reason to not do transcripts for any podcast which has a listening audience larger than "your gf and your dog" and pretensions to being more than tissue-level entertainment to be discarded after use. If a podcast is worth taking hours to do and expecting hundreds/thousands of listeners to sit through spending man-hours apiece and trying to advertise or spread it in any way, then it's almost certainly also then worth$100 to transcribe it. A transcript buys you search-engine visibility (as well as easy search... (read more)

The EMH is False - Specific Strong Evidence

There are currently high return trades (5% a month at least, possibly more) with extremely low risk (you can lose 1-2% max, probably less depending on execution).

Worth noting that a new Metaculus market estimates ~50% chance of Polymarket being a counterparty risk in some sense 2021-2022.

Article on IQ: The Inappropriately Excluded

The sample consisted of mid-level leaders from multinational private-sector companies.

This sort of pre-filtered sample suffers from issues like Berkson's paradox. For example, for those managers who have IQ>120, why are they underperforming? Perhaps for lack of leadership qualities, which they make up for on intelligence. On the flip side, for managers who have unimpressive IQs (as low as <100), why are they so successful? This is why longitudinal samples like SMPY are so much more useful when you want to talk about what high IQs are or are not go... (read more)

1Jay24dThis isn't really my field, and I see your point. The poster asked for other studies so I linked a study I'd recently seen. It's less about me endorsing the study than about trying to provide an entry point into the relevant literature.
Re: Fierce Nerds

"Fierce nerd" sounds a bit like rediscovering Eysenck's paradigm of genius: intelligence, energy, and Psychoticism (essentially, low Agreeableness).

Considering how frequent mental issues are around here, this post seems to buy entirely the wrong kinds of optionality.

EDIT: oh look what's on the main page a day later

Strongly agree. I think it was Rob Wiblin (or maybe Katja Grace) who wrote a post once about how they'd investigated the statistically-most-probable ways they could die in the next decade. And the answer (given various demographic facts) turned out to be suicide. Instead of dismissing this, they took seriously the fact that some people who haven't previously considered suicide later do so (but in a bad moment, such that following through would definitively be a mistake). So they took steps to decrease their suicide risk, the way one might take steps to dec... (read more)

Open and Welcome Thread - May 2021

The #lesswrong IRC channel has moved to Libera due to drama.

What will 2040 probably look like assuming no singularity?

The 3 babies from He Jiankui will be adults by then, definitely; one might quibble about how 'designer' they are, but most people count selection as 'designer' and GenPred claims to have at least one baby so far selected on their medical PGSes (unclear if they did any EDU/IQ PGSes in any way, but as I've always pointed out, because of the good genetic correlations of those with many diseases, any selection on complex diseases will naturally also boost those).

How to determine the value of optionality?

The value of optionality is defined by drawing out the decision tree for the scenarios with and without the option, doing backwards induction for the optimal strategy and estimating the value of each. (In financial option theory, you calculate the price of a literal option by simulating out all of the possible price trajectories and how you would respond to them, to figure out what would be a too cheap or too expensive price.) Because scenarios can be arbitrarily complex, no general answer is possible. If an option wouldn't be used at any state of the worl... (read more)

Agency in Conway’s Game of Life

OP said I can initialize a large chunk as I like (which I initialize to be empty aside from my constructors to avoid interfering with placing the pixels), and then the rest might be randomly or arbitrarily initialized, which is why I brought up the wall of still-life eaters to seal yourself off from anything that might then disrupt it. If his specific values don't give me enough space, but larger values do, then that's an answer to the general question as nothing hinges on the specific values.

4alexflint1moI was imagining that the goal configuration would be defined over the whole grid, so that it wouldn't be possible to satisfy the objective within the initial region, since that seems most analogous to constructing an AI in, say, a single room on Earth and having it eventually influence the overall arrangement of matter and energy in the Milky Way.
Agency in Conway’s Game of Life

My immediate impulse is to say that it ought to be possible to create the smiley face, and that it wouldn't be that hard for a good Life hacker to devise it.

I'd imagine it to go something like this. Starting from a Turing machine or simpler, you could program it to place arbitrary 'pixels': either by finding a glider-like construct which terminates at specific distances into a still, so the constructor can crawl along an x/y axis, shooting off the terminating-glider to create stable pixels in a pre-programmed pattern. (If that doesn't exist, then one could... (read more)

7Oscar_Cunningham1moThis sounds like you're treating the area as empty space, whereas the OP specifies that it's filled randomly outside the area where our AI starts.
Self-Predicting Markets

To update on this: Hertz stock is now worth 5-8 as it comes out of bankruptcy. I hope OP didn't short it, because he would've lost his shorts based on his belief that EMH is false and he's smarter than the markets. Challenge: know everything that the best go bot knows about go An even more pointed example: chess endgame tables. What does it mean to 'fully understand' it beyond understanding the algorithms which construct them, and is it a reasonable goal to attempt to play chess endgames as well as the tables? 4paulfchristiano1moIf you have a "lazy" version of the goal, like "have a question-answerer that can tell you anything the model knows" or "produce a locally human-legible but potentially giant object capturing everything the model knows" then chess endgame tables are a reasonably straightforward case ("position X is a win for white"). [link] If something seems unusually hard for you, see if you're missing a minor insight This reminds me of pg: If you think something's supposed to hurt, you're less likely to notice if you're doing it wrong. That about sums up my experience of graduate school. "How To Do What You Love" (Of course, there's a certain aspect of learned-helplessness here: because so many things are terrible, people often assume that something is just another broken malicious tool or workflow, when it's quite the opposite.) But really the single most important way to learn to use a search engine is this: Know people who are better at using search engines than ... (read more) gwern's Shortform 2-of-2 escrow: what is the exploding Nash equilibrium? Did it really originate with NashX? I've been looking for the history & real name of this concept for years now and have failed to refind it. Anyone? gwern's Shortform Humanities satirical traditions: I always enjoy the CS/ML/math/statistics satire in the annual SIGBOVIK and Ig Nobels; physics has Arxiv April Fools papers (like "On the Impossibility of Supersized Machines") & journals like Special Topics; and medicine has the BMJ Christmas issue, of course. What are the equivalents in the humanities, like sociology or literature? (I asked a month ago on Twitter and got zero suggestions...) gwern's Shortform Normalization-free Bayes: I was musing on Twitter about what the simplest possible still-correct computable demonstration of Bayesian inference is, that even a middle-schooler could implement & understand. My best candidate so far is ABC Bayesian inference*: simulation + rejection, along with the 'possible worlds' interpretation. Someone noted that rejection sampling is simple but needs normalization steps, which adds complexity back. I recalled that somewhere on LW many years ago someone had a comment about a Bayesian interpretation where you don't nee... (read more) 4Wei_Dai2moDoing another search, it seems I made at least one comment that is somewhat relevant, although it might not be what you're thinking of: https://www.greaterwrong.com/posts/5bd75cc58225bf06703751b2/in-memoryless-cartesian-environments-every-udt-policy-is-a-cdt-sia-policy/comment/kuY5LagQKgnuPTPYZ [https://www.greaterwrong.com/posts/5bd75cc58225bf06703751b2/in-memoryless-cartesian-environments-every-udt-policy-is-a-cdt-sia-policy/comment/kuY5LagQKgnuPTPYZ] 1eigen2moFunny that you have your great LessWrong whale as I do, and that you recall that it may be from Wei Dai as well (while him not recalling) https://www.lesswrong.com/posts/X4nYiTLGxAkR2KLAP/?commentId=nS9vvTiDLZYow2KSK Gradations of Inner Alignment Obstacles I claim that if we're clever enough, we can construct a hypothetical training regime T' which trains the NN to do nearly or exactly the same thing on T, but which injects malign behavior on some different examples. (Someone told me that this is actually an existing area of study; but, I haven't been able to find it yet.) I assume they're referring to data poisoning backdoor attacks like https://arxiv.org/abs/2010.12563 or https://arxiv.org/abs/1708.06733 or https://arxiv.org/abs/2104.09667 How can we increase the frequency of rare insights? Have you looked at the "incubation effect"? 3eigen2moA talk given by Rogen Penrose is apt here: The Problem of Modelling the Mathematical Mind [https://www.youtube.com/watch?v=gscEK5df8vE]. He tries to define how the mind of a sufficiently good mathematician may work with emphasis on parallelization of mathematical solutions. And an interesting book may be The Mathematician's Mind: The Psychology of Invention in the Mathematical Field by Jacques Hadamard. Parameter count of ML systems through time? It's not the numerical precision but the model architecture being sparse such that you only active a few experts at runtime, and only a small fraction of the model runs for each input. It may be 1.3t parameters or whatever, but then at runtime, only, I dunno, 20b parameters actually compute anything. This cheapness of forward passes/inferencing is the big selling point of MoE for training and deployment: that you don't actually ever run 1.3t parameters. But it's hard for parameters which don't run to contribute anything to the final result, whereas in GPT-... (read more) LessWrong help desk - free paper downloads and more If Reddit falls through, email me and I can order a scan for you. (Might want to delete your duplicate comments here too.) EDIT: ordered a scan Parameter count of ML systems through time? You should probably also be tracking kind of parameter. I see you have Switch and Gshard in there, but, as you can see in how they are visibly outliers, MoEs (and embeddings) use much weaker 'parameters', as it were, than dense models like GPT-3 or Turing-NLG. Plotting by FLOPS would help correct for this - perhaps we need graphs like training-FLOPS per parameter? That would also help correct for comparisons across methods, like to older architectures such as SVMs. (Unfortunately, this still obscures that the key thing about Transformers is better scaling laws than RNNs or n-grams etc, where the high FLOPS-per-parameter translates into better curves...) 1Jsevillamol2moThank you for the feedback, I think what you say makes sense. I'd be interested in seeing whether we can pin down exactly in what sense are Switch parameters "weaker". Is it because of the lower precision? Model sparsity (is Switch sparse on parameters or just sparsely activated?)? What do you think, what typology of parameters would make sense / be useful to include? March 2021 gwern.net newsletter "'Nash equilibrium strategy' is not necessarily synonymous to 'optimal play'. A Nash equilibrium can define an optimum, but only as a defensive strategy against stiff competition. More specifically: Nash equilibria are hardly ever maximally exploitive. A Nash equilibrium strategy guards against any possible competition including the fiercest, and thereby tends to fail taking advantage of sub-optimum strategies followed by competitors. Achieving maximally exploitive play generally requires deviating from the Nash strategy, and allowing for defensive leaks in one's own strategy." 2020 AI Alignment Literature Review and Charity Comparison That's interesting. I did see YC listed as a major funding source, but given Sam Altman's listed loans/donations, I assumed, because YC has little or nothing to do with Musk, that YC's interest was Altman, Paul Graham, or just YC collectively. I hadn't seen anything at all about YC being used as a cutout for Musk. So assuming the Guardian didn't screw up its understanding of the finances there completely (the media is constantly making mistakes in reporting on finances and charities in particular, but this seems pretty detailed and specific and hard to get... (read more) The best frequently don't rise to the top One of the most interesting media experiments I know of is the Yahoo Media experiments: 1. "Experimental Study of Inequality and Unpredictability in an Artificial Cultural Market", Salganik et al 2006: We investigated this paradox experimentally, by creating an artificial ‘‘music market’’ in which 14,341 participants downloaded previously unknown songs either with or without knowledge of previous participants’ choices. Increasing the strength of social influence increased both inequality and unpredictability of success. Success was also only partly determi ... (read more) 2adamzerner3moThis is fantastic. Thank you! The EMH is False - Specific Strong Evidence I knew someone was going to ask that. Yes, it's impure indexing, it's true. The reason is the returns to date on the whole-world indexes have been lower, the expense is a bit higher, and after thinking about it, I decided that I do have a small opinion about the US overperforming (mostly due to tech/AI and a general sense that people persistently underestimate the US economically) and feel pessimistic about the rest of the world. Check back in 20 years to see how that decision worked out... 1Rune3moThanks. I'm just trying to understand what people's reasons are. My portfolio is also US-weighted more than market caps would dictate. Against evolution as an analogy for how humans will create AGI gwern3mo23Ω12 As described above, I expect AGI to be a learning algorithm—for example, it should be able to read a book and then have a better understanding of the subject matter. Every learning algorithm you’ve ever heard of—ConvNets, PPO, TD learning, etc. etc.—was directly invented, understood, and programmed by humans. None of them were discovered by an automated search over a space of algorithms. Thus we get a presumption that AGI will also be directly invented, understood, and programmed by humans. For a post criticizing the use of evolution for end to end ML, t... (read more) 7Richard_Ngo3moI personally found this post valuable and thought-provoking. Sure, there's plenty that it doesn't cover, but it's already pretty long, so that seems perfectly reasonable. I particularly I dislike your criticism of it as strawmanish. Perhaps that would be fair if the analogy between RL and evolution were a standard principle in ML. Instead, it's a vague idea that is often left implicit, or else formulated in idiosyncratic ways. So posts like this one have to do double duty in both outlining and explaining the mainstream viewpoint (often a major task in its own right!) and then criticising it. This is most important precisely in the cases where the defenders of an implicit paradigm don't have solid articulations of it, making it particularly difficult to understand what they're actually defending. I think this is such a case. If you disagree, I'd be curious what you consider a non-strawmanish summary of the RL-evolution analogy. Perhaps Clune's AI-GA paper? But from what I can tell opinions of it are rather mixed, and the AI-GA terminology hasn't caught on. Thanks for all those great references! My current thinking is: (1) Outer-loop meta-learning is slow, (2) Therefore we shouldn't expect to get all that many bits of information out of it, (3) Therefore it's a great way to search for parameter settings in a parameterized family of algorithms, but not a great way to do "the bulk of the real design work", in the sense that programmers can look at the final artifact and say "Man, I have no idea what this algorithm is doing and why it's learning anything at all, let alone why it's learning things very effectively... (read more) 4adamShimi3moJust wanted to say that this comment made me add a lot of things on my reading list, so thanks for that (but I'm clearly not well-read enough to go into the discussion). The EMH is False - Specific Strong Evidence I haven't heard that claim before. My understanding was that such a claim would be improbable or cherrypicking of some sort, as a priori risk-adjusted etc returns should be similar or identical but by deliberately narrowing your index, you do predictably lose the benefits of diversification. So all else equal (such as fees and accessibility of making the investment), you want the broadest possible index. The EMH is False - Specific Strong Evidence Since we're discussing EMH and VTSAX, seems as good a place to add a recent anecdote: Chatting with someone, investments came up and they asked me where I put mine. I said 100% VTSAX. Why? Because I think the EMH is as true as it needs to be, I don't understand why markets rise and fall when they do even when I think I'm predicting future events accurately (such as, say, coronavirus), and I don't think I can beat the stock markets, at least not without investing far more effort than I care to. They said they thought it wasn't that hard, and had (unlike me) ... (read more) 1Srdjan Miletic2moOut of curiosity, why a US tracker fund instead of a global one like FTSE all-world? 4emanuele ascani3moWhy not S&P500? 3jmh3moI think this points to two very important things about investing and trading regardless of EMH. 1. psychology of the person 2. We tend to note the loses we avoided (that is the money we kept) much higher than the gains we missed (the money we actually lost by not playing in the game). Unless someone has a good plan for how to manage and overcome those two aspects of their own mind I suspect they will find it difficult to ever commit to any investing or trading program/strategy. It will not take too much to push them back into the behavior reflected in your comments. 3Rune3moIs there a reason you only invest in the US stock market and not the whole world (VTWAX)? Or is VTSAX good enough and it's not worth the effort to decide whether you should globally diversify and in what proportion? 4deluks9173moI think you need to follow good practices. Getting out with no plan is not following good practices. I will write up my opinion on best practices soon. Though it is nothing too crazy. What's a good way to test basic machine learning code? ALE is doubtless the Atari Learning Environment. I've never seen an 'ALE' in DRL discussions which refers to something else. 1Kenny3moThanks! Of course you would know :) [AN #142]: The quest to understand a network well enough to reimplement it by hand It is quite possible that CLIP “knows” that the image contains a Granny Smith apple with a piece of paper saying “iPod”, but when asked to complete the caption with a single class from the ImageNet classes, it ends up choosing “iPod” instead of “Granny Smith”. I’d caution against saying things like “CLIP thinks it is looking at an iPod”; this seems like too strong a claim given the evidence that we have right now. Yes, it's already been solved. These are 'attacks' only in the most generous interpretation possible (since it does know the difference), and ... (read more) 6rohinmshah3moAh excellent, thanks for the links. I'll send the Twitter thread in the next newsletter with the following summary: Kenshō Harper's has a new article on meditation which delves into some of these issues. It doesn't mention PNSE or Martin by name, but some of the mentioned results parallel them, at least: ...Compared with an eight-person control group, the subjects who meditated for more than thirty minutes per day experienced shallower sleep and woke up more often during the night. The more participants reported meditating, the worse their sleep became... A 2014 study from Carnegie Mellon University subjected two groups of participants to an interview with openly hostile eval ... (read more) 2Davidmanheim3moFor betting markets, the market maker may need to manage the odds differently, and for prediction markets, it's because otherwise you're paying people in lower brier scores for watching the games, rather than being good predictors beforehand. (The way that time-weighted brier scores work is tricky - you could get it right, but in practice it seems that last minute failures to update are fairly heavily penalized.) Resolutions to the Challenge of Resolving Forecasts In such cases, perhaps the rules would be to pick a probability based on the resolution of past games - with the teams tied, it resolves at 50%, and with one team up by 3 runs in the 7th inning, it resolves at whatever percentage of games where a team is up by 3 runs at that point in the game wins. Sounds like Pascal's problem of the points, where the solution is to provide the expected value of winnings, and not merely allocate all winnings to which player has the highest probability of victory. Suppose 1 team has 51% probability of winning - should the... (read more) 2Davidmanheim3moYes, that was exactly what I was thinking of, but 1) I didn't remember the name, and 2) I wanted a concrete example relevant to prediction markets. And I agree it's hard to estimate in general, but the problem can still be relevant in many cases - which is why I used my example. In the baseball game, if the market closes before the game begins - we don't have a model as good as the market, but once the game is 7/9th complete, we can do better than the pre-game market prediction. February 2021 gwern.net newsletter No; I've only seen the first season of AoT, if there are armored trains in the rest I am unaware of that. It's actually from someone on either DSL or Naval Gazing, I think, linking to a short history of Zaamurets which is patchy but interesting in its own right. The average North Korean mathematician To noodle a bit more about tails coming apart: asymptotically, no matter how large r, the probability of a 'double max' (a country being the top/max on variable A correlated r with variable B also being top/max on B) decreases to 1/n. The decay is actually quite rapid, even with small samples you need r>0.9 to get anywhere. A concrete example here: you can't get 100%, but let's say we only want a 50% chance of a double-max. And we're considering just a small sample like 192 (roughly the number of countries in the world, depending on how you count). What ... (read more) The average North Korean mathematician The tails coming apart is "Nigeria has the best Scrabble players in the world, but the persons with the richest English vocabulary in the world are probably not Nigerian" No. The tails coming apart here would be "gameplaying of game A correlates with national variable B but the top players of game A are not from the top country on variable B". I say it's borderline circular because while they aren't the same explanation, they can be made trivially the same depending on how you shuffle your definitions to save the appearances. For example, consider the hyp... (read more) The average North Korean mathematician There are many countries besides Nigeria where English is an official language, elite language, or widely taught. And language proficiency apparently has little to do with Scrabble success at pro levels where success depends on memorizing an obsolete dictionary's words (apparently even including not really-real words, to the point where I believe someone won the French Scrabble world championship or something without knowing any French beyond the memorized dictionary words). Above the Narrative I assume you're referring to the 'vault' thing WP mentions there as "Recently credited by Alan Sherman"? Then no, Chaum is irrelevant to Satoshi except inasmuch as his Digicash was a negative example to the cryptopunks about the vulnerability of trusted third parties & centralization to government interference & micromanagers (some of whom, like Szabo, worked for him). The vault thing didn't inspire Satoshi because it inspired no one; if it had, it wouldn't need any Alan Sherman to dig it up in 2018. You will not find it cited in the Bitcoin whitep... (read more) 2cousin_it3moHuh, interesting. Indeed it seems post hoc-ed in this case, I should've looked at bitcoin's history more closely before making a confident statement. Thanks! The average North Korean mathematician While the greater male variance hypothesis, and tail effects in general, are always interesting, I'm not sure if it's too illuminating here. It is not surprising that there are some weird outliers at the top of the IMO list, 'weird' in the sense of 'outperforming' what you'd expect given some relevant variable like GDP, intellectual freedom, HDI index, national IQ, or whatever. That's simply what it means for the correlation between IMO scores & that variable to be <1. If the IMO list was an exact rank-order correspondence, then the correlation woul... (read more) 2gwern3moTo noodle a bit more about tails coming apart: asymptotically [https://www.gwern.net/Order-statistics#probability-of-bivariate-maximum], no matter how large r, the probability of a 'double max' (a country being the top/max on variable A correlated r with variable B also being top/max on B) decreases to 1/n. The decay is actually quite rapid, even with small samples you need r>0.9 to get anywhere. A concrete example here: you can't get 100%, but let's say we only want a 50% chance of a double-max. And we're considering just a small sample like 192 (roughly the number of countries in the world, depending on how you count). What sort of r do we need? We turn out to need r ~ 0.93! There are not many correlations like that in the social sciences (not even when you are taking multiple measurements of the same construct). -------------------------------------------------------------------------------- Some R code to Monte Carlo estimates of the necessary r for n = 1-193 & top-p = 50%: p_max_bivariate_montecarlo <- function(n,r,iters=60000) { library(MASS) # for 'mvrnorm' mean(replicate(iters, { sample <- mvrnorm(n, mu=c(0,0), Sigma=matrix(c(1, r, r, 1), nrow=2)) which.max(sample[,1])==which.max(sample[,2]) })) } find_r_for_topp <- function(n, topp_target=0.5) { r_solver <- function(r) { topp <- p_max_bivariate_montecarlo(n, r) return(abs(topp_target - topp)) } optim(0.925, r_solver, method="Brent", lower=0, upper=1)par } library(parallel); library(plyr) # parallelism rs <- ldply(mclapply(2:193, find_r_for_topp))\$V1 # c(0.0204794413, 0.4175067131, 0.5690806174, 0.6098019663, 0.6994770020, 0.7302042200, 0.7517989571, 0.7652371794, 0.7824824776, 0.7928299227, 0.7911903664, 0.8068905240, 0.8177673342, 0.8260679686, 0.8301939461, 0.8258472869, 0.8314810573, 0.8457114147, 0.8477265340, 0.8599239760, 0.8541010795, 0.8539345369, 0.8578597015, 0.8581440013, 0.8584451493, 0.8612079626, 0.8640382310, 0
3ChristianKl3moIt might not just be specialization on Scrabble. English is the official language in Nigeria. I think it's plausible that Nigerian elite English education focuses more strongly on learning a lot of words then US English education.
6Elmer of Malmesbury3moAs far as I understand, the tails coming apart and the moment attribution are two different, superimposed problems. The tails coming apart is "Nigeria has the best Scrabble players in the world, but the persons with the richest English vocabulary in the world are probably not Nigerian". The moment attribution is "the best Scrabble players in the world are Nigerian, but Nigerians are probably not the best Scrabble players in the world". In the first case, we are talking about the distribution of country scores for two correlated variables, in the second we are talking about the distribution of individuals within a country for a single variable. Also, thank you for bringing up Nigerian Scrabble, that would have made a somehow funnier example than NK's math olympiads.
1Oskar Mathiasen4moThe comparison to chess is maybe more accurate than you think. See stuff like: Beginnings: The first IMO was held in Romania [https://en.wikipedia.org/wiki/Romania] in 1959. It was initially founded for eastern European [https://en.wikipedia.org/wiki/Europe] member countries of the Warsaw Pact [https://en.wikipedia.org/wiki/Warsaw_Pact], under the USSR [https://en.wikipedia.org/wiki/USSR] bloc of influence, but later other countries participated as well.[2] [https://en.wikipedia.org/wiki/International_Mathematical_Olympiad#cite_note-more-2] (source https://en.wikipedia.org/wiki/International_Mathematical_Olympiad) [https://en.wikipedia.org/wiki/International_Mathematical_Olympiad)] Also classic geometry is (to my knowledge) taught more generally in many eastern European countries (and make up 1/6-1/3 of the imo). Also the note about incentives being larger in North Korea also applies to much of eastern Europa to a lesser degree, where qualifying for imo is seemingly enough to get access to any university (source: Sankt Petersberg university gave an open offer at Baltic Way (a regional math competition), and i know someone who used something like that to get into Moscow university) (Romania, Serbia, Poland, Russia, Ukraine Hungary are the eastern european countries with consistently good results)
Fun with +12 OOMs of Compute

One man's a priori is another man's a posteriori, one might say; there are many places one can acquire informative priors... Learning 'tacit knowledge' can be so fast as to look instantaneous. An example here would be OA's Dactyl hand: it learns robotic hand manipulation in silico, using merely a model simulating physics, with a lot of randomization of settings to teach it to adapt on the fly, to whatever new model it finds itself in. This enables it to, without ever once training on an actual robot hand (only simulated ones), successfully run on an actual... (read more)

Takeaways from one year of lockdown

I was recently tracking down a reference in the Sequences and found that the author was so afraid of COVID that he failed to seek medical care for appendicitis and died of sepsis.

Wow! Who was that?

and the faint but pretty smell of vanilla.

I think you mean "...and a presumption that once our eyes watered." (As time passes, this is increasingly how I feel about my grandmother dying of coronavirus.)

Justin Corwin (obituary, LW account), quoted in this post. I'm sorry about your grandmother. And about Justin, and that death exists in general :(

Mentorship, Management, and Mysterious Old Wizards

Michael Nielsen calls something similar "volitional philanthropy", with some examples.

What happens to variance as neural network training is scaled? What does it imply about "lottery tickets"?
gwern4mo19Ω12

'Variance' is used in an amusing number of ways in these discussions.You use 'variance' in one sense (the bias-variance tradeoff), but "Explaining Neural Scaling Laws", Bahri et al 2021 talks about a difference kind of variance limit in scaling, while "Learning Curve Theory", Hutter 2001's toy model provides statements on yet others kinds of variances about scaling curves themselves (and I think you could easily dig up a paper from the neural tangent kernel people about scaling approximating infinite width models which only need to make infinitesimally sma... (read more)

Meetup Notes: Ole Peters on ergodicity

So, can we steelman the claims that expected utility theory is wrong? Can we find a decision procedure which is consistent with the Peters' general idea, but isn't just log-wealth maximization?

Yes. As I've pointed out before, a lot of these problems go away if you simply solve the actual problem instead of a pseudo-problem. Decision theory, and Bayesian decision theory, has no problem with multi-step processes, like POMDPs/MDPs - or at least, I have yet to see anyone explain what, if anything, of Peters/Taleb's 'criticisms' of expected-value goes away i... (read more)

4abramdemski4moI like the "Bellman did it better" retort ;p FWIW, I remain pretty firmly in the expected-utility camp; but I'm quite interested in looking for cracks around the edges, and exploring possibilities. I agree that there's no inherent decision-theory issue with multi-step problems (except for the intricacies of tiling issues!). However, the behavior of Bayesian agents with utility linear in money, on the Kelly-betting-style iterated investment game, for high number of iterations, seems viscerally wrong. I can respect treating it as a decision-theoretic counterexample, and looking for decision theories which don't "make that mistake". I'm interested in seeing what the proposals look like.

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# SSM Using Scalismo (5): Creating a point distribution model

Finally, I arrive at statistical shape models. I am going to create a point distribution model (PDM) of shapes. Reading some theory on the point distribution models, I need superimposed shapes and their mean shape. Superimposed shapes and their mean shapes are available from my previous tutorial on GPA with Scalismo. The outcomes of the full GPA is going to be utilized for creating a PDM. The PDM in this tutorial uses shapes (not forms), and it is based on the multivariate normal distribution of the points’ coordinates collected in a shape vector (Eq. 16 in this post).

This post contains code blocks; if your browser does not automatically display the code blocks, please click on the displayed dark narrow rectangular areas to display the containing code.

object PDM extends App {

import scalismo.geometry._
import scalismo.common._
import scalismo.ui.api._
import scalismo.mesh._
import scalismo.io.{StatisticalModelIO, MeshIO}
import scalismo.statisticalmodel._
import scalismo.registration._
import scalismo.statisticalmodel.dataset._

scalismo.initialize()
implicit val rng = scalismo.utils.Random(42)
// ui
val ui = ScalismoUI()

val drive = "I:/"
val meshDir = drive + "registration/Full GPA/"
val unsortedMeshFiles = new java.io.File(meshDir).listFiles()
unsortedMeshFiles.foreach(file => println(s"\n\$file"))
println(unsortedMeshFiles)
// specifying the file name of the mean mesh generated through the full GPA
val meanMeshName = "meanMesh.stl"
val meshFileNames = unsortedMeshFiles.map(file => file.getName())
val indexOfMeanMesh = meshFileNames.indexOf(meanMeshName)
// bringing the meanMesh to the head (first entry) of the meshFiles sequence
val meshFiles = unsortedMeshFiles.updated(0, unsortedMeshFiles(indexOfMeanMesh)).updated(0, unsortedMeshFiles(indexOfMeanMesh))

I have to choose a reference mesh and some training meshes. The full GPA mean mesh is chosen as the reference mes and the three (registered and superimposed) meshes as the training meshes.

// specifying the reference mesh and the training meshes; the full GPA mean mesh is set to be the reference mesh
val trainingMeshes = meshFiles.tail.map(file => MeshIO.readMesh(file).get)//the other 3 meshes

Once a reference mesh and training meshes are chosen, the deformation vector from each point of the reference mesh to its corresponding point of each training mesh should be calculated. Thereby, the deformation fields morphing the reference mesh to the training meshes are obtained.

 // [The meshes are already aligned using the full GPA, and in correspondence, so we can create a discrete GP]
// creating the deformation vector field
val deformationFields = trainingMeshes.map(mesh => {
val deformationVectors = refMesh.pointSet.pointIds.map { id =>
mesh.pointSet.point(id) - refMesh.pointSet.point(id)
}.toIndexedSeq
// a discrete deformation field needs a domain and the deformation vectors at each point of the domain
DiscreteField[_3D, UnstructuredPointsDomain[_3D], EuclideanVector[_3D]](refMesh.pointSet, deformationVectors)
})

The calculated deformation fields are samples (realizations) of a discrete random field. The domain of the random field is the reference mesh points. The random field of deformations is assumed to have a multivariate normal distribution. Here, the discrete random field is expressed by a random vector. Thereby, a shape, described by a shape vector, can be produced by adding the shape vector of the reference shape to the normally distributed random vector of deformations expanded by the deformation modes (this is explained by Eq. 16 in this post).

Technically, Scalismo works with the concept of Gaussian process (GP) which is a continuous random field whose any finite sub-collection of random variables (here deformations) has a multivariate normal (Gaussian) distribution. Therefore, continuous sample deformation fields should be approximated based on the known discrete realizations/samples of the random field. By interpolating a discrete field, Scalismo creates a continuous field. Different interpolation approaches are possible; Scalismo uses the Nearest Neighbor Interpolation method. The following code block defines a discrete GP (of the deformations) which is in fact a normally distributed random vector with elements indexed by the reference mesh points (a discrete domain). The discrete GP (random vector of deformations) is expressed in terms of its principal axes (modes). This is done using the PCA as described in this post (see Eq. 16).

  val interpolator = NearestNeighborInterpolator[_3D, EuclideanVector[_3D]]()
val continuousDefFields = deformationFields.map(field => field.interpolate(interpolator))
val gp = DiscreteLowRankGaussianProcess.createUsingPCA(refMesh.pointSet, continuousDefFields)

As I have 3 training meshes and therefore 3 training deformation fields, the rank of the sample covariance matrix of deformations is at most 3. This means that at most 3 principal axes are used to express the random vector of deformations.

I can sample the GP (random field) of deformations. By adding a sample deformation field to the reference mesh, I can generate a shape vector. This shape vector is a sample mesh from meshes whose points’ coordinates are normally distributed. Sampling some meshes from the mesh distribution, I’ll have the following realization:

  val PDMGroup = ui.createGroup("refMeshGroup")
val refMeshView = ui.show(PDMGroup, refMesh, "refMesh")
refMeshView.color = java.awt.Color.GREEN
val gpDefView = ui.addTransformation(PDMGroup, gp, "RefMeshDefrmdByGp")

The horizontal bar-like controls in the figures are proportional to the coefficients of the deformation modes (). Each coefficient varies within three standard deviations (Eq. 4 in this post).

Scalismo assembles the reference mesh and the discrete GP as a statistical mesh model:

// [creating a statistical shape model (PDM)
val model = StatisticalMeshModel(refMesh, gp.interpolate(interpolator))

This is the statistical PDM.

# Point Distribution Models of Shapes and Forms

A point distribution model (PDM) is a model that locates (or determines the geometrical (spatial) distribution of) the points of the shapes or forms (size-and-shapes) in a particular shape or form population. The word distribution refers to the spatial distribution of the points; not a probability distribution. This is my best understanding of the definition of PDM’s so far.

A PDM approximates the geometrical distribution of the points of shapes/forms in a population. This means that a PDM is intended to model/express all the possible shapes/forms in the population.

A PDM entails shapes or forms. This means we are working with superimposed configurations modulo their location, rotation, and scale information for the case of shapes, and their location and rotation information in the case of forms. Therefore, we assume that shapes or forms are available through performing a full or partial generalized Procrustes analysis (GPA) on the given configurations. In other words, a shape is an instance of the full Procrustes fits, and a form is an instance of partial Procrustes fits. Henceforward, whatever includes the concept of a shape in this article also holds for a form unless declared.

As explained, a shape can be (approximately) represented using landmarks, i.e. discrete set of points. By choosing a 3D Cartesian coordinate system (CS) with orthogonal basis vectors and x,y, and z coordinates (axes), the landmark coordinates can then be collected in a matrix. One can readily think of collecting the coordinates in a column vector (in fact every matrix is isomorphic to a vector). Therefore, a shape in a set of shapes, all having the same number of N landmarks is represented as a vector called the shape vector [1]:

X_i=\big[x_{1x}^i,x_{1y}^i,x_{1z}^i,\dots,x_{Nx}^i,x_{Ny}^i,x_{Nz}^i\big]^\text T \in\mathbb{R}^{3N\times 1}\tag{1}

See this post for linear algebra conventions a identities.

A shape vector is a member of the vector space . Therefore, there are bases, and can be written as a linear combination of the basis vectors. By default, the 3N-D vector is born in the vector space spanned by the standard basis. Indeed, other bases are possible.

Using a basis for the vector space, we can write any shape vector as a deviation from an arbitrary constant (shape) vector :

It should be noted that, is a point (position vector) and is a free vector in the concept of geometrical modelling.

On the other hand, is a random vector with with a mean and a covariance . The constant vector can be replaced with . Any other shape vector, like a known arbitrary shape vector in the population, can also be substituted for . The basis vectors can be chosen such that the linear projection of onto them generates random variables each having maximum possible variance. These vectors are obtained by PCA as a variance-maximization method. Because is symmetric, it has 3N independent and mutually orthogonal eigenvectors, and hence any shape vector can be written as:

\begin{gathered} X=\mu_X+Vb=\mu_X+\sum_{i=1}^{3N}b_iv_i \tag{2} \\ \ \\ \text{s.t. }\ \Sigma_{XX}=V\varLambda V^\text T\\ \varLambda\text{ is diagonal and }\varLambda_{ii}=\lambda_i\quad\text{s.t. } \lambda_1\ge\lambda_2\ge \dots \ge \lambda_{3N}\ge 0 \end{gathered}

where, the term is the singular value decomposition (SVD) of such that the eigenvalues, ‘s, collected in the diagonal matrix are sorted in the decreasing order. As the covariance matrix is symmetric, SVD is equivalent to the spectral/eigenvalue decomposition of the matrix.

So far we showed that a shape instance (represented by a shape vector) in the population can be generated by deforming the mean shape through a deformation field (vectors) which is determined by linear combinations of the eigenvectors of . The eigenvectors of are principal (3N dimensional) directions along which the variances of the linearly projected (random) shape vector are maximized. The principal directions are called deformation modes (deformations from the mean). The deformation modes are also called eigenshapes. The combining coefficients ‘s are called the shape model parameters. The deformation modes weighted by the shape model parameters adds up to produce a deformation vector that will be added to the mean shape vector to generate a shape. Such a representation is called the modal representation; this is the idea behind a PDM.

By choosing proper shape parameters, the model generates shapes in the shape population. It should be noted that there is generally no restrictions on the values of ‘s; in other words, the deformation modes can be combined arbitrarily. However, a generated shape out of an arbitrary values of ‘s may not fall into the shape population; this is not difficult to imagine.

If we let , then is a random vector expressing the deviation of a shape from the mean shape of the population. I call this vector the deviation or the deformation shape vector. This vector contains deformation components (in three directions of the CS) of the mean shape at each of its points. The mean of the deformation shape vector is zero, . In fact, . This means that the elements of are variances (or covariances) of the deformation components. can be linearly projected on the directions (unit vectors) defined by the eigenvectors of :

U_p=V^\text T U\ \in\mathbb{R}^{3N\times 1}

and by considering Eq. 2, we can conclude (it is simply the result of the linear projection):

U=Vb\implies U_p=V^\text T Vb=I_rb=b

Just as a side note: .

On the other hand . This is what we can conclude:

\begin{gathered}X=\mu_X+Vb=\mu_X+\sum_{i=1}^{3N}b_iv_i \tag{3} \\ \ \\ \text{s.t. }\ \Sigma_{bb}=\varLambda= \begin{bmatrix}\lambda_1&0&\dots\ &0\\ 0 &\lambda_2\ &\dots\ &0\\ \vdots&\vdots &\ddots &\vdots\\ 0&0&0&\lambda_{3N} \end{bmatrix}\quad \text{and}\quad \mu_b=0\\ \ \\ \text{with}\\ \ \\ \|b_iv_i\|=|b_i|\end{gathered}

Note that because .

It is obvious that the magnitude of controls the magnitude of deformation of the mean shape by each mode. From above, the standard deviation of is . It is assumed that a PDM generates plausible shapes, or well behaved shapes, if the shape parameters are limited as:

-3\sqrt{\lambda_i}\le b_i\le +3\sqrt{\lambda_i}\tag{4}

A plausible shape is created by adding a plausible deformation shape vector to the mean shape vector; a plausible deformation shape vector is a deformation vector that generates shapes still belong to the shape family under consideration.

The covariance matrix (and also ) is positive semi-definite; therefore, . This implies that the variance of some ‘s can be zero, i.e. the zero-variance ‘s are constant. In that case, simply because no variation about the mean. Therefore, a deformation mode associated with a shape parameter with zero variance () vanishes from the modal representation. Assuming M zero eigenvalues (including algebraic multiplicity), we can write the modal representation of a shape (vector) as:

X=\mu_X+\sum_{i=1}^{3N-M}b_iv_i \tag{5}

### Shape approximation

A PDM is based on the modal representation of shapes (Eq. 2 or 3). This kind of representation is useful when it comes to approximation of a shape. In this regard, only some of the deformation modes being relatively more effective in forming a shape is kept and the rest of the modes are ignored. We already showed (Eq. 5) that zero eigenvalue of the covariance matrix leads to zero variation of the corresponding shape parameter and consequently vanishing of the associated mode in the modal representation. This motivates us to throw out the shape parameters with relatively small variances, hence, their associated modes; thereby, approximating a shape vector by a t-term compact modal representation:

\begin{gathered} X\approx \tilde X=\mu_X+\tilde Vb=\mu_X+\sum_{i=1}^{t}b_iv_i\\ \ \\ \text{s.t. }\ b\in \mathbb{R}^{t\times 1}\ , \tilde V=[v_1\quad v_2\ \dots\quad v_t] \in \mathbb{R}^{3N\times t}\ ,\text{ and } t<3N \tag{6} \end{gathered}

where ‘s are as in Eq.2.

Moreover, by the Least-squares optimality of PCA, we can approximate the shape vector as (see Eq. 3a in this post):

\begin{gathered} X\approx\tilde X =\mu_X + \tilde V\tilde V^\text T(X-\mu_X)= \mu_X +\sum_{i=1}^{t}v_iv_i^\text T(X-\mu_X) \tag{7} \\ \ \\ \text{s.t. }\ b\in \mathbb{R}^{t\times 1}\ , \tilde V=[v_1\quad v_2\ \dots\quad v_t] \in \mathbb{R}^{3N\times t}\ ,\text{ and } t<3N \end{gathered}

This indicates that for approximating a particular shape vector, the shape model parameters should be chosen as:

But, how many modes i.e. do we need? it can be determined based on the goodness-of-fit measure defined for PCA as a dimension reduction tool (Eq. 7 in this post):

g(t) = \frac{\sum_{i=t+1}^{3N} \lambda_i}{\text{Trace}(\Sigma_{XX})=\sum_{i=1}^{3N} \lambda_i}\tag{9}

Therefore, we only need to keep the first t modes having large associated eigenvalues. Actually, we hope that the first t eigenvalues are substantially larger than the rest of the eigenvalues. As a suggestion is ok.

### Sample shapes (data set)

In practice, the mean and covariance of the shape (random) vector are unknown and should be estimated. Let be a set of sample shapes (each represented by a shape vector ) from a particular shape family population. This is, in fact, a set of independent observations of identically distributed random variables. This set is called the training set. As a remark, it should be noted that we generally start with a set of configurations, out of a configuration population; for example the set of all dinosaurs. Then, we perform the full or the partial GPA to obtain a set of shapes/forms already denoted by . Having said that, the following estimates are considered [2]:

\begin{gathered} \hat \mu_X=\bar S:=\frac{1}{n}\sum_{i=1}^{n} S_i \ \in \mathbb{R}^{3N\times 1}\tag{10}\\ \ \\ \hat\Sigma_{XX}= \frac{1}{n-1}\sum_{i=1}^n (S_i-\bar S)(S_i-\bar S)^\text T\ \in \mathbb{R}^{3N\times 3N} \end{gathered}

where n is the number of training shapes. The covariance estimate is a symmetric matrix, therefore, it has 3N independent eigenvectors, however, has at most rank n (properties of rank in this post), i.e. non-zero eigenvalues. Therefore, the modal representation of the training shapes and other shapes in the population predicted based on the model generated using sample shapes (training shapes) is:

X=\bar S+\sum_{i=1}^{n}b_i\hat v_i \tag{11}

where is an eigenvector of .

When n is large enough, then we can use the following criterion to select the first t modes with relatively large eigenvalues:

g(t) = \frac{\sum_{i=t+1}^{n} \lambda_i}{\text{Trace}(\hat \Sigma_{XX})=\sum_{i=1}^{n} \lambda_i}\tag{12}

This criterion is similar to Eq. 9.

## PDM’s with normal distributions

It can be assumed that a shape vector , being a random vector, has a multivariate normal distribution i.e. [1]:

X\sim \mathscr N(\mu_X,\Sigma_{XX})\tag{13}

This assumption can be manipulated toward a more practical form. Let’s consider the following proposition:

If has a multivariate normal distribution , , and being a constant vector, then, the linear transformation is distributed as .

Using the proposition and writing the shape vector in its modal form as , we can conclude:

X=\mu_X+Vb\sim \mathscr N(\mu_X,\Sigma_{XX})\\ \ \\ \iff X\sim \mathscr N(V\mu_b+\mu_X,V\Sigma_{bb}V^\text T)\\ \ \\ \iff b\sim \mathscr N(0,\Sigma_{bb})

where is the diagonal matrix of the eigenvalues as in Eq. 3.

By the marginalization property of normal distributions:

b_i\sim \mathscr N(0,\lambda_i)

and, we can write:

\alpha_i:=\frac{b_i}{\sqrt{\lambda_i}}\implies \alpha_i\sim \mathscr N(0,1)

Therefore, the modal representation of the shape vector becomes:

X=\mu_X+\sum_{i=1}^{3N}\alpha_i\sqrt{\lambda_i}v_i\qquad \text{s.t }\ \alpha_i\sim \mathscr N(0,1) \tag{14}

For shape vector approximation and sample shape vector, the following equations (based on Eq. 6 and 11) are readily obtainable:

\tilde X=\mu_X+\sum_{i=1}^{t}\alpha_i\sqrt{\lambda_i}v_i\qquad \text{s.t }\ \alpha_i\sim \mathscr N(0,1) \tag{15} X=\bar S+\sum_{i=1}^{n}\alpha_i\sqrt{\lambda_i}\hat v_i\qquad \text{s.t }\ \alpha_i\sim \mathscr N(0,1) \tag{16}

## Shape space spanned by the PDM

A shape vector can be represented by any of the equations 6, 11, 15, or 16. In any of the equations, a 3N dimensional shape vector is presented by a linear combination of basis vectors whose number of them is less than the dimension of the shape vector. This means that a PDM spans a space (of shape vectors) which is a subspace of . Therefore, a PDM model established by training shapes can only constitutes shapes born out of the linear combinations of the shape modes, ‘s. Such a model then misses the shapes (in the population) which are not captured by the training set of shapes or cannot be represented by linear combinations of the shape modes.

#### References

[1] Gaussian Process Morphable Models, Marcel Luthi , Thomas Gerig, Christoph Jud, and Thomas Vetter. IEEE Transactions on Pattern Analysis and Machine Intelligence (Volume: 40 , Issue: 8 , Aug. 1 2018).

[2] Image Processing and Analysis (Chapter 7: Model-Based Methods in Analysis of Biomedical Images). R. Baldock and J. Graham, Oxford University Press, 2000.


text:
# Combined parallel worksharing directives

Combined parallel worksharing directives
© PRACE and University of Ljubljana

Combined constructs are shortcuts for specifying one construct immediately nested inside another construct. Specifying a combined construct is semantically identical to specifying the first construct that encloses an instance of the second construct and no other statements. Most of the rules, clauses and restrictions that apply to both directives are in effect. The parallel construct can be combined with one of the worksharing constructs, for example for and sections.

## parallel for

When we are using a parallel region that contains only a single for directive , we can substitute the separate directives with this combined directive:

#pragma omp parallel for [clause[[,]clause]...]

This directive admits all the clauses of the parallel directive and for directive except the nowait clause is forbidden.

This combined directive must be directly in front of the for loop. An example of the combined construct is shown below:

int i;int f = 7;#pragma omp parallel for for (i = 0; i<20; i++) { c[i] = a[i] + b[i]; }
© PRACE and University of Ljubljana


text:
A carton of milk in the shape of a rectangular prism has a leak. The dimensions of the carton are 8 inches by 4 inches by 3 inches. If the rate in which

A carton of milk in the shape of a rectangular prism has a leak. The dimensions of the carton are 8 inches by 4 inches by 3 inches. If the rate in which the milk is leaking is 6 cubic inches per minute, how long would it take for the milk carton to completely empty?

9. Find the area of a circle having a circumference of 382. Round to the nearest tenth. Use 3.14 for 1. a. 1133.5 units b. 1078.6

1. corey36dylon says:

Volume of a rectangular prism = length x width x height.

Volume of milk carton = 8 x 4 x 3 = 96 cubic inches.

Divide the volume by the leak rate:

96/ 6 = 16

It will take 16 minutes.

2. Expert says:

answer: the proof is mentioned below.

step-by-step explanation:

since, given:   lines bc = ef

prove:   lines ac > ef

statements                                           reason

1. bc=ef                                           1. given

2. ac= ab+bc                                 2. because b is the point between the

line segment ac

3.   ac > bc                                       3. because if a line segment is equal to

the sum of two line segments then it is

always greater then each of these

individual line segments

4. ac > ef                                         4. by equality property.

3. Expert says:

Its d so thats how it would would back from 1.0 form three four two


text:
# 64 bit clock cipher with CFB mode. One-byte defected from ciphertext. What is the number of bits defected from plaintext

I have a question from an exam;

We encrypted a message of size 100 Bytes with CFB. In the transmission, byte number 12 got defected. How many bits defected will be in the decryption. The answer is 72 bits and I don't understand why?

• yes the block has 64 bit size.but i still don't understand... – Omer Michleviz Feb 10 '19 at 12:46
• Hint: CFB encryption and decryption are pictured here. Redraw it in your case (which uses a different block width). Examine what "transmitted byte number 12 got defected" means in this, and what that change can influence in the decryption. Notice that, contrary to what the question states, we can not tell for sure how many bits are influenced, and that 72 bits actually is extremely unlikely. The actual answer is 2 to 72 bits. – fgrieu Feb 10 '19 at 12:49
• Is "defected" errored? – kodlu Feb 10 '19 at 20:55

The decryption of CFB mode as follows;

$$P_i = E_k(C_{i-1}) \oplus C_i$$ $$C_0= \operatorname{IV}$$

Bit flipping attack in CFB

If there is a Bit Flipping attack in the ciphertext then we have two cases.

$$\color{red}{\textbf{Red case:}}$$ The last ciphertext bit is flipped. This only affects the corresponding plaintext block. So the attacker can change a plaintext bit without affecting any other bits.

$$\color{ForestGreen}{\textbf{Green case:}}$$ Assuming that the $$i$$th ciphertext bit is changed, $$0 \leq i < n$$ where $$n$$ is the total number of blocks, then this time two plaintexts blocks are affected. We can also see from the equations, too;

$$P_i = E_k(C_{i-1}) \oplus \color{blue}{C_i}$$ $$P_{i+1} = E_k(\color{blue}{C_{i}}) \oplus C_{i+1}$$

Note: Bit flipping on IV of CFB mode has no good effect as in CBC mode.

Since you have 100 Bytes, and 64-bit block cipher has 8-byte block size, the 12th-byte number falls into the second ciphertext. If it falls into the last one then only one byte is effected, $$\color{red}{\textbf{red case}}$$.

Now, since we are in the $$\color{ForestGreen}{\textbf{green case}}$$, for a defected one-byte $$C_i$$, we will see two plaintexts are effected $$P_i$$ and $$P_{i+1}$$. $$P_i$$ has one byte defection and $$P_{i+1}$$ has full block defection.

Fgrieu gave the probabilities of the defection in the comments. As noted, the defection is not clear about how many bits are flipped. It may be 1 or 8. Also, the affected full block is not under control. For example, if we assume that the cipher has avalanche criteria we expect each of the ciphertext bits has 50% probability to have a flip. Therefore we can say it is between 1 and 64. So the total number of affected bits is between 2 to 72.

Let finish this with mitigations:

The attack/defection is possible since there is no integrity on the data. A MAC or an HMAC can be used to prevent, or better use authenticated encryption which provides confidentiality, integrity, and authenticity. In TLS 1.3., CCM and GCM are standardized authenticated encryption modes.

• i'm sure if the answer in the exam(72 ) is wrong or i still dont get it.as i see it every block after number 12 will be defected but there is only 100 byes .each block size is 64 bits which is 8 bytes hence, 12 block approximately. so if there is only 12 and the last one got defected ..i mean i get it wrong.can you explain the calculation ? – Omer Michleviz Feb 10 '19 at 13:08
• For a 64-bit block cipher there will be 2 to 72 bits in error, rather than 72 bits as the question's account of the problem/solution states. 2 could occur for a 1-bit error in $C_i$ combined with the unlikely case that it causes only a 1-bit error in $E_k(C_i)$; which for an ideal cipher occurs with probability $64/(2^{64}-1)$. 72 could occur for an 8-bit error in $C_i$ combined with the unlikely case that it causes a 64-bit error in $E_k(C_i)$; which for an ideal cipher occurs with probability $1/(2^{64}-1)$. – fgrieu Feb 10 '19 at 13:09


text:
# Equal Lengths

Geometry Level 5

As shown, $$\angle DAB=15^\circ$$, $$\angle BAC =45^\circ$$ and $$AB=10$$. Given that $$B$$ is the mid-point of straight segment $$DC$$ and the $$DB=s$$. Find the value of $$\lfloor 100s\rfloor$$.

×


text:
# Can $\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx$ be evaluated?

Currently, I'm working on a problem pertaining to certain integrals involving the modified Bessel function of the first kind. On p. 59 of this book by Rosenheinrich, it is stated that

$$\int e^{-x} I_{0}(x) dx = x e^{-x}[I_{0}(x) + I_{1}(x)] .$$

Moreover, Wolfram obtains: $$\int I_{0}(2 \sqrt{x}) = \sqrt{x} I_{1}(2 \sqrt{x}) + c.$$

Definite integrals are also of interest to me. For instance , we have: $$\int_{0}^{\infty} e^{-x} I_{0}(2 \sqrt{x}) dx = e.$$

Now, I'm looking for results on the following integral:

$$\int \Big{(} \frac{1}{e^{x}-1} - \frac{1}{e^{x}} \Big{)} \Big{(} I_{0}(2 \sqrt{x}) - 1 \Big{)} dx \qquad \qquad (*)$$

The definite version for $$x$$ ranging over the nonnegative reals amounts to about $$0,761$$.

Question: have any results been obtained on the integral stated in $$(*)$$ - with regards to both the indefinite and definite version?

Just a partial answer for the definite version, name it $$\beta_0$$.

Using Laplace Transform from this Maple 2021 output

$$I(p) = \int_{0}^{\infty} e^{-px} (I_{0}(2 \sqrt{x}) - 1) dx = \frac{e^{1/p}-1}{p}$$ Since $$\frac{1}{e^{x}-1}-\frac{1}{e^x} = \frac{e^{-2x}}{1-e^{-x}}= \sum_{p=2}^{\infty} e^{-px}$$ the integral reduces to the following series $$\beta_0 = \sum_{p=2}^{\infty} \frac{e^{1/p}-1}{p} = 0.76137869727328467225018681101291207836...$$ Logarithmically convergent monotone sum that can be obtained using either Levin's-u sequence transformation or Euler-MacLaurin. In this last case PARI/GP provides this value

Expanding $$e^{1/p}-1$$ as a Taylor series we get $$\beta_0 = \sum_{p=2}^{\infty} \sum_{n=1}^{\infty}\frac{1}{n!}p^{-n-1}=\sum_{n=1}^{\infty}\frac{1}{n!}\sum_{p=2}^{\infty}\frac{1}{p^{n+1}}$$

a much faster convergent series in terms of Riemann function $$\zeta(z)$$ $$\beta_0 = \sum_{n=1}^{\infty} \frac{\zeta(n+1)-1}{n!}$$

• If you have Bessel functions $I_0$ in there, it seems confusing to call the integral also $I_0$. Nov 24, 2021 at 21:21
• @Gerald_Edgar. Corrected. Thanks a lot for pointing this out. Nov 24, 2021 at 21:30
• @JorgeZuniga thank you for you answer. Could you perhaps elaborate a little bit on why the definite integrals amounts to the sum of the laplace transform you've computed? Nov 25, 2021 at 0:08
• @Max_Muller. Ok, I will provide more details. Nov 25, 2021 at 1:38


text:
## 21.17 Build a Decision Tree Model

As seen from Rattle’s Log tab the decision tree model is built using rpart::rpart(). Once the different template variables have been defined as in Section @ref(dtrees:sec:model_setup) (form, ds, tr, and vars) we can use this template call to build the model:

library(rpart)
model <- rpart(formula=form, data=ds[tr, vars], model=TRUE)

This is essentially the same as the command used by Rattle except that some parameter settings are removed. These will be explored later.

In the above call to rpart::rpart() we have named each of the arguments. If we have a look at the structure of rpart::rpart() we see that the arguments are in their expected order, and hence the use of the argument names, and , is optional.

str(rpart)
## function (formula, data, weights, subset, na.action = na.rpart, method,
##     model = FALSE, x = FALSE, y = TRUE, parms, control, cost, ...)

Whilst the argument names are optional they can assist in reading the code, and so the use of argument names in function calls is encouraged.

A textual presentation of the model is concise and informative, once we learn how to read it. Note this tree is different to the previous one we have seen, since we are using a much larger (the full) weather dataset which includes multiple years of daily observations from many different weather stations across Australia.

model
## n= 134001
##
## node), split, n, loss, yval, (yprob)
##       * denotes terminal node
##
##  1) root 134001 28166 No (0.7898075 0.2101925)
##    2) humidity_3pm< 71.5 112640 15454 No (0.8628018 0.1371982) *
##    3) humidity_3pm>=71.5 21361  8649 Yes (0.4048968 0.5951032)
##      6) humidity_3pm< 82.5 11741  5313 No (0.5474832 0.4525168)
##       12) rainfall< 2.05 7728  2831 No (0.6336698 0.3663302) *
##       13) rainfall>=2.05 4013  1531 Yes (0.3815101 0.6184899) *
##      7) humidity_3pm>=82.5 9620  2221 Yes (0.2308732 0.7691268) *

Refer to Section @ref(dtrees:sec:explain_read_tree) for an explanation of the format of the textual presentation of the decision tree.

Your donation will support ongoing development and give you access to the PDF version of this book. Desktop Survival Guides include Data Science, GNU/Linux, and MLHub. Books available on Amazon include Data Mining with Rattle and Essentials of Data Science. Popular open source software includes rattle, wajig, and mlhub. Hosted by Togaware, a pioneer of free and open source software since 1984.


text:
# Revision history [back]

Afaik this is not supported by rosbag.

You also seem to conflate roslaunch functionality (which would be the read-from-file you suggest) with what rosbag does.

If you don't want to extend rosbag's functionality (ie: contribute reading a list of topics from a file), then I believe you can achieve what you are asking for by combining roslaunch's support for args and current rosbag functionality.

args can be set from the command line, so if you have a plain text file (no need for YAML I believe) with the list of topics on a single line, we can pass its contents as the value for a roslaunch arg, which finally gets passed on to rosbag play.

You could perhaps do something like this:

<launch>
<arg name="list_of_topics" doc="A space-separated list of topics to play, single line" />
<node
pkg="rosbag"
name="play"
type="play"
args="--bags $(find test_package)/mybag.bag --topics=$(arg list_of_topics)" />
</launch>

You would then invoke roslaunch as follows:

roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(cat /path/to/topic_list.txt)"  Alternatively write a small Python script (or whichever language you prefer), which takes in a .yaml and spits out a single line of space separated topic names. You would then do something like: roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(your_utility.py /path/to/topic_list.yaml)"

that should result again in list_of_topics to be set to the list of topics, which is then passed to rosbag.

Afaik this is not supported by rosbag.

You also seem to conflate roslaunch functionality (which would be the read-from-file you suggest) with what rosbag does.

If you don't want to extend rosbag's functionality (ie: contribute reading a list of topics from a file), then I believe you can achieve what you are asking for by combining roslaunch's support for args and current rosbag functionality.

args can be set from the command line, so if you have a plain text file (no need for YAML I believe) with the list of topics on a single line, we can pass its contents as the value for a roslaunch arg, which finally gets passed on to rosbag play.

You could perhaps do something like this:

<launch>
<arg name="list_of_topics" doc="A space-separated list of topics to play, single line" />
<node
pkg="rosbag"
name="play"
type="play"
args="--bags $(find test_package)/mybag.bag --topics=$(arg list_of_topics)" />
</launch>

You would then invoke roslaunch as follows:

roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(cat /path/to/topic_list.txt)"  Alternatively write a small Python script (or whichever language you prefer), which takes in a .yaml and spits out a single line of space separated topic names. You would then do something like: roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(your_utility.py /path/to/topic_list.yaml)"

that should result again in list_of_topics to be set to the list of topics, which is then passed to rosbag.

Afaik this is not supported by rosbag.

You also seem to conflate roslaunch functionality (which would be the read-from-file you suggest) with what rosbag does.

If you don't want to extend rosbag's functionality (ie: contribute reading a list of topics from a file), then I believe you can achieve what you are asking for by combining roslaunch's support for args and current rosbag functionality.

args can be set from the command line, so if you have a plain text file (no need for YAML I believe) with the list of topics on a single line, we can pass its contents as the value for a roslaunch arg, which finally gets passed on to rosbag play.

You could perhaps do something like this:

<launch>
<arg name="list_of_topics" doc="A space-separated list of topics to play, single line" />
<node
pkg="rosbag"
name="play"
type="play"
args="--bags $(find test_package)/mybag.bag --topics=$(arg list_of_topics)" />
</launch>

You would then invoke roslaunch as follows:

roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(cat /path/to/topic_list.txt)"  Alternatively write a small Python script (or whichever language you prefer), which takes in a .yaml and spits out a single line of space separated topic names. You would then do something like: roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(your_utility.py /path/to/topic_list.yaml)"

that should result again in list_of_topics to be set to the list of topics, which is then passed to rosbag.

Note: I've not checked whether there are extra quotes needed around the value passed to --topics. That would be something to verify.

Afaik this is not supported by rosbag.

You also seem to conflate roslaunch functionality (which would be the read-from-file you suggest) with what rosbag does.

If you don't want to extend rosbag's functionality (ie: contribute reading a list of topics from a file), then I believe you can achieve what you are asking for by combining roslaunch's support for args and current rosbag functionality.

args can be set from the command line, so if you have a plain text file (no need for YAML I believe) with the list of topics on a single line, we can pass its contents as the value for a roslaunch arg, which finally gets passed on to rosbag play.

You could perhaps do something like this:

<launch>
<arg name="list_of_topics" doc="A space-separated list of topics to play, single line" />
<node
pkg="rosbag"
name="play"
type="play"
args="--bags $(find test_package)/mybag.bag --topics=$(arg list_of_topics)" />
</launch>

You would then invoke roslaunch as follows:

roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(cat /path/to/topic_list.txt)"  Alternatively write a small Python script (or whichever language you prefer), which takes in a .yaml and spits out a single line of space separated topic names. You would then do something like: roslaunch your_pkg rosbag_wrapper.launch list_of_topics:="$(your_utility.py /path/to/topic_list.yaml)"

that should result again in list_of_topics to be set to the list of topics, which is then passed to rosbag.

Note: I've not checked whether there are extra quotes needed around the value passed to --topics. That would be something to verify.


text:
## Radon price: $4/m ?? I'm doing some research on Radon. I have to put together a PowerPoint presentation that describes its origins, properties, characteristics, etc. (Each student was assigned a different element.) One piece of information I'm required to report on is current price to purchase a sample of the element. I have found only one source that mentions a purchase price for Radon, and it says "Radon can be purchased for approximately$4/m." - without further explanation.

What is "m"?? I assume it must be a unit of volume or mass, but if so I'm not familiar with it.

Any ideas?
 PhysOrg.com science news on PhysOrg.com >> City-life changes blackbird personalities, study shows>> Origins of 'The Hoff' crab revealed (w/ Video)>> Older males make better fathers: Mature male beetles work harder, care less about female infidelity
 Recognitions: Gold Member Science Advisor Mmmmm metre, that well known unit of mass... Can we see the source?
 Yes, of course! Sorry, I should have posted that. The actual quote is "Radon is available at a cost of about $4/m." http://www.scescape.net/~woods/elements/radon.html edit: That same source has a table above the text which states that the cost is "$/100g" (no numeric value is given).

## Radon price: $4/m ?? Hmmm... I just found another source that says: "Radon is available at a cost of about$4/m Ci."

I know that Ci is a Curie, measuring radioactivity. Could "m Ci" mean mCi: a milli-Curie?

Recognitions:
Homework Help
 Quote by Bruce Bacher Hmmm... I just found another source that says: "Radon is available at a cost of about $4/m Ci." I know that Ci is a Curie, measuring radioactivity. Could "m Ci" mean mCi: a milli-Curie? That would be right. AM  Woohoo! That makes a lot more sense than "m"!  Similar discussions for: Radon price:$4/m ?? Thread Forum Replies General Discussion 46 Nuclear Engineering 10 Calculus & Beyond Homework 2 General Discussion 4


text:
# Class Information

Course Title Introduction to Differential Equations I

Instructor Kameryn Williams

Email kamerynw [ at ] hawaii ( period ) edu

Class Hours MWF 9:30–10:20

Room Bilger Hall 335

Office Hours MW 10:30–11:30, Th 1:30–2:30, or by appointment

Office Physical Science Building 305

Textbook Tennenbaum and Pollard, Ordinary Differential Equations, Dover Edition.

Course Description First order ordinary differential equations, constant coeffcient linear equations, oscillations, Laplace transform, convolution, Green’s function.

Prerequisite A grade of C or better in Math 216 or 243 (or concurrent) or 253A (or concurrent) or consent.

Grades will be assigned based upon the standard F to A scale: [93,100] A; [90,93) A-; [87-90) B+; [83,87) B; [80,83) B-; [77,80) C+; [73,77) C; [70,73) C-; [67,70) D+; [63,67) D; [60,63) D-; [0,60) F. I reserve the right to assign higher grades, based upon circumstance and student effort.

• 25% homework

• 25% midterm 1

• 25% midterm 2

• 25% final exam

# Attendance Policy

You are expected to attend every class. I advise you to exchange contact info with classmates so that you can share notes in case you have to miss class.

# Homework Policy

Assigned homework can be found on the course website. Homework will be collected once a week, on Fridays. The work you turn in is expected to be your own.

Homework will be graded 20% on completion, and 80% based upon a selection of problems from the assignment. For each homework assignment I tell you a small subset of the assigned problems that I will choose from for grading. You must still do all the assignment for the 20% completion points!

Homework should be clearly written, so that I can follow the logical steps you take. Each problem should end with the final answer appropriately formulated and circled.

There will be no acceptance of late homework. If you will miss class, make arrangements to either submit your homework early to me or have another student turn it in for you.

# Exam Policy

There will be two midterms, and a cumulative final exam. There will be no make-up exams. If you will be absent for an exam, contact me in advance so that we can schedule an alternate, earlier, time for you to take the exam.

Tentative midterm dates:

• Midterm 1: Friday, September 28.

• Midterm 2: Wednesday, November 21.

Final exams will not be given early.


text:
## anonymous 4 years ago If JK=LM then LM=JK what property is this

1. anonymous

Associative Property, "When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands" i.e $$\ 2\times4=4\times2 .$$

2. anonymous

|dw:1351661030304:dw|

3. anonymous

Okay Thanks!

4. anonymous

Wait! That would be the transitive property of congruency!

5. anonymous

I didn't know that was a congruent sign.

6. anonymous

Okay! thank you and yeah sorry wasnt sure how to do that ha

7. anonymous

So basically, if A is congruent to B, and B is congruent to C, then A is also congruent to C. Does that make sense?

8. anonymous

ehh im not to good at these...I can seem to understand it


text:
# Bull Math

Time Limit: 1000MS

Memory Limit: 65536K

[显示标签]

## Description

Bulls are so much better at math than the cows. They can multiply huge integers together and get perfectly precise answers ... or so they say. Farmer John wonders if their answers are correct. Help him check the bulls' answers. Read in two positive integers (no more than 40 digits each) and compute their product. Output it as a normal number (with no extra leading zeros). FJ asks that you do this yourself; don't use a special library function for the multiplication.

## Input

* Lines 1..2: Each line contains a single decimal number.

## Output

* Line 1: The exact product of the two input lines

## Sample Input

11111111111111
1111111111

## Sample Output

12345679011110987654321

N/A