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# Trying to draw the tautological line bundle ($\subseteq \mathbb{CP}^1\times \mathbb{C}^2$)

In order to learn about vector bundles, I would like to draw the tautological vector bundle over the complex projective line

$$E = \{(x,v) \in \mathbb{CP}^1 \times \mathbb{C}^2 : v \in x \} .$$

Identifying the complex projective line with the Riemann sphere, $\mathbb{CP}^1 \cong S^2$, I hope that it might be possible to visualize this bundle by attaching small planes to each point of the sphere, similar to how one can visualize the tangent bundle of sphere.

In other words, I'm looking for an embedding $E \hookrightarrow S^2 \times \mathbb{R}^3$ into a trivial bundle. (Obviously, $E$ has to be viewed as a 2-dimensional real vector bundle for this to make sense.) I am aware that such a thing might not exist, in which case I would like to learn why.

I claim that there is no bundle embedding of the realification of the tautological bundle $\mathcal{O}(-1)$ into the trivial (real) bundle $S^2 \times \mathbb{R}^3$. Suppose there were; then we could take the orthogonal complement $N$, and we would obtain a decomposition $$N \oplus \mathcal{O}(-1)_{\mathbb{R}} = S^2 \times \mathbb{R}^3$$ where $N$ is a real line bundle. But real line bundles on any compact CW complex $X$ are classified by the first Stiefel-Whitney class, which lives in $H^1(X, \mathbb{Z}/2)$ (since the infinite 1-Grassmannian is a $K(\mathbb{Z}/2, 1)$). However, $H^1(S^2, \mathbb{Z}/2)=0$, and so $N$ is trivial.
It follows that if such an embedding existed, then $\mathcal{O}(-1)_{\mathbb{R}}$ would be stably trivial. This is, however, not the case. Stable trivialty would imply that the Stiefel-Whitney classes were trivial, as the product formula for them shows. However, we know that the top Chern class in $H^2(\mathbb{CP}^1, \mathbb{Z})$ generates the group, and also that (Proposition 3.8 in Hatcher's Vector Bundles and K-theory, available here) implies that the top Stiefel-Whitney class is the image of the top Chern class. But the image of a generator in $H^2(\mathbb{CP}^1, \mathbb{Z})$ in $H^2(S^2, \mathbb{Z}/2)$ is nonzero.
I think you can't do much better than to visualize the Hopf fibration $S^{3} \to S^{2}$, which can be defined by restricting the tautological line bundle to $S^{3} \subset \mathbb{C}^{2}$ (in other words, the fiber over each point $x \in \mathbb{CP}^{1}$ is the circle of points of length $1$ in $x$). Most of the pictures are obtained by stereographically projecting $S^{3}\smallsetminus \{N\}$ to $\mathbb{R}^{3}$. Just google for Hopf fibration and you'll get lots of nice pictures.
I do not think this is possible. But you can try to do this over $\mathbb{S}^{2}\times \mathbb{R}^{4}$. The most basic reason for this to be impossible is the 'complex twist' involved in the tautological bundle made its rigid enough that become impossible to put into $\mathbb{S}^{3}$ (which is too tight). I doubt if one need to work with characteristic classes back and forth, this feels killing a mosquito with a big hammer.

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## Posts

The Importance of Unknown Existential Risks 2020-07-23T19:09:56.031Z · score: 60 (23 votes)
Estimating the Philanthropic Discount Rate 2020-07-03T16:58:54.771Z · score: 57 (21 votes)
How Much Leverage Should Altruists Use? 2020-01-07T04:25:31.492Z · score: 63 (20 votes)
How Can Donors Incentivize Good Predictions on Important but Unpopular Topics? 2019-02-03T01:11:09.991Z · score: 27 (13 votes)
Should Global Poverty Donors Give Now or Later? An In-Depth Analysis 2019-01-22T04:45:56.500Z · score: 22 (7 votes)
Why Do Small Donors Give Now, But Large Donors Give Later? 2018-10-28T01:51:56.710Z · score: 11 (5 votes)
Where Some People Donated in 2017 2018-02-11T21:55:09.730Z · score: 18 (18 votes)
Where I Am Donating in 2016 2016-11-01T04:10:02.389Z · score: 17 (23 votes)
Dedicated Donors May Not Want to Sign the Giving What We Can Pledge 2016-10-30T03:26:44.215Z · score: 15 (19 votes)
Altruistic Organizations Should Consider Counterfactuals When Hiring 2016-09-11T04:19:39.164Z · score: 1 (7 votes)
Why the Open Philanthropy Project Should Prioritize Wild Animal Suffering 2016-08-26T02:08:53.190Z · score: 21 (29 votes)
Evaluation Frameworks (or: When Importance / Neglectedness / Tractability Doesn't Apply) 2016-06-10T21:35:50.236Z · score: 8 (8 votes)
A Complete Quantitative Model for Cause Selection 2016-05-18T02:17:28.769Z · score: 20 (24 votes)
Quantifying the Far Future Effects of Interventions 2016-05-18T02:15:07.240Z · score: 8 (8 votes)
GiveWell's Charity Recommendations Require Taking a Controversial Stance on Population Ethics 2016-05-17T01:51:15.218Z · score: 26 (28 votes)
On Priors 2016-04-26T22:35:14.359Z · score: 9 (9 votes)
How Should a Large Donor Prioritize Cause Areas? 2016-04-25T20:46:38.304Z · score: 13 (13 votes)
Expected Value Estimates You Can (Maybe) Take Literally 2016-04-06T15:11:59.359Z · score: 19 (23 votes)
Are GiveWell Top Charities Too Speculative? 2015-12-21T04:05:07.675Z · score: 17 (20 votes)
More on REG's Room for More Funding 2015-11-16T17:31:40.493Z · score: 9 (11 votes)
Cause Selection Blogging Carnival Conclusion 2015-10-05T20:16:43.945Z · score: 7 (7 votes)
Charities I Would Like to See 2015-09-20T15:22:43.083Z · score: -5 (25 votes)
My Cause Selection: Michael Dickens 2015-09-15T23:29:40.701Z · score: 35 (30 votes)
Some Writings on Cause Selection 2015-09-08T21:56:01.033Z · score: 4 (4 votes)
EA Blogging Carnival: My Cause Selection 2015-08-16T01:07:22.005Z · score: 11 (11 votes)
Why Effective Altruists Should Use a Robo-Advisor 2015-08-04T03:37:13.789Z · score: 10 (10 votes)
Stanford EA History and Lessons Learned 2015-07-02T03:36:56.688Z · score: 25 (25 votes)
How We Run Discussions at Stanford EA 2015-04-14T16:36:05.363Z · score: 15 (14 votes)
Meetup : Stanford THINK 2014-10-23T02:10:42.641Z · score: 1 (1 votes)

Comment by michaeldickens on Replaceability Concerns and Possible Responses · 2020-08-04T01:14:02.539Z · score: 11 (4 votes) · EA · GW

I believe this only applies for certain causes, mainly global poverty. If you want to work on existential risk, movement building, or cause prioritization, basically no organizations are working on these except for EA or EA-adjacent orgs. Many non-EA orgs do cause prioritization, but they generally have a much more limited range of what causes they're willing to consider. Animal advocacy is more of a middle ground, I believe EAs make up somewhere between 10% and 50% of all factory farming focused animal advocates.

(This is just my impression, not backed up by any data.)

Comment by michaeldickens on New Top EA Causes for 2020? · 2020-07-29T19:10:27.675Z · score: 8 (4 votes) · EA · GW

I'm a bit late to the party on this one, but I figured out recently that determining the correct discount rate could be the top EA cause.

Comment by michaeldickens on The case for investing to give later · 2020-07-27T22:56:45.892Z · score: 6 (2 votes) · EA · GW

What you basically seem to be calculating is the optimal degree of free riding that you can get away with to maximize the impact of your own dollars.

If other people spend too much now and not enough later, then by investing, you do more good for the world than if you spent now. This maximizes the impact of your own dollars without reducing the impact of anyone else's, so it increases the total well-being of the world. And it's the optimal strategy if your goal is to maximize total well-being.

Comment by michaeldickens on Can High-Yield Investing Be the Most Good You Can Do? · 2020-07-27T18:12:19.700Z · score: 13 (5 votes) · EA · GW

However I have seen surprisingly little engagement from the EA community on this particular topic, possibly due to the only recent publication of Trammell’s 80,000 Hours podcast, his patient philanthropy paper, and the two blog posts by Dickens and Hoeijmakers I reference above. None of those sources directly reference the question of optimizing high-yielding investments

I did previously write about optimizing investments: https://forum.effectivealtruism.org/posts/g4oGNGwAoDwyMAJSB/how-much-leverage-should-altruists-use

The post mainly talks about using leverage, but I do talk about specific investment choices in this section. See also this followup post.

Right now, a large percent of EA money is in Facebook stock (at least according to Forbes). Holding money in a single stock has about 2x the risk of the S&P 500 and ~4x the risk of the global market portfolio, but without any additional expected return. Diversifying this money seems to me to be the most important improvement to the EA investment portfolio, although I don't know how tractable it is. I don't have any personal connections to Cari Tuna or Dustin Moskovitz, so I can't speak to their reasons for continuing to hold most of their net worth in Facebook stock. Based on a quick back-of-the-envelope calculation, moving their money from Facebook to the global market portfolio would be worth an expected >$1 billion per year. In the long run, you cannot earn greater than market returns, because eventually you will have so much money that you'll run out of above-market investing opportunities (and that's assuming you can identify above-market opportunities in the first place). So it's not obvious that changing investment returns affects whether to give now or later unless you're only talking about the next few decades. I do think it's plausible that the best time to give could be a few decades from now, plus it's still good to earn as high an investment return as possible even if you still believe you should spend your budget relatively quickly. Comment by michaeldickens on The Importance of Unknown Existential Risks · 2020-07-24T18:42:27.408Z · score: 6 (3 votes) · EA · GW Related to this, I find anthropic reasoning pretty suspect, and I don't think we have a good enough grasp on how to reason about anthropics to draw any strong conclusions about it. The same could be said about choices of priors, e.g., MacAskill vs. Ord where the answer to "are we living at the most influential time in history?" completely hinges on the choice of prior, but we don't really know the best way to pick a prior. This seems related to anthropic reasoning in that the Doomsday Argument depends on using a certain type of prior distribution over the number of humans who will ever live. My general impression is that we as a society don't know enough about this kind of thing (and I personally know hardly anything about it). However, it's possible that some people have correctly figured out the "philosophy of priors" and that knowledge just hasn't fully propagated yet. Comment by michaeldickens on The Importance of Unknown Existential Risks · 2020-07-24T18:35:32.742Z · score: 4 (2 votes) · EA · GW Thanks for this perspective! I've heard of the Doomsday Argument but I haven't read the literature. My understanding was that the majority belief is that the Doomsday Argument is wrong, we just haven't figured out why it's wrong. I didn't realize there was substantial literature on the problem, so I will need to do some reading! I think it is still accurate to claim that very few sources have considered the probability of unknown risks relative to known risks. I'm mainly basing this off the Rowe & Beard literature review, which is pretty comprehensive AFAIK. Leslie and Bostrom discuss unknown risks, but without addressing their relative probabilities (at least Bostrom doesn't, I don't have access to Leslie's book right now). If you know of any sources that address this that Rowe & Beard didn't cover, I'd be happy to hear about them. Comment by michaeldickens on Movement building and investing to give later · 2020-07-18T01:24:53.126Z · score: 17 (6 votes) · EA · GW I'm glad you wrote this! Movement-building is an important complement to financial investing, and can benefit the future in many of the same ways. Maximizing the number of longtermists at time t may require periods of spending alternated with periods of investment. I believe your model gives this result because of the constraint that you have to either spend or invest all of your salary in each period. If you allow spending greater than 0% or less than 100% of your salary, I believe you will get the result that you maximize the number of longtermists by spending some fixed proportion of your salary in each period. Alternating between periods is a way of approximating this. I added related functionality to your script here: https://github.com/michaeldickens/public-scripts/blob/master/movement-building-model.py Also, there is a bug in the invest function, money += (money + salary) * market_rate should be money = (money + salary) * market_rate. Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-08T22:03:19.533Z · score: 2 (1 votes) · EA · GW That is correct. Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-07T19:34:29.954Z · score: 4 (2 votes) · EA · GW Future utility is not less valuable, but the possibility of extinction means there is a chance that future utility will not actualize, so we should discount the future based on this chance. That's pretty much right. I would add that another reason why complete loss of capital is "special" is because it is possible to recover from any non-complete loss via sufficiently high investing returns. But if you have$0, no matter how good a return you get, you'll still have $0. Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-07T00:01:26.424Z · score: 2 (1 votes) · EA · GW Naively I would have thought that a double chance of getting half your assets expropriated would be approximately as bad as losing all of them. Diminishing marginal utility means these two events are pretty different. According to the standard assumption of constant relative risk aversion, losing all your assets produces -infinity utility. I don't think this is a realistic assumption, but it's required to make the optimal consumption problem have an analytic solution. I've done some rough numeric analysis where the utility function is bounded below at 0 instead of at -infinity, and based on what I've seen, it generally recommends about the same consumption schedule. (I only did a super preliminary analysis, so I'm not confident about this.) Similarly, organizations that avoid value drift will tend to gain power over time relative to those that don't. Perhaps it would be more accurate to say that an organization that avoids value drift and also consumes its resources slowly (more slowly than r - g) will gain resources over time. Comment by michaeldickens on A Practical Guide for Investing Money to Donate Later · 2020-07-06T23:30:40.810Z · score: 3 (2 votes) · EA · GW I think this is good advice for the most part. Allocate your assets as follows: 70% Total U.S. Market Equity Index, 20% Total International Equity Index, 10% U.S. Bond Index. (I would put my 95% confidence interval for these numbers as +/- 20 percentage points each.) Would you be willing to bet your$500 against my $100 that VTI (US total market ETF) outperforms VXUS (total international ETF) over the next 10 years? Or would you hypothetically accept this bet, even if you don't want to in practice? I realize there's a difference between being 95% confident that an investment is a good idea and being 95% confident that it will outperform over a particular time period, so I understand if you wouldn't want to take this bet. (If it's not already obvious, I strongly disagree that investors should overweight US equities relative to the global market portfolio. I wrote more about this here. I think a better default investment allocation is the global market portfolio, e.g., see section 3 here.) Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-06T20:43:50.305Z · score: 4 (2 votes) · EA · GW Do you have a transcript of your EAGx talk? Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-06T19:45:03.014Z · score: 2 (1 votes) · EA · GW My point was that we know humanity is capable of lasting 200,000 years, because it already did that. So on priors, we should expect humanity to last about another 200,000 years. We might update this prior downward based on facts like "we have nukes now" or "we might develop unfriendly AI soon". But if we assume a 0.2% annual probability of extinction, that gives a 1 in 10^174 chance of surviving 200,000 years, which requires an absurdly strong update away from the prior. Can't we just say it is unlikely - it logically must involve extremely low probabilities I find it really implausible that 10^-174 is the true probability that humanity survives 200,000 years. I don't think we are 10^-174 confident about anything ever. Personally, I see it as something like "There's a 5-90% chance that people like Toby Ord are basically right, and thus that 2 is true. I'm not very confident about that, and 1 is also very plausible. But this is enough to make the expected value of existential risk reduction very high (as long as there are tractable reduction strategies which wouldn't be adopted "by default")." The conclusion does not follow, for two reasons. The value of reducing x-risk might actually be lower if x-risk is higher. For an explanation, see the appendix of this paper: https://onlinelibrary.wiley.com/doi/full/10.1111/1758-5899.12318 (I think you need an account to download, but you can also get the paper on sci-hub.) But there are good arguments that decreasing the discount rate is more important than increasing consumption, which is also discussed in that paper. "Long-run" means "discount rate that applies after the short-run". Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-06T19:27:45.167Z · score: 4 (2 votes) · EA · GW Wouldn't one big exception be movement collapse? Yeah, that's basically an extreme form of "ties weaken within the effective altruism community". I agree that this seems like an unpleasantly plausible outcome. It was the GWWC survey. Comment by michaeldickens on Estimating the Philanthropic Discount Rate · 2020-07-06T19:24:30.827Z · score: 2 (1 votes) · EA · GW Thanks for the comments! I will respond to each of your numbered points. The possibility of, say, extinction is a discount on utility, not on money. To see this, we can extend the formula for utility at time t. Suppose there are two possibilities for the future: extinction and non-extinction. The probability that we end up in the non-extinction world is , so the expected utility due to non-extinction is . We could also add to this the utility of extinction world, call it . Then total expected utility is . Then, we can say to get the formula used in my essay. Or we can just say that we should ignore the term because there's nothing we can do to change it (that's assuming is not changeable, which is obviously not true in real life, but it's true in the standard Ramsey model). This wasn't a particularly well-thought out statement, but it was basically on the assumption that we should converge on true beliefs over time. Thanks for the link! If you dig into this a little more, it becomes apparent that the Ramsey model with constant relative risk aversion doesn't really make sense. In theory, people should accept a zero probability of losing all their assets, because that would result in negative infinity utility. But in practice, some small probability is acceptable, and in fact unavoidable. And people don't try to get the probability of bankruptcy as low as possible, either. But according to the theoretical model, asset prices move according to geometric Brownian motion, which means they can never go to 0. Therefore, losing all your assets is a distinct thing from assets having a negative return, and it has to happen due to some special event that's not part of normal asset price changes. I realize this is kind of hand-wavey, but this is a commonly-used model in economics so at least I have good company in my handwaviness. Example: Suppose . Say we have the chance to change this to . We would accept that deal, because decreasing has a bigger effect on utility than decreasing . (You can construct situations where this is false, but it's usually true.) Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-07-06T19:16:09.711Z · score: 1 (2 votes) · EA · GW Regarding implementation details, if a small donor is going to fund things that are different than what other EAs would typically fund—an approach that various EAs have advocated for and one that I personally support—then I think there's a strong argument to not "invest all your altruistic funds into a managed futures fund." I don't really understand what you're saying here. Separately, I think there's a high likelihood that this approach (i.e. 100% in QMHIX or EQCHX) will underperform a balanced portfolio, GAA That's true, I said that in the post. QMHIX/EQCHX might also have a worse ex-ante Sharpe ratio than GAA. The argument for investing in managed futures is that it has positive expected return (not guaranteed) and has basically zero correlation with stocks and bonds. I'm more enthusiastic about speculating in asset classes that have historically experienced good returns (venture capital or even gold) 1. VC is highly correlated with equities, and as an asset class, historically it has performed worse than the S&P 500. Only the top ~quarter of VC firms have outperformed the S&P 500. Investing in, say, Sequoia seems like a good idea, in the same way that investing in the Renaissance Medallion Fund seems like a good idea, but I don't believe any funds like that are open to new investors. If I could invest in the Medallion Fund, I totally would. 2. Gold has only performed well relatively recently. In the long run, there is no reason to expect gold to have a real return above 0% because it doesn't generate any cash flows like stocks and bonds do, and it doesn't gain value over time except via inflation. A quick search found this page, which shows gold averaged only a 1.2% real return from 1915 to 2020. Research Affiliates projects a -3.6% annual real return for gold over the next 10 years (I don't know their methodology for predicting gold prices, but I assume it's good because their stuff is usually good). I believe that Good Ventures' investment data may be available on their Form 990. Yep, I looked into Good Ventures' Form 990 recently. From what I saw, they don't appear to invest meaningful amounts of money, they just receive money from Cari and Dustin more or less as they donate it. According to Forbes, Cari and Dustin have almost all their money in Facebook stock (which strikes me as an extremely bad idea in principle, although presumably it's not easy for them to liquidate their stock and diversify). Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-07-03T17:05:31.984Z · score: 4 (2 votes) · EA · GW I wrote a follow-up to this post on my website, Do Theoretical Models Accurately Predict Optimal Leverage? I haven't posted it to the EA Forum because it's not directly relevant to EA. Comment by michaeldickens on EA Forum feature suggestion thread · 2020-06-20T01:43:39.768Z · score: 11 (6 votes) · EA · GW I also don't like this feature, although we should be aware that this feature is most helpful for new users, and new users are probably under-represented in this thread. Comment by michaeldickens on Are we living at the most influential time in history? · 2020-06-16T04:32:10.910Z · score: 2 (1 votes) · EA · GW A related outside-view argument for the HoH being more likely to occur in earlier centuries: 1. New things must happen more frequently in earlier centuries because over time, we will run out of new things to do. 2. HoH will probably occur due to some significant thing (or things) happening. 3. HoH must coincide with the first occurrence of this thing, because later occurrences of the same thing or similar things cannot be more important. If we accept these premises, this justifies using a diminishing prior like Laplace. Comment by michaeldickens on EA needs a cause prioritization journal · 2020-06-08T18:27:49.193Z · score: 4 (2 votes) · EA · GW The EA forum doesn't support math typesetting like LaTeX FWIW, the EA forum does now support inline LaTeX, and I have used it in a couple of essays I've published here. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-05-22T00:13:22.404Z · score: 3 (2 votes) · EA · GW I don't know much about emerging market bonds so I can't make any confident claims, but I can say how I am thinking out it for my personal portfolio. I considered holding emerging market bonds because the yield spread between them and and developed-market bonds is unusually high. I decided not to hold them because I don't think they provide enough diversification benefit in the tails. Since I invest with leverage, it doesn't necessarily make sense for me to maximally diversify, I only hold assets if I think the benefit overcomes the extra cost of leverage. But I do believe it might make sense to hold emerging bonds for someone with a less leveraged, more diversified portfolio. That said, I would consider them a "risky" asset, not a "safe" asset, and plan accordingly. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-05-17T23:43:30.393Z · score: 8 (2 votes) · EA · GW Yeah I think it probably makes sense not to hold bonds if you're using leverage and you can't get leverage at close to the risk-free rate. I personally don't hold any long-only bonds. But the argument for holding bonds is that they might be negatively correlated with stocks, in which case a negative expected return might still be worth it. Historically they've had close to 0 correlation, not a negative correlation, so I don't find this argument that persuasive. For commodities, their returns are much harder to project than bonds (where you can just look at the yield), so it's hard to say whether they're worth it. I personally don't hold any long-only commodities. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-04-28T04:22:36.119Z · score: 2 (1 votes) · EA · GW This is a totally reasonable objection, and I will try my best to respond. Sorry if this reply is a little disjointed/hard to follow. And as a result the 8x leverage recommendation is insane To be clear, 8x leverage is not a recommendation; it is the result of a particular analysis with many limitations—I tried to cover the important ones in Caveats. In light of these caveats, 8x leverage does not seem reasonable. That said, I disagree that the projected returns are insane. I agree that they look insane, and when I was writing this, I had some tension between trying to sound sane and representing my true beliefs, and I decided that the latter was more important. I don't overly trust backtests, but I trust the process behind VMOT, which is (part of the) reason to believe the cited backtest is reflective of the strategy's long-term performance.[2] VMOT projected returns were based on a 20-year backtest, but you can find similar numbers by looking at much longer data series (e.g., Value and Momentum Everywhere). VMOT backtest gives higher expected returns than generic value/momentum backtests[1], and I believe this is not due to data-mining, but I don't think there's really an efficient way for me to justify this belief other than to say read the books (Quantitative Momentum and Quantitative Value), which explain why the authors believe their particular implementations of momentum and value have (slightly) better expected return. If you assume VMOT will have returns commensurate with a generic value/momentum strategy, you might get a lower expected return than 9%, but note that Research Affiliates' estimates for generic value/momentum are still on par with my assumption for VMOT's return (I think RAFI is about 1-2% too optimistic, and VMOT's improved methodology adds about an extra 1-2% expected return). I admit that I kind of made up the returns for managed futures, but it is worth noting that the paper I cited (with a 100-year backtest, not 20-year) found a historical performance for managed futures far higher than my made-up forward-looking return estimate. Explaining why we should expect these strategies to outperform the market is much harder, and I can't really present a convincing argument in this comment, but the OP linked to a lot of sources that provide their own arguments. This is not directly relevant to the investment strategies I talked about above, but if you use the really simple (and well-supported) expected return model of earnings growth plus dividends plus P/E mean reversion and plug in the current numbers for emerging markets, you get 9-11% real return (Research Affiliates gives 9%, I've seen other sources give 11%). This is not a highly concentrated investment of 50 stocks—it's an entire asset class. So I don't think expecting a 9% return is insane. VMOT is down 20% in the last 3 years. This estimate would expect returns of 27% +- 20% over that period, so you're like 2.4 standard deviations down. This is only 2.4 standard deviations assuming returns follow a normal distribution, which they don't. By the same argument, you could look at the S&P's recent 30% decline over a one-month period (the historical monthly standard deviation is about 5%) and conclude that our 95% confidence interval for the S&P's monthly return is [-40%, -20%]. I am not disputing that VMOT has performed particularly badly in the recent past, and that this sucks if you're investing in it (which I am)—value and trendfollowing have both performed poorly over the past three years, which explains VMOT's underperformance. Value, momentum, and trendfollowing have each had many historical periods of long underperformance (often much longer than three years), which is probably part of the reason why they're all still unpopular. [1] In brief: • Quantitative Value uses EBIT/EV instead of more commonly used value metrics like P/E or P/B, and there are theoretical and empirical reasons to expect EBIT/EV to work better • Quantitative Value screens companies on quality as well as value, I'm not really convinced this helps but I doubt it hurts. The literature on the quality factor is promising but not as robust as on value or momentum • Quantitative Momentum looks for smooth returns as well as strong momentum, and there are theoretical and empirical reasons to expect smooth-return momentum to work better than "sharp" momentum [2] This actually raises some interesting epistemological questions about when backtests (or limited empirical data in general) should be trusted, it's something I need to think about more. (Edited to clean up a bit and add footnote) Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-04-28T03:19:57.313Z · score: 2 (1 votes) · EA · GW My estimates came from the book Global Asset Allocation by Meb Faber. I expect it's less rigorous than the paper you link to, so I suppose we should trust the paper more. The data in /Global Asset Allocation/ came from Global Financial Data. I don't know the details of how GFD's data is constructed, but I've seen it used by quite a few papers and a lot of institutions use it, so I assume it's pretty reliable. Without having read it, I don't expect the Doeswijk paper is unreliable either—they probably get different results because (1) they use different time horizons and (2) they don't include the same countries in their samples (I think Doeswijk includes more). While I don't doubt the specific historical results in the paper, in general I would expect the global market portfolio to outperform every individual asset class (on a risk-adjusted basis) in the long run, although it's expected that some asset classes will outperform GMP for multiple decades in a row. (Faber actually discusses this in Global Asset Allocation!) I agree it's plausible that the paper happened to pick a good time for equities. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-04-19T22:11:25.434Z · score: 4 (2 votes) · EA · GW RE international equities, I wrote about this here to explain why I think most people should underweight US equities. Looking at historical data it seems that international equities have underperformed First, if EMH is true, there is no reason to expect US equities to have a higher Sharpe ratio than international equities. Second, US outperformance is only a recent phenomenon (see this tweet and its replies), and the outperformance is pretty marginal if you look over a long time horizon. Third, the recent US outperformance is the exact reason why Research Affiliates (RAFI) projects worse returns in the future. The US stock market has outpaced its economic growth, so RAFI is counting on valuations to revert to the mean. There's a lot of historical evidence that markets tend to mean revert like this over sufficiently long time horizons (3-5 years or longer). I haven't read any of the primary research on long-term mean reversion, but apparently the original source was Jegadeesh and Titman (1993), and data is available from the Ken French Data Library (see "6 Portfolios Formed on Size and Long-Term Reversal"). (I cited RAFI for expected return projections, but the analysis is pretty easy to replicate. Just use the discounted dividend model and add in an assumption that P/E ratios will partially mean revert.) I don't see where RAFI recommends holding 0% exposure to US equities? Comment by michaeldickens on Long-term investment fund at Founders Pledge · 2020-04-18T23:31:43.606Z · score: 2 (1 votes) · EA · GW I recently wrote about how altruistic investing differs from regular investing, with some ideas about how altruists might want to invest their money. It's just some preliminary ideas and it's targeted at retail investors, but most of it is relevant to your situation. It includes some ideas that I do not commonly see discussed in EA circles or in investing circles. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-04-18T20:46:54.277Z · score: 5 (3 votes) · EA · GW There are a few ways to get something close to the global market portfolio, although they all require making small compromises. Betterment and Wealthfront will give you something like the global market, but each have skews that push them somewhat away from it. The ETF GAA holds the global market portfolio with small tilts toward value and momentum—I happen to think these tilts are a good idea, but it does move it somewhat away from a truly agnostic portfolio. You could also buy a combination of VT, BND, and BNDX, which will give you the total market in stocks and bonds, although this does exclude smaller asset classes like gold. Out of these options, I personally would probably invest in GAA, but which is best depends on which compromises you're most willing to make. (I personally don't invest in the global market portfolio, I just invest in VMOT and EQCHX, as discussed in OP.) RE government bonds: in theory, if you want to increase risk, you should hold equities and bonds in proportion to the global market portfolio and then apply leverage as desired. But this theory assumes you can borrow money at the risk-free rate, which is false. To use leverage, you will probably end up having to pay about 1% on top of short-term interest rates, and given how small the spread is between short- and long-term rates, holding bonds with leverage guarantees you a negative long-run return. For that reason, I personally do not hold any bonds. But note that it's still possible to make positive return with bonds in the short run. For example, bonds have returned >10% over the past few months, so holding stocks+bonds with leverage would have worked out better than just holding stocks. RE your last point on special opportunities: I think that's a really important question, and I don't know how to answer it. I don't know how to reconcile the general observation that almost everybody fails to beat the market with certain special cases, e.g., bitcoin like you mentioned. You invested in bitcoin in 2011; I considered investing around the same time, but didn't, in short because I didn't expect to be able to beat the market. Clearly I should have invested in bitcoin, but I don't know of any general strategy that would have led to me investing in bitcoin but wouldn't have led to me making a bunch of other stupid investing decisions. How do you think it is that you have invested in 2-3 money-making special opportunities? What distinguishes those from other, similar-looking opportunities that failed? Have you made any special investments that didn't pan out? Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-04-18T19:19:35.003Z · score: 2 (1 votes) · EA · GW Private equity is indeed a part of the global market portfolio, so on priors, it makes sense to invest in private equity. I don't know much about it, but my understanding is that the existing private equity ETFs don't do a good job of tracking the market—private equity is, well, private, so you can't really index it. Comment by michaeldickens on EA Forum 2.0 Initial Announcement · 2020-01-28T00:16:44.531Z · score: 2 (1 votes) · EA · GW Can you elaborate on how you turned off karma display? I would love to use your code if you're willing to share it. I strongly dislike posting on the EA Forum because of how the karma system works, and and my experience would be vastly improved if I couldn't see post/comment karma. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-20T18:42:44.338Z · score: 4 (3 votes) · EA · GW Both are correct. The claim made by Lifecycle Investing is not that increasing stock exposure decreases risk in general. The claim is that by increasing stock exposure early in life and decreasing late in life, while keeping net lifetime exposure the same, you decrease risk. The argument for this claim is that you have more dollars when you're older, therefore market swings in later years have a bigger effect on your portfolio than in earlier years. But you can negate this effect by using leverage when you're young, thus effectively increasing how much money you're investing with, and holding more bonds/cash when you're older. Simplified example: suppose you have$100 today and will get another $100 next year. If you invest all your money in stocks during both years, then you are exposing$100 to equity risk this year, but $200 next year. If instead you invest with 1.5:1 leverage this year, and then next year you only invest 75% of your money, that means you're exposing$150 to equity risk during both years. Either way, you're investing $150 per year on average. But in the former scenario, you are taking twice as much risk in year 2, whereas in the latter scenario, you take the same amount of risk both years, which is better. I'm not sure I'm explaining it well, so let me know if that doesn't make sense. Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-11T04:43:15.503Z · score: 2 (1 votes) · EA · GW I was reviewing my notes and I found this paper on managed futures: https://www.aqr.com/Insights/Research/White-Papers/Trend-Following-in-Focus The paper has a section on why they don't think managed futures (a.k.a. trendfollowing) will stop working in the near future. Here's the summary I wrote in my notes (of the relevant section): • Assets invested in trendfollowing peaked in mid-2008 at$210B, and have declined to $120B • All systematic hedge fund strategies have$500B AUM, or 17% of all hedge fund assets
• Futures market has grown since 2008, so trendfollowing as a % of futures markets has decreased by more than half

I don't find this super convincing, it's definitely still conceivable that trendfollowing strategies could basically stop working, but it's evidence that trendfollowing is not over-subscribed.

Comment by michaeldickens on Long-term investment fund at Founders Pledge · 2020-01-11T04:24:22.018Z · score: 10 (5 votes) · EA · GW
1. The risk-free interest is unusually low across the developed world and the US equity market looks expensive, but equity valuations in the non-US developed market are close to historical averages, and equity valuations in emerging markets are below average. So IMO it is reasonable to expect near-historical equity returns by investing outside the US. See https://mebfaber.com/2019/01/25/the-biggest-valuation-spread-in-40-years/
2. The case for giving sort-of-later depends on the current interest rate, but the case for giving much later only depends on the difference between the long-run interest rate and the long-run discount rate. As Phil Trammell argues in the post linked by OP, the long-run interest rate most likely exceeds the philanthropic discount rate, which means we should give later.
Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-09T03:42:13.827Z · score: 4 (3 votes) · EA · GW

I like this idea—a centrally-managed fund would be a lot easier than a bunch of people separately doing their own thing. But it creates a problem where the investors/donors in the fund might have unrealistic expectations about performance and could become really unhappy if the fund underperforms the S&P for several consecutive years—which is bound to happen sometimes if the fund is aiming for low correlation. This would be particularly bad from an optics perspective. So there are pros and cons to this idea.

Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-09T03:30:12.579Z · score: 4 (3 votes) · EA · GW

I'm skeptical that managed futures will continue to do as well as backtesting suggests.

Me too, I did adjust the return estimate way down from the backtest I quoted, but I can see an argument that managed futures will provide zero excess return in the future.

Regarding momentum, see AQR's Fact, Fiction and Momentum Investing—specifically, "Myth No. 4: Momentum Does Not Survive, Or Is Seriously Limited By, Trading Costs."

Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-09T03:19:16.006Z · score: 5 (4 votes) · EA · GW

Can you clarify what exactly your concern is? Which of these best describes your position?

1. Factor premia basically don't exist.
2. Factor premia are much smaller than this essay claims.
3. Factor premia are substantial, but this essay does not do enough to justify that claim, so it's not a good reference.

Initially I thought you were saying #1, but from your reply below, it sounds like you at least believe that factor premia exist, so now I'm not sure.

Edit: Or I guess a fourth option: factor premia exist, but generally should not be promoted (possibly because most readers will run into the same behavioral biases that cause the premia to exist).

Or something else entirely.

Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-09T03:13:07.794Z · score: 3 (2 votes) · EA · GW

I just updated the essay to include some more justification for (1) the claim that it's possible to beat the market and (2) the specific return estimates given. I added the new material to the sections "Improving on conventional wisdom" and "Return expectations", and have reproduced the latter below.

-------

As something of a corroboration, RAFI provides estimates of forward-looking five-year return for various long/short factors. At the time of this writing, it makes the following predictions for its long/short value and momentum factors (net of transaction costs):

• 5.7% for US large-cap value
• 1.1% for US large-cap momentum
• 8.0% for US small-cap value
• 6.4% for US small-cap momentum

(RAFI's projections for foreign developed market factors are similar but generally a bit higher.)

A concentrated long-only portfolio on a particular factor would have approximately the same expected return as the long/short factor plus the broad market (although that's not quite how the math works).

The underlying indexes used by VMOT make some improvements on RAFI's simple factor model (see Quantitative Value and Quantitative Momentum for details[^26]), so it might be reasonable to assume a higher expected return for VMOT. If we then subtract fees, we get something close to the original estimate I gave for VMOT (probably a bit higher[^27]).

RAFI believes the value and momentum premia will work as well in the future as they have in the past, and AQR makes the same claim in some of the papers I linked above. They offer good support for this claim, but in the interest of conservatism, we could justifiably subtract a couple of percentage points from expected return to account for premium degradation.

Note that RAFI's estimates use factor timing—attempting to guess how well factors will perform based on the current market environment, rather than just looking at historical behavior. This practice is not widely accepted; for example, see AQR's Factor Timing is Deceptively Difficult.

Also note that these numbers only give expected mean return. Even if these estimates are accurate, we could still see much higher or lower returns due to market volatility.

Comment by michaeldickens on How Much Leverage Should Altruists Use? · 2020-01-07T16:45:39.890Z · score: 10 (6 votes) · EA · GW

I'm not surprised that this is a sticking point. I think people should be highly skeptical of claims about EMH violations, and I didn't present a lot of evidence, because that would have dramatically increased the length of this essay. I would refer to the sources I linked in the relevant section.

If you assume EMH is broadly true, almost all of the rest of the essay still applies. When I presented some rough estimates for target leverage under full Kelly and half Kelly, I gave estimates for the global market portfolio, as well as for value/momentum/managed futures. The main exception is if EMH is fully true, you might not want to invest in zero-correlation assets because there aren't really any with positive expected real return (AFAIK), but you'd still want to seek out low correlation to the extent that it's possible.

1. The claims on value and momentum are consistent with academic factor literature and estimates from evidence-focused investing firms like Research Affiliates.
2. I didn't make any strong claims about managed futures performance, but the made-up numbers I gave are far lower than the theoretical backtest performance from the AQR paper I cited, as well as actual performance from practitioners such as Richard Dennis.

Investment writings taken as advice damaging especially valuable assets that would otherwise be used for altruism (just the flip side of the value of improvements)

I don't understand what this means, could you explain?

EDIT: I did a little more looking and found these two papers that might be persuasive to value and momentum skeptics:

The papers themselves are fairly short, but they summarize a lot of evidence from other sources.

Comment by michaeldickens on RPTP Is a Strong Reason to Consider Giving Later · 2018-10-06T22:18:39.361Z · score: 7 (4 votes) · EA · GW

I'm not clear on how RPTP fits into a general understanding of financial returns on investment. Clearly your RPTP matters, and if you have a lower RPTP than most people, that makes investing look relatively better for you. But why don't, say, financial advisors ever talk about this? Advisors largely make investment recommendations based on clients' risk tolerance, which is unrelated to RPTP.

Comment by michaeldickens on RPTP Is a Strong Reason to Consider Giving Later · 2018-10-06T17:17:50.164Z · score: 3 (2 votes) · EA · GW

I don't have a direct source on the argument that you said Elie Hassenfeld made, but I do have a quote from Scott Alexander (http://slatestarcodex.com/2013/04/05/investment-and-inefficient-charity/) who went to a live event in which Elie made this argument:

[I]n the 1960s, the most cost-effective charity was childhood vaccinations, but now so many people have donated to this cause that 80% of children are vaccinated and the remainder are unreachable for really good reasons (like they’re in violent tribal areas of Afghanistan or something) and not just because no one wants to pay for them. In the 1960s, iodizing salt might have been the highest-utility intervention, but now most of the low-iodine areas have been identified and corrected. While there is still much to be done, we have run out of interventions quite as easy and cost-effective as those. And one day, God willing, we will end malaria and maybe we will never see a charity as effective as the Against Malaria [Foundation] again.
Comment by michaeldickens on Additional plans for the new EA Forum · 2018-09-11T02:52:34.879Z · score: 5 (5 votes) · EA · GW

Another feature that could help people find old posts is to display a few random old posts on a sidebar. For example, on any of Jeff Kaufman's blog posts, five old posts display on the sidebar. I've found lots of interesting old posts on Jeff's blog via this feature.

Comment by michaeldickens on EA Forum 2.0 Initial Announcement · 2018-07-24T01:24:00.280Z · score: 0 (2 votes) · EA · GW

I think there's another downside there: we should be wary of implementing a system that doesn't have a track record. There are lots of forums that don't have voting, and reddit-style voting has a long track record as well (plus Hacker News-style, which is similar but not quite the same as reddit-style). As you start introducing extra complexity, you don't know what's going to happen. Most possible designs are bad, and most designs we come up with a priori will probably be bad, so my inclination would be to stick close to a system that has a proven track record.

That said, having multiple types of upvotes could look something like Facebook which now has multiple types of likes, and we have at least some idea of what that would look like. So it might be a good idea.

Comment by michaeldickens on EA Forum 2.0 Initial Announcement · 2018-07-23T04:57:43.249Z · score: 7 (11 votes) · EA · GW

I'm concerned with the plans to make voting/karma more significant; I would prefer to make them less significant than the status quo rather than more. Voting allows everyone's biases to influence discussion in bad ways. For example, people's votes tend to favor:

1. things they agree with over things they disagree with, which makes it harder to voice dissenting opinions
2. entertaining content over important but less-entertaining content
3. agreeable content without much substance over niche or disagreeable content with lots of substance
4. posts that raise easy questions and give strong answers over posts that raise hard questions and give weak answers

Sorting the front page by votes, and giving high-karma users more voting power, only does more to incentivize bad habits. I think the current voting system is more suited to something like reddit which is meant for entertainment, so it's reasonable for the most popular posts to appear first. If the idea is to have "all of EA’s top researchers posting and commenting regularly", I don't think votes should be such a strong driver of the UX.

About a year ago I essentially stopped making top-level posts on the EA Forum because the voting system bothers me too much, and the proposed change sounds even worse. Maybe I'm an outlier, but I'd prefer a system that more closely resembled a traditional forum without voting where all posts have equal status. That's probably not optimal and it has its own problems (the most obvious being that low-quality content doesn't get filtered out), but I'd prefer it to the current or proposed system.

Comment by michaeldickens on How to improve EA Funds · 2018-04-19T02:18:37.852Z · score: 0 (2 votes) · EA · GW

Almost all typical assets--bonds, stocks, commodities--are highly liquid, in the sense that if you decide to sell them, you can convert them into cash in a few minutes max. So even a well diversified portfolio can still be liquid. The main exceptions are real estate and private equity, but I see no reason why EA Funds need to hold those.

Comment by michaeldickens on Where Some People Donated in 2017 · 2018-02-15T03:14:23.130Z · score: 1 (1 votes) · EA · GW

I don't know, the link to Zvi's writeup works for me. But here is the URL: https://thezvi.wordpress.com/2017/12/17/i-vouch-for-miri/

Comment by michaeldickens on Four Organizations EAs Should Fully Fund for 2018 · 2017-12-12T17:21:19.882Z · score: 12 (12 votes) · EA · GW

I haven't yet gotten around to writing up where I plan on donating in 2018 (I already maxed out my 2017 donations in February), but I've been thinking along the same lines. Recently I've been leaning toward donating to these smaller, riskier organizations because I see a lot of value in helping new orgs grow and learning what they can accomplish--especially because the established charities that I like best have gotten a lot of funding recently and have room to scale up before they start to hit the limits of their funding.

Comment by michaeldickens on Discussion: Adding New Funds to EA Funds · 2017-06-03T06:46:47.334Z · score: 4 (6 votes) · EA · GW

Now that you mention it, I think this would be a much more interesting way to divide up funds. I have basically no idea whether AI safety or anti-factory farming interventions are more important; but given the choice between a "safe, guaranteed to help" fund and a "moonshot" fund I would definitely donate to the latter over the former. Dividing up by cause area does not accurately separate donation targets along the lines on which I am most confident (not sure if that makes sense). I would much rather donate to a fund run by a person who shares my values and beliefs than a fund for a specific cause area, because I'm likely to change my mind about which cause area is best, and perhaps the fund manager will, too, and that's okay.

Some possible axes:

1. live-improving vs. life-saving (or, similarly, total view vs. person-affecting view)
2. safe bets vs. moonshots
3. suffering-focused vs. "classical"
4. short-term vs. far future

Although having all possible combinations just along these axes would require 16 funds so in practice this won't work exactly as I've described.

Comment by michaeldickens on Discussion: Adding New Funds to EA Funds · 2017-06-03T06:39:32.901Z · score: 2 (4 votes) · EA · GW

RE #1, organizations doing cause prioritization and not EA community building: Copenhagen Consensus Center, Foundational Research Institute, Animal Charity Evaluators, arguably Global Priorities Project, Open Philanthropy Project (which would obviously not be a good place to donate, but still fits the criterion).

RE #2: if the point is to do what Nick wants, it should really be a "Nick Beckstead fund", not an EA Community fund.

Comment by MichaelDickens on [deleted post] 2017-05-30T14:42:42.202Z

Suppose it's 10 years in the future, and we can look back at what ACE and MIRI have been doing for the past 10 years. We now know some new useful information, such as:

• Has ACE produced research that influenced our understanding of effective charities?
• Has MIRI published new research that moved us closer to making AI safe?
• Has ACE moved more money to top animal charities?

But even then, we still don't know nearly as much as we'd like. We don't know if ACE really moved money, or if that money would have been donated to animal charities anyway. Maybe MIRI took funding away from other research avenues that would have been more fruitful. We still have no idea how (dis)valuable the far future will be.

Comment by MichaelDickens on [deleted post] 2017-05-29T23:34:34.906Z

I'm still undecided on the question of whether quantitative models can actually work better than qualitative analysis. (Indeed, how can you even ever know which works better?) But very few people actually use serious quantitative models to make decisions--even if quantitative models ultimately don't work as well as well-organized qualitative analysis, they're still underrepresented--so I'm happy to see more work in this area.

Some suggestions on ways to improve the model:

## Account for missing components

Quantitative models are hard, and it's impossible to construct a model that accounts for everything you care about. I think it's a good idea to consider which parts of reality you expect to matter most for the impact of a particular thing, and try to model those. Whatever your model is missing, try to figure out which parts of that matter most. You might decide that some things are too hard to model, in which case you should consider how those hard-to-model bits will likely affect the outcome and adjust your decision accordingly.

Examples of major things left out:

• 80K model only considers impact in terms of new donations to GWWC based on 80K's own numbers. It would be better to use your own models of the effectiveness of different cause areas and account for how many people 80K moves into/away from these cause areas using your own effectiveness estimates for different causes.
• ACE model only looks at the value from moving money among top charities. My own model includes money moved among top charities plus new money moved to top charities plus the value of new research that ACE funds.

## Sensitivity analysis

The particular ordering you found (80K > MIRI > ACE > StrongMinds) depends heavily on certain input parameters. For example, for your MIRI model, "expected value of the far future" is doing tons of work. It assumes that the far future contains about 10^17 person-years; I don't see any justification given. What if it's actually 10^11? Or 10^50? This hugely changes the outcome. You should do some sensitivity analysis to see which inputs matter the most. If any one input matters too much, break it down into less sensitive inputs.

Comment by michaeldickens on Update on Effective Altruism Funds · 2017-04-28T01:10:04.289Z · score: 1 (1 votes) · EA · GW

Alternatively, you could have global poverty and animal welfare funds that are unmanaged and just direct money to GiveWell/ACE top charities (or maybe have some light management to determine how to split funds among the top charities).


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## What was the greatest difference between

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#### What was the greatest difference between

Sat Dec 02, 2017 5:37 am
What was the greatest difference between the high and low test scores of any particular student in the third grade at PS 11?

(1) The highest test score earned by any student in the third grade at PS 11 was 98.
(2) The lowest test score earned by any student in the third grade at PS 11 was 44.

The OA is E.

I don't understand this DS question. May your experts give me some help? Please. Thanks in advanced.

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Sat Dec 02, 2017 10:04 am
M7MBA wrote:
What was the greatest difference between the high and low test scores of any particular student in the third grade at PS 11?

(1) The highest test score earned by any student in the third grade at PS 11 was 98.
(2) The lowest test score earned by any student in the third grade at PS 11 was 44.

The OA is E.

I don't understand this DS question. May your experts give me some help? Please. Thanks in advanced.
The question is a little sketchy.

From the language I guess what it is asking is "What is the greatest difference between the highest and lowest tests scores scored by any one student?" meaning the highest score of one student and the lowest score of that same student.

Obviously, without any indication that the two scores provided in the statements are the scores of the same student, we have no way of finding the answer to this question.

Not the best question ever ...

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# Residual spectrum of the adjoint on Banach space

This is a homework question and I find myself ending up with the "opposite" of what I am supposed to prove. Below is the question and my attempt:

Suppose that $\lambda \in \mathbb{R}$. Show that if $\lambda \in \sigma_p(T)$ and $\lambda \notin \sigma_p(T^*)$, then $\lambda \in \sigma_r(T^*)$. I assume $T$ is a bounded linear operator on $X$ and maps to $X$, where $X$ is Banach. $\sigma_p(T)$ is the point spectrum of $T$, and $\sigma_r(T)$ is the residual spectrum.

I thought $\lambda \in \sigma_p(T)$ implies $\lambda I - T$ is not injective and $\exists v \neq 0$ such that $\lambda v = Tv$, and $\lambda \notin \sigma_p(T^*)$ implies $\lambda I - T^*$ is injective.

Since $\sigma(T) = \sigma(T^*)$ (reference: Spectrum for a bounded linear operator and its adjoint on a Banach space are same., or Conway, "A Course in Functional Analysis", pg.208 Prop 6.1), then $\lambda$ must be in either $\sigma_c(T^*)$ or $\sigma_r(T^*)$, where $\sigma_c(T^*)$ is the continuous spectrum. However, as $Ker((\lambda I - T)^*) = Ker(\lambda I - T^*) = \{0\}$, and $Ran(\lambda I - T)^{\perp} = Ker(\lambda I - T^*)$, so $Ran(\lambda I - T)$ must be dense in $X$, so $\lambda \in \sigma_c(T^*)$ by definition. However, I'm supposed to show $\lambda \in \sigma_r(T^*)$.

Can anyone point the mistake I made?

Well I don't know how you obtained $\lambda\in\sigma_c(T^*)$, as you showed was that $\operatorname{Ran}(\lambda I-T)$ is dense, not that $\operatorname{Ran}(\lambda I-T^*)$ is dense. In any case, the calculation you want is $$^\perp\operatorname{Ran}(\lambda I-T^*)=\ker(\lambda I-T)\neq\{0\}.$$ Thus $\lambda I-T$ does not have dense range. But by hypothesis, $\lambda\notin\sigma_p(T^*)$, and thus $\lambda\in\sigma_r(T)$.

• I thought that $\lambda I - T^*$ is injective implies $Ker(\lambda I - T^*) = \{0\}$, and $Ran(\lambda I - T)^{\perp} = Ker(\lambda I - T^*)$, so $Ran(\lambda I - T)$ must be dense. And the continuous spectrum is defined as $\sigma_c(A) := \{\lambda \in \mathbb{C}: \lambda I - A \text{ is injective, } Ran(\lambda I - A) \text{ is dense, and } A \text{ is not surjective}\}$ – Yang Apr 23 '18 at 22:15
• Yes, but you showed $\operatorname{Ran}(\lambda I-T)$ is dense, not that $\operatorname{Ran}(\lambda I-T^*)$ is dense. – Aweygan Apr 23 '18 at 22:18
• Oh this is a stupid mistake. Thanks a lot! – Yang Apr 23 '18 at 22:20
• You're welcome. Glad to help! – Aweygan Apr 23 '18 at 22:20
• Something is wrong here. Didn't you have shown that $\lambda-T'$ (not $\lambda-T$) does not have dense range and hence $\lambda\in\sigma_r(T')$ (not $\lambda\in\sigma_r(T)$, which would contradict $\lambda\in\sigma_p(T)$ by the way)? – 0xbadf00d Feb 2 at 5:52


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Saturday, July 18, 2020 | History

2 edition of Tables of indefinite integrals found in the catalog.

Tables of indefinite integrals

G. Petit Bois

# Tables of indefinite integrals

## by G. Petit Bois

Written in English

Edition Notes

Originally published as Tafeln der unbestimmten Integrale. Teubner, 1906.

The Physical Object ID Numbers Statement translated from the German. Pagination 151p.,24cm Number of Pages 151 Open Library OL19455156M

Additional Physical Format: Online version: Bois, G. Petit. Tables of indefinite integrals. New York, Dover Publications [] (OCoLC) Document Type. Tables of indefinite integrals. [G Petit Bois] pages: tables ; 28 cm. Series Title: Dover books on advanced mathematics. Responsibility: by G. Petit Bois. Reviews. User-contributed reviews # Dover books on advanced mathematics.\/span>\n \u00A0\u00A0\u00A0\n schema.

TABLES OF SOME INDEFINITE INTEGRALS OF BESSEL FUNCTIONS OF INTEGER ORDER Integrals of the type Z xJ2 0(x)dx or Z [13] H. Bateman, A. Erdélyi: Tables of Integral Transforms, vol. I, McGraw-Hill Book Company, Inc., New York, Toronto, London, [14] H. Bateman, A. Erdélyi: Tables of Integral Transforms, vol. II, McGraw-Hill Book Company File Size: 2MB. : Tables of Indefinite Integrals () by Petit Bois, G. and a great selection of similar New, Used and Collectible Books available now at great prices.5/5(1).

Calculation of integrals using the linear properties of indefinite integrals and the table of basic integrals is called direct integration. Solved Problems. Click a problem to see the solution. Evaluate the indefinite integral {\int {\left ({3 {x^2} – 6x + 2\cos x} \right)dx} }. Find the indefinite integral \int {\left ({1 + x} \right)\left.   The Table of Integrals, Series, and Products is the major reference source for integrals in the English is designed for use by mathematicians, scientists, and professional engineers who need to solve complex mathematical problems. *Completely reset edition of Gradshteyn and Ryzhik reference book *New entries and sections kept in orginal .

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Petit Bois and a great selection of related books, art and collectibles available now at - Tables of Indefinite Integrals by Petit Bois, G - AbeBooks. Integrals with ax+b and px+q. Integrals with $x^n+a^n$ Integrals with $x^n-a^n$ Integrals with $\cos x$ and/or $\sin x$ Integrals with $\cos x$ and/or $\sin x$ Integrals with cotangent (cot x) Integrals with 1/cos x; Integrals with arccos, arcsin, arctan, arc cot; Integrals with $e^x$ Integrals with $\ln x$ Integrals with.

Title: Integration Tables from Stewart Calculus Textbook 4th Ed. Created Date: 8/22/ PM. INTEGRATION TABLES AND COMPLETING THE SQUARE Use integration tables to find indefinite integrals.

Use reduction formulas to find indefinite integrals. Use completing the square to find indefinite integrals. STUDY TIP A symbolic integration utility consists, in part, of a database of integration tables.

The process of finding the indefinite integral is called integration or integrating f (x). If we need to be specific about the integration variable we will say that we are integrating f (x) with respect to x. Let’s rework the first problem in light of the new terminology.

Table of Integrals∗ Basic Forms Z xndx = 1 n+ 1 xn+1 (1) Z 1 x dx= lnjxj (2) Z udv= uv Z vdu (3) Z 1 ax+ b dx= 1 a lnjax+ bj (4) Integrals of Rational Functions Z 1 (x+ a)2 dx= ln(1 x+ a (5) Z.

The book is valued by users of previous editions of the work both for its comprehensive coverage of integrals and special functions, and also for its accuracy and valuable updates. Since the first edition, published inthe mathematical content of this book has significantly increased due to the addition of new material.

The Table of Integrals, Series, and Products is the major reference source for integrals in the English language. It is essential for mathematicians, scientists, and engineers, who rely on it when identifying and subsequently solving extremely complex by: Table of Integrals, Series, and Products Seventh Edition I.S.

Gradshteyn and I.M. Ryzhik Alan Jeffrey, Editor University of Newcastle upon Tyne, England Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Translated from Russian by Scripta Technica, Inc.

AMSTERDAM •BOSTON HEIDELBERG LONDON NEW YORK •OXFORD PARIS • SAN DIEGOFile Size: KB. © BE Shapiro Page 3 This document may not be reproduced, posted or published without permission.

The copyright holder makes no representation about the accuracy, correctness, or. The constant of integration is an arbitrary constant termed as C. The variable of integration is termed as x.

Properties of Indefinite Integrals. Let us now look into some properties of indefinite integrals. Property 1: Differentiation and integration are inverse processes of each other since: and Where C is any arbitrary constant. The Table of Integrals, Series, and Products is the major reference source for integrals in the English is designed for use by mathematicians, scientists, and professional engineers who need to solve complex mathematical problems.

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Since publication of the first English-language edition init has been thoroughly revised and enlarged on a regular 5/5(3). The Table of Integrals, Series, and Products is the essential reference for integrals in the English language.

Mathematicians, scientists, and engineers, rely on it when identifying and subsequently solving extremely complex problems. Genre/Form: Tables (Data) Tables: Additional Physical Format: Online version: Brychkov, I︠U︡. (I︠U︡riĭ Aleksandrovich). Tables of indefinite integrals. The Order of Presentation of the Formulas The question of the most expedient order in which to give the formulas, in particular, in what division to include particular formulas such as the definite integrals, turned out to be quite complicated.

The thought naturally occurs to set up an order analogous to that of a dictionary.Free indefinite integral calculator - solve indefinite integrals with all the steps.

Type in any integral to get the solution, steps and graph. This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy. Learn more.Table of Integrals, Series, and Products provides information pertinent to the fundamental aspects of integrals, series, and products.

This book provides a comprehensive table of zed into 17 chapters, this book begins with an overview of elementary functions and discusses the power of binomials, the exponential function, the logarithm, the hyperbolic 3/5(2).


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# zbMATH — the first resource for mathematics

Galois representations arising from some compact Shimura varieties. (English) Zbl 1269.11053
The article under review establishes new, very deep and very important existence results for Galois representations attached to automorphic forms, as predicted by the Langlands philosophy. The main theorem is the following:
Let $$F$$ be a CM field and let $$\Pi$$ be a regular algebraic conjugate self-dual cuspidal automorphic representation of $$\mathrm{GL}_m(\mathbb{A}_F)$$ for some integer $$m \geq 2$$. If $$m$$ is even, then assume in addition a condition called ‘slight regularity’ (referred to as ‘Shin regularity’ in later work by other authors). Then for any prime number $$\ell$$ and for any choice of isomorphism $$\iota_\ell: \overline{\mathbb{Q}}_\ell \to \mathbb{C}$$ there exists a continuous semisimple Galois representation $R_{\ell,\iota_\ell}(\Pi): \mathrm{Gal}(\overline{F}/F) \to \mathrm{GL}_m(\overline{\mathbb{Q}}_\ell)$ satisfying a long list of properties, including the following ones: the representation $$\Pi$$ corresponds to $$R_{\ell,\iota_\ell}(\Pi)$$ under the local Langlands correspondence at all finite places of $$F$$ not dividing $$\ell$$, that is, ‘local-global compatibility’ is fulfilled. If $$\lambda$$ is a finite place of $$F$$ above $$\ell$$, then the Galois representation $$R_{\ell,\iota_\ell}(\Pi)$$ is potentially semistable at $$\lambda$$ with distinct and explicit Hodge-Tate weights. If, moreover, $$\Pi$$ is unramified at $$\lambda$$, then $$R_{\ell,\iota_\ell}(\Pi)$$ is crystalline at $$\lambda$$.
In fact, the author obtains the stronger result that the $$R_{\ell,\iota_\ell}(\Pi)$$ form a compatible system of Galois representations. A corollary of the main result is the Ramanujan-Petersson conjecture for $$\Pi$$, that is, $$\Pi$$ is tempered at all finite places of $$F$$. A result similar to the main theorem can be proved, when $$F$$ is a totally real field, by adapting a technique applied by R. Taylor in [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 13, No. 1, 73–119 (2004; Zbl 1074.11030)] (see Remark 7.6 of the present article).
The main theorem of the article under review strengthens previous results by R. E. Kottwitz [Invent. Math. 108, No. 3, 653–665 (1992; Zbl 0765.22011)], L. Clozel [Publ. Math., Inst. Hautes Étud. Sci. 73, 97–145 (1991; Zbl 0739.11020)], M. Harris and R. Taylor [The geometry and cohomology of some simple Shimura varieties. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)], and R. Taylor and T. Yoshida [J. Am. Math. Soc. 20, No. 2, 467–493 (2007; Zbl 1210.11118)], which need the extra assumption that $$\Pi$$ is square integrable at some finite place. However, in that case the irreducibility of $$R_{\ell,\iota_\ell}(\Pi)$$ is known (it is not established in the article under review).
Results similar to the main result were proved by S. Morel [On the cohomology of certain noncompact Shimura varieties. With an appendix by Robert Kottwitz. Princeton, NJ: Princeton University Press (2010; Zbl 1233.11069)] and L. Clozel, M. Harris and J.-P. Labesse [“Construction of automorphic Galois representations. I”, in: Stabilization of the trace formula, Shimura varieties, and arithmetic applications. Volume 1: On the stabilization of the trace formula. Somerville, MA: International Press, 497–523 (2011; Zbl 1255.11027)], however, without establishing local-global compatibility at ramified primes away from $$\ell$$.
G. Chenevier and M. Harris [“Construction of automorphic Galois representations. II”, Camb. Math. J. 1, 53–73 (2013)] managed to remove the ‘slight regularity’ assumption (in the case of even $$m$$) by an $$\ell$$-adic deformation argument, however, at the expense of having local-global compatibility only up to semisimplification at primes not above $$\ell$$.
The strategy of the proof of the main result of the article, which is technically very demanding, is summarized in quite some detail in the introduction. In this review we content ourselves with some generalities: The conditions on $$\Pi$$ imply that it gives rise to an automorphic representation of some unitary group, and $$R_{\ell,\iota_\ell}(\Pi)$$ is hence realized in the $$\ell$$-adic cohomology of certain Shimura varieties of unitary type. The extra condition in the case of even $$m$$ comes from the fact that then the Galois representation lies in the endoscopic part of the cohomology. The computation of the cohomology of the Shimura variety is through the use of certain Igusa varieties and builds on the work of Harris-Taylor cited above and E. Mantovan (e.g. [Fields Institute Communications 60, 61–83 (2011; Zbl 1258.11069)]). The computation of the relevant part of the cohomology of the Igusa variety is at the center of the present article and relies on earlier work of the author [Duke Math. J. 146, No. 3, 509–568 (2009; Zbl 1218.11061)] and [J. Inst. Math. Jussieu 9, No. 4, 847–895 (2010; Zbl 1206.22011)]. Let us finally point out that the fundamental lemma proved by B. C. Ngô enters the proof essentially [Publ. Math., Inst. Hautes Étud. Sci. 111, 1–271 (2010; Zbl 1200.22011)].

##### MSC:
 11F80 Galois representations 11G18 Arithmetic aspects of modular and Shimura varieties 11F03 Modular and automorphic functions 14G35 Modular and Shimura varieties
Full Text:
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# Thread: Confusing range of a Relation.

1. ## Confusing range of a Relation.

Okay. So our assignment is to get the domain, range, intercepts, etc. of the relation (also a function but not expressed as one)
$\displaystyle 5y(x-3)(x-1)=2(5x+3)$. I didn't have a problem at all till I realize that I got $\displaystyle (-\infty,-5]\cup[-1/5,0)\cup(0,+\infty)$
(I EXCLUDED ZERO because when I solved for x, y is in the DENOMINATOR) as the range and have an x-intercept $\displaystyle (-3/5,0)$
which is a contradiction with the range.

Now, it leaves me with a dilemma because I can't include zero in the range because x will be undefined (at least if you use quadratic formula by manipulating the given equation which I think alters its nature I presume), and I can't just simply remove the x-intercept because of the same reason.

Here's my question, do I include zero in the range OR remove the x-intercept OR just leave my answer as it is.

2. ## Re: Confusing range of a Relation.

It seems to me that $\displaystyle y \in \mathbb{R}$ and $\displaystyle x \in (-\infty, 1) \cup ( 1,3) \cup (3,\infty)$ , your x-intercept is correct. Include y=0 in the range as it gives a corresponding x-value.

What about when $\displaystyle x=0$ ?

3. ## Re: Confusing range of a Relation.

Originally Posted by pickslides
It seems to me that $\displaystyle y \in \mathbb{R}$
Huh? Do you mean that the range is all real numbers? I think you must recheck it (or maybe I'm the one who is wrong).

4. ## Re: Confusing range of a Relation.

Hello, Kaloda!

I think you're confusing yourself . . .

Find the domain, range, intercepts, etc. of the relation: .$\displaystyle 5y(x-3)(x-1)\:=\:2(5x+3)$

We have: .$\displaystyle y \;=\;\frac{2}{5}\cdot\frac{5x+3}{(x-3)(x-1)}$

If $\displaystyle y = 0\!:\;x = \text{-}\tfrac{3}{5}\;\hdots\;x\text{-intercept: }\left(\text{-}\tfrac{3}{5},\:0\right)$

If $\displaystyle x = 0\!:\;y = \tfrac{2}{5}\;\hdots\;y\text{-intercept: }\left(0,\:\tfrac{2}{5}\right)$

Horizontal asymptote: .$\displaystyle y = 0$
Vertical asymptotes: .$\displaystyle x = 1,\;x = 3$

Domain: .$\displaystyle x \ne 1,\:x \ne 3$
. . . . . . . $\displaystyle (\text{-}\infty,\,1)\,\cup\,(1,\,3)\,\cup\,(3,\,\infty)$

Range: difficult to determine without using Calculus.

The graph looks something like this:

Code:
                        |
| :         :
| :         :*
|*:         :
| :         : *
| :         :  *
oB:         :     *
A | :         :          *
- -------------------o+-:---------:--------------
*             * | :1        :3
*     *   | :         :
o      | :         :
D      | :    C    :
| :    o    :
| :  *   *  :
| : *     * :
| :         :
| :*        *:
$\displaystyle A\text{ and }B\text{ are the }x\text{- and }y\text{-intercepts, respectively.}$

$\displaystyle C\text{ has coordinates: }\,\left(\tfrac{9}{5},\text{-}5\right)$
$\displaystyle D\text{ has coordinates: }\,\left(\text{-}3,\,\text{-}\tfrac{1}{5}\right)$

Therefore, the range is: .$\displaystyle \left(\text{-}\infty,\,\text{-}5)\,\cup\,\leeft(\text{-}\tfrac{1}{5},\,\infty\right)$