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# 61

(Personal opinion; not a doctor; edited because I accidentally published the draft)

In a few months, the US should stop being careful around COVID. For the vaccinated, most of you should stop being careful now.

We’re in a new era of COVID. We have some amazingly effective vaccines that highly reduce both transmission and severity, so the emergency is effectively over for the nation (after another month or two for everyone to get a chance to take the vaccine, which they certainly all should). Cases will linger, but the point of the lockdowns and precautions of March 2020 was to buy time—to try to get a vaccine, and determine if long-term side effects were crippling—and we’ve done those things.

Now there is debate about whether we should go back to normal—both personally and in government policy. I’ve been dismayed to see headlines like “Planning on opening up? Not so fast…” Even my acquaintances sometimes make comments about whether it’s safe for me to visit a friend, despite that I’m vaccinated. I called up all my friends in March 2020 to tell them to lock down before the government did—but the people now arguing that we or the nation should stay careful are, in my estimation, missing the bigger picture.

Whether or not we take more precautionary measures, the people still susceptible to COVID are going to get it *eventually*. (Both those that didn’t take the vaccine and those that didn’t have a strong enough immune response.) We aren’t going to magically save them by continuing to be careful. We’ll eventually hit herd immunity one way or another—the only question is how much we flatten the curve. We have to flatten it enough to maintain hospital supplies, as before, but I don’t think we should do more.

And for individuals, the people who have gotten vaccinated aren't really going to get any more safe anytime soon. Rates of COVID might go down a little, but it could be years before the remaining un-immunized population hits herd immunity.

Now, there are some benefits to being careful. It won’t be *you* who gave someone COVID. If people get long COVID or die from it, it will be months later than it otherwise would have happened. But the benefits are very small in relative terms if most of the susceptible people will get it anyways, and absurdly small if the careful person is vaccinated.

The benefits are small in relative terms because the costs of continued precautions is extremely high.

# 61

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This approach to tradeoffs makes sense for the USA in 2021.

I just don't want our analysis to lose sight of the fact that facing these tradeoffs is stupid and avoidable, and that almost every country could have done so much better. Avoiding outbreaks is so much cheaper and easier than dealing with them that the choice to do so should have been overdetermined.

• The background risk rate in Australia is roughly zero. We occasionally get "outbreaks" of single-digit cases, lock down one city for a few days to trace it, and then go back to normal.
• It's not even worth wearing masks here.
• Australia is taking a (frustratingly) slow and cautious approach to the vaccine rollout. This will probably cost zero lives (though with a scary right-tail), plausibly saving some from avoided adverse reactions. IMO we should be going way faster on tail-risk and economic grounds, but...

TLDR: it's much better to be careful instead of exponential growth than after it.

I'd just like to point out that while "facing these tradeoffs is stupid and avoidable" (which I agree with), it's much, much more accurate to say instead "facing these tradeoffs is effectively impossible to avoid even though it's stupid and avoidable".  We might not like reality, but it's not going to go away no matter how much we call it stupid and avoidable.

I think it's a valuable post, and agree that as an individual in the USA in 2021 it's worth thinking carefully about these tradeoffs. In Australia though, it's trivial to avoid facing these tradeoffs, because of the different policies we followed through 2020. (I will never claim they were great policies, but they were good enough)

My broader point is that the policy playbook we learn from COVID should be how and why to avoid such situations, not about how to live with for extended periods. Just do the proper lockdown for four-six weeks at the start instead of the end, and it's over! We wouldn't even need vaccines, let alone masks!

I agree that governments should back off from trying to micromanage the lives of adults.

However, I'll personally continue to use high-quality masks in crowded situations indefinitely, because I would rather wear a mask than get any cold or flu. If you went back in time and told last-year-me that there was a single magic garment which I could wear to prevent all plane plagues, con cruds, office coughs, and myriad other droplet-borne ailments, and you'd shown me evidence of its effectiveness as compelling as what I've personally observed by avoiding the respiratory droplets emitted by strangers for the past year, past-me would have jumped at the opportunity just as enthusiastically as present-me does now.

While I'm not against this policy (I'm Chinese American and already sometimes wore surgical masks when sick), I expect your experience of the past year has given you a falsely inflated sense of the efficacy of masks. Yes, they definitely help prevent you from contracting airborne diseases, but a large part of the reduction in infectious disease transmission risk this year was due to the full battery of COVID restrictions, rather than just your own mask use. Like, flu rates were very low[1] because people were very isolated from one another, and so in any given crowded place, you'd be much less likely to encounter anyone who had the flu at all (plus of course, anyone infected would likely be wearing a mask). As things go more back to normal I'd be surprised if just wearing a mask allowed you to avoid getting sick all year, every year. But I guess it depends on the mask and how reliably you wear it?

[1] Flu deaths in the US dropped from ~30,000 in the two previous seasons, to 600 in the most recent season! (source)

I agree that last year was an uncharacteristic sample of day-to-day cold/flu prevention from masks due to the lower prevalence of other peoples' droplets. I failed to mention that my opinion of mask efficacy is also influenced by having spent a lot of time traveling, and observing a correlation between wearing a cloth mask on planes and getting sick after flying less frequently/severely. If good masking under normal germ-laden droplet loads once those return causes me to experience comparable cold and flu levels to what I had without masks, I'll need to revise my opinions, but I think even halving cold and flu frequency, duration, or severity from their prior levels would be worth the hassle.

I consider cold and flu prevention to be as much an accidental side effect of diligent mask use as it is a direct benefit. Side effects of using a mask in the way that I consider "correct" include never touching my mouth or nose after touching shared surfaces, and only eating in settings where I only touch clean/non-shared surfaces between cleaning my hands and touching my food. For instance, without respiratory droplet precautions, I might order a plate of tacos at a sit-down restaurant, get up and wash my hands at the restaurant's bathroom, touch the grimy back of my chair as I sit back down, and then put my hands (now featuring the germs of everyone else who's touched that chair today) straight onto my food. With respiratory droplet precautions, I'd order my tacos to go, take them to a park, wash my hands or use hand sanitizer, and then put my clean hands onto my food to eat it. I think the little details like that which happen to result from droplet precautions probably make just as much difference as the mask itself.

I agree. Since I have been using mask, I have rarely caught cold.

I do not see that adults are being micromanaged by my government.  I see my government doing thier best to reduce deaths and keep the hospitals from being overwhlemed.  Unfortunately their best is not good enough for the thousands who have died in my city and they do need to do more if they hope to prevent further deaths.

Which government are you talking about?

Did it fund studies about whether it makes sense for the average person to use their hands when to readjust masks when it seems it doesn't fit or did it just tell them not to touch their masks when they have them on?

Did it fund studies on all the drugs that the top Coronaexperts find promising? Did it generally give top Coronaexperts from before the pandemic a lot of say in the government response or did it prefer career burocrats to have the power?

Did it pay a price for vaccines that's corresponding to the value of the vaccine per dose if the dose gets delievered early to encourage fast vaccine production?

Did it tell people from the beginning to ventilate and meet outside instead of meeting people inside?

Did it forbid people from vaccinating with established technology before phase 3 trials ended? Did it immediately approve the vaccines.

Did it fund studies of Taffix to see whether giving it to people to reduce spread of COVID-19 make sense?

Did it science-based recommendations about gloves (gloves increase risk of spreading COVID-19 in many situations) or just recommended people to wear them?

Did it provide incentives for people to install an app for contact tracking?

Did it encourage people to install air filters? How much money did it spend to encourage the production of air filters and people installing them?

Did it recommend that people don't press buttons like elevator buttons with their hands and instead with other objects with little surface?

Did it let all people who died from Corona get autopsies like you would have done 40 years ago to understand as much as possible about how COVID-19 kills people (Germany for example didn't do that)?

Did it fund controlled trials to understand long COVID risks (or fund a vaccine trial to also look at long COVID)?

Does it have social distancing policies that incentivise people to meet outside or inside (if the rules are the same for both and it's easier to avoid detection of rule breaking if you are inside you incentivise that people meet inside)?

Did the government at any point say that masks don't work for non-medical personal?

Did it stock enough masks before 2000?

Did it pay a price for masks afterwards that was corresponding to the value provided by masks to aquire them as fast as possible and scale up production?

Does the government fund free fit testing say that people can easily learn how to wear their mask correctly?

Did it fund apps to help people diagnose whether they have COVID-19?

Did it provide people easy way to isolate in a hotel to not infect people they are living with or did they simply outlaw hotels from renting out rooms?

I have no idea of the situation in the US. I would expect the lockdowns to have a tremendous negative effect on mental health. I mean, even ignoring social isolation, lockdowns have a tremendous negative effect on the economy, and the economy has a direct impact on mental health. In France, I've read that there's a visible increase in student depression, and while the suicide rate was normal in 2020, it started to rise in 2021 possibly as a delayed effect.

Border closures: Shouldn’t be a legal mandate or recommended, except in extreme circumstances

Vaccine passports: Probably fair, seems fantastic for public health and, while highly abusable, not at all an abuse in this actual case

You seem to think that since the context that triggered vaccine passports is the right one, then the measure is not an abuse. But what matters is the use that's made of the measure.

Asking additional precautions from people who are not vaccinated when traveling could be understandable, as long as the virus is considered a big enough threat. But outright forbidding travel for people who are not even sick would be, for most western countries, a new precedent of unseen human rights breach.

Seeing the way my country has favored coercive measures to fight the pandemic, regardless of the efficiency and cost, I can confidently say that such a measure as vaccine passports will be abused. At least here.

But hey, as a major contributor on the topic here once said, I'm just a European concerned for "muh freedom".

I think the points about non-obvious costs are very important to consider and nicely evaluated. Nonetheless, I think your conclusions are a bit too strong. It's possible that I'm being too conservative and not giving adequate weight to the obvious and non-obvious costs of continued restrictions.

I think that in light of the risks of long covid from mild infections (small sample size but I haven't seen a larger study contradicting it) and the ongoing risk of vaccine-resistant mutations, saying that young healthy people are protected from overall risk by a factor of 1000x through vaccination is overly optimistic. I'd say that a 100x reduction harm might be a reasonable estimate.  This thread by Ruby which you linked to as well has lots of good analysis of vaccine efficacy for healthy young people. If I am interpreting the note at the top correctly, due to the risks of vaccine resistance, he has adjusted his harm reduction estimate from vaccination downward to 10x-100x reduction, from his initial estimate of 100x-1000x reduction. He also mentions long covid in his caveats towards the bottom. It's not clear to me why you are still saying 1000x in spite of these factors, and I'd be very interested in understanding how and why your assessment differs from Ruby's in this regard.

It's also worth considering the possibility of a tipping point. If we hold off a few months longer on full reopening, we might get enough hesitant people and people under age 16 vaccinated to avoid the rise of a highly vaccine-resistant strain in the US.

Furthermore, there's a big difference between returning mostly to normal and fully to normal. It's possible that the most extreme potential superspreader events (examples: an indoor rock concert with thousands of people pressed shoulder to shoulder and yelling over the music; a large group indoor choir singing event; a crowded bar indoor bar in Manhattan; Burning Man) are much more risky with regard to covid spread than all the other stuff combined. And at the same time these sorts of crowded and touchy-feely mass gatherings are a very important part of the human experience, and disallowing them has significant costs.

I am now more than 2 weeks past my second Pfizer shot. Considering the factors above, I am not planning to return completely to my 2019 way of life right now, but I am moving substantially in that direction while monitoring covid prevalence, vaccination rates, and new variants for the next month or more.

My current lifestyle that I've experimented with for the last several days includes eating indoors at restaurants (but being mindful of how long I spend at indoor restaurants and how often I go); hugging and hanging out indoors unmasked in small groups with some consenting friends and family who are at least one of young/healthy, vaccinated, or have recovered from covid in the past. I'm probably interested in attending large events outdoors such as concerts so long as people are mostly masked and are not crammed shoulder-to-shoulder. And also small indoor parties with an adequate combination of social distancing, ventilation, and vaccine coverage.

But so far I'm avoiding such activities as going to especially crowded restaurants/bars where I would be spending a lot of time within three feet of many unmasked people, riding on a crowded subway at rush hour, attending crowded indoor concerts, attending large indoor conventions, playing Spin the Bottle, or getting anywhere near other people who are sick and who I assign a substantial probability of having covid. I think those types of activities I will continue to wait on as I monitor the prevalence of covid and spread of variants.

I think that for the next few months, we should maintain legal restrictions on very large gatherings such as the potential superspreader type events that I discuss near the top of this post, either forbidding these sorts of gatherings or requiring most or all attendees to be masked or vaccinated or test negative. Setting guidelines that we will return further to normal once enough people are vaccinated, or allowing vaccinated people different privileges from unvaccinated, may coerce people to get vaccinated, to the benefit society. This will give us a chance at actually hitting something resembling herd immunity through vaccination and preventing a new vaccine-resistant strain from spreading while the overall vaccination rate is lower. I agree that it's problematic to have unenforced regulations. I think the optimal solution to this, both with covid and other unrelated areas of law (though it's much easier said than done) is to reduce the number of regulations and more strenuously enforce the most critical regulations.

I haven't fully grappled yet with the numbers around vaccine resistance, given that I was Fermi-ing something like 2-3x different and not 10x different. But my conclusion is mostly meant to hold up in spite of mistakes there; even if the vaccine was less effective than you thought, what does it get us if we keep social distancing for awhile? Unless we get eradication, it seems like the options are almost binary: eradicate it (~impossible) or go back to normal, unless you have a very good plan for threading the needle in some other way.

I'm pretty fine with a few more months of restrictions on superspreader events until we're more vaccinated, although I still think we have shown through the last year that we're in a not-very-cautious control system and so we shouldn't ban much stuff that will hurt some people and then just cause others to be less cautious.

I see a substantial possibility of near-eradication. Not like we eradicated smallpox, but maybe almost as well as we've done with measles? It's not clear yet what percentage immunity we need need to achieve herd immunity. But given the figures from the past month or so, there's a decent chance we're heading in that direction. Even if we can achieve herd immunity, that may not be a good enough argument to ban superspreader events for now -- we would achieve herd immunity regardless when we get R<1 with respect to normal activity. But there's the further factor that by continuing to reduce transmission while more people are vaccinated, we reduce the chance of variants.

There's also the possibility that we can eventually achieve near-eradication through a combination of immunity and good contact tracing, but we may only have few enough cases to do the contact tracing if we can get the case count down to a manageable level through continued social distancing for a while first.

Do those scenarios seem plausible to you? Why do you believe that the options are almost binary?

I think I see a larger possible downside than you do regarding variants. It seems possible that we could get a variant against which our current vaccines are less than 50% effective, and if we can reduce the chance of that happening through continued restrictions, I think it's worthwhile.

Distributing more vaccines to the rest of the world would be a bigger intervention in that regard, but perhaps through continued travel restrictions combined with some vaccine protection, we can prevent especially bad variants from spreading worldwide the way the original pandemic did.

One thing to look for in evaluating the extent to which we can return to normal: development of antivirals. I'm fairly confident that we'll be seeing some good ones. Add that to existing improvements - the vaccines, rapid home tests, PCR tests that check for covid, flu and RSV (!!!) and we could very well be indistinguishable from normal by fall. Speaking of RSV, Moderna is currently in late phase trials for an RSV vaccine. I'm guessing we'll eventually have an all-in-one vaccine for common respiratory pathogens, using the mRNA platform.

Huge if true! Do you just mean that incentives for antiviral production are now better because of COVID or that you know of specific reasons to think good ones will be produced?

Given the current vaccination rates in the US, and the fact that supply is already beginning to exceed demand, I'd recommend full open in approximately a month.  That gives most of the remaining unvaccinated people time to go get at least their first shot, and should allow us to get below exponential growth nationwide, even though we're likely to have it in sub-populations.  IMO the target should be 'not overloading hospitals'.

After that, let it burn.

From the interviews and things I've seen so far, literally the only way to change a vaccine denier's mind is for them to either 1) get sick, or 2) have a close contact get sick.  At this point, I'm kinda ok with that, if that's what it takes to get people's attention back to ground truth, instead of socially constructed fake reality.

I'm personally trying to push everyone I care about in the vaccine direction, and I would be terribly sad to lose one of my family members when I'm unsuccessful.  However, I also don't want to live in a future world where it's ok to be intentionally ignorant on decisions that affect your life.  If that's what it takes for my family to get a reminder that conspiracy theories are Not OK, that's what it takes.

[+][comment deleted]1mo 1


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Two of my friends believe insurance is a scam, for two different reasons. One is an argument from insurer profit, the other an argument from inequality. They’re sometimes right - and when they’re wrong it’s because of interesting facts about human nature.

I limit the following to for-profit, personal, noncompulsory insurance (as opposed to mutuals, insurance taken out by organisations 0, or that used to compensate people we harm, or that compelled by law). I also assume you’re honest with the insurer (don’t have risk-relevant info they don’t).

### Some jargon

• premium: money paid to the insurer for cover.
• underwriting profit  = premiums - claim amounts - operating costs - taxes.
• investment profit: return to the insurer from using premiums as free capital.

• expected value: the average payout, taking into account how likely it is.
EV = probability of win  x  payout if you win

• utility function: an abstraction about what we like; the shape of our tastes; a mapping from
event -> subjective goodness of event

• self-insurance: paying for losses out of your own savings.
• general insurance: anything but life insurance.
• property-casualty insurance: cover for personal property (cars, homes) and liabilities.

## 1. Argument from surplus profit

1. Insurance firms make profit: their revenue > their costs.
2. Their costs are *at least* the true expected loss plus operating costs.
3. Therefore average premiums are higher than the expected loss.
4. Therefore the average honest policyholder is making an expected loss.

5

This becomes normative if we accuse the consumer of inconsistent preferences:

5. People want more money.
6. People buy insurance, which is on average a loss of money, by (4).
7. Therefore people are inconsistent and should stop (5) or (6).

This has two things wrong with it: one nonfatal, and one fatal but slightly arcane.

### Insert ultimately irrelevant clarification

Premise 3 falsely equates "premiums" and "revenue".

Most insurer profit is "investment gains" - the money they make off investing the money you give them - and not the "underwriting profit", the direct flow from policyholders. (Most years have no property-casualty underwriting profit: they pay out more in claims than they get in premium. 24 )

But these capital gains are being diverted from the policyholder - so the same argument applies one level up.

(You might say that someone holding risk should be compensated for doing so - but insurers both set prices and have absolute right of refusal, so nah.)

Another fact which doesn't really change matters is targeted discounts: for various reasons (geographical diversification, or boosting the portfolio size, etc) an insurer might decide to offer people policies below the "risk price". But this isn't generally true, and is never a majority of premiums.

The real objection is that humans are more complicated than that. In particular, premise (5) obscures a fundamental fact about us: our utility is nonlinear, losses can hurt more than equal-sized gains delight, and uncertainty about losses is itself unpleasant.

### Diminishing marginal utility

The world is beautiful because it varies. 1
- Proverb

In short, we value increasing amounts of any particular thing less and less per unit. (Think how unusual it is for you to pay to see the same film in the cinema twice in succession, or any four films on the same day. Or consider the case of biscuits.) If valuing money looked like this:

6 - i.e. if getting £1000 when you’re rich was as welcome as it is when you’re poor, and if you valued bankruptcy as only a little worse than extreme poverty, then insurance wouldn’t make sense. However, people are instead something like this:

with a steep drop as you approach zero. As a result, it can be rational to purchase insurance, for things you can’t afford to replace. 7

Some people think this is foolish, because people “shouldn’t” have log utility in money: more is objectively better.3 But this is illegitimate, because economic rationality is defined relative to a given utility function (and, more practically, because one mostly doesn’t get to choose what one values). 8

This is a classic modelling mistake: to maximise x rather than U(x), to conflate the event with exposure to the event, to treat financial gain as identical with psychological gain.

### Insurance is gambling, and we are good at gambling

(Where by 'we' I mean "mathematicians".)

When reaching 'zero' (bankruptcy or death) is much worse than similar-sized losses above zero, you don't use expected value, but instead the conservative Kelly criterion. Given a few assumptions, this tells you how much you should pay for bets / insurance policies, given your current wealth:

Say your house is worth V = £100,000 and that you have other assets worth W = £120,000. Say also that you know the annual probability of a house of your vintage in your area burning down, p = 1/10000 or something. Then you buy insurance if the cost of it, C, beats $$\log(W - C) > p \times \log(W - V) + (1 - p) \times \log(W)$$ i.e. £22 is the most you'd pay per year, by one rational measure.

(Don't take this too literally. It is tricky to use the criterion properly: this post explains all the dubious assumptions involved, including that (in this case) it values losing your house as infinitely bad. But it's a good way of bounding things.)

That’s enough to kill the argument, but actually there’s more:

### Risk aversion: uncertainty hurts

Some people are willing to lose a bit to “buy peace of mind”, i.e. they prefer a fixed cost x to a random cost with the same expected value, E(θ) = x. That’s weak risk aversion, and people often act under a stronger version, paying a bit more to take a fixed cost, reducing their uncertainty.

(This is the origin of “premium”: the risk premium is the extra someone is willing to pay to mitigate a risk, over the expected value. The natural usage of “premium” is something else though…)

(NB: In utility theory this is the same phenomenon as DMU, in the sense of being implied by the same shape of utility function. But it’s psychologically distinct from the love of novelty in consumption, and I’m trying to stay close to psychological facts for these reasons.)

### Loss aversion: bad is worse than good is good

People often prefer to avoid losing £x more than they prefer to gain £x. This maybe explains why they do things like take out loans and insurance, and hold on to possessions despite not using them, even when these have a (small) negative expected value. This is to “smooth” their budgets and prevent the dismay of cuts.

Ignore the numbers, though.

(It doesn’t matter to us whether this is a distinct feature of human value, or merely a case of psychological inertia, where you want things to continue as they are.)

### How literally can we take this ‘function’ stuff?

Strictly speaking, the above is at best a useful fiction, because we don’t have “a” utility function: humans aren’t consistent enough to be described by a single payoff curve, or any specifiable set of curves - what we value depends on what mood we’re in, which depends on a host of chemical and accidental factors; and we often don’t know what we like (or more: don’t even have a preference) until we are made to choose things. So we violate the conditions of utility theory. Utility theory is a nice neat mathematical object.10 Humans are mostly buzzing blooming confusions. 9

Functions are an analogy for the (fairly solid) psychological regularities involved, not to make big claims about human simplicity. One useful part of the ‘function’ analogy is normative: utility theory reminds us that preferences should be commensurate, if you don’t want to lose systematically.

## 2. Argument from regressive burden

So, insurance only makes sense for things you can’t afford to replace (unless you have asymmetric info, unusual preferences, subsidy, regulation…). But this means that the poorer you are, the more insurance you can rationally use! Insurance can be seen as the rich selling the poor a bit of resilience.

If everyone had a chunk of savings ($10k?), rationally speaking there would be no petty insurance - for things like household contents, warranties, flights, luggage. And this would reduce deadweight (wasted economic activity). “There could be a better world than our present one,” this says. This is true and good to remember, but not helpful: a world in which the poor didn’t insure themselves against things they can’t afford to replace is worse than our present one. ## Misc notes ### Insurance and self-insurance Argument 1, where it applies, implies that one should save the money you don't spend on insurance, until you have enough to "self-insure", pay out of pocket for all relevant uninsured losses. People generally don't, though, and so they are still being economically irrational when they go without insurance. (In particular, they don't get the investment profit that the insurer would have gotten, and, again, insurers often make an underwriting loss.) ### Solve for the equilibrium Motor liability insurance is mandatory in the UK, and in many places. One of my colleages works solely on motor, which he says makes him a de facto government employee - mostly not joking. ### Ben Orlin on the grim future of perfect knowledge When we looked at the coins, we reduced our uncertainty, and without uncertainty, insurance collapses. If we know in advance who will suffer — whose boat will sink, whose employees will win the lottery, who will suffer that NFL-dream-crushing injury — then insurance becomes impossible. You can only pool risk among people who share it. In our medical system, this problem is unfolding year by year. Genetic testing and improved statistics threaten the basic logic of insurance. I see no easy solution. If you individualize the rates, then some folks pay pennies while others face premiums almost as steep as the medical bills themselves. But charging everyone the same amount turns a project of mutual benefit, with each of us hedging against risk, into a project of collective charity, with some folks subsidizing others. That’s a harder sell. It’s one reason why American healthcare remains so contentious. As a teacher, I’m inclined to think of all knowledge as a gift. But insurance complicates that story. Our ignorance of fate can force us to collaborate against it. We have built a democracy from our uncertainty—and new knowledge threatens that balance, as sure as flood or fire. Here. ### Comments Post a comment (with Markdown): Enable submit button 1. Most large endeavours couldn't exist without heavy insurance - the financial risk would scare everyone off otherwise. And being able to compensate people we harm, without ruining ourselves forever, is a clear win. 2. Il mondo è bello perché è vario. 3. US market: Not a single underwriting profit was recorded in the 25 years from 1979 through 2003. And even though that streak ended in 2004, underwriting profits are anything but the norm they were in the 1930s, 1940s, and 1950s. In the eight years from 2004 through 2011, only three tallied underwriting profits. 4. For a tiny number of people - those who give a lot to charity - this is justified, because they transfer money to help those at the sharper end of their own utility functions. 5. and the similar UK market: EY expects the industry to post a combined ratio of almost 98 per cent this year. The industry has only posted a ratio below 100 per cent — suggesting an underwriting profit — in five of the past 33 years. 6. Actuaries aren't perfect modellers by a long shot, but on average they do ok, enough to unlock the net investment profit anyway. So premise (3) is true. 7. One reason Wodehouse novels are so enchanting is that they don't get boring despite being really formulaic. I've read 20 and plan to read the other 75. 8. Strong claims about the relationship between money and wellbeing are often made, though they currently contradict each other. One large study (n=1m) found "income satiation" (no strong evidence for increased happiness) at about$95k.

But we don't need any parameter in this post. (It just amazes me that we are only just beginning to study this incredibly important subject, 200 years into the age of mature science, 100 into the age of premature social science.)
9. There could be people whose utility is linear in money - made happiest by simply following naive game theory and being risk-insensitive. But my friend is not one of them (you can tell because he isn't a crack-smuggling quant who hacks crypto exchanges in his spare time), so the argument as stated, as a general norm, fails.
10. You get further if you think of a person as a bundle of partially conflicting agents. (For a flawed prototype, consider Freud's plagiarism of Plato.)
11. Not that it's a complete solution to even rational preferences.

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Question:"Solve the equation: 4x^2 + 12x + 3 = 0

2. Originally Posted by projecterre
Question:"Solve the equation: 4x^2 + 12x + 3 = 0
you will need to apply the quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

where $a=4$, $b=12$, and $c=3$.

Does this help?

--Chris


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# Question #32052

Feb 11, 2017

#### Answer:

Domain: $\left(- \infty , 0\right) \cup \left(0 , \infty\right)$

#### Explanation:

You can easily see the domain (acceptable x-values) from the graph.
This graph is called a volcano. Even though it looks like it, the function never touches the y-axis $\left(x = 0\right)$ which is a vertical asymptote and it never touches the $y = - 25$ line since it is a horizontal asymptote.
graph{1/x^2 - 25 [-40.72, 40.4, -30.46, 10.14]}.

By default, the domain of a function is typically $\left(- \infty , \infty\right)$. When the function is a fraction with a variable in the denominator, you introduce vertical asymptotes -- locations where the function becomes undefined. Square root functions and trigonometric functions also change the domain.

In your example $x = 0$ causes the undefined condition -- the vertical asymptote.


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# Synthesis of Acetylsalicylic Acid

3414 words (14 pages) Essay

8th Feb 2020 Chemistry Reference this

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1. Introduction

The goal of this experiment is to perform a Fischer esterification reaction with salicylic acid and acetic anhydride to synthesize acetylsalicylic acid.1 The carboxyl group on the acetic acid reacts with the hydroxyl group on the salicylic acid to make an ester product of acetylsalicylic acid.2 This esterification reaction is in the presence of a phosphoric acid catalyst which speeds up the rate of the reaction without interfering with the end product.1 Acetylsalicylic acid also goes by the common name Aspirin which is a drug used for pain relief, fever, and reduction of inflammation.3

Three organic chemistry techniques are used in this experiment including thin layer chromatography (TLC), recrystallization, and vacuum filtration. Thin layer chromatography is used to monitor the reaction to ensure the reaction goes to completion.1 TLC works by initially spotting a compound on the plate which will have different affinities for the mobile and stationary phases affecting the speed at which it migrates up the plate.4 Thus, spots will appear above the initial spot. The distance the substance traveled and the distance of the solvent front will be measured to calculate the retardation/retention value (Rf).1 The number of spots and retardation/retention factor will be dependent upon the compound and can help to identify it.4 For this experiment, TLC is only used to monitor the reaction.1 Once the initial reactant dot disappears and a new one forms that is the point at which all reactants have been converted to products.1 Upon completion of the reaction, the next step was to recrystallize the product. Recrystallization utilizes an appropriate solvent to remove impurities from a solution and allows for a solid organic compound to form and grow into a crystal lattice.1 In this experiment, water was the solvent added to completely rid the solution of acetic anhydride reactant thus, forming acetylsalicylic solid product within the solution.1 The vacuum filtration technique was then used to separate and dry the crystals.1 The crystals left in the funnel of the vacuum filter was the acetylsalicylic acid product.1

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Once the experiment had concluded, different techniques were used to test the purity of the final product. The analytical techniques used were an infrared spectrum and melting point.1 Infrared spectrum was used to compare the experimental final product to the standard product. If the peaks representing different functional groups match closely then, the intended product was most likely made correctly.1 If it contained additional functional groups, then some of the reactants remained in the product and the reaction did not go to completion to obtain acetylsalicylic acid.1 The melting point was also determined to compare the product and its purity to the standard.1 A Mel-temp apparatus was used to obtain the melting point range.1 The purified product should have a higher melting point than the crude product and the temperature range should be narrow.1 If the melting point temperature is low and the range is broad, that could indicate that the product has impurities.1

1. Procedure

Once a large number of crystals were visible, a vacuum filtration technique was utilized to separate and dry the crystals from the solution. Once dry, the solid product in the funnel was weighed to determine the percent yield of the product. Next, analysis of the experimental product included obtaining an infrared spectrum and determining the melting point range and then, comparing it to the standard product.

1. Results and Data

A)    Limiting Reactant

Salicylic Acid Calculation:

Formula: (Mass of reactant) x (Molecular weight of reactant) X

2.0 g Salicylic Acid. x

x

= 0.0145 moles of Acetylsalicylic Acid

Acetic Anhydride Calculation:

Formula: (Volume of reactant) x (Density of reactant) x (Molecular weight of reactant) X

5 ml Acetic Anhydride x

x

x

= 0.0529 moles of Acetylsalicylic Acid

Limiting Reactant: Salicylic Acid

The limiting reactant was salicylic acid because it had a smaller yield of acetylsalicylic acid product. Therefore, the reaction will stop yielding product once the 2 grams of salicylic acid reactant are gone.

B)    Theoretical Yield

Formula: (Product yield from limiting reactant) x (Molar mass of product)

0.0145 moles of Acetylsalicylic Acid x

= 2.609 g of Acetylsalicylic Acid

C)    Percent Yield

Percent Yield =

x 100

x 100 = 46.2% yield

D)    Thin Layer Chromatography Results (TLC)

RF Calculation: 0

Retention/Retardation Factor (RF) = $\le 1$

TLC Plate 1: Salicylic Acid Reaction Mixture & Mixture after 5 minutes of Heating

 Initial Dot Compound and Number Dot Distance Substance Traveled (cm) Solvent Front (cm) Rf Value 1)     Salicylic Acid Mixture 1A 4.7 cm 7.6 cm 0.618 1)     Salicylic Acid Mixture 1B 5.7 cm 7.6 cm 0.750 2)     5 Minutes After Reacting 2A 4.6 cm 7.6 cm 0.605 2)     5 Minutes After Reacting 2B 5.6 cm 7.6 cm 0.737

TLC Plate 2: Salicylic Acid Reaction Mixture & Mixture after 10 minutes of Heating

 Initial Dot Compound and Number Dot Distance Substance Traveled (cm) Solvent Front (cm) Rf Value 1)     Salicylic Acid Mixture 1A 4.0 cm 7.5 cm 0.533 1)     Salicylic Acid Mixture 1B 5.3 cm 7.5 cm 0.707 2)     10 Minutes After Reacting 2A 4.4 cm 7.5 cm 0.587 2)     10 Minutes After Reacting 2B 5.3 cm 7.5 cm 0.707

E)      Infrared Spectroscopy (IR) Analysis

Literature Values

 Wavenumber (cm-1) Peak Description Functional Group 2983 cm-1 Weak, Broad O-H stretch 2872 cm-1 Weak, Broad C-H stretch 1754 cm-1 Strong, Sharp, Narrow C=O stretch 1135 cm-1 Weak, Sharp C-O stretch 918 cm-1 Strong, Sharp =C-H bending

Experimental Values

 Wavenumber (cm-1) Peak Description Functional Group 2980 cm-1 Very Weak, Broad O-H stretch 2870 cm-1 Very Weak, Broad C-H stretch 1749 cm-1 Medium, Sharp, Narrow C=O stretch 1134 cm-1 Medium, Sharp C-O stretch 914 cm-1 Strong, Sharp, Narrow =C-H bending

The previous two tables comparing the literature IR spectrum peaks to the experimental IR spectrum peaks indicate that the peaks were very similar. In comparing the wavenumbers of the peaks, they are nearly identical. The only major differences are the intensity and the width of the peaks. However, these slight differences did not impact the identity of the functional groups. All the functional groups identified by the experimental and literature IR spectrum align with the acetylsalicylic molecule.

F)     Melting Point Range

Temperature at first observable drop: 88.4 degrees Celsius

Temperature when completely liquid: 96.4 degrees Celsius

 Experimental Melting Point Range Literature Melting Point of Acetylsalicylic Acid 88.4 -96.4 degrees Celsius 135 degrees Celsius

The melting point of the product obtained in the experiment varies by a significant amount to the standard melting point. It is much lower than what was expected. This indicates that there may have been impurities in the product.

1. Discussion

It is important to note that according to the experimental procedure the initial reactants from the prepared Erlenmeyer flask were supposed to be taken and initially spotted on the TLC plates before heating. However, due to miscommunication, the reactants used to spot the TLC plates were taken after heating had started. By the time the mistake was realized, the reaction had already been heated for about 2 or 3 minutes, in which, the solution was then taken to dot the plates from the Erlenmeyer flask. So, the reaction could have already started resulting in faulty data.

In analyzing the results, there was an inconsistency in the TLC reading. In both the 5-minute test plate and the 10-minute test plate the reactant had two dots and the product had two dots in the same position as seen by the retardation/retention factor values. This indicates that the initial reactants were the same as the product in both 5 minutes and 10 minutes after reacting. This result could have been due to errors when carrying out the procedure in the lab as mentioned before. However, this error indicates that this reaction underwent rapidly to form the product in only a couple of minutes.

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The melting point of the experimental acetylsalicylic acid was very different from the standard acetylsalicylic acid. The experimental melting point was much lower than the standard and also had a broad temperature range. This could be due to impurities in the product or human error when monitoring and recording the melting point range. The infrared spectrum reading of the experimental product was similar to the standard infrared spectrum for acetylsalicylic acid. There were differences in peak intensity and width, but overall, the functional groups remained the same.

Due to inconsistencies in the data and errors occurring while conducting the experiment, the final product was not successful. Even though the experimental IR spectrum was similar to the literature IR spectrum, the melting point range differed by over 30 degrees. It is most likely that impurities remained in the product during the recrystallization process resulting in the unsatisfactory data. These impurities could also explain the differences in peak intensity and width in the infrared spectrum. In conclusion, the acetylsalicylic acid product was made in the experiment but, it was not pure.

References

1. Weldegirma, S,; Experimental Organic Chemistry: Laboratory manual, 1st Edition;

Department of Chemistry at University of Central Florida: Florida, 2018

doi:10.18411/a-2017-023

1. Introduction

The goal of this experiment is to perform a Fischer esterification reaction with salicylic acid and acetic anhydride to synthesize acetylsalicylic acid.1 The carboxyl group on the acetic acid reacts with the hydroxyl group on the salicylic acid to make an ester product of acetylsalicylic acid.2 This esterification reaction is in the presence of a phosphoric acid catalyst which speeds up the rate of the reaction without interfering with the end product.1 Acetylsalicylic acid also goes by the common name Aspirin which is a drug used for pain relief, fever, and reduction of inflammation.3

Three organic chemistry techniques are used in this experiment including thin layer chromatography (TLC), recrystallization, and vacuum filtration. Thin layer chromatography is used to monitor the reaction to ensure the reaction goes to completion.1 TLC works by initially spotting a compound on the plate which will have different affinities for the mobile and stationary phases affecting the speed at which it migrates up the plate.4 Thus, spots will appear above the initial spot. The distance the substance traveled and the distance of the solvent front will be measured to calculate the retardation/retention value (Rf).1 The number of spots and retardation/retention factor will be dependent upon the compound and can help to identify it.4 For this experiment, TLC is only used to monitor the reaction.1 Once the initial reactant dot disappears and a new one forms that is the point at which all reactants have been converted to products.1 Upon completion of the reaction, the next step was to recrystallize the product. Recrystallization utilizes an appropriate solvent to remove impurities from a solution and allows for a solid organic compound to form and grow into a crystal lattice.1 In this experiment, water was the solvent added to completely rid the solution of acetic anhydride reactant thus, forming acetylsalicylic solid product within the solution.1 The vacuum filtration technique was then used to separate and dry the crystals.1 The crystals left in the funnel of the vacuum filter was the acetylsalicylic acid product.1

Once the experiment had concluded, different techniques were used to test the purity of the final product. The analytical techniques used were an infrared spectrum and melting point.1 Infrared spectrum was used to compare the experimental final product to the standard product. If the peaks representing different functional groups match closely then, the intended product was most likely made correctly.1 If it contained additional functional groups, then some of the reactants remained in the product and the reaction did not go to completion to obtain acetylsalicylic acid.1 The melting point was also determined to compare the product and its purity to the standard.1 A Mel-temp apparatus was used to obtain the melting point range.1 The purified product should have a higher melting point than the crude product and the temperature range should be narrow.1 If the melting point temperature is low and the range is broad, that could indicate that the product has impurities.1

1. Procedure

Once a large number of crystals were visible, a vacuum filtration technique was utilized to separate and dry the crystals from the solution. Once dry, the solid product in the funnel was weighed to determine the percent yield of the product. Next, analysis of the experimental product included obtaining an infrared spectrum and determining the melting point range and then, comparing it to the standard product.

1. Results and Data

A)    Limiting Reactant

Salicylic Acid Calculation:

Formula: (Mass of reactant) x (Molecular weight of reactant) X

2.0 g Salicylic Acid. x

x

= 0.0145 moles of Acetylsalicylic Acid

Acetic Anhydride Calculation:

Formula: (Volume of reactant) x (Density of reactant) x (Molecular weight of reactant) X

5 ml Acetic Anhydride x

x

x

= 0.0529 moles of Acetylsalicylic Acid

Limiting Reactant: Salicylic Acid

The limiting reactant was salicylic acid because it had a smaller yield of acetylsalicylic acid product. Therefore, the reaction will stop yielding product once the 2 grams of salicylic acid reactant are gone.

B)    Theoretical Yield

Formula: (Product yield from limiting reactant) x (Molar mass of product)

0.0145 moles of Acetylsalicylic Acid x

= 2.609 g of Acetylsalicylic Acid

C)    Percent Yield

Percent Yield =

x 100

x 100 = 46.2% yield

D)    Thin Layer Chromatography Results (TLC)

RF Calculation: 0

Retention/Retardation Factor (RF) =

$\le 1$

TLC Plate 1: Salicylic Acid Reaction Mixture & Mixture after 5 minutes of Heating

 Initial Dot Compound and Number Dot Distance Substance Traveled (cm) Solvent Front (cm) Rf Value 1)     Salicylic Acid Mixture 1A 4.7 cm 7.6 cm 0.618 1)     Salicylic Acid Mixture 1B 5.7 cm 7.6 cm 0.750 2)     5 Minutes After Reacting 2A 4.6 cm 7.6 cm 0.605 2)     5 Minutes After Reacting 2B 5.6 cm 7.6 cm 0.737

TLC Plate 2: Salicylic Acid Reaction Mixture & Mixture after 10 minutes of Heating

 Initial Dot Compound and Number Dot Distance Substance Traveled (cm) Solvent Front (cm) Rf Value 1)     Salicylic Acid Mixture 1A 4.0 cm 7.5 cm 0.533 1)     Salicylic Acid Mixture 1B 5.3 cm 7.5 cm 0.707 2)     10 Minutes After Reacting 2A 4.4 cm 7.5 cm 0.587 2)     10 Minutes After Reacting 2B 5.3 cm 7.5 cm 0.707

E)      Infrared Spectroscopy (IR) Analysis

Literature Values

 Wavenumber (cm-1) Peak Description Functional Group 2983 cm-1 Weak, Broad O-H stretch 2872 cm-1 Weak, Broad C-H stretch 1754 cm-1 Strong, Sharp, Narrow C=O stretch 1135 cm-1 Weak, Sharp C-O stretch 918 cm-1 Strong, Sharp =C-H bending

Experimental Values

 Wavenumber (cm-1) Peak Description Functional Group 2980 cm-1 Very Weak, Broad O-H stretch 2870 cm-1 Very Weak, Broad C-H stretch 1749 cm-1 Medium, Sharp, Narrow C=O stretch 1134 cm-1 Medium, Sharp C-O stretch 914 cm-1 Strong, Sharp, Narrow =C-H bending

The previous two tables comparing the literature IR spectrum peaks to the experimental IR spectrum peaks indicate that the peaks were very similar. In comparing the wavenumbers of the peaks, they are nearly identical. The only major differences are the intensity and the width of the peaks. However, these slight differences did not impact the identity of the functional groups. All the functional groups identified by the experimental and literature IR spectrum align with the acetylsalicylic molecule.

F)     Melting Point Range

Temperature at first observable drop: 88.4 degrees Celsius

Temperature when completely liquid: 96.4 degrees Celsius

 Experimental Melting Point Range Literature Melting Point of Acetylsalicylic Acid 88.4 -96.4 degrees Celsius 135 degrees Celsius

The melting point of the product obtained in the experiment varies by a significant amount to the standard melting point. It is much lower than what was expected. This indicates that there may have been impurities in the product.

1. Discussion

It is important to note that according to the experimental procedure the initial reactants from the prepared Erlenmeyer flask were supposed to be taken and initially spotted on the TLC plates before heating. However, due to miscommunication, the reactants used to spot the TLC plates were taken after heating had started. By the time the mistake was realized, the reaction had already been heated for about 2 or 3 minutes, in which, the solution was then taken to dot the plates from the Erlenmeyer flask. So, the reaction could have already started resulting in faulty data.

In analyzing the results, there was an inconsistency in the TLC reading. In both the 5-minute test plate and the 10-minute test plate the reactant had two dots and the product had two dots in the same position as seen by the retardation/retention factor values. This indicates that the initial reactants were the same as the product in both 5 minutes and 10 minutes after reacting. This result could have been due to errors when carrying out the procedure in the lab as mentioned before. However, this error indicates that this reaction underwent rapidly to form the product in only a couple of minutes.

The melting point of the experimental acetylsalicylic acid was very different from the standard acetylsalicylic acid. The experimental melting point was much lower than the standard and also had a broad temperature range. This could be due to impurities in the product or human error when monitoring and recording the melting point range. The infrared spectrum reading of the experimental product was similar to the standard infrared spectrum for acetylsalicylic acid. There were differences in peak intensity and width, but overall, the functional groups remained the same.

Due to inconsistencies in the data and errors occurring while conducting the experiment, the final product was not successful. Even though the experimental IR spectrum was similar to the literature IR spectrum, the melting point range differed by over 30 degrees. It is most likely that impurities remained in the product during the recrystallization process resulting in the unsatisfactory data. These impurities could also explain the differences in peak intensity and width in the infrared spectrum. In conclusion, the acetylsalicylic acid product was made in the experiment but, it was not pure.

References

1. Weldegirma, S,; Experimental Organic Chemistry: Laboratory manual, 1st Edition;

Department of Chemistry at University of Central Florida: Florida, 2018

doi:10.18411/a-2017-023

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# Dimension¶

Dimension subtypes

class csdmpy.Dimension(*args, **kwargs)[source]

Bases: object

Dimension class.

An instance of this class describes a dimension of a multi-dimensional system. In version 1.0 of the CSD model, there are three subtypes of the Dimension class:

Creating an instance of a dimension object

There are two ways of creating a new instance of a Dimension class.

From a python dictionary containing valid keywords.

>>> from csdmpy import Dimension
>>> dimension_dictionary = {
...     "type": "linear",
...     "description": "test",
...     "increment": "5 G",
...     "count": 10,
...     "coordinates_offset": "10 mT",
...     "origin_offset": "10 T",
... }
>>> x = Dimension(dimension_dictionary)

Here, dimension_dictionary is the python dictionary.

From valid keyword arguments.

>>> x = Dimension(
...     type="linear",
...     description="test",
...     increment="5 G",
...     count=10,
...     coordinates_offset="10 mT",
...     origin_offset="10 T",
... )

Attributes Summary

 type The dimension subtype. description Brief description of the dimension object. application Application metadata dictionary of the dimension object. coordinates Coordinates, $${\bf X}_k$$, along the dimension. coords Alias for the coordinates attribute. absolute_coordinates Absolute coordinates, $$\bf X_k^{\rm{abs}}$$, along the dimension. count Number of coordinates, $$N_k \ge 1$$, along the dimension. increment Increment along a linear dimension. coordinates_offset Offset corresponding to the zero of the indexes array, $$\mathbf{J}_k$$. origin_offset Origin offset, $$o_k$$, along the dimension. complex_fft If true, the coordinates are the ordered as the output of a complex fft. quantity_name Quantity name associated with the physical quantities specifying dimension. label Label associated with the dimension. labels Ordered list of labels along the Labeled dimension. period Period of the dimension. axis_label Formatted string for displaying label along the dimension axis. data_structure JSON serialized string describing the Dimension class instance.

Methods Summary

 to Convert the coordinates along the dimension to the unit, unit. dict Return Dimension object as a python dictionary. to_dict Alias to the dict() method of the class. is_quantitative Return True if the dependent variable is quantitative. copy Return a copy of the Dimension object. reciprocal_coordinates Return reciprocal coordinates assuming Nyquist-Shannon theorem. reciprocal_increment Return reciprocal increment assuming Nyquist-Shannon theorem.

Attributes Documentation

type

The dimension subtype.

There are three valid subtypes of Dimension class. The valid literals are given by the DimObjectSubtype enumeration.

>>> print(x.type)
linear

Returns

A string with a valid dimension subtype.

Raises

AttributeError – When the attribute is modified.

description

Brief description of the dimension object.

The default value is an empty string, ‘’. The attribute may be modified, for example,

>>> print(x.description)
This is a test

>>> x.description = "This is a test dimension."

Returns

A string of UTF-8 allows characters describing the dimension.

Raises

TypeError – When the assigned value is not a string.

application

Application metadata dictionary of the dimension object.

>>> print(x.application)
None

The application attribute is where an application can place its metadata as a python dictionary object using a reverse domain name notation string as the attribute key, for example,

>>> x.application = {"com.example.myApp": {"myApp_key": "myApp_metadata"}}
>>> print(x.application)

Returns

A python dictionary containing dimension application metadata.

coordinates

Coordinates, $${\bf X}_k$$, along the dimension.

Example

>>> print(x.coordinates)
[100. 105. 110. 115. 120. 125. 130. 135. 140. 145.] G

For linear dimensions, the order of the coordinates also depend on the value of the complex_fft attributes. For examples, when the value of the complex_fft attribute is True, the coordinates are

>>> x.complex_fft = True
>>> print(x.coordinates)
[ 75.  80.  85.  90.  95. 100. 105. 110. 115. 120.] G

Returns

A Quantity array of coordinates for quantitative dimensions, i.e. linear and monotonic.

Returns

A Numpy array for labeled dimensions.

Raises

AttributeError – For dimensions with subtype linear.

coords

Alias for the coordinates attribute.

absolute_coordinates

Absolute coordinates, $$\bf X_k^{\rm{abs}}$$, along the dimension.

This attribute is only valid for quantitative dimensions, that is, linear and monotonic dimensions. The absolute coordinates are given as

(9)$\mathbf{X}_k^\mathrm{abs} = \mathbf{X}_k + o_k \mathbf{1}$

where $$\mathbf{X}_k$$ are the coordinates along the dimension and $$o_k$$ is the origin_offset. For example, consider

>>> print(x.origin_offset)
10.0 T
>>> print(x.coordinates[:5])
[100. 105. 110. 115. 120.] G

then the absolute coordinates are

>>> print(x.absolute_coordinates[:5])
[100100. 100105. 100110. 100115. 100120.] G

For linear dimensions, the order of the absolute_coordinates further depend on the value of the complex_fft attributes. For examples, when the value of the complex_fft attribute is True, the absolute coordinates are

>>> x.complex_fft = True
>>> print(x.absolute_coordinates[:5])
[100075. 100080. 100085. 100090. 100095.] G

Returns

A Quantity array of absolute coordinates for quantitative dimensions, i.e linear and monotonic.

Raises

AttributeError – For labeled dimensions.

count

Number of coordinates, $$N_k \ge 1$$, along the dimension.

Example

>>> print(x.count)
10
>>> x.count = 5

Returns

An Integer specifying the number of coordinates along the dimension.

Raises

TypeError – When the assigned value is not an integer.

increment

Increment along a linear dimension.

The attribute is only valid for Dimension instances with the subtype linear. When assigning a value, the dimensionality of the value must be consistent with the dimensionality of other members specifying the dimension.

Example

>>> print(x.increment)
5.0 G
>>> x.increment = "0.1 G"
>>> print(x.coordinates)
[100.  100.1 100.2 100.3 100.4 100.5 100.6 100.7 100.8 100.9] G

Returns

A Quantity instance with the increment along the dimension.

Raises
• AttributeError – For dimension with subtypes other than linear.

• TypeError – When the assigned value is not a string containing a quantity or a Quantity object.

coordinates_offset

Offset corresponding to the zero of the indexes array, $$\mathbf{J}_k$$.

When assigning a value, the dimensionality of the value must be consistent with the dimensionality of the other members specifying the dimension.

Example

>>> print(x.coordinates_offset)
10.0 mT
>>> x.coordinates_offset = "0 T"
>>> print(x.coordinates)
[ 0.  5. 10. 15. 20. 25. 30. 35. 40. 45.] G

The attribute is invalid for labeled dimensions.

Returns

A Quantity instance with the coordinates offset.

Raises
• AttributeError – For labeled dimensions.

• TypeError – When the assigned value is not a string containing a quantity or a Quantity object.

origin_offset

Origin offset, $$o_k$$, along the dimension.

When assigning a value, the dimensionality of the value must be consistent with the dimensionality of other members specifying the dimension.

Example

>>> print(x.origin_offset)
10.0 T
>>> x.origin_offset = "1e5 G"

The origin offset only affect the absolute_coordinates along the dimension. This attribute is invalid for labeled dimensions.

Returns

A Quantity instance with the origin offset.

Raises
• AttributeError – For labeled dimensions.

• TypeError – When the assigned value is not a string containing a quantity or a Quantity object.

complex_fft

If true, the coordinates are the ordered as the output of a complex fft.

This attribute is only valid for the Dimension instances with linear subtype. The value of this attribute is a boolean specifying if the coordinates along the dimension are evaluated as the output of a complex fast Fourier transform (FFT) routine. For example, consider the following Dimension object,

>>> test = Dimension(type="linear", increment="1", count=10)
>>> test.complex_fft
False
>>> print(test.coordinates)
[0. 1. 2. 3. 4. 5. 6. 7. 8. 9.]

>>> test.complex_fft = True
>>> print(test.coordinates)
[-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.]

Returns

A Boolean.

Raises

TypeError – When the assigned value is not a boolean.

quantity_name

Quantity name associated with the physical quantities specifying dimension.

The attribute is invalid for the labeled dimension.

>>> print(x.quantity_name)
magnetic flux density

Returns

A string with the quantity name.

Raises
• AttributeError – For labeled dimensions.

• NotImplementedError – When assigning a value.

label

Label associated with the dimension.

Example

>>> print(x.label)
field strength
>>> x.label = 'magnetic field strength'

Returns

A string containing the label.

Raises

TypeError – When the assigned value is not a string.

labels

Ordered list of labels along the Labeled dimension.

Consider the following labeled dimension,

>>> x2 = Dimension(type="labeled", labels=["Cu", "Ag", "Au"])

then the labels along the labeled dimension are

>>> print(x2.labels)
['Cu' 'Ag' 'Au']

Note

For Labeled dimension, the coordinates attribute is an alias of labels attribute. For example,

>>> np.all(x2.coordinates == x2.labels)
True

In the above example, x2 is an instance of the Dimension class with labeled subtype.

Returns

A Numpy array with labels along the dimension.

Raises

AttributeError – For dimensions with subtype other than labeled.

period

Period of the dimension.

The default value of the period is infinity, i.e., the dimension is non-periodic.

Example

>>> print(x.period)
inf G
>>> x.period = '1 T'

To assign a dimension as non-periodic, one of the following may be used,

>>> x.period = "1/0 T"
>>> x.period = "infinity µT"
>>> x.period = "∞ G"

Attention

The physical quantity of the period must be consistent with other physical quantities specifying the dimension.

Returns

A Quantity instance with the period of the dimension.

Raises
• AttributeError – For labeled dimensions.

• TypeError – When the assigned value is not a string containing a quantity or a Quantity object.

axis_label

Formatted string for displaying label along the dimension axis.

This attribute is not a part of the original core scientific dataset model, however, it is a convenient supplementary attribute that provides a formatted string ready for labeling dimension axes. For quantitative dimensions, this attributes returns a string, label / unit, if the label is a non-empty string, otherwise, quantity_name / unit. Here quantity_name and label are the attributes of the Dimension instances, and unit is the unit associated with the coordinates along the dimension. For examples,

>>> x.label
'field strength'
>>> x.axis_label
'field strength / (G)'

For labeled dimensions, this attribute returns label.

Returns

A formatted string of label.

Raises

AttributeError – When assigned a value.

data_structure

JSON serialized string describing the Dimension class instance.

This supplementary attribute is useful for a quick preview of the dimension object. The attribute cannot be modified.

>>> print(x.data_structure)
{
"type": "linear",
"count": 10,
"increment": "5.0 G",
"coordinates_offset": "10.0 mT",
"origin_offset": "10.0 T",
"quantity_name": "magnetic flux density",
"label": "field strength",
"description": "This is a test",
"reciprocal": {
"quantity_name": "electrical mobility"
}
}

Returns

A json serialized string of the dimension object.

Raises

AttributeError – When modified.

Method Documentation

to(unit='', equivalencies=None)[source]

Convert the coordinates along the dimension to the unit, unit.

This method is a wrapper of the to method from the Quantity class and is only valid for physical dimensions.

Example

>>> print(x.coordinates)
[100. 105. 110. 115. 120. 125. 130. 135. 140. 145.] G
>>> x.to('mT')
>>> print(x.coordinates)
[10.  10.5 11.  11.5 12.  12.5 13.  13.5 14.  14.5] mT

Parameters

unit – A string containing a unit with the same dimensionality as the coordinates along the dimension.

Raises

AttributeError – For labeled dimensions.

dict()[source]

Return Dimension object as a python dictionary.

Example

>>> x.dict()
{'type': 'linear', 'description': 'This is a test', 'count': 10,
'increment': '5.0 G', 'coordinates_offset': '10.0 mT',
'origin_offset': '10.0 T', 'quantity_name': 'magnetic flux density',
'label': 'field strength'}

to_dict()[source]

Alias to the dict() method of the class.

is_quantitative()[source]

Return True if the dependent variable is quantitative.

Example

>>> x.is_quantitative()
True

copy()[source]

Return a copy of the Dimension object.

reciprocal_coordinates()[source]

Return reciprocal coordinates assuming Nyquist-Shannon theorem.

reciprocal_increment()[source]

Return reciprocal increment assuming Nyquist-Shannon theorem.


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# Diffeomorphic Invariance implies Poincare Invariance?

1. Mar 14, 2013

### FedEx

I have been quite puzzled for some time with the concept of Diffeomorphic Invariance.

Here is what I think about it,

1) Diffeomorphic Invariance is the invariance of the theory under general coordinate transformations. For instance the Einstein-Hilbert action is diffeomorphic invariant.

2) Poincare Invariance : The invariance of the equations of physics under rotations and translations.

Should Poincare be thought of as an Active Diffeomorphism? And if it really can be, would it mean that poincare invariance is always implied when a system has diffeomorphsim invariance?

Thanks!

2. Mar 14, 2013

### NanakiXIII

This is very handwaving, but here is how I see it: classically the difference between active and passive coordinate transformations is that for the passive ones you transform the background, and therefore the metric, and then just re-express your system in terms of the new coordinates, while for active transformations you only transform whatever it is you have living on that background.

Often the difference isn't important, because the transformation you want to do is an isometry of your space anyway. For example, in flat Euclidean space, you can translate and rotate passively or actively all you want, it doesn't change your metric. On the other hand, you cannot actively scale things, because if you don't scale your metric along, you end up with a different system. In this case you can only perform your transformation passively.

Now, in GR, you promote your metric to a dynamical field, in a sense making it part of the 'foreground' rather than the background. But then there really isn't much of a distinction between passive and active transformations anymore, since the active ones also take along your metric, always.

3. Mar 14, 2013

### Staff: Mentor

Ok so far.

And boosts--i.e., Lorentz transformations (as distinct from spatial rotations and translations).

I don't think so. It's a diffeomorphism (since it's a subset of general coordinate transformations, which are diffeomorphisms), but not an active one, because it doesn't change the underlying geometry, only the coordinates you use to describe it.

4. Mar 14, 2013

### Staff: Mentor

This isn't my understanding of the standard usage of this terminology. (Although, to be fair, the standard usage of this terminology seems to be somewhat muddled; different references seem to use the terms in different ways.) My understanding is: an active diffeomorphism changes the actual geometry; a passive diffeomorphism only changes the coordinates used to describe the geometry.

For example: if I take a flat Euclidean plane and deform it into a bumpy, wiggly surface, that's an active diffeomorphism; I'm changing the actual geometry of the surface. But if I take a flat Euclidean plane and switch from Cartesian to polar coordinates, that's a passive diffeomorphism: the coordinate transformation from Cartesian to polar has to satisfy all the requirements of a diffeomorphism, but it doesn't change the underlying geometry of the plane.

There still is a distinction between passive and active in GR--at least, there is with the usage of the terms I gave above. For example: if I switch from Schwarzschild coordinates to Painleve coordinates in describing Schwarzschild spacetime, keeping the mass of the black hole constant, that's a passive diffeomorphism; I'm only changing the coordinates, not the underlying geometry. But if I take Schwarzschild spacetime and change the mass of the black hole, that's an active diffeomorphism, because I'm changing the actual geometry.

It's worth noting here as well that one of the requirements for a diffeomorphism is that the topology of the manifold does not change. That actually rules out a lot of conceivable active diffeomorphisms in GR. For example, you can't do an active diffeomorphism from Schwarzschild spacetime to Kerr spacetime, because the topologies of the underlying manifolds are different.

5. Mar 14, 2013

### WannabeNewton

There is no mathematical difference between an "active" and "passive" transformation. It is something you will only find in physics books and math books will not spend time making the distinction. They are both diffeomorphisms and all they do is map you from a smooth manifold $M$ to another smooth manifold $N$ such that the smooth structure of $M$ is preserved. There is no mention of the riemannian structure of the manifolds at all. A diffeomorphism need not preserve the metric tensor which is what I assume you mean by "geometry". If the diffeomorphism happens to be an isometry, THEN we can say the two manifolds have the same riemannian structure. One should not confuse diffeomorphisms with isometries, which are special diffeomorphisms: the former does not make any mention of the riemannian structure of the manifold whereas the latter does by definition.

If we are given a diffeomorphism $\phi :M\rightarrow N$ then we speak of active transformations of tensorial quantites by simply speaking of the pushforward and pull back $\phi _{*},\phi ^{*}$. This active point of view makes no reference to a coordinate system. The passive transformation is simply to look at the coordinate transformation $x^{\mu}\rightarrow y^{\mu}$ induced by $\phi$ and talk about tensorial quantities transforming under the usual coordinate transformation rules. Again, there is no mention of riemannian geometry and mathematically there is no difference between the notion of passive and active diffeomorphisms.

By the way, the poincare group is the group of isometries of specifically $(\mathbb{R}^{4},\eta _{ab})$ and as Peter already mentioned, includes boosts, translations, and rotations. A general space - time need not admit any isometries and as such need not have any isometry group attatched to it.

Last edited: Mar 14, 2013
6. Mar 14, 2013

### Staff: Mentor

Agreed, mathematically there is no difference, the difference is only in the physical interpretation of the math.

Yes, but if both manifolds have a riemannian structure, you can still talk about it even if the diffeomorphism between them is not an isometry.

7. Mar 14, 2013

### NanakiXIII

You're basically saying the same thing I was. If you describe a physical theory very heuristically as a set of 1) a manifold, 2) some structure on that manifold, e.g. the metric, and 3) the stuff that lives on your manifold, i.e.

$$(M, g, \psi)$$

then a passive transformation $\phi$ brings you to a theory $(\phi M, g, \psi)$ in your terminology or $(M, \phi g, \phi \psi)$ in mine. An active transformation gets you $(\phi M, \phi g, \psi)$ in your terminology or $(M, g, \phi \psi)$ in mine. I believe they're equivalent. My terminology corresponds to the good old elementary mechanics way of using these terms (see for example the Wiki page on the matter.)

I don't see how you can change the mass of the black hole through a diffeomorphism on your manifold. It involves changing the matter content of your spacetime.

8. Mar 14, 2013

### atyy

There's no standard definition of the terms. NanakiXIII is using eg. Giulini's convetion, while WannabeNewton is using something like Wald's (not exactly, since Wald says they are different, but effectively the same).

9. Mar 14, 2013

### Staff: Mentor

I understand the way you've stated my terminology, and I agree it captures what I was getting at. But I don't understand why your version transforms $\psi$ as well as $g$. A passive diffeomorphism doesn't change any of the stuff that lives on the manifold; it just changes the coordinates.

Here I don't understand either version. An active diffeomorphism changes the geometry; but physically, that means you have to change the "stuff living on the manifold" as well, at least in the context of GR, because the Einstein Field Equation links the two. So my version would have the transformation giving $(\phi M, \phi g, \phi \psi)$.

(However, even here there is another potential issue: the "manifold" should include the topology, and as I said before, I don't think an active diffeomorphism can change the topology. So there is still something--a portion of $M$--that doesn't get changed, and your notation doesn't really capture that.)

I also don't understand why your version doesn't transform $M$ or $g$.

See my comments above on an active diffeomorphism; since the matter content and the geometry are linked by the Einstein Field Equation, you can't change one without changing the other. As far as the geometry is concerned, you're just changing the M parameter in the line element; that's obviously a diffeomorphism. In the case of the maximally extended Schwarzschild spacetime, which is vacuum everywhere, there is no "matter content" to change; but in any real scenario, changing the M parameter in the line element amounts to changing the amount of matter that originally collapsed to form the black hole, so yes, it does mean changing the matter content of the spacetime.

10. Mar 14, 2013

### WannabeNewton

Sure but my point is that passive and active diffeomorphisms don't have anything to do, in general, with the geometry of the underlying manifold assuming by geometry one means the riemannian structure of the manifold. Say $\phi :M\rightarrow N$ is a diffeomorphism and $X:M\rightarrow TM$ is a vector field then the active point of view is to look at the point - wise pushforward $\phi _{*}X_{p}, p\in M$ (this works nicely since $\phi$ is a diffeomorphism). The passive point of view is to take the $\phi$ - induced coordinates $\left \{ x^{'\mu} \right \}$ and look at $X^{'\mu} = \frac{\partial x'^{\mu}}{\partial x^{\nu}}X^{\nu}$.

11. Mar 14, 2013

### atyy

Those are the Wald definitions (which are in the tradition of mathematics), let's abbreviate them WNA and WNP

These are Giulini's definitions (which are on the tradition of Anderson and MTW). http://arxiv.org/abs/gr-qc/0603087 (actually he cal's them general covariance and general invariance). Let's call them GA and GP.

I believe FAPP WNA=WNP=GP=MTW's general covariance, but GA = MTW's "no prior geometry" is different.

General covariance is not very meaningful since all theories are generally covariant (you can use any coordinates you want). The difference between SR and GR is that in SR matter does not act on the spacetime metric, but in GR matter tells spacetime how to curve.

There is one more principle in GR that is important, called the Principle of Equivalence. To show how everyone's terminology is different, Weinberg calls the Principle of Equivalence the "Principle of General Covariance", whereas he calls MTW's general covariance "general covariance".

So to summarize there are 3 things:

1) general covariance: ability to change coordinates, true for all theories
2) no prior geometry: true for GR, not true for SR (with a tiny exception)
3) principle of equivalence: also known as minimal coupling, and is sufficient to obtain covariant energy conservation in metric theories (not quite sure about the exact statement, so let me point to http://arxiv.org/abs/gr-qc/0505128, http://arxiv.org/abs/0805.1726)

Last edited: Mar 14, 2013
12. Mar 14, 2013

### Staff: Mentor

Just to make sure I understand the notation: this means that $\phi _{*}X_{p}$ is a map $\phi _{*}X_{p} : N \rightarrow TN$, correct? (That is, it's a vector field on $N$, just as $X$ is a vector field on $M$.)

Again, to make sure I understand the notation, here $X^{'\mu}$ is a representation of a vector field on $N$ in the coordinate basis $\left \{ x^{'\mu} \right \}$, and $X^{\nu}$ is a representation of "the same" vector field on $M$ in the coordinate basis $\left \{ x^{\nu} \right \}$, correct? ("The same" means that the diffeomorphism transforms one into the other.)

13. Mar 14, 2013

### WannabeNewton

The first one is $\phi _{*p}:T_p(M)\rightarrow T_{\phi(p)}(N)$ (pushes forward vectors in the tangent space to M at p to the tangent space to N at the image of p under the diffeomorphism) and yeah the second one is exactly what you said.

14. Mar 14, 2013

### Staff: Mentor

Ah, I see. (The word "pointwise" should have clued me in.) So then the vector field on $N$ that "corresponds" to $X$ on $M$ under the diffeomorphism would be a map $N \rightarrow TN$ that takes the image $\phi(p)$ of $p$ under the diffeomorphism to the vector $\phi _{*p} X(p)$, i.e., it uses the map between tangent spaces to find the "corresponding" vector at each point.

15. Mar 14, 2013

### FedEx

Thats how I think about them too.

Thanks :)

16. Mar 14, 2013

### WannabeNewton

Yeahp, and it works nicely because the map is a diffeomorphism otherwise there are more technical problems involved than one would like =D. As atyy noted, this is how Wald presents it and as you mentioned MTW has their way of saying it but unfortunately I don't have a copy of MTW (I'm far too weak to actually lift that thing) however I'm curious as to how they talk about the two notions of passive and active?

17. Mar 14, 2013

### FedEx

I understand one and three.

Two has more content that one. Agreed.

However, I do not understand what you mean by "with a tiny exception"?

18. Mar 14, 2013

### FedEx

Yeah. Transforming mixed tensors would not be possible without diffeos. Right?

I think I have understood what is that was confusing me. I was confusing isometries with diffeomorphisms.

One should actually call Poincare Invariance as Poincare Isometries :p

And I presume the way one would go about finding them would be to find the Killing vector fields.

Hence as mentioned above, it is perfectly reasonable to ask for a theory to be Diffeomorphic Invariant; since that just asks for coordinate invariance. But a theory need not necessarily be poincare invariant.

Last edited: Mar 14, 2013
19. Mar 14, 2013

### WannabeNewton

In general there is no operation of push - forward or pull - back for mixed tensor fields but yes in the case of a diffeomorphism it does work out nicely. If you want to work it out yourself, take a look at problem 11 - 6 in Lee's Smooth Manifolds book if you have access.

20. Mar 15, 2013

### atyy

To some extent, the physical content of general relativity can also be obtained by writing the theory in the form of a spin 2 field on flat spacetime. I don't know the exact limitations of this alternative formulation. It is mentioned eg. http://arxiv.org/abs/gr-qc/0411023 and http://arxiv.org/abs/1105.3735 (section 6.1).

Historically, the first relativistic theory of gravity was Nordstrom's, formulated as a field on flat spacetime. Einstein and Fokker then reformulated it as a theory of curved spacetime. It is not phenomenologically viable because it gets the perihelion of mercury wrong, but it was an important precursor to GR. http://arxiv.org/abs/gr-qc/0405030

Similarly, Newtonian gravity has its usual formulation as well as a formulation as curved spacetime. http://arxiv.org/abs/gr-qc/0506065

21. Mar 15, 2013

### NanakiXIII

I think the OP is already satisfied, but for completeness let me answer this.

You can either transform the manifold and not touch the geometry or the content, or you can transform the content and take along the metric in such a way that your physical system doesn't change. For example, if my system lives on $\mathbb{R}$ with standard geometry (i.e. $\forall x, y \in \mathbb{R}: d(x,y) = |x-y|$), I can stretch my manifold by a factor 4

$$\phi(x) = 4x$$

or I can move around my matter content and take along the metric. Suppose my content consists of two point charges at points $x$ and $y$ and a potential $V(|x-y|)$. I can affect the same change of coordinates by taking my charges and putting them closer together:

$$\phi(x,y) = (\frac{x}{4},\frac{y}{4})$$

However, this is not an isometry of the metric, so this is an active transformation that actually changes the physics (in this case the potential energy between the particles), unless we also take along the metric:

$$\phi(d(x,y)) = 4 d(x,y)$$

Again, I'm aware this is highly heuristic and it is meant to be.

That's because, in GR, you have no prior geometry, so stated in the notation I'm using it's more accurate to say GR looks like

$$(M, \emptyset, \Psi); \Psi = (g, \psi)$$

This reflects your statement that the EFE links geometry and content. I just think of it as the geometry becoming part of the content.

Doubling back, my idea of an active transformation is taking the stuff in your universe and moving it around, just the content. Like I mentioned in the above example, that's fine if your transformation is an isometry, but if it's not, you're changing your system in a nontrivial way. Using the same example:

$$\phi(x,y) = (\frac{x}{4},\frac{y}{4})$$

This is an active transformation, just acting on the content. It changes the physics because scaling is not an isometry. You end up with your particles at position $(\frac{x}{4},\frac{y}{4})$ and with a distance $\frac{|x-y|}{4}$ between them. I can of course obtain the same thing by doing

$$\phi(x) = 4x; \phi(d(x,y)) = \frac{1}{4} d(x,y)$$

The first map stretches the manifold out underneath the particles so that they end up at the desired positions and the second map makes the metric reflect this stretching in the geometry.

In GR it makes less sense to make this distinction, because you cannot transform the content and geometry separately: every active transformation necessarily also takes along the metric, getting rid of the problem altogether. Starting with $(M, \emptyset, \Psi)$, under an active transformation we get $(M, \emptyset, \phi\Psi)$ and under a passive transformation we get $(\phi M, \emptyset, \Psi) = (M, \emptyset, \phi\Psi)$.

I still think we're saying more or less the same thing.

It seems to me this is a mapping in the parameter space you additionally put into your system. What kind of diffeomorphism $\phi : M \to M$ (M the manifold now, not the mass, which I'll call m) changes this parameter m? Suppose in my simple example above, I choose my metric to be $d(x,y) = |mx-y|$. There is no change of coordinates I can perform that will change m into anything else. I can perform an arbitrary active transformation, changing x into a and y into b, and I will get $|mx-y| \to |ma-b| \neq |m'x-y|$, in general.

Of course I can change m using some mapping I define, but I don't see how you can generally write that as a diffeomorphism on your space-time manifold.

22. Mar 15, 2013

### Staff: Mentor

I don't remember them talking about it much, if at all. They have a chapter on differential topology but IIRC it's mostly about affine parametrization and the meaning of vectors and tensors (mainly to show how those concepts can *have* a meaning in the absence of a metric) and differential forms. When I have a chance I'll hook a crane to my copy and haul it out to check.

23. Mar 15, 2013

### Staff: Mentor

Your post clarifies what you meant by your notation, but I probably need to read some more references to get a better feel for it. I think some of the links that atyy posted will help.

It's the same sort of thing as the scaling transformation you gave above; m basically sets the scale of the r coordinate: more precisely, it tells you how the metric scales with the r coordinate. So a transformation that gives $\phi (m) = 2m$ and $\phi (r) = 2r$ will make the metric at $\phi (p)$, the image of the point $p$ under the diffeomorphism, be the same as the metric at $p$. I think you would call this an active transformation, not a passive one.

Last edited: Mar 15, 2013
24. Mar 15, 2013

### Staff: Mentor

I believe the key limitation is topology: since the field is on flat spacetime, this version of the theory can only describe spacetimes with the same global topology as flat spacetime. If you don't need to deal with the global topology, though, AFAIK the spin-2 field version is exactly equivalent to the curved spacetime version of GR; they have the same Lagrangian and the same field equation.

25. Mar 15, 2013

### NanakiXIII

I'm not sure this is true, since the angular part of the Schwarzschild metric just depends on r, not on M/r.


text:
Stream: maths

Topic: Use instances with local hypothesis

Filippo A. E. Nuccio (Mar 05 2021 at 13:22):

I am trying to understand how instances work and I ran across the following examples.

lemma one {S : Type*} (hs : integral_domain S) : comm_cancel_monoid_with_zero S := by apply_instance

is OK, while

lemma two {S : Type*} [ring S] (f : ℤ ≃+* S) : comm_cancel_monoid_with_zero S :=
begin
have hs : integral_domain S, sorry,
apply_instance
end

is not OK (a part from the fact that I can prove hs using f, but I've sorried it to be short). So I understand that when Lean looks for assumptions that would match instances, it looks in the hypothesis of the statement, but not on "new" ones. Is it really so? If yes, I have two questions:
1) What is the difference between an "assumed" hypothesis and one which has been "constructed" along the way?
2) How to apply_instance if the needed assumption was not part of the hypothesis, as in my lemma two?

Riccardo Brasca (Mar 05 2021 at 13:25):

For question 2, haveIor letI makes hs an instance, meaning that the system then finds it like it was written [...] in the statement.

Kevin Buzzard (Mar 05 2021 at 13:25):

The "system" which makes apply_instance work is called the type class inference system. It is initialised at the beginning of a proof in the following way: Lean looks at everything before the colon which is a term of a typeclass (like your first hs) and puts it into the system. Then, by default, it never puts anything else into the system.

To make your lemma 2 work you need to put haveI : integral_domain S (or probably letI : integral_domain S because I think it's data) -- this tactic specifically is designed to change the system.

Filippo A. E. Nuccio (Mar 05 2021 at 13:26):

Wow, perfect! Thanks (indeed, it works): the nice caveat is that it does not accept my sorry, the docs says

The syntax is the same as have, but the proof-omitted version is not supported. For this one must write have : t, { <proof> }, resetI, <proof>.

Kevin Buzzard (Mar 05 2021 at 13:27):

Note that, as Patrick reminded us yesterday (and Mario told me about a year ago), you don't need hs to be in square brackets in lemma 1 -- Lean just looks at everything before the colon. The reason you should put [integral_domain S] in square brackets instead is for people who want to use your lemma, not to help you prove it yourself.

Filippo A. E. Nuccio (Mar 05 2021 at 13:28):

Meaning it could be in curly ones? (You can also simply point at Patrick's chat)

Kevin Buzzard (Mar 05 2021 at 19:40):

Yes, any terms of type C where C is in a class, if they're defined before the colon, will end up in the instance cache. It doesn't matter what brackets are being used, I believe...

Last updated: May 14 2021 at 18:28 UTC


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# Power Series Convergence Example

Previous: Power Series Convergence

Next: Videos on Power Series

## Example

Determine the radius and interval of convergence of the infinite series

$\sum_{k=0}^{\infty} \frac{k(x+3)^k}{4^{k+1}}$

## Complete Solution

### Step 1: Apply Ratio Test

The ratio test gives us:

\begin{align} \lim_{k \rightarrow \infty} \Bigg| \frac{(k+1) (x+3)^{k+1}}{4^{k+2}} \Bigg/ \frac{k (x+3)^{k}}{4^{k+1}} \Bigg| &= \lim_{k \rightarrow \infty} \Bigg| \frac{(k+1)}{k} \cdot \frac{(x+3)^{k+1}}{ (x+3)^{k} } \cdot \frac{4^{k+1}}{ 4^{k+2}} \Bigg| \\ &= \lim_{k \rightarrow \infty} \Bigg| \frac{(k+1)}{k} \cdot (x+3) \cdot \frac{1}{4} \Bigg| \\ &= \Big| \frac{x+3}{4} \Big| \end{align}

The ratio test tells us that the power series converges only when

$\Big| \frac{x+3}{4} \Big| < 1$

or | x + 3 | < 4. Therefore, the radius of convergence is 4.

### Step 2: Test End Points of Interval to Find Interval of Convergence

The inequality | x + 3 | < 4 can be written as -7 < x < 1. By the ratio test, we know that the series converges on this interval, but we don't know what happens at the points x = -7 and x = 1.

At x = -7, we have the infinite series

$\sum_{k=0}^{\infty} \frac{k(-4)^k}{4^{k+1}} = \sum_{k=0}^{\infty} \frac{k (-1)^k}{4}$

This series diverges by the test for divergence.

At x = 1, we have the infinite series

$\sum_{k=0}^{\infty} \frac{k4^k}{4^{k+1}} = \sum_{k=0}^{\infty} \frac{k}{4}$

This series also diverges by the test for divergence.

Therefore, the interval of convergence is -7 < x < 1.

## Possible Challenges

### What Convergence Test Should Be Used?

For most problems, the ratio test can be used initially. If the ratio test yields an interval for the domain, we need to use other convergence tests to explore what the domain could be at the end points of the interval.

Previous: Power Series Convergence

Next: Videos on Power Series


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0

# I'm trying to make a trade system but I can't seem to get the PlayerGui?

So I've made the GUI and all, you can click trade and if you click trade it'll send the person you're trying to trade with a Frame saying that somebody wants to trade, but I can't seem to get the other player's PlayerGui. What I thought would fix it was to fire a RemoteEvent from the client to the server, then the server gets the player's PlayerGui and sends it back to the client, but it sends "nil" instead of the GUI. I've even printed if it gets the GUI and it does, but then it sends "nil" instead of the GUI. Please help...

Server script:

game.ReplicatedStorage.RemoteEvents.TradeEvents.GetPlayerGui.OnServerEvent:Connect(function(player,playerToTradeWith)

local realPlayer

for _,possiblePlayer in pairs(game.Players:GetPlayers()) do

realPlayer = possiblePlayer
break
end
end

local playerGui = realPlayer:FindFirstChild("PlayerGui")

end)

Client script:

script.Parent.MouseButton1Down:Connect(function()

local player = game.Players.LocalPlayer
local playerText = script.Parent.Parent.Text

if game.Players:FindFirstChild(playerText) then

script.Parent.Text = "That player isn't in the game!"
script.Parent.Frame.BackgroundColor3 = Color3.fromRGB(252, 11, 35)
script.Parent.Background.BackgroundColor3 = Color3.fromRGB(115, 12, 21)

wait(3)

script.Parent.Frame.BackgroundColor3 = Color3.fromRGB(36, 255, 99)
script.Parent.Background.BackgroundColor3 = Color3.fromRGB(16, 111, 47)

else

end
end
end)

local player = game.Players.LocalPlayer

print(plrGui.Parent)

script.Parent.Frame = Color3.fromRGB(141, 24, 255)
script.Parent.Background = Color3.fromRGB(62, 16, 108)
else
script.Parent.Text = "That player is busy right now!"
script.Parent.Frame.BackgroundColor3 = Color3.fromRGB(252, 11, 35)
script.Parent.Background.BackgroundColor3 = Color3.fromRGB(115, 12, 21)

wait(3)

script.Parent.Frame.BackgroundColor3 = Color3.fromRGB(36, 255, 99)
script.Parent.Background.BackgroundColor3 = Color3.fromRGB(16, 111, 47)
end
end)

0