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Comment by seed on Non offensive word for people who are not single-magisterium-Bayes thinkers · 2020-07-03T11:03:06.166Z · score: 3 (2 votes) · LW · GW

Why do you want to put a label on their belief? There are some adverse effects like they can get offended, or being a Toolboxist becomes a part of their identity and then it's even harder to change their mind.

Comment by seed on What are good defense mechanisms against dangerous bullet biting? · 2020-07-02T07:53:17.082Z · score: -4 (2 votes) · LW · GW

Well, looking bad leads to attracting less donor money, so it is somewhat important how you look. The argument about why Roko's basilisk won't actually be made on purpose is my central point, that's what you'd have to refute to change my mind. (While I understand how it might get created by accident, spreading awareness to prevent such an accident is more helpful than covering it up - which is now impossible to do anyway, thanks to the Streisand effect the topic comes up all the time.)

Comment by seed on On saving the world · 2020-07-01T17:35:10.755Z · score: 1 (1 votes) · LW · GW
It's unlikely that I'll be able to convince a few million people to succeed from their nations (without invoking the ire of their tax collectors) anytime soon.

What about the millions of people who are already stateless? I once thought to try and bring about anarcho-capitalism by starting a campaign for stateless people's rights, before I came up with a better plan.

Comment by seed on Industry and workers · 2020-06-26T09:28:53.661Z · score: 4 (3 votes) · LW · GW

In the previous post, you complain that you have way too many resources to spend on garbage entertainment, and wish you were poorer because then you'd spend more time reading, socializing and exercising. Now you say that Industrial Revolution failed to give workers more resources. What is it that's so bad about capitalism - that it failed to make you richer, or that it made you too rich for your own good?

Comment by seed on Industry and workers · 2020-06-26T09:01:06.352Z · score: 4 (3 votes) · LW · GW

So, you believe your scheme will fail under capitalist competition. Is there any country in the world where you believe it will work? Why don't you try it there, then tell us how it worked out?

Comment by seed on DontDoxScottAlexander.com - A Petition · 2020-06-25T15:07:58.684Z · score: 8 (5 votes) · LW · GW

I signed the petition on the assumption that it was all just a misunderstanding, but I'm willing to fight dirty if they ignore the petition and publish the name anyway.

Comment by seed on Training our humans on the wrong dataset · 2020-06-22T08:44:24.953Z · score: 1 (1 votes) · LW · GW

Em, you don't need a PhD in applied mathematics to learn to take derivatives, it's something you learn in school.

Comment by seed on You have become the supreme dictator of the United States. · 2020-06-22T08:09:16.156Z · score: 12 (3 votes) · LW · GW

Well, first of all, I'd take money from the poor and give it to the military. What? It doesn't matter how "benevolent" I am, it's just what I have to do to stay in power.

Seriously, if you're legitimately concerned for the public good, don't become a dictator. Become an entrepreneur or a scientist.

Comment by seed on $1,000 Bounty for Pro-BLM Policy Analysis · 2020-06-19T11:09:26.201Z · score: 1 (1 votes) · LW · GW
polls report public sympathy for BLM is very high.

Wow, I actually haven't expected that at all.

Maybe many years ago this turn of events would seem natural to me. People care about each other and stand up for each other when someone gets hurt, right? Well, wrong. At least in Russia, most people don't care much about victims of police violence, as I've found. And in USA it seems to only be about black people. So while I can see why Democrats are supporting their ingroop, I don't get the increase in Republican support. Could people be lying about their views because they're afraid of repercussions for expressing wrong ones? Seems like a big stretch.

My people believed in nonviolent protest, and lost. While I'd broken away from the doctrine and cheered for people who fought back against cops, I've always thought that pointless violence against innocents would make people hate me. (Or do they just hate the cops even more? I didn't notice that.) Will people like my politics more if I go loot some shops? Or is it something else they did right.

I walk away from you guys totally confused about how it all really works.

Comment by seed on Premature death paradox · 2020-06-18T18:42:18.123Z · score: 1 (1 votes) · LW · GW

It means dying before the age of 50 or so.

Comment by seed on What are good defense mechanisms against dangerous bullet biting? · 2020-06-18T17:12:12.523Z · score: -4 (2 votes) · LW · GW

Roko's basilisk was mentioned in the original comment, so I'm not doing any additional harm by mentioning it again in the same thread. I suggest you stop calling everything "infohazard", because it devalues the term and makes you look silly. Some information is really dangerous, e.g. a bioweapon recipe. Wouldn't it be good to have a term for dangerous information like this, and have people take it seriously. I think you've failed at the second part already. On this site, I've seen the term "infohazard" applied to such information as: "we are all going to die", "there is a covid pandemic" https://www.lesswrong.com/posts/zTK8rRLr6RT5yWEmX/what-is-the-appropriate-way-to-communicate-that-we-are, "CDC made some mistakes" https://www.lesswrong.com/posts/nx94BD6vBY23rk6To/thoughts-on-the-scope-of-lesswrong-s-infohazard-policies.

I'm sorry, but I can't take infohazard warnings seriously any longer. And yes, Roko's basilisk is another example of a ridiculous infohazard, because almost all AGI designs are evil anyway. What if someone creates an AI that tortures everyone who doesn't know about Roko's basilisk? Then I'm doing a public service.

I think anyone seriously anxious about some potential future AGI torturing them is ridiculously emotionally fragile and should grow up. People get tortured all the time. If you weren't born in a nice first world country, you'd live your whole life knowing you can get tortured by your government any moment. Two of my friends got tortured. Learning that my government tortures people makes one more likely to go protesting against it and end up tortured, too. Yet I don't give people any infohazard warnings before talking about it, and I'm not going to. How are you even supposed to solve a problem if you aren't allowed to discuss some of its aspects.

And if I'm mistaken somewhere, why don't you explain why, instead of just downvoting me.

Comment by seed on Open and Welcome Thread December 2018 · 2020-06-18T16:01:12.675Z · score: -16 (3 votes) · LW · GW

I've figured how not to be a racist in USA.

One of the BLM's demands is "the creation of community healing space for Black identified students." Apparently it's a real thing: you can have white skin but identify as Black. https://www.theguardian.com/us-news/2019/jul/25/race-east-jackson-ohio-appalachia-white-black

So, just identify as Black, and Blacks cannot be racist, by definition.

You're welcome.

Comment by seed on $1,000 Bounty for Pro-BLM Policy Analysis · 2020-06-18T15:49:20.024Z · score: 1 (1 votes) · LW · GW

Wow, thanks, things make more sense now.

Comment by seed on $1,000 Bounty for Pro-BLM Policy Analysis · 2020-06-18T15:48:38.554Z · score: 0 (2 votes) · LW · GW

Sorry, I've realized they have a list of demands already. https://blacklivesmatter.ca/demands/

Yes, I'm conflating "BLM movement" and "individual Americans who want to help BLM achieve its goals" because isn't it the same thing. Ok, from what you've told me, it sounds like getting Republican support is the easiest way to achieve change.

With that in mind, actionable points (for a generic BLM supporter not just for lesswrongers, I think you probably aren't bullying anyone already):

  • propose your own policy ideas, e.g. like Eliezer did on Facebook
  • stop bullying everyone who disagrees with you, so you can learn what they think and find solutions that both sides support
  • defend shops from looters so people have more sympathy for your side
Comment by seed on God and Moses have a chat · 2020-06-18T11:09:40.545Z · score: 1 (3 votes) · LW · GW

Jesus acted correctly. He might or might not be insane, but he has to figure it out some other way, like check his memories for inconsistencies, go see a psychiatrist, or go check if that person he resurrected last week is really alive and people confirm the story, so he didn't just imagine it. What the devil tells and shows him isn't reliable evidence for anything, because the devil could just be lying.

Moses failed to come up with some obvious anti-hallucination checks (see comment below), but at least he didn't go on a killing spree based on flimsy evidence, so that's pretty reasonable, too.

Comment by seed on God and Moses have a chat · 2020-06-18T10:44:23.745Z · score: 3 (2 votes) · LW · GW

Ask God to perform a miracle in front of thousands of witnesses. You cannot all be hallucinating at the same time. Or if that's not an option, ask God to give you proof of Riemann hypothesis, then publish it in a peer-reviewed journal to check that it's true. That wouldn't exclude the Matrix overlords / alien teenager prank hypotheses, but at least you'd know you're not hallucinating.

I don't think anyone could prove they're omniscient and omnipotent, rather than simply very knowledgeable and powerful, though. Seems like that would require infinite amount of evidence.

Comment by seed on $1,000 Bounty for Pro-BLM Policy Analysis · 2020-06-18T09:33:06.207Z · score: -3 (3 votes) · LW · GW

I'm not an American and don't understand in detail how your politics works, so I'm sorry if this is naive, but. It seems that to achieve its stated goals of police reform, BLM movement should do the following:

1. Launch a nationwide discussion on how exactly the police should be reformed.

2. Come to a consensus policy that BLM supports and Republicans don't find completely unreasonable.

3. Vote for politicians who support this policy.

Democrats are 50% of the population, and libertarians are just as concerned about police violence, so together you form a majority and can pass any police reform you want. Please correct me if I'm wrong.

Comment by seed on Don't call yourself a rationalist. · 2020-06-17T09:33:15.720Z · score: 1 (1 votes) · LW · GW

But smart adults are already hanging out with other smart adults in their university / workplace, and if you're a child... Why not just join a local math club or whatever it is you are interested in.

Comment by seed on Non-standard politics · 2020-06-16T16:35:48.865Z · score: 1 (1 votes) · LW · GW

I'm a Russian with Ukranian friends, and a war against Ukraine always seemed unthinkable. The war propaganda is based on the assertion that Ukrainians are our brothers and so we've got to protect them from the junta.

Comment by seed on Whining-Based Communities · 2020-06-15T02:44:49.836Z · score: 1 (1 votes) · LW · GW

No one said that exposing evil would be sufficient, and heroes of Rand's novels didn't just go around exposing evil. It helps, though.

Comment by seed on GAN Discriminators Don't Generalize? · 2020-06-09T19:59:10.879Z · score: 1 (1 votes) · LW · GW

Oh, I see, sorry.

Comment by seed on GAN Discriminators Don't Generalize? · 2020-06-09T18:59:18.185Z · score: 4 (2 votes) · LW · GW

Getting a validation accuracy of 50% in a binary classification task isn't "surprisingly well". It means your model is as good as random guessing: if you flipped a coin, you would get the right answer half the time, too. Getting 0% validation accuracy would mean that you are always guessing wrong, and would get 100% accurate results by reversing your model's prediction. So, yes, just like the article says, the discriminator does not generalize.

Comment by seed on How rational is the path of least resistance? · 2020-06-01T13:31:03.875Z · score: 2 (2 votes) · LW · GW

It depends on whether you want to end up where the flow is going, or somewhere else... I think your question is too vague to give any useful answer, though. As ch. 27 of HPMOR teaches us, it's best to know context before giving out sage life advice.

Comment by seed on Something's Wrong · 2020-05-30T18:28:19.617Z · score: 1 (1 votes) · LW · GW
Most critiques from radicals that I read don't contain an analyses of the root courses of the problem they are criticizing.

Then calling them "radicals" is a misuse of the word, I think.

Comment by seed on Ureshiku Naritai · 2020-05-25T12:12:42.506Z · score: 1 (1 votes) · LW · GW

I don't mind people talking to me on public transportation, as long as they immediately believe me when I say I'm not interested, and leave me alone.

Comment by seed on Craving, suffering, and predictive processing (three characteristics series) · 2020-05-15T19:08:30.466Z · score: 1 (1 votes) · LW · GW

So, do you not need painkillers now thanks to meditation? How did it impact your motivation, do you get more things done?

Comment by seed on A game designed to beat AI? · 2020-05-13T14:40:46.516Z · score: 1 (1 votes) · LW · GW

Each player is provided with a board and a male assistant. They're not allowed to use anyone else's help. The winner is the first one to produce a human baby.

I believe this meets the second goal, too.

Comment by seed on Teachable Rationality Skills · 2020-05-13T12:59:08.421Z · score: 1 (1 votes) · LW · GW

So, the girl challenge is to get a girl to accompany you to one of your hobbies.

Comment by seed on Mahatma Armstrong: CEVed to death. · 2020-05-11T11:58:44.576Z · score: 3 (2 votes) · LW · GW

My problem with CEV is the arbitrariness of what it means to "know more". My brain cannot hold all the knowledge about the universe, so the AI has to somehow choose what information to impart and in what order, and this would significantly influence the outcome. E.g. maybe hearing 100 heartwarming stories would make me care more about others, while hearing 100 stories about people being bastards to each other would make me care less, hearing all evidence supporting some political theory would sway me towards it, et cetera.

Comment by seed on Winning vs Truth – Infohazard Trade-Offs · 2020-05-10T11:27:49.986Z · score: 1 (1 votes) · LW · GW

It's a relief to know you aren't advocating self-deception, and you may want to choose your phrasing in the
post not to give that impression. "Epistemic rationality" means knowing the truth for yourself. Been honest with others is a different virtue.

That said, I think telling the truth almost always does more good than harm, and my policy is to only lie to defend myself or others from violence. In this particular case, I don't see how the CDC post is going to hurt the average person, since the readers are not average people, but LW community.

Comment by seed on What are good defense mechanisms against dangerous bullet biting? · 2020-05-10T09:41:16.347Z · score: -4 (2 votes) · LW · GW

I don't need any defense mechanisms against these ones, because I can just see the fallacy in the arguments.

In one description, Blaise Pascal is accosted by a mugger who has forgotten his weapon. However, the mugger proposes a deal: the philosopher gives him his wallet, and in exchange the mugger will return twice the amount of money tomorrow. Pascal declines, pointing out that it is unlikely the deal will be honoured. The mugger then continues naming higher rewards, pointing out that even if it is just one chance in 1000 that he will be honourable, it would make sense for Pascal to make a deal for a 2000 times return. Pascal responds that the probability for that high return is even lower than one in 1000. The mugger argues back that for any low probability of being able to pay back a large amount of money (or pure utility) there exists a finite amount that makes it rational to take the bet – and given human fallibility and philosophical scepticism a rational person must admit there is at least some non-zero chance that such a deal would be possible. In one example, the mugger succeeds by promising Pascal 1,000 quadrillion happy days of life. Convinced by the argument, Pascal gives the mugger the wallet.

When a mugger promises me to return twice my money tomorrow, I can see that it is almost certainly a hoax. There is maybe a one in a million chance he's saying the truth. The expected value of the wager is negative. If he promises a 2000x return, that's even less likely to be true. I estimate it as one in two billion. The expected value is still the same, and still negative. And so on, the more lavish reward the mugger promises, the less likely I am to trust him, so the expected value can always stay negative.

Roko's basilisk

Why don't I throw themselves into a research institute dedicated to building the basilisk? Because there is no such institute, and if someone seriously tried to build one, they'd just end up in prison or a mental asylum for extortion. Unless they are keeping their work secret, but then it's just an unnecessarily convoluted way of building an AI that kills everyone. So there is no reason why I would want to do that.

Comment by seed on What fraction of your lifetime (0-80 years old) egoist budget would you (want to) spend on a pill that made you live for as long as you wanted (perfect invincibility), as healthily as you wanted if you knew it would become available to you once you're 80 years old (and that you would otherwise irreversibly die)? · 2020-05-08T22:45:30.565Z · score: 1 (1 votes) · LW · GW

I'd save enough money to lead a healthy life and pay for medical emergencies so I could hopefully live to 80, and spend the rest on the pill. Hard to tell what fraction of my budget would that be, until I know how much I'm going to earn in my lifetime.

Comment by seed on Iteration Fixed Point Exercises · 2020-04-25T13:24:30.462Z · score: 1 (1 votes) · LW · GW

#1

- can show by induction.

Therefore, is a Cauchy sequence, and since (X, d) is complete, it must have a limit in X. Suppose . Then , therefore

#2

Suppose . Let's show that y is a fixed point. Indeed, for any n, , and if we take the limit in both sides, we get .

Let's show uniqueness: suppose x and y are fixed points, then , therefore d(x,y) = 0.

#3

, f(x) = x + 1/x.

#4

Suppose , where h is some convex function and . Take . Since h is convex on segment [x,y], its directional derivative is nondecreasing. Its directional derivative is a projection of gradient of g on the [x,y] line. Therefore, we have , or . Hence,

Therefore, g is a contraction mapping, and from problem 1 it follows that the gradient descent converges exponentially quickly.

#5

Suppose A is an NxN positive matrix, and e is its minimal entry. (Then e < 1/N). Then we can write A = eJ + (1 - Ne)Q, where J is a matrix whose entries are all 1, and Q is a matrix whose entries are all nonnegative and the sum of each column is 1 (because the sum of each column is 1 in A and Ne in J). Suppose x and y are probability distributions, i.e. N-dimensional vectors with nonnegative entries whose sum is 1. Then

Denote , (pointwise max/min). Then , ,

so . The space of all probability distributions with metric induced by - norm is a compact subset of , so it is a complete metrics space, therefore, the sequence converges to a unique fixed point.

#6

Let us assume (the proof for is the same). Then, from monotonicity of f, is an ascending chain. This sequence cannot have more that |P| distinct elements, so an element of this sequence is going to repeat: . Then all the inequalities in must be equalities, so , is a fixed point.

Comment by seed on Please Press "Record" · 2020-04-20T16:21:35.303Z · score: 1 (1 votes) · LW · GW

This is so sad.

Comment by seed on Premature death paradox · 2020-04-14T14:14:10.320Z · score: 1 (1 votes) · LW · GW

Life expectancy tables may overestimate on your death day, but they underestimate some people's lives on some other days, so it's not like they always overestimate. It seems like you've explained it all pretty well, I don't see any paradox left.

Comment by seed on Open & Welcome Thread - March 2020 · 2020-04-09T08:30:45.227Z · score: 1 (1 votes) · LW · GW

Lewak et al (1985) studied 81 couples and found no correlation between IQ and marital satisfation. https://www.sciencedirect.com/science/article/abs/pii/0191886985901400

And, well, everything in life is potentially value-corrupting, or value-improving, depending on whether you judge from your past self's or present self's point of view. I think the more experience you have in dating, the better judgement you can make about what makes you happy. And if a girl seduced you with great sex, that's a predictor of a good relationship, don't see anything illegitimate about that. There are known failure modes in relationships: you don't want to end up with an abuser, an alcoholic or a drug addict. If you're not in one of these, it's probably fine. From personal experience, I married a man with a lower IQ, and I'm happy.

Comment by seed on Against Dog Ownership · 2020-03-24T14:57:36.093Z · score: 5 (4 votes) · LW · GW

Okay, this essay convinced me that dogs can have depression. I also think that dogs probably have real feelings and don't just act the part like this creepy robot child, although I wonder how can one actually test this.

I am not at all convinced, though, that a dog can have preferences, long-term goals, or "a meaningful life". I don't think I've ever seen a dog work on a long-term goal. And if dogs really preferred to take their chances alone in the alien world like the author suggests, a lot more would run away.

A dog's mind is different. Just because I wouldn't enjoy being a pet, doesn't mean a dog doesn't. The author acknowledges this, but still says that "it’s reasonable to say that dogs have some sort of conception of meaning that rises above moment-by-moment pleasures, and that the unfulfillment of this meaning has a negative effect on the happiness of dogs." Well, why do you believe this?

Comment by seed on From Rationality to Power in 3 Steps · 2020-01-16T09:11:40.757Z · score: 3 (2 votes) · LW · GW

So, it's been two years... How much power did you get?

Comment by seed on Diagonalization Fixed Point Exercises · 2019-12-29T16:01:32.747Z · score: 2 (2 votes) · LW · GW

Ex 1.

Suppose there is a surjection f : S -> P(S). Consider the set . Since f is a surjection, X = f(y) for some y in S. Does y lie in X? If , then , so by definition of X, . If , then , so y must belong to X. Contradiction.

Ex 2.

Since there is a function without fixed points, T must have at least two elements. Hence, there is a surjection , which induces a surjection (a function goes to ). So, if there were a surjection , there would also be a surjection , which cannot be by previous exercise.

Ex 4.

Suppose is a computable surjective function. Consider the function defined by . The function g is computable, therefore there should exist an : .

Then . Contradiction.

Ex 5.

Suppose halt(x,y) is a computable function. Consider the function : ; T if

Suppose is a Turing code of f. Since f halts everywhere, halt(s', s') = T. But then . Contradiction.

Ex 6.

Suppose that is a continuous surjection. Consider the function (here - f(x, x) is a point diametrically opposed to f(x, x)). f is surjective, hence g = f(y), but then g(y) = f(y,y) = - f(y,y). Contradiction.

Ex 7. A quine in python3:

code = """code = {}{}{}{}{}{}{}
print(code.format('"','"','"',code,'"','"','"'))"""
print(code.format('"','"','"',code,'"','"','"'))

Ex 8. In python:

import inspect
def f(string):
return string[::-1]
def applytoself(f):
source = inspect.getsource(f)
return f(source)
applytoself(f)

'\n]1-::[gnirts nruter \n:)gnirts(f fed'

Ex 9.

The formula for is

Ex 11.

Suppose is the formula . By the diagonal lemma, there exists a formula A such that .

Therefore,

By property c,

Again by property c,

Combining previous two implications,

Since , we have

Combining this with , we get

From this we get , therefore, and . QED.


Comment by seed on Book Review: Naïve Set Theory (MIRI course list) · 2019-12-29T11:11:50.005Z · score: 1 (1 votes) · LW · GW

By the way, the Shen's book takes a different route to the Zorn's lemma: first he introduces well-ordered sets, then uses tranfinite recursion to prove Zermelo's theorem (that any set can be well-ordered), then he uses Zermelo's theorem and tranfinite recursion to prove Zorn's lemma. Thus the proof of Zorn's lemma is reduced from two pages to a few lines. I personally found it easier to follow and remember.

Comment by seed on Topological Fixed Point Exercises · 2019-12-22T11:58:00.282Z · score: 1 (1 votes) · LW · GW

Thank you!

Comment by seed on Topological Fixed Point Exercises · 2019-11-16T19:48:57.435Z · score: 2 (2 votes) · LW · GW

I am sorry because I cannot figure out how to hide big formulas in a spoiler. Also the spoiler feature is somewhat broken so it adds weird tabs around formulas.

#1:

Let's count the number of blue edge ends. Each blue point inside the segment is the end of two blue edges, and the leftmost blue vertex is the end of one. Therefore, their total number is odd. On the other hand, each bichromatic edge produces one blue edge end, and each unichromatic edge produces an odd number - zero or two - of blue edge ends. Therefore, an odd number of edges are bichromatic.

#2:

Suppose . If then and, since f is continuous, f stays positive in some neighborhood of x, and then x is not the infimum. Therefore, f(x) = 0.

#3:

Consider the function . Since and by exercise 2, there should be a point where g(x) = 0.

#8.

Consider the family of functions:

For t < 0.5, the only fixed point is of is 1; for t > 0.5, the only fixed point is 0.

#9.

Lemma:

Suppose a k-dimensional simplex is subdivided into smaller k-dimensional simplices and all vertices are colored into k+1 colors so that there are no vertices of color i on the i-th edge of the big simplex. Then there are an odd number of subdivision simplices whose vertices are colored in k+1 different colors.

Proof:

Induction by k. Base k=1 proved in exercise 1.

Induction step: supposed the lemma is proved for k-1, let's prove it for k.

Let us count the number of tuples (X, Y) where X is a k-1-dimensional simplex colored in colors 0, 1, ..., k-1,

Y is a k-dimensional subdivision simplex, and X is on the boundary of Y. Each properly colored simplex X inside the big simplex produces two tuples, and each simplex on the boundary produces one tuple. X can only be on the k-th edge of the big simplex, and by the inductional assumption, there are an odd number of simplices X there. So, the total number of tuples is odd. On the other hand, each k-dimensional simplex Y can be a part of either:

0 tuples;

1 tuple if all his vertices are different;

2 tuples if has vertices of colors 0,1,...,k-1 but not all his vertices are different.

Therefore, a number of k-simplices Y with all different vertices must be odd.

#4

Follows from 9

#5

Suppose that center is not in the image of the triangle. Let us call a set of points bichromatic if it doesn't have points of all three colors. We color each point in the triangle in the same color as its image. Then every point in the image has an open bichromatic neighborhood. Since the map is continuous, the preimage of this neighborhood is also open. So, around every point in the triangle there can be drawn an open bichromatic ball of radius r. These balls are an open cover of the triangle, let us choose a finite subcover out of them. Suppose s the minimum radius in this subcover. Split the triangle into subtriangles so that the diameter of each triangle is smaller than By Sperner's lemma, there is a trichromatic triangle, but since its diameter is smaller than it lies completely inside one of the bichromatic balls. Contradiction.

#10

First, I am going to prove that a function from a unit ball o itself has a fixed point, then that any compact convex subset of s homeomorphic to a ball.

Suppose that as no fixed point, n>1 (case n=1 was proved in exercise 3). Then I can build a retraction from nto its boundary

send x to the first intersection of the ray (f(x), x) with Let us prove that such a rectraction cannot exist. Suppose that such a rectraction exists. Denote the inclusion map. Then nd the induced homology group homorphism ust also be identity:

But this is impossible because and

Now let us prove that any compact convex subset X of s homeomorphic to a ball. Let us select a maximum set of affinely independent points in X. They form some k-dimensional simplex, all X lies in the affine space spanned by this simplex, and all the interior of this simplex belongs to X, because X is convex. I'll take a ball f radius side this simplex and build a homeomorphism between X and . Taking the center of the ball as the center of coordinates, define

where s the distance to the farthest point of X in the direction, if , 0 if

Let us prove that f and its inverse are continuous. Since X is compact, it is bounded, so there is a such that It follows that f and its inverse are continuous in zero: if if

Now let us prove that functions are continuous in all other points. It is sufficient to prove that r(x) is the continuous on the unit sphere. (Since composition and product of continuous functions is continuous, division by bounded from below (by d) continuous function r is continuous, ||x|| is a continuous function).

Since X is convex, the tangent cone from any point of X to lies in X. So if we take a point at the distance from the center, draw a tangent cone, and go down its boundary, we get the steepest possible rate of change of r(x) with respect to x. Therefore, r is continuous.

#6, #7: follow from #10.

#11:

Suppose f has no fixed point. Distance d(a, B) is a continuous function of a, and a continuous function reaches its minimum on a compact. TxT and the graph of f are nonitersecting compact sets, therefore the Hausdorff distance between them is positive. It is easy to see that Hausdorff metric is indeed a metric, i.e. that a triangle inequality holds for it. So if we take any continuous function g at a distance less than from f, its Hausdorff distance to TxT will be positive, so it can have no fixed points.

#13:

Suppose is a Kakutani function. We already know that any compact convex subset of s homeomorphic to a simplex. Denote he homeomorphism between a simplex T and S.

Denote the k-th barithentric subdivision of T. For each choose an element

Define where are the baricentric coordinates of point n its subdivision simplex. Function s continuous, and, since S is convex, the image of lies in S.

By the Brouwer fixed point theorem, as a fixed point. Since S is compact, from the infinite sequence of fixed points of e can choose a convergent subsequence.

Suppose s this subsequence, lies in the simplex and has baricentric coordinates . Then and so

(1).

Since simplices go down in diameter, as Each s a bounded sequence, so we can, sequentially, choose a convergent subsequence out of each of them, so we can assume that Similarly, we can choose a convergent subsequence out of so we assume The sequence belongs to the graph of h and converges to the point Since the graph is closed, must belong to the image of Since for every k, ince the image is convex, lso belongs to the image of On the other hand, as we remember, since equality (1) held for every k, it also holds in the limit: . Hence, So, is the fixed point of h.

Comment by seed on Does the body have an almost infinite number of potential positions? · 2019-10-19T22:42:58.970Z · score: 3 (2 votes) · LW · GW

I don't know what "almost infinite" means, but yes, the body has an infinite number of potential positions. E.g. you could raise your arm by any angle from 0 to 180 degrees. There are infinitely many real numbers from 0 to 180, hence infinitely many possible body positions.