That gave me the idea that the theorem would work for any analytic function, not just ones representable as a closed-form expression. For example, if I understand correctly, the Riemann Zeta function is analytic, so it would follow that
lim(c->a) Zeta(c)g(c) = 1
lim(c->a) g(c)Zeta(c) = 1
and even
lim(c->a) Zeta(c)Zeta(c+someoffset) = 1
for any analytic g(x), and Zeta(a) = 0, g(a) = 0, and Zeta(a+someoffset) = 0.
And I don't even need to know how to raise Zeta to some power, or something to the power of Zeta, or even what Zeta is, as long as Zeta has zeros.
It turns out (see this article) that if f(x) and g(x) are analytic everywhere and not the zero function, with the limit as x--->0 of both functions being 0, then the limit of f^g must be 1.
By they way, the paper talks about f and g being real functions. Does that mean that the theorem does not hold for complex functions?
The video makes a similar argument. It notes that the "dx" part is not merely some syntax for the integral, but in fact it represents a value that can be calculated with (more precisely, it represents the infinitesimally small width of the "columns" we are summing in order to get the are under the curve.) There is, in principle, no reason why this "width" value couldn't be somewhere else in the formula than as a multiplicand.
He proceeds to note that: xdx - 1 = ((xdx - 1) / dx) * dx, and proceeds from there to get the same result as you.
I think a distinction could be made between "goal" and "restriction".
A goal tries to achieve or reach something in the game. A restriction is a self-imposed (rather than game-imposed) rule making achieving that thing harder. The restriction in itself is not something to he achieved or reached in the game.
"Any% completion" is a goal: Complete the game as fast as possible with no or only reasonable restrictions (such as no cheat codes).
"100% completion" is a goal. Or rather, two goals. Primary goal: Achieve what can be classified as "collect everything" in a particular game. Secondary goal: Do that while completing the game as fast as possible.
"Maximum score" is likewise a goal (usually with the implied secondary goal of achieving it as fast as possible).
"No B button" is a restriction: The goal is to reach the end of the game as fast as possible, but a self-imposed restriction is imposed to make achieving that goal more difficult. (Not pressing the B button is not a goal itself, because it's not something that you achieve or reach in-game, or enticed in any way by the game.)
Other categories can be fuzzier in whether they are "goals" or "restrictions". For example using an alternate (slower) route could be said to be a goal (because it's something that you aim to reach in-game, just like "100%"). Using an alternate (slower) character could be classified as either (you could see it as reaching game completion with that character, of you could see it as a self-imposed restriction to make the actual goal harder).
I think that "arbitrariness" can also be classified into two: Arbitrary goals and arbitrary restrictions. Whether a goal or restriction is "arbitrary" is subjective, but I would say that most often arbitrary goals usually feel "less arbitrary" than arbitrary restrictions. After all, a goal is a goal, something that you can achieve or reach in-game. A restriction is self-imposed, not something imposed by the game itself, and therefore more up to one's own opinion and taste.
"Arbitrariness" is often something that's very game-specific, and/or has no rhyme or reason (for example "collect 15 coins" in a game with hundreds of coins, and where there's absolutely nothing special or particular about collecting precisely 15 of them, and it would make no difference whether it would be 14, 16, or 27.) That's not to say that all game-specific goals or restrictions are arbitrary, but arbitrary goals and restrictions often tend to be.
Ultimately it's up to subjective opinion what is "arbitrary" and what isn't, and how much.
Yes, I think that the "limit laws" do state that lim(A(x)/B(x)) = lim(A(x))/lim(B(x)) only if lim(B(x)) is not 0. In other words, the law is conditional.
However, the post I saw stated that lim(A(x)^B(x)) = lim(A(x))^lim(B(x)) can never be assumed, regardless of any conditionals, which I don't think is true. The only situation where it can't be assumed, I think, is when both limits are zero.
Yes, and I think we can use the same example.
f(x) = exp(-1/x^2)
g(x) = x^2
Then lim(x->0) [ f(x)^g(x) ] = lim(x->0) [ exp(-1) ] = 1/e
but
[ lim(x->0) f(x) ] ^ [ lim(x->0) g(x) ] = 0^0 is undefined
I don't think that's a valid argument because you could use the exact same argument to argue that the rule
lim(a/b) = lim(a) / lim(b)
is invalid (even though it's one of the basic laws of limits).
Btw, a somewhat related question:
I saw someone in another forum claim that this equality does not hold, and can't be assumed:
lim(x->c)(f(x)g(x)) = lim(x->c)(f(x))lim(x->c)(g(x))
Is that the case?
It turns out (see this article) that if f(x) and g(x) are analytic everywhere and not the zero function, with the limit as x--->0 of both functions being 0, then the limit of f^g must be 1.
Dammit, I knew this to be true for some large family of functions, but for something like 30 years I couldn't prove (or disprove) it, or find the proper constraints or terminology, because I'm not a mathematical expert. Turns out it's a result known at least as far back as the 1970's.
When I was in high school (what now feels like an eternity ago), in calculus class the teacher presented us with the classical problem of finding the limit of xx when x approaches 0 (from the positive side). Since back then I was really into math, I found the proof that the limit is 1 to be marvelously interesting. (I have presented the question here too, years ago.)
I don't remember exactly when, but at some point I noticed that the same seemed to be true not only for xx, but in fact for a whole lot of different kinds of functions f(x) and g(x), for the limit of f(x)g(x) when both approach 0. I tried to come up with a proof of this, using similar techniques as with the proof of xx, but it wasn't as simple (back then I thought I had a solid proof, but it turned out later to be flawed and thus invalid. I don't think the result can be proven in the exact same manner as the more simplistic case.)
At university I presented the hypothesis to a maths professor who was immediately very dismissive of it, and in fact acted in a bit of a rude manner. He later apologized for his dismissive attitude and considered the hypothesis more seriously, and presented a counter-example. I don't remember what it was exactly, but it was a function that had a discontinuity at x=0, and he simply used a case definition (ie. like "f(x) { (that function) when x≠0, 0 when x=0 }").
I wasn't satisfied with that counter-example because it just felt like "cheating". Back then I didn't know the proper mathematical terminology for what I was trying to express. Asking around I came to the conclusion that what I was looking for were so-called "smooth" functions. Although, apparently, that isn't enough of a restriction either, and what I was actually looking for were analytic functions.
I just intuitively knew, all these decades, that there had to be a concrete family of functions for which that hypothesis is true, and it turns out that I was right, all these years. I wish I had known about that result a long time ago.
Is it possible to have two functions f(x) and g(x) such that
lim(x->a) f(x) = 0,
lim(x->a) g(x) = 0
lim(x->a) f(x)g(x) = 0
and f(x) is not just the constant 0 (nor reduces to the constant zero, ie. eg. "f(x)=x-x" would be too boring of an answer)?
I asked this question many years ago, and one of the answers suggested was:
f(x) = exp(-x-4),
g(x) = x2
a = 0
I was satisfied with the answer back then. However, I recently realized that f(x) isn't actually smooth (at least in the sense that it has a discontinuity at the critical point we are interested in, ie. x = 0.)
I suppose I have to update my question a bit:
Is it possible to have two functions f(x) and g(x) such that
f(a) = 0,
g(a) = 0
lim(x->a) f(x)g(x) ≠ 1
with the conditions that f(x) cannot be the null function, and both functions must be smooth (or analytic, I'm not sure about the difference) around a? Ie. there cannot be any discontinuity in either function at x=a.
(I'm using the term "smooth function" meaning that the functions are continuous and all of their derivatives are also continuous at x=a. I'm not sure if the requirement that they be analytic is also required.)
Also, no case trickery (ie. "f(x) = { (some function) if x ≠ 0, 0 if x = 0 }".)
Also "hostility" seriously?
Either make a valid point here about why you're for or against a massive list or just don't participate in the thread in the first place.
You're not a Publisher or an Encoder, you seriously lack the information.
That sounds like a rather hostile attitude to me. Just calm down.
For a project I would need to solve this:
Given x and a, solve y:
tan(y) * ((1 - a*sin(y)) / (1 + a*sin(y)) ^ (a/2) = x
If it's not possible with a closed-form expression, then an approximation (eg. using Newton's method) would be fine too.
Because of my work I have had to acquaint myself with geographic coordinate systems. I find the subject more interesting than I thought.
One could naively think (and probably the vast majority of people do) that geographic coordinates are quite simple: The poles are unambiguous, fix the zero meridian somewhere, and then just use degrees of latitude (north-south angle) and longitude (angle around the equator). Simple and clear.
But it's not that simple. Why? Because continents move, that's why.
In numerous applications it's useful, even necessary, to know the exact coordinates, to the centimeter level, even millimeter level, of particular points eg. inside a city, or overall inside a country. The problem is that if you used global coordinates for this, what you measure today exactly will be off by several centimeters in a year's time, because of continental drift. This is problematic. Because continents move many centimeters, even tens of centimeters per year, if you needed the exact coordinates of a point at the millimeter level, you would need to be constantly adjusting and re-measuring, which isn't practical.
The zero meridian used in global coordinates suffers from the same problem: Where should this zero meridian be? And, inevitably, the zero meridian also moves because of continental drift. Other continents move in different directions relative to it, so their distance to the zero meridian is constantly changing.
This is why continents and individual counties have local coordinate systems, which are fixed to the soil of the country, rather than the entirety of the Earth. The local coordinates move with the country, as continents drift. This suffers significantly less cumulative errors over the years. It's also the reason why if you ask the corresponding city or country department for the exact coordinates of a particular point, they will usually give it using a local coordinate system rather than universal coordinates. The local coordinates are more useful.
An even more problematic measurement is that of height. In some situations height is actually even more important than latitude and longitude. Many applications need to know also height at the centimeter, even millimeter level. The problem is: What should height be relative to? This is actually not an unambiguous or easy question.
Relative to sea level? Problem is that sea level isn't actually constant. It's at different distances from the center of the Earth at different parts of Earth. Global coordinate systems (such as GRS80) express height relative to a hypothetical ellipsoid that approximates the surface of the Earth. Problem is, again, that the Earth isn't actually a perfect ellipsoid (even ignoring mountains etc), and that approximation is just that: An approximation. It can differ from actual sea and ground level by quite a margin in some places. In fact, if you use the GRS80 ellipsoid as your height system, you'll get seemingly paradoxical situations where rivers flow uphill, according to GRS80 heights, because gravity on an ellipsoid is a bit wonky like that.
This is why, once again, most countries will use a local height system as well. This local height can differ by several meters from a global coordinate system like GRS80 (for example the current standardized height system in Finland differs from GRS80 by something like 16 meters or so). They usually also fix the rivers-flowing-uphill problem.
One attempt at a completely neutral and universal coordinate system is the ECEF XYZ coordinate system. Rather than use degrees of latitude and longitude, it just uses Cartesian coordinates: The origin is at the center of the Earth, the Z axis goes directly through the north pole (which is unambiguous) and the X axis goes through the (internationally agreed) zero meridian. The unit of distance is 1 meter. All positions anywhere on, in or above the Earth can be thus expressed as x, y and z coordinates.
Of course this suffers from the drift of the zero meridian, so its coordinates can only be used relatively temporarily for anything, and cannot be really relied on staying the same long-term, especially if used for local coordinates.
But this got me thinking, and finally after this gigantic wall of text, comes the physics question: Does the gravitational center of the Earth stay the same, or does it change over time? The poles in this system are the two points on earth that stay stationary as it rotates. But do these two points also stay the same over time, or do they drift?
It's completely fine. There is no "necroposting" rule of any sort here. If you have something new to add, it doesn't matter how old the thread may be. It's welcome.
It actually takes a lot of conscious effort to do the 4-7th gyms in order that most don't take. Sabrina is typically the 5th for people, but many also don't even see the Celadon Gym until later when they're stumped since it's on the dead-end part of Celadon.
If it's reasonably clear that the game developers were OK with the player not doing all the gyms in a precise order, ie. they didn't put any hurdles in the way that would need to be skipped by abusing glitches or clearly unintentional design flaws, then it should be obvious that the runner can complete the gyms in any order (at least those gyms that can be reached in such a non-glich-or-design-error manner).
"Intended route" in this context means that nothing is skipped by abusing glitches or level design flaws (it's usually quite clear if something is a level design flaw or not, as it usually requires a lot of effort and trying to find them, and it's usually quite clear that the skip was not intended by the level designers). Doing things "out of order", or even skipping some goals, is completely ok if it's relatively clear that the game was actually designed for that to be perfectly possible by a player who's playing the game normally.
Of course this still leaves open the question of whether glitches, or even quirks in the game engine, can be abused to traverse the intended route faster than would normally be possible. I would say that for the most part yes: The aim is to show the intended route, the intended actions in a normal playthrough. This route does not need to be traversed at a "normal" speed, as long as it's traversed, and as long as nothing is skipped that would normally have to be played through.
No glitches are to be used in the run, these include: screen warps, guard jumps, clipping corners while falling, etc.
I am wondering if it would be good to actually specifically list all known glitches that are known, and avoided in this run. Also, if possible, techniques that are allowed but which might perhaps be interpreted by some to be "glitches".
I ask this because this kind of would have to be a requirement for any run that intends to obsolete this. Anybody making a run intending to obsolete this would need to follow the exact same restrictions, or else it wouldn't be comparable, and we would enter this whole problematic issue of what counts as "glitchless" and what doesn't. Maybe someone makes a new run that's faster than yours, which abuses a glitch that you merely collated into that "etc" part. If you didn't specify it, who's to say it can't be used?
(Of course even then there's the question of what exactly is the "official" list of banned glitches for the "glitchless" category of this game.)
The latest Numberphile video brought up an interesting pattern in the prime numbers. Or, rather, in the list of sums of two successive primes. This list is:
5, 8, 12, 18, 24, 30, 36, 42, 52, 60, 68, 78, 84, 90, 100, 112, 120, 128, 138, 144, 152, 162, 172, 186, 198, 204, 210, 216, 222, 240, 258, and so on.
Curiously, the segment 12, 18, 24, 30, 36, 42 consists of successive multiples of 6.
Then there are multiples of 6 at semi-regular intervals until we reach another surprisingly long segment of consecutive multiples of 6: 198, 204, 210, 216, 222.
(The next two numbers, 240 and 258, are also multiples of 6, but not the next consecutive ones after 222.)
I wonder if this is just coincidence.
I have sometimes thought of the Ship of Theseus thought experiment, and how it applies to us.
The original question is: Suppose there's a seaship. Over the years and decades, parts of it get broken and replaced. As enough time has passed, every single original part of the ship has been replaced with a new one (perhaps even several times in some cases). Can it still be considered the same ship?
A more modern variant: You buy a brand new computer. However, over time parts of it break and you replace them. First the internal parts, and at some point even the computer case itself. After some years not a single part of the computer is original. Even the main hard drive has been replaced with a new one, just with the data from the original copied to it. Can it still be considered the same computer?
How does this relate to us?
Well, consider that it's estimated that every single molecule in your body will have been replaced with a new one in the span of about 7 years. This means that 7 years from now it's likely that not even a single original molecule remains, and your entire body has been replaced.
Can you still be considered the same person?
Or are you just a "copy" of your own self, from 7 years ago? And an imperfect copy at that.
Suddenly, the Ship of Theseus thought experiment doesn't sound so silly or trivial anymore.
Being an anti-vaxxer is not an 'activity', like driving a car or practicing a sport, that involves taking some risk. It's just a risk on itself without any action coupled.
Note that being hesitant with the covid-19 vaccine does not necessarily mean one is an "anti-vaxxer". Some people may have zero problems with other vaccines, but may have doubts about this one because its development was rushed, and many brands are very special type of vaccines that completely different from traditional vaccines.
This is nothing new or special. For example, many people may have zero problems in getting for example their tetanus vaccine at the recommended intervals (10-20 years), but may be hesitant about getting the influenza vaccine. That's because influenza is special, and influenza vaccines are a bit different from other typical vaccines.
The problem with influenza is that there are many strains of it, the vaccine only protects against a particular strain, and every year the influenza vaccine is just an educated guess of which particular strain will become a pandemic that particular year. This guess is only sometimes right. If it's not, then you can still get influenza regardless of the vaccine, because you weren't vaccinated for that particular strain.
However, that's not even the main problem. The main problem is that you would need to get vaccinated every year (with something like a 50-50 chance that they guess the strain correctly), and the influenza vaccine in particular can have more severe side effects than other vaccines usually. There's a reason why (at least in most places) they tell you to wait in the lobby for 10-15 minutes before leaving, after you have got vaccinated. They aren't doing it for the fun of it, nor are they being ridiculously cautious. They are doing it because the influenza vaccine in particular has a relatively high chance of causing a very strong immune system reaction soon after administered, that's very similar to influenza itself. Typically you may get a sudden onset of high fever, dizziness, weakness and even fainting. People have been hospitalized because of this, and it may take days for them to fully recover.
So every year you are taking the risk of being hospitalized because of a vaccine that has like a 50-50 chance of protecting you from influenza that particular year.
No wonder some people are a bit hesitant about it, while not being hesitant about other vaccines (such as tetanus).
