it's just tough to quickly recall a word's meaning when i have to think of it's exact definition. much easier to just think about how it's defined in terms of other things i'm talking about at the moment
e.g. i'd immediately recognize how open and closed sets are defined when i'm talking about boundaries. but if i'm talking about limit points it becomes a lot more of a headache
then trying to define limit points/isolated points nicely in terms of the easy interior/boundary/exterior points becomes really messy
hell i even found an asterisk when trying to define isolated points in terms of limit points (i.e. it must not be a limit point, and it must also be in the set)
I think the problem is that you're expecting topology to make intuitive sense
But our intuition operates mostly on "nice" topological spaces like metric spaces, where much of the finer points of topology are completely irrelevant
In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset A of a topological space X, is a point x in X such that every neighbourhood of x (i.e. every open set containing x) contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A, thus
x
∈
Cl
(
A
)
⟺
∀
U
⊂
X
open
:
x
∈
U...
Like why do we need this, we already have limit and isolated point
Is it too hard to say "in the closure"
"Adherent" isn't really a useful way to think about it, not more than "limit". these names are terrible
In mathematics, a subset
A
{\displaystyle A}
of a topological space is said to be dense-in-itself if
A
{\displaystyle A}
contains no isolated points.
Every dense-in-itself closed set is perfect. Conversely, every perfect set is dense-in-itself.
A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers (considered as a subset of the real numbers). This set is dense-in-itself because every neighborhood of an irrational number...
Which is distinct from the definition of "dense" (i.e. everything is an element or limit point of X)
Since everything would be literally dense in itself (everything's an element of it's own set)
Come on at least choose a better name, I spent 30 mins confused before realizing it was a new term
I think much of this terminology stems from the fact that topologists often try to look at definitions that are equivalent in "nice" settings and then try to figure out how much you need to weaken the axioms in order to make the two notions diverge
That is, the terminology is not intended to be pedagogical in any way - many of these notions are irrelevant to any applications of topology to other fields, it's terminology purely invented for some papers exploring the distinction between these notions in exotic settings most non-topologists won't ever encounter
@SirCumference Or you could make a really big deal about it; make a blog with like a million posts on the topic; and also post it in the comment sections everywhere else even slightly related. Just become "that guy" who always brings it up. There's nothing to lose but your sanity!
This question is closed as being opinion-based. Perhaps I do not understand English well, could someone explain in more detail what opinion-based is on this question?
I'm confused about P/E ratios. For example, a P/E ratio of 30 means that investors are willing to pay 30$ for 1$ they earn? Like... wtf? Aren't they loosing 29$ each year then. I'm confused.
I wonder if there's a polite way to tell a prof their 9am mandatory attendance is extremely destructive for sleeping with insomnia and studying during the day
"A Wikipedia factoid also compares the Sun's heat-per-unit-volume to the heat produced by an active compost pile, although the energy production from compost varies with temperature—since a hot compost pile kills off the organisms that do the composting."
Is it legal if I put Steve Job's picture and name on my website?
The website will make money.
Is there anything illegal if I put Steve Jobs or Bill Gates' picture, name and description on my website which I plan to commercialize and make money from?
I'm planning to build a website where I'm going to make money from their faces and names. Is that legally fine?
@YuvrajSingh... Are you hovering at a fixed distance from the planet or falling freely towards it? This makes a difference because if you are falling freely you will experinece a tidal force, though it's a very small one.
Th gravitational force always points towards the centre of the planet, so the forces at your head and at your feet are at slightly different angles. They converge slightly.
And that means they have a non-zero component along your length. So the gravitational forces are slight compressing you.
So, yes, you do feel a tidal force in this situation, though unlike the usual tidal forces it is compressing you not stretching you.
[N.B.: I use a Mac, but I'm assuming the following is not Mac-specific.]
I've noticed that, font-wise, SE sites can be divided into two categories:
1) Those that employ a sans-serif font (e.g., TeX SE, superuser SE, and this one). These display numerals as "lining figures" (~full-height number...
And the prof gave this explanation for osmosis: higher solute concentration dictates lower water concentration, and diffusion across a semipermeable membrane produces a flow of water from the higher water concentration to the lowe
@NovaliumCompany I really enjoyed Dune. It's a very well known sci-fi book. There's also like a bunch of sequels; but the advice for those is just keep reading until you stop enjoying it, apparently they just keep getting slightly worse.
@Semiclassical I think I've always only heard the simplified explanation as well. I was personally never tested on it or had to apply it since HS chemistry though; but I don't remember anyone clearing up that it wasn't diffusion. I don't think we ever covered it in physics though, so I'm not overly surprised chemistry had a misleading interpretation.
If I read the blog post (and some of the other sources) correctly, then one should have the following simple counter example: sodium fluoride, dissolved in water, should increase the water concentration rather than increase it
@JMac that said, this is supposed to be a physics course
If all he asks them to do is compute the osmotic pressure or predict the direction of flow, then fine—they just need to know that osmosis directs solvent flow from higher solute concentration to lower
But if there’s any concept multiple-choice questions...
Yeah, I'm fine with what I learned because in the context, we weren't necessarily good at physics and able to describe it in an accurate way; since it was a chemistry class. If the professor asked about it an expected that wrong answer; that's definitely quite bad.
That said, I'm going to break out my usual rant; that I had a "Advanced Math 11" teacher try to convince our class that 'double rounding' was reasonable. I.e. 0.45 would round up to 1 in the nearest integer because 0.45 rounds to 0.5 which rounds to 1. So teaching a common misconception about osmosis barely phases me compared to that.
Description of the system
Assume two point masses one at the point $C$ and the other on the circumference of the circle with radius $R$. They are attracting one another gravitationally and no external forces are acting on them. The point mass at C has very large mass such that it is at the COM...
Does anyone think that my derivation of the equation for angular momentum of the external particle is wrong (I ask this because I have got two answer which are implying that $\ell =rp$ and not the one I have derived).
@JMac I think I'll try to read only the first book. I'm really excited because there's a movie and I want to see how close my imagination is to the movie.
@AaronStevens do you think the central mass has an angular momentum of $\ell _{central} = -Rp \sin {\omega t}$? It's quite hard to derive it for central mass!
Is the global economy currently in an expansion, boom, recession or depression?
I feel like just how after the first world war people called just "world war", never assuming there would be a second one, the same way we call just "great depression". xDD
@JohanLiebert I am still thinking about it, but I think JEB's answer is correct. Although some unsaid assumptions in his analogy might be bypassing your issues. I am not entirely sure yet, and I don't know if I will have the time in the near future to think more carefully about it.
@AaronStevens I have derived it and am going to post the answer in some time. But I would need your help to check if I am doing anything wrong somewhere.
During an economic recession, whom are the people selling their stocks to? After all, everyone panics and everyone wants to cash out. Who the hell buys their stocks at that time?
I have a couple of Eloy albums somewhere, if they haven't got lost over the years, but I would have put them in the prog mainstream as in pleasant to listen to but not terribly innovative.
there was a particular assertion made ("The semipermeable membrane exerts the force that drives solvent flow") which the prof thought was clearly wrong
The good prog bands tend to be the well known ones - they became well known for a reason. Though I would have to concede this isn't universally true as the inexplicable popularity of Emerson, Lake and Palmer testifies.
@AaronStevens @JohnRennie can the universe have a net angular momentum? (I mean does this make any sense for the universe to have a non-zero angular momentum?)
@JohnRennie while making an argument in my answer to my own question I assume that the total angular momentum cannot be a non-zero (and this simplifys the situation). So what should I do?
Thanks for the answers they gave me motivation to try deriving angular momentum for the central mass.
The following picture says most of it. (A description is coming ahead):
Description
The outer mass is at a distance $R$ from the com and has a linear momentum as $P$. The inner mass has is ...
@AaronStevens you can have a look at my answer. Is it correct?
Well, before trying to read the book I decided to read a quick summary. The annoyingly complicated names and political style of the book are not my type.
It uses unnecessarily twisted linguistics, which pushed me away as well. I'm a fan of simplicity.
(or at least not overcomplicating things unnecessarily)
Fair enough I guess, though I tend to avoid reading summaries for books before I read them and usually just stick to the blurbs to see if that interests me. Personally, I also wouldn't give up on books just because of naming conventions; but I guess I kinda grew up reading fantasy so I'm just really used to that.
The way I see it, it's not much different than describing anything, besides that the people and place names are just not ones you're used to; but that can happen without fictional place names.
I think you might like "harder" science fiction. Dune does actually have some pretty heavy fantasy elements, and it sounds like those are a bit off-putting for you.
Have you read The Martian? I haven't read the book; but it's supposed to be really good, and I thought the movie was really good. It's pretty much all real science, so quite different than Dune.
Another one where I've seen the movie but haven't read the book is Contact en.wikipedia.org/wiki/Contact_(novel) by Carl Sagan. The movie was pretty cool again; has to do with aliens and SETI stuff.