You are not logged in.
Let com(A) = complement of A and clo(A) = closure of A. The key thing here is com(clo(com(A))) can produce a new set.
In your first example, com(clo(com(A))) = (a, b), which is not in your list. Now clo(com(clo(com(A)))) = [a,b], which you already have.
For latex, the command \overline is almost always better than \bar.
You'll find that trying random sets won't get you very far, at least not very quickly. What you want to do is look at things in a more general setting.
If we do com(clo(A)), we get every point which is not a limit point of A. On the other hand, clo(com(A)) will give us every point which is not an interior point of A.
If we want to keep getting "new stuff", what has to happen?
That little bit cleared up almost everything for me, Thanks! Now I'm thinking of indices like I think dictionaries in Python, is that correct?
Correct. Just remember that the name you call something is important. U_0 conveys information about what thing I'm talking about, here the interval located around the point 0.
Just one last question, how would you relate your example:
to this notation:
I would write:
An alternate way to write this set is:
And just to wet your appetite, here is something you'll probably see in complex analysis, called a residue integral.
You use complex analysis to calculate the value of that integral. Another cool theorem is known as Liouville's theorem:
If f is a complex differentiable function defined on the entire complex plane and f is bounded, then f is a constant function.
Being a fan of calculus and complex numbers, I thought that'd be the coolest thing ever, but just a few weeks in I'm already fried out.
Keep going. I am not a fan of calculus (for the most part), and I found it was really cool. Your professor is entirely right of course, you do need a good background in topology when learning complex analysis.
Not having a background in proofs however, things may get a bit hard. One thing to keep in mind is that it is supposed to be frustrating... One of my professors described point-set topology as "...the gates of Hell. Everything you thought was true turns out to have a counter-example."
Is there an example you could give, I think it's all the theory without any examples that's overwhelming me.
Unfortunately that is an example... and this is something you're going to have to get used to over time. In more advanced mathematics, examples can (at times) be simpler theorems...
But let's start slower. Let's say I have a set {1, 2, 3}. When I index one set with another set, what I do is a set up a 1-1 and onto correspondence. So for example, I can index {1, 2, 3} with the set {x_a, x_b, x_c}. So now:
x_a means 1
x_b means 2
x_c means 3
And in this example, what I've just done is a bit silly. There is really no need to index this set, or at least I can't think of one. But when you go to bigger sets, indexing makes referring to elements a lot easier.
Let's go to the real line. Now I'm doing a problem, and for every integer I want to associate with it an interval of size 1/2. So for example, I associate the interval (-1/2, 1/2) with the integer 0, and the interval (1/2, 3/2) with the integer 1. Now the number 0 corresponds to my open set (-1/2, 1/2). But it would be just awkward and confusing if I were to just write the number 0 for my interval. So I introduce a letter, say U, by which I will index with my number 0.
So for example:
U_0 = (-1/2, 1/2)
U_1 = (1/2, 3/2)
....
Now writing U_15 is much easier than writing (29/2, 31/2), and much more clear to the reader.
What's the difference between Point Set Topology and Algabraic Topology?
Point-set topology can be viewed as a generalization of real-analysis, where you study a space of points and sequences/functions on them. Algebraic-topology is using the tools of algebra to study (point-set) topological spaces.
Can someone kindly explain to me what indices are? The book show the notation, but doesn't really explain it. Some examples would be great.
Indices is the plural of index, so I'm not entirely sure what it is you're referring to. My best guess is that a lot of times in topology you use an indexing set. For example, if I have an open set O (in the reals), and a point alpha is in O, then there exists an open ball around alpha contained in O, denoted by U_\alpha. Then
Now my collection of these balls is indexed by alpha. Is this what you mean?
Am I supposed to be providing a counter example?
Yes.
And super bonus points to anyone who can explain to me why I shouldn't just give up now and do something easier.
I miss calculus.
Because all math is hard and it will fight you. If you have patience, you'll get it. The great thing about mathematics is that the only way you can ever really lose is to throw in the towel.
Why are you studying topology in the first place? Do you plan to go on to grad school? Or are you simply interested in the subject?
Algebra won't be enough. You have the points A = 0, A = -1, and A = -2 all give the value S = 0. This means the inverse (i.e. solving for A) won't be a function. You can locally come up with approximations for the inverse where the derivative with respect to A is nonzero, but this is about all you can do.
Another great problem published by the Association of Math Teachers Obsessed with Fruit.
In the Fox TV series House, House and Wilson are named after Holmes and Watson. Also, the band Jethro Tull has an awesome song "Baker Street Muse".
3. Lake Erie has a maximum depth of 210 ft.
That is precisely the point, we don't understand much about anything. If the law of gravity were suddenly repealed, I don't think duct tape would help.
You seem to be missing the point. We make decisions every day based on incomplete knowledge. We go with our best understanding, which is entirely rational (since by definition there isn't anything better). Unfortunately we must do this in science as well. To "wait until all the facts are in" is silly if there is enough evidence to make a decision, even if the knowledge is still incomplete.
What I was getting at is this: Cryptozoology is not your or my profession. Studying up on medicine does not a doctor make. Your opinions may be invalid.
And a doctor's opinions (on medicine) may be invalid! Being an expert does not suddenly make you into an all-knowing creature. I feel sorry for you if you think that just because you are not an chemist you can't conclude that homeopathy is bunk.
Ricky wrote:
"They laughed at Columbus, they laughed at Fulton, they laughed at the Wright Brothers. But they also laughed at Bozo the Clown." - Carl SaganI wish I knew what was worse, that statement, or how it gets interpreted. Why do we have to focus on Bozo while missing the tragedy of laughing at the others. Experts laughed at Columbus,Fulton,the Wright brothers,Leeuwenhoek, Pasteur, Darwin, Freud, Cantor and a host of others. Experts are still laughing at David Deutsch, Doron Zeilberger, Linus Pauling. Nikola Tesla and many more.
Because that's the point of what he was saying! Yes, it is well known that many scientists were laughed at and turned out to be right. It is so well known it has become cliche. But the point of the quote is that just because someone is laughed at does not make them right.
Some eyewitnesses are in error, but how does that prove they are all in error, Some eyewitnesses might be relating the truth.
It doesn't prove that any are in error! But the main thing is that it doesn't show with any reasonable amount of certainty that any are right either. It can't count as positive evidence, only suggestive. If there are a lot of eyewitnesses, then this could suggest the need for more research. But we can not accept eyewitness testimony as evidence.
This was the point I was trying to make before. If we were to accept eyewitness testimony as evidence, then we would have evidence for fairies, UFOs, Elvis, and so on.
How reliable is it when we say we know? Can we assign a probability to the statement? Would it be accurate? How do we determine the error bounds? Your sense of knowing is constantly evolving, refining itself. It is not unusual for people to make abrupt changes or even reversals in their most precious beliefs.
You're going down the path of the Bayesians, which personally I do not recommend (from a philosophic standpoint). Everyone's judgment of how much is enough is different. When you have the entire field of zoology not recognizing cryptozoology as a valid discipline of science, this to me is enough.
As for abrupt changes in precious beliefs, it happens, but it doesn't happen as much as it should.
My sister has one of those problems. It does not make her see bigfoot.
Either there is a serious misunderstanding here, or you're pulling my leg. I'll assume it is the later for now...
when sometimes the exercises ask to use the Cauchy integral theorem how is work!?
There are multiple versions of Cauchy's Integral Theorem, so you need to be more specific. The simplest version says that if f is analytic in ball of radius R, then the integral of f around that ball is going to be 0. Actually, it really says a bit more than that, it says that f has a primitive (and so the integral around any closed curve in that ball is zero).
But as this example demonstrates, f may very well have a primitive and the CIT need not be applicable.
Open-mindedness means not making a decision until all the facts are in. Unfortunately neither you or I know anything about this field. We are both amateurs. We are not qualified to judge it one way or the other.
If you are going to wait for all the facts, then you are going to be waiting forever. There will never be a time where all the facts are in. That's the way science works: there are no absolutes. But even you are guilty of not waiting till all the facts are in. We don't understand much about gravity, yet you are not afraid of gravity suddenly reversing itself in the middle of the night. Why do you not duct tape yourself to bed before you go to sleep?
Also, don't presume my knowledge on the subject. One may not be an expert but still know enough to make informed decisions. I have studied it quite a bit, and have come to my conclusions because of it.
Historically there have been many opinions that scientists have ridiculed and were forced to accept later.
"They laughed at Columbus, they laughed at Fulton, they laughed at the Wright Brothers. But they also laughed at Bozo the Clown." - Carl Sagan
I think with that list your getting away from what cryptozoology is supposed to be. Some of them we might be able to rule out. One of those topics is still a big area of research. I do have to say that I am not even a talented amateur on any of those subjects. To be open-minded I have to say I don't know. Can you say more? I am willing to listen.
The list was not aimed at cryptozoology, but rather at the reliability of eyewitnesses. If you think we can rule out some of them, then I take it you agree that eyewitness testimony is not a reliable source of information.
Saying you don't know is a great answer. But remember that even when you say you do know, there is always a chance to be wrong. When I say I know something, I am not being dogmatic. It is only to the best of my knowledge. In that sense, I know bigfoot doesn't exist.
Ricky wrote:But experts would not call an eyewitness crazy. Such an idea that they would is rather... crazy.
Are you calling my ideas crazy? If so you can hardly expect me to agree with you about that.
No... it was a joke.
Aren't you calling the eyewitnesses liars or hinting that they have some sort of mental problem?
How the heck did you get that? Obviously some eyewitnesses are liars, that should come as no surprise to anyone. As for the mental problem, let me try to be a bit more clear. False Memory Syndrome and confirmation bias are things that afflict even the most healthy of brains. FMS tends to afflict only those who have gone through traumatic events. Though some are more susceptible to hypnosis than others, this is no indication of a mental problem.
They are just well documented cases that our brains are not always perfect.
I have not implied that there is a conspiracy here. Just a lot of excited people. I do think that believing that there are no conspiracies is a bit naive. Scientists again, are people. If some of them were covering up anything it would most likely be for money (I guess).
The way scientists make money (i.e. get grants) is to beat other scientists to new discoveries, or show that other scientists are wrong. Science is an extremely competitive discipline, and money is made by exposing new things, not hiding them.
I have always found it interesting, because I'm open-minded.
"Open-minded" is not about accepting or not rejecting things. It is about considering them. When you claim you have evidence for (in the this example) bigfoot, an open minded person will say "Show me" where as a close minded person will say "No you don't" or perhaps even "Awesome, I believe it!" You see, a close minded person is someone who accepts or rejects things without consideration.
In contrast, I reject the notion that crypto-zoology (how it is commonly defined) is a valid field of scientific study. However I have given rather thorough consideration to it, so I claim that I am still open-minded.
It is the study of legendary creatures (creatures that modern science won't admit exist).
bobbym, this wording suggests that you think there is some conspiracy to deny the existence of these creatures. Is that correct? If so, what do scientists have to gain by doing so?
Have experts denied its existence? Have they called the eyewitnesses crazy or fame seekers?
It's important to note here the unreliability of eyewitnesses. If you were to accept eyewitness testimony, then you would have to accept:
Aliens
Bigfoot
Loch Ness
Cold fusion
Fairies
Leprechauns
Perpetual motion devices
Ghosts
And the list goes on. But science has shown time and time again that eyewitness testimony is unreliable. There are many various reasons for this, some as simply as "He's lying" to more advanced psychological explanations such as confirmation bias. There are also various phenomenon related with memory (e.g. False Memory Syndrome) and especially hypnosis. These are all well documented and studied phenomenon.
But experts would not call an eyewitness crazy. Such an idea that they would is rather... crazy.
Let x be the number of eggs he ordered. What equation can you use x in which describes this problem?
The first half of the Jordan-Holder program is estimated at 10,000 pages.
There. That was not really heavy machinery, was it?
That solvability of composite order implies nonsimplicity is not, no. In fact, it becomes more obvious when you use the decomposition series that comes from being solvable. On the other hand, Feit-Thompson's theorem required 255 pages of heavy machinery to prove. I'm not sure of any proofs discovered since then, so perhaps it has become shorter with time.
The number of Sylow p groups must divide q and be congruent to 1 (mod p). The only such number is 1, and thus the Sylow p-subgroup is normal.
Jane, let's just continually modify your opening post, keep them all in one place. If we have a general argument that works for multiple orders, or if an argument for a certain order is long, make a post below it and then just reference that post number. Also, please write the prime factorization for each order in the list!
I will be cleaning up this thread periodically, removing posts which don't contain material used in the proofs. Let me know if you object to this.
The minimum modulus principle holds, but the theorem must add the case that mathsyperson pointed out:
If f attains it's minimum inside a region G, and f is never zero, then f is constant.
It is a good exercise to prove this. As for "in" and "on", they are used interchangeably.
There's no similar way to pull a third dimension from the complex numbers, because the complex numbers are algebraically closed.
It is important to keep in mind that this is one way to look at the complex numbers, but not the only way. The complex numbers form a 2 dimensional vector space over the real numbers, with multiplication defined in a funny way. They actually have a lot more properties than that, but this is just one view.
The question is can we do the same thing with a third component, extending the complex numbers? The answer is a rather surprising no. We would have to lose a fundamental algebraic property to do so (I think it was the associative property, but I need to look this up). However, as discovered by Hamilton, we can do such a thing with four components, and these are known as the Quaternions. And so we have 1, 2, 4, and the next number is the Octonions.
http://en.wikipedia.org/wiki/Quaternion
http://en.wikipedia.org/wiki/Octonion
I would suggest to try to come up with arguments that handle a large number of orders, rather than go one by one. For example, any group of order pq will not be simple.
I didnt know what counted as heavy machinery.
Heavy machinery to me would be the theorem of Burnside and FeitThompson. With those, the task becomes rather trivial.
So, now that we have completed this assignment, whats next?
I'd be impressed if we did all groups of order up to 1000.
Now that you're done, the theorem I was speaking of was:
If G is a simple group, and H is a subgroup with [G : H] = n, then there exists an injective homomorphism
. That is, G may be identified with a subgroup of A_n.Using this, here is how I did it. First I proved this lemma:
Let p be a prime, and |G| = p^k * m, where it may be that p | m. If |G| does not divide m!/2, then G is not simple.
Now using 120 as an upper bound for |G|, this may be now used to conclude statements:
If 2^4 divides |G| and |G| < 120, then G is not simple.
If 3^3 divides |G| and |G| < 120, then G is not simple.
If 5^2 divides |G| and |G| < 120, then G is not simple.
If 11 divides |G| and |G| < 120, then G is not simple.
If 13 divides |G| and |G| < 120, then G is not simple.
If 17 divides |G| and |G| < 120, then G is not simple.
If 19 divides |G| and |G| < 120, then G is not simple.
These divisibility conditions will get 22 group orders. We can squeeze more out of it if we forgo the use of the upper bound and just use a particular order. This will get groups of order 12, 18, 20, 24, 28, 36, 40, 42, 45, 74, 92, 98, 116. Each argument is just finding the right p^k so that |G| = p^k * m where p^k does not divide m!/2.
Most of the rest are prime powered or groups of order pq (with p ≠ q), all of which of course not simple. This only left 8 groups which were rather manageable. On each of these, either the counting arguments or a direct application of the theorem I gave above worked. That is, on all except 90. Here was my proof for 90:
Let G be a group of order 90, and assume G is simple. The number of Sylow 5-subgroups must divide 90/5 = 18, and is congruent to 1 modulo 5. As G is simple, we conclude this number must be 6. Let H be a Sylow 5-subgroup, and so [G : N(H)] = 6. Thus, we may identify G as a subgroup of A_6. But now |A_6| = 360, |G| = 90, and so [A_6, G] = 4, with A_6 being a simple subgroup. Thus, A_6 may be identified with a subgroup of A_4, a contradiction.
I do not believe you explain your sorting technique thoroughly enough. You assign numbers A=1, !A = 2, B = 3, !B = 4, and so on, but how does this assign a number to a triple such as:
A + B + !C
Intuition suggests what you're going for is 1 + 3 + 6 = 10, but then I don't see how sorting this makes reducible triples adjacent to one another.
da_coolest, find the intersection points, this is the area you want to be calculating. Now over this area, you want to calculate the integral of f(x) - g(x), or alternatively g(x) - f(x), which ever is positive.