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@mgmartin51 12 күн бұрын
At 6:50, because all the mathematicians would switch their votes.
@David__U 13 күн бұрын
I have a question about the first problem presented in this lecture at 0:30 (but don't feel obligated to answer; I'm not a student in your class, or even at Fairfield!): As a first step, you asked the students to find the dominant terms in the numerator and denominator, and they were to do so without showing any proof. For the numerator, n+2^n they were to recognize that 2^n dominates the n. What does this mean? I think it means that 2^n grows faster than n, i.e., lim [2^n / n ] goes to infinity, or alternatively lim [ n/2^n] = 0. Is this correct? Or maybe it would be preferable to say lim [(n+2^n)/2^n] = 1, but either way.... Then I don't understand your reply to the student who asked (at 4:26 ) if we could avoid using l'Hopital's rule by noting that lim [(n+2^n)/2^n] clearly equals 1.... since we already established that 2^n is the dominate term in n+2^n. Why can we accept (without proof) that 2^n dominates (n+2^n) but not that lim [(n+2^n)/2^n] = 1? They seem the same to me. Thanks for posting your lectures. I really enjoy them!
@profstaecker6579 13 күн бұрын
This is a good question- I am using the phrase "dominating term" somewhat informally. Like you say, if I say "a_n dominates b_n", this could mean that a_n > b_n when n is large, or it could mean that (a_n)/(b_n) approaches infinity, and there are other ways to interpret it as well. Generally I mean just that a_n > b_n for large n. In this video I'm talking about the limit comparison test, in which you compare the given sequence to a simpler one. You can choose the simpler one to be whatever you like, and then you must prove that the limit of the ratios exists and is positive. So the only step that needs to be justified by a proof is the limit of the ratios. When I stated that "2^n is the dominant term", I'm OK saying this without proof because that step is only used to help me choose the comparison sequence- it's not part of the logical argument. The only proof I really need is when we prove that the limit exists and is positive. And when doing that limit, we really DO need specific proof, which I why I didn't like the suggestion to simply say that (n+2^n)/(2^n) goes to 1 because "2^n is the dominant term". Because that's informal, but being used during a part which really needs to be fully justified. I hope that's clear! sorry for the long answer.
@David__U 12 күн бұрын
@@profstaecker6579 Ah, I get it now, but I'll review the previous lecture again anyway. Thank you!
@kryptosteel9809 13 күн бұрын
missing episode 27?
@mgmartin51 14 күн бұрын
You made me laugh at around 38:00 when you said "Isn't than in the Bible? Either that or Jurassic Park."
@วัยรุ่นยุค90-ล4ฃ 15 күн бұрын
What does calculus do with real number? Why do we need to understand real number before understand limit, or derivatives?
@mgmartin51 21 күн бұрын
At around 12:00, when you made the denominator 9n^2 - 3n^2, couldn't you just throw away the -3 to make the denominator bigger? Then you would have (21n)/ (9n^2) and wind up with 7/(3n).
@profstaecker6579 21 күн бұрын
Yes this would work too! Probably your suggestion is easier than what I did. These arguments generally have many possible sequences of steps which would lead to a good answer. (Lots of students don't like this, since it means they have to be creative!)
@FawadzMaddan 23 күн бұрын
Aloo sir
@gvvivekagvviveka6425 25 күн бұрын
Hey i am from India i found this video while search for calculus videos you know your videos are not bad they are helpful
@Nicole-xx8ll 27 күн бұрын
Thank you for the explanation 😊❤
@anaheim869 28 күн бұрын
For the epsilon-delta definition of continuity in R, I believe you do not restrict |x-a| to be strictly greater than 0, i.e, x different from a.
@profstaecker6579 28 күн бұрын
Yes you're correct- that was a mistake. Thanks!
@MrCEO-jw1vm 29 күн бұрын
My respect to you Prof Staecker!!! Pointset Topology is such an abstract topic and my university only requires linear alg and multivarible calc to take this. So, I have been having a tough time in this course until I found your course. You have made things very well and I hope many students struggling could find your course soon. I will follow through and take notes very closely with your lectures to make sure I finish this semester with confidence in this material! God bless you !!
@zhiqingchen632 29 күн бұрын
As a master student & someone switched to math in junior year I never had a chance to take topology during my undergraduate. Topology was a notoriously hard course at my undergrad school. Now I encounter it here&there in commutative algebra class…this series really saved my life. If I had a topology professor as an undergraduate student then I’d definitely like this subject instead of fearing it…prof Staecker you deserve a Nobel prize for teaching topology
@Theeasternmen Ай бұрын
Great Professor. Now I can tell people what Real analysis really is as well as point set topology.
@juyifan7933 Ай бұрын
There is a mistake in the classes of the torus. There should be a four point equivalence class with all the corners. In the way it was described the classes would not form a partition since the corners are both in the B_1 or 0 and in C_1 or 0.
@AimeePlaysMSM Ай бұрын
16:35 (hiking and distance measurement by apps) The way they actually work with timed intervals aside, this seems like one that should be reasonably measurable (with an error range), since every step taken is from a specific position to another specific position, and the nature of your hiking motion is one that even if not linear, does have a limit in the distance traveled between those two points. The exact positions of those points can be refined, but the distance between them when calculated would not go to infinity as the positions and trajectory remain static. I think
@profstaecker6579 Ай бұрын
Yes I think I agree- a person walking naturally has a sequence of footsteps, and you can define the distance as the sum of the straight distances between the footsteps. This assumes that the person is walking normally, not sliding or rolling, not jumping with 2 feet at once, etc. At some level, the specific location of a footstep is not a specific well-defined quantity. The app (I assume) is measuring individual straight hops of my phone's GPS receiver. I imagine the figures you obtain depend strongly on the time interval between each location sample, which corresponds to measuring the distance with different "stick lengths".
@CaylaHenderson-m6f Ай бұрын
She is all that 2:4:2:6
@CaylaHenderson-m6f Ай бұрын
Brea
@DashrathGurjar3.14 Ай бұрын
That's good 👍👍
@CaylaHenderson-m6f Ай бұрын
Yes
@AimeePlaysMSM Ай бұрын
32:00 you were fine 😊 🟥🟥| 🟩🟩🟩 🟥🟥| 🟩🟩🟩 🟦🟥| 🟩🟩🟩 🟦🟥| 🟥🟥🟩 🟦🟦| 🟦🟥🟥 🟦🟦| 🟦🟪🟪 🟦🟪| 🟪🟪🟨 🟦🟪| 🟨🟨🟨 🟪🟪| 🟨🟨🟨 🟪🟪| 🟨🟨🟨
@theskepticai3653 Ай бұрын
where is ep05?
@profstaecker6579 Ай бұрын
This is it! Sorry it was mislabeled- i fixed it.
@mgmartin51 Ай бұрын
Math can't explain why we are here? Wait, math explains why I am here!
@mgmartin51 2 ай бұрын
At 4:40: gotcha! log base 2 of 3 is not 8, but log base 2 of 8 is 3.
@profstaecker6579 2 ай бұрын
yikes! You're right- thanks
@mgmartin51 Ай бұрын
@@profstaecker6579 I really enjoy your videos.
@AimeePlaysMSM 2 ай бұрын
I love seeing these voting system-related math explorations so close to elections ☝One minor note 5:00 ... "C = 3 · 0 + 2 · 1 = 1" ? ( I kept waiting for the correction, or a student to catch it. It doesn't affect the IIA issue you were demonstrating. )
@profstaecker6579 2 ай бұрын
Sheesh! The students are supposed to notice those! Thanks for tuning in
@EastBurningRed 2 ай бұрын
Is a revolving door open or closed?
@juyifan7933 2 ай бұрын
I think a distinction needs to be made between a basis for a topology and a basis for the topology T (already established on X). The conditions given were for B to generate a topology, not for it to generate the topology T.
@walterpu8357 2 ай бұрын
This video made me cry. It saved my Econ PhD.
@sadianahar8820 3 ай бұрын
Such a put-together video!! i have been trying to understand this concept since few days and this video helped me grasp the concept quiet well.
@EthanDills 3 ай бұрын
Thank you so much for having these lectures and the course website publicly available!! This was great
@dansheppard2965 3 ай бұрын
I see from math overflow someone says that in the projective plane, the corner cells of tic-tac-toe are diagonally self-adjacent, so you only need two moves. I can kind of see where they're coming from. Say you had the top left corner cell and you break the diagonal move into two parts: an up (which takes you to the bottom right) and then a right which takes you back to the top-left! I suppose this is an "up-left" diagonal from the top left because the second part of the move, to the right, would have been to the left before flipping over. So going diagonally from the top left takes you to the top left, so you can use that one twice! The question is called "Tic-Tac-Toe on the Real Projective Plane is a trivial first-player win in three moves", and the answer is by Parcly Taxel. I won't post a link because then KZbin eats the comment!
@MattMcIrvin 3 ай бұрын
I never actually took Real Analysis but I did take a class on topology, so I approached this whole subject of open and closed sets from the other direction, with the general definition of a topological space, where you just arbitrarily take certain subsets of it as open, to define the topology, as long as they satisfy certain axioms. And then it was kind of a mental stretch to see how this very general definition corresponded to the ideas of open and closed that I learned back in junior-high algebra class. But it's interesting seeing the stuff I learned as the *axioms* of open sets derived as theorems from the standard topology on the real line. I guess from a topological perspective, these theorems constitute proofs that the real line with the standard topology is a topological space.
@MattMcIrvin 3 ай бұрын
(probably I actually did that as a homework exercise or something, but it was a long long time ago in a year that began with 1)
@MattMcIrvin 3 ай бұрын
This pattern of "introduce the theorem at the end of the lecture, talk about it less formally, then prove it at the beginning of the next lecture" is an interesting one that I don't think I ever tried when I was teaching stuff. Gives the class some time for the statement to sink in.
@Carmenifold 3 ай бұрын
when talking about which of those diamond sets are open in the given topologies, (around 26:00) i was confused as to why the diamond with only its lower left edge included was not open in the lower limit X lower limit topology. obviously the interior points of the diamond can be completely contained within a basis set, but since the basis sets include their lower left corners, couldn't a point on the diamond's lower left edge stay completely inside a basis set if you put it at that set's lower left corner?
@juyifan7933 Ай бұрын
Indeed, i think he made a mistake on that one.
@vector8310 3 ай бұрын
I saw the cone but not S^2. But for me that's progress.
@MattMcIrvin 3 ай бұрын
There's an extra wrinkle to watch out for in the diagonal proof: it's best to construct your "choose another digit" formula so that it avoids choosing 9s, just to make sure you can't get caught by the .9999... = 1 ambiguity in decimal numerals, which implies that some different decimal expansions can represent the same number. That's easy to do, though. You can just choose 0 if the digit is either 8 or 9, for example. And assume that in your list, you never use the representation that ends in an infinite string of nines. I think that as long as you do that, you can show that the diagonal argument is really the same thing as the nested-intervals proof. Choosing an nth digit that is different from the nth number is just a specific procedure for choosing nested closed intervals that avoid the sequence of numbers. Another quibble: I think alef-one is defined to be the "next" cardinality after alef-zero, so the cardinality of the real numbers is only alef-one IF the continuum hypothesis is true.
@MattMcIrvin 3 ай бұрын
26:30 I think the "when added to itself" is specifically where the natural numbers come in. He's talking about taking some fixed interval, then adding it to itself, that is, *multiplying* it some natural number of times. And that can be made to exceed any other magnitude, that is, any real number. So Archimedes is really saying that for any real numbers a and b, an > b for some n in N. That implies the modern version of the Archimedean property if we take a=1.
@MattMcIrvin 3 ай бұрын
Sup of the evening, beautiful sup
@MattMcIrvin 3 ай бұрын
I keep wondering about the first proof in this one, particularly the reverse case. We can imagine sort of reasonable definitions of |x| and > and < on the line with two origins: there's a second origin 0* which is less than any positive number and has absolute value 0*. Then the theorem is incorrect because the right side is true for 0*. So there must be some property of the real numbers we're implicitly assuming and using here that excludes the line with two origins, and I can think of a lot of them, but most of them are kind of close to assuming the thing we're trying to prove. Maybe that isn't so bad because it's not supposed to be a complicated proof. I guess the proof assumes that if x is not 0, |x| must be greater than zero, which is not true for the definitions I came up with for the line with two origins (|0*| must not be greater than 0, or else the proof will still work because we could just set epsilon to 0*).
@hansolo6831 4 ай бұрын
Euler wasn't a real person. Can't believe this is still being spread
@spenxerbdp9809 5 ай бұрын
Gem of a series, awesome teacher
@dispersoincella 5 ай бұрын
thank you very much for these lectures! I found them perfect and very useful... Thanks!!!
@aaryantejus1491 5 ай бұрын
Thanks professor❤😊
@dispersoincella 5 ай бұрын
About the equivalence classes of torus, I can't have corners point in two different equivalent classes, the equivalent classes are partitions of the set. I have to define a separate equivalent class for the corner [(0,0)] = \{(0,0), (1,0), (0,1), (1,1)\}, and exclude the extreme points from the interval of y and x.
@profstaecker6579 5 ай бұрын
Yes you're right- At 15:00 you can see I was a bit confused in the moment about how to handle the edges, and I did it wrong. I should've said something like: B_y is defined for 0<y<1, and C_x is defined for 0<x<1, and like you say there is ANOTHER class D = {(0,0),(1,0),(0,1),(1,1)}. Thanks for the correction-
@devanhill2330 4 ай бұрын
@@profstaecker6579 I am a little confused on this correction. If they are all in the same eq class then they have been realized to be the same thing, essentially, but we are not identifying (0,0) as (1,1) - for example. On the other hand, if we were to separate it so that we only connect the corners that we are "gluing" we would double count everything. Could you explain to me why the class D is correct?
@profstaecker6579 3 ай бұрын
@@devanhill2330 actually we do want to identify (0,0) with (1,1).
@CantTouchMe7 6 ай бұрын
I'm self teaching myself math as a hobbyist and the fact that you upload your courses is a blessing.Thank you man.
@aaaaaaaaaaaaa. 6 ай бұрын
Thank you for all those invaluable series of lectures. After attending all ToC contents, I became convinced that there could be a breakthrough that would outstrip the capability of the Turing Machine by expanding dimensions, just as the TM did on more rudimentary automatons. Difficulty(subjective): 10/10
@aaaaaaaaaaaaa. 6 ай бұрын
Thank you, extremely abstract however. Difficulty(subjective): 10/10
@aaaaaaaaaaaaa. 6 ай бұрын
Thank you for letting me know invaluable knowledge that deepened my understanding of computing! Difficulty(subjective): 3/10
@aaaaaaaaaaaaa. 6 ай бұрын
Thank you Prof.Staecker! Liked and subscribed. Question since the last lecture. How would the TM for a^n*b^n*c^n(and rest of them) act if n = 0? I wonder if there was a premise that n is greater than zero, as it seems that other machines handle n = 0 condition. Difficulty(subjective): 8/10
@profstaecker6579 5 ай бұрын
(Sorry taking so long to reply- I don't check this account during the summer.) You are correct that my example didn't correctly handle the n=0 case. To fix it, I would just add an arrow labeled B/B,R from the start state to an accepting state. Thanks for the correction!
@aaaaaaaaaaaaa. 6 ай бұрын
Thank you for the lecture, TM for balanced brackets is pretty hard. Difficulty(subjective): 8/10
@aaaaaaaaaaaaa. 6 ай бұрын
I didn't know I would learn about Turing machines! Difficulty(subjective): 5/10