I believe the last math class I took was a statistics class in freshman year of undergrad… I definitely do not understand a thing about this test or any of these problems, but your passion for the subject keeps me watching! It comes out in how you speak about it - you love this stuff. Good for you, man.
This is amazing, I’m currently in my Civil Engineering curriculum taking courses such as CE Statics and Mechanics of Deformable Bodies. I only took math through calc 3, and then differential equations and linear algebra. Just watching your videos is insane to me, and I can tell you put so much time into studying and broadening your intelligence towards these topics , congrats! I can’t ever imagine learning what you’re learning!
Thank you for sharing your experiences and congrats on passing! I'm starting my maths major next month so your channel reminds me to work hard and do my best 👍
For 3, I would take compact neighborhoods (closed balls) of all the rational points in U, their union is U. The image of each compact is compact, and then f(U) is a countable union of compact sets so it is a Borel set.
For 2. b): Assuming that a_k -> 0 (otherwise just set f_k = a_k) a typewriter-like sequence should do it, i.e. in the first iteration pick some n s.t. sum_{k = 1}^{n} a_k >= 1 and set f_1 = 1_[0, a_1], f_2 = [a_1, a_1 + a_2], ..., f_n = 1_[...] * 1_[0, 1]. Continue in this fashion so that f_k(x) fails to converge for any x. Just like a typewriter, the indicator function wanders from left to right along the interval over and over again.
I don't agree with what you wrote but I think you have the correct idea. Let S_1 be the first time the partial sum is greater than 1 (so it is for the n you found). Then, the trick you did works until then: define f_1 = 1_[0,a_1], and f_k = 1_[summation until k_1, summation until k] (so that the interval length is a_k). For f_n, we can do the same thing, and notice that since summation k=1 to n goes outside [0,1], the interval is smaller than a_n, so the integral still holds. For f_(n+1), we should continue as before, but make sure to subtract S_1 (to map back into [0,1]. Eventually, we will go above 1 again, and have to subtract S_2, etc. There's no need to multiply by 1_[0,1] since the f_ks are defined only for that interval anyway.
Hey man... Congratulations... Not many people in your life will understand what a great achievement this is, but trust that we in the comments acknowledge the significance of such an accomplishment! All the best,
@@joelfalco8735 when you laugh at such crass humor as a reference to testicles, you are asserting your entertainment is more important than interrupting class. Save those type of jokes for your weekly circle jerks with your friend group clowns. Cheating is so widespread these days that reporting good grades is not particularly noteworthy.
congratulations! I've really enjoyed the videos you have been posting but I'm wondering what it is you want to do after you PhD, or have you not thought that far down the line.
What do you want to work after finishing your studies , like afyer your PhD and after you have evry certificate you want which field are you going to work in ?
Yup, I got the same. I assume you meant using the integral test to show that `sum n^{-1.5}` converges (we need that sum |1/b_k - 1/b_{k + 1}| converges)
Is the term "complex analysis" interchangeable with "complex variables" in a similar way to the interchangeability of "advanced calculus" and "real analysis"?
From my understanding, they are. But each institution has their own name for the course. They may call complex variables the undergrad version and the grad version complex analysis.
Hey man any tips or advice how can i get into a IMO team and also how to learn Combinatorics, Geometry, Algebra and Number Theory fast lol like any website that is challenging and yeah worth it
Hey, im a second years math student (bachelor). Taking (real) analysis 2 rn and I feel overwhelmed by the proofs, do you have tips on how to study it? Also, would you be interested in reviewing my university's syllabus? They write the theory themselves and I feel like their expectations are a bit high for second years students.
Did you switch majors yet? If you made it this far to real analysis 2 then i’d say just stick with it. Re read and re do proofs over and over and over again.
For Q.3, it seems that we may try in this direction: Let $\{K_n \mid n\in\mathbb{N} \}$ be a countable collection of compact sets such that K_n \subseteq U and U = union of these compact sets. Then f(U) = \cup_n f(K_n), which is a F_\sigma set and hence it is Borel. I have not worked out the details for the construction of {K_n}. If the domain of $f$ is R, this is easy...
I'm assistant professor of mathematics I suggest you try question paper NTA CSIR MATHEMATICAL SCIENCE This is national Eligibility test in India for assistant professor certificate
I tried to solve the third question for the case when n=m=1. If f is a real valued continuous fn. on R, prove that f(U) is Borel. Since U is an open set, we can write it as a countable union of disjoint open intervals, and since f is continuous it will map an open interval to some sort of an interval (maybe open, close, half-open, all of them are Borel that's what we care about). Therefore the image of U under f will be a countable union of intervals OR a countable union of Borel sets. Which is a again a Borel set. Is this the right approach?? How did you do it for general n and m??
Yes, that's correct for the 1D case (the rigorous argument would be that an open interval is connected and connectedness is preserved under continuous transformations, hence also its image will be connected and thus an interval). For the general case you want to argue by means of compact sets since compactness is preserved under continuous transformations. This means all you have to do is to show that an open subset of R^n can be written as the countable union of compact sets (e.g. dyadic decomposition).
@@jeecrack3396 That's most definitely not a lie, year 2 sem 2 here we have complex functions (which is complex analysis) and we go deep into this subject, we end up in conformal mappings and mostly talk about the Mobius mapping but also others. In year 3 sem 1 we have what we call "modern analysis" which is basically real analysis but we really go deeply into banach spaces and hilbert spaces as well of course measure theory. The exam had as question 1 literally to find a set of outer measure 1/2 that isn't lebesgue measurable and I think there was also a condition that it's in [0, 1] (I still have no idea what you were supposed to do there btw). Most integral problems in analysis you would use Tonellis theorem because its for some reason way more useful than Fubini for many problems and there were really difficult double integrals to prove things about using it!
I think working through math books and daily practice. You can't really skip to much when talking about the fundamentals but after that all of math opens up for you. (pre-)Algebra -> Trig -> (pre-)Calc -> complex numbers. For analysis you need to know logic, sets and proof writing. Which is basically building math up from logic to proof the stuff you use in Calculus. There are many more subjects in mathematics but basically if you're doing Real Analysis and you understand the way of thinking then you are well equipped to tackle any other subject.
So essentially, you have some balls, you pick some of them, the balls grow to 3 times their size, and the balls create a family.
You got it
This dude
So essentially, you have oversimplified.
@@MyOneFiftiethOfADollar
So essentially, you don't get jokes
Congratulations man! Great to see this perspective of the process
I believe the last math class I took was a statistics class in freshman year of undergrad… I definitely do not understand a thing about this test or any of these problems, but your passion for the subject keeps me watching! It comes out in how you speak about it - you love this stuff. Good for you, man.
This is amazing, I’m currently in my Civil Engineering curriculum taking courses such as CE Statics and Mechanics of Deformable Bodies. I only took math through calc 3, and then differential equations and linear algebra.
Just watching your videos is insane to me, and I can tell you put so much time into studying and broadening your intelligence towards these topics , congrats! I can’t ever imagine learning what you’re learning!
Yo CE gang
All engineering is maths so you're no less , don't underestimate yourself.
@@BadAss_691Lol just no
I dig the whiteboard that's been in these past few vids!
Thank you for sharing your experiences and congrats on passing! I'm starting my maths major next month so your channel reminds me to work hard and do my best 👍
Congratulations! Keep going
For 3, I would take compact neighborhoods (closed balls) of all the rational points in U, their union is U. The image of each compact is compact, and then f(U) is a countable union of compact sets so it is a Borel set.
Congratulations on passing the exam. Would you mind throwing a video together really quick for number 5 on the real analysis?
bn is bounded on that interval so since an converges you can use ratio test
For 2. b): Assuming that a_k -> 0 (otherwise just set f_k = a_k) a typewriter-like sequence should do it, i.e. in the first iteration pick some n s.t. sum_{k = 1}^{n} a_k >= 1 and set f_1 = 1_[0, a_1], f_2 = [a_1, a_1 + a_2], ..., f_n = 1_[...] * 1_[0, 1]. Continue in this fashion so that f_k(x) fails to converge for any x. Just like a typewriter, the indicator function wanders from left to right along the interval over and over again.
I don't agree with what you wrote but I think you have the correct idea. Let S_1 be the first time the partial sum is greater than 1 (so it is for the n you found). Then, the trick you did works until then: define f_1 = 1_[0,a_1], and f_k = 1_[summation until k_1, summation until k] (so that the interval length is a_k). For f_n, we can do the same thing, and notice that since summation k=1 to n goes outside [0,1], the interval is smaller than a_n, so the integral still holds. For f_(n+1), we should continue as before, but make sure to subtract S_1 (to map back into [0,1]. Eventually, we will go above 1 again, and have to subtract S_2, etc. There's no need to multiply by 1_[0,1] since the f_ks are defined only for that interval anyway.
I think there should be a cleaner way to do this
Congratz my man. I feel kind of curious of wanting to try those questions myself lol.
Loved these videos even though I don't understand the problems😅
Hey man... Congratulations... Not many people in your life will understand what a great achievement this is, but trust that we in the comments acknowledge the significance of such an accomplishment! All the best,
At the end of the week he will be broke AF..papi.....
@@jamesromano3288 you clrarlh don't know anything about job opportunities for math PhDs, cope + brainlet
However he will be broke with a Phd while you will be broke without an education @@jamesromano3288
@@jamesromano3288tf
Thankfully, I only have to go as far as Calculus 3 for my major, but I am extremely proud of your success and passion for math.
Congratulations, man!❤
I have zero interest in pursuing a PHD in math but I like this guy so I’m subscribing 🤣
Congratulations!!!
Congrats
Congratulations and best wishes. Fun analysis comic.
Congrats, I came to check this thing after I saw your first video.
How much time did you get for these 5 questions in real analysis? Over 7 hours seems too much for me...
Every time the teacher said Balls in class, my whole friend group would start laughing. You wouldn't think we're math students lol
@@MyOneFiftiethOfADollar we're respectful and get good grades. having a little fun doesn't hurt anyone
@@joelfalco8735 when you laugh at such crass humor as a reference to testicles, you are asserting your entertainment is more important than interrupting class.
Save those type of jokes for your weekly circle jerks with your friend group clowns.
Cheating is so widespread these days that reporting good grades is not particularly noteworthy.
@@MyOneFiftiethOfADollarno
@@MyOneFiftiethOfADollar you're weird, man
Congratulations!
congrats
For me it's shocking that this is not a take-home exam. Everything we do is take-home because that is how you do it in research
🎉Congrats🎉
congratulations! I've really enjoyed the videos you have been posting but I'm wondering what it is you want to do after you PhD, or have you not thought that far down the line.
He will be working at Home Depot, papi.
@@jamesromano3288 why are you commenting so much about this guy going broke? why a random math phd student of all people? you're weird, man.
this guy is the sam sulek of maths
Lmaooo what a comparison
What do you want to work after finishing your studies , like afyer your PhD and after you have evry certificate you want which field are you going to work in ?
By the sound of it, I think he might continue to be academics as prof and continue with his post doc research
I laughed every time he said balls.
Congratulations !!
You should input your information into an a.i. Prompt and charge for the file used in a compiler.
Congratulations. I just came across your channel. I am a freshman in math major in an undergrad.
For no.5 I used the Abel summation formula and then show the cauchy condition is true(by suitable bounding)....I think it works can someone verify?
yes. I think the key idea is to use Abel summation formula.
Yup, I got the same. I assume you meant using the integral test to show that `sum n^{-1.5}` converges (we need that sum |1/b_k - 1/b_{k + 1}| converges)
@@robertschumann6977 I rechecked my work..my bound for |1/b_k+1 - 1/b_k| was ugly...sum of 1/n^1.5 works fine.
great job man, nice.
fucking love your videos
Congrats!!
Congratz man!
Well done!
lets go congrats man
balls 1:23 , 1:26, 1:28, 1:38, 1:44, 1:52 , 1:55
standard for analysis but whatever
you think people who enjoy the simple pleasures of "balls" don't know analysis? how very assumptionous of you willy
the *union* of the balls
@@joelwillis2043 pee balls
HUGE CONGRATS
Is the term "complex analysis" interchangeable with "complex variables" in a similar way to the interchangeability of "advanced calculus" and "real analysis"?
From my understanding, they are. But each institution has their own name for the course. They may call complex variables the undergrad version and the grad version complex analysis.
@@PhDVlog777 Thank you!
Congrats!
holy shit, congrats
I'm soo early that the vid isn't even loading lmaoo😅
If I want to learn calculus,,,,,is all that trig stuff really needed, papi ?
Yeah it's building block, trig with you mean trigonometry na?
Hey man any tips or advice how can i get into a IMO team and also how to learn Combinatorics, Geometry, Algebra and Number Theory fast lol like any website that is challenging and yeah worth it
Hey, im a second years math student (bachelor). Taking (real) analysis 2 rn and I feel overwhelmed by the proofs, do you have tips on how to study it? Also, would you be interested in reviewing my university's syllabus? They write the theory themselves and I feel like their expectations are a bit high for second years students.
switch majors
@@MyOneFiftiethOfADollar yeah im thinking of quitting math and cs and pursue a econometrics degree
Where do you go to college?
@@bennoarchimboldi6245 Radboud University, the Netherlands
Did you switch majors yet? If you made it this far to real analysis 2 then i’d say just stick with it. Re read and re do proofs over and over and over again.
Hi there....are these concepts and theorems covered/proved in your class ???
All the theorems and tools used in these problems were developed in the analysis course.
New to this channel. What are you studying for? Academia or what job prospects are you using this math for?
He will be working at Home Depot, papi.
g(x) = f(x) + f(-x) is an even function if that matters?
where do you go to school?
Go Golden Flash!
Can you suggest me a good analysis book for problem. Solving?
Pugh - Real Mathematical Analysis
For Q.3, it seems that we may try in this direction:
Let $\{K_n \mid n\in\mathbb{N} \}$ be a countable collection of compact sets
such that K_n \subseteq U and U = union of these compact sets.
Then f(U) = \cup_n f(K_n), which is a F_\sigma set and hence it is Borel.
I have not worked out the details for the construction of {K_n}.
If the domain of $f$ is R, this is easy...
Interesting
The fifth problem is quite trivial, without any integration:
From convergence you have that for eps>0 there is N for that if n,p>N then |S(n,p)|
worst math student is the one who uses trivial and trivially
“quite trivial” 🤓
Is it that I'm delirium or 3 is probably the most easy one in the test?
1:22 Haha, Bj 😂
I'm assistant professor of mathematics
I suggest you try question paper NTA CSIR MATHEMATICAL SCIENCE
This is national Eligibility test in India for assistant professor certificate
I tried to solve the third question for the case when n=m=1. If f is a real valued continuous fn. on R, prove that f(U) is Borel. Since U is an open set, we can write it as a countable union of disjoint open intervals, and since f is continuous it will map an open interval to some sort of an interval (maybe open, close, half-open, all of them are Borel that's what we care about). Therefore the image of U under f will be a countable union of intervals OR a countable union of Borel sets. Which is a again a Borel set. Is this the right approach?? How did you do it for general n and m??
Yes, that's correct for the 1D case (the rigorous argument would be that an open interval is connected and connectedness is preserved under continuous transformations, hence also its image will be connected and thus an interval). For the general case you want to argue by means of compact sets since compactness is preserved under continuous transformations. This means all you have to do is to show that an open subset of R^n can be written as the countable union of compact sets (e.g. dyadic decomposition).
@@robertschumann6977 this guy has it
I'm taking math grad RA right now, and these questions seem decent. I was surprised to hear you say hour seven. 😅
hehe bawlz
early view gang
These are easier than the exams I had as an undergraduate and we had way less time (2.5 hours for each exam) I am so confused 😢
Stop lying bro
@@jeecrack3396 That's most definitely not a lie, year 2 sem 2 here we have complex functions (which is complex analysis) and we go deep into this subject, we end up in conformal mappings and mostly talk about the Mobius mapping but also others. In year 3 sem 1 we have what we call "modern analysis" which is basically real analysis but we really go deeply into banach spaces and hilbert spaces as well of course measure theory. The exam had as question 1 literally to find a set of outer measure 1/2 that isn't lebesgue measurable and I think there was also a condition that it's in [0, 1] (I still have no idea what you were supposed to do there btw). Most integral problems in analysis you would use Tonellis theorem because its for some reason way more useful than Fubini for many problems and there were really difficult double integrals to prove things about using it!
@@yanntal954What university?
what was harvard like?
At UofT i had exam problems that looked easy at first glance but ended up stumping me for an hour.
Is this guy like Terrance Tao?
I want to be like this how can I do this as quick as possible
I think working through math books and daily practice.
You can't really skip to much when talking about the fundamentals but after that all of math opens up for you. (pre-)Algebra -> Trig -> (pre-)Calc -> complex numbers.
For analysis you need to know logic, sets and proof writing. Which is basically building math up from logic to proof the stuff you use in Calculus.
There are many more subjects in mathematics but basically if you're doing Real Analysis and you understand the way of thinking then you are well equipped to tackle any other subject.
@@BboyKenythank you for the advice.
How I feel hearing a math problem:🥱😴
Congrats!!