"I know I'm not the strongest mathematician, but hopefully I'm better than what I was last year" This is a great mindset, my dude, but also don't get too hard on yourself. Remember that you're already in the top 0.1% of the world in terms of math, not even accounting for your peak :). I wish you the best of luck in your academic advancement and remember to stay confident. [Edited grammar]
@@one4runner435 it is correct but it is the Indian number system. Here, it’s written 8 lakh which is 8,00,000 ie 800,000. You know even 800,000 might be weird for some other people. For example, if a french person were to read it, he would think that it is written 800.000 which is equal to 800 because the . becomes , and vice versa. Each country has its own maths symbols and formalism though the maths stays the same
Speaking from the perspective of a theoretical physics PhD candidate, an exam where it’s expected that most people would fail is a terribly written exam. You’re not looking for basic competence of the students at that point. Seems like the instructor is more interested in flexing how hard they can write their exams more than anything.
Actually, sometimes I think the instructors should give the students some ultra hard exams just to see who are the extraordinary ones. I don't know, I think that someone like that is always something else, as if they were chosen or sth. I am not good at math though. Maybe I have been reading too much Webtoon manhwas that I started fantasizing about being a superhero/MC with super powers lol.
@@subliminalfalllenangel2108 I think it’s fine if you’re trying to pick out who already knows a lot of things but this is not how you would go about doing this. The qualifying exam is there to judge basic competence, so _every_ student is judged according to this standard. It wouldn’t make sense to judge every student to a standard of an exceptional person since that person, by definition, is the exception. On top of that, if you want to pick out the ones who already know a lot, you can just have them test out of the 1st and 2nd year work load. Qualifying exams are stupid and outdated anyway. My school and many schools across the country have started to fade them out entirely because they are in no way predictive of how a grad student can do research. It’s just an exam that needlessly stresses out students and it doesn’t even simulate anything that is expected from a researcher. There will be no other context where you have to sit in a room and solve a bunch of hard problems in a small amount of time. This is an antiquated way of doing things and I hope their department will rethink their approach to evaluating grad students.
This is why I hate ppl who love pure theory. Not to be trusted, hence why I did Computer Science because at least I know the maths is still going to be interesting but still grounded in reality.
Having more than a handful people failing your exams, just shows that you sucked at teaching them to begin with. Which to be fair is not that big news. Universities are notorious for sucking at teaching people properly. But it surely is nothing to brag about.
I graduated in pure math less than 2 years ago. I failed once on the conprehensive oral exam, which is required to be a PhD candidate, and you are allowed to fail the oral exam only once. In total, I spent the whole year with just the oral exam. Once I passed, one of my profs just said it's done, let's just do the research. Now I'm a postdoc and so far research ability is not so bad at all. I never thought of myself as a competent math researcher. I just do what I can, until these days. Trust me, there were many times I compared myself with other people and I feel like I didn't deserve to be where I am. Now I say, forget about it, go with your own pace. Anyway, wish you luck and rock the next written exam!
@@Linterna001 Not really, when I did my oral math exam it was about the structure of certain topics and how you understand them. You get asked about some definitions and have to explain how to solve certain problems step by step. You are also asked about certain relations between Lemmas and how to use those to solve a given question. There are some problems that can be solved orally and on the spot directly, like for set theory and mathematical logic. Typical questions go something like this: "How would you define ...?" "How would you find ... if I give you / you know that ...?" "What is the difference in measuring ... and ...?" "Can you tell me an example for ...?" "What kind of structure is ...?" "What procedure is used for ...?" "What does ... tell you about ...?" "In which case do ..., ... and ... apply?" "Can you tell me what ... is and what it's used for?" etc. You don't typically find an equation to solve on the board, but I used the board to solve or show certain problems and concepts and I also used it to be sure I answered an oral example correctly, since I needed to see how it looked written down to be completely sure of my answer.
@@CIA466 For the university where I came from, what written was that you will be kicked out of the program. However, it was rarely the case. Most of them will pass from the 1st attempt. I heard some of my friends who didn't even manage to do the oral exam in time, but the department is still quite lenient about it. The dept. will consider based on the overall performance, and as long as you don't violate some strict rules nor went missing from the duty for a whole semester, it should be fine. This depends on each university and department though.
@@Linterna001 Exactly what Hagen L. said. It quite depends on the prof's style of question. First time I was asked to prove the theorem that requires the knowledge equivalent to the whole semester's material. I went on for 5 minutes and the prof seemed ok, so they moved onto the next question. Sometimes they asked you to recite the proofs, or came up with new problems where you have to think right away.
For some reason the RUclips algorithm recommended your channel even though I rarely watch math videos and when I do, it’s pretty much entry level stuff. I think this is definitely an interesting take into the math part of RUclips and hopefully you can get some traction. Great video btw
Im a pure math undergrad right now and I really want to get into a PhD program for math. Seeing what I will have to do if I get into one is a bit intimidating but your commentary really reassured me. Good luck in your endeavors and I’m gonna keep following you while you go through your journey
"I don't think I wrote total garbage." That's how I feel doing some homeworks in EE courses lately 😂. I'm a music major who decided to minor in EE because I like learning about how sound works. The challenge is, there's a lot of quirky math behind sound and electrical circuits. It's a good mindset that you have with problems like these. I've learned a lot of the time my instincts on problems that look weird actually turn out to be a good instinct if I studied correctly. If I second guess myself, then I tend to get stuck on the problem and usually come out with the wrong answer. Point being, we know more than we think we do. It's a matter of trusting ourselves. But also, yeah this exam looks scary and I could imagine it being even more difficult on that last problem. Good luck on the rest of them, and I hope you pass it! It sounds like you've studied a lot. I wouldn't have the foggiest idea where to even start with these lol.
RUclips recommended this to me out of nowhere, but it was really cool! I just want to say that I have a lot of respect for your passion and dedication. I'm just an undergrad at the moment, but from what I can read about measure theory and the likes, that stuff really gets tough as hell. I wish you luck, and even if you didn't pass this one, you have half a year to improve. That's plenty of time to master the subject. I believe in you!
Q2 and Q3 can be solved by Egorov's theorem. For Q2, suppose by contradiction that f_n tends pointwise to 0, then by Egorov's theorem one has E of measure eps, such that in the limit of both integrals concentrate on the set E. But from there it is possible to show that liminf_n an \geq eps^{-1/2}. Sending \eps to 0, one obtains the contradiction. For Q3, it suffices to show that f_n converges to 0 in L2. But this is a consequence of the pointwise convergence to 0 and the Lp boundedness for p>2, by Egorov's theorem. For Q5, one easily shows that |a_j|\leq 1/2, then by Taylor expansion and the inequality \sum_j |a_j|^k\leq 2^{-k+1} , the bound can be established.
I think 5 is a super reasonable question, by the fact the magnitude sum is 1 and sum is 0, then the sum of positive a's = .5, same for negative a's negated, Taylor series of sin First two terms sum to zero by the sum of a =sum of a³=0 Third term Sum x⁵/5!= < .5⁴/5! (.5⁵ actually but longer argument doesn't matter) Because Sum |a⁵_j|=
For Q2 if an is bounded let's apply payley zigmund inequality. Then we got m(f_n>1)>(1-1/n)²xE[f]²/E[f²] wich is in our case bigger than (1-1/n)²*1/an and that bigger than 1/4 * 1/M for n>2 and M>an for all n. Hence fn cannot go to zero everywhere If we take a fonction wich is zero every wher except in [-1/n,1/n] , wich have a triangular form then if it's height is n² it will have a integral of n and an integral square of n*3 so we can take an=n. But this goes to zero eveywhere (except in 0) hence the result Q5) can be done by substracting the sum of an+an^3/6 to the sum (it's zero by assumption !) of the sin and using taylor theorem We get that the sum is less tan sum(|an|^5) over 5! and because |an|
I was a math major that has since moved on to grad school in applied math and man this brings back memories. I miss the magic but boy do I not miss being stuck with a real analysis problem fruitlessly banging my head against the wall for hours on end.
Q3 looks like a Holder question. You are given that the functions are L^p integrable. The trick would be the apply Holders with one of the functions as the identity/characteristic function on a set that would be small (using the a.e. convergence to 0 property). Q4 looks like a similar trick. You are given some integrability requirements and some a.e. convergence, so |\{x: |g_n(x)| < \ep\}| will be small. Note that Holders gives you one function in a "lower" Lp space but the other function will be a "higher" integrabiltiy for which the given integrability bounds will fail. Q5 you should use the Taylor theorem with the integral remainder term.
i just want to tell you that i find you really really freaking cool. i notice many people pursing such degrees and always feel awe. i may not know you, but i am amazed and am so proud of you. please keep pushing forward, you are amazing.
It'd be nice to learn the tools required to solve these questions, as a math major I didn't understand any of the questions, and I'm applying this year for a masters program. Thanks for sharing the experience
You are at the end of math bachelors and you didnt understand any of the questions? How is that even possible dude. I am in 3rd semester engineering bachelor and i understand what all of these questions mean, though i havent tried to solve them yet and i might not be able to. But understand kek
@@richard_darwin Although some of the question statements aren't too hard to understand, I don't think you could even attempt to solve them with only an undergrad engineering background. Even if you are a real analysis hobbyist at the same time, you would have to be more of a mathematician than an engineer at that point. Maybe Kevin meant that they had no idea where to start, which is very understandable.
@@richard_darwin That is absolutely bullshit. Unless you study math outside of engineering, there's no way you can solve these with only engineering knowledge.
Some of my undergrad physics exams went over 12hrs (thankfully not in one sitting), and qualifying exams seem less scary now. I'm hoping to study more analysis, so it's neat to see some of your experience with it.
I say from the bottom of my heart I am so sorry that you have chosen such a difficult field. I hate dimensional analysis so this stuff would drive me to deep depths of depression and sorrow
I took my qualifying exams in 2009. A Russian professor who proctored the Algebra exam told us, “there are six questions. You must get three correct to pass. You have three hours. Good luck.” I passed, but just barely, I think. Same with Analysis and Topology. Glad those are far in the past.
Totally impressed. The only thing I followed was Taylor Series, seeing Q5 that would be all I’d think to do and just fumble through it. I have literally forgotten all my measure theory. That was one exam in 1999 which I walked out of in third year undergraduate and then unenrolled from the next term’s algebra and left pure math for good. Managed first class honours in measure theory but remember none of it. You my friend are totally kickass at math - go get that PhD. 👊
In my opinion, there is a fine line between a good mentor and someone that is egotistical and discouraging. Is it the case that the opportunity to try a practice problem set representative of the style of questions on the actual exam is standard practice of the professor but there are simply very few students who choose to do it? Also, do they go through solutions of the practice set in a way that is not intimidating and in a way that is encouraging so students are better prepared for the actual test? If so, even though the questions seem difficult and some seem contrived, I think this may be the type of professor that has the right idea and their heart is the right place. This can be a great way to learn how to solve difficult problems especially in a PhD program that requires savvy, fresh, and slick ways of tackling new types of problems. On the other hand, if that's not the case or if they go through the practice problem solutions, but it is in a manner that is intimidating, discouraging, and demoralizing, then it seems to me like it is an unnecessary and vexatious exercise that is far more of a disservice than it is helpful. I hope this professor does the former and not the latter.
@@manya7903if he’s 17 then good chance he is a junior or a senior already. it’s perfectly acceptable for a high school junior or senior to have a major picked out. No shit he’s not majoring in cs in high school, op is referring to college
@@lowcoolant6131 In my country or school specifically, students are categorized into certain classes which are specialized in their favorite subject (math, CS, bio, etc). Of course there is an entrance exam which decides which class you end up in or even fail! So it’s not entirely wrong to say that I’ve majored in CS.
@@lowcoolant6131 And yes I’m referring to high school! These classes help students to get familiar with their future career. I’d even say some are taught even more thoroughly and widely than some uni courses!
I do not envy you. I got my Degree in Astrophysics. Exoplanets specifically with the Kepler Satellite Data. I remember taking qualifying exams. It was by far the most stressful point in my academic career. Jesus christ, the nightmares it brings back. I did pass of course, but god damn... Good luck to you.
Your video randomly came up somehow but I can tell you how to do #5. The key is to use Taylor's series for sin x with remainder (in Lagrange form for example). This says that sin a_j = a_j - a_j^3/6 + remainder, where the remainder is at most |a_j|^5/120 in absolute value. So |sum_j sin a_j| = |sum_j a_j - 1/6 sum_j a_j^3 + sum_j (remainder for a_j)|. (One can reorder the sum here without concern since all sums are absolutely convergent.) Since the sum of the a_j and the sum of the a_j^3 are zero, you can use the above bounds on the remainder to say that |sum_j sin a_j| is at most 1/120 \sum_j |a_j|^5. Next let the positive a_j be denoted by b_j and the negative a_j be denoted by c_j. Your first two conditions are that sum_j b_j = sum_j c_j and sum_j b_j + sum_j c_j = 1. Hence sum_j b_j = sum_j c_j = 1/2. In particular, each |a_j|
I don't know why professors put questions like question 1 on the exam. No one needs to remember a proof in real life. Also, after three or four years of math research the quals become super easy, so grad students, please don't worry if you fail your quals the first time around haha Anyways, here are some hints Question 2: Egoroff + Cauchy Schwarz on f_n/n Question 3: Reduce to the case where g_n = f_n using Cauchy Schwarz and then Egoroff+Holders Question 4: Reduce the problem to g_n being supported on [-1,1]. Use definition of convergence in measure and use Holders on (|x|^{1/2}g_n)(1/|x|^{1/2}) with p = 4, q = 4/3 Question 5: Taylor series plus the trick that |a_j|
Just from looking at it I guess that in the 2nd problem the weak convergence would do the job. In the 3rd problem I would try with Young inequality and maybe with weak convergence too. The 4th question is not for me because I don't know much about measure convergence. The 5th problem follows from Taylor series expansion
Which institute uses this qualification test? Where I live, PhD-students are chosen via an application for which the grade list and master thesis have to been handed in. Then the best candidates are interviewed to see who fits best with the project.
This is probably a US university - a common procedure in the US is for a qualifying exam to be taken after you have been there for two years and this exam allows to to commence on a final 3 years of research. This is probably not an exam to get onto the PhD program - as I said above it is a hurdle taken after two years to allow you to carry on.
I'm a grad student at a different uni, and I can tell you, I've been struggling to pass the quals for all these years. I've taken my final qual only a few days ago, and I hope I passed it. Wish me luck!
Interesting to see, I am a computer science PhD student and we do a lot of math, but I always feel out of my depth mathematically. I guess it's no different for math PhDs 😂
Same 😢 I used the inequality : For all k>=0, |a_n|^(5+2k) =< |a_n| We got this inequality because Sum for all n>=0 of |a_n|=1 thus for all n>=0, |a_n|=
I didn't appreciate of the test author how much emphasis was put on Lebesgue Integration theory and how little on continuity, differentiation, measure, and a lot else that is escaping me now, years after my PhD.
very normal for a PhD student. I am doing real analysis in university i am undergrad, the level of that test is just one more from ours. I would say that the guy who wrote that is very good at creating real analysis excercises, you should ask him the bibliography of all of that.
I’m an undergrad mathematics major so I may not have much of a say here, but… (11:52) if you knew they might give you the time you needed to finish the problem, don’t let your pride hold you back from getting it. Get that extra time, even if you need to embarrass yourself in front of a few snooty proctors to do it and especially if you know how to do/approach the problem; the proctors won’t be there with you after you graduate with your PhD so screw them, complain for your extra time, get those points in. I was never fast with math either. I do things my own way and take my time too. I’m sure that guy you mentioned in a different video who has been in the PHD program for years would give you the same talk.
Haha i have i got this video recommended while studying for my complex analysis test, i suck at math and hopefully will be done with it by next semester
Hang on Bro...hang on. How well did you do on the test? (I am not a math major. I took my civil engineering PhD qualifying exam not too long ago.. in the summer of 1992).
Are you supposed to use Faton's Lemma on problem 2? For problem 3, I would use one of the Lp normed inequalities (I can't remember the name), remembering to put ^p. I am following the Real Analysis course right now, and I want to convince myself that I have a general idea of how to solve those questions.
The only contribution I can bring to this discussion that tries to justify the exam is: maybe mathematicians are conditioned to find an answer rather than consider the process of unwinding a solution. What do I mean by that? Michael Penn has an excellent (in my opinion) approach to solving the unknowns. He often uses a 'scratch' pad to work various attempts at finding a solution. Start at the end and work backwards. Identify a weak spot if any, see if there is an intuitive approach. What equivalences, bounded equivalences can be drawn in? I suppose in applied maths it would be called something like "working methods" I am not very good at math but when I see bounded by descriptors sometimes intermediate value theories spring to mind, add in distance and measure raises the epsilon delta stuff maybe with approximations added into the estimation of bounds where an easier upper bound rather than an analytically derived horrible upper bound happens to be bounded on the same interval in the same event space. Saying things just follow their own geodesic seems a nicer method? So maybe, just maybe (?) it is an attempt to see how solving the unknown while under stress shows the character of the examinee. I suppose at PhD level & research levels coping with the unknowns happens to be a day-to-day activity
i hate real analysis. it's like intentionally confusing and torturing yourself. i once even tried to re-write my own form of real functions analysis. one that would help physicists with their work and proofs. i study physics myself and you can't see any shortage of skipping mathematical elegance and diligence in arriving at solutions and formulas for physics problems. there are invented ways to avoid and not do real analysis in physics almost everywhere.
About the timing thing, are they testing you on how well you know the material or how quickly you can recall it? Why should it matter how quickly you complete the test, so long as you get the questions right?
I'm a pure math undergrad and I want to go into a PhD program. This is.. terrifying. Is it a pass/fail system? Or are you graded and ranked among the other candidates?
These are generally pass/fail across the US. This exam has questions that are extremely similar to the questions found in the textbooks foe the material, so it’s really not so bad. You’re also generally expected to have memorized the proofs and use of a ton of different theorems. If you did properly learn Egarov’s theorem for problem 1, then problems 2 and 3 should be far more approachable. That on its own gets you up to the 30 point threshold. In conclusion, this test is a way to sneak theorem memorization and utilizations into being the primary component of the test. It’s not as bad as it looks for being tricky; it’s far worse than it looks for being an extreme amount of preparation and memorization.
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
As a 14 year old, I can conclude that this is rather difficult.
yeah because you're stupid
as a 16 year old, i can conclude that this is quite difficult
As a 24 year old, I can confirm that this is quite difficult
As an 18 year old, I can affirm that this is quite difficult
As a 19 year old, I can conclude that this is quite difficult
"I know I'm not the strongest mathematician, but hopefully I'm better than what I was last year"
This is a great mindset, my dude, but also don't get too hard on yourself. Remember that you're already in the top 0.1% of the world in terms of math, not even accounting for your peak :). I wish you the best of luck in your academic advancement and remember to stay confident.
[Edited grammar]
@Wasnovak 0.01 percent is still 8,00,000
@@yashrawat9409 With the way you wrote that number, you're not in it
@@one4runner435 it is correct but it is the Indian number system. Here, it’s written 8 lakh which is 8,00,000 ie 800,000. You know even 800,000 might be weird for some other people. For example, if a french person were to read it, he would think that it is written 800.000 which is equal to 800 because the . becomes , and vice versa.
Each country has its own maths symbols and formalism though the maths stays the same
@@one4runner435😂😂😂 killed him
@@nid7819 That is actually fascinating. I'm not familiar with that number system and I was mistaken. My general point was that YASH was being a D,IKH.
Speaking from the perspective of a theoretical physics PhD candidate, an exam where it’s expected that most people would fail is a terribly written exam. You’re not looking for basic competence of the students at that point. Seems like the instructor is more interested in flexing how hard they can write their exams more than anything.
The ego of mathematics professors extends from -infinity to infinity.
Actually, sometimes I think the instructors should give the students some ultra hard exams just to see who are the extraordinary ones. I don't know, I think that someone like that is always something else, as if they were chosen or sth. I am not good at math though. Maybe I have been reading too much Webtoon manhwas that I started fantasizing about being a superhero/MC with super powers lol.
@@subliminalfalllenangel2108 I think it’s fine if you’re trying to pick out who already knows a lot of things but this is not how you would go about doing this. The qualifying exam is there to judge basic competence, so _every_ student is judged according to this standard. It wouldn’t make sense to judge every student to a standard of an exceptional person since that person, by definition, is the exception. On top of that, if you want to pick out the ones who already know a lot, you can just have them test out of the 1st and 2nd year work load.
Qualifying exams are stupid and outdated anyway. My school and many schools across the country have started to fade them out entirely because they are in no way predictive of how a grad student can do research. It’s just an exam that needlessly stresses out students and it doesn’t even simulate anything that is expected from a researcher. There will be no other context where you have to sit in a room and solve a bunch of hard problems in a small amount of time. This is an antiquated way of doing things and I hope their department will rethink their approach to evaluating grad students.
This is why I hate ppl who love pure theory. Not to be trusted, hence why I did Computer Science because at least I know the maths is still going to be interesting but still grounded in reality.
Having more than a handful people failing your exams, just shows that you sucked at teaching them to begin with.
Which to be fair is not that big news. Universities are notorious for sucking at teaching people properly.
But it surely is nothing to brag about.
I graduated in pure math less than 2 years ago. I failed once on the conprehensive oral exam, which is required to be a PhD candidate, and you are allowed to fail the oral exam only once. In total, I spent the whole year with just the oral exam. Once I passed, one of my profs just said it's done, let's just do the research. Now I'm a postdoc and so far research ability is not so bad at all.
I never thought of myself as a competent math researcher. I just do what I can, until these days. Trust me, there were many times I compared myself with other people and I feel like I didn't deserve to be where I am. Now I say, forget about it, go with your own pace.
Anyway, wish you luck and rock the next written exam!
What happens if u fail more then once
Question, how the hell do you make an oral math exam? Do they write an equation in whiteboard and you have to say what to do step by step?
@@Linterna001 Not really, when I did my oral math exam it was about the structure of certain topics and how you understand them. You get asked about some definitions and have to explain how to solve certain problems step by step. You are also asked about certain relations between Lemmas and how to use those to solve a given question. There are some problems that can be solved orally and on the spot directly, like for set theory and mathematical logic.
Typical questions go something like this:
"How would you define ...?"
"How would you find ... if I give you / you know that ...?"
"What is the difference in measuring ... and ...?"
"Can you tell me an example for ...?"
"What kind of structure is ...?"
"What procedure is used for ...?"
"What does ... tell you about ...?"
"In which case do ..., ... and ... apply?"
"Can you tell me what ... is and what it's used for?"
etc.
You don't typically find an equation to solve on the board, but I used the board to solve or show certain problems and concepts and I also used it to be sure I answered an oral example correctly, since I needed to see how it looked written down to be completely sure of my answer.
@@CIA466 For the university where I came from, what written was that you will be kicked out of the program. However, it was rarely the case. Most of them will pass from the 1st attempt. I heard some of my friends who didn't even manage to do the oral exam in time, but the department is still quite lenient about it. The dept. will consider based on the overall performance, and as long as you don't violate some strict rules nor went missing from the duty for a whole semester, it should be fine. This depends on each university and department though.
@@Linterna001 Exactly what Hagen L. said. It quite depends on the prof's style of question. First time I was asked to prove the theorem that requires the knowledge equivalent to the whole semester's material. I went on for 5 minutes and the prof seemed ok, so they moved onto the next question. Sometimes they asked you to recite the proofs, or came up with new problems where you have to think right away.
For some reason the RUclips algorithm recommended your channel even though I rarely watch math videos and when I do, it’s pretty much entry level stuff. I think this is definitely an interesting take into the math part of RUclips and hopefully you can get some traction. Great video btw
Im a pure math undergrad right now and I really want to get into a PhD program for math. Seeing what I will have to do if I get into one is a bit intimidating but your commentary really reassured me. Good luck in your endeavors and I’m gonna keep following you while you go through your journey
Me too! ^-^ good luck fellow math lover!!!
Me too. I hope you’re doing well :>
"I don't think I wrote total garbage." That's how I feel doing some homeworks in EE courses lately 😂. I'm a music major who decided to minor in EE because I like learning about how sound works. The challenge is, there's a lot of quirky math behind sound and electrical circuits. It's a good mindset that you have with problems like these. I've learned a lot of the time my instincts on problems that look weird actually turn out to be a good instinct if I studied correctly. If I second guess myself, then I tend to get stuck on the problem and usually come out with the wrong answer. Point being, we know more than we think we do. It's a matter of trusting ourselves. But also, yeah this exam looks scary and I could imagine it being even more difficult on that last problem. Good luck on the rest of them, and I hope you pass it! It sounds like you've studied a lot. I wouldn't have the foggiest idea where to even start with these lol.
RUclips recommended this to me out of nowhere, but it was really cool! I just want to say that I have a lot of respect for your passion and dedication. I'm just an undergrad at the moment, but from what I can read about measure theory and the likes, that stuff really gets tough as hell. I wish you luck, and even if you didn't pass this one, you have half a year to improve. That's plenty of time to master the subject. I believe in you!
Q2 and Q3 can be solved by Egorov's theorem. For Q2, suppose by contradiction that f_n tends pointwise to 0, then by Egorov's theorem one has E of measure eps, such that in the limit of both integrals concentrate on the set E. But from there it is possible to show that liminf_n an \geq eps^{-1/2}. Sending \eps to 0, one obtains the contradiction. For Q3, it suffices to show that f_n converges to 0 in L2. But this is a consequence of the pointwise convergence to 0 and the Lp boundedness for p>2, by Egorov's theorem.
For Q5, one easily shows that |a_j|\leq 1/2, then by Taylor expansion and the inequality \sum_j |a_j|^k\leq 2^{-k+1} , the bound can be established.
I think 5 is a super reasonable question,
by the fact the magnitude sum is 1 and sum is 0, then the sum of positive a's = .5, same for negative a's negated,
Taylor series of sin
First two terms sum to zero by the sum of a =sum of a³=0
Third term
Sum x⁵/5!= < .5⁴/5! (.5⁵ actually but longer argument doesn't matter)
Because
Sum |a⁵_j|=
I'm about to cry.. 😭✋
As an engineering student, I felt like I stumbled upon an alien language
@@redgefleming1535 fr
No way I just witnessed someone complete a PhD qualifying exam while making it look simple in a RUclips comments section
For Q2 if an is bounded let's apply payley zigmund inequality.
Then we got m(f_n>1)>(1-1/n)²xE[f]²/E[f²] wich is in our case bigger than (1-1/n)²*1/an and that bigger than 1/4 * 1/M for n>2 and M>an for all n. Hence fn cannot go to zero everywhere
If we take a fonction wich is zero every wher except in [-1/n,1/n] , wich have a triangular form then if it's height is n² it will have a integral of n and an integral square of n*3 so we can take an=n. But this goes to zero eveywhere (except in 0) hence the result
Q5) can be done by substracting the sum of an+an^3/6 to the sum (it's zero by assumption !) of the sin and using taylor theorem
We get that the sum is less tan sum(|an|^5) over 5! and because |an|
You and anyone attempting this have my respect. Good luck on your journey!
I was a math major that has since moved on to grad school in applied math and man this brings back memories. I miss the magic but boy do I not miss being stuck with a real analysis problem fruitlessly banging my head against the wall for hours on end.
Q3 looks like a Holder question. You are given that the functions are L^p integrable. The trick would be the apply Holders with one of the functions as the identity/characteristic function on a set that would be small (using the a.e. convergence to 0 property). Q4 looks like a similar trick. You are given some integrability requirements and some a.e. convergence, so |\{x: |g_n(x)| < \ep\}| will be small. Note that Holders gives you one function in a "lower" Lp space but the other function will be a "higher" integrabiltiy for which the given integrability bounds will fail.
Q5 you should use the Taylor theorem with the integral remainder term.
i just want to tell you that i find you really really freaking cool. i notice many people pursing such degrees and always feel awe. i may not know you, but i am amazed and am so proud of you. please keep pushing forward, you are amazing.
I don’t wanna go back to qualifying exam preparation days. Glad I’m done with them. Good luck 👍
I can't comment on the math here but this is sure making me feel better about my Calculus 2 class!
It'd be nice to learn the tools required to solve these questions, as a math major I didn't understand any of the questions, and I'm applying this year for a masters program. Thanks for sharing the experience
You are at the end of math bachelors and you didnt understand any of the questions? How is that even possible dude. I am in 3rd semester engineering bachelor and i understand what all of these questions mean, though i havent tried to solve them yet and i might not be able to. But understand kek
@@richard_darwin pretentious😂
@@richard_darwin Although some of the question statements aren't too hard to understand, I don't think you could even attempt to solve them with only an undergrad engineering background. Even if you are a real analysis hobbyist at the same time, you would have to be more of a mathematician than an engineer at that point. Maybe Kevin meant that they had no idea where to start, which is very understandable.
@@richard_darwin That is absolutely bullshit. Unless you study math outside of engineering, there's no way you can solve these with only engineering knowledge.
@@richard_darwin I doubt measure theory is covered very much in an undergraduate mathematics course
Some of my undergrad physics exams went over 12hrs (thankfully not in one sitting), and qualifying exams seem less scary now. I'm hoping to study more analysis, so it's neat to see some of your experience with it.
I say from the bottom of my heart I am so sorry that you have chosen such a difficult field. I hate dimensional analysis so this stuff would drive me to deep depths of depression and sorrow
I took my qualifying exams in 2009. A Russian professor who proctored the Algebra exam told us, “there are six questions. You must get three correct to pass. You have three hours. Good luck.” I passed, but just barely, I think. Same with Analysis and Topology. Glad those are far in the past.
I hope you do well on your exam. ^^
Totally impressed. The only thing I followed was Taylor Series, seeing Q5 that would be all I’d think to do and just fumble through it. I have literally forgotten all my measure theory. That was one exam in 1999 which I walked out of in third year undergraduate and then unenrolled from the next term’s algebra and left pure math for good. Managed first class honours in measure theory but remember none of it. You my friend are totally kickass at math - go get that PhD. 👊
Despite the tests seeming quite stressful, it was a great insight on what math your expected for at PhD, thanks!😄
From 1+1 = 2 to These hard questions 😂😅
In my opinion, there is a fine line between a good mentor and someone that is egotistical and discouraging.
Is it the case that the opportunity to try a practice problem set representative of the style of questions on the actual exam is standard practice of the professor but there are simply very few students who choose to do it? Also, do they go through solutions of the practice set in a way that is not intimidating and in a way that is encouraging so students are better prepared for the actual test? If so, even though the questions seem difficult and some seem contrived, I think this may be the type of professor that has the right idea and their heart is the right place. This can be a great way to learn how to solve difficult problems especially in a PhD program that requires savvy, fresh, and slick ways of tackling new types of problems.
On the other hand, if that's not the case or if they go through the practice problem solutions, but it is in a manner that is intimidating, discouraging, and demoralizing, then it seems to me like it is an unnecessary and vexatious exercise that is far more of a disservice than it is helpful. I hope this professor does the former and not the latter.
As a 17 year-old high schooler majoring in CS, I find this incredibly beautiful and fascinating even though I understand nothing.
Only college students have majors
@@manya7903if he’s 17 then good chance he is a junior or a senior already. it’s perfectly acceptable for a high school junior or senior to have a major picked out. No shit he’s not majoring in cs in high school, op is referring to college
@@lowcoolant6131 In my country or school specifically, students are categorized into certain classes which are specialized in their favorite subject (math, CS, bio, etc). Of course there is an entrance exam which decides which class you end up in or even fail! So it’s not entirely wrong to say that I’ve majored in CS.
@@lowcoolant6131 And yes I’m referring to high school! These classes help students to get familiar with their future career. I’d even say some are taught even more thoroughly and widely than some uni courses!
Nice questions - i am too out of practice nowadays for this - but it is very interesting - hope you pass when you sit it
lets go baby you are getting better everyday
I do not envy you. I got my Degree in Astrophysics. Exoplanets specifically with the Kepler Satellite Data. I remember taking qualifying exams. It was by far the most stressful point in my academic career. Jesus christ, the nightmares it brings back. I did pass of course, but god damn... Good luck to you.
Your video randomly came up somehow but I can tell you how to do #5. The key is to use Taylor's series for sin x with remainder (in Lagrange form for example). This says that sin a_j = a_j - a_j^3/6 + remainder, where the remainder is at most |a_j|^5/120 in absolute value. So |sum_j sin a_j| = |sum_j a_j - 1/6 sum_j a_j^3 + sum_j (remainder for a_j)|. (One can reorder the sum here without concern since all sums are absolutely convergent.) Since the sum of the a_j and the sum of the a_j^3 are zero, you can use the above bounds on the remainder to say that |sum_j sin a_j| is at most 1/120 \sum_j |a_j|^5.
Next let the positive a_j be denoted by b_j and the negative a_j be denoted by c_j. Your first two conditions are that sum_j b_j = sum_j c_j and sum_j b_j + sum_j c_j = 1. Hence sum_j b_j = sum_j c_j = 1/2. In particular, each |a_j|
Love your channel.
why do you put urself through this honestly is it worth it
Good luck man!
I don't know why professors put questions like question 1 on the exam. No one needs to remember a proof in real life. Also, after three or four years of math research the quals become super easy, so grad students, please don't worry if you fail your quals the first time around haha
Anyways, here are some hints
Question 2: Egoroff + Cauchy Schwarz on f_n/n
Question 3: Reduce to the case where g_n = f_n using Cauchy Schwarz and then Egoroff+Holders
Question 4: Reduce the problem to g_n being supported on [-1,1]. Use definition of convergence in measure and use Holders on (|x|^{1/2}g_n)(1/|x|^{1/2}) with p = 4, q = 4/3
Question 5: Taylor series plus the trick that |a_j|
fingers crossed on you; mine is in the next year
3. Apply Holder's inequality
5. Expand in Maclauren series, bound the terms (a_j)^5
Ah yes, how did I miss that?
Exactly.
Just from looking at it I guess that in the 2nd problem the weak convergence would do the job. In the 3rd problem I would try with Young inequality and maybe with weak convergence too. The 4th question is not for me because I don't know much about measure convergence. The 5th problem follows from Taylor series expansion
Which institute uses this qualification test? Where I live, PhD-students are chosen via an application for which the grade list and master thesis have to been handed in. Then the best candidates are interviewed to see who fits best with the project.
This is probably a US university - a common procedure in the US is for a qualifying exam to be taken after you have been there for two years and this exam allows to to commence on a final 3 years of research. This is probably not an exam to get onto the PhD program - as I said above it is a hurdle taken after two years to allow you to carry on.
@@ngc-fo5te should also mention that we generally go straight into the PhD program and get our masters en route.
@@innocentsmith6091 Are you not taking a Qualifier after two years to be allowed to progress?
wish you a very good Luke.
I'm a grad student at a different uni, and I can tell you, I've been struggling to pass the quals for all these years. I've taken my final qual only a few days ago, and I hope I passed it. Wish me luck!
You have my sympathy, I've had many friends talk to me about the horrors of Real Analysis qualifying exams! Hope you do well on your future attempt!
I learned about these 10 years ago trying to get a math phd. Now I'm a programmer...
Hi, math major here. I am not going for a PhD now.
It all depends on what you want to do with your education. PhD's are so specific to a certain job market that in general they are not recommended.
Lmao thanks for letting us know
I have a Phd in physics and I couldn't answer those questions.
physics is more algebra than analysis, right?
loving these videos
Scrolling through the comments to find someone working the problems was maybe a bit too ambitious 😂
Yeah me too. I’d be interested in seeing the solution to #5. I have a vague intuition about the question but no idea how to begin.
I'm about to defend my PhD dissertation, STEM but not math, i swear this gave me so much anxiety even though my field is definitely not math.
Interesting to see, I am a computer science PhD student and we do a lot of math, but I always feel out of my depth mathematically. I guess it's no different for math PhDs 😂
And I cry about the German Abitur here.. massive respect for the people who deal with this stuff..
I did a bachelor in pure math. To date I have never used it at all... But then again I went into chemistry 🙂
Theoretical chemistry?
gibberish to me, I was good at math too till I hit calculus. It did me in. Found a different love, I still wonder why profs make math so difficult
This is why I went to medical school and self study math as a hobby. I will say though that my initial board certification exam was 10 hours long.
10 hours 🙀
Same bruh
And that's why I don't do analysis :D but I'm sure you'll pass the qual!
If you are not struggling, failing, and trying again then you are not doing mathematics. Keep going.
As a 32 year old I can conclude I gave up on page 1
For number 5 i can only prove that the series is
Same 😢
I used the inequality :
For all k>=0, |a_n|^(5+2k) =< |a_n|
We got this inequality because
Sum for all n>=0 of |a_n|=1
thus for all n>=0, |a_n|=
We can show for all n>=0, |a_n| =
Never mind, proving that for all n>=0, |a_n|=< 1/2 is enough
I didn't appreciate of the test author how much emphasis was put on Lebesgue Integration theory and how little on continuity, differentiation, measure, and a lot else that is escaping me now, years after my PhD.
He did tell me the questions were randomly selected from a list of problems he wrote, so it might be a little lop-sided.
As a 27 year old Computer Science undergrad, I can conclude that Linear Algebra will be as far as I take my mathematics career
I’m a biology student and I just took my calc test. I know it isn’t anything like this but still, they make it hard for no apparent reason
very normal for a PhD student. I am doing real analysis in university i am undergrad, the level of that test is just one more from ours. I would say that the guy who wrote that is very good at creating real analysis excercises, you should ask him the bibliography of all of that.
Good luck mate
I have a BS in Applied Math. The REAL math guys are the pure math guys. I wish you the best!!
I’m an undergrad mathematics major so I may not have much of a say here, but… (11:52) if you knew they might give you the time you needed to finish the problem, don’t let your pride hold you back from getting it. Get that extra time, even if you need to embarrass yourself in front of a few snooty proctors to do it and especially if you know how to do/approach the problem; the proctors won’t be there with you after you graduate with your PhD so screw them, complain for your extra time, get those points in. I was never fast with math either. I do things my own way and take my time too. I’m sure that guy you mentioned in a different video who has been in the PHD program for years would give you the same talk.
omg this is making me afraid before even opening the envelope.
That kid in Goodwill Hunting finished test in an hour and got 100% correct.
Maybe, because it is dumb fiction, completely unrelated to real life?
@@crabcrab2024 Relax Francis.
@@swingger9674 Citing the movie is the best you can, dude? 😂
1, 2, 3, 4 are standard real analysis problems. 5 seems to be a tough calculus problem that needs a smart trick
im so fascinated by grad school math, all the symbols and theorems, im only doing precalculus now and just want to jump to the fun part already
Stay curious
oh blissful naivety
"I could have taken it in January, but I did not take it" - my life as a grad student
Can't he just use a calculator?
Can you do a follow up with the correct solutions?
This is just on my Feed out of nowhere and I'm just watching even though I don't understand a thing 😂
This video was really great. Keep it up. Share more cool math stuff! Turn some notes into paper airplanes and toss them out the window for us :)
Can we please have the PDF?
Watching this i feel like im a chimp 🗿
I am with you
I only took undergrad real analysis. I love it’s so difficult but so extremely vague at the same time.
Haha i have i got this video recommended while studying for my complex analysis test, i suck at math and hopefully will be done with it by next semester
Hang on Bro...hang on. How well did you do on the test? (I am not a math major. I took my civil engineering PhD qualifying exam not too long ago.. in the summer of 1992).
I was not told my score, just that I did not pass.
So what happens next? Will you get another chance?
@@ILoveMaths07 this was just his practice test. He planned to take the real test in August. Work hard, study well, and good luck Vincent!
@@profraihan Usually you don't get a practice for the qualifying exam.
@@ngc-fo5te well, if you are nice to the professors and telling them of your plight, they will be more than willing to help you.
good luck!!!!
"I dont know how he's going to grade it, but I dont think I wrote total garbage" Me after every analysis exam
Wow what a W, i think i dont know im still taking algebra 2
Are you supposed to use Faton's Lemma on problem 2? For problem 3, I would use one of the Lp normed inequalities (I can't remember the name), remembering to put ^p.
I am following the Real Analysis course right now, and I want to convince myself that I have a general idea of how to solve those questions.
The only contribution I can bring to this discussion that tries to justify the exam is: maybe mathematicians are conditioned to find an answer rather than consider the process of unwinding a solution.
What do I mean by that?
Michael Penn has an excellent (in my opinion) approach to solving the unknowns. He often uses a 'scratch' pad to work various attempts at finding a solution. Start at the end and work backwards. Identify a weak spot if any, see if there is an intuitive approach. What equivalences, bounded equivalences can be drawn in?
I suppose in applied maths it would be called something like "working methods"
I am not very good at math but when I see bounded by descriptors sometimes intermediate value theories spring to mind, add in distance and measure raises the epsilon delta stuff maybe with approximations added into the estimation of bounds where an easier upper bound rather than an analytically derived horrible upper bound happens to be bounded on the same interval in the same event space.
Saying things just follow their own geodesic seems a nicer method?
So maybe, just maybe (?) it is an attempt to see how solving the unknown while under stress shows the character of the examinee.
I suppose at PhD level & research levels coping with the unknowns happens to be a day-to-day activity
As someone who's taken Calc 2 i dont know what tf is going on
lol isn’t calculus stuff u learn at like high school
Yup, that looks difficult alright (i don't know anything in the paper)
See, this is why I took engineering 😉 Best of luck to you and your brilliant 🧠
Damn, I mentally checked out after calc 3 & discrete math 2
i hate real analysis. it's like intentionally confusing and torturing yourself. i once even tried to re-write my own form of real functions analysis. one that would help physicists with their work and proofs. i study physics myself and you can't see any shortage of skipping mathematical elegance and diligence in arriving at solutions and formulas for physics problems. there are invented ways to avoid and not do real analysis in physics almost everywhere.
those real analysis questions are pretty challenging. however, in all of math there is nothing as hard as number theory.
@user-xy4jz9ct8v why?
About the timing thing, are they testing you on how well you know the material or how quickly you can recall it?
Why should it matter how quickly you complete the test, so long as you get the questions right?
I’m about to graduate with my BS in mathematics and I’m scared to death of math grad school 😢
Then switch fields immediately. Maybe consider the more lucrative vocation of engineering
Question five is actually totally reasonable until right after it says “show that”
I might be stupid but 5 seems like a taylor series error bound? Especially since you have the aj and aj^3?
And here I am complaining about AP Calc and Physics
Hmmm, reminds me of Folland, the book I studied for my measure theory course in college
This is literally an alien language to me and I want it to stay that way
Wow. That looks way harder than any of my undergrad real analysis exams.
That's because it's measure theory and not metric theory
@@adfriedman I was studying measure theory at 19 years old, but there's no way I could pass that exam. At least, not without serious study.
They gave you 5 hours?!?! I had 6 tough questions for only 3 hours 😅.
That was tough finishing on time, I wish I had that luxury.
9 questions in 4 hours on my end. Didn't completely finish but still passed.
I'm a pure math undergrad and I want to go into a PhD program. This is.. terrifying. Is it a pass/fail system? Or are you graded and ranked among the other candidates?
These are generally pass/fail across the US. This exam has questions that are extremely similar to the questions found in the textbooks foe the material, so it’s really not so bad. You’re also generally expected to have memorized the proofs and use of a ton of different theorems. If you did properly learn Egarov’s theorem for problem 1, then problems 2 and 3 should be far more approachable. That on its own gets you up to the 30 point threshold.
In conclusion, this test is a way to sneak theorem memorization and utilizations into being the primary component of the test. It’s not as bad as it looks for being tricky; it’s far worse than it looks for being an extreme amount of preparation and memorization.
Is he speaking in English?
wow this is too hard for me to look at. I dont think i will ever touch this
Fascinating! Many top-notch high schoolers could probably solve 5 in an hour. Me? It would take me months.