So how did I do? Real Analysis PhD Qualifying exam review

[@JorgeOchoaLions]

The grades seem to reflect your own sense from the first video, it seems to me. I would like to share with you this: part 1. I once took a Linear Algebra (undergrad) final which I knew I had aced. Professor graded it as 99 out of 100; there were no errors marked on the exam. When I asked why 99, Prof said: "Well, you don't know everything." The exam was perfect though. I made a mental note. part 2. At a different university, I took an Algebraic Logic class (grad level). Final Exam was handed out the first day; it was one single problem, could be turned in anytime during the semester. The Final could be answered in the classroom for those of us who had not answered it by the end of the semester. After two hours of staring, I turned in a blank sheet of paper. I had never even understood what the problem statement was saying, much less did I have a sense for how to solve the problem. Dr P did not accept my blank sheet and said "sit down and pass this exam, I know you can." I sat down and for an hour I wrote everything I had learned in the course, everything, trying to hitch key concepts to the problem statement, clearly stating that I was only guessing, but articulating my reasoning in the best way I could. I got the only 100 in the course. At some point, inspired by Dr P, I decided that I would never turn in a homework or an exam that did not clearly reflect my love of Math. From that point forward, everything I did would have to reflect my love of Math, or I would not be satisfied, and therefore I would not turn it in. I began to see my life as a Math student in a whole different light. I began to notice that the professors whom I admired the most were more like artists, rather than Mathematicians. From the way they wrote, the way the talked, the way they treated me, the way they prepared for their classes, the assignments they gave and the exams they gave (extremely hard exams), everything was a work of art, everything. Once I noticed that, I decided for myself that my relationship with Mathematics was going to be a relationship of Love. Pure & Humble Love. I graduated Magna Cum Laude; that never meant anything to me. I am now 67 years old, and I love Mathematics more than I ever have. Your videos touched a nerve, in some way. They have taken me back to days long gone; I am very deeply grateful.

[@FanOfLiberty1776]

Great story!! I had a similar epiphany with my education when I returned to school (junior college first, then a university) as an adult and parent, nearly 20 years after barely graduating HS. Graduated university with honors. I needed some math for my major, but not much. Could barely pass pre-algebra in HS, but received an A in pre-calculus in college. It was a big boost for me personally, and gave confidence that I carried through my education.

[@zidenzz]

i aint reading allat 😂

[@varadthube4851]

monitor younsmonitors you seem like a tiktok guy with short span attention disorder.......

[@varadthube4851]

@@zidenzz ok then...go to doctor...easy..

[@youtubepremium9253]

Thank you for sharing

[@danielfromca]

So much respect for you for studying pure math on a grad school level. I wouldn't have the patience in a million years.

[@Tiqerboy]

Freshman undergrad courses in math ruled that out as a major for me. The great thing about university and learning as you go, figuring out what you want to do. So, every math course after that was application over theory and that was enough.

[@creatorofimages7925]

@@Tiqerboy I wish I could say the same with "application over theory" but for my location that is not the case.

[@nhitc6832]

Yeah. I quit math major after Real Analysis class because I was struggling so hard. Wasn't used to everything being theoretical instead of computational.

[@ASOT666]

I absolutely love pure math but am in no way good enough at it, which always makes me a little sad lol. I can only ever seem to get Bs-B+s in my 2nd/3rd year courses, and same with in my masters/phd level economics/econometrics/game theory classes.

[@youMatterItDoesGetBetter]

Would be pretty cool to know you're Good Will Hunting though.

[@therealhotwatertunes]

I watched this video and its prequel. I really appreciate what you're doing and the fact you're vlogging something that many students would really benefit from seeing. Not that math is hard or that profs are brutal but that even if you feel mediocre with time and effort you get better and trying is the important part. I made the mistake of working mostly alone at home, and rarely saw that my peers struggled too, and that brought my spirits down. These days I study sometimes around peers and even though they take different courses, seeing them mull over things for a while puts my struggles into perspective. I'd note that at some point it might make sense to have a disclaimer that math education differs between institutions and countries. I study in the Netherlands, and I haven't had an exam longer than 3 hours for my BSc nor for my MSc. I would not expect exceptional results from a 6 hour long examination. It's a miracle anyone can focus for that long to begin with. Here in NL there are some unreasonable profs too (like anywhere really) but at least the system requires 3 hour long exams haha. thanks again for sharing!!

[@a.b3203]

Which Uni? Eindhoven, Delft?

[@AndresRuiz007]

This shit just called me dumb in 10 languages

[@mattm9069]

I was a pure math undergrad. I know this is phD, but the explanations of your thought processes and ways of attacking the problem brought back memories that I thought I forgot. I decided to study statistics in grad school. Now, I'm doing data science in industry.

[@mattm9069]

@@philscimath9489 I'm glad I chose statistics because it was easy for me to see how it's used in various industries i.e. biostatistics, economics/finance, etc. and still get a good dose of theoretical math. Statistics is applicable in many different areas. My advice is to take some computer science electives or a minor you can pair with statistics. Or even just computer science as a major. A lot of what I'm teaching myself now is how to automate code to be able to stand up algorithms. It's not necessarily just about understanding the math, but making the computer work efficiently as well.

[@zxynj]

Math undergrad, stats master (originally want to be an actuary), ds in finance now. If you want to do analysis type of ds, learn how to tell stories using cool interactive visualizations (learn some dashboards). If you go for predictive model building type of ds, one shortcut is taking ml coding bootcamp if you don’t have time to take cs class in college. I attended TDI. Make sure you spend time on data structure and algorithm. Regardless of what type of ds you choose, the most important skill is to be able to explain complex concept in a way that 5 year old can understand. Selling of your work is the most difficult part. 😂

[@magicmulder]

I graded a couple exams back at university, and some of the other people did make some massive grading mistakes (from completely not getting that the answer was correct to giving full points to a crappy solution). The hardest to grade were the handful of geniuses who always came up with alternate solutions to the preferred way, and where I struggled whether to allow them to skip some steps (because I knew they were trivial to them) or apply the strict common standard (where you can’t omit too much because otherwise it’s not clear whether you actually understood what you wrote).

[@FanOfLiberty1776]

Don’t lose faith in yourself!! You are trying to enter the realm of the big dogs of math(Ph.Ds) and the guy measuring your efforts is making you prove yourself to be like him. You have gotten this far, you can make the rest of the way!! Ask questions and use this experience to study smart.

[@vander9678]

the realm of pyramid scheme

[@EzBz982]

He’s preparing you to publish. Good enough isn’t going to cut it for journals. Best of luck on your journey- I don’t miss these days.

[@mrpanda5783]

I'm barely passing Calc 1, this is some next level shit.

[@yinyang9508]

I have an A in calc 1 and I can’t understand shit.

[@orang1921]

@@yinyang9508 you have no reason to understand something you've never encountered

[@nikanevskaya5567]

I’m undergraduate math student and I really enjoy your videos. Thank you.

[@trevornoble337]

BSc in Math here from Canada who is trying to become a high school math teacher. I think what you said about the effort and writing what you can is justified! It’s not that you will always get a mark, but you are giving yourself a chance for a mark. I wouldn’t change that or teach any differently. On another note, I really enjoyed going through this content! I still practice questions here and there from my undergraduate studies to see what I have retained. All the best for when you take the exam for real!

[@supasugaman]

I whole heartedly disagree. The common practice of writing things that you know are complete nonsense in the hope that you get sympathy points and, even worse, the practice of GIVING these points is dentrimental to math education. You yourself, me, and this professor have all suffered guessing an appropriate amount of partial credit to give for scibbles that have no meaning. This practice imposes additional labor on the part of the grader (sometimes dealing with this can take more than twice as long) and these scribbles do not reflect care, effort or insights. I want to distinguish this from writing scratchwork, thoughts, and (clearly indicated) guesses which is valuable and should be worth credit.

[@artum1s_550]

In education, learning is what comes first. Grading is an arbitrary, subjective way of understanding what work the student has done. When encouraging students to just "spew" things they have a hunch about (without certainty) for a better grade, you prioritize something that isn't the most important (i.e. actually learning and being certain of things before you say them). The extra work shown is a positive from a grading standpoint, however, it does not show learning, nor does it encourage it. If you are not certain of what you are doing, then you aren't practicing. And worse than this, you may be logically proving in your head that something is true, even when it is not, simply because you have a hunch. For example, a language teacher might read words to the class and encourage students to repeat the words back to the teacher. Theoretically this is showing effort on the students part and is making the students better at the language. However, what it is really doing is building false associations in the students' heads on how certain words are pronounced. If you don't actually know how to pronounce a word (which you can only know by having a lot of input in that language), then you fundamentally can't get it right. It's analogous to blindfolding someone and making them shoot hoops from the 3 point line. Sure you will sometimes get the ball in, but you aren't practicing. Learning is about memorizing things, and understanding why things are true. I don't think I gave a particularly good example, though want I wanted to get across is that learning is when you understand something and through repetition memorize it. If you don't actually know something and try giving fake answers you are counterproductively in part memorizing false information. my opinion ;-;

[@supasugaman]

@@artum1s_550 I agree with the spirit of this. The important thing is to ask yourself what you want grades to reflect. Should it reflect how well the student understands the subject? How much effort the student put in? Should it be an incentive to get your students to study more? When professors or employers see an A or a C, what should this indicate to them? You said you wanted the grade to reflect what 'work the student has done' and then went on to describe how such work should reflect what the student has learned, outlined what 'learned' means to you, and indicated that certain 'work' should not count. For others, it seems like the effort and incentives perspective is popular. My point was that no matter what you wish for the grade to indicate- you do not wish for it to indicate your students willingness to write things that make no sense in order to get a better grade- or maybe I'm wrong since you are teaching your students valuable grade optimization techniques for future courses. One point where I disagree with you, however, is what it means to learn. Our struggling friend is in a doctoral program which aims to train him to be a professional mathematician. There are many skills involved and many resemble (are?) guessing and hunches. Certainly there are skills the program wishes to train that are far from memorizing theorems and their proofs. For example, a common strategy is to assume a hypothesis that you suspect is true to see where it leads, deferring its proof until later if it turns out to be useful. If it does not seem to be useful you cross it out. You have done nothing but convince yourself not to do something, but what you did was a sensible part of being a mathematician. It is certainly not unreasonable to give some credit for this.

[@trevornoble337]

@@supasugaman I hear you, but I think you may have also misunderstood how exactly I deal with this. When I mark the math work of students, I would always prefer a student give a question their best shot and be wrong than leave a question blank. This then lets me see where exactly they go wrong so I can give them direct feedback as opposed to leaving it blank because they left it blank. Obviously, if the student is immediately wrong at the start, I’m not going to give them sympathy marks, hence the comment about not necessarily getting a mark, but having a chance at one. That said, if they can show any sort of semblance of writing a solution on the right path, that is worth something at the minimum; the amount would then depend on how correct their solution is. The extra time to write these comments is indeed more of a time sacrifice than leaving an X and moving on, but it is something that I am willing to do in the hopes that it will later allow for future conversations and ultimately the betterment of the student’s pursuit in understanding mathematics.

[@abud9933]

I remember when the prof said he’d subtract grades if we make up stuff. Still made stuff up and didn’t lose anything 😉

[@theboombody]

I remember being in real analysis class and deciding at that point I wouldn't ever be able to contribute anything meaningful to the field of mathematics. I got my bachelor's degree and then went straight to the working world. Well, it was actually a bit more bumpy of a transition than that.

[@treebattery2901]

could you elaborate? I'm interested

[@bingbong2179]

@@treebattery2901 I did a bachelor of mathematics and felt similarly. While real analysis is by no means the only area of mathematics using proofs, I found that it did not at all come naturally to me and very rarely did I enjoy the end result after spending hours and hours whacking my head against a brick wall. I won't say that I simply wasn't talented enough but I don't think I had the temperament needed for pure maths nor the passion. I ended up favouring statistics courses as they were much more applied and I could see the value (real-world) of the work more clearly. To put it simply there's a big difference between enjoying high school solving equations maths, and university proofsy maths and you may well find that you enjoy one but not the other.

[@speedspeed121]

The same with me and physics

[@Twizz993]

​@@bingbong2179 Couldn't have expressed that better myself! I had the same realizations about my situation (bachelors in physics, minor in math and CS). Ended up getting a software dev role after graduating 2 years ago and couldn't be happier that I gave up on physics as a career.

[@jabr0nicus]

I relate, very similar thing happened to me in physics

[@alvydagr8108]

Im an barely a sophomore in my mid 20s but working on physics, this just blew my mind. I subscribed immediately. Thanks for putting this on the internet.

[@lsad8733]

Yah same. Doing calc 2 rn and I thought that math was hard LOL

[@orang1921]

@@lsad8733 continue, it'll only get more intuitive as you go along

[@carstenmeyer7786]

For the last (calculus style) question, I'd say there are two key facts to solve it: (1) |ak|

[@bryanmata8319]

Shows how much common-sense professors lack haha. "On the real thing I would give you a 30 but I gave you a 21" This is why I hated college and flourished in the real world.

[@josephkeres4604]

Yes. I’m sure that’s why.

[@skivvytv6229]

@@josephkeres4604 🤡

[@BlastinRope]

​@@josephkeres4604 theres no top dog winners staying at uni longer than they have to

[@carlosbenavides3023]

There's no way to get a job in the area he's looking for without graduate education. Also, is just a way to push people to be proficient in a topic.

[@franckplanks]

Yeah academia seems to be full of these kinda of theatrics and other bullshit. I enjoy college but I doubt going past a masters degree will do much good

[@sambo6959]

“Hey I really liked your answers they were good” 8/10

[@orang1921]

8/10 is fantastic when you get to these levels of math

[@lezley9548]

Seeing the half full pack of pens and the stacks of paper and books on your desk really just strikes a nerve. My desk is the same, and my room is covered with books, pens, and papers, and seeing it here makes me feel really happy (: good luck with the rest of your studies!

[@stranglehold4713]

21:00 He mentioned this to you because he is trying to push you that last little bit. It isn't meant to discourage as much as it is to challenge you. Giving you the full 30 would give you a false sense of security while this practice test is going to push you to sharpen your skills further. Best of luck

[@gunpixelgun]

He needs to see this

[@FenShen-us9tv]

That's great and all but you're paying thousands of dollars for his class. The professor should have a consistent grading system.

[@CrustyJoeMC]

@@FenShen-us9tv that is not how graduate school works

[@Duffyyy94]

This would make me become an alcoholic. How depressing and defeated I would feel all the time.

[@nanoc.2103]

As a former math teacher, it pains me to hear that there was no rubric and it goes ‘on feel’. That implies so much variability, subjectivity, and (unintentional) gatekeeping. This is such a game on how the grader feels in the time of day.

[@Sukkulents_]

you literally could get them when they’re in a bad mood and they just flat out refuse to give you any marks for wiggle room/ benefit of the doubt

[@franckplanks]

This sums up a major problem with academia beautifully. Professors say that your work is graded objectively by clear standards, as it should be in math. Then they fuck you backwards and you get a 42% just because they “felt you could do better”. Bitch, math is worst place to be judged based on your toothless opinion after your tenth glass of bourbon.

[@CD-yf4wm]

But it’s so hard to have a rubric for this exam

[@nanoc.2103]

@@CD-yf4wm It need not be a specific rubric with boxes to check off. There needs to be at least some sort of (math department) established guidelines to follow to determine whether full credit or partial credit can be granted or not. If you grade a person's work on the basis of feel, you then gate-keep depending on what your personal views are of their written work (ie you introduce your own subjectivity) From an educational standpoint, there needs to be at least a level of objectivity when grading. As a former math major, I know and have seen proofs vary greatly, especially for analysis and topology! Quick possible examples: Are there logic errors? Misuse of definitions? Was the theorem stated/written correctly?

[@CD-yf4wm]

@@nanoc.2103 Yeah that makes sense. I'm also grad student in math, I guess I'm just too used to my department's policy.. My department grades qual with no partial credits, (i.e., full credit for correct solution w/ minor errors and 0 otherwise) and I guess that's also not unreasonable.

[@nmayes1984]

I just finished my oral candidacy exam on non-Archimedean quantum mechanics via quantum groups and have passed my three comprehensive exams a year ago. For the record, I think you did amazing man and your Analysis exam was more difficult than mine other than some really messed up problem about completeness that got me. I think you have studied hard for this and definitely look forward to hearing about your research area. PS I also had the same problem about V = Img direct sum Kernal for the linear algebra part of my algebra exam (Axler book as well) :P

[@MadScientyst]

Like many other comments I watched both parts of your 'tutorial' so to speak, which I found to be immensely interesting. When I did R.A. at Undergrad level, I failed Course-I 4 times & Course-II 2 times. BUT, the interesting thing is that afterwards, I flew through C.A. Courses I&II with Grade A's as a consequence of those past failures. I also had a Lecturer who came from Waterloo University with an M.Phil in Analysis & your Prof reminded me of him. His standard for exposition was so high that on many of his corrected Papers, had this uncanny habit of writing 'I.S.W.' (Ignored Subsequent Working) anytime a Student's argument or reasoning went askew. To his credit though, once your 'optional' insight added logic or coherence to the problem, he'd give some marks in true 'I.F.F.' (If & Only If) reasoning. He also suggested to us Students to do just as u did with some notes on important Axioms, Corollaries & Theorems, etc. I'm now at Grad level hoping to pursue a full (Pure Math) Master's & I actually noticed some of these questions are familiar from that degree of Mathematics. My personal arsenal in proofs usually tend from the following methods & (fairly memorized) techniques: *M.V.T. & I.V.T. (Calc) *Induction /Triangle Inequality /Series & Sequences /Convergence & Intervals *epsilon\ε - δ\delta /Bounds, Lim Inf & Sup *Integration Theory >> (Riemann-Stieltjes-Lebesgue) *Cauchy-Schwarz /Heine-Borel /Rolle /Bolzano-Weierstrass For reference material, the following Authors are tops: Rudin /Tao /Polya /Bartle /Halmos I have a similar list for C.A., but I'll reserve that to comment on any Vlog u post on it (as a new subscriber...lol). Practice makes perfect & 'Mathematics is not a spectator Sport' as often quoted. I apologize this comment is so long, BUT being able to express points crystal clear is a consequence of my Real Analysis training & experience. So good luck & please don't be deterred in your PhD pursuit, fellow Mathematician! I look fwd to exploring your Channel's content indeed!

[@Grova]

You took real analysis I **four times**?

[@oooBASTIooo]

@@Grova no, he failed it 4 times.. so i guess he took it 5 times.

[@elijahr_1998]

I'm in year 3 of my math PhD, passed all my quals in my first year on my first try. What you're going through used to be my greatest nightmare. Qual grading is notoriously bullshit. Some just pass, and some just don't. It's terrifying. The fact your instructor casually admits his grading is variable by a whopping 20% or more, enrages me. These tests can be career-making or career-shattering. They should be treated with MORE delicacy than regular in-term undergrad/grad exams, not LESS! It's all backward. Every test should be reviewed by multiple graders, and those graders shouldn't know whose test they're grading. That's the MINIMUM. Beyond that, more careful question selection, a complete overhaul of the preparation process (ambiguity, insufficient practice materials), more standardization across tests written by different instructors within the same subject... the list goes on. You put in a great effort on your exam. A zero should never be given unless the page is blank, your instructor is a dick.

[@ILoveMaths07]

You deserve to pass! That's a 60% as he said!

[@Monadoabyss]

You've probably figured it out already but I think you're supposed to use the theorem in question 1 (Egorov) for question 2. I think it's quite common in exams for the first part to ask you to regurgitate some theorem shown in the course then the question immediately following it to apply that theorem in some way, so you probably want to watch out for this in the future. (I haven't worked through the details but I think the proof goes like this: Assume f_n converges to 0 almost everywhere on [0,1]. Then use Egorov to split the integral of f_n into a set E that has measure e and a set [0,1]\E that has measure 1-e where the f_n converges uniformly to 0, so the integral on [0,1]\E is very small whereas the integral on E is very close to n. Then use Holder's inequality to show the square of the integral of f_n (which you just showed to be close to n^2) must be

[@arghyadeepchatterjee6100]

You can directly see that f_{n}/n is L^{2} bounded if a_{n} are bounded. Thus f_{n}/n is uniformly integrable and almost sure convergence of f_{n} to 0 would mean that f_{n}/n converges in L^1 to 0 which contradicts that f_{n}/n has first moments identically equal to 1. That is why I always view ([0,1],B,\lambda) as a probability space and the most important probability space out there.

[@TheArtofFugue]

@@arghyadeepchatterjee6100 ^

[@Adam-io6oz]

@@arghyadeepchatterjee6100 The fact that uniformly integrable a.s converging subsequences converge in L1 (Vitali convergence theorem) is proved afaik using Egorov's theorem. And the fact that f_n/n is uniformly integrable is shown via 123 456's use of Hölder's inequality. So your proofs are really the same 😀

[@arghyadeepchatterjee6100]

​@@Adam-io6oz For finite measure spaces, the proof is NOT by Egorov's Theorem and is much much easier than the general case. In fact, I can fit it in this particular comment itself and it is much used in Probability Theory as I said. Consider a finite measure space (Omega, F, P) Let X_{n} converge to X in measure(I'll use P) and let X_{n} be uniformly integrable Consider Z_{n}=X_{n}-X Then Z_{n} is uniformly integrable. So for epsilon>0, choose M such that \int 1_{|Z_{n}|>M} |Z_{n}| M} dP + \int |Z_{n}| 1_{|Z_{n}|\leq M} dP \leq epsilon + \int |Z_{n}| 1_{|Z_{n}|\leq M} dP Now, as Z_{n} goes to 0 in measure pick N such that P(|Z_{n}|>epsilon)\epsilon, |Z_{n}|\leq M} dP + \int |Z_{n} 1_{|Z_{n}|\epsilon) + \epsilon P(|Z_{n}|>\epsilon, |Z_{n}|\leq M) \leq (M.epsilon)/M + \epsilon P(Omega) \leq \epsilon (1+P(Omega)) And the above holds for all epsilon>0. Note that this is the step where we use finiteness of measure P. This gives us that |Z_{n}|\to 0 in L^{1} sense. Also note that we only need convergenece in measure which is guaranteed by almost sure convergence. So no Egorov's theorem is required.

[@arghyadeepchatterjee6100]

@@Adam-io6oz Also note that by 1_{A} , I mean the indicator of a set. The notation would have been shorter if I'd have used Expectation instead of \int but I did it to show that it is true for all finite measure spaces and not necessarily only probability spaces.

[@TheeMelloMan]

What am i looking at? No, why am i here???

[@finmat95]

You know why.

[@joefuentes2977]

I think you did great, you had the backwards E, the upsidedown A, and even the square at the end! You should have gotten more points for that

[@brandongroth4569]

For #5, I think you needed to do a Taylor expansion of sin(x) = x + x^3/3! + x^5/5! + ... For sin(a_n), inside the sum then a_n + a_n^3/3! = 0 by the assumption, so you only have a_n^k for k>=5 to deal with. Likely can do some argument where you choose n>N s.t |a_n|^k < 1/2000.

[@SPIDERGHOST99]

5:40 totally accurate - definitely better to put down anything rather than nothing!

[@franckplanks]

This

[@arsalannss]

​@franckplanks what do you mean by "This"?

[@arghyadeepchatterjee6100]

I always look at [0,1] with Borel sigma algebra and Lebesgue Measure as a Probability space(The most important one) and hence for the second question you might as well consider f_{n} to be random variables. What you need to see is that if an is bounded, then f_{n}/n is a sequence which is L^2 bounded and hence Uniformly Integrable. Now if f_{n} converged almost surely to 0 then f_{n}/n also converges a.s. to 0 , then by uniform integrability theorem, you have f_{n}/n converges in L^1 to 0. Which is a contradiction as the first moments of f_{n}/n are identically 1. Easy proof . Also uniform integrability seems to be a recurring theme. In question 3 for example, you just need to show that f_{n}g_{n} are bounded in L^{1+delta} for some delta>0 which would again imply uniform integrability and hence you can deduce L^1 convergence from almost sure convergence . To do that let p=2+delta. Then both f_{n} and g_{n} are in L^{2+delta} and are uniformly bounded in the L^{2+delta} norm. Then if we consider |f_{n}|^{1+delta/2} and |g_{n}|^{1+delta/2} then By Holders Inequality taking integrals(or Expectation) on both sides E[|f_{n}||g_{n}|)^{1+delta/2}] \leq (||f_{n}||_{2+delta} )^1/2+delta times (||g_{n}||_{2+delta})^{1/2+delta} and both the terms on the RHS are bounded uniformly. Hence you again have uniform integrability and you can conclude L^{1} convergence of f_{n}g_{n} to 0. For a counter example for p=2 , you just need a function (random variable) which have uniformly bounded variance but higher moments are not bounded. So let f_{n}=g_{n}=\sqrt{n/2} Indicator_{[0,1/n]} . Thats it . Also in the 4th problem you also need to conclude convergence of first moment from convergence in Measure. So that is also using the notion of uniform integrability but sadly here we cannot use the finiteness of the measure space. But like you did, using DCT for [1,infinity) and [-infinity,-1) we directly have L^{1} convergence to 0. So we only need to focus on $[-1,1]$ as which is again a finite measure space. So again, you need to prove Uniform Integrability which is easy enough. |g_{n}^3 g_{n} x^{2}|> |g_{n}^{3}| for all x in (-1,1) due to the fact that |g_{n}(x)|

[@arghyadeepchatterjee6100]

@@vxidh_0771 Those are from the past days when Fifa was still a good game. Now I only play patched and modded versions of Pes 2021 (SP Football life 2023) . Also I was in high school back then . Now I am studying for a PhD in Mathematics (Probability) . Anyways thanks for your comment.

[@mycrushisachicken]

probability theory is the best

[@xslickrickx2103]

As a grad student you are there to learn enough to contribute original knowledge to your field. While professors are there to help guide you there evaluation / opinion is irrelevant other than it’s potential hindrance to your progress. I’d consider trying to let yourself be salty for a weekend then let it go. If you really are that bitter, graduate and come back to do their job better / the way you think it should be done.

[@NotThatSocially]

My goodness, you clearly knew what you were talking about about in the first problem. Mathematicians are so damn picky

[@zhangkevin6748]

i remember my real analysis exam, i got a 30/60, it was very brutal

[@PhDVlog777]

My brother in analysis, I feel your pain

[@zhangkevin6748]

@@PhDVlog777 thanks man, I feel like sometimes I am right and the prof gives me 2/10 for the question 😂

[@potatogaming6613]

@@PhDVlog777 do you you play games or watch Netflix like normal people do

[@PhDVlog777]

@@potatogaming6613 Not as much these days. I play Dark Souls when I have some time.

[@Jim-be8sj]

Good luck with it. I've been in that situation and understand the work and stress related to it.

[@drogoff9059]

Really impressed that you made this video.

[@abebuckingham8198]

Grades are not a measure of your achievement, they're feedback to help you improve. If he's grading you more harshly it's because he fully believes you can achieve that higher standard. I would consider this an excellent effort and a bright point towards your further success. Being open to criticism is half of being a mathematician.

[@50Steaks68]

If future educational institutions see a poor GPA, do they accept you because they see how much potential they see you have to improve?

[@mightylock3223]

@@50Steaks68 yeah, i agree with your statement.

[@embern3372]

@@50Steaks68 Feedback that counts towards the final result, not the final result itself.

[@bluecollar8525]

​@@50Steaks68this is something in surprised no one seems to get. If some people are marked more harshly because "they have higher potential" or some bullshit. Then the next idiot who got marked easier is gonna get into the postdoc or some other position in their place. Grades matter

[@finmat95]

I could agree with the the first and last line of the comment, not the rest.

[@worldblazin4209]

Thank you through showing your performance on the Qualifying Real Analysis exam! Just finished Calculus 2 last week and hearing terms like absolute convergence and summation gave me good times as I viewed your interpretation. Maybe I'm not the target audience, since I didn't view all 10 Questions, but thanks for showing us your hardships! Questions 1 to 5 --> 0:00 to 15:36

[@NewWesternFront]

yes same here and it gave me some smartz feelz

[@nikosneely1558]

i am an undergrad college dropout with dyscalculia and for some reason find these videos fascinating

[@abood5813]

I don't understand anything of this, but I believe in you to improve.

[@deltalima6703]

Upside down letters, bunch of greek letters, some backwards letters. Sheesh! Gibberish.

[@millyyeasmin7904]

Why are you here then

[@howardlam6181]

Yep, at grad level, it's either right or wrong. No credit for wrong stuff. Full credit for right stuff. No partial grade lol It's either your paper getting published, or not getting published.

[@finmat95]

That sucks.

[@nicos1097]

Good luck dude you’ll get that Phd

[@julienbongars4287]

Thanks for reminding me why I dropped out of University. Like you said though, don't let this get to you, your prof obviously seems like he has an enormous ego that requires him to prove to everyone how smart he is by proving to others how dumb they are because his job security depends on it.

[@tmann986]

Under grade engineering major here. One thing I want to throw my two cents is I have a habit now because of my engineering professor doesn’t mind if we write something belligerent such as, wtf am I suppose to do with this or how tf does this value help me?? I feel a little better writing my frustration for some reason. Probably why journaling is great but the point is i like to do my exams with a little of my personality or comments what im thinking. I think it gives the grader a funny but real understanding of what i was thinking at that time. Probably helps too when giving feedback and maybe a chuckle at the frustration of not knowing what to do. I recently got a few books on proof writing ever since I learned how to do an Epsilon-delta proof when my professor didn’t want to show us how that worked in multivariable calculus. Looking at your exam brings a lot of excitement to me because just knowing some of the jargon and symbols i can read what you’re saying (not that i understand it) is so cool! I love math! I wish i could do more which is why self study with textbooks and KZbin seems to be my choice.

[@priyanshukalal1195]

Same story here. Whenever I get time from engineering studies I try to focus on pure math

[@stevenlam6163]

Really appreciate you sharing this with us!

[@LilJollyJoker]

Stumbled across thei video randomly. Literally have no idea what any of this means, but sure looks interesting! 9.97K Subs, SO CLOSE TO 10K!

[@bigbunn_]

Im only 20 and in my second year but this is fascinating and at the same time terrifying. I hope i dont have to do classes like this lmao.

[@samuelokon8842]

My God. I'm really enjoying your content. I can't believe I spent about an hour try to solve question 5 on my own. Were you finally able to redo the problem? For me seems one can prove that the series has a tighter upper bound (i.e instead of proving for 1/1000, I was able to prove it for with an upper bound of 1 using the mean value theorem trick). I might be wrong, but I can't find (yet) a counter example to suggest otherwise.

[@PhDVlog777]

He said that the Taylor series expansion of sine is necessary and the cancellation of the first few terms comes in to play. Unfortunately, I haven’t found time to solve it but I believe I can.

[@aghileslounis]

I hold a Bachelor in Physics, but didn't have enough courage to continue that academic path, even though I love Physics and Math, but it was VERY difficult for me, the abstraction is very high, especially in Quantum Mechanics. I decided to switch to computer science, and like it so far, I work in the field, it's 1000x easier than physics tbh. But I will never ever know if I would succeed if I followed the academic physics path... I do miss the excitement of solving problems, sometimes I just want to practice Math and Physics again, just for fun. But It's probably a bad idea, since I much rather be better at programming instead, haha.

[@Fabiandur]

Start doing algorithms! You can get pretty high-up with them as well.

[@ziko317]

"He likes pictures so I drew him a picture." Yup. 😂

[@rocketman1104]

I remember actually watching your first video before you took this test and I'm only now just seeing the results 9 months after the fact. But your mood about it embodies my opinion quite heavily haha. I have had a tough go at Calc 3 and Diff and had a similar argue with the prof for points regarding a certain question, i had all the parts written down for this particular question asking for a system of equations in a first order derivative. Well this did not do, especially since even though IMO the instructions should have explicitly stated as such(its inferred I know) that i needed to write a matrices of the system as well with it.

[@nathanspreitzer6738]

Very weird that the grading is so vague, especially considering it’s a math course

[@neerajnongmaithem392]

You are right about our school teachers telling us to write even if we didnt knew the exact answer as effort does count. But i assume your phd prof dont think in that regard, he is strict with your grading to seek perfection from you because at the end of day in the real world your papers wont get published or get revered much if they are anything short then perfect. All the best mate, afterall you have choosen to pursue a phd in a subject which is all about being exact and perfect.

[@joetursi9573]

Re. problem #1, it's easy to get order of quantifiers backwards as it definition of limit of a function.

[@JohnSmith-rr3jt]

The best part about education is that the higher level you achieve, the more you realise no one has any fucking idea what anyone else is doing. Hell half the people can't even explain what they are doing lol...

[@Badsniperarmy]

Dude I'm just learning about slope fields... BUT I CAN CONFIRM THAT WHEN X IS NEAR 0: X (similar) = sin x

[@zzayy8282]

Something I learned at my time in college is that if you complain, you will almost always get points for it. If not full credit or any credit at all, you get some very valuable information from it.

[@Pommes736]

You are right, you can't grade anything which isn't written on the paper. My prof would also agree with you.

[@Konchok_Dawa]

Your handwriting is very satisfying

[@johnrussell3067]

And here I am struggling in Calculus 2 lol.

[@generalmalaise2930]

same

[@russellkamback1905]

good video and hella respect for doing a phd in real analysis, but, respectfully, why would anyone do a phd in real analysis ever? seems like needless pain

[@timothynolan7250]

I know the struggle! Keep trucking!

[@salvatorebellino-jw4jk]

First of all the best for your future! Can I ask you on which exercises notes you studied for Real Analysis? I'm looking for notes that help me to get ready for exercises like these. Again, good luck!

[@NewWesternFront]

try the satanic bible.

[@86_beans]

How about doing the questions in a different order? Leave the easy one til later so you have some confidence of time up your sleeve? Do the hard ones when you are fresher. I never trust the order of an examination.

[@hypnogri5457]

greatest comedy channel on youtube

[@aviberezovskiy7633]

Currently binging all your videos! I am currently studying to get my math BSc in Israel. We already have a course on measure theory 🥴. Don’t let your spirit fall, math is incredibly hard and we all struggle :( :)

[@maryamabdal-jalil]

Where is Israel? I'm confused

[@Umbresp]

Watching this instead of studying for my real analysis exam tomorrow morning

[@melvin3459]

how did you do? :)

[@Umbresp]

@@melvin3459 I think I did fine, it's undergrad so nowhere near as hard as this. The questions were over bounded functions and suprema

[@melvin3459]

@@Umbresp nice!

[@PhDVlog777]

Hope you aced it!

[@Umbresp]

@@PhDVlog777 We shall see, I'll provide an update once I get my grade back

[@dimimegesis]

I agree that you should always write something if you possibly can! that grader was mean

[@franckplanks]

This

[@kreyton3312]

something about a teacher marking wrong answers on an exam with question marks makes me angry

[@randomcandy1000]

LETS GOOOOO

[@simonreiff3889]

Is the trick for P5 to use a Taylor expansion for sin for more terms than you did? Feel like that probably would have gotten you to the finish line.

[@dmwalker24]

I have no issue with strict grading. Some of my favorite professors were those with the most exacting standards. The worst professor I ever had, on the other hand, was for Organic Chemistry. His lectures consisted of "Here is a mechanism. Reagents, conditions, caveats (steric hindrance, etc). Now on to the next mechanism". It was quick, and pretty vague, and there was little to no connection between his description, and why some given scenario would react one way versus another. Exams were then, "here are some reagents, show a mechanism". That class wrecked me and 70% of the other students. At the same time, I excelled at doing the chemistry in the lab. The professor was simply not giving us enough to produce what he wanted in return.

[@TripleKmafia]

Damn, tell him it’s his fault.

[@bluecollar8525]

Those are the worst. You have to teach material first if you want students to learn it. If not, then you have to provide them with all the material they need to teach themselves. I've studied at both a brick & mortar and online university. In online studies, they give you everything you need. Lecture notes, text readings, practice problems, solutions, etc. You're provided whatever you need to learn the material, and if you can't figure something out, you email the prof to ask for help. By the time you sit the exam, you can feel confident that you know it solidly. If you don't, it's because you didn't prepare, not because information was omitted/ambiguous during the study phase. I hate the brick & mortar belief that you have to waste time and effort getting to and from lectures to be given information that is only in the lecture and not in the readings, while you speedily try to take notes, which end up either being incomplete because you're listening trying to learn, or complete but nonsense because you get what was being said because you were just trying to get it down in the notes. Profs are incentivized to put artificial road blocks in place to create the illusion of difficult material and get a bell curve, but instead of measuring how well a student has studied, they're just measuring which person could tolerate an early morning lecture better, or which student could write notes fastest, or which student could make sense of the thick accent better. It was such a waste of time and energy. Studying online is so much more efficient and it makes it so much easier to thoroughly learn the material and go into an exam confident that you know your studf

[@blandconstant5548]

not gonna lie, im not a huge fan of these questions and needing to remember a proof of a randomly chosen theorem. Would make more sense to me if this exam didnt exist and instead they actually looked at how you did in your, in this case, real analysis courses and exams. And im not saying you shouldnt be required to know the theorems and how to use them, but i dont think needing to remember them from memory is necessary. the questions, in terms of difficulty seemed ok if not a little strange. and yes 5 wasnt exactly hard but just weird.

[@evionlast]

If you're not learning the theorems then what is the point?

[@blandconstant5548]

@@evionlast im not saying dont learn the theorems, but imo they shouldnt just ask to memorize a theorem just because. I would prefer the theorems to be tested in a larger amount of different exercises and not just memorization. For me maths and maths exams are not for memorizing things but understanding them and knowing how to apply them and this remember them. Its not like learning a new language where you just memorize how to write "do" in the present and past and etc form.

[@Alex-je5eu]

@@blandconstant5548 How can you know to use a theorem without memorising it?

[@vlix123]

@@blandconstant5548 You don't have to memorise them. Just understand what proof technique is used to prove it and why and how to use it is enough to recreate the proof, exactly as you said. So I don't really understand your problem with the first question.

[@royalfinest]

I was thinking about further maths study until I saw that epsilon-delta thingy still clinging there. Sorry 😂

[@nestorv7627]

It never goes away

[@leidecarvalhodemello4594]

It only took me 2 semesters of applied maths to decide i wanted to be an engineer, best decision i ever took, id kill myself over going through that

[@melvin3459]

I love your handwriting!

[@w__a__l__e]

i know so little about this.. you could be lying to me and i would have no idea. lol.. it really feels to the more i learn to more i realize how little i know.. but keep up the work dude.. its interesting to see :D

[@jarednewton9238]

Just like with all exams, this doesn't test how well you know things at all. In my Physics PhD program, I am lucky enough to be in the first cohort of grad students that weren't required to pass a qualifying exam. This is because a professor in my department ran a study that found that the only thing evaluated in a student's test is how good they are at taking tests, not how much they actually understand everything. Just saying all that so you don't get too discouraged!

[@nady2296]

It's the first video from you I see but I relate a lot. I'm a French student in mathematics and informatic and I faced similar struggles during the two years of college I already did. Although the level was not as high as your paper since I'm not as advanced in my studies yet, my professor of probability / measure theory did grade me harshly. I got a 3/20 at my first exam with him and 6/20 at the second one while working a lot. During the second exam, the class median was at 6 too, just to paint the picture. It pained me a lot and I felt some of the notation was unfair. Although to be frank, the professor didn't have much time for all the notions we had to learn. Some of them were from the third year while we where at the beginning of the second. It was harsh. But despite my grades, I know how much I learned about the subject. And this year I feel like I can do a lot better because we got ahead of the program last year and because I know what to expect. But it doesn't erase the fact that it was a real struggle and I like being able to relate with people about mathematics :)

[@Yetiman2141]

Shoulda just used PEMDAS bro

[@thorneto2742]

I only went as far as fourth year undergrad real analysis but this really brought me back. Math 411 I think? I still don't really understand what a Lebesgue measure is but I make video games now so it is what it is lol

[@trollol_]

i noticed in my math classes that the profs seemed to enjoy making class and exam difficult for the students. my theory on that is most of them are either bitter about something, or have something to prove or compensate for. my take. only undergraduate math classes.

[@LeNoLi.]

This is like receiving a grade from a preceptor in medical school that came out of nowhere and comments that don't apply to you. I saw one doc once the while rotation and his feedback was so long and critical. Sir, you are confusing me with another student.

[@UlricGrim]

Can’t we just solve for x and y and call it a day?

[@diamondlong23]

me who doesn't know a lick of this stuff

[@tomlavelle8340]

Keep on slugging! You are way smarter than me.

[@ChiIeboy]

I have never felt so unintelligent in my life as I do watching this video and reading some of these comments.

[@adamrubinson6875]

I agree with your professor's philosophy. But I would say, "Don't ever write anything that you aren't sure is true (unless it is scrapwork/ where you are testing to see whether or not it is true)".

[@frankbank8720]

This is why I do complex analysis it’s so much easier than analysis 😂

[@Alex95334]

Great video! In my opinion, I believe you need to work on your formatting. Even if you have some of the answers correct, your writing could be better presented to make it a pleasant read. All the lines seem to be given equal weight, there seems to be no division between paragraphs of ideas and you could maybe condense some of the lines. Cheers!

[@bruh60297]

no

[@gabrielsantana3861]

Thanks for the video

[@TiredGradStudent]

Christ on a bike. Any exam longer than 2 hours seems like more of a test of endurance than intellect and competency. Hoping my ME dept. doesn't make my prelim this awful. Best of luck to you in your future endeavors!

[@EncleHumpaKev]

A university academic returning marked work on the same day??? Video must be fake.

[@MaJia-ed4fn]

I am a math phd and am also struggling with qualifying.

[@johndarland3633]

Yeah I suppose if you write something and even if it's not right then you at least convey your thought process somewhat to allow the assessor/teacher to identify gaps and support you. It's not even about the grade like it's about learning.

[@aminkhafizi4364]

this is the way brain get their work..some time we know the answer but we forget the CLUE but when we saw the SIGN it remind about the past but the enviroment see like don't care ..

[@ghausiyunus121]

You should be proud of yourself…I have tried to enroll for PhD program but have not been qualified yet….I wondering to know which one of state’s the college is located…thanks

[@josh1234567892]

5:14 I totally agree with you here

[@cyclon1337]

So that’s how an enchantment table works

[@stevencook9884]

Leaving nothing blank has saved me many times. Try to make something make sense.

[@tmo314]

Next try Fake Analysis. It’s much easier 👍🏻