Here is a solution for problem 6. Let M = {A : A = A^T, A p.d., det(A) = 1}. Evidently we have to consider the set of values V = {det(A + 2 C) : A, C in M}. We claim that V = [27, +\infty). To see that V includes this set, let A be the identity and let C be diagonal with entries (a, a, 1/(a^2)) where a >= 1. The determinant of the sum A + 2C is then (1+2a)^2(1 + 2/a^2). Consider this as a function of a on the interval [1, \infty). It is easily verified to be strictly increasing (differentiate), and equal to 27 at a = 1, and tending to infinity as a->infinity. To see that V is included in [27, \infty), apply the Minkowski determinant inequality (or prove it directly for dimension 3, which is actually straightforward). It immediately implies det(A + 2C) >= 27 for any A, C in M, which means we're done.
I just graduated from my Master's and will be attending a PhD program for Mathematics. A review of Measure Theory would help alot because I have taken a measure theory class that only talks about the Lebesgue measure, while the classes I will be taking my first semester of my PhD will just be abstract measures, and these will be on my qualifying exams. I also am a bit intimidated by my options of what I want to focus on, like Functional or Harmonic Analysis, Differential Geometrty, or PDE'S. Looking forward to start my PhD journey!
Maybe you can try problem 8 if it doesn't seem that bad to you yourself :) Maybe your instructor wanted to spook you! Often easy questions are propped up to be spooky lol.😂
Here is a solution for problem 5. Lemma: For any pair of unit vectors (u, v) there exists a matrix B = B^T with operator norm (w.r.t. Euclidean norm) at most 1 such that Bu = v. Proof (assuming Lemma): Write x = |x| u, where u is a unit vector. It is straightforward to see that we can write {Ax : A = A^T, A psd, ||A||
Check out Donald L. Cohn's book on Measure Theory. Helped me out a lot when I first took it. For my graduate course on Measure and Probability Theory, Williams' book "Probability with Martingales" was exceptionally good. Lots of examples, and a nice, clean approach to the construction of measures.
Hey grad, since the video series is not a lecture series, you can go through the basic/less important stuff quicker or just skip it. I guess that undergraduates who are interested in just getting a feel won't mind, and masters students will appreciate that as well. That way you can make fewer videos and also make them more enjoyable for yourself. Big fan
Hey, how would i do (software engineering masters), in a pure math masters, thinking of jumping to a different branch of STEM. Also stay in EU or try to apply in USA?
How are you at writing proofs? The proof writing and abstract theory are a part of the challenge. Notorious classes are Abstract Algebra, which looks at structure; furthermore, Real and Complex Analysis, which looks at the theory which supports calculus. Look for a reputable mathematics university, which exists in many areas of the world. Best wishes; Cheerful Calculations.
I studied environmental science before entering into pure math, and I struggled to understand proof writing but manage to overcome it and finish my masters. If you can overcome proof writing, then I think you can do it. As far as EU or USA, I am pretty ignorant on the subject, but I would do what is most comfortable for you. Moving to the USA may be stressful but many of my close friends in the dept. are international students and they like it here. I hope this helps :)
Here is a solution for problem 6.
Let M = {A : A = A^T, A p.d., det(A) = 1}. Evidently we have to consider the set of values V = {det(A + 2 C) : A, C in M}.
We claim that V = [27, +\infty). To see that V includes this set, let A be the identity and let C be diagonal with entries (a, a, 1/(a^2)) where a >= 1. The determinant of the sum A + 2C is then (1+2a)^2(1 + 2/a^2). Consider this as a function of a on the interval [1, \infty). It is easily verified to be strictly increasing (differentiate), and equal to 27 at a = 1, and tending to infinity as a->infinity.
To see that V is included in [27, \infty), apply the Minkowski determinant inequality (or prove it directly for dimension 3, which is actually straightforward). It immediately implies det(A + 2C) >= 27 for any A, C in M, which means we're done.
"bro are you for real not enriching right now?"
fr bro I gotta lock in rn
"Look at this fellow, a highly enriched individual. Get enriched, chump!"
Mr. Frog for president 2024.
Hello
I just graduated from my Master's and will be attending a PhD program for Mathematics. A review of Measure Theory would help alot because I have taken a measure theory class that only talks about the Lebesgue measure, while the classes I will be taking my first semester of my PhD will just be abstract measures, and these will be on my qualifying exams. I also am a bit intimidated by my options of what I want to focus on, like Functional or Harmonic Analysis, Differential Geometrty, or PDE'S. Looking forward to start my PhD journey!
Always a good day when Struggling Grad Student posts a video 😃
Looking forward to the series!
Your videos are always eagerly awaited for.
Prof: "Pick which end of the gator pond you want to swim in."
Thanks for another great video Mr. Student.
Excited for your measure theory videos!!
hell yeahhh a measure theory series would b great
Maybe you can try problem 8 if it doesn't seem that bad to you yourself :)
Maybe your instructor wanted to spook you! Often easy questions are propped up to be spooky lol.😂
I'm inclined to believe him when he says a problem is too difficult for us haha But I might just be brave enough to try it!
Here is a solution for problem 5.
Lemma: For any pair of unit vectors (u, v) there exists a matrix B = B^T with operator norm (w.r.t. Euclidean norm) at most 1 such that Bu = v.
Proof (assuming Lemma): Write x = |x| u, where u is a unit vector. It is straightforward to see that we can write {Ax : A = A^T, A psd, ||A||
My man is saving me right now!
Check out Donald L. Cohn's book on Measure Theory. Helped me out a lot when I first took it. For my graduate course on Measure and Probability Theory, Williams' book "Probability with Martingales" was exceptionally good. Lots of examples, and a nice, clean approach to the construction of measures.
Thanks, I will check it out :)
thanks for the video Mr.
Great video. Do more
Hey grad, since the video series is not a lecture series, you can go through the basic/less important stuff quicker or just skip it.
I guess that undergraduates who are interested in just getting a feel won't mind, and masters students will appreciate that as well.
That way you can make fewer videos and also make them more enjoyable for yourself.
Big fan
pls make a video roadmap for study applied and pure mathematics . thank you
If i may ask, what is candidacy exam?
great video
I like stein and sakarchi.
Is there any youTube channel as good as this one but focused on physics?
Yes there is .Parth G , check it out .
Andrew Dotson
Andrew Dotson has been going through a nuclear physics PhD program right now and posting his progress
After graduating grad school what field of career are you planning going into
I love teaching so I am leaning toward university instructor or community college instructor.
Do vectors calculus
Excited bro
Hey, how would i do (software engineering masters), in a pure math masters, thinking of jumping to a different branch of STEM. Also stay in EU or try to apply in USA?
How are you at writing proofs? The proof writing and abstract theory are a part of the challenge. Notorious classes are Abstract Algebra, which looks at structure; furthermore, Real and Complex Analysis, which looks at the theory which supports calculus. Look for a reputable mathematics university, which exists in many areas of the world. Best wishes; Cheerful Calculations.
I studied environmental science before entering into pure math, and I struggled to understand proof writing but manage to overcome it and finish my masters. If you can overcome proof writing, then I think you can do it.
As far as EU or USA, I am pretty ignorant on the subject, but I would do what is most comfortable for you. Moving to the USA may be stressful but many of my close friends in the dept. are international students and they like it here. I hope this helps :)
it's better to expect making 10 videos and then giving up, you'll def feel better for continuing
Below, I think I can get you 5+6+7 = 18 points!! I don't know how to do problem 8 yet. I'm thinking about it still. Seems a little more challenging.
Can you do 4?
@@iyadelkhoury1573 Argh. Still I can't figure this one or 8 out. Did you have any luck?
@@iyadelkhoury1573 Argh. not yet, can you?
Out of curiousity, why do you write your notes on legal pads?
big = good
The dept. supplies us with legal pads so it is cheaper than buying my own paper. Plus, I've come to like the legal pads.