As a grad student in pde analysis, the most useful inequality to me is the so-called "Cauchy inequality with epsilon". It's the very simple: ab \leq \epsilon*a^2+\frac{1}{4\epsilon}b^2, for a,b real. Very useful for bounding products and converting them into sums to absorb one of the terms with the epsilon into the left hand side. At a considerably more advanced level, the Sobolev and Poincare inequalities are also totally indispensable.
definitely checking out that book on nonlinear control of engineering systems. This was really neat. I've been wanting to dive back into a lot of this stuff since I really started diving into applications of inner product spaces in engineering
Thanks Prof for this detailed video. It still baffles me how these inequalities came about. Sure I could find neat proofs for them in textbooks. But I'm more concerned with how can I get a feel (Intuition) for why these inequalities are true? Something as simple as the analogy you gave for triangle inequality. Right now, I pretty much cram these inequalities or look them up when I need them. I there a better way of remembering them?
Hi great vid btw, at 12:44 could when you took the Logarithim on both sides are the bases the same because to my understanding the natural logarithm on the left hand side has a base e and the logarithm on the right hand side has a Logarithim of base 10 could you clear that up voe me please
Hey prof, recently finished my BSc in maths and stats and am going into industry. What are some interesting areas of math to self study? Not planning on going much deeper education-wise but I do enjoy math as a hobby. Cheers
Personally, I’m partial to Functional Analysis. It’s a great area to study that intersects both and applied math. Conway’s A Course in Functional Analysis is a pretty standard text for it. Nachtergale (sp?) and his coauthor has a book called Applied Analysis that also has a good amount of coverage. Then there is Hardy’s An Introduction to the Theory of Numbers. It’s a classic text that covers some Analytic Number Theory. Measure Theory is also usually the next class after Baby Rudin level intro analysis. Folland’s Real Analysis text is good for that. Is this the sort of things you were looking for? I could figure out other things depending on your personal interests.
@@JoelRosenfeld these sound look good suggestions, thanks. In terms of the pure stuff I’ve done, I did all the calculus modules, 2 modules in real analysis, ODEs, PDEs, linear programming, network theory, and one or two others that escape me. Any other suggestions given my experience? I’m leaning towards number theory as that’s one of the more ‘iconic’ areas of study.
@@lemonflavouredtnt6969 A Hilbert Space Problem Book by Paul Halmos might be up your alley. It presents Hilbert Space Theory through a series of problems for you to solve. For number theory, there are several avenues. Hardy goes the way of Analytic number theory. But if you want a different angle, there is George Andrew’s works on Partition Theory which is much more combinatorial. And Rosen’s Elementary Number Theory is a huge Book that also has applications.
I think you have great potential to make educational math videos. I feel you didn't start low enough the mathematical education spectrum. I had no idea what you were talking about. Try and follow Veritasms method. He gives a VERY simple example of what he's going to talk about then takes steps to the more complex parts of the topic. The method with the circle amc vectors was a good example... But how does that help me in real life? From what i got from the video it is used to make estimates. But what's wrong with just spit balling a number and calling it an estimate?
Hi Joshua, I appreciate your input! The level I’m aiming at really varies between videos. Some are made for those who are really in the weeds of research, and others are targeting beginning undergrads. This one was for someone sorta in the beginning of grad school or upper division undergraduate studies. It’s a good point, though, giving concrete examples. I will keep that in mind as I move forward. Thank you
Hey Joshua, You know the more I thought about what you said, the more I realized you are right. Even if I'm aiming at researchers, I should still put a hook or a lifeboat to give everyone an idea of what's going on. Thanks for taking the time to comment. Best, Joel
TH-cam Math videos are a vast, but for the most part ankle deep lake. If we want math videos of all levels on the platform, then everyone has to be able to accept that there will be content they find incomprehensible. The only critique I might or even could give of the video is that it perhaps under utilizes the medium past the vector inequality, but I have no idea what it would take to make complementary visuals for the other inequalities in the video, so even that might be a useless critique.
@@paulfoss5385 yeah, I tried to think of how I could best express the visuals for the others, but I couldn’t come up with anything compelling. Young’s inequality has a nice graphical representation, so I used that. I try to make videos that are at least interesting for everyone, but that eventually go into enough depth that everyone can take something new away from it.
I want to become an analyst, currenty teaching myself measure theory and topology, and after that i want to learn some complex analysis. Should i go with stein shakarchi, lang or something else?
Nice video. I love inequalities. Jensen's is my favorite (currently).
love the review. I am a brave measure theory survivor and having everything so nicely highlighted just refreshes things in your head
13:34 That is one of the most dangerous stunts I've seen someone perform on TH-cam. Would also love a video discussing concentration inequalities.
As a grad student in pde analysis, the most useful inequality to me is the so-called "Cauchy inequality with epsilon". It's the very simple:
ab \leq \epsilon*a^2+\frac{1}{4\epsilon}b^2, for a,b real.
Very useful for bounding products and converting them into sums to absorb one of the terms with the epsilon into the left hand side. At a considerably more advanced level, the Sobolev and Poincare inequalities are also totally indispensable.
I can definitely see how that would be useful. Not only changing a product into a sum, but also killing off a term in the process!
Incredible videos! I wish there were more content like this on youtube
Thanks! I’m glad you like it! I’ll work hard to make sure there is more content like this!
definitely checking out that book on nonlinear control of engineering systems. This was really neat. I've been wanting to dive back into a lot of this stuff since I really started diving into applications of inner product spaces in engineering
Thanks Prof for this detailed video.
It still baffles me how these inequalities came about. Sure I could find neat proofs for them in textbooks. But I'm more concerned with how can I get a feel (Intuition) for why these inequalities are true? Something as simple as the analogy you gave for triangle inequality.
Right now, I pretty much cram these inequalities or look them up when I need them. I there a better way of remembering them?
Hi great vid btw, at 12:44 could when you took the Logarithim on both sides are the bases the same because to my understanding the natural logarithm on the left hand side has a base e and the logarithm on the right hand side has a Logarithim of base 10 could you clear that up voe me please
Good catch. Should be natural log on both sides.
Hey prof, recently finished my BSc in maths and stats and am going into industry. What are some interesting areas of math to self study? Not planning on going much deeper education-wise but I do enjoy math as a hobby. Cheers
Personally, I’m partial to Functional Analysis. It’s a great area to study that intersects both and applied math. Conway’s A Course in Functional Analysis is a pretty standard text for it. Nachtergale (sp?) and his coauthor has a book called Applied Analysis that also has a good amount of coverage.
Then there is Hardy’s An Introduction to the Theory of Numbers. It’s a classic text that covers some Analytic Number Theory.
Measure Theory is also usually the next class after Baby Rudin level intro analysis. Folland’s Real Analysis text is good for that.
Is this the sort of things you were looking for? I could figure out other things depending on your personal interests.
@@JoelRosenfeld these sound look good suggestions, thanks. In terms of the pure stuff I’ve done, I did all the calculus modules, 2 modules in real analysis, ODEs, PDEs, linear programming, network theory, and one or two others that escape me. Any other suggestions given my experience? I’m leaning towards number theory as that’s one of the more ‘iconic’ areas of study.
@@lemonflavouredtnt6969 A Hilbert Space Problem Book by Paul Halmos might be up your alley. It presents Hilbert Space Theory through a series of problems for you to solve.
For number theory, there are several avenues. Hardy goes the way of Analytic number theory. But if you want a different angle, there is George Andrew’s works on Partition Theory which is much more combinatorial. And Rosen’s Elementary Number Theory is a huge Book that also has applications.
@@JoelRosenfeld thanks prof, sounds good I’ll defo look into those. I’ll give you a progress report in a few months haha.
@@lemonflavouredtnt6969 Sounds great! Keep me apprised!
I think you have great potential to make educational math videos.
I feel you didn't start low enough the mathematical education spectrum. I had no idea what you were talking about.
Try and follow Veritasms method. He gives a VERY simple example of what he's going to talk about then takes steps to the more complex parts of the topic.
The method with the circle amc vectors was a good example... But how does that help me in real life? From what i got from the video it is used to make estimates. But what's wrong with just spit balling a number and calling it an estimate?
Hi Joshua,
I appreciate your input! The level I’m aiming at really varies between videos. Some are made for those who are really in the weeds of research, and others are targeting beginning undergrads.
This one was for someone sorta in the beginning of grad school or upper division undergraduate studies.
It’s a good point, though, giving concrete examples. I will keep that in mind as I move forward.
Thank you
Hey Joshua,
You know the more I thought about what you said, the more I realized you are right. Even if I'm aiming at researchers, I should still put a hook or a lifeboat to give everyone an idea of what's going on.
Thanks for taking the time to comment.
Best,
Joel
TH-cam Math videos are a vast, but for the most part ankle deep lake. If we want math videos of all levels on the platform, then everyone has to be able to accept that there will be content they find incomprehensible. The only critique I might or even could give of the video is that it perhaps under utilizes the medium past the vector inequality, but I have no idea what it would take to make complementary visuals for the other inequalities in the video, so even that might be a useless critique.
@@paulfoss5385 yeah, I tried to think of how I could best express the visuals for the others, but I couldn’t come up with anything compelling. Young’s inequality has a nice graphical representation, so I used that.
I try to make videos that are at least interesting for everyone, but that eventually go into enough depth that everyone can take something new away from it.
I want to become an analyst, currenty teaching myself measure theory and topology, and after that i want to learn some complex analysis. Should i go with stein shakarchi, lang or something else?
Personally, I learned from Conway’s text. Lang is a solid option too, and Ahlfors is also excellent.
Though if you don’t have much experience with Complex Numbers and Calculus, Brown and Churchill is a great undergrad text.
@@Anonymous-cw4yd 💯%
❤❤❤
Can you solve this equation
What equation?
@@JoelRosenfeld this th-cam.com/users/shorts0KJk1DdRVB0?feature=share