Mathemaniac | Indefinite integrals? Cavalieri’s quadrature? Complex analysis? | DIw/oI #6 @mathemaniac | Uploaded 5 years ago | Updated 2 minutes ago
In this last video of this video series, we are going to investigate a bunch of ideas that fall within the umbrella of "Doing integrals without integration", including indefinite integrals, an elementary way to derive Cavalieri's quadrature formula, as well as complex analysis as a tool.
Links:
(1) Fundamental theorem of calculus: en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
(2) Quora answer (Cavalieri’s quadrature):
quora.com/How-do-mathematicians-come-up-with-proofs-seemingly-out-of-nowhere-Can-I-come-up-with-my-own-unique-Proof-of-the-Power-Rule/answer/Dean-Rubine
[which is based on Wildberger, N. J. (2002). "A new proof of Cavalieri's quadrature formula". The American Mathematical Monthly. 109 (9): 843–845. doi:10.2307/3072373. JSTOR 3072373.]
(3) Complex analysis (HLMA - a high school student academic research competition):
youtube.com/watch?v=8QiW6FSGUnE&list=PLKJV8qXE3Zwx9gAWBIDUOfgV0LbWC6wr_&index=4
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
In this last video of this video series, we are going to investigate a bunch of ideas that fall within the umbrella of "Doing integrals without integration", including indefinite integrals, an elementary way to derive Cavalieri's quadrature formula, as well as complex analysis as a tool.
Links:
(1) Fundamental theorem of calculus: en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
(2) Quora answer (Cavalieri’s quadrature):
quora.com/How-do-mathematicians-come-up-with-proofs-seemingly-out-of-nowhere-Can-I-come-up-with-my-own-unique-Proof-of-the-Power-Rule/answer/Dean-Rubine
[which is based on Wildberger, N. J. (2002). "A new proof of Cavalieri's quadrature formula". The American Mathematical Monthly. 109 (9): 843–845. doi:10.2307/3072373. JSTOR 3072373.]
(3) Complex analysis (HLMA - a high school student academic research competition):
youtube.com/watch?v=8QiW6FSGUnE&list=PLKJV8qXE3Zwx9gAWBIDUOfgV0LbWC6wr_&index=4
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
![Alternative to SIR: Modelling coronavirus (COVID-19) with stochastic process [PART I]](https://i.ytimg.com/vi/ZYGcSYKdGHM/hqdefault.jpg "Alternative to SIR: Modelling coronavirus (COVID-19) with stochastic process [PART I]
A stochastic process approach to model the spread of coronavirus (COVID-19) as opposed to the compartmental deterministic SIR model. This model is generally known as branching process, but this video only focuses on the simplest type, called Bienaymé-Galton-Watson (BGW) process, because the math involved will get a LOT more complicated if we relax a few constraints that we impose for this simple process.
This video will involve concepts like distribution, independence, expected value, generating function and so on, but a basic understanding of the concept of probabilities will suffice, because these will all be introduced in this video. It also demonstrates some problem-solving strategy of mathematicians (generating function: encoding [countably] infinite data into one single thing, and this can be decoded uniquely, which is justified with standard tools in real analysis).
By the way, during the rigour part of the video where the video displayed a lot of text, the potential generating function needs to also be equal to its own Taylor series itself, because this is not necessarily guaranteed for a smooth function.
The next videos on the likelihood that coronavirus will be eradicated (extinction probability), limitations / improvement and the historical context of the BGW process should be released in a couple of days, because they are already done.
REFERENCES / SOURCES (which I will explain in much more detail in later videos that I promised, but if you are impatient, you can read these):
(1) Branching Processes: Their Role in Epidemiology [basis / limitations / improvement of the model]
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2872325/pdf/ijerph-07-01186.pdf
(2) Branching Processes Since 1873 [historical context of the Bienaymé-Galton-Watson process]
https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/jlms/s1-41.1.385
(3) The Genealogy of Genealogy Branching Processes before (and after) 1873 [historical context of the Bienaymé-Galton-Watson process]
https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/blms/7.3.225
(4) The Educational Times (March 1873) [Galtons original famous Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://web.archive.org/web/20170123114453/http://ioearc.da.ulcc.ac.uk/9344/1/Educational%20Times%20Vol%2025%20Iss%20143.PDF
(5) The Educational Times (August 1873) [Watsons (brief) answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://web.archive.org/web/20161201212518/http://ioearc.da.ulcc.ac.uk/9309/1/Educational%20Times%20Vol%2026%20Iss%20148.PDF
(6) On the probability of extinction of families [Watsons paper / more detailed albeit partially incorrect answer to the Problem 4001; historical context of the Bienaymé-Galton-Watson process]
https://www.jstor.org/stable/pdf/2841222.pdf
Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:
https://forms.gle/QJ29hocF9uQAyZyH6
If you want to know more interesting Mathematics, stay tuned for the next video!
SUBSCRIBE and see you in the next video!
If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I dont use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!
#mathemaniac #math #coronavirus #COVID_19 #SIRmodel #statistics #probability #epidemic #pandemic #mask
Stay safe everyone! Please do wear a mask to protect ourselves, and stay at home as much as possible. We will get through this together.
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See you next time!")
