Comments on current projects are welcome
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Comments on current projects are welcome. Please attach a PDF file if you have formatted text as your comments. Your comments will be always read, and answered when necessary. iWeb 3.0.4The draft version of the book is done.
http://pangu2002.org/Zigang_Pans_Home_Page/Comments/Entries/2019/10/9_The_draft_version_of_the_book_is_done..html
321ac9fc-7284-4697-95dc-31ba84ac1660Wed, 9 Oct 2019 19:59:00 -0400I have completed the draft version of the book, and would like to have you send me feedback on the book, the style, typos, or mistakes. Please, send your comments to me, it will be appreciated. I will try to incorporate your feedback in the final version of the book. Thank you for your patronage over the years. It has been a long journey and mostly enjoyable doing the writing. I am already onto my next book on dynamical systems. Also, I will work out the optimality guided robust adaptive control research. These tasks are next on my to do list. I am still working on probability theory.
http://pangu2002.org/Zigang_Pans_Home_Page/Comments/Entries/2019/8/28_I_am_still_working_on_probability_theory..html
fa501083-8345-4a63-a52a-635e26064d94Wed, 28 Aug 2019 15:23:16 -0400I am still working on Chapter 14 and have corrected a couple of mistakes. The Law of Large Numbers was not proved corrected. The earlier proof only yields the Weak Law of Large Numbers. Now, I have added the Strong Law of Large Numbers at the end of Section 14.4. The proof for the Central Limit Theorem was not entirely correct. The proof is now done correctly and had to involve the logarithm function on the complex plane. The result has to be done properly via a trip to the theory of manifold. I have add Section 12.10 on manifolds. My treatment of manifold is slightly different from the usual exposure of manifold. I think that my treatment is more general. It allows me to resolve the logarithm function and prove the Jacobi identity for vector fields. The proof of the Central Limit Theorem has been corrected accordingly, which I haven’t fully reviewed yet. I am working on the Chapter 14 for a complete review and hopefully finish the book by the end of this year. A new update.
http://pangu2002.org/Zigang_Pans_Home_Page/Comments/Entries/2019/1/19_A_new_update..html
2687bb65-6b7e-4b4a-9942-f6657e0b5437Sat, 19 Jan 2019 14:12:29 -0500While working on Chapter 14, I need some general result on Riemann-Stieltjes integral in general spaces. So, I am adding some basic results (generalized from Bartle (1976) book) into Sections A.3 and 12.9. These added stuff is still under development. But, most importantly, I decided to set the title of the book to “Measure Theoretic Calculus in Abstract Spaces'', which I find is more transparent, relevant, appropriate, and also appealing. So, I think that this is the title of the book then. Chapter 14 is now taking shape
http://pangu2002.org/Zigang_Pans_Home_Page/Comments/Entries/2018/10/20_Chapter_14_is_now_taking_shape.html
93e21c16-25de-4ecf-9246-1ed00bc8cff8Sat, 20 Oct 2018 15:06:52 -0400I have spent much time on Probability Theory, Chapter 14. Now, it is taking shape given my research in the previous chapters. The definition of stochastic process has taken a significant update, now I require a stochastic process to be measurable in the product measure space of the time index measure space and the underlying probability measure space. In Professor Burkholder’s notes, these processes are called progressively measurable. For discrete-time processes, this updated version of definition is equivalent to the old definition. For continuous-time processes, this updated version adds significant constraints on what can be called a stochastic process. My past research had been a bit misguided since I never go into the depth of defining a stochastic process, and just take other researchers’ work at their face value. Now, I thought deep and hard into this problem and decided that this definition is the right way to go forward. I hope that this hiccup is going to be forgiven by all my readers. Currently, I am thinking deep and hard into the definition of Itô integral. Hope that I can make something out of it. Chapters 12 and 13 are now citable.
http://pangu2002.org/Zigang_Pans_Home_Page/Comments/Entries/2017/10/21_Chapters_12_and_13_are_now_citable..html
45ac0dbb-fffe-405e-b78f-317014b45a0eSat, 21 Oct 2017 15:11:31 -0400I have added in Chapter 12 the final version of Change of Variable Theorem for finite-dimensional Euclidean spaces. I decided against including surface measure in Chapter 12, since I don’t have a general theory. I have proofread Chapter 13, and I added a result that the Fourier series is complete in L2((-π,π],IK). I think that I will leave Chapters 12 and 13 as is now, and I may add to these chapters in the future if I think of something interesting. One topic I definitely want to add is Fourier transform. Another would be Laplace transform. But they may not be fit in Chapter 13. But, my current top priority is Chapter 14, Probability Theory. I want to lay the foundation for my teaching and research in stochastic systems.