Category: Mathematics

Problem rendering equations

Some of the equations in my Linear Algebra page and other longer pages on this website do not render properly. Such pages just shows the LaTex code I used to build the equations. Reloading the page usually eventually causes the site to render the equations properly so I’ll leave the offending pages up for now. (Note that this may take more than one reload to work.) Equations seem to look like they should on short pages. Therefore, I may ultimately need to convert these problematic long pages into a page containing links to shorter pages (like I did on my page on Calculus). I’m currently looking into this and hope to have a better solution soon.

Linear Algebra

A simple explanation of Linear algebra for the interested amateur

I’ve finally published a page on linear algebra. It’s a bit long. It’s a compilation of traditional textbooks as well as several online sources. The major source for this page is Gilbert Strang’s Introduction to Linear Algebra. Personally, I think it’s an excellent resource. I highly recommend it for anyone looking to learn linear algebra on their own. The fourth edition can be bought on Amazon, at the link above, for around $50. What’s great about it is that it’s coupled to a free online video course – specifically, the undergraduate linear algebra course that Strang teaches at MIT. This course, in turn, is part of a whole slew of great free online courses that are part of MIT’s OpenCourseWare series. I strongly urge to reader to check out this site.

Note that the current page is far from an exhaustive treatise on linear algebra. Rather, it summarizes the high points, especially those that I anticipate will be necessary for subsequent pages on physics, particularly those dealing with quantum mechanics. As in other articles on this site, there are topics in this article that are spelled out in detail or proven – discussions that professionals in these respective areas might find trivial. However, as stated in the “about” section and elsewhere, this site is geared toward interested amateurs like myself. Hopefully, these discussions will help this audience to understand these topics better.

I expect that I may add to or modify this page at some point or points in the future.

This linear algebra page that I have written can be reached by clicking on the following link:

Linear Algebra

Update to The Formula, Chapter 62 (Long Version)

Just a short detour before moving on to an introduction to quantum encryption. I’ve updated the long version of Chapter 62 of my novel The Formula. You can navigate to that chapter by clicking on the following link

Chapter 62 Long Version

Also, here are some acknowledgements/references for that chapter.

The general overviews from which I generated most of this article can be found at the following sites:

https://www.khanacademy.org/computing/computer-science/cryptography/modern-crypt/v/the-fundamental-theorem-of-arithmetic-1

https://en.wikipedia.org/wiki/RSA_(cryptosystem)

The discussions on Euler’s Totipotent Theorem were derived from the following sources:

http://artofproblemsolving.com/wiki/index.php?title=Euler%27s_Totient_Theorem

https://www.chegg.com/homework-help/definitions/eulers-theorem-33

http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/fermatlittletheorem.pdf

Information on Euler’s Totipotent Function was gleaned from several sources. The main one was:

http://mathworld.wolfram.com/TotientFunction.html

Proof of the multiplicity rule in modular arithmetic was largely taken from:

https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/modular-multiplication

The proof of phi function multiplicity follows the arguments outlined in:

http://www.oxfordmathcenter.com/drupal7/node/172

The discussion of Euclid’s Algorithm was taken from:

https://en.wikipedia.org/wiki/Euclidean_algorithm

The proof that the Euclidean Algorithm works come from this source (document will be downloaded by clicking on the link)

www.cs.ucf.edu/~dmarino/ucf/cot3100h/lectures/COT3100Euclid01.doc

 

Next 4 topics

The next 4 topics that I would like to discuss on this blog expand upon subjects described in my novel, The Formula. They are:

  1. RSA encryption
  2. Quantum encryption
  3. Bell’s Theorem
  4. Bohmian mechanics

An explanation of RSA encryption is given in the long version of Chapter 62 from The Formula. It can be found elsewhere on this website. My page on RSA encryption will be, for the most part, the discussion in Chapter 62 (Long Version) presented in expository form. In keeping with my attempt on this site to avoid “black boxes,” I have included a detailed but slow and step-by-step derivation of the mathematical formulas used in RSA encryption. I’ve done this so the reader isn’t left scratching his or her head, wondering where those formulas came from. Because mathematical proofs have never been my forte, I’ve relied heavily on information gleaned from several on-line sources.

Likewise, my page on quantum encryption is largely an expository version of the treatment on this subject given in Chapter 79 (Long Version) from The Formula. This chapter can also be found on this website and can be reached by clicking here.

Because my treatments of RSA  and quantum encryption are essentially reformulations of the descriptions found in the above-mentioned chapters, it shouldn’t take too long to produce them. On the other hand, I plan to expand considerably on the discussion of Bell’s Theorem given in Chapter 79 (Long Version). Thus, I expect that my page on Bell’s Theorem will take a little longer to produce. Finally, the last page I wish to create that relates to The Formula has to do with Bohmian mechanics. Development of this page, quite frankly, will take some doing.

So RSA encryption should be coming up next. Slightly before or after my page on this subject is released, I anticipate posting news of a promotion involving The Formula: for 5 days, The Formula will be given away-free! On amazon.com.

So stay tuned.