## Some Observations about Roots with Multiplicities.

One of the subjects which I had visited in my past, was to write a C++ program that approximates the roots of polynomials, but that needed to deal with ‘Doubled Roots’ in a special way, just because the search algorithm within that program, which searches for the roots, cannot deal with doubled roots. And what I found was, roots cannot only be doubled, but can have multiplicities higher than (2). After not having pondered that programming project from the past, for some time, I now come back to rethink it, and find that it can be given a lecture of sorts, all to itself.

So, this is a link to a work-sheet, in which I try to explain Roots with Multiplicities, maybe to people with limited knowledge in Calculus:

Link to EPUB File, for viewing on Mobile Devices

And as those two documents mention, the following is a link to the earlier blog posting, from which readers can download the C++ program, as well as to find instructions on how to compile it, on Linux computers:

Hence, the C++ program linked to in the earlier blog posting, needed to make use of the subject, that the two PDF Files describe.

N.B.:

(Updated 5/06/2019, 13h15 … )

## A simplified way to insert TikZ graphics into actual documents.

One of the observations I seem to be making is, that the technology-intensive part of the world is still using such tools as LaTeX, to typeset documents and vector-based graphics, such as Mathematical equations which I’ve included in previous parts of this blog, even though we know that GUI-based applications have existed for some time, that also typeset and do graphics…

I like ‘LaTeX’, even though I never learned the full syntax. LaTeX defines the document in a type of textual syntax. And there is even a system of TeX macros, which defines a sub-language called “TikZ”, that defines entire figures, sketches and plots, using a textual syntax.

Edit: Alternatively, there is a sub-language called “PSTricks”, which offers yet another way to do the same thing

And so a question which strikes me as important is, how to use these languages, even though I don’t know the syntax. And the answer is, ‘I use a GUI.’ That GUI mainly consists of the application ‘LyX’.

(Updated 3/07/2019, 17h55 … )

## How the EPUB2 and MOBI formats can be used for typeset Math.

According to This preceding posting, I was experiencing some frustration over trying to typeset Math, for publication in EPUB2 format. EPUB3 format with MathML support was a viable alternative, though potentially hard on any readers I might have.

Well a situation exists in which either EPUB2 or MOBI can be used to publish typeset Math: Each lossless image can claim the entire width of a column of text, and each image can represent an entire equation. That way, the content of the document can alternate vertically between Text and Typeset Math.

In fact, if an author was to choose to do this, he or she could also use the Linux-based solutions ‘LyX’ , ‘ImageMagick’ , and ‘tex4ebook’ .

(Edited 1/9/2019, 15h35 … )

## Trying to bridge the gap to mobile-friendly reading of typeset equations, using EPUB3?

One of the sad facts about this blog is, that it’s not very mobile-friendly. The actual WordPress Theme that I use is very mobile-friendly, but I have the habit of inserting links into postings, that open typeset Math, in the form of PDF Files. And the real problem with those PDF Files is, the fact that when people try to view them on, say, smart-phones, the Letter-Sized page format forces them to pinch-zoom the document, and then to drag it around on their phone, not getting a good view of the overall document.

And so eventually I’m going to have to look for a better solution. One solution that works, is just to output a garbled PDF-File. But something better is in order.

A solution that works in principle, is to export my LaTeX -typeset Math to EPUB3-format, with MathML. But, the other EPUB and/or MOBI formats just don’t work. But the main downside after all that work for me is, the fact that although there are many ebook-readers for Android, there are only very few that can do everything which EPUB3 is supposed to be able to do, including MathML. Instead, the format is better-suited for distributing prose.

One ebook-reader that does support EPUB3 fully, is called “Infinity Reader“. But if I did publish my Math using EPUB3 format, then I’d be doing the uncomfortable deed, of practically requiring that my readers install this ebook-reader on their smart-phones, for which they’d next need to pay a small in-app purchase, just to get rid of the ads. I’d be betraying all those people who, like me, prefer open-source software. For many years, some version of ‘FBReader’ has remained sufficient for most users.

Thus, if readers get to read This Typeset Math, just because they installed that one ebook-reader, then the experience could end up becoming very disappointing for them. And, I don’t get any kick-back from ImeonSoft, for having encouraged this.

I suppose that this cloud has a silver lining. There does exist a Desktop-based / Laptop-based ebook-reader, which is capable of displaying all these EPUB3 ebooks, and which is as free as one could wish for: The Calibre Ebook Manager. When users install this either under Linux or under Windows, they will also be able to view the sample document I created and linked to above.

(Updated 1/6/2019, 6h00 … )