An Observation, about How My Blog Gets Spidered

If a blog was just to consists of a bunch of Web-pages, then one side-effect of that would be, that the blogger would add pages on a daily basis, but that his latest pages would not be spidered by the major search engines, potentially for a long time.

If we use ‘WordPress.org’ as our blogging engine, then one feature is, the very first time we add a new posting, doing so is announced or broadcast, to a service that lets search engines know, that a posting has been added to the blog. But one way in which this service does not work, is to reannounce, every time an existing posting has been edited by the blogger. This is fair, because bloggers like me may edit postings as many as 10 or even 20 times eventually. Reannouncing these edits would put an unfair burden on a free service.

But as it happens, the way this service works can have side-effects for me. For example, one of my recent postings links to the following URL:

http://dirkmittler.homeip.net/Plot_Complex_Points.pdf

The problem with this URL is, that in the original posting, it ended with ‘.html’, and not with ‘.pdf’ . I edited that posting after I had first created it, to use a ‘…pdf’ URL, instead of an ‘…html’ URL. What tends to happen is that Google will spider my new postings within seconds of their being created, while certain other search-engines will take maybe a few hours to spider the same posting. And this can entirely be a performance issue, with each search engine. But unfortunately for me, this suggests that Google caught a version of this earlier posting, that contains a broken link, just because the ‘…html’ URL no longer exists on my server. And if the posting contains any apparent, broken links, obviously, the search engine penalizes their ranking.

So that may be one reason, why the posting in question, has not received as many clicks, as it should according to how often other postings of mine receive clicks.

Oops.

Dirk

 

Complex Roots

One question which has fascinated me in recent years, was the question of what exactly happens, if we start with a complex number as a base, and if we then raise that base to either a rational exponent, or an irrational exponent.

In the following worksheet, I explored that subject for all to see:

http://dirkmittler.homeip.net/Plot_Complex_Points.pdf

(Note : )

The worksheet above was made using a graphical front-end to the open-source Computer Algebra System named ‘Maxima’. That graphical front-end is named ‘wxMaxima’, and adds as features, to be able to convert textual output from Maxima, into typeset equations, as well as to make certain presentations and plots nicer.

Enjoy,

Dirk

 

The General Solution to a Cubic Equation

According to “Maxima”, or more specifically, according to “wxMaxima”, the three Roots to a Cubic Equation are generally as shown below, assuming that there exists one solution entirely in Real numbers:

http://dirkmittler.homeip.net/cubic.pdf

(Edit 2/7/2016 : ) There are two observations which need to be made about the solution shown above, which are related to the fact that a cubic equation can sometimes have three Real roots, or two, but that it always has at least one.

1) The expression which we’re told to find the cube root of could be equal to zero. And while finding the cube root of zero represents no obstacle, a division by zero does, and a division by zero ensues.

2) The expression we’re asked to find the square root of can become negative. In that case the solution shown above finds no Real numbers. Further, this output from ‘Maxima’ does not elucidate, how to process the fact that radicals are usually both negative and positive. An entire expression gets repeated, in which the radical could be negative. And there is no easy way to know, whether this radical is allowed to be negative in only one occurrence, or in both occurrences…

When using ‘Maxima’, a frequent goal is to eliminate extraneous complex numbers, by applying the sequence [‘rectform’, ‘trigsimp’] to an already-formed solution which is capable of producing Real numbers. But in this example, the sequence does not produce meaningful results. And one main reason is the fact that this sequence has no magic, by which to output information which was not input. So this trick does not produce an inverse-trigonometric function whose angle is naturally divided by three, so that a multiple of (2π/3) Radians could simply be added to it, before a trig function is taken again. That ‘Maxima’ can recognize.

(End of Edit 2/7/2016)

If we need to find three existing real roots, then we must apply the system of Reduction To A Depressed Cubic as shown here:

Step 1

Followed by Trigonometric Method For Three Real Roots as shown here:

Step 2