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 Myth of Wave / Particle Duality

This posting describes some of the History, which many people may be bypassing, in their appreciation of Quantum Mechanics.

About until the 1920s, ‘light’ was largely thought to consist of waves. But a problem with that was, how to explain, why light can travel through apparently empty space. After all, the light that reaches us from distant stars is not fundamentally different, from light that originates on Planet Earth. And until the 1920s, it was believed that there exists a mysterious “Aether“, which transmitted light through space.

A basic premise of wave-propagation, such as in the case of sound-waves, is that there must first be some sort of medium, to conduct the waves, which in the case of sound may be air. But the need for the existence of a medium, also explains why there is no sound in space.

But during the 1920s, the existence of an aether was disproved. Decisively. And so another explanation was needed, of what constitutes light. And the thought seemed more logical, that particles can easily travel through empty space – hence, photons. Even though this was not actually the first form in which photons were theorized.

But then obviously, this raises questions, about how these particles are supposed to relate to waves, where waves were at first easier to observe.

I think that the way many people today are presented, what Quantum-Mechanics consists of, is just, “Wave / Particle Duality”. But then what many students believe – and what I once believed myself – is, that Quantum Mechanics holds some sort of secret key, as to how Matter and Energy might simultaneously consist of particles and waves. And in reality, QM holds no such decisive, secret answers. The only real secret which QM may hold, is a detail that could be embarrassing to the present way in which QM works.

Continue reading The Myth of Wave / Particle Duality

Self-Educating about Perpendicular Matrices with Complex Elements

One of the key reasons for which my class was taught Linear Algebra, including how to compute Eigenvalues and Eigenvectors of Matrices, was so that we could Diagonalize Symmetrical Matrices, in Real Numbers. What this did was to compute the ‘Perpendicular Matrix’ of a given matrix, in which each column was one of its Eigenvectors, and which was an example of an Orthogonal Matrix.  (It might be the case that what was once referred to as a Perpendicular Matrix, may now be referred to as the Orthogonal Basis of the given matrix,?)

(Edit 07/04/2018 :

In fact, what we were taught, is now referred to as The Eigendecomposition of a matrix. )

Having computed the perpendicular matrix P of M, it was known that the matrix product

PT M P = D,

which gives a Diagonal Matrix ‘D’. But, a key problem my Elementary Linear class was not taught to solve, was what to do if ‘M’ had complex Eigenvalues. In order to be taught that, we would need to have been taught in general, how to combine Linear Algebra with Complex Numbers. After that, the Eigenvectors could have been computed as easily as before, using Gauss-Jordan Elimination.

I have brushed up on this in my old Linear Algebra textbook, where the last chapter writes about Complex Numbers. Key facts which need to be understood about Complex Vector Spaces, is

  • The Inner Product needs to be computed differently from before, in a way that borrows from the fact that complex numbers naturally have conjugates. It is now the sum, of each element of one vector, multiplied by the conjugate, of the corresponding element of the other vector.
  • Orthogonal and Symmetrical Matrices are relatively unimportant with Complex Elements.
  • A special operation is defined for matrices, called the Conjugate Transpose, A* .
  • A Unitary Matrix now replaces the Orthogonal Matrix, such that A-1 = A* .
  • A Hermitian Matrix now replaces the Symmetrical Matrix, such that A = A* , and the elements along the main diagonal are Real. Hermitian Matrices are also easy to recognize by inspection.
  • Not only Hermitian Matrices can be diagonalized. They have a superset, known as Normal Matrices, such that A A* = A* A . Normal Matrices can be diagonalized.

This could all become important in Quantum Mechanics, considering the general issue known to exist, by which the bases that define how particles can interact, somehow need to be multiplied by complex numbers, to describe accurately, how particles do interact.

Continue reading Self-Educating about Perpendicular Matrices with Complex Elements