In certain situations, Maxima can actually solve a sextic equation.

For readers who don’t know, a sextic equation is a polynomial of the 6th degree. As the subject line suggests, recent versions of Maxima can find symbolic solutions to those, if used correctly, and, if the sextic actually has ‘an exact, analytical solution’, which is also referred to sometimes as ‘a symbolic solution’.

Whether these analytical solutions are actually more useful than numeric approximations, remains an unanswered question.

What has happened to me is, that I’ve tried to use the method shown below, to cause Maxima to display the solution, and that due to what amounted to a typo, I had given it a polynomial which was visually similar to the one shown, but which was also different in some small way, so that the only solution which Maxima displayed, was the original polynomial, thus implying that Maxima was not able to solve an altered one. The reason this happened is easy to explain…

Not all polynomials of the 6th degree actually have an analytical solution. If given an example that does not, Maxima will fail to display one. All polynomials of the 4th degree actually have an analytical solution, but it may easily be too complex for consumer-grade Computer Algebra Systems (CAS) to output. But, by the time the user is asking a CAS to solve a cubic, he should be able to expect this form of a solution to be output.

The sextic below is actually the product of two cubics, which also explains why Maxima was able to solve it. The reader will need to enable JavaScript:

  • From my site, And
  • From MathJax.org,

To be able to view the worksheet:

 


 

(Updated 7/04/2020, 13h30… )


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The current state of Augmented Reality

In spite of the subject line of this posting, my state of knowledge on that topic may be somewhat incomplete. I apologize to the reader if this is so.

A decade or more ago, Augmented Reality started to make its way to mobile devices, mainly smart-phones, and it consisted of an application that would record the scene which the forward-facing camera was viewing, and would perform some amount of image recognition, or of decoding of humanly imperceptible optical codes, and the app would then overlay CG content over that camera video, to give an enhanced experience to the user on the phone’s display, such as, to highlight certain geographical features of tourist sites, or, to highlight certain commercial products which the user was also seen as a potential customer of…

I think that one phenomenon which has taken place with AR has been, that that market was not lucrative, so that AR apps which did exactly those things have become less frequent. However, some such apps still exist, such as the Android app named ‘ROAR‘. This app does not require that regular users create accounts, but does require that content designers do so. Yet, content designers for this app have expressed pessimism in the question of, whether to create content of this form will pay off in increased profits, because of the simple fact that a customer would need to have this exact app installed, and be running it, either, when visiting a certain store, or, when pointing the app at a product which has already been purchased, say, in order to obtain instructions on how to set up the product. One might say that there is less of a will to invest as much money as was done decades ago, into content creation.

But while this assessment sounds rather bleak, there now exist some newer forms of AR, that have led to more apps. In one new form of AR, the user points his phone at an inside room, and places virtual furniture into it, in order to preview what the best arrangements of furniture would be, before actually spending the money and committing to buy said furniture.

The main reason I don’t want to link directly to such apps is the fact, that they sometimes belong to one specific furniture company, and I don’t want to play favours.

Additionally, AR apps now exist, which do exactly one thing: To act as rulers, i.e., to give a measuration of the distance between two points in 3D, when the 3D scene has been video-recorded in 2D. This would be one example. I can’t really tell how accurate such a ruler app finally is. I’m only documenting that it exists. It seems to have averaged more than 4 stars in user satisfaction reviews.

Another type of AR which exists now, takes the form of “Google Lens“. This was once integrated into ‘Google Photos’, but has been made a separate app. It acts as a search engine, but in order to use it, instead of typing in a search, the user points his phone-cam at a scene or object.

But then, there is also a form of AR, the only purpose of which seems to be, to allow the user to start with a photo, and to create an animation from it himself, that has some sort of fictitious- or fantasy- aspect, and which is simply supposed to look interesting. The resulting animation can then be posted on social media, to impress friends. In one case, the result is an animation in which parts of the photo end up seemingly to move, while in another case, random objects which the user uploaded to a Web-site, are placed ‘in front of’ an actual scene, just like virtual furniture was, except that no intention ever existed, to place those objects physically.

Therefore, in some forms, AR still exists. I suppose that yet another, big context in which it exists is, gaming.

Dirk

 

An Observation About Chrome OS

There is a list of criticisms I could make about Chrome OS, but which I will simply skip for the moment.

I recently bought a new Chromebook.

I have an important piece of information – and praise – about Chrome OS. Its version of the Chrome Web browser – which is simply referred to as ‘Chrome’ – is capable of doing everything that full, desktop Web browsers can do, including, to install extensions from the Chrome Web store. This differs obviously from what the Chrome browser under Android did, which was only a small fraction of that.

Because of this, it’s unnecessary to install numerous Android apps, that just used to be front-ends of sorts, for services that were already available from Web-sites. Four apps which I did not need to install because I was just able to point Chrome to the relevant site, bookmark each site, and log in, are:

  1. Skype
  2. Netflix
  3. Amazon Prime Video
  4. Cineplex Store

And I’m writing this, even though there exist reports that the Android Netflix app runs fine under Chrome OS.

I’m sure my list will grow.

Dirk

 

About a minor (Home-Screen) nuisance I’ve experienced on Android deviceS.

I have owned several Android-based devices, and some of those were purchased from Samsung, those being:

  • A Galaxy Tab S, First Generation,
  • (An earlier Smart-Phone),
  • A Galaxy S6 Smart-Phone,
  • A Galaxy S9 Smart-Phone.

A feature which all these devices have, is the Touchwiz Home-Screen (program). This is the default of what the devices display, when not displaying a specific app, when not displaying the app drawer, and when not displaying ‘Bixby’ (most recently). An unfortunate behaviour of the devices is, that Touchwiz will sometimes crash. In my experience, when it does, this is no big deal, because it restarts automatically, and after a few minutes, even my Notification-Bar Entries will reappear. If certain apps fail to make their notifications reappear by themselves, then launching those apps from the application groups will make their notifications reappear.

I tend to rate each Android device, according to how rarely its Home-Screen will crash in this way. According to that, my Google Pixel C Tablet fared better because its home-screen has never crashed on me. My S9 Phone fared almost as well, in that Touchwiz seldom crashed. But now I think I’ve identified a situation which will frequently cause Touchwiz to crash on the S9 Phone.

Firstly, as I’m writing this, the firmware on that phone is at its latest version, that being the October 1 patch, of 2019, of Android 9.

I discovered that I can trigger this situation, as I was experimenting with the Super-Slow-Mo camera recording mode, in which the camera can record up to 0.4 seconds of video at 960FPS, at a resolution of 1280×720. When the camera does this, it generates a 20MB video, after that has been compressed via a standard H.264 CODEC into an .MP4 container-file. I have the default set, to record all camera footage to the external Micro SD Card. Having recorded the super-slow-mo video once, triggered this behaviour.

There is a simple way to interpret what has caused this, that does not seem to lay any blame on Samsung: When the camera is recording video that fast, it’s generating data faster than the external SD Card can store it. Therefore, the data takes up RAM, until some later point in time, when the O/S has transferred the data to the SD Card, by writing it out. This moment was reached several seconds later.

Here’s where the news gets a bit worse. I can download This 3rd-party app, that’s designed to test what speed of external SD Card I have. The reason I need to do this is the fact that I never seem to remember exactly what type of SD Card I purchased, for use with one specific device.

According to this app, my external SD Card can be written to sequentially at ~12MBytes/Sec. That makes it a Class 10 card. Yet, 20MB of data are to be stored in 0.4 seconds. In fact, simply running the benchmarking app caused a second Touchwiz crash, which was just as inconsequential as the first, that I was trying to investigate. What this seems to suggest is, that virtually no SD Card that I can buy, can really be fast enough to be written to at the speed with which the camera app can generate its data. The camera app will need to cache its footage in RAM, before that footage has been written to the SD Card.

Further, the footage is certainly being stored in RAM in an uncompressed form of data (384 raw frames), while what’s to be written to the SD Card is finally compressed. (:1)

And yet, either of these two apps will cause the Touchwiz crash. Hmm… I think that for the moment, I’ll just hold my horses, and record a maximum of 0.2 seconds of Super-Slow-Mo. Thankfully, this is a parameter that I can choose, with the little icon in the upper-right-hand corner of the view, before shooting.

(Updated 11/17/2019, 12h10 … )

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