A recent posting of Mario Lemelin showed that Maple's default numerical methods produced wrong results for a certain differential equation.  Further investigation revealed that the problem stemmed from the fact that the fourth and fifth order Runge-Kutta methods used within the rkf45 method both produce the same (exactly correct) result at any step size, causing the adaptive error analysis to go badly wrong.  This leads to the question: when do Runge-Kutta methods produce exact results for arbitrary step sizes and initial conditions?

For a partial answer, see this worksheet: View 4541_runge.mw on MapleNet or Download 4541_runge.mw

It's not often that a funny music video is made about Fractals. Here is something very special:

 Here is an interesting look at the tetrations (infinitely repeating powers) of some fractions.

As a side note, we see that infinitely repeating powers of 1/4 could equal one product of repeating powers of 1/16

. Download 565_multi powers 2.mw

http://www.mapleprimes.com/files/565_multi%20powers%202.mw

How do I locate a particular theorem in Linear Algebra that I need for my research?

I have been to conferences which seriously discussed a unified and universal bank of all known math theorems. Theoretically, all proven math theorems could be connected logically: A implies B. But, in reality, most proven math theorems are scattered throughout the literature.

I have no access to a university with math journals.
I might be able to do inter-library loan at my local community county college here in the United States. But, that may take a long time. I have no paid job. My earned income is only from social security disability.

View 2640_Stratum_Compactness_Maplet.mw on MapleNet or Download 2640_Stratum_Compactness_Maplet.mw
View file details

Convex Optimization and Distance Geometry:

I came across this interesting piece of reading, and thought I'd share it.

It's an online version of the title, "The Mathematics of Gambling", by Dr. Edward Thorp.

Although after browsing the web it was implied that Dr. Thorp was world renowned, I honestly knew nothing of him.

You may also look him up on Wikipedia to find a link to this publication.

Maybe some of you may find ways of applying Maple to some of the content in this book.

Dear all,

I'm a beginner user of Mapple. I'm seeking an analytic solution of:

integral{ 1/(ax^3 + bx + c) dx}

Would you like to help me...

thanks.

regards,
agus

Get this Quaternion Package from Maple's Application Center. Make sure you get the March 2007 version not the March 2005 version.

Overview on Hamilton Quaternions

A Hamilton Quaternion is a hypercomplex number with one real part (the scalar) and three imaginary parts (the vector).

This is an extension of the concept of numbers. We have found that a real number is a one-part number that can be represented on a number line and a complex number is a two-part number that can be represented on a plane. Extending that logic, we have also found that we can produce more numbers by adding more parts.
Quaternion --> a + b*i + c*j + d*k, where the coefficients a, b, c, d are elements of the reals

In 1998 I felt compelled to research a certain number I felt was over looked. The inspiration actually came from a dream. I quickly began writing friends and telling them I was going to discover a new constant. I first called the constant rc for root constant. Later it became the MRB Constant for the Marvin Ray Burns Constant. My first tools were a Casio programmable graphing calculator and a Sony hand-held "computer-organizer" and a makeshift internet connection. I went to the Inverse Symbolic Calculator Site and used a very old form of Maple. Here is the story surrounding my compulsion: Download 565_final1.doc
From the autobiography in the above final1.doc, you see that I experienced something mystical while researching that constant with maple. As you read in final1.doc, I am a tradesman; however, the intoxicating power of numbers left me hung-over for knowledge. Particularly, there was the Irresistible draw on my mind that there could be something special to be discovered about that constant I mentioned in the last paragraph. It was in my search for something special about that constant,from an alternating series, that I enrolled in college as a 40 year old.
However, in my second semester of calculus, I got the sad impression from my books that that many alternating series do not converge and thus you can not rely on them to give you any particular values. These series do not converge. They have nothing to do with convergence. They are valueless (have no defined sum). Having already explored some of those alternating series, I was left with a bitter taste in my mouth. Here is a worksheet about a family of so called "non converging" alternating series’ that we can rely on to converge upon at least one constant value. Download 565_mrbgraphs.mws

For more information of the constant shown in the above worksheet you may study this file.
It has a link to a third part that might be of some interest.

Download 565_QA-1.doc
I make no claim to rigor or expertise; the above is just my findings thus far. Click the above MRB Constant Link to continue reading marvinrayburns.com