Presenters:

Affiliated Research Professor Mohammad Khoshnevisan

Physics Department, Northeastern University

United States of America

&

Behzad Mohasel Afshari, Admitted Ph.D. student

School of Advanced Manufacturing & Mechanical Engineering

University of South Australia

Australia

Note: This webinar is free. For registration, please send an e-mail to m.khoshnevisan@ieee.org. The registration link will be sent to all the participants on April 26, 2021. Maximum number of participants =60

Vibrational Mechanics - Practical Applications- Animation 1

Vibrational Mechanics - Practical Applications- Animation 2

Vibrational Mechanics - Practical Applications- Animation 3

Rotating Disk(Click to Play the Movie)

Natural Frequency(Click to Play the Movie)

Hanging Mass(Click to Play the Movie)

Rotating Pendulum(Click to Play the Movie)

Cross Over a Bumper(Click to Play the Movie)

I make a maple worksheet for generating Pythagorean Triples Ternary Tree :

Around 10,000 records in the matrix currently !

You can set your desire size or export the Matrix as text ...

But yet ! I wish to understand from you better techniques If you have some suggestion ?

the mapleprimes Don't load my worksheet for preview so i put a screenshot !

Pythagoras_ternary.mw

Pythagoras_ternary_data.mw

Pythagoras_ternary_maple.mw

            We announce the release of a new book, of title Fourier Transforms for Chemistry, which is in the form of a Maple worksheet.  This book is freely available through Maple Application Centre, either as a Maple worksheet with no output from commands or as a .pdf file with all output and plots.

            This interactive electronic book in the form of a Maple worksheet comprises six chapters containing Maple commands, plus an overview 0 as an introduction.  The chapters have content as follows.

  -   1    continuous Fourier transformation

  -   2    electron diffraction of a gaseous sample

  -   3    xray diffraction of a crystal and a powder

  -   4    microwave spectrum of a gaseous sample

  -   5    infrared and Raman spectra of a liquid sample

  -   6    nuclear magnetic resonance of various samples

            This book will be useful in courses of physical chemistry or devoted to the determination of molecular structure by physical methods.  Some content, duly acknowledged, has been derived and adapted from other authors, with permission.

Hi all,

Look at my pretty plot.  It is defined by

x=sin(m*t);
y=sin(n*t);

where n and m are one digit positive integers.

You can modify my worksheet with different values of n and m.

pillow_curve.mw

pillow_curve.pdf

The name of the curve may be something like Curve of Lesotho.  I saw this first in one of my father's books.

Regards,
Matt

The order in which expressions evaluate in Maple is something that occasionally causes even advanced users to make syntax errors.

I recently saw a single line of Maple code that provided a good example of a command not evaluating in the order the user desired.

The command in question (after calling with(plots):) was

animate(display, [arrow(<cos(t), sin(t)>)], t = 0 .. 2*Pi)

This resulted in the error:

Error, (in plots/arrow) invalid input: plottools:-arrow expects its 3rd argument, pv, to be of type {Vector, list, vector, complexcons, realcons}, but received 0.5000000000e-1*(cos(t)^2+sin(t)^2)^(1/2)
 

This error indicates that the issue in the animation is the arrow command

arrow(<cos(t), sin(t)>)

on its own, the above command would give the same error. However, the animate command takes values of t from 0 to 2*Pi and substitutes them in, so at first glance, you wouldn't expect the same error to occur.

What is happening is that the command 

arrow(<cos(t), sin(t)>)

in the animate expression is evaluating fully, BEFORE the call to animate is happening. This is due to Maple's automatic simplification (since if this expression contained large expressions, or the values were calculated from other function calls, that could be simplified down, you'd want that to happen first to prevent unneeded calculation time at each step).

So the question is how do we stop it evaluating ahead of time since that isn't what we want to happen in this case?

In order to do this we can use uneval quotes (the single quotes on the keyboard - not to be confused with the backticks). 

animate(display, ['arrow'(<cos(t), sin(t)>)], t = 0 .. 2*Pi)

By surrounding the arrow function call in the uneval quotes, the evaluation is delayed by a level, allowing the animate call to happen first so that each value of t is then substituted before the arrow command is called.

Maple_Evaluation_Order_Example.mw

Download pendulumprojection.mw

Most of the time in multivariable calculus the vectors we are decomposing with respect to a given direction are attached to a specific point and this "orthogonal decomposition" should be visualized as a triplet of vectors with initial point at that given point such that the two projections sum to the original vector. Keep in mind our high school physics pendulum example where we need to evaluate both the component of the downward gravitational force along the tangential direction of motion of the pendulum for the equations of motion as well as along the perpendicular direction to the motion to evaluate the tension in the string or wire holding the pendulum mass.

We create an Explorable graphic to visualize this idea and then create an animation which requires a bit of Maple expert user knowledge to accomplish (thanks Maplesoft!). A delayed evaluation of the arrow command from the plot package is required in order to animate that arrow so that it waits for the animation command to feed it actual numerical values for the vector to be plotted.

My worksheet URL:
http://www34.homepage.villanova.edu/robert.jantzen/courses/mat2500/handouts/pendulumprojection.mw

As a continuation of the posts:
https://www.mapleprimes.com/posts/208958-Determination-Of-The-Angles-Of-The-Manipulator
https://www.mapleprimes.com/posts/209255-The-Use-Of-Manipulators-As-Multiaxis
https://www.mapleprimes.com/posts/210003-Manipulator-With-Variable-Length-Of
But this time without Draghilev's method.
Motion along straight lines can replace motion along any spatial path (with any practical precision), which means that solving the inverse problem of the manipulator's kinematics can be reduced to solving the movement along a sequential set of segments. Thus, another general method for solving the manipulator inverse problem is proposed.
An example of a three-link manipulator with 5 degrees of freedom. Its last link, like the first link, geometrically corresponds to the radius of the sphere. We calculate the coordinates of the ends of its links when passing a straight line segment. We do this in a loop for interior points of the segment using the procedure for finding real roots of polynomial systems of equations RootFinding [Isolate]. First, we “remove” two “extra” degrees of freedom by adding two equations to the system. There can be an infinite set of options for additional equations - if only they correspond to the technical capabilities of the device. In this case, two maximally easy conditions were taken: one equation corresponds to the perpendicularity of the last (third) link directly to the segment of the trajectory itself, and the second equation corresponds to the perpendicularity to the vector with coordinates <1,1,1>. As a result, we got four ways to move the manipulator for the same segment. All of these ways are selected as one of the RootFinding [Isolate] solutions (in text  jj=1,2,3,4).
In this text jj=4
without_Draghilev_method.mw       

As you can see, everything is very simple, there is practically no programming and is performed exclusively by Maple procedures.

 

Here is a very nice (but not easy) elementary problem.
The equality
ceil(2/(2^(1/n)-1)) = floor(2*n/ln(2));

is not an identity, it does not hold for each positive integer n.
How to find such a number?

[This is a re-post, because the original vanished when trying a conversion Question-->Post]

The problem appears in the recent book:
Richard P. Stanley - Conversational Problem Solving. AMS, 2020. 

The problem is related to a n-dimensional tic-tac-toe game. The first counterexample (2000) was wrong due to a multiprecision arithmetic error.
The  author of the book writes 
"To my knowledge, only eight values of n are known for which the equation fails,
and it is not known whether there are infinitely many such values",
but using Maple it will be easy to find more.

A brute-force solution is problematic because the smallest counterexample is > 7*10^14.

restart;
a := 2/(2^(1/n)-1): b := 2*n/ln(2):
asympt(b-a, n);

It results:  b - a → 1 (for n →oo);
So, to have a counterexample, b must be close to an integer
m ≈ 2*n/ln(2)  ==> n/m ≈ ln(2)/2

The candidates for n/m will be obviously the convergents of the continued fraction of the irrational number ln(2)/2.
 

convert(ln(2)/2, confrac, 200, 'L'):
Digits:=500:
for n in numer~(L[3..]) do
  if not evalf(ceil(a)=floor(b)) then printf("n=%d\n", n) fi;
od:

n=777451915729368
n=140894092055857794
n=1526223088619171207
n=54545811706258836911039145
n=624965662836733496131286135873807507
n=1667672249427111806462471627630318921648499
n=36465374036664559522628534720215805439659141
n=2424113537313479652351566323080535902276508627
n=123447463532804139472316739803506251988903644272
n=97841697218028095572510076719589816668243339678931971
n=5630139432241886550932967438485653485900841911029964871
n=678285039039320287244063811222441860326049085269592368999
n=312248823968901304612135523777926467950572570270886324722782642817828920779530446911
n=5126378297284476009502432193466392279080801593096986305822277185206388903158084832387
n=1868266384496708840693682923003493054768730136715216748598418855972395912786276854715767
n=726011811269636138610858097839553470902342131901683076550627061487326331082639308139922553824778693815
 

So, we have obtained 16 counterexamples. The question whether there are an infinity of such n's remains open.

With this application, the differential equation of forced systems is studied directly. It comes with embedded components and also with native Maple code. Soon I will develop this same application with MapleSim. Only for engineering students.

Damped_Forced_Movement.mw

Lenin AC

Ambassador of Maple

One of the most interesting help page about the use of the Physics package is Physics,Examples. This page received some additions recently. It is also an excellent example of the File -> Export -> LaTeX capabilities under development.

Below you see the sections and subsections of this page. At the bottom, you have links to the updated PhysicsExample.mw worksheet, together with PhysicsExamples.PDF.

The PDF file has 74 pages and is obtained by going File -> Export -> LaTeX (FEL) on this worksheet to get a .tex version of it using an experimental version of Maple under development. The .tex file that results from FEL (used to get the PDF using TexShop on a Mac) has no manual editing. This illustrates new automatic line-breaking, equation labels, colours, plots, and the new LaTeX translation of sophisticated mathematical physics notation used in the Physics package (command Latex in the Maplesoft Physics Updates, to be renamed as latex in the upcoming Maple release). 

In brief, this LaTeX project aims at writing entire course lessons or scientific papers directly in the Maple worksheet that combines what-you-see-is-what-you-get editing capabilities with the Maple computational engine to produce mathematical results. And from there get a LaTeX version of the work in two clicks, optionally hiding all the input (View -> Show/Hide -> Input).

PhysicsExamples.mw   PhysicsExamples.pdf

PS: MANY THANKS to all of you who provided so-valuable feedback on the new Latex here in Mapleprimes.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

If you are interested in experimenting with simple antenna arrays, this worksheet may prove useful.  I have provided a few examples of arrays that repeat in the x, y and z directions, but it will be very easy to tweak this tool if you are more interested in circular or triangular arrays.

This is one of the example arrays:

Antenna_arrays.mw

antenna_arrays.pdf

In two recent questions

• What might be the best way to convey calculated re...
• Can I redirect Maple output that includes tables ...

dnaviaux  raised concerns about saving informations he could include in a typeset report.

In the first question these informations were mainly printf terminal outputs, while the second question was oriented to informations of type Table (in the sense of DocumentTools:-Layout:-Table).

Acer has given a direct answer to this last question.
On my side I proposed a less direct one based on a LaTeXapproach.

Since several years I use to generate a LaTeX source code for typeset reports directly from Maple (this is not very difficult once you know LaTeX).
These reports often contain data tables Mbut latex(M)is not the good way to code these data tables into LaTeX.
That is why I turned towards the LaTeX/tabularstructure

\begin{tabular}{...}
...
\end{tabular]

Data tables are very simple structures and converting them programmatically from Maple Matrix into  LaTeX tabular is rather simple: the most important thing you must remember is that some characters have a special role in LaTeX  (for instance "\" , "_" and "$") while having a special meaning in Maple too: so beware of conflicts.

The procedure Tabular in the attached file takes as argument a Matrix and returns the LaTeX code of its tabular counterpart.
Note that it could be easily extendend to accept a DataFrameas input.
This is a very simple simple version and fonts (\rm, \bf, \it... or others), styles (\scriptsize, \large, ...) and much more customization features could be accounted for without any difficulty.

For this particular example inspired by dnaviaux first question, it would be probably better to replace the LaTeX code by one of thes two:

\begin{tabular}{|c @{ $\pm$  } c|}
....
Asdfsdg & +5.190e+01 & 2.5950e+00 \\ 

or :

\begin{tabular}{|c|c|}
....
Asdfsdg & \multicolumn{2}{c|}{$+8.680e-01 \pm 0.9190$} \\ 

dnaviaux second question concerned a DocumentTools:-Layout:-Table object.
For what I understood this table has a simple structure (no rowspan neither columnspan used).

But, knowing that tabular can manage merged columns (\multicolumn) and merged rows (\multirow, provided the ad hoc pasckage is used), I wondered if it would be possible to generate the LaTeX/tabular code corresponding to a Table using rowspan and columnspan?

This is done by the procedure Tabular_Table in the attached file.
Here again this is a very simple procedure which doesn't exploit all the informations a Table structure contains (backgroundstyle, fillcolor, align, alignment, separator [Tabular_Table separates all columns and rows] ...).

The "raw" rendering is not bad but could be improved:

• adjust the way \multirowcenters the content of a cell (Xis badly placed in example 3)
• adjust the spaces between line/clineand the text
  • by using the package cellspace 
  • or my modufying \arraystretch with \renewcommand.
• one could also add a legend with \caption
• parameterize Tabular_Table to accept other column separators
• manage the colors (for instance a blue for a math expression and a black for a text)
• ...

Finally Tabularand Tabular_Tablecould have an optional argument file:

{file::{symbol, string}:= terminal} 

set by default to terminalor wich could be the name of a  .texfile 


This work is still under developpement and I would be happy to exchange with you on this topic.
Happy New Year to all of you
 

Tabular_from_Maple.mw

Here is the content of the .tex file (I used Welcome-to-CoCalc.texto create/compile it)

\documentclass{article}

% set font encoding for PDFLaTeX, XeLaTeX, or LuaTeX
\usepackage{ifxetex,ifluatex}
\if\ifxetex T\else\ifluatex T\else F\fi\fi T%
  \usepackage{fontspec}
\else
  \usepackage[T1]{fontenc}
  \usepackage[utf8]{inputenc}
  \usepackage{lmodern}
\fi

\usepackage{hyperref}

% DO NOT FORGET THIS PACKAGE !!!
\usepackage{multirow}

\title{Tabulars from Maple}
\author{mmcdara}


\begin{document}
\maketitle

\begin{itemize}
\item[\bf{Example 1}]

\begin{tabular}{|c|c|c|}
\hline 
\hline 
Quantity & Nominal value & Uncertainty \\ 
\hline 
Asdfsdg & +9.060e+01 & 4.5298e+00 \\ 
\hline 
Bdfg & +1.437e+01 & 7.1870e-01 \\ 
\hline 
C123 & +8.025e+01 & 4.0125e+00 \\ 
\hline 
Ddf sdfg dsfg  & +9.614e+00 & 4.8072e-01 \\ 
\hline 
\hline 
\end{tabular}

\item[]\vspace*{10mm}

\item[\bf{Example 3}]

\begin{tabular}{| c | c | c | c | c |}
\hline 
\multicolumn{1}{|c|}{\multirow{4}{*}{X}} & \multicolumn{2}{c|}{\multirow{1}{*}{A1}} & \multicolumn{1}{c|}{\multirow{2}{*}{A2}} & \multicolumn{1}{c|}{\multirow{1}{*}{A3}} \\ 
\cline{2-2} 
\cline{3-3} 
\cline{5-5} 
 & \multicolumn{1}{c|}{\multirow{2}{*}{B1}} &\multicolumn{1}{c|}{\multirow{1}{*}{B2}} & & \multicolumn{1}{c|}{\multirow{3}{*}{B3}} \\ 
\cline{3-3} 
\cline{4-4} 
 &  & \multicolumn{2}{c|}{${\cos \left( \omega\,t+\phi \right) }^{\mathstrut}_{\atop{}{\mathstrut}}$} &  \\ 
\cline{2-2} 
\cline{3-3} 
\cline{4-4} 
 & \multicolumn{3}{c|}{${{\frac {1}{\Gamma  \left( x \right) }\sqrt {{{\rm e}^{-{\frac {{t}^{2}}{\pi }}}}}}}^{\mathstrut}_{\atop{}{\mathstrut}}$} &  \\ 
\hline 
\end{tabular} 

\end{itemize}

\end{document}

Image of the resulting PDF:

 

For comparison here is the Tableof Example 3 that Mapledisplays in the worksheet

Recently I came back on the general problem of drawing the syntactic graph of a mathematical expression.
Probably some of you have already done this as students for it is a classic when you learn recursive procedures, chained lists or graphs.

I wasn't interested in doing this with Maple, because Maple had already done  a part of the job thanks to the procedure ToInert.
More of this, the package GraphTheory seemed to possess all the required features to obtain quickly this syntactic graph.
Nevertheless it took me a lot of time to fix (almost all) the problems.
The issues are mainly of two orders:

1. ToInert is very verbose: a necessary feature when you want to have a non ambiguous syntax of an expression, but partly useless for simple visualization.
   Here is an example

   ToInert(f(x))
   _Inert_FUNCTION(_Inert_NAME("f"), _Inert_EXPSEQ(_Inert_NAME("x")))

    

2. GraphTheory 
   Once the inert form of the expression is known, it is necessary to put it in a form that can be manipulated by the procedures of the GraphTheory package.
   More precisely one needs to transform this inert form into a set of lists [a, b], where a and b are two neighboring vertices of the syntactic graph and [a, b] the directed arc from a to b.
   As the syntactic graph is a tree, this implies using edges {a, b} instead of arcs [a, b].
   The problem is that some operators are commutative while others are not: for the latter this means that the edges and vertices on the syntactic graph must appear in an order that respects the non-commutativity.
   Here his a toy example where I manually buid the syntactic graphs of a/b and b/a: the two graphs are identical and this comes from the fact that edges in Graph( edges )  must be a set, thus an ordered structure whose order doesn't care about non-commutativity.

   restart:
   with(GraphTheory):
   # The first is aimed to represent the expression a/b
   # while the second is aimed to represent the expression b/a
   Gdiv := Graph({{"/", "a"}, {"/", "b"}}):
   g1 := DrawGraph(Gdiv, style=tree, root="/", title=a/b):
   
   Gdiv := Graph({{"/", "b"}, {"/", "a"}}):
   g2 :=DrawGraph(Gdiv, style=tree, root="/", title=b/a):
   
   plots:-display(<g1 | g2>)
   

   Download ab_ba.mw

    

After several attempts, I decided to discard the GraphTheory package, that is to deprive myself of all the interesting features one needs to manipulate a graph.

The result is given on the attached file (... and the content of the worksheet can't be loaded as usual).

Download Syntactic_Graph.mw

Here is an example

Twelve test cases are given, all the corresponding syntactic graphs are correct, but one of them (test case iexpr=1) seems incorrect because the right child of a parent P is located to the right of the left child of a parent P', even though P' is to the right of P.
This could be corrected by modifying the way the posiitons are computed in procedure Place.

PS : It doesn't seem that Maple has a built-in procedure to construct the syntactic graph of a mathematical expression.
But maybe I'm wrong?

 

In the two examples below (in the second example, the range for the roots is simply expanded), we see bugs in both examples (Maple 2018.2). I wonder if these errors are fixed in Maple 2020?
 

I am glad that  Student:-Calculus1:-Roots  command successfully handles both examples.

Download bugs-in-solve.mw