Research work

The fractal structure’s researching.

Modeling of the fractal sets in the Maple program.

Municipal Budget Educational Establishment “School # 57” of Kirov district of Kazan

    Author:  Ibragimova Evelina

    Scientific advisor:  Alsu Gibadullina - mathematics teacher

    Translator:  Aigul Gibadullina

In Russian

ИбрагимоваЭ_Фракталы.docx

In English

Fractals_researching.doc

     ( Images - in attached files )

Table of contents:

Introduction

I. Studying of principles of fractals construction

II. Applied meaning of fractals

III. Researching of computer programs of fractals construction

Conclusion

Introduction

We don’t usually think about main point of things, which we have to do with every day. Environmental systems are many-sided, ever–changing and compli­cated, but they are formed by a little number of rules. Fractals are apt example of this – they are complicated, but based on simple regulations. Self – similarity is the main attribute of them.  Just one fractal element contains genetically information about all system.  This information have a forming role for all system. But some­times self – similarity is partial.

Hypothesis of the research. Fractals and various elements of the Universe have general principles of structural organization. It is a reason why the theory of frac­tals is instrument for cognition of the world.

Purpose of the research. Studying  of genetic analogy  between fractals and alive and non-living Universe systems with computer-based mathematical mod­eling in the Mapel’s computer space.

Problems of the research. 

1. studying of principles of fractal’s construction;
2. Detection of  general fractal content of physical, biological and artificial sys­tems;
3. Researching of applied meaning of fractals;
4. Searching of computer programs which can generate all of known fractals;
5. Researching of fractals witch was assigned by complex variables;
6. Formation of innovative ideas of using of fractals in different spaces;

The object of research.  Fractal structures, nature and society objects.

The subject of research. Manifestation of fractality in different objects of the Universe.

Methods of the researching.

1. Studying and analysis of literature of research’s problem;
2. Searching of computer programs which can generate fractals and experimenta­tion with them;
3. Comparative analysis of principles of generating of fractals and structural or­ganizations of physical, biological and artificial systems;
4. Generation and formulation of innovation ways to applied significance of fractals.

Applied significance.

Researching of universality of fractals gives general academic way of cognition of nature and society.

I. Studying of principles of fractal construction

We can see fractal constructions everywhere – from crystals and different accu­mulations (clouds, rivers, mountains, stars etc.) to complex ecosystems and bio­logical objects like fern leafs or human brain. Actually, the idea that frac­tal principles are genetic code of our Universe has been discussed for about fifteen years. The first attempt of modeling of the process of the Universe construction was done by A.D. Linde. We also know that young Andrey Saharov had solved “fractal” calculation problem – it was already half a century ago.

Now therefore, fractal picture of the world was intuitively anticipated by human genius and it inevitably manifested in its activity.

Fractals are divided into four groups in the traditional way: geometric (constructive), algebraical (dynamical), stochastical and natural.

The first group of fractals is geometric. It is the most demonstrative type of fractals, because we can instantly observe the self-similarity in it. This type of fractals is constructed in the basis of original figure by her fragmentation and real­izing of different transformations. Geometrical fractals ensue on repeating of this procedure. They are using in computer-generated graphics for generating the pic­tures of leafs, bush, dimensional structures, etc.

The second large group of fractals -  algebraical. This fractals are constructed by iteration of nonlinear displays, which set by simple formulas. There are two types of algebraical fractals – linear and nonlinear. The first of them are determined by first order equates (linear equates), and the second by nonlinear equates, their na­ture significantly brighter, richer and more diverse than first order equates.

The third known group of fractals – stochastical. It is generated by method of random modification of options in iterative process. Therefore, we get an objects which is similar to nature fractals – asymmetrical trees, rugged coasts, mountain scenery etc. Such fractals are useful in modeling of land topography, sea–surface and electrolysis process etc.

The fourth group of fractals is nature, they are dominate in our life. The main difference of such fractals is that they can’t demonstrate infinite self-similarity. There is “physical fractals” term in the classification concept for nature fractals, this term notes their naturalness. These fractals are created with two simple opera­tions: copy and scaling. We can indefinitely list examples of nature fractals: hu­man’s circulatory system, crowns and leafs of trees, lungs, etc.  It is impossible to show all diversity of nature fractals.

II. Applied meaning of fractals

Fractals are having incredibly widespread application nowadays.

In the medicine. Human’s organism is consists from fractal structures: circulatory system, bronchus, muscle, neuron system, etc. So it’s naturally that fractal algorithms are useful in the medicine. For example, assessment of rhythm of fractal dimension while electric diagrams analyzing allows to make more infor­mative and accurate view on the beginning of specific illnesses. Also fractals are using for high–quality processing of  X–ray images (in the experimental way). There are designing of new methods in the gastroenterology which allows to ex­plore gastrointestinal tract organs qualitative and painlessly. Actually, there are discoveries of application of fractal methods for diagnosis and treatment of cancer.

In the science. There are no scientific and technical areas without fractal calcu­lations nowadays. It happens due to the fact that majority of nature objects have fractal structures and dimension: coasts of the continents; natural resources alloca­tion; magnetic field anomaly; dissemination of surges and vibrations in an elastic environments; porous, solid and fungal bodies; crystals; turbulence; dynamic of complicated systems in general, etc. Fractals are useful in geology, geophysics, in the oil sciences… It’s impossible to list all the spaces of adaptability.

Modeling of chaotic processes, particularly, in description of population models.

In telecommunications. It’s naturally that fractals are popular in this area too. Natan Coen is person, who had started to use fractal antennas. Fractal antenna has very compact form which provides high productivity. Due to this, such antennas are used in marine and air transport, in personal devises. The theory of fractal an­tennas has become an independent, well-developed apparatus of synthesis and analysis of electric small antenna (ESA) nowadays. There are developments of possibility of fractal compression of the traffic which is transmitted through the web. The goal of this is more effective transfer of information.

In the visual effects. The theory of fractals has penetrated area of formation of different kinds of visualizations and creation of special effects in the computer graphics soon. This theory are very useful in modeling of nature landscapes in computer games. The film industry also has not been without fractal geometry. All the special effects are based in fractal structure: mountain landscape, lava, flame, fog, large flows and the same. In the modern level of the cinema creation of the special effects is impossible without modeling of fractals.

In the economics. The Veirshtrass’s function is famous example of stochastic fractals. Analysis of graph of the function in interactive mathematic environment Maple allows to make sure in fractal structure of function by way of entry of dif­ferent ranges of graphic visualization. In any indefinitely small area of the part graph of the function absolutely looks like area of this part in the all . The property of function is used in analysis of stock markets.

In the architect. Notably, fractal structures have become useful in the architect more earlier than B. Mandelbrode had discovered them. S.B. Pomorov, Doctor of Architecture, Professor, member of Russian Architect Union, talks about applica­tion of fractal theory in the architect in his article. Let’s see on the part of this arti­cle:

“Fractal structures were found in configuration of African tribal villages, in an­cient Vavilon’s ziggurats, in iconic buildings of ancient India and China, in gothic temples of ancient Russia .

We can see the high fractal level in Malevich’s Architectons. But they were cre­ated long before emergence of the notion of fractals in the architect. People started to use fractal algorithms on the architect morphogenesis consciously after Mandel­brot’s publications. It was made possible to use fractal geometry for analyzing of architectural forms.

Fractals had become available to the majority of specialists due to the comput­erization.  They had been incredibly attractive for architectors, designers and town planners in aesthetic, philosophical and psychological way. Fractal theory was per­ceived on emotional, sensual level in the first phase. The constant repression lead­ing to loss of sensuality.

Application of fractal structures is effective on the microenvironment designing level: interior, household items and their elements. Fractal structures introduction allows creating a new surroundings for people with fractal properties on all levels. It corresponds to nesting spaces.

Fractal formations are not a panacea or a new era in the architect history. But it’s a new way to design architect forms which enriches the architectural theory and practice language. The understanding of na­ture fractal impacts on architectural view of urban environment. An attempt to de­velop the method of architectural designing which will base in an in-depth fractal forms is especially interesting. Will this method base only on mathematics? Will it be different methods and features symbiosis? The practice experiments and re­searches will show us. It’s safe to say that modern fractal approach can be useful not only for analysis, but also for harmonic order and nature’s chaos, architect which may be semantic dominant in nature and historic context.”

Computer systems. Fractal data compression is the most useful fractal applica­tion in the computer science. This kind of compression is based on the fact that it’s easy to describe the real world by fractal geometry. Nevertheless, pictures are compressed better than by other methods (like jpeg or gif). Another one advantage is that picture isn’t pixelateing while compressing. Often picture looks better after increase in fractal compressing.

Basic concept for fractal computer graphics is “Fractal triangle”. Also there are “Fractal figure”, “Fractal object”, “Fractal line”, “Fractal composition”, “Parent object” and “Heir object”. However, it should be noted that fractal computer graphics has recently received as a kind of computer graphics of 21th century.

 The opportunities of fractal computer graphics cannot be overemphasized. It allows creating abstract compositions where we can realize a lot of moves: hori­zontal and vertical, diagonal directions, symmetry and asymmetry etc. Only a few programmers from all over the world know about fractal graphics today.  To what can we compare fractal picture? For example, with complex structure of crystal or with snowflake, the elements of which line up in the one complex composition. This property of fractal object can be useful in ornament creating or designing of decorative composition. Algorithms of synthesis of fractal rates which allows to reproduce copy of any picture too close to the original are developed today.

III. Researching of computer programs of fractal construction

Strict algorithms of fractals are really good for programming. There are a lot of computer programs which introduce fractals nowadays. Computer mathematic systems are stand out from over programs, especially, Maple. Computer mathe­matics is mathematic modeling tool. So programming represents genetic structure of fractal in these systems and we can see precise submission of fractal structure in the picture while we enter a number of iterations . This is the reason why mathematic fractals should be studied with computer mathematics.  The last dis­covery in fractal geometry has been made possible by powerful, modern com­puters. Fractal property researching is almost completely based on computer cal­culations. It allows making computer experiments which reproduce processes and phenomenon which we can’t experiment in the real world with.

Our school has been worked with computer mathematics Maple package more than 10 years. So we have unique opportunity to experiment with mathematic fractals, thanks to that we can understand how initial values impact on outcome   (it is stochastic fractal). For example, we have understood the meaning of the fact that color is the fourth dimension: color changing leads to changing of physical char­acteristics. That is what astrophysics mean talking about “multicolored” of the Universe. While fractal constructing in interactive mathematic environment we re­ceived graphic models which was like A. D. Linde’s model of the Universe. Perhaps, it demonstrates that Universe has fractal structure.

Conclusion

Scientists and philosophers argue, can we talk about universality of fractals in recent years. There are two groups of two opposite positions. We agree with the fact that fractals are universal. Due to the fact that movement is inherent property of material also we always have fractals wherever we have movement.  

We are convinced that fractal is genetic property of the Universe, but it is not mean that all the Universe elements to the one fractal organization. In deployment process fractal structure is undergoing a lot of fluctuations (deviations) and a lot of points of bifurcation (branching) lead grate number of fractal development varie­ties.  

Therefore, we think that fractals are general academic method of real world re­searching. Fractals give the methodology of nature and community researching.

In transitional, chaotic period of society development social life become harder. Different social systems clash. Ancient values are exchanged for new values literally in all spaces. So it’s vitally important for science to develop behavior strategies which allow to avoid tragic mistakes. We think that fractals play important role in developing of such technologies. And – synergy is theory of evolving systems self- organization. But evolution happens on fractal principles, as we know now.

P.S.  Images - in attached files

Since we are getting many questions on how to create Math apps to add to the Maple Cloud. I wanted to go over the different GUI aspects of how you go about creating a Math App in Maple. The following Document also includes some code examples that are used in the the Math App but doesn't go into them in detail. For more details on the type of coding you do in a Math App see the DocumentTools package help page.

Some of the graphical features of the Math app don't display on Maple Primes so I'd recommend downloading this worksheet from here: HowToMathApp.mw to follow along.

How to make a Math App (An example of using the Document Tools).   This Document will provide a beginners guide on one way to make a Math app in Maple. It will contain some coding examples as well as where to find different options in the user interface. Step 1 Insert a Table     •  When making a Math App in Maple I often start with a table. You can enter a table by going to Insert > Table...      •  I often make the table 1 x 2 to start with as this gives an area for input and an area for the output (such as plots).   Add a plot component to one of the cells of the table     •  From the Components  Palette you can add a Plot Component . Add it to the table by clicking and dragging it over.     Add another table inside the other cell     •  In the other cell of the table I'll add another table to organize my use of buttons, sliders, and other components. Add some components to the new table     •  From the Components Palette I'll add a slider, or dial, or something else for interaction.   •  You may also want a Math region for an area to enter functions and a button to tell Maple to do something with it.   Arrange the Components to look nice     •  You can change how the components are placed either by resizing the tables or changing the text orientation of the contents of the cells.   Write some code for the interaction of the buttons.     •  Using the DocumentTools  package there are lots of ways you can use the components. I often will start writing my code using a code edit region  as it provides better visualization for syntax. On MaplePrimes these display as collapsed so I will also include code blocks for the code.   Let's write something that takes the value of the slider and applies it to the dial     •  Note that the names of the components will change in each section as they are copies of the previous section.   with(DocumentTools): with(DocumentTools): sv:=GetProperty('Slider2',value); SetProperty('Dial2',value,sv); •  This code will only execute when run using the  button. Change the value of the slider below then run the code above to see what happens.   Move the code 'inside' the slider     •  Instead of putting the code inside the code edit region where it needs to be executed, we'll next add the code to the value changed code of the slider.   •  Right click the Slider then select "Edit Value Changed Code".     •  This will open the code editor for the Slider     •  Enter your code (ensuring you're using the correct name for the slider and dial).   •  Notice that you don't need to use the with(DocumentTools): command as "use DocumentTools in ... end use;" is already filled in for you.   •  Save the code in the Slider and hit the  button inside it once. •  Now move the slider.   •  On future uses of the App you won't need to hit  as the code will be run on startup. Add some more details to your App     •  Let's make this app do something a bit more interesting than change the contents of a dial when a slider moves.   •  The plan in the next few steps is to make this app allow a user to explore parameters changing in a sinusoidal expression.   •  I'm going to add a second Math Component, put the expression into both then uncheck the box in the context panel that says "Editable".   •  To make the Math containers fit nicely I'll check the Auto-fit container box and set the Minimum Width Pixels to 200.   Add code to change the value of phi in the second Math Container when the Slider changes     Note: Maple uses Radians for trigonometric functions so we should convert the value of phi to Radians. use DocumentTools in   use DocumentTools in phi_s:=GetProperty(Slider5,value); expr:= GetProperty(MathContainer6,expression); new_expr:=algsubs(phi=phi_s*Pi/180,expr); SetProperty(MathContainer7,expression,new_expr); end use: Make the Dial go from 0 to 360°     •  Click the Dial and look at the options in the context panel on the right.   •  Update the values in the Dial so that the highest position is 360 and the spacing makes sense for the app.   Have the Dial update the theta value of the expression     •  Add the following code to the Dial   use DocumentTools in use DocumentTools in theta_d:=GetProperty(Dial7,value); phi_s:=GetProperty(Slider7,value); #This is added so that phi also has the value updated expr:= GetProperty(MathContainer10,expression); new_expr0:=algsubs(theta=theta_d*Pi/180,expr); new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0); #This is added so that phi also has the value updated SetProperty(MathContainer11,expression,new_expr); end use:   •  Update the value in the slider to include the value from the dial   use DocumentTools in   use DocumentTools in theta_d:=GetProperty(Dial7,value); #This is added so that theta also has the value updated phi_s:=GetProperty(Slider7,value); expr:= GetProperty(MathContainer10,expression); new_expr0:=algsubs(theta=theta_d*Pi/180,expr); #This is added so that theta also has the value updated new_expr:=algsubs(phi=phi_s*Pi/180,new_expr0); SetProperty(MathContainer11,expression,new_expr); end use:   Notice that the code in the Dial and Slider are the same     •  Since the code in the Dial and Slider are the same it makes sense to put the code into a procedure that can be called from multiple places.   Note: The changes in the code such as local and the single quotes are not needed but make the code easier to read and less likely to run into errors if edited in the future (for example if you create a variable called dial8 it won't interfere now that the names are in quotes).     UpdateMath:=proc()  UpdateMath:=proc() local theta_d, phi_s, expr, new_expr, new_expr0; use DocumentTools in theta_d:=GetProperty('Dial8','value'); #Get value of theta from Dial phi_s:=GetProperty('Slider8','value'); #Get value of phi from slider expr:= GetProperty('MathContainer12','expression'); new_expr0:=algsubs('theta'=theta_d*Pi/180,expr); # Put value of theta in expression new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0); # Put value of phi in expression SetProperty('MathContainer13','expression',new_expr); # Update expression end use: end proc:   •  Now change the code in the components to call the function using .   •  Since the code above is only defined there it will need to be run once (but only once) before moving the components. Instead of leaving it here you can add it to the Startup code by clicking  or going to Edit > Startup code.  This code will run every time you open the Math App ensuring that it works right away.   •  The startup code isn't defined in this document to allow progression of these steps.   Make the button initialize the app     •  Since the startup code isn't defined in this document we are going to move this function into the button.   UpdateMath:=proc()   UpdateMath:=proc() local theta_d, phi_s, expr, new_expr, new_expr0; use DocumentTools in theta_d:=GetProperty('Dial9','value'); #Get value of theta from Dial phi_s:=GetProperty('Slider9','value'); #Get value of phi from slider expr:= GetProperty('MathContainer14','expression'); new_expr0:=algsubs('theta'=theta_d*Pi/180,expr); # Put value of theta in expression new_expr:=algsubs('phi'=phi_s*Pi/180,new_expr0); # Put value of phi in expression SetProperty('MathContainer15','expression',new_expr); # Update expression end use: end proc: •  First click the button to rename it, you'll see the  option in the context panel on the right. Then add the code above to the button in the same way as the Slider an Dial (Right click and select Edit Click Code).   Now it is easy to add new components     •  Now if we want to add new components we just have to change the one procedure.  Let's add a Volume Gauge to change the value of A.   •  Click in the cell containing the Dial, the context panel will show the option to Insert a row below the Dial. •  Now drag a Volume Gauge into the new cell.   •  Click in the cell and choose the alignment (from the context panel) that looks best to you. In this case I chose center:     Update the procedure code for the Gauge     •  Add two lines for the volume gauge to get the value and sub it into the expression UpdateMath:=proc() UpdateMath:=proc() local theta_d, phi_s, expr, new_expr, new_expr0; use DocumentTools in theta_d:=GetProperty('Dial11','value'); #Get value of theta from the Dial phi_s:=GetProperty('Slider11','value'); #Get value of phi from the Slider A_g:=GetProperty('VolumeGauge1','value'); #Get value of A from the Guage expr:= GetProperty('MathContainer18','expression'); new_expr0:=algsubs('theta'=theta_d*Pi/180,expr); # Put value of theta in expression new_expr1:=algsubs('phi'=phi_s*Pi/180,new_expr0); # Put value of phi in expression new_expr:=algsubs('A'=A_g,new_expr1); # Put value of A in expression SetProperty('MathContainer19','expression',new_expr); # Update expression end use: end proc: •  Now add UpdateMath();   to the Gauge.   Plot the changing expression     •  Make a procedure to get the value in the second Math Container and plot it   PlotMath:=proc() PlotMath:=proc() local expr, p; use DocumentTools in expr:=GetProperty('MathContainer21','expression'); p:=plot(expr,'t'=-Pi/2..Pi/2,'view'=[-Pi/2..Pi/2,-100..100]): SetProperty('Plot14','value',p) end use: end proc: •  Put this procedure in the Initialize button and the call to it in the components.   Tidy up the app     •  Now that we have an interactive app let's tidy it up a bit.   •  The first thing I'd recommend in your own app is moving the code from the initialize button to startup code. In this document we choose to use the button instead to preserve earlier versions.   •  You can also remove the borders around the components by clicking in the table and selecting "Interior Borders" > "None" and "Exterior Borders" > "None" from the context panel. Now you have a Math App     •  You can upload your Math App to the Maple Cloud to share with others by going to "File" > "Save to Cloud".   •  I'd recommend also including a description of what your app does. You can do this nicely using another table and Text mode.

HowToMathApp.mw

I’m extremely pleased to introduce the newest update to the Maple Companion. In this time of wide-spread remote learning, tools like the Maple Companion are more important than ever, and I’m happy that our efforts are helping students (and some of their parents!) with at least one small aspect of their life.  Since we’ve added a lot of useful features since I last posted about this free mobile app, I wanted to share the ones I’m most excited about. 

(If you haven’t heard about the Maple Companion app, you can read more about it here.) 

If you use the app primarily to move math into Maple, you’ll be happy to hear that the automatic camera focus has gotten much better over the last couple of updates, and with this latest update, you can now turn on the flash if you need it. For me, these changes have virtually eliminated the need to fiddle with the camera to bring the math in focus, which sometimes happened in earlier versions.

If you use the app to get answers on your phone, that’s gotten much better, too. You can now see plots instantly as you enter your expression in the editor, and watch how the plot changes as you change the expression. You can also get results to many numerical problems results immediately, without having to switch to the results screen. This “calculator mode” is available even if you aren’t connected to the internet.  Okay, so there aren’t a lot of students doing their homework on the bus right now, but someday!

Speaking of plots, you can also now view plots full-screen, so you can see more of plot at once without zooming and panning, squinting, or buying a bigger phone.

Finally, if English is not you or your students’ first language, note that the app was recently made available in Spanish, French, German, Russian, Danish, Japanese, and Simplified Chinese. 

As always, we’d love you hear your feedback and suggestions. Please leave a comment, or use the feedback forms in the app or our web site.

Visit Maple Companion to learn more, find links to the app stores so you can download the app, and access the feedback form. If you already have it installed, you can get the new release simply by updating the app on your phone.

One of the forums asked a question: what is the maximum area of a triangle inscribed in a given ellipse x^2/16 + y^2/3 - 1 = 0? It turned out to be 9, but there are infinitely many such triangles. There was a desire to show them in one of the possible ways. This is a complete (as far as possible) set of such triangles.
(This is not an example of Maple programming; it is just an implementation of a Maple-based algorithm and the work of the Optimization package).
MAX_S_TRIAN_ANINATION.mw

This app shows the modeling and simulation of DNA carried out entirely in Maple. The mathematical model is inserted through the combination of trigonometric functions. It shows the graphs of the curvature vs time for its interpretation. Made for engineering and health science students.

MV_AC_R3_UNI_2020.mw

Lenin Araujo Castillo

Ambassador of Maple

With the new features added to the Student[LinearAlgebra] package I wanted to go over some of the basics on how someone can do Linear Algebra in Maple without require them to do any programming.  I was recently asked about this and thought that the information may be useful to others.
 

This post will be focussed towards new Maple users. I hope that this will be helpful to students using Maple for the first time and professors who want their students to use Maple without needing to spend time learning the language.
 

In addition to the following post you can find a detailed video on using Maple to do Linear Algebra without programming here. You can also find some of the tools that are new to Maple 2020 for Linear Algebra here.

The biggest tools you'll be using are the Matrix palette on the left of Maple, and the Context Panel on the right of Maple.

First you should load the Student[LinearAlgebra] package by entering:

with(Student[LinearAlgebra]);

Once you have a Matrix you can use the context panel on the right to apply a variety of operations to it:

The Student Linear Algebra Menu will give you many linear algebra commands.

You can also access Maple's Tutors from the Tools Menu

Tools > Tutors > Linear Algebra

?Student[LinearAlegbra]

Maple_for_Beginners.mw

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

Over the past weeks, we have spoken with many of our academic customers throughout the world, many of whom have decided to continue their academic years online. As you can imagine, this is a considerable challenge for instructors and students alike. Academia has quickly had to pivot to virtual classrooms, online testing and other collaborative technologies, while at the same time dealing with the stress and uncertainty that has resulted from this crisis.

We have been working with our customers to help them through this time in a variety of ways, but we know that there are still classes and students out there who are having trouble getting all the resources they need to complete their school year. So starting today, Maple Student Edition is being made free for every student, anywhere in the world, until the end of June. It is our hope that this action will remove a barrier for instructors to complete their Maple-led math instruction, and will help make things a bit more simple for everyone.

If you are a student, you can get your free copy of Maple here.

In addition, many of you have asked us about the best way to work on your engineering projects from home and/or teaching and learning remotely during this global crisis. We have put together resources for both that you can use as a starting point, and I invite you to contact us if you have any questions, or are dealing with challenges of your own. We are here to support you, and will be very flexible as we work together through these uncertain times.

I wish you all the best,

Laurent
President & CEO

Download Vectors_and_Spherical_coordinates_in_tensor_notation.mw

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

A way of cutting holes on an implicit plot. This is from the field of numerical parameterization of surfaces. On the example of the surface  x3 = 0.01*exp (x1) / (0.01 + x1^4 + x2^4 + x3^4)  consider the approach to producing holes. The surface is locally parameterized in some suitable way and the place for the hole and its size are selected. In the first example, the parametrization is performed on the basis of the section of the initial surface by perpendicular planes. In the second example, "round"  parametrization. It is made on the basis of the cylinder and the planes passing through its axis. Holes can be of any size and any shape. In the figures, the cut out surface sections are colored green and are located above their own holes at an equidistant to the original surface.
HOLE_1.mwHOLE_2.mw

In maple plot, very many symbols like, diamond, star, solidcircle are available. Many of them may have been used also for teaching purposes.

Recently, someone encountered the need to draw graphs with arrowheads and many solutions may be available as well. But it requires a thorough understanding of maple's features which are infinitely many. My feeling was that an arrow symbol also could be added in the symbol feature so that the option can be used as a plot point in the graph at the graph end points very easily. It can be just like adding a solidbox symbol at any point on the curve.

Hope my suggestions are in order.

Thanks.

Ramakrishnan V

The following puzzle prompted me to write this post: "A figure is drawn on checkered paper that needs to be cut into 2 equal parts (the cuts must pass along the sides of the squares.)" (parts are called equal if, after cutting, they can be superimposed on one another, that is, if one of them can be moved, rotated and (if need to) flip so that they completely coincide) (see the first picture below). 
I could not solve it manually and wrote a procedure called  CutTwoParts  that does this automatically (of course, this procedure applies to other similar puzzles). This procedure uses my procedure  AreIsometric  published earlier  https://www.mapleprimes.com/posts/200157-Testing-Of-Two-Plane-Sets-For-Isometry  (for convenience, I have included its text here). In the procedure  CutTwoParts  the figure is specified by the coordinates of the centers of the squares of which it consists).

I advise everyone to first try to solve this puzzle manually in order to feel its non-triviality, and only then load the worksheet with the procedure for automatic solution.


For some reason, the worksheet did not load and I was only able to insert the link.

Cuttings.mw



 

With this application our students of science and engineering in the areas of physics will check the first condition of balance using Maple technology. Only with entering mass and angles we obtain graphs and data for a better interpretation.

First_equilibrium_condition.zip

Lenin AC

Ambassador of Maple

When discussing Maple programming, we often refer to for-loops, while-loops, until-loops, and do-loops (the latter being an infinite loop). But under the hood, Maple has only two kinds of loop, albeit very flexible and powerful ones that can combine the capabilities of any or all of the above, making it possible to write very concise code in a natural way.

Before looking at some actual examples, here is the formal definition of the loops' syntax, expressed in Wirth Syntax Notation, where "|" denotes alternatives, "[...]" denotes an optional part, "(...)" denotes grouping, and Maple keywords are in boldface:

[ for  ] [ from  ] [ by  ] [ to  ]
    [ while  ]
do
    
( end do | until  )

[ for  [ , variable ] ] in 
    [ while  ]
do
    
( end do | until  )

In the first form, every part of the loop syntax is optional, except the do keyword before the body of the loop, and either end do or an until clause after the body. (For those who prefer it, end do can also be written as od.) In the second form, only the in clause is required.

The simplest loop is just:

do
    
end do

This will repeat the forever, unless a break or return statement is executed, or an error occurs.

One or two loop termination conditions can be added:

• A while clause can be written before the do, specifying a condition that is tested before each iteration begins. If the condition evaluates to false, the loop ends.
• An until clause can be written instead of the end do, specifying a condition that is tested after each iteration finishes. If the condition evaluates to true, the loop ends.

A so-called for-loop is just a loop to which iteration clauses have been added. These can take one of two forms:

• Any combination of for (with a single variable), from, by, and to clauses. The last three can appear in any order.
• A for clause with one or two variables, followed by an in clause.

The following for-loop executes 10 times:

for  from 1 to 10 do
    
end do

However, if the doesn't depend on the value of , both the for and from clauses can be omitted:

to 10 do
    
end do

In this case, Maple supplies an implicit for clause (with an inaccessible internal variable), as well as an implicit "from 1" clause. In fact, all of the clauses are optional, and the infinite loop shown earlier is understood by Maple in exactly the same way as:

for  from 1 by 1 to infinity while true do
    
until false

When looping over the contents of a container, such as a one-dimensional array A, there are several possible approaches. The one closest to how it would be done in most other programming languages is (this example and those that follow can be copied and pasted into a Maple session):

 := Array([,"foo",42]);
for  from lowerbound() to upperbound() do
    print([],[])
end do;

If only the entries in the container are of interest, it is not necessary to loop over the indices. Instead, one can write:

 := Array([,"foo",42]);
for  in  do
    print()
end do;

If both the indices and values are needed, one can write:

 := Array([,"foo",42]);
for ,  in  do
    print([],)
end do;

For a numerically indexed container such as an Array, this is equivalent to the for-from-to example. However, this method also works with arbitrarily indexed containers such as a Matrix or table:

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    print([],)
end do;

 := table({1="one","hello"="world",=42});
for ,  in eval() do
    print([],)
end do;

(The second example requires the call to eval due to last-name evaluation of tables in Maple, a topic for another post.)

As with a simple do-loop, a while and/or until clause can be added. For example, the following finds the first negative entry, if any, in a Matrix (traversing the Matrix in storage order):

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    # nothing to do here
until  < 0;
if  < 0 then
    print([],)
end if;

Notice that the test, < 0, is written twice, since it is possible that the Matrix has no negative entry. Another way to write the same loop but only perform the test once is as follows:

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    if  < 0 then
	print([],);
	break
    end if;
end do;

Here, we perform the test within the loop, perform the desired processing on the found value (just printing in this case), and use a break statement to terminate the loop.

Sometimes, it is useful to abort the current iteration of the loop and move on to the next one. The next statement does exactly that. The following loop prints all the indices but only the positive values in a Matrix:

 := LinearAlgebra:-RandomMatrix(2,3);
for ,  in  do
    print(=[]);
    if  < 0 then
	next
    end if;
    print(=);
end do;

(Note that a simple example like this would be better written by enclosing the printing of the value in an if-statement instead of using next. The latter is generally only used if the former is not possible.)

Maple's loop statements are very flexible and powerful, making it possible to write loops with complex combinations of termination conditions in a concise yet readable way. The ability to use while and/or until in conjunction with for means that break statements are often unnecessary, further improving clarity.

The binary search algorithm is used to obtain the index of a given number by dividing the search bound in half over iteration. If the value entered in the array a message pop up telling that ''value is not present in the array". Please see the code. 
 

Download BINARY_SEARCH1.mw