Algebra: New Applications
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en-us2018 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 17 Jan 2018 02:57:33 GMTWed, 17 Jan 2018 02:57:33 GMTNew applications in the Algebra categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgAlgebra: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=173
Mathematics for Chemistry
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This interactive electronic textbook in the form of Maple worksheets comprises two parts.
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Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
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Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.<img src="https://www.maplesoft.com/view.aspx?si=154267/molecule.PNG" alt="Mathematics for Chemistry" style="max-width: 25%;" align="left"/>This interactive electronic textbook in the form of Maple worksheets comprises two parts.
<BR><BR>
Part I, mathematics for chemistry, is supposed to cover all mathematics that an instructor of chemistry might hope and expect that his students would learn, understand and be able to apply as a result of sufficient courses typically, but not exclusively, presented in departments of mathematics. Its nine chapters include (0) a summary and illustration of useful Maple commands, (1) arithmetic, algebra and elementary functions, (2) plotting, descriptive geometry, trigonometry, series, complex functions, (3) differential calculus of one variable, (4) integral calculus of one variable, (5) multivariate calculus, (6) linear algebra including matrix, vector, eigenvector, vector calculus, tensor, spreadsheet, (7) differential and integral equations, and (8) probability, distribution, treatment of laboratory data, linear and non-linear regression and optimization.
<BR><BR>
Part II presents mathematical topics typically taught within chemistry courses, including (9) chemical equilibrium, (10) group theory, (11) graph theory, (12a) introduction to quantum mechanics and quantum chemistry, (14) applications of Fourier transforms in chemistry including electron diffraction, x-ray diffraction, microwave spectra, infrared and Raman spectra and nuclear-magnetic-resonance spectra, and (18) dielectric and magnetic properties of chemical matter.
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Other chapters are in preparation and will be released in due course.https://www.maplesoft.com/applications/view.aspx?SID=154267&ref=FeedTue, 30 May 2017 04:00:00 ZProf. John OgilvieProf. John OgilvieLong Division
https://www.maplesoft.com/applications/view.aspx?SID=154229&ref=Feed
This procedure produces a LaTeX code which illustrates the process of dividing `a` by `b` using long division.<img src="https://www.maplesoft.com/view.aspx?si=154229/longdev.png" alt="Long Division" style="max-width: 25%;" align="left"/>This procedure produces a LaTeX code which illustrates the process of dividing `a` by `b` using long division.https://www.maplesoft.com/applications/view.aspx?SID=154229&ref=FeedTue, 21 Feb 2017 05:00:00 ZDmitry DemskoyDmitry DemskoyClassroom Tips and Techniques: Norm of a Matrix
https://www.maplesoft.com/applications/view.aspx?SID=1430&ref=Feed
The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.<img src="https://www.maplesoft.com/view.aspx?si=1430/thumb.jpg" alt="Classroom Tips and Techniques: Norm of a Matrix" style="max-width: 25%;" align="left"/>The greatest benefits from bringing Maple into the classroom are realized when the static pedagogy of a printed textbook is enlivened by the interplay of symbolic, graphic, and numeric calculations made possible by technology. Getting Maple to compute the correct answer is just the first step. Using Maple to bring insights not easily realized with by-hand calculations should be the goal of everyone who sets a hand to improving the learning experiences of students. In this article we will show how Maple can be used to gain insight on what the norm of a matrix means.https://www.maplesoft.com/applications/view.aspx?SID=1430&ref=FeedMon, 13 Feb 2017 05:00:00 ZDr. Robert LopezDr. Robert LopezAplicativo de Ecuaciones en primer orden
https://www.maplesoft.com/applications/view.aspx?SID=154139&ref=Feed
With this application you can develop your equations without the need to worry about the difficult calculation. Save calculation time and you will increase the time in interpreting the results. It was developed in Maple 2016 and can be executed in maple player.
In Spanish.<img src="https://www.maplesoft.com/view.aspx?si=154139/appec.png" alt="Aplicativo de Ecuaciones en primer orden" style="max-width: 25%;" align="left"/>With this application you can develop your equations without the need to worry about the difficult calculation. Save calculation time and you will increase the time in interpreting the results. It was developed in Maple 2016 and can be executed in maple player.
In Spanish.https://www.maplesoft.com/applications/view.aspx?SID=154139&ref=FeedSun, 07 Aug 2016 04:00:00 ZProf. Lenin Araujo CastilloProf. Lenin Araujo CastilloTips and Techniques: Working with Finitely Presented Groups in Maple
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This Tips and Techniques article introduces Maple's facilities for working with finitely presented groups. A finitely presented group is a group defined by means of a finite number of generators, and a finite number of defining relations. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute with many infinite groups.<img src="https://www.maplesoft.com/view.aspx?si=153852/thumb.jpg" alt="Tips and Techniques: Working with Finitely Presented Groups in Maple" style="max-width: 25%;" align="left"/>This Tips and Techniques article introduces Maple's facilities for working with finitely presented groups. A finitely presented group is a group defined by means of a finite number of generators, and a finite number of defining relations. It is one of the principal ways in which a group may be represented on the computer, and is virtually the only representation that effectively allows us to compute with many infinite groups.https://www.maplesoft.com/applications/view.aspx?SID=153852&ref=FeedTue, 25 Aug 2015 04:00:00 ZMaplesoftMaplesoftGroebner Bases: What are They and What are They Useful For?
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Since they were first introduced in 1965, Groebner bases have proven to be an invaluable contribution to mathematics and computer science. All general purpose computer algebra systems like Maple have Groebner basis implementations. But what is a Groebner basis? And what applications do Groebner bases have? In this Tips and Techniques article, I’ll give some examples of the main application of Groebner bases, which is to solve systems of polynomial equations.<img src="https://www.maplesoft.com/view.aspx?si=153693/thumb.jpg" alt="Groebner Bases: What are They and What are They Useful For?" style="max-width: 25%;" align="left"/>Since they were first introduced in 1965, Groebner bases have proven to be an invaluable contribution to mathematics and computer science. All general purpose computer algebra systems like Maple have Groebner basis implementations. But what is a Groebner basis? And what applications do Groebner bases have? In this Tips and Techniques article, I’ll give some examples of the main application of Groebner bases, which is to solve systems of polynomial equations.https://www.maplesoft.com/applications/view.aspx?SID=153693&ref=FeedFri, 17 Oct 2014 04:00:00 ZProf. Michael MonaganProf. Michael MonaganDescartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies
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<p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies" style="max-width: 25%;" align="left"/><p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p>https://www.maplesoft.com/applications/view.aspx?SID=153515&ref=FeedFri, 07 Mar 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyzoMbi
https://www.maplesoft.com/applications/view.aspx?SID=129642&ref=Feed
<p>Higher Mathematics for external students of biological faculty.<br />Solver-practicum.<br />1st semester.<br />300 problems (15 labs in 20 variants).<br />mw.zip</p>
<p>Before use - Shake! <br />(Click on the button and activate the program and Maplet).<br />Full version in html: <a href="http://webmath.exponenta.ru/zom/index.html">http://webmath.exponenta.ru/zom/index.html</a></p><img src="https://www.maplesoft.com/view.aspx?si=129642/zombie_3.jpg" alt="zoMbi" style="max-width: 25%;" align="left"/><p>Higher Mathematics for external students of biological faculty.<br />Solver-practicum.<br />1st semester.<br />300 problems (15 labs in 20 variants).<br />mw.zip</p>
<p>Before use - Shake! <br />(Click on the button and activate the program and Maplet).<br />Full version in html: <a href="http://webmath.exponenta.ru/zom/index.html">http://webmath.exponenta.ru/zom/index.html</a></p>https://www.maplesoft.com/applications/view.aspx?SID=129642&ref=FeedSun, 15 Jan 2012 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoHollywood Math
https://www.maplesoft.com/applications/view.aspx?SID=6611&ref=Feed
Over its storied and intriguing history, Hollywood has entertained us with many mathematical moments in film. John Nash in “A Beautiful Mind,” the brilliant janitor in “Good Will Hunting,” the number theory genius in “Pi,” and even Abbott and Costello are just a few of the Hollywood “mathematicians” that come to mind. This document highlights just a few examples of mathematics in film, and how Maple can work with them.<img src="https://www.maplesoft.com/view.aspx?si=6611/thumb.jpg" alt="Hollywood Math" style="max-width: 25%;" align="left"/>Over its storied and intriguing history, Hollywood has entertained us with many mathematical moments in film. John Nash in “A Beautiful Mind,” the brilliant janitor in “Good Will Hunting,” the number theory genius in “Pi,” and even Abbott and Costello are just a few of the Hollywood “mathematicians” that come to mind. This document highlights just a few examples of mathematics in film, and how Maple can work with them.https://www.maplesoft.com/applications/view.aspx?SID=6611&ref=FeedFri, 23 Sep 2011 04:00:00 ZMaplesoftMaplesoftWhy I Needed Maple to Make Cream Cheese Frosting
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<p>A recipe for cream cheese frosting I was making called for 8 oz. (about 240 grams) of cream cheese. Unfortunately, I didn't have a kitchen scale, and the product I bought came in a 400 gram tub in the shape of a<strong> truncated cone</strong>, which has a rather cumbersome volume formula. <br />Given the geometry of this tub, how deep into the tub should I scoop to get 240 grams? The mathematics is trickier than you might think but lots of fun! And the final, tasty result is worth the effort!</p><img src="https://www.maplesoft.com/view.aspx?si=125069/philly_thumb.png" alt="Why I Needed Maple to Make Cream Cheese Frosting" style="max-width: 25%;" align="left"/><p>A recipe for cream cheese frosting I was making called for 8 oz. (about 240 grams) of cream cheese. Unfortunately, I didn't have a kitchen scale, and the product I bought came in a 400 gram tub in the shape of a<strong> truncated cone</strong>, which has a rather cumbersome volume formula. <br />Given the geometry of this tub, how deep into the tub should I scoop to get 240 grams? The mathematics is trickier than you might think but lots of fun! And the final, tasty result is worth the effort!</p>https://www.maplesoft.com/applications/view.aspx?SID=125069&ref=FeedTue, 23 Aug 2011 04:00:00 ZDr. Jason SchattmanDr. Jason SchattmanMapler. 05. Аlgebraic equations. Part II. Variants
https://www.maplesoft.com/applications/view.aspx?SID=102312&ref=Feed
<p>Mathematical program-controlled multivariate Workshop. Problems 01-49 (including 12 versions of each problem with a complete solution)</p>
<p><span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">Application</span><span title="Нажмите, чтобы увидеть альтернативный перевод">.</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">Unzip the</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">folder ex.zip</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">and</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">place</span> together<span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод"> with </span><span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">the file</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">index.mw</span></p><img src="https://www.maplesoft.com/view.aspx?si=102312/mrs.jpg" alt="Mapler. 05. Аlgebraic equations. Part II. Variants" style="max-width: 25%;" align="left"/><p>Mathematical program-controlled multivariate Workshop. Problems 01-49 (including 12 versions of each problem with a complete solution)</p>
<p><span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">Application</span><span title="Нажмите, чтобы увидеть альтернативный перевод">.</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">Unzip the</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">folder ex.zip</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">and</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">place</span> together<span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод"> with </span><span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">the file</span> <span class="hps" title="Нажмите, чтобы увидеть альтернативный перевод">index.mw</span></p>https://www.maplesoft.com/applications/view.aspx?SID=102312&ref=FeedTue, 08 Mar 2011 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoMapler. 05. Аlgebraic equations & Index
https://www.maplesoft.com/applications/view.aspx?SID=102285&ref=Feed
<p>Mathematical program-controlled multivariate Workshop.<br />Version without maplets and test problems. <br />Further depends on community interest.</p><img src="https://www.maplesoft.com/view.aspx?si=102285/mrs.jpg" alt="Mapler. 05. Аlgebraic equations & Index" style="max-width: 25%;" align="left"/><p>Mathematical program-controlled multivariate Workshop.<br />Version without maplets and test problems. <br />Further depends on community interest.</p>https://www.maplesoft.com/applications/view.aspx?SID=102285&ref=FeedMon, 07 Mar 2011 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoExotic EIE-course
https://www.maplesoft.com/applications/view.aspx?SID=102076&ref=Feed
<p>Ukraine. <br />Exotic training course for the entrance examination in mathematics.<br /><strong>External independent evaluation</strong> <br />Themes:<br />0101 Goals and rational number <br />0102 Interest. The main problem of interest <br />0103 The simplest geometric shapes on the plane and their properties <br />0201 Degree of natural and integral indicator <br />0202 Monomial and polynomials and operations on them <br />0203 Triangles and their basic properties <br />0301 Algebraic fractions and operations on them <br />0302 Square root. Real numbers <br />0303 Circle and circle, their properties <br />0401 Equations, inequalities and their systems <br />0402 Function and its basic properties <br />0403 Described and inscribed triangles <br />0501 Linear function, linear equations, inequalities and their systems <br />0502 Quadratic function, quadratic equation, inequality and their systems <br />0503 Solving square triangles <br />0601 Rational Equations, Inequalities and their sysytemy <br />0602 Numerical sequence. Arithmetic and geometric progression <br />0603 Solving arbitrary triangles <br />0701 Sine, cosine, tangent and cotangent numeric argument <br />0702 Identical transformation of trigonometric expressions <br />0703 Quadrilateral types and their basic properties <br />0801 Trigonometric and inverse trigonometric functions, their properties <br />0802 Trigonometric equations and inequalities <br />0803 Polygons and their properties <br />0901 The root of n-th degree. Degree of rational parameters <br />0902 The power functions and their properties. Irrational equations, inequalities and their systems <br />0903 Regular polygons and their properties <br />1001 Logarithms. Logarithmic function. Logarithmic equations, inequalities and their systems <br />1002 Exponential function. Indicator of equations, inequalities and their systems <br />1003 Direct and planes in space <br />1101 Derivative and its geometric and mechanical content <br />1102 Derivatives and its application <br />1103 Polyhedron. Prisms and pyramids. Regular polyhedron <br />1201 Initial and definite integral <br />1202 Application of certain integral <br />1203 Body rotation <br />1301 Compounds. Binomial theorem <br />1302 General methods for solving equations, inequalities and their systems <br />1303 Coordinates in the plane and in space <br />1401 The origins of probability theory <br />1402 Beginnings of Mathematical Statistics <br />1403 Vectors in the plane and in space <br /><strong>Maple </strong>version<br /><strong>Html-interactive</strong> version</p><img src="https://www.maplesoft.com/view.aspx?si=102076/ell.jpg" alt="Exotic EIE-course" style="max-width: 25%;" align="left"/><p>Ukraine. <br />Exotic training course for the entrance examination in mathematics.<br /><strong>External independent evaluation</strong> <br />Themes:<br />0101 Goals and rational number <br />0102 Interest. The main problem of interest <br />0103 The simplest geometric shapes on the plane and their properties <br />0201 Degree of natural and integral indicator <br />0202 Monomial and polynomials and operations on them <br />0203 Triangles and their basic properties <br />0301 Algebraic fractions and operations on them <br />0302 Square root. Real numbers <br />0303 Circle and circle, their properties <br />0401 Equations, inequalities and their systems <br />0402 Function and its basic properties <br />0403 Described and inscribed triangles <br />0501 Linear function, linear equations, inequalities and their systems <br />0502 Quadratic function, quadratic equation, inequality and their systems <br />0503 Solving square triangles <br />0601 Rational Equations, Inequalities and their sysytemy <br />0602 Numerical sequence. Arithmetic and geometric progression <br />0603 Solving arbitrary triangles <br />0701 Sine, cosine, tangent and cotangent numeric argument <br />0702 Identical transformation of trigonometric expressions <br />0703 Quadrilateral types and their basic properties <br />0801 Trigonometric and inverse trigonometric functions, their properties <br />0802 Trigonometric equations and inequalities <br />0803 Polygons and their properties <br />0901 The root of n-th degree. Degree of rational parameters <br />0902 The power functions and their properties. Irrational equations, inequalities and their systems <br />0903 Regular polygons and their properties <br />1001 Logarithms. Logarithmic function. Logarithmic equations, inequalities and their systems <br />1002 Exponential function. Indicator of equations, inequalities and their systems <br />1003 Direct and planes in space <br />1101 Derivative and its geometric and mechanical content <br />1102 Derivatives and its application <br />1103 Polyhedron. Prisms and pyramids. Regular polyhedron <br />1201 Initial and definite integral <br />1202 Application of certain integral <br />1203 Body rotation <br />1301 Compounds. Binomial theorem <br />1302 General methods for solving equations, inequalities and their systems <br />1303 Coordinates in the plane and in space <br />1401 The origins of probability theory <br />1402 Beginnings of Mathematical Statistics <br />1403 Vectors in the plane and in space <br /><strong>Maple </strong>version<br /><strong>Html-interactive</strong> version</p>https://www.maplesoft.com/applications/view.aspx?SID=102076&ref=FeedMon, 28 Feb 2011 05:00:00 ZTIMOTIMOHow Fast Does An Advent Candle Burn?
https://www.maplesoft.com/applications/view.aspx?SID=100332&ref=Feed
<p>Any kid who's ever been entranced by an advent wreath knows that a tapered advent candle shrinks faster on Sunday night when it's new and slender than on Saturday night when it's old, stubby and caked with melted wax. How much faster? As an apropos application of math during this Christmas season, <strong>we derive a formula for the height of a burning tapered candle as a function of time</strong>. Assuming the candle has the shape of a cone when it is new and that it loses volume at a constant rate as it burns, we show that the height of the candle shrinks roughly in proportion to the cube root of time.</p><img src="https://www.maplesoft.com/view.aspx?si=100332/thumb.jpg" alt="How Fast Does An Advent Candle Burn?" style="max-width: 25%;" align="left"/><p>Any kid who's ever been entranced by an advent wreath knows that a tapered advent candle shrinks faster on Sunday night when it's new and slender than on Saturday night when it's old, stubby and caked with melted wax. How much faster? As an apropos application of math during this Christmas season, <strong>we derive a formula for the height of a burning tapered candle as a function of time</strong>. Assuming the candle has the shape of a cone when it is new and that it loses volume at a constant rate as it burns, we show that the height of the candle shrinks roughly in proportion to the cube root of time.</p>https://www.maplesoft.com/applications/view.aspx?SID=100332&ref=FeedMon, 20 Dec 2010 05:00:00 ZDr. Jason SchattmanDr. Jason Schattmanvan Roomen Problem
https://www.maplesoft.com/applications/view.aspx?SID=96978&ref=Feed
<p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p><img src="https://www.maplesoft.com/view.aspx?si=96978/maple_icon.jpg" alt="van Roomen Problem" style="max-width: 25%;" align="left"/><p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p>https://www.maplesoft.com/applications/view.aspx?SID=96978&ref=FeedSat, 18 Sep 2010 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyThe CayleyDickson Algebra from 4D to 256D
https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=Feed
<p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="The CayleyDickson Algebra from 4D to 256D" style="max-width: 25%;" align="left"/><p>There are higher dimensional numbers besides complex numbers. There are also hypercomplex numbers, such as, quaternions (4 D), octonions (8 D), sedenions (16 D), pathions (32 D), chingons (64 D), routons (128 D), voudons (256 D), and so on, without end. These names were coined by Robert P.C. de Marrais and Tony Smith. It is an alternate naming system providing relief from the difficult Latin names, such as:<br /> trigintaduonions (32 D), sexagintaquatronions (64 D), centumduodetrigintanions (128 D), and ducentiquinquagintasexions (256 D).</p>https://www.maplesoft.com/applications/view.aspx?SID=35420&ref=FeedFri, 23 Apr 2010 04:00:00 ZMichael CarterMichael CarterQuaternions, Octonions and Sedenions
https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=Feed
<p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Quaternions, Octonions and Sedenions" style="max-width: 25%;" align="left"/><p>This Hypercomplex package provides the algebra of the quaternion, octonion and sedenion hypercomplex numbers.</p>https://www.maplesoft.com/applications/view.aspx?SID=35196&ref=FeedFri, 16 Apr 2010 04:00:00 ZDr. Michael Angel Carter
Dr. Michael Angel Carter
Classroom Tips and Techniques: Jordan Normal Form of a Matrix
https://www.maplesoft.com/applications/view.aspx?SID=33195&ref=Feed
<p>A square matrix is similar either to a diagonal matrix or to a Jordan matrix, that is, a diagonal matrix with a mix of 1s and 0s on the superdiagonal. This month's article explores computations by which the matrix of transition to Jordan form can be constructed, and in so doing, illustrates some of the underlying theory.</p><img src="https://www.maplesoft.com/view.aspx?si=33195/thumb.png" alt="Classroom Tips and Techniques: Jordan Normal Form of a Matrix" style="max-width: 25%;" align="left"/><p>A square matrix is similar either to a diagonal matrix or to a Jordan matrix, that is, a diagonal matrix with a mix of 1s and 0s on the superdiagonal. This month's article explores computations by which the matrix of transition to Jordan form can be constructed, and in so doing, illustrates some of the underlying theory.</p>https://www.maplesoft.com/applications/view.aspx?SID=33195&ref=FeedMon, 06 Jul 2009 04:00:00 ZDr. Robert LopezDr. Robert LopezStandard Subscripts vs Literal Subscripts
https://www.maplesoft.com/applications/view.aspx?SID=6846&ref=Feed
Subscripts in Maple can be interpreted in different ways. This Tips and Techniques document explores the different types of subscripts you are able to use in Maple.
<P>
Note that this document uses Maple versions 16 and prior.
<P>
For more information on subscripts in Maple 17 and later, see:
<A HREF="/support/help/Maple/view.aspx?path=updates/Maple17/ImprovedSubscriptHandling">http://www.maplesoft.com/support/help/Maple/view.aspx?path=updates/Maple17/ImprovedSubscriptHandling</A><img src="https://www.maplesoft.com/view.aspx?si=6846/THUMBSTIENER.JPG" alt="Standard Subscripts vs Literal Subscripts" style="max-width: 25%;" align="left"/>Subscripts in Maple can be interpreted in different ways. This Tips and Techniques document explores the different types of subscripts you are able to use in Maple.
<P>
Note that this document uses Maple versions 16 and prior.
<P>
For more information on subscripts in Maple 17 and later, see:
<A HREF="/support/help/Maple/view.aspx?path=updates/Maple17/ImprovedSubscriptHandling">http://www.maplesoft.com/support/help/Maple/view.aspx?path=updates/Maple17/ImprovedSubscriptHandling</A>https://www.maplesoft.com/applications/view.aspx?SID=6846&ref=FeedThu, 30 Oct 2008 04:00:00 ZMaplesoftMaplesoftRegression and Data Fitting in Maple
https://www.maplesoft.com/applications/view.aspx?SID=6685&ref=Feed
This application demonstrates Maple's ability to fit data using the LeastSquares command in the CurveFitting package. Several examples are shown, using linear and nonlinear regression.<img src="https://www.maplesoft.com/view.aspx?si=6685/thumb.GIF" alt="Regression and Data Fitting in Maple" style="max-width: 25%;" align="left"/>This application demonstrates Maple's ability to fit data using the LeastSquares command in the CurveFitting package. Several examples are shown, using linear and nonlinear regression.https://www.maplesoft.com/applications/view.aspx?SID=6685&ref=FeedMon, 22 Sep 2008 00:00:00 ZMaplesoftMaplesoft