Geometry: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=140
en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemFri, 09 Dec 2016 11:36:26 GMTFri, 09 Dec 2016 11:36:26 GMTNew applications in the Geometry categoryhttp://www.mapleprimes.com/images/mapleapps.gifGeometry: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=140
Bodies with fixed volume and minimal surface
http://www.maplesoft.com/applications/view.aspx?SID=154057&ref=Feed
We would like to find r/h when a surface is minimal.
Can you solve more complex problem for others bodies<img src="/view.aspx?si=154057/1e22ebea45ed480d944eaae36fe7be0e.gif" alt="Bodies with fixed volume and minimal surface" align="left"/>We would like to find r/h when a surface is minimal.
Can you solve more complex problem for others bodies154057Mon, 21 Mar 2016 17:44:27 ZProf. Valery OchkovProf. Valery OchkovPerimeter, area and visualization of a plane figure
http://www.maplesoft.com/applications/view.aspx?SID=146470&ref=Feed
<p>The work contains three procedures that allow symbolically to calculate the perimeter and area of any plane figure bounded by <span>non-selfintersecting piecewise smooth curve</span>, and to portray this figure together with its boundary in a suitable design.</p><img src="/view.aspx?si=146470/planefigure_thumb.png" alt="Perimeter, area and visualization of a plane figure" align="left"/><p>The work contains three procedures that allow symbolically to calculate the perimeter and area of any plane figure bounded by <span>non-selfintersecting piecewise smooth curve</span>, and to portray this figure together with its boundary in a suitable design.</p>146470Tue, 30 Apr 2013 04:00:00 ZDr. Yury ZavarovskyDr. Yury ZavarovskyClassroom Tips and Techniques: Custom and Task Palettes
http://www.maplesoft.com/applications/view.aspx?SID=132914&ref=Feed
New in Maple 16, the Custom palette is a palette added to Maple by the user. It is populated with task templates that are already in the Task Browser Table of Contents. A separate Tasks palette can be populated with task templates created by the "Create Task" option in the Context Menu for any selection in a worksheet. This article sheds light on these new functionalities, and gives an example of a Custom palette developed to capture part of the geom3d package in task templates.<img src="/view.aspx?si=132914/thumb.jpg" alt="Classroom Tips and Techniques: Custom and Task Palettes" align="left"/>New in Maple 16, the Custom palette is a palette added to Maple by the user. It is populated with task templates that are already in the Task Browser Table of Contents. A separate Tasks palette can be populated with task templates created by the "Create Task" option in the Context Menu for any selection in a worksheet. This article sheds light on these new functionalities, and gives an example of a Custom palette developed to capture part of the geom3d package in task templates.132914Thu, 12 Apr 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Caustics for a Plane Curve
http://www.maplesoft.com/applications/view.aspx?SID=131655&ref=Feed
This article shows how to construct and visualize a <i>caustic</i>, the envelope of lines emanating from a fixed point, and reflecting off a plane curve.<img src="/view.aspx?si=131655/thumb.jpg" alt="Classroom Tips and Techniques: Caustics for a Plane Curve" align="left"/>This article shows how to construct and visualize a <i>caustic</i>, the envelope of lines emanating from a fixed point, and reflecting off a plane curve.131655Mon, 12 Mar 2012 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Sliders for Parameter-Dependent Curves
http://www.maplesoft.com/applications/view.aspx?SID=130674&ref=Feed
Methods for building slider-controlled graphs are explored, and used to show the variations in the limaçon. Then, the conchoid of a cubic is explored with the same set of tools.<img src="/view.aspx?si=130674/thumb.jpg" alt="Classroom Tips and Techniques: Sliders for Parameter-Dependent Curves" align="left"/>Methods for building slider-controlled graphs are explored, and used to show the variations in the limaçon. Then, the conchoid of a cubic is explored with the same set of tools.130674Tue, 14 Feb 2012 05:00:00 ZDr. Robert LopezDr. Robert LopezzoMbi
http://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="/view.aspx?si=129642/zombie_3.jpg" alt="zoMbi" 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>129642Sun, 15 Jan 2012 05:00:00 ZDr. Valery CyboulkoDr. Valery CyboulkoComputational procedures for weighted projective spaces
http://www.maplesoft.com/applications/view.aspx?SID=127621&ref=Feed
<p>This file provides procedures which are able to produce the toric data associated with a (polarized) wps i.e. fans , polytopes and their equivalences. More originally it provides procedures which are able to detect a weights vector Q starting from either a fan or a polytope: we will call this process the recognition of a (polarized) weighted projective space. Moreover it gives procedures connecting polytopes of a polarized weighted projective space with an associated fan and viceversa.</p><img src="/view.aspx?si=127621/fan.jpg" alt="Computational procedures for weighted projective spaces" align="left"/><p>This file provides procedures which are able to produce the toric data associated with a (polarized) wps i.e. fans , polytopes and their equivalences. More originally it provides procedures which are able to detect a weights vector Q starting from either a fan or a polytope: we will call this process the recognition of a (polarized) weighted projective space. Moreover it gives procedures connecting polytopes of a polarized weighted projective space with an associated fan and viceversa.</p>127621Thu, 10 Nov 2011 05:00:00 ZProf. Michele RossiProf. Michele RossiWhy 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="/view.aspx?si=125069/philly_thumb.png" alt="Why I Needed Maple to Make Cream Cheese Frosting" 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>125069Tue, 23 Aug 2011 04:00:00 ZDr. Jason SchattmanDr. Jason SchattmanMapler. Practicum. Analytic Geometry.
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<p>Examples. 2 x 30 variants.<br />The main thing - the idea.<br />To activate the solution, press the appropriate button, and then - Enter or !!!</p><img src="/view.aspx?si=102180/ag1.jpg" alt="Mapler. Practicum. Analytic Geometry." align="left"/><p>Examples. 2 x 30 variants.<br />The main thing - the idea.<br />To activate the solution, press the appropriate button, and then - Enter or !!!</p>102180Thu, 03 Mar 2011 05:00:00 ZDonetsk National UniversityDonetsk National UniversityExotic EIE-course
http://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="/view.aspx?si=102076/ell.jpg" alt="Exotic EIE-course" 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>102076Mon, 28 Feb 2011 05:00:00 ZTIMOTIMOTerminator circle with animation
http://www.maplesoft.com/applications/view.aspx?SID=100509&ref=Feed
<p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p><img src="/view.aspx?si=100509/thumb.jpg" alt="Terminator circle with animation" align="left"/><p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p>100509Tue, 28 Dec 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyCoon's Patch Examples
http://www.maplesoft.com/applications/view.aspx?SID=100369&ref=Feed
<p><span id="ctl00_ctl00_basicMain_main_viewer_body" class="mainBody document">
<p>Coon's patch defines a patch from 4 curves forming a closed chain. This application contains 4-examples to Coon's patch. Boundary curves in the examples are constructed by parametric cubic curves (in Bezier and Hermite form). First example is in 3D and comparable to parametric bicubic surface. Rest of examples are in 2D (abstract surface) which can be described shortly as:</p>
<ul>
<li>2D Example#1: 4 circular-arc like curves are combined by Coon's patch to form a circular-like region</li>
<li>2D Example#2: vertices of a square are assigned to rotating tangents</li>
<li>2D Example#3: boundaries corresponding to nozzle like geometry</li>
</ul>
</span></p><img src="/view.aspx?si=100369/maple_icon.jpg" alt="Coon's Patch Examples" align="left"/><p><span id="ctl00_ctl00_basicMain_main_viewer_body" class="mainBody document">
<p>Coon's patch defines a patch from 4 curves forming a closed chain. This application contains 4-examples to Coon's patch. Boundary curves in the examples are constructed by parametric cubic curves (in Bezier and Hermite form). First example is in 3D and comparable to parametric bicubic surface. Rest of examples are in 2D (abstract surface) which can be described shortly as:</p>
<ul>
<li>2D Example#1: 4 circular-arc like curves are combined by Coon's patch to form a circular-like region</li>
<li>2D Example#2: vertices of a square are assigned to rotating tangents</li>
<li>2D Example#3: boundaries corresponding to nozzle like geometry</li>
</ul>
</span></p>100369Tue, 21 Dec 2010 05:00:00 ZHakan TiftikciHakan TiftikciClassroom Tips and Techniques: Maple Meets Marden's Theorem
http://www.maplesoft.com/applications/view.aspx?SID=99069&ref=Feed
<p>Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial <em>p(z)</em>, the zeros of <em>p'(z)</em> are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.<br /></p><img src="/view.aspx?si=99069/thumb.jpg" alt="Classroom Tips and Techniques: Maple Meets Marden's Theorem" align="left"/><p>Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial <em>p(z)</em>, the zeros of <em>p'(z)</em> are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.<br /></p>99069Tue, 16 Nov 2010 05:00:00 ZRobert LopezRobert LopezSome Fractals with Maple
http://www.maplesoft.com/applications/view.aspx?SID=97624&ref=Feed
<p>This paper contains some procedures concerning the fractals. In particular we describe the procedures in order to draw the triadic Cantor dust, the Koch curve, the Snowflake and the procedures in order to calculate the area and the perimeter of the Snowflake curve.</p><img src="/view.aspx?si=97624/maple_icon.jpg" alt="Some Fractals with Maple" align="left"/><p>This paper contains some procedures concerning the fractals. In particular we describe the procedures in order to draw the triadic Cantor dust, the Koch curve, the Snowflake and the procedures in order to calculate the area and the perimeter of the Snowflake curve.</p>97624Fri, 08 Oct 2010 04:00:00 ZProf. Marina MarchisioProf. Marina MarchisioASTROLABE MATHEMATICS
http://www.maplesoft.com/applications/view.aspx?SID=35214&ref=Feed
<p>After a very short introduction about the origin & the evolution of the astrolabe through ages, we go into the basic geometry behind its design and construction. <br />
This involves: <br />
1- the therory of STEREOGRAPHIC PROJECTION with two simple properties that we prove, <br />
2- simple ANALYTIC GEOMETRY,<br />
3- OBLIQUE CONE THEORY according to LEGENDRE. <br />
We then explore the details of designing an astrolabe PLATE according to ancient constructors technics and today's technology using Maple powerful plotting engine. This resuls in a beautiful PLATE and on top of it the RETE. (See Figure below). <br />
As a whole the article is a good exercise in Maple programming and a way to stimulate further interest in Astronomy.</p><img src="/view.aspx?si=35214/Rete_on_top_of_plate.jpeg" alt="ASTROLABE MATHEMATICS" align="left"/><p>After a very short introduction about the origin & the evolution of the astrolabe through ages, we go into the basic geometry behind its design and construction. <br />
This involves: <br />
1- the therory of STEREOGRAPHIC PROJECTION with two simple properties that we prove, <br />
2- simple ANALYTIC GEOMETRY,<br />
3- OBLIQUE CONE THEORY according to LEGENDRE. <br />
We then explore the details of designing an astrolabe PLATE according to ancient constructors technics and today's technology using Maple powerful plotting engine. This resuls in a beautiful PLATE and on top of it the RETE. (See Figure below). <br />
As a whole the article is a good exercise in Maple programming and a way to stimulate further interest in Astronomy.</p>35214Sat, 27 Feb 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyClassroom Tips and Techniques: Spatial Transformations of a Triangle
http://www.maplesoft.com/applications/view.aspx?SID=35050&ref=Feed
<p>The <em>geom3d</em> package is used to implement translations and rotations that bring two congruent triangles into coincidence in R<sup>3</sup>.</p><img src="/view.aspx?si=35050/thumb.jpg" alt="Classroom Tips and Techniques: Spatial Transformations of a Triangle" align="left"/><p>The <em>geom3d</em> package is used to implement translations and rotations that bring two congruent triangles into coincidence in R<sup>3</sup>.</p>35050Tue, 12 Jan 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezEichler Orders
http://www.maplesoft.com/applications/view.aspx?SID=5567&ref=Feed
<p>The aim of this worksheet is to give an explicit description in term of bases over Z, of some Eichler orders of an indefinite quaternion algebra B defined over Q. Following a work of Hashimoto, we give a procedure which returns a basis of the Eichler orders of level N. This construction provides a very useful tool for working with Eichler orders. Let q be a prime number not dividing the discriminant of B; it is well known that there are two natural inclusions of R(Nq) in R(N). We provide a new procedure in Maple which returns a basis of an Eichler order on level Nq in B and a basis of the other copy of R(Nq) in R(N).</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Eichler Orders" align="left"/><p>The aim of this worksheet is to give an explicit description in term of bases over Z, of some Eichler orders of an indefinite quaternion algebra B defined over Q. Following a work of Hashimoto, we give a procedure which returns a basis of the Eichler orders of level N. This construction provides a very useful tool for working with Eichler orders. Let q be a prime number not dividing the discriminant of B; it is well known that there are two natural inclusions of R(Nq) in R(N). We provide a new procedure in Maple which returns a basis of an Eichler order on level Nq in B and a basis of the other copy of R(Nq) in R(N).</p>5567Fri, 18 Dec 2009 05:00:00 ZDr. Miriam CiavarellaDr. Miriam CiavarellaComparison of Multivariate Optimization Methods
http://www.maplesoft.com/applications/view.aspx?SID=1718&ref=Feed
<p>The worksheet demonstrates the use of Maple to compare methods of unconstrained nonlinear minimization of multivariable function. Seven methods of nonlinear minimization of the n-variables objective function f(x1,x2,.,xn) are analyzed:</p>
<p>1) minimum search by coordinate and conjugate directions descent; 2) Powell's method; 3) the modified Hooke-Jeeves method; 4) simplex Nelder-Meed method; 5) quasi-gradient method; 6) random directions search; 7) simulated annealing. All methods are direct searching methods, i.e. they do not require the objective function f(x1,x2,.,xn) to be differentiable and continuous. Maple's Optimization package efficiency is compared with these programs. Optimization methods have been compared on the set of 21 test functions.</p><img src="/view.aspx?si=1718/SearchPaths2.PNG" alt="Comparison of Multivariate Optimization Methods" align="left"/><p>The worksheet demonstrates the use of Maple to compare methods of unconstrained nonlinear minimization of multivariable function. Seven methods of nonlinear minimization of the n-variables objective function f(x1,x2,.,xn) are analyzed:</p>
<p>1) minimum search by coordinate and conjugate directions descent; 2) Powell's method; 3) the modified Hooke-Jeeves method; 4) simplex Nelder-Meed method; 5) quasi-gradient method; 6) random directions search; 7) simulated annealing. All methods are direct searching methods, i.e. they do not require the objective function f(x1,x2,.,xn) to be differentiable and continuous. Maple's Optimization package efficiency is compared with these programs. Optimization methods have been compared on the set of 21 test functions.</p>1718Tue, 15 Sep 2009 04:00:00 ZMilnor and Tyurina numbers
http://www.maplesoft.com/applications/view.aspx?SID=19341&ref=Feed
<p>We give algorithms, based on local and global monomial orders, for computing Milnor and Tyurina numbers in the case of an isolated algebraic hypersurface singularity.</p><img src="/view.aspx?si=19341/milnor.jpg" alt="Milnor and Tyurina numbers" align="left"/><p>We give algorithms, based on local and global monomial orders, for computing Milnor and Tyurina numbers in the case of an isolated algebraic hypersurface singularity.</p>19341Fri, 03 Apr 2009 04:00:00 ZProf. Lea TerraciniProf. Lea Terracini