Astrophysics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=185
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemMon, 01 Sep 2014 19:12:01 GMTMon, 01 Sep 2014 19:12:01 GMTNew applications in the Astrophysics categoryhttp://www.mapleprimes.com/images/mapleapps.gifAstrophysics: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=185
Hohmann Elliptic Transfer Orbit with Animation
http://www.maplesoft.com/applications/view.aspx?SID=151351&ref=Feed
<p>Abstract<br /><br />The main purpose of this article is to show how to use Hohmann elliptic transfer in two situations:<br />a- When one manned spaceship is trying to catch up with an other one <br />on the same circular orbit around Earth.<br />b- When delivering a payload from Earth to a space station on a circular <br />orbit around Earth using 2-stage rocket .<br /><br />The way we set up the problem is as follows:<br />Consider two manned spaceships with astronauts Sally & Igor , the latter<br />lagging behind Sally by a given angle = 4.5 degrees while both are on the same<br />circular orbit C2 about Earth. A 2d lower circular orbit C1 is given. <br />Find the Hohmann elliptic orbit that is tangent to both orbits which allows<br />Sally to maneuver on C1 then to get back to the circular orbit C2 alongside Igor.<br /><br />Though the math was correct , however the final result we found was not !! <br />It was somehow tricky to find the culprit!<br />We have to restate the problem to get the correct answer. <br />The animation was then set up using the correct data. <br />The animation is a good teaching help for two reasons:<br />1- it gives a 'hand on' experience for anyone who wants to fully understand it,<br />2- it is a good lesson in Maple programming with many loops of the type 'if..then'.<br /><br />Warning<br /><br />This particular animation is a hog for the CPU memory since data accumulated <br />for plotting reached 20 MB! This is the size of this article when animation is <br />executed. For this reason and to be able to upload it I left the animation <br />procedure non executed which drops the size of the article to 300KB.<br /><br />Conclusion<br /><br />If I can get someone interested in the subject of this article in such away that he or <br />she would seek further information for learning from other sources, my efforts<br />would be well rewarded.</p><img src="/view.aspx?si=151351/Elliptic_image1.jpg" alt="Hohmann Elliptic Transfer Orbit with Animation" align="left"/><p>Abstract<br /><br />The main purpose of this article is to show how to use Hohmann elliptic transfer in two situations:<br />a- When one manned spaceship is trying to catch up with an other one <br />on the same circular orbit around Earth.<br />b- When delivering a payload from Earth to a space station on a circular <br />orbit around Earth using 2-stage rocket .<br /><br />The way we set up the problem is as follows:<br />Consider two manned spaceships with astronauts Sally & Igor , the latter<br />lagging behind Sally by a given angle = 4.5 degrees while both are on the same<br />circular orbit C2 about Earth. A 2d lower circular orbit C1 is given. <br />Find the Hohmann elliptic orbit that is tangent to both orbits which allows<br />Sally to maneuver on C1 then to get back to the circular orbit C2 alongside Igor.<br /><br />Though the math was correct , however the final result we found was not !! <br />It was somehow tricky to find the culprit!<br />We have to restate the problem to get the correct answer. <br />The animation was then set up using the correct data. <br />The animation is a good teaching help for two reasons:<br />1- it gives a 'hand on' experience for anyone who wants to fully understand it,<br />2- it is a good lesson in Maple programming with many loops of the type 'if..then'.<br /><br />Warning<br /><br />This particular animation is a hog for the CPU memory since data accumulated <br />for plotting reached 20 MB! This is the size of this article when animation is <br />executed. For this reason and to be able to upload it I left the animation <br />procedure non executed which drops the size of the article to 300KB.<br /><br />Conclusion<br /><br />If I can get someone interested in the subject of this article in such away that he or <br />she would seek further information for learning from other sources, my efforts<br />would be well rewarded.</p>151351Wed, 04 Sep 2013 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyPeriodicity of Sunspots
http://www.maplesoft.com/applications/view.aspx?SID=144592&ref=Feed
<p>This application finds the periodicity of sunspots with two independent approaches</p>
<ul>
<li>a frequency domain transformation of the data, </li>
<li>and autocorrelation. </li>
</ul>
<p>If implemented and interpreted correctly, both approaches should give the same sunspot period. The application uses routines from Maple 17’s new <a href="/products/maple/new_features/signal_processing.aspx">Signal Processing package</a>, and uses historical sunspot data from the National Geophysical Data Center. Additionally, an embedded video component demonstrates how you can zoom into a plot.</p><img src="/view.aspx?si=144592/sunspots.jpg" alt="Periodicity of Sunspots" align="left"/><p>This application finds the periodicity of sunspots with two independent approaches</p>
<ul>
<li>a frequency domain transformation of the data, </li>
<li>and autocorrelation. </li>
</ul>
<p>If implemented and interpreted correctly, both approaches should give the same sunspot period. The application uses routines from Maple 17’s new <a href="/products/maple/new_features/signal_processing.aspx">Signal Processing package</a>, and uses historical sunspot data from the National Geophysical Data Center. Additionally, an embedded video component demonstrates how you can zoom into a plot.</p>144592Wed, 13 Mar 2013 04:00:00 ZMaplesoftMaplesoftAlexander Friedmann's Cosmic Scenarios
http://www.maplesoft.com/applications/view.aspx?SID=142459&ref=Feed
<p>The Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p><img src="/view.aspx?si=142459/friedmannscenario.jpg" alt="Alexander Friedmann's Cosmic Scenarios" align="left"/><p>The Russian mathematician and physicist Alexander Friedmann (1888-1925) is well known among relativists, but his contributions to cosmology are largely misunderstood. Even the Royal Swedish Academy of Sciences misrepresented Friedmann's work in the 2011 Nobel Prize scientific background essay. Friedmann was the first physicist who demonstrated that Albert Einstein's general relativity admits non-static solutions, and the universe can expand, oscillate, and be born in a singularity. Friedmann's conclusion was based on his analysis of an elliptic integral; this worksheet employs Maple's utility of handling elliptic integrals to present Friedmann's results graphically. Friedmann's differential equation governing the evolution of the universe based on Einstein's general theory of relativity is also derived using Maple's tensor package. </p>142459Sun, 20 Jan 2013 05:00:00 ZDr. Frank WangDr. Frank WangCoriolis Effect
http://www.maplesoft.com/applications/view.aspx?SID=1437&ref=Feed
The Coriolis effect is a force that modifies the trajectory of falling object on Earth. It is due to the rotation of the referential and, thereby, it is not a real force. The mathematical expression of this effect is obtained from the crossproduct of Earth's angular velocity (omega) with the object's linear velocity (v). The exact equation is F = 2m(v x omega). This worksheet demonstrates the action of the Coriolis effect on a projectile launched from our planet. It includes a graphic of the projectile's path as well as a procedure that determines how far the projectile will travel.<img src="/view.aspx?si=1437/coriolis_sm.jpg" alt="Coriolis Effect" align="left"/>The Coriolis effect is a force that modifies the trajectory of falling object on Earth. It is due to the rotation of the referential and, thereby, it is not a real force. The mathematical expression of this effect is obtained from the crossproduct of Earth's angular velocity (omega) with the object's linear velocity (v). The exact equation is F = 2m(v x omega). This worksheet demonstrates the action of the Coriolis effect on a projectile launched from our planet. It includes a graphic of the projectile's path as well as a procedure that determines how far the projectile will travel.1437Fri, 09 Dec 2011 05:00:00 ZPascal Thériault LauzierPascal Thériault LauzierThe Orbit of Kepler 16b
http://www.maplesoft.com/applications/view.aspx?SID=126766&ref=Feed
<p>NASA's Kepler space telescope recently made the news by finding a planet that orbits a double-star system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. <br /><br />This worksheet explores the orbital mechanics of such a system.</p><img src="/view.aspx?si=126766/kepler16b.png" alt="The Orbit of Kepler 16b" align="left"/><p>NASA's Kepler space telescope recently made the news by finding a planet that orbits a double-star system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. <br /><br />This worksheet explores the orbital mechanics of such a system.</p>126766Tue, 18 Oct 2011 04:00:00 ZDr. Robert IsraelDr. Robert IsraelTerminator 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 BaroudyProfessional Tips & Techniques: Using Scientific Constants in Maple
http://www.maplesoft.com/applications/view.aspx?SID=1724&ref=Feed
Maple includes an extensive ScientificConstants package that provides access to the values of various constant physical quantities.
Such values can be used to solve equations in fields such as chemistry and physics. The ScientificConstants package also provides the units for each of the constant values, allowing for greater understanding of the equation as well as units matching for error checking of the solution.
The quantities available in the ScientificConstants package are divided into two distinct categories.
1. physical constants
2. properties of the chemical elements (and their isotopes)
This document will highlight the uses of the ScientificConstants package in Maple, and provides some examples of its use.<img src="/view.aspx?si=1724/TTMarApr.jpg" alt="Professional Tips & Techniques: Using Scientific Constants in Maple" align="left"/>Maple includes an extensive ScientificConstants package that provides access to the values of various constant physical quantities.
Such values can be used to solve equations in fields such as chemistry and physics. The ScientificConstants package also provides the units for each of the constant values, allowing for greater understanding of the equation as well as units matching for error checking of the solution.
The quantities available in the ScientificConstants package are divided into two distinct categories.
1. physical constants
2. properties of the chemical elements (and their isotopes)
This document will highlight the uses of the ScientificConstants package in Maple, and provides some examples of its use.1724Mon, 27 Mar 2006 00:00:00 ZMaplesoftMaplesoftTime Dilation and Length Contraction Formulas
http://www.maplesoft.com/applications/view.aspx?SID=1697&ref=Feed
This Worksheet demonstrates the Time Dilation and Length Contraction Formulas.<img src="/view.aspx?si=1697/Light Box At Motion small.JPG" alt="Time Dilation and Length Contraction Formulas" align="left"/>This Worksheet demonstrates the Time Dilation and Length Contraction Formulas.1697Tue, 13 Dec 2005 00:00:00 ZMohammed MokbelMohammed MokbelKinematics of Our Earth-Moon System
http://www.maplesoft.com/applications/view.aspx?SID=1501&ref=Feed
Apart from Newtonian forces of attraction between masses and Einstein's theory of gravitation, we can get an insight into the movements of our moon by using Maple's Linear Algebra Package and some key data about our only natural satellite.<img src="/view.aspx?si=1501/EarthMoon.gif" alt="Kinematics of Our Earth-Moon System" align="left"/>Apart from Newtonian forces of attraction between masses and Einstein's theory of gravitation, we can get an insight into the movements of our moon by using Maple's Linear Algebra Package and some key data about our only natural satellite.1501Wed, 25 May 2005 00:00:00 ZDr. Friedrich FUTSCHIKDr. Friedrich FUTSCHIKCreating Maple Documents: Deriving the Rocket Equation
http://www.maplesoft.com/applications/view.aspx?SID=1402&ref=Feed
Maple 10 lets you create sophisticated live documents that seamlessly integrate the mathematical results with derivations, explanations, plots, and other forms of technical knowledge. This document develops a simple model of a rocket to illustrate the use of many of these new features.
The following Maple features are highlighted:
<ul><li>Document mode and Document blocks</li><li>2-D math editor and palettes</li><li>Autoexecute regions</li><li>Plotting</li><li> Tables</li><li>Embedded Components</li></ul><img src="/view.aspx?si=1402/thumb.jpg" alt="Creating Maple Documents: Deriving the Rocket Equation" align="left"/>Maple 10 lets you create sophisticated live documents that seamlessly integrate the mathematical results with derivations, explanations, plots, and other forms of technical knowledge. This document develops a simple model of a rocket to illustrate the use of many of these new features.
The following Maple features are highlighted:
<ul><li>Document mode and Document blocks</li><li>2-D math editor and palettes</li><li>Autoexecute regions</li><li>Plotting</li><li> Tables</li><li>Embedded Components</li></ul>1402Thu, 19 May 2005 00:00:00 ZMaplesoftMaplesoftFitting a Sine Curve to Data
http://www.maplesoft.com/applications/view.aspx?SID=1422&ref=Feed
This worksheet demonstrate making a best sine curve fit to a set of sparse data from observations of the star 51 Pegasi. Also shown is making a best sine curve fit to a set of sparse data from observation of the tides in the Bay of Fundy. The worksheet illustrates nonlinear curve fitting with Maple, using both elementary commands and sophisticated tools.<img src="/view.aspx?si=1422/sinefit_24.gif" alt="Fitting a Sine Curve to Data" align="left"/>This worksheet demonstrate making a best sine curve fit to a set of sparse data from observations of the star 51 Pegasi. Also shown is making a best sine curve fit to a set of sparse data from observation of the tides in the Bay of Fundy. The worksheet illustrates nonlinear curve fitting with Maple, using both elementary commands and sophisticated tools.1422Mon, 10 Jan 2005 00:00:00 ZSteven DunbarSteven DunbarRelativistic Orbits and Black Holes
http://www.maplesoft.com/applications/view.aspx?SID=4508&ref=Feed
Using Maple to derive equations of motion from the Lagrangian, to solve differenctial equations numerically, and to form graphs based on numerical solutions, we present trajectries of a particle under gravity based on Newtonian theory and general relativity with the relativity with the Schwarzschild metric and the Kerr metric.<img src="/view.aspx?si=4508//applications/images/app_image_blank_lg.jpg" alt="Relativistic Orbits and Black Holes" align="left"/>Using Maple to derive equations of motion from the Lagrangian, to solve differenctial equations numerically, and to form graphs based on numerical solutions, we present trajectries of a particle under gravity based on Newtonian theory and general relativity with the relativity with the Schwarzschild metric and the Kerr metric.4508Thu, 20 May 2004 14:01:13 ZFrank WangFrank WangA Virtual 3D Solar System
http://www.maplesoft.com/applications/view.aspx?SID=4484&ref=Feed
In solar system, all planetary orbits, except Pluto 's, are nearly in a plane and circular. Here, we use a little knowledge of Celestial Mechanics and power of Maple to make a virtual 3D solar system and illustrate these characteristics. <img src="/view.aspx?si=4484//applications/images/app_image_blank_lg.jpg" alt="A Virtual 3D Solar System " align="left"/>In solar system, all planetary orbits, except Pluto 's, are nearly in a plane and circular. Here, we use a little knowledge of Celestial Mechanics and power of Maple to make a virtual 3D solar system and illustrate these characteristics. 4484Mon, 26 Jan 2004 10:24:38 ZYi XieYi XieModeling of Atmospheric Motion and Astrodynamics
http://www.maplesoft.com/applications/view.aspx?SID=4309&ref=Feed
Equations of motion are systems of differential equations. Most practical systems are only solvable numerically. We show applications to movement in the atmosphere and the dynamics of space ships.<img src="/view.aspx?si=4309/1095.jpg" alt="Modeling of Atmospheric Motion and Astrodynamics" align="left"/>Equations of motion are systems of differential equations. Most practical systems are only solvable numerically. We show applications to movement in the atmosphere and the dynamics of space ships.4309Fri, 20 Sep 2002 14:42:03 ZStanislav BartonStanislav BartonSatellite Motion in a Multipolar Gravitational Field
http://www.maplesoft.com/applications/view.aspx?SID=4268&ref=Feed
This worksheet demonstrates the use of Maple for studying the path of a satellite. This worksheet uses the idea of gravitation, gravitational force field, and Newton's second law ( F = m a ) to describe the motion of any object or satellite in a gravitational field. If we have only one star, and a planet orbiting that star, we can derive the motions for equation of that planet quiet simply. BUT what happens when a planet orbits say 2 or maybe 3 stars (stars which are stationary) each with their own gravitational field? Then the planet doesn't orbit the stars in simple elliptical or hyperbolic (or parabolic... etc.) orbits. This is because there is more than one gravitational field present and the force that the satellite experiences is the summation of the gravitational fields. Using this idea, we can write differential equations that will give us a view of the path of a planet or satellite in a multipolar inverse-square gravitational field.<img src="/view.aspx?si=4268/motion.gif" alt="Satellite Motion in a Multipolar Gravitational Field" align="left"/>This worksheet demonstrates the use of Maple for studying the path of a satellite. This worksheet uses the idea of gravitation, gravitational force field, and Newton's second law ( F = m a ) to describe the motion of any object or satellite in a gravitational field. If we have only one star, and a planet orbiting that star, we can derive the motions for equation of that planet quiet simply. BUT what happens when a planet orbits say 2 or maybe 3 stars (stars which are stationary) each with their own gravitational field? Then the planet doesn't orbit the stars in simple elliptical or hyperbolic (or parabolic... etc.) orbits. This is because there is more than one gravitational field present and the force that the satellite experiences is the summation of the gravitational fields. Using this idea, we can write differential equations that will give us a view of the path of a planet or satellite in a multipolar inverse-square gravitational field.4268Tue, 23 Apr 2002 15:33:17 ZForhad AhmedForhad AhmedThe ScientificConstants Package
http://www.maplesoft.com/applications/view.aspx?SID=4260&ref=Feed
The new ScientificConstants package in Maple 8 provides the values, symbols, uncertainty and units of over 13,000 physical constants. Never again do you have to search the internet or grab a reference book to look up such values. This extensive package gives you access to the basic chemical properties of any element or isotope from the Periodic Table. Also available are 70 physical constants, such as Planck's constant, the Newtonian constant of gravitation, magnetic flux quantum, and speed of light in a vacuum. If a specific physical constant that you need is not included, simply add the constants you commonly use to Maple's database. This flexible package also allows you to modify properties or change the system associated with a constant. The ScientificConstants package in conjunction with the comprehensive Units package introduced in Maple 7 together provide valuable tools for scientists and engineers.<img src="/view.aspx?si=4260//applications/images/app_image_blank_lg.jpg" alt="The ScientificConstants Package" align="left"/>The new ScientificConstants package in Maple 8 provides the values, symbols, uncertainty and units of over 13,000 physical constants. Never again do you have to search the internet or grab a reference book to look up such values. This extensive package gives you access to the basic chemical properties of any element or isotope from the Periodic Table. Also available are 70 physical constants, such as Planck's constant, the Newtonian constant of gravitation, magnetic flux quantum, and speed of light in a vacuum. If a specific physical constant that you need is not included, simply add the constants you commonly use to Maple's database. This flexible package also allows you to modify properties or change the system associated with a constant. The ScientificConstants package in conjunction with the comprehensive Units package introduced in Maple 7 together provide valuable tools for scientists and engineers.4260Mon, 15 Apr 2002 16:08:32 ZMaplesoftMaplesoftTrajectory Near a Black Hole: an application of Lagrangian mechanics
http://www.maplesoft.com/applications/view.aspx?SID=4240&ref=Feed
The lagrangian formulation of mechanics has great advantages in practical use. Calculations of this type require finding the derivative of a function with respect to another function. Although our method is a pedagogic approach that might involve longer steps, it is a straightforward attack on this problem, and practically all problems in classical mechanics can be solved once the lagrangian is found. In most real physics problems, there are no analytic solutions to differential equations. We particularly emphasize forming plots numerically. We introduce an example in general relativity, to find the trajectory of a particle near a black hole, which corresponds to the shortest path between two points in a curved space.<img src="/view.aspx?si=4240//applications/images/app_image_blank_lg.jpg" alt="Trajectory Near a Black Hole: an application of Lagrangian mechanics " align="left"/>The lagrangian formulation of mechanics has great advantages in practical use. Calculations of this type require finding the derivative of a function with respect to another function. Although our method is a pedagogic approach that might involve longer steps, it is a straightforward attack on this problem, and practically all problems in classical mechanics can be solved once the lagrangian is found. In most real physics problems, there are no analytic solutions to differential equations. We particularly emphasize forming plots numerically. We introduce an example in general relativity, to find the trajectory of a particle near a black hole, which corresponds to the shortest path between two points in a curved space.4240Tue, 12 Mar 2002 11:24:18 ZProf. J. OgilvieProf. J. OgilvieTwo-body gravitation in 3-D
http://www.maplesoft.com/applications/view.aspx?SID=4212&ref=Feed
This maplet computes the 2-D or 3-D path of two bodies under the influence of gravity, given the bodies' masses, initial positions and initial velocities. The paths are computed by numeric solutions of systems of differential equations.<img src="/view.aspx?si=4212//applications/images/app_image_blank_lg.jpg" alt="Two-body gravitation in 3-D" align="left"/>This maplet computes the 2-D or 3-D path of two bodies under the influence of gravity, given the bodies' masses, initial positions and initial velocities. The paths are computed by numeric solutions of systems of differential equations.4212Thu, 24 Jan 2002 12:10:05 ZSylvain MuiseSylvain MuiseAstronomie
http://www.maplesoft.com/applications/view.aspx?SID=4098&ref=Feed
Simulation der Bewegung des Mondes um die Erde, bei<img src="/view.aspx?si=4098/astro.gif" alt="Astronomie" align="left"/>Simulation der Bewegung des Mondes um die Erde, bei4098Fri, 17 Aug 2001 13:34:06 ZJan KästleJan KästlePlanetary motion
http://www.maplesoft.com/applications/view.aspx?SID=4013&ref=Feed
Parameterizing planetary motion under gravity. It solves for planetary motion with a flow line solution of a vector field in 4 dimensions. The orbits of Mars and Halley's comet are studied as examples.
<img src="/view.aspx?si=4013//applications/images/app_image_blank_lg.jpg" alt="Planetary motion " align="left"/>Parameterizing planetary motion under gravity. It solves for planetary motion with a flow line solution of a vector field in 4 dimensions. The orbits of Mars and Halley's comet are studied as examples.
4013Thu, 02 Aug 2001 13:39:03 ZProf. Michael MayProf. Michael May