Topology: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=152
en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemSun, 01 Feb 2015 16:42:23 GMTSun, 01 Feb 2015 16:42:23 GMTNew applications in the Topology categoryhttp://www.mapleprimes.com/images/mapleapps.gifTopology: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=152
The Extremal and Non-Trivial Minimal Topologies by Definitions
http://www.maplesoft.com/applications/view.aspx?SID=153625&ref=Feed
<p> by </p>
<p> <br /> MS.C Taha Guma el turki </p>
<p> Benghazi University department of Mathematics</p>
<p> email: taha 1978_2002@yahoo.com </p>
<p> <strong><em>Definition </em>[1]:-</strong> </p>
<p>Let X be any set, τ is not a discrete topology on X then τ is said to be an extremal topology if every topology strictly finer than τ is discrete.<br /> <br />A non-trivial minimal topology is a topology which is not Indiscrete and does not contain any other topology over X .<br /><br /><em>References</em><br /><br />[1] http://www.damascusuniversity.edu.sy/mag/asasy/images/stories/e19.pdf .</p>
<p>[2] http://www.maplesoft.com/applications/view.aspx?SID=153617 .</p><img src="/applications/images/app_image_blank_lg.jpg" alt="The Extremal and Non-Trivial Minimal Topologies by Definitions" align="left"/><p> by </p>
<p> <br /> MS.C Taha Guma el turki </p>
<p> Benghazi University department of Mathematics</p>
<p> email: taha 1978_2002@yahoo.com </p>
<p> <strong><em>Definition </em>[1]:-</strong> </p>
<p>Let X be any set, τ is not a discrete topology on X then τ is said to be an extremal topology if every topology strictly finer than τ is discrete.<br /> <br />A non-trivial minimal topology is a topology which is not Indiscrete and does not contain any other topology over X .<br /><br /><em>References</em><br /><br />[1] http://www.damascusuniversity.edu.sy/mag/asasy/images/stories/e19.pdf .</p>
<p>[2] http://www.maplesoft.com/applications/view.aspx?SID=153617 .</p>153625Thu, 17 Jul 2014 04:00:00 ZTaha Guma el turkiTaha Guma el turkiThe Extremal and Non-Trivial Minimal Topologies Over a Finite Set with Maple
http://www.maplesoft.com/applications/view.aspx?SID=153617&ref=Feed
<p><strong> <br /> M.Sc .Taha Guma el turki , Prof. Al mabrouk Ali sola</strong><strong> </strong><br /><br /><br />There was a beautiful mathematical work done by Kherie Mohamed mera & Prof.Al mabrouk Ali sola .Related to extremal topologies and how to extract the extremal topologies and their numbers by a formula .<strong> </strong></p>
<p><strong>Definition :-</strong></p>
<p>Let X be a set and ,T is not a discreet topology on X then T is said to be an extremal topology if every topology strictly finer than T is discreet.</p>
<p><strong>Theorem 1-2 of [1] :-</strong> <br /><br /> If X is any set with more than one element , x , y ∈ X , x ≠y , and<br />T<sub>{x,y}</sub>= P(X\{x}) U {{x} U A , A ∈P(X\{x}),y ∈ A} ,then T<sub>{x,y}</sub> is <br />an extremal topology on X [1] .<br /><br /><strong>Remark<br /></strong><br />i-Notice that if X is a set x,y ∈ X , x≠ y , then T<sub>{x,y} ≠</sub>T<sub>{y,x}</sub> [1].</p>
<p><strong>Theorem 2-1 of [1] :-</strong></p>
<p> Any extremal topology on a finite set with more than one element is in the form T<sub>{x,y}</sub> for some x,y ∈ X , x ≠y [1] .<br /><br /><strong>Theorem 2-2 of [1] :-<br /><br /></strong>If X is a set has n elements then the number of extremal topologies defined on X is n(n-1) [1].</p>
<p><strong>Theorem 2-3 of [1] :-</strong><br /><br />If X is a set with n elements then any extremal topology has 3(2<sup>n-2</sup>) elements [1] .<br /><br />Also we compute in this application the Non-trivial minimal topologies and there number which is equal to 2<sup>n</sup>-2 ;<br /><br /><strong>Notes </strong>:-<br /><br />1- The Author of the procedures: Taha Guma el turki uses low speed computer with 1.7 GH processor.<br /><br />2- If you use such or lower portable then replace ; by : at the end of procedure calling To compute issues for n>10.<br /><br />3- The users can easily remove #Example(2) and #Example(3) and use the application for arbitrary n depending on their computer options .</p>
<p> <strong>References<br /></strong><br />[1] A lmabrouk Ali Sola , Extremal Topologies ,Damascus University Journal of BASIC SCIENCES,2005,Vol.21,No 1,19-25 . </p><img src="/applications/images/app_image_blank_lg.jpg" alt="The Extremal and Non-Trivial Minimal Topologies Over a Finite Set with Maple" align="left"/><p><strong> <br /> M.Sc .Taha Guma el turki , Prof. Al mabrouk Ali sola</strong><strong> </strong><br /><br /><br />There was a beautiful mathematical work done by Kherie Mohamed mera & Prof.Al mabrouk Ali sola .Related to extremal topologies and how to extract the extremal topologies and their numbers by a formula .<strong> </strong></p>
<p><strong>Definition :-</strong></p>
<p>Let X be a set and ,T is not a discreet topology on X then T is said to be an extremal topology if every topology strictly finer than T is discreet.</p>
<p><strong>Theorem 1-2 of [1] :-</strong> <br /><br /> If X is any set with more than one element , x , y ∈ X , x ≠y , and<br />T<sub>{x,y}</sub>= P(X\{x}) U {{x} U A , A ∈P(X\{x}),y ∈ A} ,then T<sub>{x,y}</sub> is <br />an extremal topology on X [1] .<br /><br /><strong>Remark<br /></strong><br />i-Notice that if X is a set x,y ∈ X , x≠ y , then T<sub>{x,y} ≠</sub>T<sub>{y,x}</sub> [1].</p>
<p><strong>Theorem 2-1 of [1] :-</strong></p>
<p> Any extremal topology on a finite set with more than one element is in the form T<sub>{x,y}</sub> for some x,y ∈ X , x ≠y [1] .<br /><br /><strong>Theorem 2-2 of [1] :-<br /><br /></strong>If X is a set has n elements then the number of extremal topologies defined on X is n(n-1) [1].</p>
<p><strong>Theorem 2-3 of [1] :-</strong><br /><br />If X is a set with n elements then any extremal topology has 3(2<sup>n-2</sup>) elements [1] .<br /><br />Also we compute in this application the Non-trivial minimal topologies and there number which is equal to 2<sup>n</sup>-2 ;<br /><br /><strong>Notes </strong>:-<br /><br />1- The Author of the procedures: Taha Guma el turki uses low speed computer with 1.7 GH processor.<br /><br />2- If you use such or lower portable then replace ; by : at the end of procedure calling To compute issues for n>10.<br /><br />3- The users can easily remove #Example(2) and #Example(3) and use the application for arbitrary n depending on their computer options .</p>
<p> <strong>References<br /></strong><br />[1] A lmabrouk Ali Sola , Extremal Topologies ,Damascus University Journal of BASIC SCIENCES,2005,Vol.21,No 1,19-25 . </p>153617Wed, 02 Jul 2014 04:00:00 ZProf.Al mabrouk Ali solaProf.Al mabrouk Ali solaMaple in Finite Topological Spaces-Connectedness
http://www.maplesoft.com/applications/view.aspx?SID=150631&ref=Feed
<p><strong>Maple in Finite Topological Spaces-Connectedness </strong></p>
<p><strong><br /> </strong>Taha Guma el turki , Kahtan H. Alzubaidy</p>
<p>Department of Mathematics , Faculty of Science ,University of Benghazi</p>
<p> e_mails: <a href="mailto:Taha1978_2002@yahoo.com">Taha1978_2002@yahoo.com</a> , <a href="mailto:Kahtanalzubaidy@yahoo.com">Kahtanalzubaidy@yahoo.com</a> <strong><br /> </strong><strong><br /> </strong><strong>Introduction <br /> <br /> </strong>Connectedness and path connectedness are equivalent on finite topological spaces [2] .However , one definition of a connected space X is that ∅ and X are the only clopen subsets of X . For <em>x</em>∈ X , the connected component C<em><sub>x </sub></em>of <em>x </em> is the largest connected subset of X containing <em>x</em> .{C<sub>x</sub>}<sub>x∈X</sub> is a partition of X and C<sub>x</sub> is clopen for any <em>x</em>∈ X .<br /> We have presented :<br /> i) a new procedure to list all topologies on a finite set .<br /> ii) a procedure to list the connected topologies on a finite set .<br /> iii) a procedure to find the connected components of a finite space .<br /> Maple 15 has been used .</p>
<p><strong>References <br /> <br /> </strong>[1] Dider Deses : Math-Page http : // student.vub.ac.be./~diddesen/math.html (2001) .<strong> </strong></p>
<p><a href="http://www.maplesoft.com/applications/view.aspx?SID=4122&view=html">www.maplesoft.com/applications/view.aspx?SID=4122&view=html</a> (2001).<br /> <br /> [2] J.P. May : Finite Topological Spaces<br /> <a href="http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf">http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf</a> (2008).<br /> <br /> <br /></p><img src="/view.aspx?si=150631/1746821e7396d35a8f4c65fa55f2e8d9.gif" alt="Maple in Finite Topological Spaces-Connectedness" align="left"/><p><strong>Maple in Finite Topological Spaces-Connectedness </strong></p>
<p><strong><br /> </strong>Taha Guma el turki , Kahtan H. Alzubaidy</p>
<p>Department of Mathematics , Faculty of Science ,University of Benghazi</p>
<p> e_mails: <a href="mailto:Taha1978_2002@yahoo.com">Taha1978_2002@yahoo.com</a> , <a href="mailto:Kahtanalzubaidy@yahoo.com">Kahtanalzubaidy@yahoo.com</a> <strong><br /> </strong><strong><br /> </strong><strong>Introduction <br /> <br /> </strong>Connectedness and path connectedness are equivalent on finite topological spaces [2] .However , one definition of a connected space X is that ∅ and X are the only clopen subsets of X . For <em>x</em>∈ X , the connected component C<em><sub>x </sub></em>of <em>x </em> is the largest connected subset of X containing <em>x</em> .{C<sub>x</sub>}<sub>x∈X</sub> is a partition of X and C<sub>x</sub> is clopen for any <em>x</em>∈ X .<br /> We have presented :<br /> i) a new procedure to list all topologies on a finite set .<br /> ii) a procedure to list the connected topologies on a finite set .<br /> iii) a procedure to find the connected components of a finite space .<br /> Maple 15 has been used .</p>
<p><strong>References <br /> <br /> </strong>[1] Dider Deses : Math-Page http : // student.vub.ac.be./~diddesen/math.html (2001) .<strong> </strong></p>
<p><a href="http://www.maplesoft.com/applications/view.aspx?SID=4122&view=html">www.maplesoft.com/applications/view.aspx?SID=4122&view=html</a> (2001).<br /> <br /> [2] J.P. May : Finite Topological Spaces<br /> <a href="http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf">http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf</a> (2008).<br /> <br /> <br /></p>150631Tue, 20 Aug 2013 04:00:00 ZProf. Kahtan H.A lzubaidyProf. Kahtan H.A lzubaidyMaple in Finite Topological Spaces – Special Points
http://www.maplesoft.com/applications/view.aspx?SID=145571&ref=Feed
<p ><strong>Kahtan H.Alzubaidy, Taha Guma Elturki</strong>
<BR>
<em>Department of Mathematics, Faculty of Science, University of Benghazi</em></p>
<p ><em>E-mail: </em><a href="mailto:kahtanalzubaidy@yahoo.com"><em>kahtanalzubaidy@yahoo.com</em></a>, <a href="mailto:taha1978_2002@yahoo.com"><em>taha1978_2002@yahoo.com</em></a><strong>
<p><strong>Introduction</strong></p>
<p>The special points of a set in a topological space are limit points, closure points, interior points, boundary points, exterior points, and isolated points. Except limit points and isolated points the other special points can be computed by implementing simple formulas. For limit points we have to resort to the very definition to find them. On the other hand all special points can be derived from limit points. We have found computer procedures to compute the limit points of a set in finite space. Upon these procedures we have created other procedures to find the other special points. The software used is Maple 15. Some ready-made procedures are also used.</p>
<p>Let X be a finite topological space and A is a sub set of X. If the limit points, closure, interior, boundary, exterior and isolated points of A are denoted by LimitPoints(A), ClosurePoints(A), BoundaryPoints(A), InteriorPoints(A), ExteriorPoints(A) and IsolatedPoints(A) respectively, then we have :</p>
<p>ClosurePoints(A) = A U LimitPoints(A) .</p>
<p>BoundaryPoints(A) = ClosurePoints(A) ∩ ClosurePoints(X−A) .</p>
<p>InteriorPoints(A) = ClosurePoints(A) − BoundaryPoints(A) .</p>
<p>ExteriorPoints(A) = InteriorPoints(X − A) .</p>
<p>IsolatedPoints(A) = A − LimitPoints(A) .</p>
<p><strong>References</strong></p>
<p> [1] Dider Deses : Math-Page <br /> http : // student.vub.ac.be./~diddesen/math.html (2001).<br /> <br /> [2] <a href="http://www.maplesoft.com/applications/view.aspx?SID=4122&view=html">www.maplesoft.com/applications/view.aspx?SID=4122&view=html</a> (2001).</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Maple in Finite Topological Spaces – Special Points" align="left"/><p ><strong>Kahtan H.Alzubaidy, Taha Guma Elturki</strong>
<BR>
<em>Department of Mathematics, Faculty of Science, University of Benghazi</em></p>
<p ><em>E-mail: </em><a href="mailto:kahtanalzubaidy@yahoo.com"><em>kahtanalzubaidy@yahoo.com</em></a>, <a href="mailto:taha1978_2002@yahoo.com"><em>taha1978_2002@yahoo.com</em></a><strong>
<p><strong>Introduction</strong></p>
<p>The special points of a set in a topological space are limit points, closure points, interior points, boundary points, exterior points, and isolated points. Except limit points and isolated points the other special points can be computed by implementing simple formulas. For limit points we have to resort to the very definition to find them. On the other hand all special points can be derived from limit points. We have found computer procedures to compute the limit points of a set in finite space. Upon these procedures we have created other procedures to find the other special points. The software used is Maple 15. Some ready-made procedures are also used.</p>
<p>Let X be a finite topological space and A is a sub set of X. If the limit points, closure, interior, boundary, exterior and isolated points of A are denoted by LimitPoints(A), ClosurePoints(A), BoundaryPoints(A), InteriorPoints(A), ExteriorPoints(A) and IsolatedPoints(A) respectively, then we have :</p>
<p>ClosurePoints(A) = A U LimitPoints(A) .</p>
<p>BoundaryPoints(A) = ClosurePoints(A) ∩ ClosurePoints(X−A) .</p>
<p>InteriorPoints(A) = ClosurePoints(A) − BoundaryPoints(A) .</p>
<p>ExteriorPoints(A) = InteriorPoints(X − A) .</p>
<p>IsolatedPoints(A) = A − LimitPoints(A) .</p>
<p><strong>References</strong></p>
<p> [1] Dider Deses : Math-Page <br /> http : // student.vub.ac.be./~diddesen/math.html (2001).<br /> <br /> [2] <a href="http://www.maplesoft.com/applications/view.aspx?SID=4122&view=html">www.maplesoft.com/applications/view.aspx?SID=4122&view=html</a> (2001).</p>145571Sun, 07 Apr 2013 04:00:00 ZTaha Guma el turkiTaha Guma el turkiFractal Dimension and Space-Filling Curves (with iterated function systems)
http://www.maplesoft.com/applications/view.aspx?SID=4869&ref=Feed
By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.<img src="/view.aspx?si=4869/thumb.png" alt="Fractal Dimension and Space-Filling Curves (with iterated function systems)" align="left"/>By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.4869Fri, 16 Feb 2007 00:00:00 ZProf. Mark MeyersonProf. Mark MeyersonTopology with Maple
http://www.maplesoft.com/applications/view.aspx?SID=4122&ref=Feed
This is a set of procedures to do finite topology in Maple. The procedures were chosen to illustrate basic topological constructions and properties. The only limitation is of course that it can only be done for relatively small finite sets. <img src="/view.aspx?si=4122//applications/images/app_image_blank_lg.jpg" alt="Topology with Maple" align="left"/>This is a set of procedures to do finite topology in Maple. The procedures were chosen to illustrate basic topological constructions and properties. The only limitation is of course that it can only be done for relatively small finite sets. 4122Tue, 21 Aug 2001 13:11:09 ZDidier DesesDidier DesesMOISE - A Topology Package for Maple
http://www.maplesoft.com/applications/view.aspx?SID=3703&ref=Feed
Welcome to the worksheet version of MOISE, a Maple library for doing calculations with simplicial complexes. MOISE started out as just the procedure "homology", which computes the homology of a simplicial complex. The algorithm used by "homology" follows the one given in James Munkres' book "Algebraic Topology". It is an old and straightforward method for computing homology groups, but one that is hard to do by hand. In addition to computing the homology groups of a complex, MOISE also contains a number of functions that operate on simplicial complexes.<img src="/view.aspx?si=3703//applications/images/app_image_blank_lg.jpg" alt="MOISE - A Topology Package for Maple" align="left"/>Welcome to the worksheet version of MOISE, a Maple library for doing calculations with simplicial complexes. MOISE started out as just the procedure "homology", which computes the homology of a simplicial complex. The algorithm used by "homology" follows the one given in James Munkres' book "Algebraic Topology". It is an old and straightforward method for computing homology groups, but one that is hard to do by hand. In addition to computing the homology groups of a complex, MOISE also contains a number of functions that operate on simplicial complexes.3703Tue, 19 Jun 2001 00:00:00 ZR. HicksR. HicksCool knots drawn using Maple
http://www.maplesoft.com/applications/view.aspx?SID=3624&ref=Feed
Knot theory is a mathematical study of knots. It is mainly used to distinguish between different types of knots and links. There are some practical applications of knots that are discussed in more detail later. Looking at the history of Knot theory, before people used to try to twist wires or threads and try to classify different types of knots. Nowadays, this can be done on the computer. The knots shown in this page are all drawn using Maple software<img src="/view.aspx?si=3624/knots.gif" alt="Cool knots drawn using Maple" align="left"/>Knot theory is a mathematical study of knots. It is mainly used to distinguish between different types of knots and links. There are some practical applications of knots that are discussed in more detail later. Looking at the history of Knot theory, before people used to try to twist wires or threads and try to classify different types of knots. Nowadays, this can be done on the computer. The knots shown in this page are all drawn using Maple software3624Mon, 18 Jun 2001 00:00:00 ZRohit ChaudharyRohit Chaudhary