Group Theory: New Applications
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en-us2017 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemMon, 27 Mar 2017 12:31:51 GMTMon, 27 Mar 2017 12:31:51 GMTNew applications in the Group Theory categoryhttp://www.mapleprimes.com/images/mapleapps.gifGroup Theory: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=142
Tips 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="/view.aspx?si=153852/thumb.jpg" alt="Tips and Techniques: Working with Finitely Presented Groups in Maple" 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.153852Tue, 25 Aug 2015 04:00:00 ZMaplesoftMaplesoftSymmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE
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<p>Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal ﬁlm corrugation [1,2].</p>
<p>The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field.</p><img src="/view.aspx?si=151383/440fb9a2994e797b26c18564d860131b.gif" alt="Symmetry of two-dimensional hybrid metal-dielectric photonic crystal within MAPLE" align="left"/><p>Hybrid structures were made by assembling monolayers (MLs) of closely packed colloidal microspheres on a metal-coated glass substrate . In fact, this architecture is one of several realizations of hybrid plasmonic-photonic crystals (PHs), which differ in photonic crystals dimensionality and metal ﬁlm corrugation [1,2].</p>
<p>The main challenge to us were exploring of those properties of structures which are caused by their space symmetry. In particular, it was necessary to establish the so-called "rules of selection", i.e. the list of the allowed transitions between electronic states of different symmetry and energy that can be induced by light of varying polarization. Additional interest for us was to demonstrate the possibilities of MAPLE within this specific field.</p>151383Thu, 05 Sep 2013 04:00:00 ZOlga V. DvornikOlga V. DvornikVisualizing a Parallel Field in a Curved Manifold
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<p>My PhD thesis was in relativistic cosmology, a study that took me into differential geometry, continuous group theory, and tensor calculus. One of the most difficult concepts in all this was the notion of parallel transport of a vector from one tangent space to another. Of course, the image I had in my head was a basketball for a manifold, and a vector in a tangent plane on this (unit) sphere. The manifold sat in an enveloping R<sup>3</sup>, and I struggled mightily to visualize the difference between the transported field appearing parallel to the surface observer and Euclidean parallelism as seen by an external observer. The Kantian imperative is true - it's natural to imagine the vectors in R<sup>3</sup>, but devilishly difficult to visualize the difference between Euclidean parallelism and parallel transport in an intrinsically curved space.</p><img src="/view.aspx?si=35113/thumb.jpg" alt="Visualizing a Parallel Field in a Curved Manifold" align="left"/><p>My PhD thesis was in relativistic cosmology, a study that took me into differential geometry, continuous group theory, and tensor calculus. One of the most difficult concepts in all this was the notion of parallel transport of a vector from one tangent space to another. Of course, the image I had in my head was a basketball for a manifold, and a vector in a tangent plane on this (unit) sphere. The manifold sat in an enveloping R<sup>3</sup>, and I struggled mightily to visualize the difference between the transported field appearing parallel to the surface observer and Euclidean parallelism as seen by an external observer. The Kantian imperative is true - it's natural to imagine the vectors in R<sup>3</sup>, but devilishly difficult to visualize the difference between Euclidean parallelism and parallel transport in an intrinsically curved space.</p>35113Thu, 28 Jan 2010 05:00:00 ZDr. Robert LopezDr. Robert LopezTinygroups
http://www.maplesoft.com/applications/view.aspx?SID=6566&ref=Feed
The tinygroups package can be used to access basic information about small algebraic groups in Maple. It is a Maple version of the basic information of the "smallgroups" package for the GAP computer algebra system. Whereas the "smallgroups" package provides for groups up to size 2000, the tinygroups package only provides for groups up to size 60.
To install the package, download the file "tinygroups.m" into your Maple library directory.<img src="/view.aspx?si=6566//applications/images/app_image_blank_lg.jpg" alt="Tinygroups" align="left"/>The tinygroups package can be used to access basic information about small algebraic groups in Maple. It is a Maple version of the basic information of the "smallgroups" package for the GAP computer algebra system. Whereas the "smallgroups" package provides for groups up to size 2000, the tinygroups package only provides for groups up to size 60.
To install the package, download the file "tinygroups.m" into your Maple library directory.6566Fri, 22 Aug 2008 00:00:00 ZDavid PritchardDavid PritchardInterface to the QaoS databases of algebraic objects
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<p>The Maple package QaoS contains procedures for querying the QaoS databases for algebraic objects in Berlin. Currently a database of all transitive groups up to degree 30 and a database of more than a million number fields of degree up to 9 are available. There are procedures for accessing invariants of the query results. The results of a query can then be used in Maple.</p><img src="/view.aspx?si=1668/Command line.JPG" alt="Interface to the QaoS databases of algebraic objects" align="left"/><p>The Maple package QaoS contains procedures for querying the QaoS databases for algebraic objects in Berlin. Currently a database of all transitive groups up to degree 30 and a database of more than a million number fields of degree up to 9 are available. There are procedures for accessing invariants of the query results. The results of a query can then be used in Maple.</p>1668Thu, 29 Sep 2005 00:00:00 ZSebastian PauliSebastian PauliThe Application of Quasigroup Fields in Designing Efficient Hash Functions
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Two eminent specialists on quasigroups, Dénes and Keedwell, onced augured the advent of a new era in cryptology, consisting in the application of non-associative algebraic systems. Nevertheless, at present, very few researchers use these tools and in many cases it seems unreasonable and reckless. For example, constructing one-way functions, algorithms are usually designed, in which computations are performed using regular algebraic systems as groups, rings and fields, simple boolean operations, modular arithmetic, and cyclic permutations. Such an approach may simplify cryptanalysis. However, computations of the value of cryptographic one-way function should be easy, but taking into account the security, an algorithm describing the hash function should involve rather an algebraic system, which is strongly recalcitrant, and the behaviour of which is unforeseeable. A quasigroup field, as an algebraic system, has such favorable properties. Thus quasigroup fields can be easily applied in designing both unkeyed and keyed hash functions, iterated or not-iterated as well, that is why in this contribution it will be shown how to do it.<img src="/view.aspx?si=4435/1237.jpg" alt="The Application of Quasigroup Fields in Designing Efficient Hash Functions" align="left"/>Two eminent specialists on quasigroups, Dénes and Keedwell, onced augured the advent of a new era in cryptology, consisting in the application of non-associative algebraic systems. Nevertheless, at present, very few researchers use these tools and in many cases it seems unreasonable and reckless. For example, constructing one-way functions, algorithms are usually designed, in which computations are performed using regular algebraic systems as groups, rings and fields, simple boolean operations, modular arithmetic, and cyclic permutations. Such an approach may simplify cryptanalysis. However, computations of the value of cryptographic one-way function should be easy, but taking into account the security, an algorithm describing the hash function should involve rather an algebraic system, which is strongly recalcitrant, and the behaviour of which is unforeseeable. A quasigroup field, as an algebraic system, has such favorable properties. Thus quasigroup fields can be easily applied in designing both unkeyed and keyed hash functions, iterated or not-iterated as well, that is why in this contribution it will be shown how to do it.4435Tue, 04 Nov 2003 14:27:14 ZProf. Czeslaw KoscielnyProf. Czeslaw KoscielnySubgroup lattice plotting in 3-D
http://www.maplesoft.com/applications/view.aspx?SID=3907&ref=Feed
This routine is intended to show how one can program one's own plotting routines using the PLOT3D structure in Maple. The example we have chosen is to plot a subgroup lattice in 3-dimensions. But this code can serve as an example for how to plot an arbitrary graph G = V,E
<img src="/view.aspx?si=3907//applications/images/app_image_blank_lg.jpg" alt="Subgroup lattice plotting in 3-D" align="left"/>This routine is intended to show how one can program one's own plotting routines using the PLOT3D structure in Maple. The example we have chosen is to plot a subgroup lattice in 3-dimensions. But this code can serve as an example for how to plot an arbitrary graph G = V,E
3907Wed, 20 Jun 2001 00:00:00 ZMichael MonaganMichael MonaganElliptic group procedures
http://www.maplesoft.com/applications/view.aspx?SID=3541&ref=Feed
This worksheet covers the procedures for computing the order of certain EC groups, and of points in such groups. It also covers the Hellman-Pohlig-Silver discrete logarithm algorithm for elliptic curve groups with an example. <img src="/view.aspx?si=3541//applications/images/app_image_blank_lg.jpg" alt="Elliptic group procedures " align="left"/>This worksheet covers the procedures for computing the order of certain EC groups, and of points in such groups. It also covers the Hellman-Pohlig-Silver discrete logarithm algorithm for elliptic curve groups with an example. 3541Mon, 18 Jun 2001 00:00:00 ZMarion SheepersMarion SheepersGroup theory via Rubiks Cube
http://www.maplesoft.com/applications/view.aspx?SID=3449&ref=Feed
An application of Maple's group package to answering various questions about the small Rubik's cube. Maple has a group theory package that can be used to generate and manipulate finite groups. Do you remember the pocket Rubik's cube? If you don't, it's a Rubik's cube with only two smaller cubes on each edge. We can represent the permutation group generated by the three cube twist on the coloured stickers as follows. Maple uses a permutation notation to represent groups. Each list of lists on the next command line is a permutation, with each sublist a cycle. The group is generated by these 3 permutations on 21 elements.
<img src="/view.aspx?si=3449//applications/images/app_image_blank_lg.jpg" alt="Group theory via Rubiks Cube" align="left"/>An application of Maple's group package to answering various questions about the small Rubik's cube. Maple has a group theory package that can be used to generate and manipulate finite groups. Do you remember the pocket Rubik's cube? If you don't, it's a Rubik's cube with only two smaller cubes on each edge. We can represent the permutation group generated by the three cube twist on the coloured stickers as follows. Maple uses a permutation notation to represent groups. Each list of lists on the next command line is a permutation, with each sublist a cycle. The group is generated by these 3 permutations on 21 elements.
3449Mon, 18 Jun 2001 00:00:00 ZMaplesoftMaplesoft