advanced_tools:group_theory
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advanced_tools:group_theory [2018/03/28 16:23] jakobadmin [Intuitive] |
advanced_tools:group_theory [2018/03/28 16:24] jakobadmin |
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| Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$, $270^\circ$, and of course $0^\circ$. | Obvious examples of such transformations are rotations, by $90^\circ$, $180^\circ$, $270^\circ$, and of course $0^\circ$. | ||
| - | {{:advanced_tools:einheitsquadrat-gedreht22-150x150.png?nolink|}} | + | {{ :advanced_tools:einheitsquadrat-gedreht22-150x150.png?nolink|}} |
| A counter-example is a rotation by, say $5^\circ$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^\circ$ is no symmetry of the square. | A counter-example is a rotation by, say $5^\circ$. The upper-right corner point $A$ of the square is obviously mapped to a point $A'$ outside of the initial set. Of course, a square still looks like a square after a rotation by $5^\circ$, but, by definition, this is a different square, mathematically a different set of points. Hence, a rotation by $5^\circ$ is no symmetry of the square. | ||
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| <tabbox Abstract> | <tabbox Abstract> | ||
| - | <blockquote>The case I have chosen to treat in this paper concerns the question as to whether | + | |
| - | the group concept should be extended to, or even subsumed by, the [[advanced_tools:category_theory:groupoids|groupoid concept]]. | + | |
| - | Over a period stretching from at least as long ago as the early nineteenth | + | |
| - | century, the group concept has emerged as the standard way to measure the degree | + | |
| - | of invariance of an object under some collection of transformations.4 The informal | + | |
| - | ideas codified by the group axioms, an axiomatisation which even Lakatos thought | + | |
| - | unlikely to be challenged, relate to the composition of reversible processes | + | |
| - | revealing the symmetry of a mathematical entity. Two early manifestations of | + | |
| - | groups were as the permutations of the roots of a polynomial, later re-interpreted | + | |
| - | as the automorphisms of the algebraic number field containing its roots, in Galois | + | |
| - | theory, and as the structure-preserving automorphisms of a geometric space in the | + | |
| - | Erlanger Programme. Intriguingly, it now appears that there is a challenger on the | + | |
| - | scene. In some situations, it is argued, groupoids are better suited to extracting the | + | |
| - | vital symmetries. And yet there has been a perception among their supporters— | + | |
| - | who include some very illustrious names—of an unwarranted resistance in some | + | |
| - | quarters to their use, which is only now beginning to decline.<cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote> | + | |
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| * [[advanced_tools:group_theory:group_contraction]] | * [[advanced_tools:group_theory:group_contraction]] | ||
| + | <tabbox Research> | ||
| + | <blockquote>The case I have chosen to treat in this paper concerns the question as to whether | ||
| + | the group concept should be extended to, or even subsumed by, the [[advanced_tools:category_theory:groupoids|groupoid concept]]. | ||
| + | Over a period stretching from at least as long ago as the early nineteenth | ||
| + | century, the group concept has emerged as the standard way to measure the degree | ||
| + | of invariance of an object under some collection of transformations.4 The informal | ||
| + | ideas codified by the group axioms, an axiomatisation which even Lakatos thought | ||
| + | unlikely to be challenged, relate to the composition of reversible processes | ||
| + | revealing the symmetry of a mathematical entity. Two early manifestations of | ||
| + | groups were as the permutations of the roots of a polynomial, later re-interpreted | ||
| + | as the automorphisms of the algebraic number field containing its roots, in Galois | ||
| + | theory, and as the structure-preserving automorphisms of a geometric space in the | ||
| + | Erlanger Programme. Intriguingly, it now appears that there is a challenger on the | ||
| + | scene. In some situations, it is argued, groupoids are better suited to extracting the | ||
| + | vital symmetries. And yet there has been a perception among their supporters— | ||
| + | who include some very illustrious names—of an unwarranted resistance in some | ||
| + | quarters to their use, which is only now beginning to decline.<cite>[[http://www.sciencedirect.com/science/article/pii/S0039368101000073|The importance of mathematical conceptualisation]] by David Corfield</cite></blockquote> | ||
| <tabbox FAQ> | <tabbox FAQ> | ||
advanced_tools/group_theory.txt · Last modified: 2020/09/07 05:18 by 14.161.7.200
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