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Category Archives: Uncategorized
Free Groups and Tiling
Introduction Consider the following simple problem. Prove that the shape on the left cannot be completely tiled by 20 polygons of the types shown on the right. The solution is rather simple: colour the shape in the following manner. This … Continue reading
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Tagged combinatorics, free groups, group theory, groups, polyominoes, tiling, words
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Solving Permutation-Based Puzzles
Introduction In the previous article, we described the Schreier-Sims algorithm. Given a small subset which generates the permutation group G, the algorithm constructs a sequence such that for: we have a small generating set for each Specifically, via the Sims … Continue reading
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Tagged group actions, group theory, permutations, rubik's cube, schreier-sims, symmetric group
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Schreier-Sims Algorithm
Introduction Throughout this article, we let G be a subgroup of generated by a subset We wish to consider the following questions. Given A, how do we compute the order of G? How do we determine if an element lies in G? Assuming , how … Continue reading
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Tagged group actions, group theory, permutations, programming, rubik's cube, schreier-sims, symmetries
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Primality Tests III
Solovay-Strassen Test This is an enhancement of the Euler test. Be forewarned that it is in fact weaker than the Rabin-Miller test so it may not be of much practical interest. Nevertheless, it’s included here for completeness. Recall that to … Continue reading
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Tagged cryptography, elementary, jacobi symbol, legendre symbol, number theory, primality tests, primes, programming
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Primality Tests II
In this article, we discuss some ways of improving the basic Fermat test. Recall that for Fermat test, to test if n is prime, one picks a base a < n and checks if We also saw that this method would utterly fail … Continue reading
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Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
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Primality Tests I
Description of Problem The main problem we wish to discuss is as follows. Question. Given n, how do we determine if it is prime? Prime numbers have opened up huge avenues in theoretical research – the renowned Riemann Hypothesis, for … Continue reading
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Tagged carmichael numbers, cryptography, elementary, number theory, primality tests, primes, pseudoprimes
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Polynomials and Representations XXXIX
Some Invariant Theory We continue the previous discussion. Recall that for we have a -equivariant map which induces an isomorphism between the unique copies of in both spaces. The kernel Q of this map is spanned by for various fillings T with shape and entries … Continue reading
