On the Subgroup Distance Problem

We investigate the computational complexity of finding an element of a permutation group H subset S_n with a minimal distance to a given pi in S_n , for different metrics on S_n . We assume that H is given by a set of generators, such that the problem cannot be solved in polynomial time by exhaustive enumeration. For the case of the Cayley Distance, this problem has been shown to be NP-hard, even if H is abelian of exponent two Pinch, 2006. We present a much simpler proof for this result, which also works for the Hamming Distance, the l\_p distance, Lee's Distance, Kendall's tau, and Ulam's Distance. Moreover, we give an NP-hardness proof for the l\_oo distance using a different reduction idea. Finally, we settle the complexity of the corresponding fixed-parameter and maximization problems

PERBANDINGAN KEMAMPUAN MEMECAHKAN MASALAH STRUKTUR ALJABAR ANTARA MAHASISWA JARAK JAUH DAN MAHASISWA TATAP MUKA

Learning to solve problems is the principal reason for studying mathematics. Teachers play important roles to enhance students’ ability to solve problems in mathematics. While teachers’ presence is common situation in face-to-face education system, it’s hardly the case in distance education system.  The situation often post questions on the quality of distance education system, especially in the field of mathematics.  The aim of this research was to compare students’ form distance education system and face-to-face system in term of the students’ achievement in problem solving.  Data collected through  questionnaires and written test are analyzed quantitatively to compare the algebra structure problem solving competences between the groups of students based on Polya's phase guide for solving problem.  Analysis shows that  the average scores of face-to-face students are slightly higher than distance education students. However, on one out of four competences -how to prove the subgroup- there is no differences in the two groups average score

Weight Distributions, Automorphisms, and Isometries of Cyclic Orbit Codes

Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup Fqn* on an Fq-subspace U of Fqn. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference space in the code. My dissertation investigates the structure of the weight distribution for cyclic orbit codes. We show that for full-length orbit codes with maximal possible distance the weight distribution depends only on q,n and the dimension of U. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the weight distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. We also briefly address the weight distribution of a union of full-length orbit codes with maximum distance. A related problem is to find the automorphism group of a cyclic orbit code, which plays a role in determining the isometry classes of the set of all cyclic orbit codes. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of Fqn. We then generalize to orbits under the normalizer of the Singer subgroup, although in this setup there is a remaining exceptional case. Finally, we can characterize linear isometries between such codes

Stallings graphs for quasi-convex subgroups

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the conjugacy or the almost malnormality problems. Our results extend earlier algorithmic results for the more restricted class of virtually free groups. We also extend our construction to relatively quasi-convex subgroups of relatively hyperbolic groups, under certain additional conditions.Comment: 40 pages. New and improved versio