2,213 research outputs found
Invariable generation and the chebotarev invariant of a finite group
A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si
> for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the
minimal size of an invariable generating set for an arbitrary finite group G.
In response to a question in [KZ] we also bound the size of a randomly chosen
set of elements of G that is likely to generate G invariably. Along the way we
prove that every finite simple group is invariably generated by two elements.Comment: Improved versio
ESTSS at 20 years: "a phoenix gently rising from a lava flow of European trauma"
Roderick J. Ørner, who was President between 1997 and 1999, traces the phoenix-like origins of the European Society for Traumatic Stress Studies (ESTSS) from an informal business meeting called during the 1st European Conference on Traumatic Stress (ECOTS) in 1987 to its emergence into a formally constituted society. He dwells on the challenges of tendering a trauma society within a continent where trauma has been and remains endemic. ESTSS successes are noted along with a number of personal reflections on activities that give rise to concern for the present as well as its future prospects. Denial of survivors' experiences and turning away from survivors' narratives by reframing their experiences to accommodate helpers' theory-driven imperatives are viewed with alarm. Arguments are presented for making human rights, memory, and ethics core elements of a distinctive European psycho traumatology, which will secure current ESTSS viability and future integrity
Learning what matters - Sampling interesting patterns
In the field of exploratory data mining, local structure in data can be
described by patterns and discovered by mining algorithms. Although many
solutions have been proposed to address the redundancy problems in pattern
mining, most of them either provide succinct pattern sets or take the interests
of the user into account-but not both. Consequently, the analyst has to invest
substantial effort in identifying those patterns that are relevant to her
specific interests and goals. To address this problem, we propose a novel
approach that combines pattern sampling with interactive data mining. In
particular, we introduce the LetSIP algorithm, which builds upon recent
advances in 1) weighted sampling in SAT and 2) learning to rank in interactive
pattern mining. Specifically, it exploits user feedback to directly learn the
parameters of the sampling distribution that represents the user's interests.
We compare the performance of the proposed algorithm to the state-of-the-art in
interactive pattern mining by emulating the interests of a user. The resulting
system allows efficient and interleaved learning and sampling, thus
user-specific anytime data exploration. Finally, LetSIP demonstrates favourable
trade-offs concerning both quality-diversity and exploitation-exploration when
compared to existing methods.Comment: PAKDD 2017, extended versio
Generalized Shortest Path Kernel on Graphs
We consider the problem of classifying graphs using graph kernels. We define
a new graph kernel, called the generalized shortest path kernel, based on the
number and length of shortest paths between nodes. For our example
classification problem, we consider the task of classifying random graphs from
two well-known families, by the number of clusters they contain. We verify
empirically that the generalized shortest path kernel outperforms the original
shortest path kernel on a number of datasets. We give a theoretical analysis
for explaining our experimental results. In particular, we estimate
distributions of the expected feature vectors for the shortest path kernel and
the generalized shortest path kernel, and we show some evidence explaining why
our graph kernel outperforms the shortest path kernel for our graph
classification problem.Comment: Short version presented at Discovery Science 2015 in Banf
On the length and depth of finite groups
An unrefinable chain of a finite group is a chain of subgroups = 0> 1>⋯> =1 , where each is a maximal subgroup of −1 . The length (respectively, depth) of is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth 3. Here, we go much further by determining the finite groups of depth 3 and 4. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most 9, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a number‐theoretic result of Heath‐Brown, which implies that there are infinitely many non‐abelian simple groups of length at most 9. Finally, we study the chain difference of (namely the length minus the depth). We obtain results on groups with chain differences 1 and 2, including a complete classification of the simple groups with chain difference 2, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of / ( ) in terms of the chain difference of , where ( ) is the soluble radical of
McKay graphs for alternating and classical groups
Let G be a finite group, andαa nontrivial character of G. The McKay graph M (G,α) has the irreducible characters of Gas vertices, with an edge fromχ1toχ2ifχ2is a constituent ofαχ1. We study the diameters of McKay graphs for finite simple groups G. For alternating groups G=An, we prove a conjecture made in [20]: there is an absolute constant C such that diam M (G,α)≤ C log | G| log α (1)for all nontrivial irreducible characters α of G. Also for classical groups of symplectic or orthogonal type of rank r, we establish a linear upper bound Cr on the diameters of all nontrivial McKay graphs. Finally, we provide some sufficient conditions for a productχ1χ2···χlof irreducible characters of some finite simple groups G to contain all irreducible characters of G as constituents
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