2,086 research outputs found
Recognising the Suzuki groups in their natural representations
Under the assumption of a certain conjecture, for which there exists strong
experimental evidence, we produce an efficient algorithm for constructive
membership testing in the Suzuki groups Sz(q), where q = 2^{2m + 1} for some m
> 0, in their natural representations of degree 4. It is a Las Vegas algorithm
with running time O{log(q)} field operations, and a preprocessing step with
running time O{log(q) loglog(q)} field operations. The latter step needs an
oracle for the discrete logarithm problem in GF(q).
We also produce a recognition algorithm for Sz(q) = . This is a Las Vegas
algorithm with running time O{|X|^2} field operations.
Finally, we give a Las Vegas algorithm that, given ^h = Sz(q) for some h
in GL(4, q), finds some g such that ^g = Sz(q). The running time is O{log(q)
loglog(q) + |X|} field operations.
Implementations of the algorithms are available for the computer system
MAGMA
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Recent advances in algorithmic problems for semigroups
In this article we survey recent progress in the algorithmic theory of matrix
semigroups. The main objective in this area of study is to construct algorithms
that decide various properties of finitely generated subsemigroups of an
infinite group , often represented as a matrix group. Such problems might
not be decidable in general. In fact, they gave rise to some of the earliest
undecidability results in algorithmic theory. However, the situation changes
when the group satisfies additional constraints. In this survey, we give an
overview of the decidability and the complexity of several algorithmic problems
in the cases where is a low-dimensional matrix group, or a group with
additional structures such as commutativity, nilpotency and solvability.Comment: survey article for SIGLOG New
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