4,341 research outputs found
Elements in finite classical groups whose powers have large 1-Eigenspaces
We estimate the proportion of several classes of elements in finite classical
groups which are readily recognised algorithmically, and for which some power
has a large fixed point subspace and acts irreducibly on a complement of it.
The estimates are used in complexity analyses of new recognition algorithms for
finite classical groups in arbitrary characteristic
Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs
The chemical kinetics ODEs arising from operator-split reactive-flow
simulations were solved on GPUs using explicit integration algorithms. Nonstiff
chemical kinetics of a hydrogen oxidation mechanism (9 species and 38
irreversible reactions) were computed using the explicit fifth-order
Runge-Kutta-Cash-Karp method, and the GPU-accelerated version performed faster
than single- and six-core CPU versions by factors of 126 and 25, respectively,
for 524,288 ODEs. Moderately stiff kinetics, represented with mechanisms for
hydrogen/carbon-monoxide (13 species and 54 irreversible reactions) and methane
(53 species and 634 irreversible reactions) oxidation, were computed using the
stabilized explicit second-order Runge-Kutta-Chebyshev (RKC) algorithm. The
GPU-based RKC implementation demonstrated an increase in performance of nearly
59 and 10 times, for problem sizes consisting of 262,144 ODEs and larger, than
the single- and six-core CPU-based RKC algorithms using the
hydrogen/carbon-monoxide mechanism. With the methane mechanism, RKC-GPU
performed more than 65 and 11 times faster, for problem sizes consisting of
131,072 ODEs and larger, than the single- and six-core RKC-CPU versions, and up
to 57 times faster than the six-core CPU-based implicit VODE algorithm on
65,536 ODEs. In the presence of more severe stiffness, such as ethylene
oxidation (111 species and 1566 irreversible reactions), RKC-GPU performed more
than 17 times faster than RKC-CPU on six cores for 32,768 ODEs and larger, and
at best 4.5 times faster than VODE on six CPU cores for 65,536 ODEs. With a
larger time step size, RKC-GPU performed at best 2.5 times slower than six-core
VODE for 8192 ODEs and larger. Therefore, the need for developing new
strategies for integrating stiff chemistry on GPUs was discussed.Comment: 27 pages, LaTeX; corrected typos in Appendix equations A.10 and A.1
On the frequency of permutations containing a long cycle
A general explicit upper bound is obtained for the proportion of
elements of order dividing , where for some constant ,
in the finite symmetric group . This is used to find lower bounds for the
conditional probabilities that an element of or contains an
-cycle, given that it satisfies an equation of the form where
. For example, the conditional probability that an element is an
-cycle, given that , is always greater than 2/7, and is greater than
1/2 if does not divide 24. Our results improve estimates of these
conditional probabilities in earlier work of the authors with Beals,
Leedham-Green and Seress, and have applications for analysing black-box
recognition algorithms for the finite symmetric and alternating groups
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