16,448 research outputs found
Difference Balanced Functions and Their Generalized Difference Sets
Difference balanced functions from to are closely related
to combinatorial designs and naturally define -ary sequences with the ideal
two-level autocorrelation. In the literature, all existing such functions are
associated with the -homogeneous property, and it was conjectured by Gong
and Song that difference balanced functions must be -homogeneous. First we
characterize difference balanced functions by generalized difference sets with
respect to two exceptional subgroups. We then derive several necessary and
sufficient conditions for -homogeneous difference balanced functions. In
particular, we reveal an unexpected equivalence between the -homogeneous
property and multipliers of generalized difference sets. By determining these
multipliers, we prove the Gong-Song conjecture for prime. Furthermore, we
show that every difference balanced function must be balanced or an affine
shift of a balanced function.Comment: 17 page
Ackermann Encoding, Bisimulations, and OBDDs
We propose an alternative way to represent graphs via OBDDs based on the
observation that a partition of the graph nodes allows sharing among the
employed OBDDs. In the second part of the paper we present a method to compute
at the same time the quotient w.r.t. the maximum bisimulation and the OBDD
representation of a given graph. The proposed computation is based on an
OBDD-rewriting of the notion of Ackermann encoding of hereditarily finite sets
into natural numbers.Comment: To appear on 'Theory and Practice of Logic Programming
Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of
large-scale linear systems that arise from the discretization of elliptic PDEs
amenable to rank compression. The preconditioner is based on hierarchical
low-rank approximations and the cyclic reduction method. The setup and
application phases of the preconditioner achieve log-linear complexity in
memory footprint and number of operations, and numerical experiments exhibit
good weak and strong scalability at large processor counts in a distributed
memory environment. Numerical experiments with linear systems that feature
symmetry and nonsymmetry, definiteness and indefiniteness, constant and
variable coefficients demonstrate the preconditioner applicability and
robustness. Furthermore, it is possible to control the number of iterations via
the accuracy threshold of the hierarchical matrix approximations and their
arithmetic operations, and the tuning of the admissibility condition parameter.
Together, these parameters allow for optimization of the memory requirements
and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics,
Dec 201
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