164 research outputs found
Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
An alternating sign matrix, or ASM, is a -matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an hypermatrix is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page
Loopy, Hankel, and Combinatorially Skew-Hankel Tournaments
We investigate tournaments with a specified score vector having additional
structure: loopy tournaments in which loops are allowed, Hankel tournaments
which are tournaments symmetric about the Hankel diagonal (the anti-diagonal),
and combinatorially skew-Hankel tournaments which are skew-symmetric about the
Hankel diagonal. In each case, we obtain necessary and sufficient conditions
for existence, algorithms for construction, and switches which allow one to
move from any tournament of its type to any other, always staying within the
defined type
A generalization of Alternating Sign Matrices
In alternating sign matrices the first and last nonzero entry in each row and
column is specified to be +1.
Such matrices always exist. We investigate a generalization by specifying
independently the sign of the first and last nonzero entry in each row and
column to be either a +1 or a -1. We determine necessary and sufficient
conditions for such matrices to exist.Comment: 14 page
Preface
It has been shown by Freris, Graham and Kumar that clocks in distributed networks cannot be synchronized precisely in the presence of asymmetric time delays even in idealized situations. Motivated by that impossibility result, we test under similar settings the performance of some existing clock synchronization protocols and show that the synchronization errors between neighboring nodes can be bounded within an acceptable level of accuracy that is determined by the degree of asymmetry in time delays. After studying the basic case of synchronizing two clocks in the two-way message passing process, we first analyze the directed ring networks, in which neighboring clocks are likely to experience severe asymmetric time delays. We then discuss connected undirected networks with two-way message passing between each pair of adjacent nodes. In the end, we expand the discussions to networks with directed topologies that are strongly connected
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