133 research outputs found
Convergence theorems for some layout measures on random lattice and random geometric graphs
This work deals with convergence theorems and bounds on the
cost of several layout measures for lattice graphs, random
lattice graphs and sparse random geometric graphs. For full
square lattices, we give optimal layouts for the problems
still open. Our convergence theorems can be viewed as an
analogue of the Beardwood, Halton and Hammersley theorem for
the Euclidian TSP on random points in the -dimensional
cube. As the considered layout measures are
non-subadditive, we use percolation theory to obtain our
results on random lattices and random geometric graphs. In
particular, we deal with the subcritical regimes on these
class of graphs.Postprint (published version
Fault Tolerance in Cellular Automata at High Fault Rates
A commonly used model for fault-tolerant computation is that of cellular
automata. The essential difficulty of fault-tolerant computation is present in
the special case of simply remembering a bit in the presence of faults, and
that is the case we treat in this paper. We are concerned with the degree (the
number of neighboring cells on which the state transition function depends)
needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We
consider both the traditional transient fault model (where faults occur
independently in time and space) and a recently introduced combined fault model
which also includes manufacturing faults (which occur independently in space,
but which affect cells for all time). We also consider both a purely
probabilistic fault model (in which the states of cells are perturbed at
exactly the fault rate) and an adversarial model (in which the occurrence of a
fault gives control of the state to an omniscient adversary). We show that
there are cellular automata that can tolerate a fault rate (with
) with degree , even with adversarial combined
faults. The simplest such automata are based on infinite regular trees, but our
results also apply to other structures (such as hyperbolic tessellations) that
contain infinite regular trees. We also obtain a lower bound of
, even with purely probabilistic transient faults only
Progress on the Strong Eshelby's Conjecture and Extremal Structures for the Elastic Moment Tensor
We make progress towards proving the strong Eshelby's conjecture in three
dimensions. We prove that if for a single nonzero uniform loading the strain
inside inclusion is constant and further the eigenvalues of this strain are
either all the same or all distinct, then the inclusion must be of ellipsoidal
shape. As a consequence, we show that for two linearly independent loadings the
strains inside the inclusions are uniform, then the inclusion must be of
ellipsoidal shape. We then use this result to address a problem of determining
the shape of an inclusion when the elastic moment tensor (elastic
polarizability tensor) is extremal. We show that the shape of inclusions, for
which the lower Hashin-Shtrikman bound either on the bulk part or on the shear
part of the elastic moment tensor is attained, is an ellipse in two dimensions
and an ellipsoid in three dimensions
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