19,958 research outputs found
Repetitive Delone Sets and Quasicrystals
This paper considers the problem of characterizing the simplest discrete
point sets that are aperiodic, using invariants based on topological dynamics.
A Delone set whose patch-counting function N(T), for radius T, is finite for
all T is called repetitive if there is a function M(T) such that every ball of
radius M(T)+T contains a copy of each kind of patch of radius T that occurs in
the set. This is equivalent to the minimality of an associated topological
dynamical system with R^n-action. There is a lower bound for M(T) in terms of
N(T), namely N(T) = O(M(T)^n), but no general upper bound.
The complexity of a repetitive Delone set can be measured by the growth rate
of its repetitivity function M(T). For example, M(T) is bounded if and only if
the set is a crystal. A set is called is linearly repetitive if M(T) = O(T) and
densely repetitive if M(T) = O(N(T))^{1/n}). We show that linearly repetitive
sets and densely repetitive sets have strict uniform patch frequencies, i.e.
the associated topological dynamical system is strictly ergodic. It follows
that such sets are diffractive. In the reverse direction, we construct a
repetitive Delone set in R^n which has
M(T) = O(T(log T)^{2/n}(log log log T)^{4/n}), but does not have uniform
patch frequencies. Aperiodic linearly repetitive sets have many claims to be
the simplest class of aperiodic sets, and we propose considering them as a
notion of "perfectly ordered quasicrystal".Comment: To appear in "Ergodic Theory and Dynamical Systems" vol.23 (2003). 37
pages. Uses packages latexsym, ifthen, cite and files amssym.def, amssym.te
Estimate of the number of one-parameter families of modules over a tame algebra
The problem of classifying modules over a tame algebra A reduces to a block
matrix problem of tame type whose indecomposable canonical matrices are zero-
or one-parameter. Respectively, the set of nonisomorphic indecomposable modules
of dimension at most d divides into a finite number f(d,A) of modules and
one-parameter series of modules.
We prove that the number of m-by-n canonical parametric block matrices with a
given partition into blocks is bounded by 4^s, where s is the number of free
entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page
Estimate of the number of one-parameter families of modules over a tame algebra
The problem of classifying modules over a tame algebra A reduces to a block
matrix problem of tame type whose indecomposable canonical matrices are zero-
or one-parameter. Respectively, the set of nonisomorphic indecomposable modules
of dimension at most d divides into a finite number f(d,A) of modules and
one-parameter series of modules.
We prove that the number of m-by-n canonical parametric block matrices with a
given partition into blocks is bounded by 4^s, where s is the number of free
entries (which is at most mn), and estimate the number f(d,A).Comment: 23 page
Self-repairing Homomorphic Codes for Distributed Storage Systems
Erasure codes provide a storage efficient alternative to replication based
redundancy in (networked) storage systems. They however entail high
communication overhead for maintenance, when some of the encoded fragments are
lost and need to be replenished. Such overheads arise from the fundamental need
to recreate (or keep separately) first a copy of the whole object before any
individual encoded fragment can be generated and replenished. There has been
recently intense interest to explore alternatives, most prominent ones being
regenerating codes (RGC) and hierarchical codes (HC). We propose as an
alternative a new family of codes to improve the maintenance process, which we
call self-repairing codes (SRC), with the following salient features: (a)
encoded fragments can be repaired directly from other subsets of encoded
fragments without having to reconstruct first the original data, ensuring that
(b) a fragment is repaired from a fixed number of encoded fragments, the number
depending only on how many encoded blocks are missing and independent of which
specific blocks are missing. These properties allow for not only low
communication overhead to recreate a missing fragment, but also independent
reconstruction of different missing fragments in parallel, possibly in
different parts of the network. We analyze the static resilience of SRCs with
respect to traditional erasure codes, and observe that SRCs incur marginally
larger storage overhead in order to achieve the aforementioned properties. The
salient SRC properties naturally translate to low communication overheads for
reconstruction of lost fragments, and allow reconstruction with lower latency
by facilitating repairs in parallel. These desirable properties make
self-repairing codes a good and practical candidate for networked distributed
storage systems
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