14,346 research outputs found
Universal Spatiotemporal Sampling Sets for Discrete Spatially Invariant Evolution Systems
Let be a finite abelian group and be a circular
convolution operator on . The problem under consideration is how to
construct minimal and such that is
a frame for , where is the canonical
basis of . This problem is motivated by the spatiotemporal sampling
problem in discrete spatially invariant evolution systems. We will show that
the cardinality of should be at least equal to the largest geometric
multiplicity of eigenvalues of , and we consider the universal
spatiotemporal sampling sets for convolution operators
with eigenvalues subject to the same largest geometric
multiplicity. We will give an algebraic characterization for such sampling sets
and show how this problem is linked with sparse signal processing theory and
polynomial interpolation theory
The universal Glivenko-Cantelli property
Let F be a separable uniformly bounded family of measurable functions on a
standard measurable space, and let N_{[]}(F,\epsilon,\mu) be the smallest
number of \epsilon-brackets in L^1(\mu) needed to cover F. The following are
equivalent:
1. F is a universal Glivenko-Cantelli class.
2. N_{[]}(F,\epsilon,\mu)0 and every probability
measure \mu.
3. F is totally bounded in L^1(\mu) for every probability measure \mu.
4. F does not contain a Boolean \sigma-independent sequence.
It follows that universal Glivenko-Cantelli classes are uniformity classes
for general sequences of almost surely convergent random measures.Comment: 26 page
Aggregation of Votes with Multiple Positions on Each Issue
We consider the problem of aggregating votes cast by a society on a fixed set
of issues, where each member of the society may vote for one of several
positions on each issue, but the combination of votes on the various issues is
restricted to a set of feasible voting patterns. We require the aggregation to
be supportive, i.e. for every issue the corresponding component of
every aggregator on every issue should satisfy . We prove that, in such a set-up, non-dictatorial
aggregation of votes in a society of some size is possible if and only if
either non-dictatorial aggregation is possible in a society of only two members
or a ternary aggregator exists that either on every issue is a majority
operation, i.e. the corresponding component satisfies , or on every issue is a minority operation, i.e.
the corresponding component satisfies We then introduce a notion of uniformly non-dictatorial
aggregator, which is defined to be an aggregator that on every issue, and when
restricted to an arbitrary two-element subset of the votes for that issue,
differs from all projection functions. We first give a characterization of sets
of feasible voting patterns that admit a uniformly non-dictatorial aggregator.
Then making use of Bulatov's dichotomy theorem for conservative constraint
satisfaction problems, we connect social choice theory with combinatorial
complexity by proving that if a set of feasible voting patterns has a
uniformly non-dictatorial aggregator of some arity then the multi-sorted
conservative constraint satisfaction problem on , in the sense introduced by
Bulatov and Jeavons, with each issue representing a sort, is tractable;
otherwise it is NP-complete
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