6,194 research outputs found
Construction of scaling partitions of unity
Partitions of unity in formed by (matrix) scales of a fixed
function appear in many parts of harmonic analysis, e.g., wavelet analysis and
the analysis of Triebel-Lizorkin spaces. We give a simple characterization of
the functions and matrices yielding such a partition of unity. For invertible
expanding matrices, the characterization leads to easy ways of constructing
appropriate functions with attractive properties like high regularity and small
support. We also discuss a class of integral transforms that map functions
having the partition of unity property to functions with the same property. The
one-dimensional version of the transform allows a direct definition of a class
of nonuniform splines with properties that are parallel to those of the
classical B-splines. The results are illustrated with the construction of dual
pairs of wavelet frames
Causes of Ineradicable Spurious Predictions in Qualitative Simulation
It was recently proved that a sound and complete qualitative simulator does
not exist, that is, as long as the input-output vocabulary of the
state-of-the-art QSIM algorithm is used, there will always be input models
which cause any simulator with a coverage guarantee to make spurious
predictions in its output. In this paper, we examine whether a meaningfully
expressive restriction of this vocabulary is possible so that one can build a
simulator with both the soundness and completeness properties. We prove several
negative results: All sound qualitative simulators, employing subsets of the
QSIM representation which retain the operating region transition feature, and
support at least the addition and constancy constraints, are shown to be
inherently incomplete. Even when the simulations are restricted to run in a
single operating region, a constraint vocabulary containing just the addition,
constancy, derivative, and multiplication relations makes the construction of
sound and complete qualitative simulators impossible
Finite state verifiers with constant randomness
We give a new characterization of as the class of languages
whose members have certificates that can be verified with small error in
polynomial time by finite state machines that use a constant number of random
bits, as opposed to its conventional description in terms of deterministic
logarithmic-space verifiers. It turns out that allowing two-way interaction
with the prover does not change the class of verifiable languages, and that no
polynomially bounded amount of randomness is useful for constant-memory
computers when used as language recognizers, or public-coin verifiers. A
corollary of our main result is that the class of outcome problems
corresponding to O(log n)-space bounded games of incomplete information where
the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio
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