9,765 research outputs found
Thatched Partial Orders
A thatch is a type of partial ordering which is most naturally described as a geometric construction. The geometry of these constructions is outlined in this note. A wait-and-see concurrent process may include any number of sub-processes, each regarded as a separate thread of execution. However if at any point the process launches a collection of concurrent threads it must wait and see how they terminate before proceeding. Each sub-process may also launch further sub-processes, which are also subject to the wait and see restriction. A wait-and-see process can be regarded as a thatch. This note describes some interesting mathematical properties of thatching that are relevant to these concurrent processes
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Partial orders on partial isometries
This paper studies three natural pre-orders of increasing generality on the
set of all completely non-unitary partial isometries with equal defect indices.
We show that the problem of determining when one partial isometry is less than
another with respect to these pre-orders is equivalent to the existence of a
bounded (or isometric) multiplier between two natural reproducing kernel
Hilbert spaces of analytic functions. For large classes of partial isometries
these spaces can be realized as the well-known model subspaces and
deBranges-Rovnyak spaces. This characterization is applied to investigate
properties of these pre-orders and the equivalence classes they generate.Comment: 30 pages. To appear in Journal of Operator Theor
On multivariate quantiles under partial orders
This paper focuses on generalizing quantiles from the ordering point of view.
We propose the concept of partial quantiles, which are based on a given partial
order. We establish that partial quantiles are equivariant under
order-preserving transformations of the data, robust to outliers, characterize
the probability distribution if the partial order is sufficiently rich,
generalize the concept of efficient frontier, and can measure dispersion from
the partial order perspective. We also study several statistical aspects of
partial quantiles. We provide estimators, associated rates of convergence, and
asymptotic distributions that hold uniformly over a continuum of quantile
indices. Furthermore, we provide procedures that can restore monotonicity
properties that might have been disturbed by estimation error, establish
computational complexity bounds, and point out a concentration of measure
phenomenon (the latter under independence and the componentwise natural order).
Finally, we illustrate the concepts by discussing several theoretical examples
and simulations. Empirical applications to compare intake nutrients within
diets, to evaluate the performance of investment funds, and to study the impact
of policies on tobacco awareness are also presented to illustrate the concepts
and their use.Comment: Published in at http://dx.doi.org/10.1214/10-AOS863 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Extending Utility Representations of Partial Orders
The problem is considered as to whether a monotone function defined on a
subset P of a Euclidean space can be strictly monotonically extended to the
whole space. It is proved that this is the case if and only if the function is
{\em separably increasing}. Explicit formulas are given for a class of
extensions which involves an arbitrary bounded increasing function. Similar
results are obtained for monotone functions that represent strict partial
orders on arbitrary abstract sets X. The special case where P is a Pareto
subset is considered.Comment: 15 page
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