4,625 research outputs found
Modeling the variability of rankings
For better or for worse, rankings of institutions, such as universities,
schools and hospitals, play an important role today in conveying information
about relative performance. They inform policy decisions and budgets, and are
often reported in the media. While overall rankings can vary markedly over
relatively short time periods, it is not unusual to find that the ranks of a
small number of "highly performing" institutions remain fixed, even when the
data on which the rankings are based are extensively revised, and even when a
large number of new institutions are added to the competition. In the present
paper, we endeavor to model this phenomenon. In particular, we interpret as a
random variable the value of the attribute on which the ranking should ideally
be based. More precisely, if items are to be ranked then the true, but
unobserved, attributes are taken to be values of independent and
identically distributed variates. However, each attribute value is observed
only with noise, and via a sample of size roughly equal to , say. These
noisy approximations to the true attributes are the quantities that are
actually ranked. We show that, if the distribution of the true attributes is
light-tailed (e.g., normal or exponential) then the number of institutions
whose ranking is correct, even after recalculation using new data and even
after many new institutions are added, is essentially fixed. Formally, is
taken to be of order for any fixed , and the number of institutions
whose ranking is reliable depends very little on .Comment: Published in at http://dx.doi.org/10.1214/10-AOS794 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Effective Scalar Products for D-finite Symmetric Functions
Many combinatorial generating functions can be expressed as combinations of
symmetric functions, or extracted as sub-series and specializations from such
combinations. Gessel has outlined a large class of symmetric functions for
which the resulting generating functions are D-finite. We extend Gessel's work
by providing algorithms that compute differential equations these generating
functions satisfy in the case they are given as a scalar product of symmetric
functions in Gessel's class. Examples of applications to k-regular graphs and
Young tableaux with repeated entries are given. Asymptotic estimates are a
natural application of our method, which we illustrate on the same model of
Young tableaux. We also derive a seemingly new formula for the Kronecker
product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2:
corrections from original submission, improved clarity; now formatted for
journal + bibliograph
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
Constructing equivariant vector bundles via the BGG correspondence
We describe a strategy for the construction of finitely generated
-equivariant -graded modules over the exterior algebra for a
finite group . By an equivariant version of the BGG correspondence,
defines an object in the bounded derived category of
-equivariant coherent sheaves on projective space. We develop a necessary
condition for being isomorphic to a vector bundle that can be
simply read off from the Hilbert series of . Combining this necessary
condition with the computation of finite excerpts of the cohomology table of
makes it possible to enlist a class of equivariant vector bundles
on that we call strongly determined in the case where is the
alternating group on points
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