4,813 research outputs found
Algebraic foundations for qualitative calculi and networks
A qualitative representation is like an ordinary representation of a
relation algebra, but instead of requiring , as
we do for ordinary representations, we only require that , for each in the algebra. A constraint
network is qualitatively satisfiable if its nodes can be mapped to elements of
a qualitative representation, preserving the constraints. If a constraint
network is satisfiable then it is clearly qualitatively satisfiable, but the
converse can fail. However, for a wide range of relation algebras including the
point algebra, the Allen Interval Algebra, RCC8 and many others, a network is
satisfiable if and only if it is qualitatively satisfiable.
Unlike ordinary composition, the weak composition arising from qualitative
representations need not be associative, so we can generalise by considering
network satisfaction problems over non-associative algebras. We prove that
computationally, qualitative representations have many advantages over ordinary
representations: whereas many finite relation algebras have only infinite
representations, every finite qualitatively representable algebra has a finite
qualitative representation; the representability problem for (the atom
structures of) finite non-associative algebras is NP-complete; the network
satisfaction problem over a finite qualitatively representable algebra is
always in NP; the validity of equations over qualitative representations is
co-NP-complete. On the other hand we prove that there is no finite
axiomatisation of the class of qualitatively representable algebras.Comment: 22 page
Bayesian Poisson process partition calculus with an application to Bayesian L\'evy moving averages
This article develops, and describes how to use, results concerning
disintegrations of Poisson random measures. These results are fashioned as
simple tools that can be tailor-made to address inferential questions arising
in a wide range of Bayesian nonparametric and spatial statistical models. The
Poisson disintegration method is based on the formal statement of two results
concerning a Laplace functional change of measure and a Poisson Palm/Fubini
calculus in terms of random partitions of the integers {1,...,n}. The
techniques are analogous to, but much more general than, techniques for the
Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
(1984) 351-357] and [Ann. Inst. Statist. Math. 41 (1989) 227-245]. In order to
illustrate the flexibility of the approach, large classes of random probability
measures and random hazards or intensities which can be expressed as
functionals of Poisson random measures are described. We describe a unified
posterior analysis of classes of discrete random probability which identifies
and exploits features common to all these models. The analysis circumvents many
of the difficult issues involved in Bayesian nonparametric calculus, including
a combinatorial component. This allows one to focus on the unique features of
each process which are characterized via real valued functions h. The
applicability of the technique is further illustrated by obtaining explicit
posterior expressions for L\'evy-Cox moving average processes within the
general setting of multiplicative intensity models.Comment: Published at http://dx.doi.org/10.1214/009053605000000336 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Infinite-body optimal transport with Coulomb Cost
We introduce and analyze symmetric infinite-body optimal transport (OT)
problems with cost function of pair potential form. We show that for a natural
class of such costs, the optimizer is given by the independent product measure
all of whose factors are given by the one-body marginal. This is in striking
contrast to standard finite-body OT problems, in which the optimizers are
typically highly correlated, as well as to infinite-body OT problems with
Gangbo-Swiech cost. Moreover, by adapting a construction from the study of
exchangeable processes in probability theory, we prove that the corresponding
-body OT problem is well approximated by the infinite-body problem.
To our class belongs the Coulomb cost which arises in many-electron quantum
mechanics. The optimal cost of the Coulombic N-body OT problem as a function of
the one-body marginal density is known in the physics and quantum chemistry
literature under the name SCE functional, and arises naturally as the
semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results
imply that in the inhomogeneous high-density limit (i.e. with
arbitrary fixed inhomogeneity profile ), the SCE functional converges
to the mean field functional.
We also present reformulations of the infinite-body and N-body OT problems as
two-body OT problems with representability constraints and give a dual
characterization of representable two-body measures which parallels an
analogous result by Kummer on quantum representability of two-body density
matrices.Comment: 22 pages, significant revision
The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings
Let G be a profinite group which is topologically finitely generated, p a
prime number and d an integer. We show that the functor from rigid analytic
spaces over Q_p to sets, which associates to a rigid space Y the set of
continuous d-dimensional pseudocharacters G -> O(Y), is representable by a
quasi-Stein rigid analytic space X, and we study its general properties. Our
main tool is a theory of "determinants" extending the one of pseudocharacters
but which works over an arbitrary base ring; an independent aim of this paper
is to expose the main facts of this theory. The moduli space X is constructed
as the generic fiber of the moduli formal scheme of continuous formal
determinants on G of dimension d. As an application to number theory, this
provides a framework to study the generic fibers of pseudodeformation rings
(e.g. of Galois representations), especially in the "residually reducible"
case, and including when p <= d.Comment: 56 pages. v2 : final version, to appear in the Proceedings of the LMS
Durham Symposium "Automorphic forms and Galois representations" (2011
Towards a Law of Invariance in Human Concept Learning
Invariance principles underlie many key theories in modern science. They provide the explanatory and predictive framework necessary for the rigorous study of natural phenomena ranging from the structure of crystals, to magnetism, to relativistic mechanics. Vigo (2008, 2009)introduced a new general notion and principle of invariance from which two parameter-free (ratio and exponential) models were derived to account for human conceptual behavior. Here we introduce a new parameterized \ud
exponential “law” based on the same invariance principle. The law accurately predicts the subjective degree of difficulty that humans experience when learning different types of concepts. In addition, it precisely fits the data from a large-scale experiment which examined a total of 84 category structures across 10 category families (R-Squared =.97, p < .0001; r= .98, p < .0001). Moreover, it overcomes seven key challenges that had, hitherto, been grave obstacles for theories of concept learning
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