45,650 research outputs found
A Representation Theorem for Second-Order Functionals
Representation theorems relate seemingly complex objects to concrete, more
tractable ones.
In this paper, we take advantage of the abstraction power of category theory
and provide a general representation theorem for a wide class of second-order
functionals which are polymorphic over a class of functors. Types polymorphic
over a class of functors are easily representable in languages such as Haskell,
but are difficult to analyse and reason about. The concrete representation
provided by the theorem is easier to analyse, but it might not be as convenient
to implement. Therefore, depending on the task at hand, the change of
representation may prove valuable in one direction or the other.
We showcase the usefulness of the representation theorem with a range of
examples. Concretely, we show how the representation theorem can be used to
show that traversable functors are finitary containers, how parameterised
coalgebras relate to very well-behaved lenses, and how algebraic effects might
be implemented in a functional language
A variational method for second order shape derivatives
We consider shape functionals obtained as minima on Sobolev spaces of
classical integrals having smooth and convex densities, under mixed
Dirichlet-Neumann boundary conditions. We propose a new approach for the
computation of the second order shape derivative of such functionals, yielding
a general existence and representation theorem. In particular, we consider the
p-torsional rigidity functional for p grater than or equal to 2.Comment: Submitted paper. 29 page
Second-order properties and central limit theorems for geometric functionals of Boolean models
Let be a Boolean model based on a stationary Poisson process of
compact, convex particles in Euclidean space . Let denote a
compact, convex observation window. For a large class of functionals ,
formulas for mean values of are available in the literature.
The first aim of the present work is to study the asymptotic covariances of
general geometric (additive, translation invariant and locally bounded)
functionals of for increasing observation window , including
convergence rates. Our approach is based on the Fock space representation
associated with . For the important special case of intrinsic volumes,
the asymptotic covariance matrix is shown to be positive definite and can be
explicitly expressed in terms of suitable moments of (local) curvature measures
in the isotropic case. The second aim of the paper is to prove multivariate
central limit theorems including Berry-Esseen bounds. These are based on a
general normal approximation result obtained by the Malliavin--Stein method.Comment: Published at http://dx.doi.org/10.1214/14-AAP1086 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Hecke Modules from Metaplectic Ice
We present a new framework for a broad class of affine Hecke algebra modules,
and show that such modules arise in a number of settings involving
representations of -adic groups and -matrices for quantum groups.
Instances of such modules arise from (possibly non-unique) functionals on
-adic groups and their metaplectic covers, such as the Whittaker
functionals. As a byproduct, we obtain new, algebraic proofs of a number of
results concerning metaplectic Whittaker functions. These are thus expressed in
terms of metaplectic versions of Demazure operators, which are built out of
-matrices of quantum groups depending on the cover degree and associated
root system
Poisson approximation of Rademacher functionals by the Chen-Stein method and Malliavin calculus
New bounds on the total variation distance between the law of integer valued
functionals of possibly non-symmetric and non-homogeneous infinite Rademacher
sequences and the Poisson distribution are established. They are based on a
combination of the Chen-Stein method and a discrete version of Malliavin
calculus. We give some applications to shifted discrete multiple stochastic
integrals.Comment: Comments on version 2: An error in the bound of Theorem 3.4 was
corrected. Corollary 3.9, Remark 3.10 and Remark 3.14 were added to discuss
some concrete applications to Theorem 3.7 and Theorem 3.13 (formerly Theorem
3.11
Functional It\^{o} calculus and stochastic integral representation of martingales
We develop a nonanticipative calculus for functionals of a continuous
semimartingale, using an extension of the Ito formula to path-dependent
functionals which possess certain directional derivatives. The construction is
based on a pathwise derivative, introduced by Dupire, for functionals on the
space of right-continuous functions with left limits. We show that this
functional derivative admits a suitable extension to the space of
square-integrable martingales. This extension defines a weak derivative which
is shown to be the inverse of the Ito integral and which may be viewed as a
nonanticipative "lifting" of the Malliavin derivative. These results lead to a
constructive martingale representation formula for Ito processes. By contrast
with the Clark-Haussmann-Ocone formula, this representation only involves
nonanticipative quantities which may be computed pathwise.Comment: Published in at http://dx.doi.org/10.1214/11-AOP721 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Second order analysis of geometric functionals of Boolean models
This paper presents asymptotic covariance formulae and central limit theorems
for geometric functionals, including volume, surface area, and all Minkowski
functionals and translation invariant Minkowski tensors as prominent examples,
of stationary Boolean models. Special focus is put on the anisotropic case. In
the (anisotropic) example of aligned rectangles, we provide explicit analytic
formulae and compare them with simulation results. We discuss which information
about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their
Applications in Stochastic Geometry and Imaging" in Lecture Notes in
Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second
version mainly resolves minor LaTeX problems.
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