49,559 research outputs found
A Localization Algorithm for -modules
We present a method to compute the holonomic extension of a -module from a
Zariski open set in affine space to the whole space. A particular application
is the localization of coherent -modules which are holonomic on the
complement of an affine variety.Comment: 9 pages, amslate
Computing the support of local cohomology modules
For a polynomial ring , we present a method to compute the
characteristic cycle of the localization for any nonzero polynomial that avoids a direct computation of as a -module. Based on this
approach, we develop an algorithm for computing the characteristic cycle of the
local cohomology modules for any ideal using the
\v{C}ech complex. The algorithm, in particular, is useful for answering
questions regarding vanishing of local cohomology modules and computing
Lyubeznik numbers. These applications are illustrated by examples of
computations using our implementation of the algorithm in Macaulay~2.Comment: 15 page
Doctor of Philosophy
dissertationThis dissertation develops the structure theory of the category Whittaker modules for a complex semisimple Lie algebra. We establish a character theory that distinguishes isomorphism classes of Whittaker modules in the Grothendieck group of the category, then use the localization functor of Beilinson and Bernstein to realize Whittaker modules geometrically as certain twisted D-modules on the associated flag variety (so called "twisted Harish-Chandra sheaves"). The main result of this document is an algorithm for computing the multiplicities of irreducible Whittaker modules in the composition series of standard Whittaker modules, which are generalizations of Verma modules. This algorithm establishes that the multiplicities are determined by a collection of polynomials we refer to as Whittaker Kazhdan--Lusztig polynomials
An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
In this paper we develop an axiomatic setup for algorithmic homological
algebra of Abelian categories. This is done by exhibiting all existential
quantifiers entering the definition of an Abelian category, which for the sake
of computability need to be turned into constructive ones. We do this
explicitly for the often-studied example Abelian category of finitely presented
modules over a so-called computable ring , i.e., a ring with an explicit
algorithm to solve one-sided (in)homogeneous linear systems over . For a
finitely generated maximal ideal in a commutative ring we
show how solving (in)homogeneous linear systems over can be
reduced to solving associated systems over . Hence, the computability of
implies that of . As a corollary we obtain the computability
of the category of finitely presented -modules as an Abelian
category, without the need of a Mora-like algorithm. The reduction also yields,
as a by-product, a complexity estimation for the ideal membership problem over
local polynomial rings. Finally, in the case of localized polynomial rings we
demonstrate the computational advantage of our homologically motivated
alternative approach in comparison to an existing implementation of Mora's
algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu
Towards Full Automated Drive in Urban Environments: A Demonstration in GoMentum Station, California
Each year, millions of motor vehicle traffic accidents all over the world
cause a large number of fatalities, injuries and significant material loss.
Automated Driving (AD) has potential to drastically reduce such accidents. In
this work, we focus on the technical challenges that arise from AD in urban
environments. We present the overall architecture of an AD system and describe
in detail the perception and planning modules. The AD system, built on a
modified Acura RLX, was demonstrated in a course in GoMentum Station in
California. We demonstrated autonomous handling of 4 scenarios: traffic lights,
cross-traffic at intersections, construction zones and pedestrians. The AD
vehicle displayed safe behavior and performed consistently in repeated
demonstrations with slight variations in conditions. Overall, we completed 44
runs, encompassing 110km of automated driving with only 3 cases where the
driver intervened the control of the vehicle, mostly due to error in GPS
positioning. Our demonstration showed that robust and consistent behavior in
urban scenarios is possible, yet more investigation is necessary for full scale
roll-out on public roads.Comment: Accepted to Intelligent Vehicles Conference (IV 2017
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