5,520 research outputs found
Finiteness in derived categories of local rings
New homotopy invariant finiteness conditions on modules over commutative
rings are introduced, and their properties are studied systematically. A number
of finiteness results for classical homological invariants like flat dimension,
injective dimension, and Gorenstein dimension, are established. It is proved
that these specialize to give results concerning modules over complete
intersection local rings. A noteworthy feature is the use of techniques based
on thick subcategories of derived categories.Comment: 40 pages. Minor revisions. To appear in Commentarii Math. Helvetic
Some recent results on Anosov representations
In this note we give an overview of some of our recent work on Anosov
representations of discrete groups into higher rank semisimple Lie groups.Comment: 16 page
The orbit rigidity matrix of a symmetric framework
A number of recent papers have studied when symmetry causes frameworks on a
graph to become infinitesimally flexible, or stressed, and when it has no
impact. A number of other recent papers have studied special classes of
frameworks on generically rigid graphs which are finite mechanisms. Here we
introduce a new tool, the orbit matrix, which connects these two areas and
provides a matrix representation for fully symmetric infinitesimal flexes, and
fully symmetric stresses of symmetric frameworks. The orbit matrix is a true
analog of the standard rigidity matrix for general frameworks, and its analysis
gives important insights into questions about the flexibility and rigidity of
classes of symmetric frameworks, in all dimensions.
With this narrower focus on fully symmetric infinitesimal motions, comes the
power to predict symmetry-preserving finite mechanisms - giving a simplified
analysis which covers a wide range of the known mechanisms, and generalizes the
classes of known mechanisms. This initial exploration of the properties of the
orbit matrix also opens up a number of new questions and possible extensions of
the previous results, including transfer of symmetry based results from
Euclidean space to spherical, hyperbolic, and some other metrics with shared
symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure
Liaison classes of modules
We propose a concept of module liaison that extends Gorenstein liaison of
ideals and provides an equivalence relation among unmixed modules over a
commutative Gorenstein ring. Analyzing the resulting equivalence classes we
show that several results known for Gorenstein liaison are still true in the
more general case of module liaison. In particular, we construct two maps from
the set of even liaison classes of modules of fixed codimension into stable
equivalence classes of certain reflexive modules. As a consequence, we show
that the intermediate cohomology modules and properties like being perfect,
Cohen-Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module
liaison classes. Furthermore, we prove that the module liaison class of a
complete intersection of codimension one consists of precisely all perfect
modules of codimension one
Group actions on 1-manifolds: a list of very concrete open questions
This text focuses on actions on 1-manifolds. We present a (non exhaustive)
list of very concrete open questions in the field, each of which is discussed
in some detail and complemented with a large list of references, so that a
clear panorama on the subject arises from the lecture.Comment: 21 pages, 2 figure
Projective simulation with generalization
The ability to generalize is an important feature of any intelligent agent.
Not only because it may allow the agent to cope with large amounts of data, but
also because in some environments, an agent with no generalization capabilities
cannot learn. In this work we outline several criteria for generalization, and
present a dynamic and autonomous machinery that enables projective simulation
agents to meaningfully generalize. Projective simulation, a novel, physical
approach to artificial intelligence, was recently shown to perform well in
standard reinforcement learning problems, with applications in advanced
robotics as well as quantum experiments. Both the basic projective simulation
model and the presented generalization machinery are based on very simple
principles. This allows us to provide a full analytical analysis of the agent's
performance and to illustrate the benefit the agent gains by generalizing.
Specifically, we show that already in basic (but extreme) environments,
learning without generalization may be impossible, and demonstrate how the
presented generalization machinery enables the projective simulation agent to
learn.Comment: 14 pages, 9 figure
Geometric auxetics
We formulate a mathematical theory of auxetic behavior based on one-parameter
deformations of periodic frameworks. Our approach is purely geometric, relies
on the evolution of the periodicity lattice and works in any dimension. We
demonstrate its usefulness by predicting or recognizing, without experiment,
computer simulations or numerical approximations, the auxetic capabilities of
several well-known structures available in the literature. We propose new
principles of auxetic design and rely on the stronger notion of expansive
behavior to provide an infinite supply of planar auxetic mechanisms and several
new three-dimensional structures
- …