71,861 research outputs found
Quillen cohomology of enriched operads
A modern insight due to Quillen, which is further developed by Lurie, asserts
that many cohomology theories of interest are particular cases of a single
construction, which allows one to define cohomology groups in an abstract
setting using only intrinsic properties of the category (or -category)
at hand. This universal cohomology theory is known as Quillen cohomology. In
any setting, Quillen cohomology of a given object is classified by its
cotangent complex. The main purpose of this paper is to study Quillen
cohomology of enriched operads, when working in the model categorical
framework. Our main result provides an explicit formula for computing Quillen
cohomology of enriched operads, based on a procedure of taking certain
infinitesimal models of their cotangent complexes. There is a natural
construction of twisted arrow -category of a simplicial operad, which
extends the notion of twisted arrow -category of an -category
introduced by Lurie. We assert that the cotangent complex of a simplicial
operad can be represented as a spectrum valued functor on its twisted arrow
-category.Comment: 70 pages, substantial modifications, section 8 has been remove
Design of State-based Schedulers for a Network of Control Loops
For a closed-loop system, which has a contention-based multiple access
network on its sensor link, the Medium Access Controller (MAC) may discard some
packets when the traffic on the link is high. We use a local state-based
scheduler to select a few critical data packets to send to the MAC. In this
paper, we analyze the impact of such a scheduler on the closed-loop system in
the presence of traffic, and show that there is a dual effect with state-based
scheduling. In general, this makes the optimal scheduler and controller hard to
find. However, by removing past controls from the scheduling criterion, we find
that certainty equivalence holds. This condition is related to the classical
result of Bar-Shalom and Tse, and it leads to the design of a scheduler with a
certainty equivalent controller. This design, however, does not result in an
equivalent system to the original problem, in the sense of Witsenhausen.
Computing the estimate is difficult, but can be simplified by introducing a
symmetry constraint on the scheduler. Based on these findings, we propose a
dual predictor architecture for the closed-loop system, which ensures
separation between scheduler, observer and controller. We present an example of
this architecture, which illustrates a network-aware event-triggering
mechanism.Comment: 17 pages, technical repor
The ghosts of forgotten things: A study on size after forgetting
Forgetting is removing variables from a logical formula while preserving the
constraints on the other variables. In spite of being a form of reduction, it
does not always decrease the size of the formula and may sometimes increase it.
This article discusses the implications of such an increase and analyzes the
computational properties of the phenomenon. Given a propositional Horn formula,
a set of variables and a maximum allowed size, deciding whether forgetting the
variables from the formula can be expressed in that size is -hard in
. The same problem for unrestricted propositional formulae is
-hard in . The hardness results employ superredundancy: a
superirredundant clause is in all formulae of minimal size equivalent to a
given one. This concept may be useful outside forgetting
Curve-counting invariants for crepant resolutions
We construct curve counting invariants for a Calabi-Yau threefold
equipped with a dominant birational morphism . Our invariants
generalize the stable pair invariants of Pandharipande and Thomas which occur
for the case when is the identity. Our main result is a PT/DT-type
formula relating the partition function of our invariants to the
Donaldson-Thomas partition function in the case when is a crepant
resolution of , the coarse space of a Calabi-Yau orbifold
satisfying the hard Lefschetz condition. In this case, our partition function
is equal to the Pandharipande-Thomas partition function of the orbifold
. Our methods include defining a new notion of stability for
sheaves which depends on the morphism . Our notion generalizes slope
stability which is recovered in the case where is the identity on .
Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence
via the Hall algebra and Joyce's integration map.Comment: In this version, Jim Bryan has been added as an author and the
required boundedness result for our stability condition has been added. arXiv
admin note: text overlap with arXiv:1002.4374 by other author
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