7,365 research outputs found
Cubic complexes and finite type invariants
Cubic complexes appear in the theory of finite type invariants so often that
one can ascribe them to basic notions of the theory. In this paper we begin the
exposition of finite type invariants from the `cubic' point of view. Finite
type invariants of knots and homology 3-spheres fit perfectly into this
conception. In particular, we get a natural explanation why they behave like
polynomials.Comment: Published by Geometry and Topology Monographs at
http://www.maths.warwick.ac.uk/gt/GTMon4/paper14.abs.htm
13/2 ways of counting curves
In the past 20 years, compactifications of the families of curves in
algebraic varieties X have been studied via stable maps, Hilbert schemes,
stable pairs, unramified maps, and stable quotients. Each path leads to a
different enumeration of curves. A common thread is the use of a 2-term
deformation/obstruction theory to define a virtual fundamental class. The
richest geometry occurs when X is a nonsingular projective variety of dimension
3.
We survey here the 13/2 principal ways to count curves with special attention
to the 3-fold case. The different theories are linked by a web of conjectural
relationships which we highlight. Our goal is to provide a guide for graduate
students looking for an elementary route into the subject.Comment: Typo fixed, In "Moduli spaces", LMS Lecture Note Series, 411 (2014),
282-333. Cambridge University Pres
Petri nets for systems and synthetic biology
We give a description of a Petri net-based framework for
modelling and analysing biochemical pathways, which uni¯es the qualita-
tive, stochastic and continuous paradigms. Each perspective adds its con-
tribution to the understanding of the system, thus the three approaches
do not compete, but complement each other. We illustrate our approach
by applying it to an extended model of the three stage cascade, which
forms the core of the ERK signal transduction pathway. Consequently
our focus is on transient behaviour analysis. We demonstrate how quali-
tative descriptions are abstractions over stochastic or continuous descrip-
tions, and show that the stochastic and continuous models approximate
each other. Although our framework is based on Petri nets, it can be
applied more widely to other formalisms which are used to model and
analyse biochemical networks
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