146,925 research outputs found
Characteristic varieties and Betti numbers of free abelian covers
The regular \Z^r-covers of a finite cell complex X are parameterized by the
Grassmannian of r-planes in H^1(X,\Q). Moving about this variety, and recording
when the Betti numbers b_1,..., b_i of the corresponding covers are finite
carves out certain subsets \Omega^i_r(X) of the Grassmannian.
We present here a method, essentially going back to Dwyer and Fried, for
computing these sets in terms of the jump loci for homology with coefficients
in rank 1 local systems on X. Using the exponential tangent cones to these jump
loci, we show that each \Omega-invariant is contained in the complement of a
union of Schubert varieties associated to an arrangement of linear subspaces in
H^1(X,\Q).
The theory can be made very explicit in the case when the characteristic
varieties of X are unions of translated tori. But even in this setting, the
\Omega-invariants are not necessarily open, not even when X is a smooth complex
projective variety. As an application, we discuss the geometric finiteness
properties of some classes of groups.Comment: 40 pages, 2 figures; accepted for publication in International
Mathematics Research Notice
Stochastic networks with multiple stable points
This paper analyzes stochastic networks consisting of a set of finite
capacity sites where different classes of individuals move according to some
routing policy. The associated Markov jump processes are analyzed under a
thermodynamic limit regime, that is, when the networks have some symmetry
properties and when the number of nodes goes to infinity. An intriguing
stability property is proved: under some conditions on the parameters, it is
shown that, in the limit, several stable equilibrium points coexist for the
empirical distribution. The key ingredient of the proof of this property is a
dimension reduction achieved by the introduction of two energy functions and a
convenient mapping of their local minima and saddle points. Networks with a
unique equilibrium point are also presented.Comment: Published in at http://dx.doi.org/10.1214/009117907000000105 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Fundamental groups, Alexander invariants, and cohomology jumping loci
We survey the cohomology jumping loci and the Alexander-type invariants
associated to a space, or to its fundamental group. Though most of the material
is expository, we provide new examples and applications, which in turn raise
several questions and conjectures.
The jump loci of a space X come in two basic flavors: the characteristic
varieties, or, the support loci for homology with coefficients in rank 1 local
systems, and the resonance varieties, or, the support loci for the homology of
the cochain complexes arising from multiplication by degree 1 classes in the
cohomology ring of X. The geometry of these varieties is intimately related to
the formality, (quasi-) projectivity, and homological finiteness properties of
\pi_1(X).
We illustrate this approach with various applications to the study of
hyperplane arrangements, Milnor fibrations, 3-manifolds, and right-angled Artin
groups.Comment: 45 pages; accepted for publication in Contemporary Mathematic
Open system dynamics and quantum jumps: Divisibility vs. dissipativity
Several key properties of quantum evolutions are characterized by
divisibility of the corresponding dynamical maps. In particular, a Markovian
evolution respects CP-divisibility, whereas breaking of P-divisibility provides
a clear sign of non-Markovian effects. We analyze a class of evolutions which
interpolates between CP- and P-divisible classes and is characterized by
dissipativity -- a long known but so far not widely used formal concept to
classify open system dynamics. By making a connection to stochastic jump
unravellings of master equations, we demonstrate that there exists inherent
freedom in how to divide the terms of the underlying master equation into the
deterministic and jump parts for the stochastic description. This leads to a
number of different unravelings, each one with a measurement scheme
interpretation and highlighting different properties of the considered open
system dynamics. Starting from formal mathematical concepts, our results allow
us to get fundamental insights in open system dynamics and to ease their
numerical simulations.Comment: 5+7 pages, 1+1 figures, 1 tabl
Notes on the "Ramified" Seiberg-Witten Equations and Invariants
In these notes, we carefully analyze the properties of the "ramified"
Seiberg-Witten equations associated with supersymmetric configurations of the
Seiberg-Witten abelian gauge theory with surface operators on an oriented
closed four-manifold X. We find that in order to have sensible solutions to
these equations, only surface operators with certain parameters and embeddings
in X, are admissible. In addition, the corresponding "ramified" Seiberg-Witten
invariants on X with positive scalar curvature and b^+_2 > 1, vanish, while if
X has b^+_2 = 1, there can be wall-crossings whence the invariants will jump.
In general, for each of the finite number of basic classes that corresponds to
a moduli space of solutions with zero virtual dimension, the perturbed
"ramified" Seiberg-Witten invariants on Kahler manifolds will depend - among
other parameters associated with the surface operator - on the monopole number
"l" and the holonomy parameter "alpha". Nonetheless, the (perturbed) "ramified"
and ordinary invariants are found to coincide, albeit up to a sign, in some
examples.Comment: 21 pages. Published versio
Fine properties of metric space-valued mappings of bounded variation in metric measure spaces
Here we consider two notions of mappings of bounded variation (BV) from the
metric measure space into the metric space; one based on relaxations of
Newton-Sobolev functions, and the other based on a notion of AM-upper
gradients. We show that when the target metric space is a Banach space, these
two notions coincide with comparable energies, but for more general target
metric spaces, the two notions can give different function-classes. We then
consider the fine properties of BV mappings (based on the AM-upper gradient
property), and show that when the target space is a proper metric space, then
for a BV mapping into the target space, co-dimension -almost every point in
the jump set of a BV mapping into the proper space has at least two, and at
most , number of jump values associated with it, and that the preimage of
balls around these jump values have lower density at least at that
point. Here and depend solely on the structural constants
associated with the metric measure space, and jump points are points at which
the map is not approximately continuous
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