6,286 research outputs found
Counting Solutions to Binomial Complete Intersections
We study the problem of counting the total number of affine solutions of a
system of n binomials in n variables over an algebraically closed field of
characteristic zero. We show that we may decide in polynomial time if that
number is finite. We give a combinatorial formula for computing the total
number of affine solutions (with or without multiplicity) from which we deduce
that this counting problem is #P-complete. We discuss special cases in which
this formula may be computed in polynomial time; in particular, this is true
for generic exponent vectors.Comment: Several minor improvements. Final version to appear in the J. of
Complexit
Higher Dimensional Transition Systems
We introduce the notion of higher dimensional transition systems as a model of concurrency providing an elementary, set-theoretic formalisation of the idea of higher dimensional transition. We show an embedding of the category of higher dimensional transition systems into that of higher dimensional automata which cuts down to an equivalence when we restrict to non-degenerate automata. Moreover, we prove that the natural notion of bisimulation for such structures is a generalisation of the strong history preserving bisimulation, and provide an abstract categorical account of it via open maps. Finally, we define a notion of unfolding for higher dimensional transition systems and characterise the structures so obtained as a generalisation of event structures
Frobenius Modules and Hodge Asymptotics
We exhibit a direct correspondence between the potential defining the H^{1,1}
small quantum module structure on the cohomology of a Calabi-Yau manifold and
the asymptotic data of the A-model variation of Hodge structure. This is done
in the abstract context of polarized variations of Hodge structure and
Frobenius modules.Comment: Updated bibliography. Final version published in Commun. Math. Phy
Asymptotic Hodge theory and quantum products
Assuming suitable convergence properties for the Gromov-Witten potential of a
Calabi-Yau manifold one may construct a polarized variation of Hodge
structure over the complexified K\"ahler cone of . In this paper we show
that, in the case of fourfolds, there is a correspondence between ``quantum
potentials'' and polarized variations of Hodge structures that degenerate to a
maximally unipotent boundary point. Under this correspondence, the WDVV
equations are seen to be equivalent to the Griffiths' trasversality property of
a variation of Hodge structure.Comment: References and comments added. To appear in "Advances in Algebraic
Geometry Motivated by Physics", Ed. E. Previatto, Contemporary Mathematic
Restriction of A-Discriminants and Dual Defect Toric Varieties
We study the -discriminant of toric varieties. We reduce its computation
to the case of irreducible configurations and describe its behavior under
specialization of some of the variables to zero. We prove a Gale dual
characterization of dual defect toric varieties and deduce from it the
classsification of such varieties in codimension less than or equal to four.
This classification motivates a decomposition theorem which yields a sufficient
condition for a toric variety to be dual defect. For codimension less than or
equal to four, this condition is also necessary and we expect this to be the
case in general.Comment: 22 pages; In addition to minor corrections, Section 5 has been
expanded and rewritten to include a Gale dual characterization of dual defect
toric varietie
The derivation of continuum limits of neuronal networks with gap-junction couplings
We consider an idealized network, formed by N neurons individually described
by the FitzHugh-Nagumo equations and connected by electrical synapses. The
limit for N to infinity of the resulting discrete model is thoroughly
investigated, with the aim of identifying a model for a continuum of neurons
having an equivalent behaviour. Two strategies for passing to the limit are
analysed: i) a more conventional approach, based on a fixed nearest-neighbour
connection topology accompanied by a suitable scaling of the diffusion
coefficients; ii) a new approach, in which the number of connections to any
given neuron varies with N according to a precise law, which simultaneously
guarantees the non-triviality of the limit and the locality of neuronal
interactions. Both approaches yield in the limit a pde-based model, in which
the distribution of action potential obeys a nonlinear
reaction-convection-diffusion equation; convection accounts for the possible
lack of symmetry in the connection topology. Several convergence issues are
discussed, both theoretically and numerically
A construction of Frobenius manifolds with logarithmic poles and applications
A construction theorem for Frobenius manifolds with logarithmic poles is
established. This is a generalization of a theorem of Hertling and Manin. As an
application we prove a generalization of the reconstruction theorem of
Kontsevich and Manin for projective smooth varieties with convergent
Gromov-Witten potential. A second application is a construction of Frobenius
manifolds out of a variation of polarized Hodge structures which degenerates
along a normal crossing divisor when certain generation conditions are
fulfilled.Comment: 46 page
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