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Smallness problem for quantum affine algebras and quiver varieties

David Hernandez (2008)

Annales scientifiques de l'École Normale Supérieure

The geometric small property (Borho-MacPherson [2]) of projective morphisms implies a description of their singularities in terms of intersection homology. In this paper we solve the smallness problem raised by Nakajima [37, 35] for certain resolutions of quiver varieties [37] (analogs of the Springer resolution): for Kirillov-Reshetikhin modules of simply-laced quantum affine algebras, we characterize explicitly the Drinfeld polynomials corresponding to the small resolutions. We use an elimination...

Soft local times and decoupling of random interlacements

Serguei Popov, Augusto Teixeira (2015)

Journal of the European Mathematical Society

In this paper we establish a decoupling feature of the random interlacement process ℐ u ⊂ ℤ d at level u , d ≥ 3 . Roughly speaking, we show that observations of ℐ u restricted to two disjoint subsets A 1 and A 2 of ℤ d are approximately independent, once we add a sprinkling to the process ℐ u by slightly increasing the parameter u . Our results differ from previous ones in that we allow the mutual distance between the sets A 1 and A 2 to be much smaller than their diameters. We then provide an important application of this...

Solutions to the XXX type Bethe ansatz equations and flag varieties

E. Mukhin, A. Varchenko (2003)

Open Mathematics

We consider a version of the A N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl N+1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B N and...

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

Luise Blank, Martin Butz, Harald Garcke (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading...

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

Luise Blank, Martin Butz, Harald Garcke (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space...

Some spectral properties of the streaming operator with general boundary conditions

Mohamed Boulanouar (2008)

Applications of Mathematics

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K . We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C 0 -semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ∥ K ∥ ≥ 1 . We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the...

Spatial stochastic predator-prey models

Mauro Mobilia, Ivan T. Georgiev, Uwe C. Täuber (2008)

Banach Center Publications

We consider a broad class of stochastic lattice predator-prey models whose main features are overviewed. In particular, this article aims at drawing a picture of the influence of spatial fluctuations, which are not accounted for by the deterministic rate equations, on the properties of the stochastic models. Here, we outline the robust scenario obeyed by most of the lattice predator-prey models with an interaction à la Lotka-Volterra. We also show how a drastically different behavior can emerge...

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