On the Rapoport-Zink space for GU(2,4)\mathrm{GU}(2, 4) over a ramified prime

In this work, we study the supersingular locus of the Shimura variety associated to the unitary group $\mathrm{GU}(2,4)$ over a ramified prime. We show that the associated Rapoport-Zink space is flat, and we give an explicit description of the irreducible components of the reduction modulo $p$ of the basic locus. In particular, we show that these are universally homeomorphic to either a generalized Deligne-Lusztig variety for a symplectic group or to the closure of a vector bundle over a classical Deligne-Lusztig variety for an orthogonal group. Our results are confirmed in the group-theoretical setting by the reduction method \`a la Deligne and Lusztig and the study of the admissible set

Chow rings of toric varieties defined by atomic lattices

We study a graded algebra $D=D(\mathcal{L},\mathcal{G})$ over ℤ defined by a finite lattice ℒ and a subset $\mathcal{G}$ in ℒ, a so-called building set. This algebra is a generalization of the cohomology algebras of hyperplane arrangement compactifications found in work of De Concini and Procesi [2]. Our main result is a representation of D, for an arbitrary atomic lattice ℒ, as the Chow ring of a smooth toric variety that we construct from ℒ and $\mathcal{G}$ . We describe this variety both by its fan and geometrically by a series of blowups and orbit removal. Also we find a Gröbner basis of the relation ideal of D and a monomial basis of

Tropical totally positive cluster varieties

We study the relation between the integer tropical points of a cluster variety (satisfying the full Fock-Goncharov conjecture) and the totally positive part of the tropicalization of an ideal presenting the corresponding cluster algebra. Suppose we are given a presentation of the cluster algebra by a Khovanskii basis for a collection of ${\bf g}$-vector valuations associated with several seeds related by mutations. In presence of a full rank fully extended exchange matrix we construct the rays of a subfan of the totally positive part of the tropicalization of the ideal that coincides combinatorially with the subgraph of the exchange graph of the cluster algebra corresponding to the collection of seeds. Moreover, geometric information about Gross-Hacking-Keel-Kontsevich's toric degenerations associated with seeds gets identified with the Gr\"obner toric degenerations obtained from maximal cones in the tropicalization. As application we prove a conjecture about the relation between Rietsch-Williams' valuations for Grassmannians arising from plabic graphs \cite{RW17} to Kaveh-Manon's work on valuations from the tropicalization of an ideal \cite{KM16}. In a second application we give a partial answer to the question if the Feigin-Fourier-Littelmann-Vinberg degeneration of the full flag variety in type $\mathtt A$ is isomorphic to a degeneration obtained from the cluster structure.Comment: Comments are very welcom

How to obtain lattices from (f,σ,δ)-codes via a generalization of Construction A

We show how cyclic (f,σ,δ)-codes over finite rings canonically induce a Z-lattice in RN by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial f. This construction generalizes the one using certain σ-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by f, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases

Constraint-Driven Fault Diagnosis

Constraint-Driven Fault Diagnosis (CDD) is based on the concept of constraint suspension [6], which was proposed as an approach to fault detection and diagnosis. In this chapter, its capabilities are demonstrated by describing how it might be applied to hardware systems. With this idea, a model-based fault diagnosis problem may be considered as a Constraint Satisfaction Problem (CSP) in order to detect any unexpected behavior and Constraint Satisfaction Optimization Problem (COP) constraint optimization problem in order to identify the reason for any unexpected behavior because the parsimony principle is taken into accountMinisterio de Ciencia y Tecnología TIN2015-63502-C3-2-