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The shape stability of the reaction interface for reactive flow in a layered porous medium is studied. This is done using a complete linearized stability analysis in the setting of a free boundary model of this phenomenon. The spectrum of the linearized problem is obtained in the general case, and it is compared with that obtained for homogeneous media in two typical cases.

Let R be a not necessarily commutative ring with 1. In the present paper we first introduce a notion of quasi-orderings, which axiomatically subsumes all the orderings and valuations on R. We proceed by uniformly defining a coarsening relation ≤ on the set Q(R) of all quasi-orderings on R. One of our main results states that (Q(R),≤′) is a rooted tree for some slight modification ≤′ of ≤, i.e. a partially ordered set admitting a maximum such that for any element there is a unique chain to that maximum. As an application of this theorem we obtain that (Q(R),≤′) is a spectral set, i.e. order-isomorphic to the spectrum of some commutative ring with 1. We conclude this paper by studying Q(R) as a topological space.

Motivated by new explicit positive Ricci curvature metrics on the four-sphere which are also Einstein-Weyl, we show that the dimension of the Einstein-Weyl moduli near certain Einstein metrics is bounded by the rank of the isometry group and that any Weyl manifold can be embedded as a hypersurface with prescribed second fundamental form in some Einstein-Weyl space. Closed four-dimensional Einstein-Weyl manifolds are proved to be absolute minima of the L²-norm of the curvature of Weyl manifolds and a local version of the Lafontaine inequality is obtained. The above metrics on the four-sphere are shown to contain minimal hypersurfaces isometric to S¹S² whose second fundamental form has constant length.

The cocenter of an affine Hecke algebra plays an important role in the study of representations of the affine Hecke algebra and the geometry of affine DeligneLusztig varieties (see for example, He and Nie in Compos Math 150(11):19031927, 2014; He in Ann Math 179:367404, 2014; Ciubotaru and He in Cocenter and representations of affine Hecke algebras, 2014). In this paper, we give a BernsteinLusztig type presentation of the cocenter. We also obtain a comparison theorem between the class polynomials of the affine Hecke algebra and those of its parabolic subalgebras, which is an algebraic analog of the HodgeNewton decomposition theorem for affine DeligneLusztig varieties. As a consequence, we present a new proof of the emptiness pattern of affine DeligneLusztig varieties (Grtz et al. in Compos Math 146(5):13391382, 2010; Grtz et al. in Ann Sci cole Norm Sup, 2012).