Locally anisotropic toposes

This paper continues the investigation of isotropy theory for toposes. We develop the theory of isotropy quotients of toposes, culminating in a structure theorem for a class of toposes we call locally anisotropic. The theory has a natural interpretation for inverse semigroups, which clarifies some aspects of how inverse semigroups and toposes are related.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Counter-examples to non-noetherian Elkik's approximation theorem

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies.     

On the homology groups of the Brauer complex for a triquadratic field extension

The homology groups , , and of the Brauer complex for a triquadratic field extension are studied. In particular, given , we find equivalent conditions for the image of D in to be zero. We consider as well the second divided power operation , and show that there are nonstandard elements with respect to γ₂. Further, a natural transformation , which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension such that the cohomology group of the Brauer complex contains the images of prescribed elements of , provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions with nontrivial are given. As a consequence we show that the homology group can be arbitrarily big.     

Densely defined non‐closable curl on carpet‐like metric measure spaces

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1‐forms on metric measure spaces. The main examples we consider are the non self‐similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one‐dimensional, they may have positive two‐dimensional Lebesgue measure and carry nontrivial 2‐forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1‐forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1‐forms into 2‐forms cannot be closable if the martingale dimension is larger than one.     

Modularity results for interpolation,amalgamation and superamalgamation

Wolter in [38] proved that the Craig interpolation property transfers to fusion of normal modal logics. It is well-known [21] that for such logics Craig interpolation corresponds to an algebraic property called superamalgamability. In this paper, we develop model-theoretic techniques at the level of first-order theories in order to obtain general combination results transferring quantifier-free interpolation to unions of theories over non-disjoint signatures. Such results, once applied to equational theories sharing a common Boolean reduct, can be used to prove that superamalgamability is modular also in the non-normal case. We also state that, in this non-normal context, superamalgamability corresponds to a strong form of interpolation that we call “comprehensive interpolation property” (which consequently transfers to fusions).     

Initial-boundary problem for the 1-D Euler-Boltzmann equations in radiation hydrodynamics

We study the initial-boundary value problem for the one dimensional EulerBoltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions(U_(△t,d), I_(△t,d)) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.     

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.     

Existence of multiple solutions for a fractional p-Laplacian system with concave-convex term

In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions{(-?)_p~su = a(x)|u|~(q-2) u +2α/α + βc(x)|u|~(α-2) u|v|~β, in ?,(-?)_p~sv = b(x)|v|~(q-2) v +2β/α + βc(x)|u|α|v|~(β-2) v, in ?,u = v = 0, in Rn\?,(0.1) where Ω is a smooth bounded domain in Rn, n ps with s ∈(0,1) fixed, a(x), b(x), c(x) ≥ 0 and a(x),b(x),c(x) ∈L∞(Ω), 1 q p and α,β 1 satisfy pα + βp*,p* =np/n-ps.By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem(0.1).?????     

Strong comparison principles for some nonlinear degenerate elliptic equations

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ▽~2ψ + L(x, ▽ψ), including the conformal hessian operator.