Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl

Assessing the volcanic hazard for Rome. 40Ar/39Ar and In-SAR constraints on the most recent eruptive activity and present-day uplift at Colli Albani Volcanic District

We present new 40Ar/39Ar data which allow us to refine the recurrence time for the most recent eruptive activity occurred at Colli Albani Volcanic District (CAVD) and constrain its geographic area. Time elapsed since the last eruption (36 kyr) overruns the recurrence time (31 kyr) in the last 100 kyr. New interferometric synthetic aperture radar data, covering the years 1993–2010, reveal ongoing inflation with maximum uplift rates (>2 mm/yr) in the area hosting the most recent (<200 ka) vents, suggesting that the observed uplift might be caused by magma injection within the youngest plumbing system. Finally, we frame the present deformation within the structural pattern of the area of Rome, characterized by 50 m of regional uplift since 200 ka and by geologic evidence for a recent (<2000 years) switch of the local stress-field, highlighting that the precursors of a new phase of volcanic activity are likely occurring at the CAVD

Binary Fluids with Long Range Segregating Interaction I: Derivation of Kinetic and Hydrodynamic Equations

We study the evolution of a two component fluid consisting of ``blue'' and ``red'' particles which interact via strong short range (hard core) and weak long range pair potentials. At low temperatures the equilibrium state of the system is one in which there are two coexisting phases. Under suitable choices of space-time scalings and system parameters we first obtain (formally) a mesoscopic kinetic Vlasov-Boltzmann equation for the one particle position and velocity distribution functions, appropriate for a description of the phase segregation kinetics in this system. Further scalings then yield Vlasov-Euler and incompressible Vlasov-Navier-Stokes equations. We also obtain, via the usual truncation of the Chapman-Enskog expansion, compressible Vlasov-Navier-Stokes equations.Comment: TeX, 50 page

Idempotent generated algebras and Boolean powers of commutative rings

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker R-algebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module. For an indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces. For a totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is equivalent to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.Comment: 18 page

De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces

We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space $X$ and a totally ordered algebra $A$, we introduce the concept of a finitely valued normal function $f:X\to A$. We show that the operations of $A$ lift to the set $FN(X,A)$ of all finitely valued normal functions, and that there is a canonical proximity relation $\prec$ on $FN(X,A)$. This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain $A$ are axiomatized as proximity Baer Specker $A$-algebras, those pairs $(S,\prec)$, where $S$ is a torsion-free $A$-algebra generated by its idempotents that is a Baer ring, and $\prec$ is a proximity relation on $S$. We introduce the category of proximity Baer Specker $A$-algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker $A$-algebras.Comment: 34 page

Electromagnetic wave absorption and structural properties of wide-band absorber made of graphene-printed glass-fibre composite

Lightweight composites combining electromagnetic wave absorption and excellent mechanical properties are required in spacecraft and aircraft. A one- dimensional metamaterial absorber consisting of a stack of glass fibre/epoxy layers and graphene nanoplatelets/epoxy films was proposed and fabricated through a facile air-spraying based printing technology and a liquid resin infusion method. The production process allows an optimum dispersion of graphene nanoplatelets, promoting adhesion and mechanical integration of the glass fibre/epoxy layers with the graphene nanoplatelets/epoxy films. According to experimental results, the proposed wide-band absorber provides a reflection coefficient lower than −10 dB in the range 8.5–16.7 GHz and an improvement of flexural modulus of more than 15%, with a total thickness of ∼1 mm. Outstanding electromagnetic wave absorption and mechanical performance make the proposed absorber more competitive in aeronautical and aerospace applications

Seal accommodating thermal expansion between adjacent casings in gas turbine engine

A casing around a turbine and a casing around discharge nozzles have a concentrically arranged shell portion. The seal contains internal pressure while accommodating eccentric, expansion and axial travel. Arcuate seal segments have one leg sealing against a radial surface extending from the inner shell and the other leg against the outer shell. A linkage guides travel of the segments

Kinetics of a Model Weakly Ionized Plasma in the Presence of Multiple Equilibria

We study, globaly in time, the velocity distribution $f(v,t)$ of a spatially homogeneous system that models a system of electrons in a weakly ionized plasma, subjected to a constant external electric field $E$. The density $f$ satisfies a Boltzmann type kinetic equation containing a full nonlinear electron-electron collision term as well as linear terms representing collisions with reservoir particles having a specified Maxwellian distribution. We show that when the constant in front of the nonlinear collision kernel, thought of as a scaling parameter, is sufficiently strong, then the $L^1$ distance between $f$ and a certain time dependent Maxwellian stays small uniformly in $t$. Moreover, the mean and variance of this time dependent Maxwellian satisfy a coupled set of nonlinear ODE's that constitute the ``hydrodynamical'' equations for this kinetic system. This remain true even when these ODE's have non-unique equilibria, thus proving the existence of multiple stabe stationary solutions for the full kinetic model. Our approach relies on scale independent estimates for the kinetic equation, and entropy production estimates. The novel aspects of this approach may be useful in other problems concerning the relation between the kinetic and hydrodynamic scales globably in time.Comment: 30 pages, in TeX, to appear in Archive for Rational Mechanics and Analysis: author's email addresses: [email protected], [email protected], [email protected], [email protected], [email protected]