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

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

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

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

Tuned mass damper for integrally bladed turbine rotor

The invention is directed to a damper ring for damping the natural vibration of the rotor blades of an integrally bladed rocket turbine rotor. The invention consists of an integral damper ring which is fixed to the underside of the rotor blade platform of a turbine rotor. The damper ring includes integral supports which extend radially outwardly therefrom. The supports are located adjacent to the base portion and directly under each blade of the rotor. Vibration damping is accomplished by action of tuned mass damper beams attached at each end to the supports. These beams vibrate at a predetermined frequency during operation. The vibration of the beams enforce a local node of zero vibratory amplitude at the interface between the supports and the beam. The vibration of the beams create forces upon the supports which forces are transmitted through the rotor blade mounting platform to the base of each rotor blade. When these forces attain a predetermined design frequency and magnitude and are directed to the base of the rotor blades, vibration of the rotor blades is effectively counteracted

Identity in second language learning: Access to the target language group

In recent years the role of identity in SLA has emerged in the literature as the linguistic community develops an understanding of the significance of identity in language learning (McKay & Wong, 1996). Norton (2000) refers to identity as “how a person understands his or her relationship to the world, how that relationship is constructed across time and space, and how the person understands possibilities for the future” (p. 5). An understanding of identity in language learning and the relationship of the language learner to the learning environment helps to illuminate what previous theories about the role of affective variables failed to explain, namely, why some learners succeed and others fail with seemingly similar abilities and motivation levels and under similar circumstances. Previous research on learner differences attempted to identify characteristics in learners which explained why some individuals are ‘good language learners’ and others are not (Norton Peirce, 1995). Those studies focused on factors such as anxiety levels, extroversion vs. introversion, age, aptitude, and motivation. Motivated individuals were found more likely to seek opportunities to practice in the target language than individuals who were not motivated (Gardner & MacIntyre, as cited in Norton Peirce, 1995). However, theories about motivation carry an implication that learners who avoid interaction fail to create such opportunities for themselves (Norton, 2000). Moreover, motivation theories have not explained why an individual can be motivated and seek out interaction with target language speakers in some circumstances, but be reticent, unmotivated, and uninvolved under other circumstances (Norton Peirce, 1995). Research on identity is beginning to shed light on this seeming dichotomy in second language learning. Drawing on recent literature, this paper seeks to show that second language learners do not always have agency to create language learning opportunities in circumstances in which they might find themselves, and to further show that power differential between interlocutors in language learning contexts can affect the ability of Identity in SLA Page 3 language learners to interact with target language speakers, ultimately affecting their ability to become proficient in the target language. The question raised for second language acquisition is, “Can learners access interaction through development of a social identity which empowers them to take social risks?” The implication of that question for second language teaching is, “What can be done to effectively foster a social identity in students that facilitates their learning of the target language?