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.

Admissibility via natural dualities

It is shown that admissible clauses and quasi-identities of quasivarieties generated by a single finite algebra, or equivalently, the quasiequational and universal theories of their free algebras on countably infinitely many generators, may be characterized using natural dualities. In particular, axiomatizations are obtained for the admissible clauses and quasi-identities of bounded distributive lattices, Stone algebras, Kleene algebras and lattices, and De Morgan algebras and lattices.

Exact Unification and Admissibility

A new hierarchy of "exact" unification types is introduced, motivated by the study of admissible rules for equational classes and non-classical logics. In this setting, unifiers of identities in an equational class are preordered, not by instantiation, but rather by inclusion over the corresponding sets of unified identities. Minimal complete sets of unifiers under this new preordering always have a smaller or equal cardinality than those provided by the standard instantiation preordering, and in significant cases a dramatic reduction may be observed. In particular, the classes of distributive lattices, idempotent semigroups, and MV-algebras, which all have nullary unification type, have unitary or finitary exact type. These results are obtained via an algebraic interpretation of exact unification, inspired by Ghilardi's algebraic approach to equational unification.

Paracrine or virus-mediated induction of decorin expression by endothelial cells contributes to tube formation and prevention of apoptosis in collagen lattices

Resting endothelial cells express the small proteoglycan biglycan, whereas sprouting endothelial cells also synthesize decorin, a related proteoglycan. Here we show that decorin is expressed in endothelial cells in human granulomatous tissue. For in vitro investigations, the human endothelium-derived cell line, EA.hy 926, was cultured for 6 or more days in the presence of 1% fetal calf serum on top of or within floating collagen lattices which were also populated by a small number of rat fibroblasts. Endothelial cells aligned in cord-like structures and developed cavities that were surrounded by human decorin. About 14% and 20% of endothelial cells became apoptotic after 6 and 12 days of co-culture, respectively. In the absence of fibroblasts, however, the extent of apoptosis was about 60% after 12 days, and cord-like structures were not formed nor could decorin production be induced. This was also the case when lattices populated by EA.hy 926 cells were maintained under one of the following conditions: 1) 10% fetal calf serum; 2) fibroblast-conditioned media; 3) exogenous decorin; or 4) treatment with individual growth factors known to be involved in angiogenesis. The mechanism(s) by which fibroblasts induce an angiogenic phenotype in EA.hy 926 cells is (are) not known, but a causal relationship between decorin expression and endothelial cell phenotype was suggested by transducing human decorin cDNA into EA.hy 926 cells using a replication-deficient adenovirus. When the transduced cells were cultured in collagen lattices, there was no requirement of fibroblasts for the formation of capillary-like structures and apoptosis was reduced. Thus, decorin expression seems to be of special importance for the survival of EA.hy 926 cells as well as for cord and tube formation in this angiogenesis model.

Amalgamation and interpolation in ordered algebras

The first part of this paper provides a comprehensive and self-contained account of the interrelationships between algebraic properties of varieties and properties of their free algebras and equational consequence relations. In particular, proofs are given of known equivalences between the amalgamation property and the Robinson property, the congruence extension property and the extension property, and the flat amalgamation property and the deductive interpolation property, as well as various dependencies between these properties. These relationships are then exploited in the second part of the paper in order to provide new proofs of amalgamation and deductive interpolation for the varieties of lattice-ordered abelian groups and MV-algebras, and to determine important subvarieties of residuated lattices where these properties hold or fail. In particular, a full description is given of all subvarieties of commutative GMV-algebras possessing the amalgamation property.

Atomic Quantum Simulation of U(N) and SU(N) Abelian Lattice Gauge Theories

Using ultracold alkaline-earth atoms in optical lattices, we construct a quantum simulator for U(N) and SU(N) lattice gauge theories with fermionic matter based on quantum link models. These systems share qualitative features with QCD, including chiral symmetry breaking and restoration at nonzero temperature or baryon density. Unlike classical simulations, a quantum simulator does not suffer from sign problems and can address the corresponding chiral dynamics in real time.

Ultracold Quantum Gases and Lattice Systems: Quantum Simulation of Lattice Gauge Theories

Abelian and non-Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, AbelianU(1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev’s toric code is a Z(2) lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non-Abelian SU(3) gauge theory of the strong interactions between quarks and gluons, is nonperturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real-time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should allow us to address very challenging problems, ranging from confinement and deconfinement, or chiral symmetry breaking and its restoration at finite baryon density, to color superconductivity and the real-time evolution of heavy-ion collisions, first in simpler model gauge theories and ultimately in QCD.

Synchrotron microbeam radiation therapy induces hypoxia in intracerebral gliosarcoma but not in the normal brain

PURPOSE Synchrotron microbeam radiation therapy (MRT) is an innovative irradiation modality based on spatial fractionation of a high-dose X-ray beam into lattices of microbeams. The increase in lifespan of brain tumor-bearing rats is associated with vascular damage but the physiological consequences of MRT on blood vessels have not been described. In this manuscript, we evaluate the oxygenation changes induced by MRT in an intracerebral 9L gliosarcoma model. METHODS Tissue responses to MRT (two orthogonal arrays (2 × 400Gy)) were studied using magnetic resonance-based measurements of local blood oxygen saturation (MR_SO2) and quantitative immunohistology of RECA-1, Type-IV collagen and GLUT-1, marker of hypoxia. RESULTS In tumors, MR_SO2 decreased by a factor of 2 in tumor between day 8 and day 45 after MRT. This correlated with tumor vascular remodeling, i.e. decrease in vessel density, increases in half-vessel distances (×5) and GLUT-1 immunoreactivity. Conversely, MRT did not change normal brain MR_SO2, although vessel inter-distances increased slightly. CONCLUSION We provide new evidence for the differential effect of MRT on tumor vasculature, an effect that leads to tumor hypoxia. As hypothesized formerly, the vasculature of the normal brain exposed to MRT remains sufficiently perfused to prevent any hypoxia.