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.

Remarks on the convergence of pseudospectra

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting operator has constant resolvent norm on an open set. We extend the class of operators for which it is known that the latter cannot happen by showing that if the resolvent norm is constant on an open set, then this constant is the global minimum. We present a number of examples exhibiting various resolvent norm behaviours and illustrating the applicability of this characterisation compared to known results.

On restricted families of projections in ℝ3

We study projections onto non-degenerate one-dimensional families of lines and planes in R 3 . Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most 12 -dimensional sets [Math Processing Error] is typically preserved under one-dimensional families of projections onto lines. We improve the result by an ε , proving that if [Math Processing Error], then the packing dimension of the projections is almost surely at least [Math Processing Error]. For projections onto planes, we obtain a similar bound, with the threshold 12 replaced by 1 . In the special case of self-similar sets [Math Processing Error] without rotations, we obtain a full Marstrand-type projection theorem for 1-parameter families of projections onto lines. The [Math Processing Error] case of the result follows from recent work of M. Hochman, but the [Math Processing Error] part is new: with this assumption, we prove that the projections have positive length almost surely.

Content-Aware Delivery of Scalable Video in Network Coding Enabled Named Data Networks

In this work, we propose a novel network coding enabled NDN architecture for the delivery of scalable video. Our scheme utilizes network coding in order to address the problem that arises in the original NDN protocol, where optimal use of the bandwidth and caching resources necessitates the coordination of the forwarding decisions. To optimize the performance of the proposed network coding based NDN protocol and render it appropriate for transmission of scalable video, we devise a novel rate allocation algorithm that decides on the optimal rates of Interest messages sent by clients and intermediate nodes. This algorithm guarantees that the achieved flow of Data objects will maximize the average quality of the video delivered to the client population. To support the handling of Interest messages and Data objects when intermediate nodes perform network coding, we modify the standard NDN protocol and introduce the use of Bloom filters, which store efficiently additional information about the Interest messages and Data objects. The proposed architecture is evaluated for transmission of scalable video over PlanetLab topologies. The evaluation shows that the proposed scheme performs very close to the optimal performance

Stringy origin of diboson and dijet excesses at the LHC

Very recently, the ATLAS and CMS Collaborations reported diboson and dijet excesses above standard model expectations in the invariant mass region of 1.8–2.0 TeV. Interpreting the diboson excess of events in a model independent fashion suggests that the vector boson pair production searches are best described by WZ or ZZ topologies, because states decaying into W+W− pairs are strongly constrained by semileptonic searches. Under the assumption of a low string scale, we show that both the diboson and dijet excesses can be steered by an anomalous U(1) field with very small coupling to leptons. The Drell–Yan bounds are then readily avoided because of the leptophobic nature of the massive Z′ gauge boson. The non-negligible decay into ZZ required to accommodate the data is a characteristic footprint of intersecting D-brane models, wherein the Landau–Yang theorem can be evaded by anomaly-induced operators involving a longitudinal Z. The model presented herein can be viewed purely field-theoretically, although it is particularly well motivated from string theory. Should the excesses become statistically significant at the LHC13, the associated Zγ topology would become a signature consistent only with a stringy origin.

Dimensions of projections of sets on Riemannian surfaces of constant curvature

We apply the theory of Peres and Schlag to obtain generic lower bounds for Hausdorff dimension of images of sets by orthogonal projections on simply connected two-dimensional Riemannian manifolds of constant curvature. As a conclusion we obtain appropriate versions of Marstrand's theorem, Kaufman's theorem, and Falconer's theorem in the above geometrical settings.