Distributive inverse semigroups and non-commutative Stone dualities

We develop the theory of distributive inverse semigroups as the analogue of distributive lattices without top element and prove that they are in a duality with those etale groupoids having a spectral space of identities, where our spectral spaces are not necessarily compact. We prove that Boolean inverse semigroups can be characterized as those distributive inverse semigroups in which every prime filter is an ultrafilter; we also provide a topological characterization in terms of Hausdorffness. We extend the notion of the patch topology to distributive inverse semigroups and prove that every distributive inverse semigroup has a Booleanization. As applications of this result, we give a new interpretation of Paterson's universal groupoid of an inverse semigroup and by developing the theory of what we call tight coverages, we also provide a conceptual foundation for Exel's tight groupoid.Comment: arXiv admin note: substantial text overlap with arXiv:1107.551

Invariant means on Boolean inverse monoids

The classical theory of invariant means, which plays an important role in the theory of paradoxical decompositions, is based upon what are usually termed `pseudogroups'. Such pseudogroups are in fact concrete examples of the Boolean inverse monoids which give rise to etale topological groupoids under non-commutative Stone duality. We accordingly initiate the theory of invariant means on arbitrary Boolean inverse monoids. Our main theorem is a characterization of when a Boolean inverse monoid admits an invariant mean. This generalizes the classical Tarski alternative proved, for example, by de la Harpe and Skandalis, but using different methods

Uniform existence of the integrated density of states for random Schr\"odinger operators on metric graphs over Zd\mathbb{Z}^d

We consider ergodic random magnetic Schr\"odinger operators on the metric graph $\mathbb{Z}^d$ with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin-Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss percolation models.Comment: 17 pages; typos removed, references updated, definition of subgraph densities explaine

LpL^p-approximation of the integrated density of states for Schr\"odinger operators with finite local complexity

We study spectral properties of Schr\"odinger operators on \RR^d. The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in \ZZ^d, with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e.the normalised eigenvalue counting functions. The convergence holds in the space $L^p(I)$ where $I$ is any finite energy interval and $1\leq p< \infty$ is arbitrary.Comment: 15 pages; v2 has minor fixe

Transcriptomics and metatranscriptomics in zooplankton: wave of the future?

Abstract Molecular tools have changed the understanding of zooplankton biodiversity, speciation, adaptation, population genetics and global patterns of connectivity. However, the molecular resources needed to capitalize on these advances continue to be limited in comparison with those available for other eukaryotic plankton. This deficiency could be addressed through an Ocean Zooplankton Open 'Omics Project (Ocean ZOOP) that would generate de novo assembled transcriptomes for hundreds of metazoan plankton species. A collection of comparable reference transcriptomes would generate a new framework for ecological and physiological studies. Defining species niches, identifying optimal habitats, assessing adaptive capacity and predicting changes in phenology are just a few examples of how such a resource could transform studies on zooplankton ecology

Lenz-Majewski mutations in PTDSS1 affect phosphatidylinositol 4-phosphate metabolism at ER-PM and ER-golgi junctions

Lenz-Majewski syndrome (LMS) is a rare disease characterized by complex craniofacial, dental, cutaneous, and limb abnormalities combined with intellectual disability. Mutations in the PTDSS1 gene coding one of the phosphatidylserine (PS) synthase enzymes, PSS1, were described as causative in LMS patients. Such mutations render PSS1 insensitive to feedback inhibition by PS levels. Here we show that expression of mutant PSS1 enzymes decreased phosphatidylinositol 4-phosphate (PI4P) levels both in the Golgi and the plasma membrane (PM) by activating the Sac1 phosphatase and altered PI4P cycling at the PM. Conversely, inhibitors of PI4KA, the enzyme that makes PI4P in the PM, blocked PS synthesis and reduced PS levels by 50% in normal cells. However, mutant PSS1 enzymes alleviated the PI4P dependence of PS synthesis. Oxysterol-binding protein-related protein 8, which was recently identified as a PI4P-PS exchanger between the ER and PM, showed PI4P-dependent membrane association that was significantly decreased by expression of PSS1 mutant enzymes. Our studies reveal that PS synthesis is tightly coupled to PI4P-dependent PS transport from the ER. Consequently, PSS1 mutations not only affect cellular PS levels and distribution but also lead to a more complex imbalance in lipid homeostasis by disturbing PI4P metabolism

Mesons on a transverse lattice

The meson eigenstates of the light-cone Hamiltonian in a coarse transverse lattice gauge theory are investigated. Building upon previous work in pure gauge theory, the Hamiltonian and its Fock space are expanded in powers of dynamical fields. In the leading approximation, the couplings appearing in the Hamiltonian are renormalised by demanding restoration of space-time symmetries broken by the cut-off. Additional requirements from chiral symmetry are discussed and difficulties in imposing them from first principles in the leading approximation are noted. A phenomenological calculation is then performed, in which chiral symmetry in spontaneously broken form is modelled by imposing the physical pion-rho mass splitting as a constraint. The light-cone wavefunctions of the resulting Hamiltonian are used to compute decay constants, form factors and quark momentum and spin distributions for the pion and rho mesons. Extensions beyond leading order, and the implications for first principles calculations, are briefly discussed.Comment: 31 pages, 7 figure

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v