10,675 research outputs found

    A strong operator topology adiabatic theorem

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    We prove an adiabatic theorem for the evolution of spectral data under a weak additive perturbation in the context of a system without an intrinsic time scale. For continuous functions of the unperturbed Hamiltonian the convergence is in norm while for a larger class functions, including the spectral projections associated to embedded eigenvalues, the convergence is in the strong operator topology.Comment: 15 pages, no figure

    Integrable RCS as a proposed replacement for Fermilab Booster

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    Integrable optics is an innovation in particle accelerator design that potentially enables a greater betatron tune spread and damps collective instabilities. An integrable rapid-cycling synchrotron (RCS) would be an effective replacement for the Fermilab Booster, as part of a plan to reach multi-MW beam power at 120 GeV for the Fermilab high-energy neutrino program. We provide an example integrable lattice with features of a modern RCS - dispersion-free drifts, low momentum compaction factor, superperiodicity, chromaticity correction, bounded beta functions, and separate-function magnets.Comment: arXiv admin note: substantial text overlap with arXiv:1703.0095

    Bridge trisections of knotted surfaces in 4--manifolds

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    We prove that every smoothly embedded surface in a 4--manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4--manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors' definition of bridge trisections for surfaces in S4S^4. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4--manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside CP2\mathbb{CP}^2. Using these examples, we prove that there exist exotic 4--manifolds with (g,0)(g,0)--trisections for certain values of gg. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.Comment: 17 pages, 5 figures. Comments welcom

    Characterizing Dehn surgeries on links via trisections

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    We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In addition, we describe how the potential counterexamples to the Generalized Property R Conjecture given by Gompf, Scharlemann, and Thompson yield genus four trisections of the standard four-sphere that are unlikely to be standard. Finally, we give an analog of the Casson- Gordon Rectangle Condition for trisections that can be used to obstruct reducibility of a given trisection.Comment: 15 pages, 4 color figures. Comments welcome
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