607,538 research outputs found

    Chern forms of singular metrics on vector bundles

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    We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-P{\u{a}}un. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.Comment: 18

    Adiabatic limits of eta and zeta functions of elliptic operators

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    We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator δ\delta, constructed from an elliptic family of operators indexed by S1S^1. We show that the regularized values η(δt,0){\eta}(\delta_t,0) and tζ(δt,0)t{\zeta}(\delta_t,0) are smooth functions of tt at t=0t=0, and we identify their values at t=0t=0 with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions η(δt,s){\eta}(\delta_t,s) and tζ(δt,s)t{\zeta}(\delta_t,s) are shown to extend smoothly to t=0t=0 for all values of ss. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms.Comment: 32 pages, final versio

    Catastrophe Models for Cognitive Workload and Fatigue

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    We reconceptualised several problems concerning the measurement of cognitive workload – fixed versus variable limits on channel capacity, work volume versus time pressure, adaptive strategies, resources demanded by tasks when performed simultaneously, and unclear distinctions between workload and fatigue effects – as two cusp catastrophe models: buckling stress resulting from acute workload, and fatigue resulting from extended engagement. Experimental participants completed a task that was intensive on non-verbal episodic memory and had an automatically speeded component. For buckling stress, the epoch of maximum (speeded) performance was the asymmetry parameter; however, anxiety did not contribute to bifurcation as expected. For fatigue, the bifurcation factor was the total work accomplished, and arithmetic, a compensatory ability, was the asymmetry parameter; R2 for the cusp models outperformed the linear comparison models in both cases. A research programme is outlined that revolves around the two models with different types of task and resource configurations

    Asymptotic expansions, LL-values and a new Quantum Modular Form

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    In 2010 Zagier introduced the notion of a quantum modular form. One of his first examples was the "strange" function F(q)F(q) of Kontsevich. Here we produce a new example of a quantum modular form by making use of some of Ramanujan's mock theta functions. Using these functions and their transformation behaviour, we also compute asymptotic expansions similar to expansions of F(q)F(q).Comment: 7 page
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