Stringy K-theory and the Chern character

We construct two new G-equivariant rings: \(\mathcal{K}(X,G)\), called the stringy K-theory of the G-variety X, and \(\mathcal{H}(X,G)\), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack \(\mathcal{X}\), we also construct a new ring \(\mathsf{K}_{\mathrm{orb}}(\mathcal{X})\) called the full orbifold K-theory of \(\mathcal{X}\). We show that for a global quotient \(\mathcal{X} = [X/G]\), the ring of G-invariants \(K_{\mathrm{orb}}(\mathcal{X})\) of \(\mathcal{K}(X,G)\) is a subalgebra of \(\mathsf{K}_{\mathrm{orb}}([X/G])\) and is linearly isomorphic to the “orbifold K-theory” of Adem-Ruan [AR] (and hence Atiyah-Segal), but carries a different “quantum” product which respects the natural group grading.We prove that there is a ring isomorphism \(\mathcal{C}\mathbf{h}:\mathcal{K}(X,G)\to\mathcal{H}(X,G)\), which we call the stringy Chern character. We also show that there is a ring homomorphism \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}:\mathsf{K}_{\mathrm{orb}}(\mathcal{X}) \rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\), which we call the orbifold Chern character, which induces an isomorphism \(Ch_{\mathrm{orb}}:K_{\mathrm{orb}}(\mathcal{X})\rightarrow H^\bullet_{\mathrm{orb}}(\mathcal{X})\) when restricted to the sub-algebra \(K_{\mathrm{orb}}(\mathcal{X})\). Here \(H_{\mathrm{orb}}^\bullet(\mathcal{X})\) is the Chen–Ruan orbifold cohomology. We further show that \(\mathcal{C}\mathbf{h}\) and \(\mathfrak{C}\mathfrak{h}_\mathrm{orb}\) preserve many properties of these algebras and satisfy the Grothendieck–Riemann–Roch theorem with respect to étale maps. All of these results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.We further prove that \(\mathcal{H}(X,G)\) is isomorphic to Fantechi and Göttsche’s construction [FG, JKK]. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results greatly simplify the definitions of the Fantechi–Göttsche ring, Chen–Ruan orbifold cohomology, and the Abramovich–Graber–Vistoli orbifold Chow ring.We conclude by showing that a K-theoretic version of Ruan’s Hyper-Kähler Resolution Conjecture holds for the symmetric product of a complex projective surface with trivial first Chern class.     

Short-term changes in delta(13)C and delta(15)N signatures of water discharged from grazed grasslands

The composition of dissolved organic matter (DOM) in a soil is the product of a variety of soil processes. Changes in the composition of DOM in water discharged from soil should, therefore, give an important insight into modifications in these soil processes. We hypothesise that these processes in soils, under different grassland management regimes, would be affected to different extents by the short-term disturbance of a storm event and that evidence of this could be detected in delta(13)C and delta(15)N signatures in drainage and surface runoff waters. During a storm event we collected discharge waters from 1 ha grassland lysimeters, with or without artificial drainage, which received contrasting fertiliser inputs, and delta(13)C and delta(15)N signatures were determined. Changes in (13)C enrichment during the storm event were clearly identifiable, as were differences between plots for (13)C and (15)N, illustrating that this technique has potential to be a useful tool for identifying and investigating short- and long-term changes in soil organic matter dynamics. Copyright 1999 John Wiley & Sons, Ltd.     

Thermodynamic evaluation of the stability of the bone-seeking radiopharmaceutical [Lu]Lu(III)–DOTP under simulated blood plasma conditions

The stability and in vivo robustness of [¹⁷⁷Lu]Lu–DOTP as a potential bone-targeting radiopharmaceutical was determined with the aid of thermodynamic blood plasma modeling simulations. Glass electrode potentiometry was employed to measure the stability constants of the complexes of Lu³⁺ with DOTP. Similarly, the complexes of DOTP with a selection of the important physiological metal ions: Ca²⁺, Mg²⁺, and Cu²⁺ were determined, representing the typical interactions that the ligand would encounter upon administration. This made possible the construction of a blood plasma model of DOTP, aiding in establishing the potential susceptibility of the radiopharmaceutical. The ligand binds predominantly to calcium in vivo, accounting for 59.6% of that initially introduced as a component of the Lu–DOTP complex. Furthermore, due to a preference of the DOTP to bind to Cu²⁺ it causes mobilization of the ions in blood plasma, and would therefore indicate a deficiency if the ligand is administered at a concentration of 8.5 × 10⁻⁵ mol dm⁻³. The lutetium-ions are preferentially bound to DOTP, with as much as 98.1% of the Lu³⁺ occupying the ligand under physiological conditions.     

Multiobjective evolutionary optimisation for surface-enhanced Raman scattering

In most optimisation experiments, a single parameter is first optimised before a second and then third one are subsequently modified to give the best result. By contrast, we believe that simultaneous multiobjective optimisation is more powerful; therefore, an optimisation of the experimental conditions for the colloidal SERS detection of l-cysteine was carried out. Six aggregating agents and three different colloids (citrate, borohydride and hydroxylamine reduced silver) were tested over a wide range of concentrations for the enhancement and the reproducibility of the spectra produced. The optimisation was carried out using two methods, a full factorial design (FF, a standard method from the experimental design literature) and, for the first time, a multiobjective evolutionary algorithm (MOEA), a method more usually applied to optimisation problems in computer science. Simulation results suggest that the evolutionary approach significantly out-performs random sampling. Real experiments applying the evolutionary method to the SERS optimisation problem led to a 32% improvement in enhancement and reproducibility compared with the FF method, using far fewer evaluations.     

Toggling bistable atoms via mechanical switching of bond angle

We reversibly switch the state of a bistable atom by direct mechanical manipulation of bond angle using a dynamic force microscope. Individual buckled dimers at the Si(100) surface are flipped via the formation of a single covalent bond, actuating the smallest conceivable in-plane toggle switch (two atoms) via chemical force alone. The response of a given dimer to a flip event depends critically on both the local and nonlocal environment of the target atom-an important consideration for future atomic scale fabrication strategies.     

Direct submolecular scale imaging of mesoscale molecular order in supported dipalmitoylphosphatidylcholine bilayers

Supported dipalmitoylphosphatidylcholine (DPPC) bilayers are widely used membrane systems in biophysical and biochemical studies. Previously, short-range positional and orientational order of lipid headgroups of supported DPPC bilayers was observed at room temperature using low deflection noise frequency modulation atomic force microscopy (FM-AFM). While this ordering was supported by X-ray diffraction studies, it conflicted with diffusion coefficient measurements of gel-phase bilayers determined from fluorescence photobleaching experiments. In this work, we have directly imaged mica-supported DPPC bilayers with submolecular resolution over scan ranges up to 146 nm using low deflection noise FM-AFM. Both orientational and positional molecular ordering were observed in the mesoscale, indicative of crystalline order. We discuss these results in relation to previous biophysical studies and propose that the mica support induces mesoscopic crystalline order of the DPPC bilayer at room temperature. This study also demonstrates the recent advance in the scan range of submolecular scale AFM imaging.     

Torsion-Free Sheaves and Moduli of Generalized Spin Curves

This article treats compactifications of the space of generalized spin curves. Generalized spin curves, or r-spin curves, are pairs (X,L) with X a smooth curve and L a line bundle whose rth tensor power is isomorphic to the canonical bundle of X. These are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), which have been of interest recently, in part because of their applications to fermionic string theory. Three different compactifications over Z[1/r], all using torsion-free sheaves, are constructed. All three yield algebraic stacks, one of which is shown to have Gorenstein singularities that can be described explicitly, and one of which is smooth. All three compactifications generalize constructions of Deligne and Cornalba done for the case when r=2.