The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

A note on the regularity of the free boundaries in the optimal partial transport problem

In this work we study the global regularity of the free boundaries arising in the optimal partial transport problem. Assuming the supports of both the source and the target measure to be convex, we show that the free boundaries of the active regions are globally C ^0,1/2.      

Crystal-field spectroscopy of Eu3+ doped silica glasses

Fourier transform absorption spectroscopy in the 500–6000 cm^? 1 and 9–300 K ranges is applied to monitor the effects produced by Eu³⁺ incorporation into sol–gel silica samples doped with concentration increasing from 0.001 to 10 mol%. The aim is to investigate the formation of aggregates by exploiting the Eu³⁺ crystal-field transitions. Complementary microreflectance and Raman spectra are also measured in the range of silica intrinsic vibrational modes to confirm the hypothesis of matrix modification induced by increasing doping levels. Evidences of clustering are found for high Eu³⁺ concentrations. Up to 3 mol% the crystal-field line intensities gradually increase and the OH^? content smoothly decreases. A further increase to 10 mol% causes drastic, remarkable changes, i.e. sharp crystal-field lines appear which narrow by lowering the temperature. Furthermore, the OH^? related bands are no longer detectable. For concentrations up to 3 mol% the aggregates are amorphous as the silica matrix, while for the Eu³⁺ 10 mol% sample they show a rather ordered structure.     

Three equivalent conjectures on the birational geometry of Fano 3-folds

I propose three equivalent conjectures on the birational geometry of Fano 3-folds. Roughly speaking, they suggest that ergodic, or chaotic, behaviour does not occur for Fano 3-folds.     

On groups whose subgroups are closed in the profinite topology

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two

We prove C ¹ regularity of c-convex weak Alexandrov solutions of a Monge–Ampère type equation in dimension two, assuming only a bound from above on the Monge–Ampère measure. The Monge–Ampère equation involved arises in the optimal transport problem. Our result holds true under a natural condition on the cost function, namely non-negative cost-sectional curvature, a condition introduced in Ma et al. (Arch Ration Mech Anal 177(2):151–183, 2005), that was shown in Loeper (Acta Math, to appear) to be necessary for C ¹ regularity. Such a condition holds in particular for the case “cost = distance squared” which leads to the usual Monge–Ampère equation det D ² u = f. Our result is in some sense optimal, both for the assumptions on the density [thanks to the regularity counterexamples of Wang (Proc Am Math Soc 123(3):841–845, 1995)] and for the assumptions on the cost-function [thanks to the results of Loeper (Acta Math, to appear)].     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

Understanding the inelastic electron-tunneling spectra of alkanedithiols on gold

We present results for a simulated inelastic electron-tunneling spectra (IETS) from calculations using the "gDFTB" code. The geometric and electronic structure is obtained from calculations using a local-basis density-functional scheme, and a nonequilibrium Green's function formalism is employed to deal with the transport aspects of the problem. The calculated spectrum of octanedithiol on gold(111) shows good agreement with experimental results and suggests further details in the assignment of such spectra. We show that some low-energy peaks, unassigned in the experimental spectrum, occur in a region where a number of molecular modes are predicted to be active, suggesting that these modes are the cause of the peaks rather than a matrix signal, as previously postulated. The simulations also reveal the qualitative nature of the processes dominating IETS. It is highly sensitive only to the vibrational motions that occur in the regions of the molecule where there is electron density in the low-voltage conduction channel. This result is illustrated with an examination of the predicted variation of IETS with binding site and alkane chain length.