Simple recurrence formulas to count maps on orientable surfaces

We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of faces of the map into account, or equivalently a simple recurrence formula for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and faces. These formulas give by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. In the very particular case of one-face maps, we recover the Harer-Zagier recurrence formula. Our main formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It is similar in look to the one discovered by Goulden and Jackson for triangulations, and indeed our method to go from the KP equation to the recurrence formula can be seen as a combinatorial simplification of Goulden and Jackson's approach (together with one additional combinatorial trick). All these formulas have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved.Comment: Version 3: We changed the title once again. We also corrected some misprints, gave another equivalent formulation of the main result in terms of vertices and faces (Thm. 5), and added complements on bivariate generating functions. Version 2: We extended the main result to include the ability to track the number of faces. The title of the paper has been changed accordingl

On localization in holomorphic equivariant cohomology

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.Comment: 16 pages. Completely rewritten, new title. v3: Minor changes in the exposition. v4: final version to appear in Centr. Eur. J. Mat

Combining DLC, Shot blasting, chemical dip and nano fullerene surface treatments to reduce wear and friction when used with bio-lubricants in automotive contacts

The interaction of three bio-lubricant base oil candidates with seventeen combinations of surface treatment was studied, comparing wear scar volumes and coefficient of friction results. Substrates were initially ground, then a combination of superfinished, Dymon-iC™ DLC, an impact technique of ultra-fine shot blasting method doped with Tin and Molybdenum Disulfide, a calcium based chemical dip containing calcium sulfate and nano fullerene, were used.DLC is well reported to reduce friction. Some reports suggest wear in coated contacts is independent of the type of lubricant used, whilst others report that bio-lubricants offer reduced friction and wear in combination with DLC. Shot blasting can also reduce wear and friction, due to the surface dimples acting as lubricant reservoirs, making hydrodynamic lubrication more likely. Previous work has also explored the performance of surface texturing in combination with coatings, some reporting higher friction when surface texturing and DLC is used. As a surface coating, fullerene has been shown to have significantly lower wear and friction than DLC coatings. The calcium based chemical treatment used has no published data.A ball on flat reciprocating wear tester was used with bio-lubricant base oil candidates, jojoba and soybean oil, with a mineral base oil used for comparison. Wear scars were analysed using a scanning electron microscope.Coefficient of friction results from testing with bio-lubricant base oil candidates’ soybean and jojoba oil were lower than tests with mineral base oil. A hybridized coating combination of superfinish, diamond like carbon and chemical dip gave the highest wear protection for tests with the mineral base oil and bio-lubricant base oil candidate soybean oil. A hybridized coating combination of superfinish, impact technique and chemical dip gave highest wear protection when tested with bio-lubricant base oil candidate jojoba oil. Results showed no overall improvement in wear protection when substrates were processed with the impact technique. Superfinishing substrates improved the performance of both the chemical dip and DLC

Residues and World-Sheet Instantons

We reconsider the question of which Calabi-Yau compactifications of the heterotic string are stable under world-sheet instanton corrections to the effective space-time superpotential. For instance, compactifications described by (0,2) linear sigma models are believed to be stable, suggesting a remarkable cancellation among the instanton effects in these theories. Here, we show that this cancellation follows directly from a residue theorem, whose proof relies only upon the right-moving world-sheet supersymmetries and suitable compactness properties of the (0,2) linear sigma model. Our residue theorem also extends to a new class of "half-linear" sigma models. Using these half-linear models, we show that heterotic compactifications on the quintic hypersurface in CP^4 for which the gauge bundle pulls back from a bundle on CP^4 are stable. Finally, we apply similar ideas to compute the superpotential contributions from families of membrane instantons in M-theory compactifications on manifolds of G_2 holonomy.Comment: 47 page

The nil Hecke ring and singularity of Schubert varieties

We give a criterion for smoothness of a point in any Schubert variety in any G/B in terms of the nil Hecke ring.Comment: AMSTE

GALEX J201337.6+092801: The lowest gravity subdwarf B pulsator

We present the recent discovery of a new subdwarf B variable (sdBV), with an exceptionally low surface gravity. Our spectroscopy of J20136+0928 places it at Teff = 32100 +/- 500, log(g) = 5.15 +/- 0.10, and log(He/H) = -2.8 +/- 0.1. With a magnitude of B = 12.0, it is the second brightest V361 Hya star ever found. Photometry from three different observatories reveals a temporal spectrum with eleven clearly detected periods in the range 376 to 566 s, and at least five more close to our detection limit. These periods are unusually long for the V361 Hya class of short-period sdBV pulsators, but not unreasonable for p- and g-modes close to the radial fundamental, given its low surface gravity. Of the ~50 short period sdB pulsators known to date, only a single one has been found to have comparable spectroscopic parameters to J20136+0928. This is the enigmatic high-amplitude pulsator V338 Ser, and we conclude that J20136+0928 is the second example of this rare subclass of sdB pulsators located well above the canonical extreme horizontal branch in the HR diagram.Comment: 5 pages, accepted for publication in ApJ Letter