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

Hamster oocyte membrane potential and ion permeability vary with preantral cumulus cell attachment and developmental stage

BACKGROUND: In vitro maturation of mammalian oocytes is an area of great interest due to its potential application in the treatment of infertility. The morphological and physiological changes that occur during oocyte development are poorly understood, and further studies are needed investigating the physiological changes associated with oocyte maturation. In this study we evaluated the membrane potential and the sodium/potassium permeability ratio of oocytes acutely isolated, and cumulus-oocyte complexes in metaphase II and preantral follicle stages. RESULTS: Intracellular electrical recordings revealed that cumulus-enclosed oocytes have a membrane potential significantly more negative at the preantral follicle stage than at metaphase II stage (-38.4 versus -19.7 mV, p < 0.0005). The membrane potential of the cumulus-free oocytes was not different between the preantral and metaphase II stages. The membrane potential of the cumulus cells forming preantral stage follicles was shown to be significantly different from that of the oocyte within the follicle (-28.6 versus -38.4 mV, p < 0.05). The sodium/potassium permeability measured in cumulus-enclosed oocytes at the preantral stage equaled a mean value of 0.33. The ratio was significantly lower when measured in oocytes denuded of cumulus cells or cumulus-enclosed metaphase II oocytes, 0.76, 0.79, 0.77 respectively (p < 0.001). CONCLUSIONS: These data show a change in the membrane potential and Na(+)/K(+) permeability ratio during ooycte development from the preantral stage oocyte to the metaphase II stage. We have also demonstrated a change in the preantral oocyte membrane potential when surrounding cumulus cells are removed; either due to membrane changes or loss of cumulus cells

Vowel Recognition from Articulatory Position Time-Series Data

A new approach of recognizing vowels from articulatory position time-series data was proposed and tested in this paper. This approach directly mapped articulatory position time-series data to vowels without extracting articulatory features such as mouth opening. The input time-series data were time-normalized and sampled to fixed-width vectors of articulatory positions. Three commonly used classifiers, Neural Network, Support Vector Machine and Decision Tree were used and their performances were compared on the vectors. A single speaker dataset of eight major English vowels acquired using Electromagnetic Articulograph (EMA) AG500 was used. Recognition rate using cross validation ranged from 76.07% to 91.32% for the three classifiers. In addition, the trained decision trees were consistent with articulatory features commonly used to descriptively distinguish vowels in classical phonetics. The findings are intended to improve the accuracy and response time of a real-time articulatory-to-acoustics synthesizer