Towards the Intersection Theory on Hurwitz Spaces

Moduli spaces of algebraic curves and closely related to them Hurwitz spaces, that is, spaces of meromorphic functions on the curves, arise naturally in numerous problems of algebraic geometry and mathematical physics, especially in relationship with the string theory and Gromov--Witten invariants. In particular, the classical Hurwitz problem about enumeration of topologically distinct ramified coverings of the sphere with prescribed ramification type reduces to the study of geometry and topology of these spaces. The cohomology rings of such spaces are complicated even in the simplest cases of rational curves and functions. However, the cohomology classes that are the most important from the point of view of applications (namely, the classes Poincar\'e dual to the strata of functions with given singularities) can be expressed in terms of relatively simple ``basic'' classes (which are, in a sense, tautological). The aim of the present paper is to identify these basic classes, to describe relations among them, and to find expressions for the strata in terms of these classes. Our approach is based on R. Thom's theory of universal polynomials of singularities, which has been extended to the case of multisingularities by the first author. Although the general Hurwitz problem still remains open, our approach allows one to achieve a significant progress in its solution, as well as in the understanding of the geometry and topology of Hurwitz spaces.Comment: 29 pages, AMSTe

Mini-laparotomy cholecystectomy: Technique, outcomes: A prospective study

AbstractBackgroundThe last decades have been characterized by a rapid growth in minimally invasive techniques for acute and chronic cholecystitis. The aim of our study was to analyze 10years of experience with the mini-laparotomy cholecystectomy.MethodsFrom 1994 to 2004, we performed 2295 mini-laparotomy cholecystectomies, including 1028 patients with acute and 1267 patients with chronic cholecystitis. There were 1780 women and 515 men. We utilized a special surgical tool kit with a system of circular and small hook-retractors incorporating an illuminator and long surgical instruments. Our surgical approach was carried out using a 3–5cm longitudinal incision located immediately above the gallbladder with a muscle splitting technique.ResultsThe mean time of operation was 64.5±24.5min and the conversion rate was 3.7%. Intraoperative complications occurred in 25 cases (1.1%), including 4 cases (0.17%) of biliary tract injury. Cholecystectomy was combined with intervention on the choledochus and the papilla of Vater in 133 patients with choledocholithiasis. Postoperative complications developed in 4.1%. Five hundred and five patients (22%) required opioid analgesics on the first postoperative day. The mortality rate was 0.17%. The mortalities involved patients who had severe concomitant diseases and required urgent surgery for acute cholecystitis. Patients operated for acute cholecystitis had significantly higher rates of postoperative complications (5.8% vs. 2.8%), need for opioids (25.5% vs. 19.2%) and mortality (0.39% vs. 0%).ConclusionsMini-laparotomy cholecystectomy is an alternative to laparoscopic approach in the surgical treatment of acute and chronic cholecystitis

The charge-dyon bound system in the spherical quantum well

The spherical wave functions of charge-dyon bounded system in a rectangular spherical quantum dot of infinitely and finite height are calculated. The transcendent equations, defining the energy spectra of the systems are obtained. The dependence of the energy levels from the wall sizes is found.Comment: 8 pages, 5 figure

On double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers \CP^1\to\CP^1 with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.Comment: 17 pages, 3 figure

Electron States and Light Absorption in Strongly Oblate and Strongly Prolate Ellipsoidal Quantum Dots in Presence of Electrical and Magnetic Fields

In framework of the adiabatic approximation the energy states of electron as well as direct light absorption are investigated in strongly oblate and strongly prolate ellipsoidal quantum dots (QDs) at presence of electric and magnetic fields. Analytical expressions for particle energy spectrum are obtained. The dependence of energy levels’ configuration on QD geometrical parameters and field intensities is analytically obtained. The energy levels of electrons are shown to be equidistant both for strongly oblate and prolate QDs. The effect of the external fields on direct light absorption of a QD was investigated. The dependence of the absorption edge on geometrical parameters of QDs and intensities of the electric and magnetic fields is obtained. Selection rules are obtained at presence as well as absence of external electric and magnetic fields. In particular, it is shown that the presence of the electric field cancels the quantum numbers selection rules at the field direction, whereas in radial direction the selection rules are preserved. Perspectives of practical applications for device manufacturing based on ellipsoidal quantum dots are outlined