Towards the geometry of double Hurwitz numbers

AbstractDouble Hurwitz numbers count branched covers of CP1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ specified by partitions of the degree (with m and n parts, respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. The remarkable ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande's proof of Witten's conjecture.In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewise-polynomiality result analogous to that implied by the ELSV formula. In the case m=1 (complete branching over one point) and n is arbitrary, we conjecture an ELSV-type formula, and show it to be true in genus 0 and 1. The corresponding Witten-type correlation function has a richer structure than that for single Hurwitz numbers, and we show that it satisfies many geometric properties, such as the string and dilaton equations, and an Itzykson–Zuber-style genus expansion ansatz. We give a symmetric function description of the double Hurwitz generating series, which leads to explicit formulae for double Hurwitz numbers with given m and n, as a function of genus. In the case where m is fixed but not necessarily 1, we prove a topological recursion on the corresponding generating series, which leads to closed-form expressions for double Hurwitz numbers and an analogue of the Goulden–Jackson polynomiality conjecture (an early conjectural variant of the ELSV formula). In a later paper (Faber's intersection number conjecture and genus 0 double Hurwitz numbers, 2005, in preparation), the formulae in genus 0 will be shown to be equivalent to the formulae for “top intersections” on the moduli space of smooth curves Mg. For example, three formulae we give there will imply Faber's intersection number conjecture (in: Moduli of Curves and Abelian Varieties, Aspects of Mathematics, vol. E33, Vieweg, Braunschweig, 1999, pp. 109–129) in arbitrary genus with up to three points

AYURVEDIC APPROACH IN THE MANAGEMENT OF CEREBRAL PALSY- A CASE STUDY

Cerebral Palsy is the most common form of childhood disability. It is a group of non progressive, non contagious condition that causes motor impairment syndrome characterized by abnormalities of movement, muscle control, muscle co-ordination, muscle tone, reflex, posture and balance. It arises from an injury to the brain or abnormal development during the brain’s formation. Ayurveda mentioned two main therapeutic measures i.e. Shodhana and Shamana. Panchakarma (Shodhana Chikitsa) is the best available treatment for the removal of disease from its root and also for better outcome. In this present study, child suffering from Spastic Hemiplegic type of CP treated by Panchakarma (Shodhana) as well as Shamana Chikitsa. 80 days of treatment protocol includes 3 course of Panchakarma therapy at the interval of 10 days along with internal medication (Medhya Churna). Each course includes 5 days Udavartana, 7 days Sarvanga Abhyanga (Bala Taila) and Nadi Swedana along with Pratimarsha Nasya (Panchabhautika taila) and Yoga Basti. The result showed reduced spasticity of right upper and lower limbs, proper neck holding achieved, child started sitting without support, standing and walking with support as well as general health condition improved

Development and Evaluation of an Electronic Vertical Situation Display

Current advanced commercial transport aircraft, such as the Boeing B777/B747-400, the Airbus A320/A340 and the McDonnell Douglas MD-11, rely on AutoFlight Systems (AFS) for flight management, navigation and inner loop control. These systems have evolved from straightforward autopilots into multiple computers capable of sophisticated and interrelated tasks. These tasks span the range from high level flight management to low level inner loop control. In addition, these systems provide envelope protection to prevent pilots from committing mistakes such as stalling the aircraft or lowering flaps at high speeds. Unfortunately, as these systems have become more complex and interconnected, a new class of problems has developed associated with pilots’ interaction with the automation. Many incidents have been reported where there exists some confusion between the pilots’ expectations of the AFS and what the system is actually doing (Corwin, 1995). This confusion has been termed a Mode Awareness Problem (MAP). After a description of the AFS, a formal definition of mode awareness problems is presented in Section 1.3 followed by representative incidents in which mode awareness problems are suspected as being a contributory factor

Accuracy of impressions with different impression materials in angulated implants

Purpose: To evaluate the dimensional accuracy of the resultant (duplicative) casts made from two different impression materials (polyvinyl siloxane and polyether) in parallel and angulated implants.Materials and Methods: Three definitive master casts (control groups) were fabricated in dental stone with three implants, placed at equi-distance. In first group (control), all three implants were placed parallel to each other and perpendicular to the plane of the cast. In the second and third group (control), all three implants were placed at 10° and 15° angulation respectively to the long axis of the cast, tilting towards the centre. Impressions were made with polyvinyl siloxane and polyether impression materials in a special tray, using a open tray impression technique from the master casts. These impressions were poured to obtain test casts. Three reference distances were evaluated on each test cast by using a profile projector and compared with control groups to determine the effect of combined interaction of implant angulation and impression materials on the accuracy of implant resultant cast.Results: Statistical analysis revealed no significant difference in dimensional accuracy of the resultant casts made from two different impression materials (polyvinyl siloxane and polyether) by closed tray impression technique in parallel and angulated implants.Conclusion: On the basis of the results of this study, the use of both the impression materials i.e., polyether and polyvinyl siloxane impression is recommended for impression making in parallel as well as angulated implants.Key words: Angulated implants, implant impression, impression accuracy, impression materia

OX40L gene polymorphism and breast cancer in Iranian population

Background and Aim: The immune system plays an important role in the breast cancer development. OX40L (also known as TNFSF4), a membrane protein, which is a member of the tumor necrosis factor super family binds to its receptor OX40 and this co-stimulation has a crucial role in T cell proliferation, survival and cytokine release. Previous studies have shown that OX40L is associated with cancer. Due to the importance of the T cells in antitumor activities of OX40L we studied the association of rs3850641 (A→G) polymorphism of OX40L gene with the breast cancer. Materials and Methods: The study included 123 women with breast cancer and 126 healthy volunteers with no signs of cancer. Genomic DNA was extracted from blood leucocytes. Genotype and allele frequencies were determined in patients and control cases with the method of polymerase chain reaction-restriction fragment length polymorphism and the statistical analysis was performed by Med Calc. Results: The prevalence of genotype frequencies of AA/AG/GG were 60.9%, 30.08% and 8.9% in patients with breast cancer cases and 74.6%, 18.25% and 7.14% in healthy volunteers while the A and G allelic frequency was 76.01 and 23.98% in patients and 83.73 and 16.26% in healthy controls, respectively. Statistical analysis has shown significant difference from the comparison of genotype (p = 0.03). Conclusion: It is concluded that the rs3850641 SNP is significantly associated with the breast cancer susceptibility in Iranian population. However, further studies in larger populations including other genetic and environmental factors are required to achieve conclusion. Key Words: OX40L, gene polymorphism, breast cancer, PCR-RFLP

Analysis of Complexity Evolution Management and Human Performance Issues in Commercial Aircraft Automation Systems

Autoflight systems in the current generation of aircraft have been implicated in several recent incidents and accidents. A contributory aspect to these incidents may be the manner in which aircraft transition between differing behaviours or “modes.” The current state of aircraft automation was investigated and the incremental development of the autoflight system was tracked through a set of aircraft to gain insight into how these systems developed. This process appears to have resulted in a system without a consistent global representation.National Aeronautics and Space Administration under grants NAG1-1581 and NAG-1-1857. Support was also provided by FAA and NASA Grant FAA-95-G-017

Tunable hybrid surface waves supported by a graphene layer

We study surface waves localized near a surface of a semi-infinite dielectric medium covered by a layer of graphene in the presence of a strong external magnetic field. We demonstrate that both TE-TM hybrid surface plasmons can propagate along the graphene surface. We analyze the effect of the Hall conductivity on the disper- sion of hybrid surface waves and suggest a possibility to tune the plasmon dispersion by the magnetic field.Comment: 3 pages, 3 figure