Dynamical Belyi maps

We study the dynamical properties of a large class of rational maps with exactly three ramification points. By constructing families of such maps, we obtain infinitely many conservative maps of degree $d$; this answers a question of Silverman. Rather precise results on the reduction of these maps yield strong information on the rational dynamics.Comment: 21 page

Minimal surfaces bounded by elastic lines

In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and then removing it; the shape of the spanning soap film is a solution to the Plateau problem. But what happens if a soap film spans a loop of inextensible but flexible wire? We consider this simple query that couples Plateau's problem to Euler's Elastica: a special class of twist-free curves of given length that minimize their total squared curvature energy. The natural marriage of two of the oldest geometrical problems linking physics and mathematics leads to a quest for the shape of a minimal surface bounded by an elastic line: the Euler-Plateau problem. We use a combination of simple physical experiments with soap films that span soft filaments, scaling concepts, exact and asymptotic analysis combined with numerical simulations to explore some of the richness of the shapes that result. Our study raises questions of intrinsic interest in geometry and its natural links to a range of disciplines including materials science, polymer physics, architecture and even art.Comment: 14 pages, 4 figures. Supplementary on-line material: http://www.seas.harvard.edu/softmat/Euler-Plateau-problem

On Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations

We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.Comment: To appear in Comm. Partial Differential Equations; 10 page

Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras.Comment: 32 pages, 1 figure; changes: major revision of presentation, clarifications adde

Curvature homogeneous spacelike Jordan Osserman pseudo-Riemannian manifolds

Let s be at least 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s,s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of order 3; they are not timelike Jordan Osserman. They are k-spacelike higher order Jordan Osserman for $2\le k\le s$; they are k-timelike higher order Jordan Osserman for $s+2\le k\le 2s$, and they are not k timelike higher order Jordan Osserman for $2\le s\le s+1$.Comment: Update bibliography, fix minor misprint

Late Onset Myasthenia Gravis Is Associated with HLA DRB1*15:01 in the Norwegian Population

BACKGROUND: Acquired myasthenia gravis (MG) is a rare antibody-mediated autoimmune disease caused by impaired neuromuscular transmission, leading to abnormal muscle fatigability. The aetiology is complex, including genetic risk factors of the human leukocyte antigen (HLA) complex and unknown environmental factors. Although associations between the HLA complex and MG are well established, not all involved components of the HLA predisposition to this heterogeneous disease have been revealed. Well-powered and comprehensive HLA analyses of subgroups in MG are warranted, especially in late onset MG. METHODOLOGY/PRINCIPAL FINDINGS: This case-control association study is of a large population-based Norwegian cohort of 369 MG patients and 651 healthy controls. We performed comprehensive genotyping of four classical HLA loci (HLA-A, -B, -C and -DRB1) and showed that the DRB1*15:01 allele conferred the strongest risk in late onset MG (LOMG; onset ≥ 60 years) (OR 2.38, p(c)7.4 × 10(-5)). DRB1*13:01 was found to be a protective allele for both early onset MG (EOMG) and LOMG (OR 0.31, p(c) 4.71 × 10(-4)), a finding not previously described. No significant association was found to the DRB1*07:01 allele (p(nc) = 0.18) in a subset of nonthymomatous anti-titin antibody positive LOMG as reported by others. HLA-B*08 was mapped to give the strongest contribution to EOMG, supporting previous studies. CONCLUSION: The results from this study provide important new information concerning the susceptibility of HLA alleles in Caucasian MG, with highlights on DRB1*15:01 as being a major risk allele in LOMG

Myasthenia gravis

Myasthenia gravis (MG) is a rare, autoimmune neuromuscular junction disorder. Contemporary prevalence rates approach 1/5,000. MG presents with painless, fluctuating, fatigable weakness involving specific muscle groups. Ocular weakness with asymmetric ptosis and binocular diplopia is the most typical initial presentation, while early or isolated oropharyngeal or limb weakness is less common. The course is variable, and most patients with initial ocular weakness develop bulbar or limb weakness within three years of initial symptom onset. MG results from antibody-mediated, T cell-dependent immunologic attack on the endplate region of the postsynaptic membrane. In patients with fatigable muscle weakness, the diagnosis of MG is supported by: 1. pharmacologic testing with edrophonium chloride that elicits unequivocal improvement in strength; 2. electrophysiologic testing with repetitive nerve stimulation (RNS) studies and/or single-fiber electromyography (SFEMG) that demonstrates a primary postsynaptic neuromuscular junctional disorder; and 3. serologic demonstration of acetylcholine receptor (AChR) or muscle-specific tyrosine kinase (MuSK) antibodies. Differential diagnosis includes congenital myasthenic syndromes, Lambert Eaton syndrome, botulism, organophosphate intoxication, mitochondrial disorders involving progressive external ophthalmoplegia, acute inflammatory demyelinating polyradiculoneuropathy (AIDP), motor neuron disease, and brainstem ischemia. Treatment must be individualized, and may include symptomatic treatment with cholinesterase inhibitors and immune modulation with corticosteroids, azathioprine, cyclosporine, and mycophenolate mofetil. Rapid, temporary improvement may be achieved for myasthenic crises and exacerbations with plasma exchange (PEX) or intravenous immunoglobulin (IVIg). Owing to improved diagnostic testing, immunotherapy, and intensive care, the contemporary prognosis is favorable with less than five percent mortality and nearly normal life expectancy

Effect of Restricted Preen-Gland Access on Maternal Self Maintenance and Reproductive Investment in Mallards

As egg production and offspring care are costly, females should invest resources adaptively into their eggs to optimize current offspring quality and their own lifetime reproductive success. Parasite infections can influence maternal investment decisions due to their multiple negative physiological effects. The act of preening--applying oils with anti-microbial properties to feathers--is thought to be a means by which birds combat pathogens and parasites, but little is known of how preening during the reproductive period (and its expected disease-protecting effects) influences maternal investment decisions at the level of the egg.Here, we experimentally prevented female mallards (Anas platyrhynchos) from accessing their preen gland during breeding and monitored female immunoresponsiveness (e.g., plasma lysozyme concentration) as well as some egg traits linked to offspring quality (e.g., egg mass, yolk carotenoid content, and albumen lysozyme levels). Females with no access to their preen gland showed an increase in plasma lysozyme level compared to control, normally preening females. In addition, preen-gland-restricted females laid significantly lighter eggs and deposited higher carotenoid concentrations in the yolk compared to control females. Albumen lysozyme activity did not differ significantly between eggs laid by females with or without preen gland access.Our results establish a new link between an important avian self-maintenance behaviour and aspects of maternal health and reproduction. We suggest that higher yolk carotenoid levels in eggs laid by preen-gland-restricted females may serve to boost health of offspring that would hatch in a comparatively microbe-rich environment