Ullemar's formula for the Jacobian of the complex moment mapping

The complex moment sequence m(P) is assigned to a univalent polynomial P by the Cauchy transform of the P(D), where D is the unit disk. We establish the representation of the Jacobian det dm(P) in terms of roots of the derivative P'. Combining this result with the special decomposition for the Hurwitz determinants, we prove a formula for the Jacobian which was previously conjectured by C. Ullemar. As a consequence, we show that the boundary of the class of all locally univalent polynomials in $U$ is contained in the union of three irreducible algebraic surfaces.Comment: 14 pages, submitted for "Complex Variables. Theory and Application

Gravitating discs around black holes

Fluid discs and tori around black holes are discussed within different approaches and with the emphasis on the role of disc gravity. First reviewed are the prospects of investigating the gravitational field of a black hole--disc system by analytical solutions of stationary, axially symmetric Einstein's equations. Then, more detailed considerations are focused to middle and outer parts of extended disc-like configurations where relativistic effects are small and the Newtonian description is adequate. Within general relativity, only a static case has been analysed in detail. Results are often very inspiring, however, simplifying assumptions must be imposed: ad hoc profiles of the disc density are commonly assumed and the effects of frame-dragging and completely lacking. Astrophysical discs (e.g. accretion discs in active galactic nuclei) typically extend far beyond the relativistic domain and are fairly diluted. However, self-gravity is still essential for their structure and evolution, as well as for their radiation emission and the impact on the environment around. For example, a nuclear star cluster in a galactic centre may bear various imprints of mutual star--disc interactions, which can be recognised in observational properties, such as the relation between the central mass and stellar velocity dispersion.Comment: Accepted for publication in CQG; high-resolution figures will be available from http://www.iop.org/EJ/journal/CQ

Quantum field theory and Hopf algebra cohomology

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.Comment: 39 pages, no figures, LaTeX + AMS macros; title changed, minor corrections, references update

Management of retinal vascular diseases: a patient-centric approach

Retinal vascular diseases are a leading cause of blindness in the Western world. Advancement in the clinical management of these diseases has been fast-paced, with new treatments becoming available as well as license extensions of existing treatments. Vascular endothelial growth factor (VEGF) has been implicated in certain retinal vascular diseases, including wet age-related macular degeneration (AMD), diabetic macular oedema (DMO), and retinal vein occlusion (RVO). Treatment of wet AMD and visual impairment due to either DMO or macular oedema secondary to RVO with an anti-VEGF on an as needed basis, rather than a fixed schedule, allows an individualised treatment approach; providing treatment when patients are most likely to benefit from it, while minimising the number of unnecessary intravitreal injections. Thus, an individualised treatment regimen reduces the chances of over-treatment and under-treatment, optimising both the risk/benefit profile of the treatment and the efficient use of NHS resource. Streamlining of treatment for patients with wet AMD and visual impairment due to either DMO or macular oedema secondary to RVO, by using one treatment with similar posology across all three diseases, may help to minimise burden of clinic capacity and complexity and hence optimise patient outcomes. Informed treatment decisions and efficient clinic throughput are important for optimal patient outcomes in the fast-changing field of retinal vascular diseases

Supersymmetric QCD corrections to e+e−→tbˉH−e^+e^-\to t\bar{b}H^- and the Bernstein-Tkachov method of loop integration

The discovery of charged Higgs bosons is of particular importance, since their existence is predicted by supersymmetry and they are absent in the Standard Model (SM). If the charged Higgs bosons are too heavy to be produced in pairs at future linear colliders, single production associated with a top and a bottom quark is enhanced in parts of the parameter space. We present the next-to-leading-order calculation in supersymmetric QCD within the minimal supersymmetric SM (MSSM), completing a previous calculation of the SM-QCD corrections. In addition to the usual approach to perform the loop integration analytically, we apply a numerical approach based on the Bernstein-Tkachov theorem. In this framework, we avoid some of the generic problems connected with the analytical method.Comment: 14 pages, 6 figures, accepted for publication in Phys. Rev.

25th RCOphth Congress, President's Session paper:25 years of progress in medical retina

The quarter century since the foundation of the Royal College of Ophthalmologists has coincided with immense change in the subspecialty of medical retina, which has moved from being the province of a few dedicated enthusiasts to being an integral, core part of ophthalmology in every eye department. In age-related macular degeneration, there has been a move away from targeted, destructive laser therapy, dependent on fluorescein angiography to intravitreal injection therapy of anti-growth factor agents, largely guided by optical coherence tomography. As a result of these changes, ophthalmologists have witnessed a marked improvement in visual outcomes for their patients with wet age-related macular degeneration (AMD), while at the same time developing and enacting entirely novel ways of delivering care. In the field of diabetic retinopathy, this period also saw advances in laser technology and a move away from highly destructive laser photocoagulation treatment to gentler retinal laser treatments. The introduction of intravitreal therapies, both steroids and anti-growth factor agents, has further advanced the treatment of diabetic macular oedema. This era has also seen in the United Kingdom the introduction of a coordinated national diabetic retinopathy screening programme, which offers an increasing hope that the burden of blindness from diabetic eye disease can be lessened. Exciting future advances in retinal imaging, genetics, and pharmacology will allow us to further improve outcomes for our patients and for ophthalmologists specialising in medical retina, the future looks very exciting but increasingly busy

Branch Retinal Vein Occlusion: Pathogenesis, Visual Prognosis, and Treatment Modalities

In branch retinal vein occlusion (BRVO), abnormal arteriovenous crossing with vein compression, degenerative changes of the vessel wall and abnormal hematological factors constitute the primary mechanism of vessel occlusion. In general, BRVO has a good prognosis: 50–60% of eyes are reported to have a final visual acuity (VA) of 20/40 or better even without treatment. One important prognostic factor for final VA appears to be the initial VA. Grid laser photocoagulation is an established treatment for macular edema in a particular group of patients with BRVO, while promising results for this condition are shown by intravitreal application of steroids or new vascular endothelial growth factor inhibitors. Vitrectomy with or without arteriovenous sheathotomy combined with removal of the internal limiting membrane may improve vision in eyes with macular edema which are unresponsive to or ineligible for laser treatment

Fluids and barriers of the CNS: a historical viewpoint

Tracing the exact origins of modern science can be a difficult but rewarding pursuit. It is possible for the astute reader to follow the background of any subject through the many important surviving texts from the classical and ancient world. While empirical investigations have been described by many since the time of Aristotle and scientific methods have been employed since the Middle Ages, the beginnings of modern science are generally accepted to have originated during the 'scientific revolution' of the 16th and 17th centuries in Europe. The scientific method is so fundamental to modern science that some philosophers consider earlier investigations as 'pre-science'. Notwithstanding this, the insight that can be gained from the study of the beginnings of a subject can prove important in the understanding of work more recently completed. As this journal undergoes an expansion in focus and nomenclature from cerebrospinal fluid (CSF) into all barriers of the central nervous system (CNS), this review traces the history of both the blood-CSF and blood-brain barriers from as early as it was possible to find references, to the time when modern concepts were established at the beginning of the 20th century

Erratum to: 36th International Symposium on Intensive Care and Emergency Medicine

[This corrects the article DOI: 10.1186/s13054-016-1208-6.]