Perturbative expansion of N<8 Supergravity

We characterise the one-loop amplitudes for N=6 and N=4 supergravity in four dimensions. For N=6 we find that the one-loop n-point amplitudes can be expanded in terms of scalar box and triangle functions only. This simplification is consistent with a loop momentum power count of n-3, which we would interpret as being n+4 for gravity with a further -7 from the N=6 superalgebra. For N=4 we find that the amplitude is consistent with a loop momentum power count of n, which we would interpret as being n+4 for gravity with a further -4 from the N=4 superalgebra. Specifically the N=4 amplitudes contain non-cut-constructible rational terms.Comment: 13 pages. v2 adds analytic expression for rational parts of 5-pt 1-loop N=4 SUGRA amplitude; v3 normalisations clarifie

Obtaining One-loop Gravity Amplitudes Using Spurious Singularities

The decomposition of a one-loop scattering amplitude into elementary functions with rational coefficients introduces spurious singularities which afflict individual coefficients but cancel in the complete amplitude. These cancellations create a web of interactions between the various terms. We explore the extent to which entire one-loop amplitudes can be determined from these relationships starting with a relatively small input of initial information, typically the coefficients of the scalar integral functions as these are readily determined. In the context of one-loop gravity amplitudes, of which relatively little is known, we find that some amplitudes with a small number of legs can be completely determined from their box coefficients. For increasing numbers of legs, ambiguities appear which can be determined from the physical singularity structure of the amplitude. We illustrate this with the four-point and N=1,4 five-point (super)gravity one-loop amplitudes.Comment: Minor corrections. Appendix adde

The n-point MHV one-loop Amplitude in N=4 Supergravity

We present an explicit formula for the n-point MHV one-loop amplitude in a N=4 supergravity theory. This formula is derived from the soft and collinear factorisations of the amplitude.Comment: 8 pages; v2 References added. Minor changes to tex

Andreev's Theorem on hyperbolic polyhedra

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing $C$ with the assigned dihedral angles. Andreev's Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev's Theorem is both an interesting statement about the geometry of hyperbolic 3-dimensional space, as well as a fundamental tool used in the proof for Thurston's Hyperbolization Theorem for 3-dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev's Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev's proof of existence and also provide a readable new proof of the other parts of the proof of Andreev's Theorem, because Andreev's paper has the reputation of being ``unreadable''.Comment: To appear les Annales de l'Institut Fourier. 47 pages and many figures. Revision includes significant modification to section 4, making it shorter and more rigorous. Many new references include

La Porta D\u27Oro

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Glaucoma and the Optic Nerve

In summary, the anatomic characteristics of the optic nerve head have been described along the ophthalmoscopic interpretation of these characteristics. One hopes that this information combined with a knowledge of what constitutes glaucomatous abnormality in the optic disc will encourage the ophthalmic as well as non-ophthalmic practitioner to evaluate the optic nerve head and recognize those which are suspicious of glaucoma

Analytic results for two-loop Yang-Mills

Recent Developments in computing very specific helicity amplitudes in two loop QCD are presented. The techniques focus upon the singular structure of the amplitude rather than on a diagramatic and integration approachComment: Talk presented at 13th International Symposium on Radiative Corrections, 24-29 September, 2017,St. Gilgen, Austria, 9 page

Constructing Gravity Amplitudes from Real Soft and Collinear Factorisation

Soft and collinear factorisations can be used to construct expressions for amplitudes in theories of gravity. We generalise the "half-soft" functions used previously to "soft-lifting" functions and use these to generate tree and one-loop amplitudes. In particular we construct expressions for MHV tree amplitudes and the rational terms in one-loop amplitudes in the specific context of N=4 supergravity. To completely determine the rational terms collinear factorisation must also be used. The rational terms for N=4 have a remarkable diagrammatic interpretation as arising from algebraic link diagrams.Comment: 18 pages, axodraw, Proof of eq. 4.3 adde

Clinical Perspectives and Trends: Microperimetry as a trial endpoint in retinal disease

Endpoint development trials are underway across the spectrum of retinal disease. New validated endpoints are urgently required for the assessment of emerging gene therapies and in preparation for the arrival of novel therapeutics targeting early stages of common sight-threatening conditions such as age-related macular degeneration. Visual function measures are likely to be key candidates in this search. Over the last two decades, microperimetry has been used extensively to characterize functional vision in a wide range of retinal conditions, detecting subtle defects in retinal sensitivity that precede visual acuity loss and tracking disease progression over relatively short periods. Given these appealing features, microperimetry has already been adopted as an endpoint in interventional studies, including multicenter trials, on a modest scale. A review of its use to date shows a concurrent lack of consensus in test strategy and a wealth of innovative disease and treatment-specific metrics which may show promise as clinical trial endpoints. There are practical issues to consider, but these have not held back its popularity and it remains a widely used psychophysical test in research. Endpoint development trials will undoubtedly be key in understanding the validity of microperimetry as a clinical trial endpoint, but existing signs are promising