Realization of functions on the symmetrized bidisc

We prove a realization formula and a model formula for analytic functions with modulus bounded by 1 on the symmetrized bidisc G = def {( z + w , z w ) : | z | < 1 , | w | < 1} . As an application we prove a Pick-type theorem giving a criterion for the existence of such a function satisfying a finite set of interpolation conditions

Pinch Technique and the Batalin-Vilkovisky formalism

In this paper we take the first step towards a non-diagrammatic formulation of the Pinch Technique. In particular we proceed into a systematic identification of the parts of the one-loop and two-loop Feynman diagrams that are exchanged during the pinching process in terms of unphysical ghost Green's functions; the latter appear in the standard Slavnov-Taylor identity satisfied by the tree-level and one-loop three-gluon vertex. This identification allows for the consistent generalization of the intrinsic pinch technique to two loops, through the collective treatment of entire sets of diagrams, instead of the laborious algebraic manipulation of individual graphs, and sets up the stage for the generalization of the method to all orders. We show that the task of comparing the effective Green's functions obtained by the Pinch Technique with those computed in the background field method Feynman gauge is significantly facilitated when employing the powerful quantization framework of Batalin and Vilkovisky. This formalism allows for the derivation of a set of useful non-linear identities, which express the Background Field Method Green's functions in terms of the conventional (quantum) ones and auxiliary Green's functions involving the background source and the gluonic anti-field; these latter Green's functions are subsequently related by means of a Schwinger-Dyson type of equation to the ghost Green's functions appearing in the aforementioned Slavnov-Taylor identity.Comment: 45 pages, uses axodraw; typos corrected, one figure changed, final version to appear in Phys.Rev.

Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions.Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u=(u1,…,uN) : EL[u,X]=⎧⎩⎨⎪⎪Δu=div(P(x)cof∇u)det∇u=1u≡φinX,inX,on∂X, where X is a finite, open, symmetric N -annulus (with N≥2 ), P=P(x) is an unknown hydrostatic pressure field and φ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N=2 , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N≥4 and discuss a number of closely related issues

Nevanlinna representations in several variables

We generalize to several variables the classical theorem of Nevanlinna that characterizes the Cauchy transforms of positive measures on the real line. We show that for the Loewner class, a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space. We identify four types of such representations, and we obtain function-theoretic conditions that are necessary and sufficient for a given function to possess a representation of each of the four types

On a class of stationary loops on SO(n) and the existence of multiple twisting solutions to a nonlinear elliptic system subject to a hard incompressibility constraint

For full abstract please refer to http://dx.doi.org/10.1186/s13661-018-1047-

Finite Blaschke products and the construction of rational Γ-inner functions

A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bΓ of Γ. A rational Γ-inner function h induces a continuous map h|T from T to b Γ. The latter set is topologically a Möbius band and so has fundamental group Z. The degree of h is defined to be the topological degree of h|T. In a previous paper the authors showed that if h=(s,p) is a rational Γ-inner function of degree n then s2−4p has exactly n zeros in the closed unit disc D−, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n zeros of s2−4p prescribed

The value of carbon sequestration and storage in coastal habitats

Coastal margin habitats are globally significant in terms of their capacity to sequester and store carbon, but their continuing decline, due to environmental change and human land use decisions, is reducing their capacity to provide this ecosystem service. In this paper the UK is used as a case study area to develop methodologies to quantify and value the ecosystem service of blue carbon sequestration and storage in coastal margin habitats. Changes in UK coastal habitat area between 1900 and 2060 are documented, the long term stocks of carbon stored by these habitats are calculated, and the capacity of these habitats to sequester CO2 is detailed. Changes in value of the carbon sequestration service of coastal habitats are then projected for 2000–2060 under two scenarios, the maintenance of the current state of the habitat and the continuation of current trends of habitat loss. If coastal habitats are maintained at their current extent, their sequestration capacity over the period 2000–2060 is valued to be in the region of £1 billion UK sterling (3.5% discount rate). However, if current trends of habitat loss continue, the capacity of the coastal habitats both to sequester and store CO2 will be significantly reduced, with a reduction in value of around £0.25 billion UK sterling (2000–2060; 3.5% discount rate). If loss-trends due to sea level rise or land reclamation worsen, this loss in value will be greater. This case study provides valuable site specific information, but also highlights global issues regarding the quantification and valuation of carbon sequestration and storage. Whilst our ability to value ecosystem services is improving, considerable uncertainty remains. If such ecosystem valuations are to be incorporated with confidence into national and global policy and legislative frameworks, it is necessary to address this uncertainty. Recommendations to achieve this are outlined

On the Behavior of the Effective QCD Coupling alpha_tau(s) at Low Scales

The hadronic decays of the tau lepton can be used to determine the effective charge alpha_tau(m^2_tau') for a hypothetical tau-lepton with mass in the range 0 < m_tau' < m_tau. This definition provides a fundamental definition of the QCD coupling at low mass scales. We study the behavior of alpha_tau at low mass scales directly from first principles and without any renormalization-scheme dependence by looking at the experimental data from the OPAL Collaboration. The results are consistent with the freezing of the physical coupling at mass scales s = m^2_tau' of order 1 GeV^2 with a magnitude alpha_tau ~ 0.9 +/- 0.1.Comment: 15 pages, 4 figures, submitted to Physical Review D, added references, some text added, no results nor figures change

Search for the standard model Higgs boson decaying into two photons in pp collisions at sqrt(s)=7 TeV

A search for a Higgs boson decaying into two photons is described. The analysis is performed using a dataset recorded by the CMS experiment at the LHC from pp collisions at a centre-of-mass energy of 7 TeV, which corresponds to an integrated luminosity of 4.8 inverse femtobarns. Limits are set on the cross section of the standard model Higgs boson decaying to two photons. The expected exclusion limit at 95% confidence level is between 1.4 and 2.4 times the standard model cross section in the mass range between 110 and 150 GeV. The analysis of the data excludes, at 95% confidence level, the standard model Higgs boson decaying into two photons in the mass range 128 to 132 GeV. The largest excess of events above the expected standard model background is observed for a Higgs boson mass hypothesis of 124 GeV with a local significance of 3.1 sigma. The global significance of observing an excess with a local significance greater than 3.1 sigma anywhere in the search range 110-150 GeV is estimated to be 1.8 sigma. More data are required to ascertain the origin of this excess.Comment: Submitted to Physics Letters