Algebras generated by two bounded holomorphic functions

We study the closure in the Hardy space or the disk algebra of algebras generated by two bounded functions, of which one is a finite Blaschke product. We give necessary and sufficient conditions for density or finite codimension of such algebras. The conditions are expressed in terms of the inner part of a function which is explicitly derived from each pair of generators. Our results are based on identifying z-invariant subspaces included in the closure of the algebra. Versions of these results for the case of the disk algebra are given.Comment: 22 pages ; a number of minor mistakes have been corrected, and some points clarified. Conditionally accepted by Journal d'Analyse Mathematiqu

A phase-I trial of pre-operative, margin intensive, stereotactic body radiation therapy for pancreatic cancer: the 'SPARC' trial protocol.

BACKGROUND: Standard therapy for borderline-resectable pancreatic cancer in the UK is surgery with adjuvant chemotherapy, but rates of resection with clear margins are unsatisfactory and overall survival remains poor. Meta-analysis of single-arm studies shows the potential of neo-adjuvant chemo-radiotherapy but the relative radio-resistance of pancreatic cancer means the efficacy of conventional dose schedules is limited. Stereotactic radiotherapy achieves sufficient accuracy and precision to enable pre-operative margin-intensive dose escalation with the goal of increasing rates of clear resection margins and local disease control. METHODS/DESIGN: SPARC is a "rolling-six" design single-arm study to establish the maximum tolerated dose for margin-intensive stereotactic radiotherapy before resection of pancreatic cancer at high risk of positive resection margins. Eligible patients will have histologically or cytologically proven pancreatic cancer defined as borderline-resectable per National Comprehensive Cancer Network criteria or operable tumour in contact with vessels increasing the risk of positive margin. Up to 24 patients will be recruited from up to 5 treating centres and a 'rolling-six' design is utilised to minimise delays and facilitate ongoing recruitment during dose-escalation. Radiotherapy will be delivered in 5 daily fractions and surgery, if appropriate, will take place 5-6 weeks after radiotherapy. The margin-intense radiotherapy concept includes a systematic method to define the target volume for a simultaneous integrated boost in the region of tumour-vessel infiltration, and up to 4 radiotherapy dose levels will be investigated. Maximum tolerated dose is defined as the highest dose at which no more than 1 of 6 patients or 0 of 3 patients experience a dose limiting toxicity. Secondary endpoints include resection rate, resection margin status, response rate, overall survival and progression free survival at 12 and 24 months. Translational work will involve exploratory analyses of the cytological and humoral immunological responses to stereotactic radiotherapy in pancreatic cancer. Radiotherapy quality assurance of target definition and radiotherapy planning is enforced with pre-trial test cases and on-trial review. Recruitment began in April 2015. DISCUSSION: This prospective multi-centre study aims to establish the maximum tolerated dose of pre-operative margin-intensified stereotactic radiotherapy in pancreatic cancer at high risk of positive resection margins with a view to subsequent definitive comparison with other neoadjuvant treatment options

Calcium-Dependent Increases in Protein Kinase-A Activity in Mouse Retinal Ganglion Cells Are Mediated by Multiple Adenylate Cyclases

Neurons undergo long term, activity dependent changes that are mediated by activation of second messenger cascades. In particular, calcium-dependent activation of the cyclic-AMP/Protein kinase A signaling cascade has been implicated in several developmental processes including cell survival, axonal outgrowth, and axonal refinement. The biochemical link between calcium influx and the activation of the cAMP/PKA pathway is primarily mediated through adenylate cyclases. Here, dual imaging of intracellular calcium concentration and PKA activity was used to assay the role of different classes of calcium-dependent adenylate cyclases (ACs) in the activation of the cAMP/PKA pathway in retinal ganglion cells (RGCs). Surprisingly, depolarization-induced calcium-dependent PKA transients persist in barrelless mice lacking AC1, the predominant calcium-dependent adenylate cyclase in RGCs, as well as in double knockout mice lacking both AC1 and AC8. Furthermore, in a subset of RGCs, depolarization-induced PKA transients persist during the inhibition of all transmembrane adenylate cyclases. These results are consistent with the existence of a soluble adenylate cyclase that plays a role in calcium-dependent activation of the cAMP/PKA cascade in neurons

Spectral reconstruction and representations of finitely generated groups

It is well-known that characters classify linear representations of finite groups, that is if characters of two representations of a finite group are the same, these representations are equivalent. It is also well-known that, in general, this is not true for representations of infinite groups, even if they are finitely generated. The goal of this paper is to establish a characterization of representations of finitely generated groups in terms of projective joint spectra. This approach has a clear advantage compared to character classification as it is valid for a much wider family of groups and for both finite and infinite dimensional representations. The main tool in establishing our spectral characterization is a reconstruction of an operator acting on a separable Hilbert space from the proper projective joint spectrum of the quadruple containing this operator along with a certain triple of bounded operators acting on the same space.Comment: The old version of the paper has to be withdrawn. Once a new version is ready, it will be posted to ArXi

Generalized Factorization in Hardy Spaces and the Commutant of Toeplitz Operators

Abstract. Every classical inner function ϕ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when ϕ(z) = z. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.

Composition operators on the polydisc induced by smooth symbols

AbstractWe study composition operators CΦ on the Hardy spaces Hp and weighted Bergman spaces Aαp of the polydisc Dn in Cn. When Φ is of class C2 on Dn¯, we show that CΦ is bounded on Hp or Aαp if and only if the Jacobian of Φ does not vanish on those points ζ on the distinguished boundary Tn such that Φ(ζ)∈Tn. Moreover, we show that if ε>0 and if CΦ:Aαp→Aα+12n−εp, then CΦ is bounded on Aαp