Ricci flows with unbounded curvature

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).Comment: 12 pages, 1 figure; updated reference

Study of research and development requirements of small gas-turbine combustors

A survey is presented of the major small-engine manufacturers and governmental users. A consensus was undertaken regarding small-combustor requirements. The results presented are based on an evaluation of the information obtained in the course of the study. The current status of small-combustor technology is reviewed. The principal problems lie in liner cooling, fuel injection, part-power performance, and ignition. Projections of future engine requirements and their effect on the combustor are discussed. The major changes anticipated are significant increases in operating pressure and temperature levels and greater capability of using heavier alternative fuels. All aspects of combustor design are affected, but the principal impact is on liner durability. An R&D plan which addresses the critical combustor needs is described. The plan consists of 15 recommended programs for achieving necessary advances in the areas of liner thermal design, primary-zone performance, fuel injection, dilution, analytical modeling, and alternative-fuel utilization

Ricci flows with bursts of unbounded curvature

Given a completely arbitrary surface, whether or not it has bounded curvature, or even whether or not it is complete, there exists an instantaneously complete Ricci flow evolution of that surface that exists for a specific amount of time [GT11]. In the case that the underlying Riemann surface supports a hyperbolic metric, this Ricci flow always exists for all time and converges (after scaling by a factor 1/2t ) to this hyperbolic metric [GT11], i.e. our Ricci flow geometrises the surface. In this paper we show that there exist complete, bounded curvature initial metrics, including those conformal to a hyperbolic metric, which have subsequent Ricci flows developing unbounded curvature at certain intermediate times. In particular, when coupled with the uniqueness from [Top13], we find that any complete Ricci flow starting with such initial metrics must develop unbounded curvature over some intermediate time interval, but that nevertheless, the curvature must later become bounded and the flow must achieve geometrisation as t → ∞, even though there are other conformal deformations to hyperbolic metrics that do not involve unbounded curvature. Another consequence of our constructions is that while our Ricci flow from [GT11] must agree initially with the classical flow of Hamilton and Shi in the special case that the initial surface is complete and of bounded curvature, by uniqueness, it is now clear that our flow lasts for a longer time interval in general, with Shi’s flow stopping when the curvature blows up, but our flow continuing strictly beyond in these situations. All our constructions of unbounded curvature developing and then disappearing are in two dimensions. Generalisations to higher dimensions are then immediate

Flowing maps to minimal surfaces: Existence and uniqueness of solutions

We study the new geometric flow that was introduced in [11] that evolves a pair of map and (domain) metric in such a way that it changes appropriate initial data into branched minimal immersions. In the present paper we focus on the existence theory as well as the issue of uniqueness of solutions. We establish that a (weak) solution exists for as long as the metrics remain in a bounded region of moduli space, i.e. as long as the flow does not collapse a closed geodesic in the domain manifold to a point. Furthermore, we prove that this solution is unique in the class of all weak solutions with non-increasing energy. This work complements the paper [11] of Topping and the author where the flow was introduced and its asymptotic convergence to branched minimal immersions is discussed

Developing a Culture of Publication: a joint enterprise writing retreat

Purpose: Many students irrespective of level of study produce excellent course work which, if given support and encouragement could clearly be of a publishable standard. Academic staff are expected to produce quality publications meeting peer review standards although they may be relatively novice authors. All are engaged in some aspects of academic writing practices but not as frequently involved in co-production of publications emanating from student work. This activity is still at the margins of much of the student experience. Design/methodology: Mindful of these issues, we designed and offered a writing programme including a writing retreat. This brought together undergraduate and postgraduate students from a range of applied disciplines (health and art, design and architecture) and their supervisors with the aim of co-producing publications and participating in a community of scholarly practice. The project was delivered over nine months. It involved four days ‘compulsory’ attendance and included a preparatory workshop, a two day off campus writing retreat and a dissemination event. Student and supervisors applied to participate as a team. Kirkpatrick’s (2006) four-stage classic model: reaction, learning, changes in behaviour and real world results was used as a framework for the educational evaluation. Key findings organised thematically were: Supervisor-supervisee relationships; space and time; building confidence enabling successful writing and publication. Originality/Value: This paper will provide an overview of the design, content and approaches used for successful delivery of this innovative project. It will draw on examples that illustrate the different types of joint enterprise that emerged, illuminate experiences of co-production and co-authorship along with recommendations for future ventures

Inverse modelling of Köhler theory – Part 1: A response surface analysis of CCN spectra with respect to surface-active organic species

This is the final version of the article. Available from European Geosciences Union (EGU) and Copernicus Publications via the DOI in this record.In this study a novel framework for inverse modelling of cloud condensation nuclei (CCN) spectra is developed using Köhler theory. The framework is established by using model-generated synthetic measurements as calibration data for a parametric sensitivity analysis. Assessment of the relative importance of aerosol physicochemical parameters, while accounting for bulk–surface partitioning of surface-active organic species, is carried out over a range of atmospherically relevant supersaturations. By introducing an objective function that provides a scalar metric for diagnosing the deviation of modelled CCN concentrations from synthetic observations, objective function response surfaces are presented as a function of model input parameters. Crucially, for the chosen calibration data, aerosol–CCN spectrum closure is confirmed as a well-posed inverse modelling exercise for a subset of the parameters explored herein. The response surface analysis indicates that the appointment of appropriate calibration data is particularly important. To perform an inverse aerosol–CCN closure analysis and constrain parametric uncertainties, it is shown that a high-resolution CCN spectrum definition of the calibration data is required where single-valued definitions may be expected to fail. Using Köhler theory to model CCN concentrations requires knowledge of many physicochemical parameters, some of which are difficult to measure in situ on the scale of interest and introduce a considerable amount of parametric uncertainty to model predictions. For all partitioning schemes and environments modelled, model output showed significant sensitivity to perturbations in aerosol log-normal parameters describing the accumulation mode, surface tension, organic : inorganic mass ratio, insoluble fraction, and solution ideality. Many response surfaces pertaining to these parameters contain well-defined minima and are therefore good candidates for calibration using a Monte Carlo Markov Chain (MCMC) approach to constraining parametric uncertainties. A complete treatment of bulk–surface partitioning is shown to predict CCN spectra similar to those calculated using classical Köhler theory with the surface tension of a pure water drop, as found in previous studies. In addition, model sensitivity to perturbations in the partitioning parameters was found to be negligible. As a result, this study supports previously held recommendations that complex surfactant effects might be neglected, and the continued use of classical Köhler theory in global climate models (GCMs) is recommended to avoid an additional computational burden. The framework developed is suitable for application to many additional composition-dependent processes that might impact CCN activation potential. However, the focus of this study is to demonstrate the efficacy of the applied sensitivity analysis to identify important parameters in those processes and will be extended to facilitate a global sensitivity analysis and inverse aerosol–CCN closure analysis.This work was supported by the UK Natural Environment Research Council grants NE/I020148/1 (AerosolCloud Interactions – A Directed Programme to Reduce Uncertainty in Forcing) and NE/J024252/1 (Global Aerosol Synthesis And Science Project). P. Stier would like to acknowledge funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC project ACCLAIM (grant agreement no. FP7-280025)

Existence of Ricci flows of incomplete surfaces

We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.Comment: 20 pages; updated to reflect galley proof correction

Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2

We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m=2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even "eternal oscillation".Comment: 34 page

Type-Decomposition of a Pseudo-Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.Comment: 18 page