The use of singular functions for the approximate conformal mapping of doubly-connected domains

Let f be the function which maps conformally a given doubly- connected domain onto a circular annulus. We consider the use of two closely related methods for determining approximations to f of the form fn (z) = z exp, ⎪⎩⎪⎨⎧⎭⎬⎫Σ−(z)uan1jjj where {uj} is a set of basis functions. The two methods are respectively a variational method, based on an extremum property of the function H(z) = f′(z)/f(z) - 1/z, and an orthononnalization method, based on approximating the function H by a finite Fourier series sum. The main purpose of the paper is to consider the use of the two methods for the mapping of domains having sharp corners, where corner singularities occur. We show, by means of numerical examples, that both methods are capable of producing approximations of high accuracy for the mapping of such "difficult" doubly-connected domains. The essential requirement for this is that the basis set {uj} contains singular functions that reflect the asymptotic behaviour of the function H in the neighbourhood of each "singular" corner

Numerical conformal mapping of exterior domains

The work of the present paper is closely related to the two numerical procedures described in [11], for determining approximations to the function which maps conformally a bounded simply-connected domain Ω1 , with boundary ∂Ω, onto the unit disc. Here, we consider the use of these procedures for the solution of the corresponding exterior problem, i.e. the problem of determining approximations to the mapping function which maps conformally the exterior domain Ω = compl(ΩI⋃∂Ω) onto the unit disc

Numerical techniques for conformal mapping onto a rectangle

This paper is concerned with the problem of determining approximations to the function F which maps conformally a simply-connected domain onto a rectangle R, so that four specified points on are mapped Ω∂respectively onto the four vertices of R. In particular, we study the following two classes of methods for the mapping of domains of the form . (i) Methods which approximate where f is an approximation to the conformal map of Q onto the unit disc, and S is a simple Schwarz-Christoffel transformation. (ii) Methods based on approximating the conformal map of a certain symmetric doubly-connected domain onto a circular annulus. Keywords: Conforma

An orthonormalization method for the approximate conformal mapping of multiply-connected domains

We consider the use of an orthonormalization method for constructing approximations to one of the standard conformal maps for multiply-connected domains. The method has been used successfully in [12], but only for the mapping of doubly-connected domains. Our purpose here is to consider its application to the mapping of domains whose connectivity is greater than two

Patients' unvoiced agendas in general practice consultations.

Objective: To investigate patients' agendas before consultation and to assess which aspects of agendas are voiced in the consultation and the effects of unvoiced agendas on outcomes. Design: Qualitative study. Setting: 20 general practices in south east England and the West Midlands. Participants: 35 patients consulting 20 general practitioners in appointment and emergency surgeries. Results: Patients' agendas are complex and multifarious. Only four of 35 patients voiced all their agendas in consultation. Agenda items most commonly voiced were symptoms and requests for diagnoses and prescriptions. The most common unvoiced agenda items were: worries about possible diagnosis and what the future holds; patients' ideas about what is wrong; side effects; not wanting a prescription; and information relating to social context. Agenda items that were not raised in the consultation often led to specific problem outcomes (for example, major misunderstandings), unwanted prescriptions, non-use of prescriptions, and non-adherence to treatment. In all of the 14 consultations with problem outcomes at least one of the problems was related to an unvoiced agenda item. Conclusion: Patients have many needs and when these are not voiced they can not be addressed. Some of the poor outcomes in the case studies were related to unvoiced agenda items. This suggests that when patients and their needs are more fully articulated in the consultation better health care may be effected. Steps should be taken in both daily clinical practice and research to encourage the voicing of patients' agenda

Misunderstandings in general practice prescribing decisions: a qualitative study

Objectives: To identify and describe misunderstandings between patients and doctors associated with prescribing decisions in general practice. Design: Qualitative study. Setting: 20 general practices in the West Midlands and south east England. Participants: 20 general practitioners and 35 consulting patients. Main outcome measures: Misunderstandings between patients and doctors that have potential or actual adverse consequences for taking medicine. Results: 14 categories of misunderstanding were identified relating to patient information unknown to the doctor, doctor information unknown to the patient, conflicting information, disagreement about attribution of side effects, failure of communication about doctor's decision, and relationship factors. All the misunderstandings were associated with lack of patients' participation in the consultation in terms of the voicing of expectations and preferences or the voicing of responses to doctors' decisions and actions. They were all associated with potential or actual adverse outcomes such as non-adherence to treatment. Many were based on inaccurate guesses and assumptions. In particular doctors seemed unaware of the relevance of patients' ideas about medicines for successful prescribing. Conclusions: Patients' participation in the consultation and the adverse consequences of lack of participation are important. The authors are developing an educational intervention that builds on these findings

What does the future hold for utility electricity efficiency programs?

This study develops projections of future spending and savings from electricity efficiency programs funded by electric utility customers in the United States through 2030 based on three scenarios. Our analysis relies on detailed bottom-up modeling of current state energy efficiency policies, demand-side management and integrated resource plans, and regulatory decisions. The three scenarios represent a range of potential outcomes given the policy environment at the time of the study and uncertainties in the broader economic and state policy environment in each state. We project spending to increase to $8.6 billion in 2030 in the medium scenario, about a 45 percent increase relative to 2016 spending. In the high case, annual spending increases to$11.1 billion in 2030 and remains relatively flat in the low case ($6.8 billion in 2030). Our analysis suggests that electricity efficiency programs funded by utility customers will continue to impact load growth significantly at least through 2030, as savings as a percent of retail sales are forecast at 0.7 percent in the medium scenario and 0.98 percent in the high scenario

Palladium-catalyzed reactions

Palladium is probably the most versatile and exploited transition metal in catalysis due to its capability to promote a myriad of organic transformations both at laboratory and industrial scales (alkylation, arylation, cyclization, hydrogenation, oxidation, isomerization, cross-coupling, cascade, radical reactions, etc [...