We prove Lp norm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed in light of Lamé’s Theorem and the image method.

This paper investigates the distributions of triangle counts per vertex and edge, as a means for network description, analysis, model building, and other tasks. The main interest is in estimating these distributions through sampling, especially for large networks. A novel sampling method tailored for the estimation analysis is proposed, with three sampling designs motivated by several network access scenarios. An estimation method based on inversion and an asymptotic method are developed to recover the entire distribution. A single method to estimate the distribution using multiple samples is also considered. Algorithms are presented to sample the network under the various access scenarios. Finally, the estimation methods on synthetic and real-world networks are evaluated in a data study.

We examine a model of network formation in single-layer and multiplex networks in which individuals have positive incentives for social ties, closed triangles, and spillover edges. In particular, we investigate the influence of shocks to the network in which the cost of social ties changes after an initial equilibrium. We highlight the emergence of structural entrenchment: the retention of structural features, such as closed triangles and spillover edges, which are formed under historically different conditions from those currently driving network evolution. This work has broad implications for understanding path dependence in the structure and dynamics of single-layer and multiplex networks.

Bowen Family Systems Theory is most commonly used to understand and predict family process. It is also applied to other potentially intense relationship systems, especially the workplace. It has been used by workplace consultants and by individuals to understand, and to manage, their own workplace functioning. This paper will draw from several key Bowen concepts to analyse common workplace dilemmas and to suggest responses that may assist the functioning of the individual and the system. The author's own experience as a manager in a child and adolescent mental health programme will be used as the case study to illustrate the theory's application.

A Bowen Family Systems therapist employs concepts of triangles and the family projection process to view a child's symptoms as embedded in the broader family patterns. This article will examine the dynamics of two family therapy cases where parents anxiously asked for their children's symptoms to be fixed. These cases will be used to explore the common presentation in child and adolescent mental health, where the parents are concerned for their children but are also keen not to open their own ‘can of worms’. The presenting problem in the first case was violent hostility between adolescent sisters and in the second case was an adolescent's anorexia. Drawing on client feedback, I reflect on the therapy process behind the divergent outcomes. In case one, the parents were willing to address their own troubled relationship and family of origin, while in case two, the parents discontinued therapy when family of origin dynamics began to be explored. The article suggests how the therapist can evoke parents' curiosity about their role in anxious family patterns, without them feeling blamed.

Let T be a triangle. P be a parallelogram, E be an ellipse, A, B be concentric circles, C, D be concentric dartboard regions, R, S be rectangles of the same orientation, U, V be two finite unions and/or differences of convex regions in the Euclidean plane. Given a function f on [0,∞), let E[/(r), U, V] denote the mean value of f(|u–v|), where |u–v| is the distance between u∊U and v∊V. Using Borel’s overlap technique, a specific distance weight function and a specific equivalence relation, we obtain formulae expressing E[f(r), U, V] in terms of triple integrals, expressing E(rn, U, V), E[f(r), A, V] and E[f(r), R, V] in terms of double integrals, expressing E[f(r), A, B], E[f(r), R, S], E[f(r), T, T], E[f(r), P, P], E(rn, C, D) and E(rn, R, V) in terms of single integrals, and expressing E(rn, R, S), E(rn, P, P), E(rn, T, T), E(rn, E, E) in terms of elementary functions, where n is an integer ≧−1. Many other related results are also given.