Abstract:
We provide a method for obtaining upper estimates of the resolvent kernel of the Laplacian on a post-critically finite self-similar fractal that relies on a self-similar series decomposition of the resolvent. Decay estimates on the positive real axis are proved by analyzing functions satisfying an interior eigenfunction condition with positive eigenvalue. These lead to estimates on the complement of the negative real axis via the Phragmen-Lindelof theorem. Applications are given to kernels for functions of the Laplacian, including the heat kernel, and to proving the existence of a self-similar series decomposition for the Laplacian resolvent on fractal blowups.

Abstract:
The aim of this study is to devise numerical methods for dealing with very high-dimensional Bermudan-style derivatives. For such problems, we quickly see that we can at best hope for price bounds, and we can only use a simulation approach. We use the approach of Barraquand & Martineau which proposes that the reward process should be treated as if it were Markovian, and then uses this to generate a stopping rule and hence a lower bound on the price. Using the dual approach introduced by Rogers, and Haugh & Kogan, this approximate Markov process leads us to hedging strategies, and upper bounds on the price. The methodology is generic, and is illustrated on eight examples of varying levels of difficulty. Run times are largely insensitive to dimension.

Abstract:
This paper presents an approach to estimating a hidden process in a continuous-time setting, where the hidden process is a diffusion. The approach is simply to minimize the negative log-likelihood of the hidden path, where the likelihood is expressed relative to Wiener measure. This negative log-likelihood is the action integral of the path, which we minimize by calculus of variations. We then perform an asymptotic maximum-likelihood analysis to understand better how the actual path is distributed around the least-action path; it turns out that the actual path can be expressed (approximately) as the sum of the least-action path and a zero-mean Gaussian process which can be specified quite explicitly. Numerical solution of the ODEs which arise from the calculus of variations is often feasible, but is complicated by the shooting nature of the problem, and the possibility that we have found a local but not global minimum. We analyze the situations when this happens, and provide effective numerical methods for studying this. We also show how the methodology works in a situation where the hidden positive diffusion acts as the random intensity of a point process which is observed; here too it is possible to estimate the hidden process.

Abstract:
A six-year-old girl presented with gradual loss of vision in the left eye a year after sustaining a lightning strike while in her home. Examination revealed healed burns to her cheek, left arm, and right leg and a dense left cataract. There was no evidence of other ocular sequelae, and her right eye was normal. Cataract surgery and lens implantation were performed on the left eye with good results. Isolated, unilateral, paediatric cataract due to lightning is discussed.

Abstract:
Two canonical models of statistical mechanics, the fully-connected voter and Glauber-Ising models, are modified to incorporate growth via the addition or replication of spins. The resulting behaviour is examined in a regime where the timescale of expansion cannot be separated from that of the internal dynamics. Depending on the model specification, growth radically alters the long-time dynamical behaviour by breaking or unbreaking ergodicity.

Abstract:
We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and construct all possible resistance (Dirichlet) forms, and the corresponding Laplacians, for which the topology in the effective resistance metric coincides with the usual topology. Then we concentrate on two particular cases. One is a self-similar harmonic structure, for which the energy renormalization factor is 2, the spectral dimension is $\log9/\log6$, and we can compute all the eigenvalues and eigenfunctions by a spectral decimation method. The other is graph-directed self-similar under the map $z\mapsto P(z)$; it has energy renormalization factor $\sqrt2$ and spectral dimension 4/3, but the exact computation of the spectrum is difficult. The latter Dirichlet form and Laplacian are in a sense conformally invariant on the basilica Julia set.

Abstract:
The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into $\mathbb{R}$ and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling.

Abstract:
Anti-predator behaviour of magpies was investigated, using five species of model predators, at times of raising offspring. We predicted differences in mobbing strategies for each predator presented and also that raising juveniles would affect intensity of the mobbing event. Fourteen permanent resident family groups were tested using 5 different types of predator (avian and reptilian) known to be of varying degrees of risk to magpies and common in their habitat. In all, 210 trials were conducted (across three different stages of juvenile development). We found that the stage of juvenile development did not alter mobbing behaviour significantly, but predator type did. Aerial strategies (such as swooping) were elicited by taxidermic models of raptors, whereas a taxidermic model of a monitor lizard was approached on the ground and a model snake was rarely approached. Swooping patterns also changed according to which of the three raptors was presented. Our results show that, in contrast to findings in other species, magpies vary mobbing strategy depending on the predator rather than varying mobbing intensity.

Abstract:
The degree of bitterness and pungency of a virgin olive oil largely defines its style, and therefore how it is most appropriately used by consumers. In order to assess how Australian olive oil producers interpret the style of their oils, 920 Australian virgin olive oils were classified by their producers as either being mild, medium or robust in style. Although in general, the classifications by producers were associated with the oils’ total phenol concentration, significant variability in phenol concentration within each style category was observed. The perceived styles of a subset of these oils were further assessed by panels of expert tasters. The expert panels were more discriminating when assigning oils to style categories based on total phenol levels. The producers and the expert panels were in moderate agreement with respect to oil style, with the interpretation of what constitutes a mild oil being the most contentious. El grado de amargor y picante de un aceite de oliva define en gran manera su tipo, y, por tanto, también su uso más apropiado por el consumidor. Para evaluar cómo los productores australianos de aceite de oliva interpretan el tipo de sus aceites, 920 aceites de oliva virgen australianos fueron clasificados por sus productores en tipo suave, medio o fuerte. Aunque, en general, la clasificación de los productores estuvo asociada a la concentración de fenoles totales de los aceites, se observó una variabilidad significativa en la concentración de fenoles en cada tipo de aceite. Los tipos percibidos en un subgrupo de estos aceites fueron además evaluados por paneles de catadores expertos. Los paneles de expertos fueron más discriminantes cuando asignaron los aceites a los diversos tipos basándose en el nivel de fenoles totales. Los productores y los paneles de expertos estuvieron en relativamente de acuerdo con respecto al tipo de aceite, si bien la interpretación de qué es un aceite suave fue la más conflictiva.

Abstract:
This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-consistency properties for a firm seeking to spread its risk across a group of subsidiaries.