Differential Recursion Relations for Laguerre Functions on Symmetric Cones

Let $\Omega$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $\Omega$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(\Omega,d\mu_\nu)$, where $d\mu_\nu$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $\Omega$ invariant and fix the identity in $\Omega$. The space $L^2(\Omega,d\mu_\nu)$ supports a highest weight representation of the group $G$ of holomorphic diffeomorphisms that act on the tube domain $T(\Omega)=\Omega + iV.$ In this article we give an explicit formula for the action of the Lie algebra of $G$ and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on $\mathbb{R}^+$

Project Based Learning: Are There Any Academic Benefits for the Teacher or Students?

In this paper, I raise an issue often neglected in Project Based Learning (PBL) literature. What academic benefits, if any, does the teacher or the student gain by adopting PBL pedagogy in college? I argue that PBL by its structure yields little academic benefits for the teacher or the students, and this could affect motivation as well. I present some examples from my personal teaching experience in mathematics. And thus, as I explain, a more “traditional” project-based approach could be better for both teacher and students

Irrationality and human reasoning

In his account of intentional interpretation, Donald Davidson assumes that people are mostly rational. Several psychological experiments though, reveal that human beings deviate drastically from the normative standards of rationality. Therefore, some psychologists arrive to the conclusion that humans are mostly irrational. In this thesis, I raise some objections to both points of view. On the one hand, ascribing rationality to humans in an a priori manner seems a suspicious position to adopt, considering the empirical data that show otherwise. On the other hand, the validity of the experiments and what exactly they test can also be put in question, since the position that humans are in general irrational is also unacceptable intuitively. In this thesis, I suggest that the discrepancy is due to the notion of rationality we adopt, which I bring into question. I do not find convincing reasons that humans should be thought a priori as rational and I do not also see why humans should be called irrational just because they fail certain tests. Many of the alleged irrationalities in the tests can be explained if we adopt different styles of reasoning than the traditional ones. Hence, humans can count as rational in another way. But, is this what Davidson thinks of rational, or does he think of rationality in the traditional sense? I think the type of rationality that Davidson endorses relies on Classic Logical conditions, which makes it inflexible. A type of rationality that relies on Fuzzy Logical conditions, as I claim, is more appropriate to describe human rationality

Some Thoughts on the Epicurean Critique of Mathematics

In this paper, we give a comprehensive summary of the discussion on the Epicurean critique of mathematics and in particular of Euclid\u27s geometry. We examine the methodological critique of the Epicureans on mathematics and we assess whether a \u27mathematical atomism\u27 was proposed, and its implications. Finally, we examine the Epicurean philosophical stance on mathematics and evaluate whether it was on target or not

Tracking and modelling motion for biomechanical analysis

This thesis focuses on the problem of determining appropriate skeletal configurations for which a virtual animated character moves to desired positions as smoothly, rapidly, and as accurately as possible. During the last decades, several methods and techniques, sophisticated or heuristic, have been presented to produce smooth and natural solutions to the Inverse Kinematics (IK) problem. However, many of the currently available methods suffer from high computational cost and production of unrealistic poses. In this study, a novel heuristic method, called Forward And Backward Reaching Inverse Kinematics (FABRIK), is proposed, which returns visually natural poses in real-time, equally comparable with highly sophisticated approaches. It is capable of supporting constraints for most of the known joint types and it can be extended to solve problems with multiple end effectors, multiple targets and closed loops. FABRIK was compared against the most popular IK approaches and evaluated in terms of its robustness and performance limitations. This thesis also includes a robust methodology for marker prediction under multiple marker occlusion for extended time periods, in order to drive real-time centre of rotation (CoR) estimations. Inferred information from neighbouring markers has been utilised, assuming that the inter-marker distances remain constant over time. This is the first time where the useful information about the missing markers positions which are partially visible to a single camera is deployed. Experiments demonstrate that the proposed methodology can effectively track the occluded markers with high accuracy, even if the occlusion persists for extended periods of time, recovering in real-time good estimates of the true joint positions. In addition, the predicted positions of the joints were further improved by employing FABRIK to relocate their positions and ensure a fixed bone length over time. Our methodology is tested against some of the most popular methods for marker prediction and the results confirm that our approach outperforms these methods in estimating both marker and CoR positions. Finally, an efficient model for real-time hand tracking and reconstruction that requires a minimum number of available markers, one on each finger, is presented. The proposed hand model is highly constrained with joint rotational and orientational constraints, restricting the fingers and palm movements to an appropriate feasible set. FABRIK is then incorporated to estimate the remaining joint positions and to fit them to the hand model. Physiological constraints, such as inertia, abduction, flexion etc, are also incorporated to correct the final hand posture. A mesh deformation algorithm is then applied to visualise the movements of the underlying hand skeleton for comparison with the true hand poses. The mathematical framework used for describing and implementing the techniques discussed within this thesis is Conformal Geometric Algebra (CGA)

Laguerre functions associated to Euclidean Jordan algebras

Certain differential recursion relations for the Laguerre functions, defined on a symmetric cone Ω, can be derived from the representations of a specific Lie algebra on L2(Ω,dμv). This Lie algebra is the corresponding Lie algebra of the Lie group G that acts on the tube domain T(Ω)=Ω+iV, where V is the associated Euclidean Jordan algebra of Ω. The representations involved are the highest weight representations of G on L2(Ω,dμv). To obtain these representations, we start from the highest weight representations of G on Hv(T(Ω)), the Hilbert space of holomorphic functions on T(Ω), and we transfer the representations to L2(Ω,dμv) via the Laplace transform. The Laguerre functions correspond to an orthogonal set of functions in Hv(T(Ω)) and they form an orthogonal basis in L2(Ω,dμv)L, where L is a specific subgroup of G. The recursion relations result by restricting the representation to a distinguished 3-dimensional subalgebra which is isomorphic to sl2(C). First, we construct the differential recursion relations for Laguerre functions defined on Ω = Sym+(n,R), the cone of positive definite real symmetric matrices, from the highest weight representations of Sp(2n,R). These relations generalize the \u27classical\u27 relations for Laguerre functions on R+. Then, we consider highest weight representations of any simple Lie group G to construct general differential recursion relations, for Laguerre functions defined on any symmetric cone, that generalize both the \u27classical\u27 recursion relations for Laguerre functions on Ω = R+ and the ones for Laguerre functions on Ω = Sym+(n,R)

Understanding the environmental degradation of methylammonium lead iodide Perovskite

Hybrid lead halide perovskite semiconductors have accelerated to the forefront of pho- tovoltaics. These materials possess highly desirable features including, fast charge trans- port, high extinction coefficients, large spectral overlap and solution process-ability. As a result of these, low-cost devices have emerged boasting impressive power conversion ef- ficiencies in excess of 20%. The rapid development of this technology is in part due to the materials versatility allowing numerous device configurations and fabrication techniques to be employed. Unfortunately, the excitement surrounding perovskites is hampered by their inability to withstand environmental stress. These systems have been found to ex- hibit significant performance losses and undergo irreversible material degradation when ex- posed to oxygen and light. Highlighted from these initial findings is that under these condi- tions, the reactive oxygen species superoxide can form and breakdown the perovskite crystal. A greater understanding of the mechanistic action leading to the generation of superoxide has been achieved through a powerful combination of experimental and computational results. The work has examined the role of material selection in the fabrication of devices. In addition the role of morphology of the perovskite has also been examined, where electron extraction from the perovskite layer is critical in achieving long term stability. The driving force for separation and the velocity at which electrons can be extracted are critical components in the effective- ness of an electron extraction layer in aiding stability enhancements. Rapid oxygen diffusion and iodide vacancies have been identified as key contributors to the mechanistic formation of superoxide. In order to achieve this a unique combination of isothermal gravimetric analysis and Time-of-Flight secondary ion mass spectrometry were employed. Critically these showed the rapid uptake of oxygen and the ubiquitous presence of these species after exposure to air. Inspired by these results, new methods have been developed to generate perovskite solar cells with increased performance life-time. The work herein, has also identified the impact of the selection of the organic cation and exchanging the halide upon the stability towards oxygen and light. Furthermore, the consequence of introducing moisture into the equation has been consid- ered and revealed greater detail about the mechanistic formation of superoxide from these species. The generation of superoxide in perovskite materials for photovoltaic applications is highly undesirable and persists as a key issue regarding their commercial employment. However, inspired by the fact photo-absorbers can generate superoxide a new application where the generation of the species could be used in a productive way is explored. To this end, the generation of superoxide from the organic polymer P3HT is explored. The production of superoxide from films is then harnessed to react with another species in a solution media. This simulation, leads to potential application where a contaminated solution, for example with a biological species, could be cleaned by addition of a P3HT film, oxygen and light. Here, the superoxide species would form from the film and then react and denature the contaminant.Open Acces

Accelerating the computation of critical eigenvalues with parallel computing techniques

Eigenanalysis of power systems is frequently used to study the effect and tune the response of existing controllers, or to guide the design of new controllers. However, recent developments in the area lead to the necessity of studying larger power system models, resulting from the interconnection of transmission networks or the joint consideration of transmission and distribution networks. Moreover, these models include new types of controls, mainly based on power electronic interfaces, which are expected to provide significant support in the future. The consequence is that the size and complexity of these models challenge the computational efficiency of existing eigenanalysis methods. In this paper, a procedure is proposed that uses domain decomposition and parallel computing methods, to accelerate the computation of eigenvalues in a selected region of the complex plane with iterative eigenanalysis methods. The proposed algorithm is validated on a small transmission system and its performance is assessed on a large-scale, combined transmission and distribution system

Dynamic Simulations of Combined Transmission and Distribution Systems using Parallel Processing Techniques

peer reviewedSimulating a power system with both transmission and distribution networks modeled in detail is a huge computational challenge. In this paper, we propose a Schur-complement-based domain decomposition algorithm to provide accurate, detailed dynamic simulations of the combined system. The simulation procedure is accelerated with the use of parallel programming techniques, taking advantage of the parallelization opportunities inherent in domain decomposition algorithms. The proposed algorithm is general, portable and scalable on inexpensive, shared-memory, multicore machines. A large-scale test system is used for its performance evaluation

Research disruption during PhD studies and its impact on mental health: Implications for research and university policy

Research policy observers are increasingly concerned about the impact of the disruption caused by the Covid-19 pandemic on university research. Yet we know little about the effect of this disruption, specifically on PhD students, their mental health, and their research progress. This study drew from survey responses of UK PhD students during the Covid-19 pandemic. We explored evidence of depression and coping behaviour (N = 1780), and assessed factors relating to demographics, PhD characteristics, Covid-19-associated personal circumstances, and significant life events that could explain PhD student depression during the research disruption (N = 1433). The majority of the study population (86%) reported a negative effect on their research progress during the pandemic. Results based on eight mental health symptoms (PHQ-8) showed that three in four PhD students experienced significant depression. Live-in children and lack of funding were among the most significant factors associated with developing depression. Engaging in approach coping behaviours (i.e., those alleviating the problem directly) related to lower levels of depression. By assessing the impact of research disruption on the UK PhD researcher community, our findings indicate policies to manage short-term risks but also build resilience in academic communities against current and future disruptions