Three-dimensional description and mathematical characterization of the parasellar internal carotid artery in human infants

Inside the `cavernous sinus` or `parasellar region` the human internal carotid artery takes the shape of a siphon that is twisted and torqued in three dimensions and surrounded by a network of veins. The parasellar section of the internal carotid artery is of broad biological and medical interest, as its peculiar shape is associated with temperature regulation in the brain and correlated with the occurrence of vascular pathologies. The present study aims to provide anatomical descriptions and objective mathematical characterizations of the shape of the parasellar section of the internal carotid artery in human infants and its modifications during ontogeny. Three-dimensional (3D) computer models of the parasellar section of the internal carotid artery of infants were generated with a state-of-the-art 3D reconstruction method and analysed using both traditional morphometric methods and novel mathematical algorithms. We show that four constant, demarcated bends can be described along the infant parasellar section of the internal carotid artery, and we provide measurements of their angles. We further provide calculations of the curvature and torsion energy, and the total complexity of the 3D skeleton of the parasellar section of the internal carotid artery, and compare the complexity of this in infants and adults. Finally, we examine the relationship between shape parameters of the parasellar section of the internal carotid artery in infants, and the occurrence of intima cushions, and evaluate the reliability of subjective angle measurements for characterizing the complexity of the parasellar section of the internal carotid artery in infants. The results can serve as objective reference data for comparative studies and for medical imaging diagnostics. They also form the basis for a new hypothesis that explains the mechanisms responsible for the ontogenetic transformation in the shape of the parasellar section of the internal carotid artery.

Identifying the borders of mathematical knowledge

Based on a divide and conquer approach, knowledge about nature has been organized into a set of interrelated facts, allowing a natural representation in terms of graphs: each `chunk` of knowledge corresponds to a node, while relationships between such chunks are expressed as edges. This organization becomes particularly clear in the case of mathematical theorems, with their intense cross-implications and relationships. We have derived a web of mathematical theorems from Wikipedia and, thanks to the powerful concept of entropy, identified its more central and frontier elements. Our results also suggest that the central nodes are the oldest theorems, while the frontier nodes are those recently added to the network. The network communities have also been identified, allowing further insights about the organization of this network, such as its highly modular structure.

On Aspects of Mathematical Reasoning : Affect and Gender

This thesis explores two aspects of mathematical reasoning: affect and gender. I started by looking at the reasoning of upper secondary students when solving tasks. This work revealed that when not guided by an interviewer, algorithmic reasoning, based on memorising algorithms which may or may not be appropriate for the task, was predominant in the students reasoning. Given this lack of mathematical grounding in students reasoning I looked in a second study at what grounds they had for different strategy choices and conclusions. This qualitative study suggested that beliefs about safety, expectation and motivation were important in the central decisions made during task solving.  But are reasoning and beliefs gendered? The third study explored upper secondary school teachers conceptions about gender and students mathematical reasoning. In this study I found that upper secondary school teachers attributed gender symbols including insecurity, use of standard methods and imitative reasoning to girls and symbols such as multiple strategies especially on the calculator, guessing and chance-taking were assigned to boys. In the fourth and final study I found that students, both male and female, shared their teachers view of rather traditional feminities and masculinities. Remarkably however, this result did not repeat itself when students were asked to reflect on their own behaviour: there were some discrepancies between the traits the students ascribed as gender different and the traits they ascribed to themselves. Taken together the thesis suggests that, contrary to conceptions, girls and boys share many of the same core beliefs about mathematics, but much work is still needed if we should create learning environments that provide better opportunities for students to develop beliefs that guide them towards well-grounded mathematical reasoning. 

Teachers' conceptions about students' mathematical reasoning : Gendered or not?

This study looks at how upper secondary school teachers gender stereotype aspects of students' mathematical reasoning. Girls were attributed gender symbols including insecurity, use of standard methods and imitative reasoning. Boys were assigned the symbols such as multiple strategies especially on the calculator, guessing and chance-taking. 

Mathematical games with the alphabet

John Gough once again makes the ordinary fascinating as he piques our curiosity about the origins of the alphabet and turns the various mathematical properties of letters into bases for games.

Mathematical modelling of the refractive index reconstruction through global optimization

We examine a mathematical model of non-destructive testing of planar waveguides, based on numerical solution of a nonlinear integral equation. Such problem is ill-posed, and the method of Tikhonov regularization is applied. To minimize Tikhonov functional, and find the parameters of the waveguide, we use two new optimization methods: the cutting angle method of global optimization, and the discrete gradient method of nonsmooth local optimization. We examine how the noise in the experimental data influences the solution, and how the regularization parameter has to be chosen. We show that even with significant noise in the data, the numerical solution is of high accuracy, and the method can be used to process real experimental da.ta..

Developing formal mathematical assessment for 4-to-8-year-olds

The assessment of children in their years before school and their first years of school has been, traditionally, informal. Further, assessment of children's mathematical skills at this level has been infrequent compared to social, emotional and physical assessments. However, there are contexts where reliable, valid, standardised data from assessment in mathematics are required. This paper outlines the development of two assessment tools for mathematics that were originally developed for such contexts. Item Response Theory (IRT) analyses enabled the construction of assessment forms that address the range of abilities of 4- to 8-year-old children, and provided the scales used for constructing formative and summative reports of achievement. A description of the development of the assessment tools and the IRT analysis that provides the reporting formats are presented together with some research uses of the tools.

On the EA-style integrated processing of self-contained mathematical texts

A sound and complete first-order goal-oriented sequent-type calculus is developed with ``large-block'' inference rules. In particular, the calculus contains formal analogues of such natural proof-search techniques as handling definitions and applying auxiliary propositions.

Evidence Algorithm and Processing Formalized Mathematical Texts

The goal of a research programme Evidence Algorithm is a development of an open system of automated proving that is able to accumulate mathematical knowledge and to prove theorems in a context of a self-contained mathematical text. By now, the first version of such a system called a System for Automated Deduction, SAD, is implemented in software. The system SAD possesses the following main features: mathematical texts are formalized using a specific formal language that is close to a natural language of mathematical publications; a proof search is based on special sequent-type calculi formalizing natural reasoning style, such as application of definitions and auxiliary propositions. These calculi also admit a separation of equality handling from deduction that gives an opportunity to integrate logical reasoning with symbolic calculation.