Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of $n\times n$ Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is $2n-3$.Comment: v1 14 pages; v2 19 page

Existence of positive periodic solutions for higher order singular functional difference equations

We study a higher order singular functional difference equation on $\mathbb{Z}$. Sufficient conditions are obtained for the existence of at least one positive periodic solution of the equation. Our proof utilizes the nonlinear alternative of Leray-Schauder

The moving frame method for iterated-integrals: orthogonal invariants

Geometric features, robust to noise, of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply the Fels-Olver's moving frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in $\mathbb{R}^d$ from the iterated-integral signature. In particular we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations which yields a characterization of a curve in $\mathbb{R}^d$ under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations.Comment: 37 pages, 4 figure

The moving frame method for iterated-integrals: Orthogonal invariants

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Kiraly and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties

A Discursive Analysis of Cancer Survivors' Identities

This thesis explores how people who have experienced cancer construct themselves in talk of their experience of the illness. Semi structured interviews with six adults who had completed cancer treatment and were considered to be cancer free or in remission were analysed using a discursive psychological approach. Cancer has been characterised as being both an acute and chronic illness. The sequelae of treatment and uncertainty over possible recurrence of the disease mean that there is not a discrete point at which one ceases to be a cancer patient. As such the challenges to identities that may arise following a cancer diagnosis can be considered different to those that may arise following other serious illnesses. Cancer also carries with it cultural associations about the culpability of the patient for their illness, as well as a number of culturally legitimate narratives that are centred around metaphors of heroic battles or journeys of self discovery. Previous research has suggested that disparity between the identity challenges being faced by the individual with cancer and the identities into which others may altercast them into can result in cancer patients and survivors not receiving the support that they need. The analysis revealed that a consistent feature of the accounts was maintaining continuity between pre- and postcancer identities. Where post cancer changes were acknowledged these were assimilated into pre-cancer identities. Contrary to some previous research none of the participants oriented to the identity of being a cancer survivor. This finding is discussed in relation to methodological differences between the current and previous research and in relation the identity concerns present the participants' talk. I also discuss the constraints that discourses of austerity politics place on the identities available to cancer survivors who find themselves unable to work following treatment