Abstract


We introduce the notion of twisted generalized complex submanifolds and describe an
equivalent characterization in terms of Poisson–Dirac submanifolds. Our
characterization recovers a result of Vaisman (2007). An equivalent characterization
is also given in terms of spinors. As a consequence, we show that the fixed locus of an
involution preserving a twisted generalized complex structure is a twisted generalized
complex submanifold. We also prove that a twisted generalized complex
manifold has a natural Poisson structure. We also discuss generalized Kähler
submanifolds.


Keywords

generalized complex geometry, Poisson bivector, Poisson–Dirac submanifold

Mathematical Subject Classification

Primary: 53C56, 53D17, 53D35

Abstract

We study positive solutions of an equation with singular nonlinearities.
The equation arises in the study of equilibrium states of thin films.
Under weak assumptions on the nonlinearity, we show that for
N ≥ 3 there exists a family
of radial solutions {uα}α>0
with uα(0) = α and each of
them is oscillatory in (0,∞).
We obtain then a singular radial solution in
(0,∞) by taking the limit
α → 0. Meanwhile, using the
solutions obtained in (0,∞),
we show some existence results for the corresponding Neumann eigenvalue problem
on a ball.


Keywords
elliptic equation with singular nonlinearity, rupture solutions, oscillation

Mathematical Subject Classification

Primary: 35J60, 35B45, 35B05

With Reference to Cortical Magnification and Dipole Source Localisation in the Visual Cortex

Mathematical and computer models are important tools that are available to investigate natural phenomena. They can be used to model many systems. In this thesis, mathematical models are developed, implemented and applied to research involving the human brain and in particular, the human visual cortex. The visual cortex constitutes a relatively large part of the cerebral cortex. It is often used in investigations of the human brain because conclusions regarding the visual cortex can be extended to other regions of the brain. Virtually all information from the visual system is recognised as first being processed by the primary visual cortex and is then passed to other regions of the brain involved in more complex processing.

The primary visual cortex has a retinotopic mapping in that one spot in the retinal visual field maps directly to a spot on the primary visual cortex. However, there is disagreement as to the amount of cortex that is allocated to the representation of central vision or other portions of the visual field. A mathematical formulation of this mapping is presented and mapping functions which transform the surface representing the retina to the surface representing the visual cortex are developed.

If the head is modelled as three concentric spherical shells and neural sources of brain activity are modelled as dipoles, then a mathematical model which incorporates biophysical properties can be used to estimate the location of sources which generate a set of electrical potentials measured on the surface of the scalp. This model is known as dipole source localisation. The forward problem, which is the prediction of a potential distribution due to a given electrical source is implemented, and the inverse problem, which is to determine a dipole source that is the best generator of a given potential distribution is solved in the least squares sense. Monte Carlo simulations and mathematical analysis show that the optimal reference electrode to use in dipole analysis is a weighted version of the common average reference. Monte Carlo simulations are also used to investigate the accuracy of confidence regions surrounding the estimated dipole parameters.

Subsequently, a methodology for modelling a region of cortex from magnetic resonance images is developed. This methodology is applied to the calcarine fissure and surrounding grey matter to produce a three dimensional surface reconstruction of the visual cortex. This model is used to provide anatomical constraints in the dipole source localisation model. These models are then applied to visual evoked potential data obtained from an experiment which uses a chromatic grating stimulus. Results reveal that these mathematical and computer models, combined with imaging and experimental approaches, elicit new information and improved results in investigations of the human brain.