Personal Identity, Survival And What Matters

Since the entire discussion of personal identity revolves around the identity of a person it is difficult to address these issues without presupposing that identity is maintained. In this dissertation, I propose an alternative approach to discussing the topic of personal identity (at least initially). This alternative approach is from the perspective of what I call ‘continuance’. I use ‘continuance’ to refer to some kind of ‘continuing life’ that is embodied in some person or persons. The term will be used as a neutral term for discussing the continuity of a person without any implications of identity. That is, in some cases, continuance will refer to a continuing life of one person but in other cases it will refer to a continuing life of some other person. In the literature on personal identity up until now, there has been no such neutral term. As a result, when considering the various cases, it is difficult to talk about the resulting person in those cases without presupposing that he is identical with the original person. The use of the term ‘continuance’ will allow us to talk of the resulting person without such presuppositions. Most theories of personal identity can be grouped into two main types: physical and psychological. These theories place the determining factor of whether or not a person persists on physical continuity and psychological continuity, respectively. We can address them in terms of continuance in the same manner. The question then becomes: what kinds of continuance are necessary and sufficient for persistence? I argue that neither a purely physical nor a purely psychological continuance theory are sufficient for persistence. Rather, persistence requires both a physical component as well as a psychological component. I argue that the physical substratum requirement is satisfied by continuance of at least part of the brain. Regarding the psychological component, I argue that memory, although sufficient, is not necessary. This is because I believe that what I call ‘Frankfurtian continuance’ (a continuance theory inspired by Harry Frankfurt’s “self” involving higher-order desires and volitions) is also sufficient. I then address split-brain (fission) cases. Most psychological continuity theorists take what is called a “non-branching” approach to split-brain cases. This allows them to claim that a person will survive if only one half of their brain is transplanted, but if both halves are, then neither of the resulting persons are identical to the original. My view is slightly different. I argue that branching is acceptable provided that transplanting the other half takes place after a sufficiently long period of time has passed. Finally, I address Parfit’s very influential work on personal identity. He argues that identity itself does not matter. Although I concede this point, I offer many extrinsic considerations for why identity matters. These considerations are not intended to show that identity itself matters, but they do show that there are numerous contingent reasons why one may prefer a scenario in which they do survive over one in which they do not

On the Linearization of the Painleve' III-VI Equations and Reductions of the Three-Wave Resonant System

We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3x3 matrix Fuchs--Garnier pairs for the third and fifth Painleve' equations, together with the previously known Fuchs--Garnier pair for the fourth and sixth Painleve' equations. These Fuchs--Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2x2 matrix Fuchs--Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3x3 matrix pairs, we found an integral auto-transformation for the standard Fuchs--Garnier pair for the fifth Painleve' equation. It generates an Okamoto-like B\"acklund transformation for the fifth Painleve' equation. Another application is an integral transformation relating two different 2x2 matrix Fuchs--Garnier pairs for the third Painleve' equation.Comment: Typos are corrected, journal and DOI references are adde

On a q-difference Painlev\'e III equation: II. Rational solutions

Rational solutions for a $q$-difference analogue of the Painlev\'e III equation are considered. A Determinant formula of Jacobi-Trudi type for the solutions is constructed.Comment: Archive version is already official. Published by JNMP at http://www.sm.luth.se/math/JNMP

Exact vortex solutions of the complex sine-Gordon theory on the plane

We construct explicit multivortex solutions for the first and second complex sine-Gordon equations. The constructed solutions are expressible in terms of the modified Bessel and rational functions, respectively. The vorticity-raising and lowering Backlund transformations are interpreted as the Schlesinger transformations of the fifth Painleve equation.Comment: 10 pages, 1 figur

Transformations of ordinary differential equations via Darboux transformation technique

A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of the solutions of the overdetermined linear systems are derived in the frameworks of the Darboux transformation technique.Comment: 7 pages, LaTeX2

On a Schwarzian PDE associated with the KdV Hierarchy

We present a novel integrable non-autonomous partial differential equation of the Schwarzian type, i.e. invariant under M\"obius transformations, that is related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered as the generating equation for the entire hierarchy of Schwarzian KdV equations. We present its Lax pair, establish its connection with the SKdV hierarchy, its Miura relations to similar generating PDEs for the modified and regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate that its similarity reductions lead to the {\it full} Painlev\'e VI equation, i.e. with four arbitary parameters.Comment: 11 page

Power expansions for solution of the fourth-order analog to the first Painlev\'{e} equation

One of the fourth-order analog to the first Painlev\'{e} equation is studied. All power expansions for solutions of this equation near points $z=0$ and $z=\infty$ are found by means of the power geometry method. The exponential additions to the expansion of solution near $z=\infty$ are computed. The obtained results confirm the hypothesis that the fourth-order analog of the first Painlev\'{e} equation determines new transcendental functions.Comment: 28 pages, 5 figure

Multivortex Solutions of the Weierstrass Representation

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlev\'e test and analyse the possibility of deriving the B\"acklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlev\'{e} equations to construct explicit formulae in terms of the Umemura polynomials which are $\tau$-functions for rational solutions of the third Painlev\'{e} equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex Sine-Gordon equation is reduced to coupled Riccati equations.Comment: 27 pages LaTeX2e, 1 encapsulated Postscript figur

Rational Solutions of the Painleve' VI Equation

In this paper, we classify all values of the parameters $\alpha$, $\beta$, $\gamma$ and $\delta$ of the Painlev\'e VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlev\'e VI equation.Comment: 13 pages, 1 Postscript figure Typos fixe