Ktisis Cyprus University of Technologyhttps://ktisis.cut.ac.cyThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 02 Aug 2021 22:46:27 GMT2021-08-02T22:46:27Z5051The octapolic ellipsoidal term in magnetoencephalographyhttps://ktisis.cut.ac.cy/handle/10488/7480Title: The octapolic ellipsoidal term in magnetoencephalography
Authors: Dassios, George; Hadjiloizi, Demetra; Kariotou, Fotini
Abstract: The forward problem of magnetoencephalography (MEG) in ellipsoidal geometry has been studied by Dassios and Kariotou ["Magnetoencephalography in ellipsoidal geometry," J. Math. Phys. 44, 220 (2003)] using the theory of ellipsoidal harmonics. In fact, the analytic solution of the quadrupolic term for the magnetic induction field has been calculated in the case of a dipolar neuronal current. Nevertheless, since the quadrupolic term is only the leading nonvanishing term in the multipole expansion of the magnetic field, it contains not enough information for the construction of an effective algorithm to solve the inverse MEG problem, i.e., to recover the position and the orientation of a dipole from measurements of the magnetic field outside the head. For this task, the next multipole of the magnetic field is also needed. The present work provides exactly this octapolic contribution of the dipolar current to the expansion of the magnetic induction field. The octapolic term is expressed in terms of the ellipsoidal harmonics of the third degree, and therefore it provides the highest order terms that can be expressed in closed form using long but reasonable analytic and algebraic manipulations. In principle, the knowledge of the quadrupolic and the octapolic terms is enough to solve the inverse problem of identifying a dipole inside an ellipsoid. Nevertheless, a simple inversion algorithm for this problem is not yet known.
Thu, 01 Jan 2009 00:00:00 GMThttps://ktisis.cut.ac.cy/handle/10488/74802009-01-01T00:00:00ZGravitational solitons and $C^0$ vacuum metrics in five-dimensional Lovelock gravityhttps://ktisis.cut.ac.cy/handle/10488/14219Title: Gravitational solitons and $C^0$ vacuum metrics in five-dimensional Lovelock gravity
Authors: Garraffo, C.; Giribet, G.; Gravanis, Elias; Willison, S.
Abstract: Junction conditions for vacuum solutions in five-dimensional
Einstein-Gauss-Bonnet gravity are studied. We focus on those cases where two
spherically symmetric regions of space-time are joined in such a way that the
induced stress tensor on the junction surface vanishes. So a spherical vacuum
shell, containing no matter, arises as a boundary between two regions of the
space-time. A general analysis is given of solutions that can be constructed by
this method of geometric surgery. Such solutions are a generalized kind of
spherically symmetric empty space solutions, described by metric functions of
the class $C^0$. New global structures arise with surprising features. In
particular, we show that vacuum spherically symmetric wormholes do exist in
this theory. These can be regarded as gravitational solitons, which connect two
asymptotically (Anti) de-Sitter spaces with different masses and/or different
effective cosmological constants. We prove the existence of both static and
dynamical solutions and discuss their (in)stability under perturbations that
preserve the symmetry. This leads us to discuss a new type of instability that
arises in five-dimensional Lovelock theory of gravity for certain values of the
coupling of the Gauss-Bonnet term. The issues of existence and uniqueness of
solutions and determinism in the dynamical evolution are also discussed.
Tue, 15 Apr 2008 00:00:00 GMThttps://ktisis.cut.ac.cy/handle/10488/142192008-04-15T00:00:00ZSingular sources in gravity and homotopy in the space of connectionshttps://ktisis.cut.ac.cy/handle/10488/14235Title: Singular sources in gravity and homotopy in the space of connections
Authors: Gravanis, Elias; Willison, Steven
Abstract: Suppose a Lagrangian is constructed from its fields and their derivatives.
When the field configuration is a distribution, it is unambiguously defined as
the limit of a sequence of smooth fields. The Lagrangian may or may not be a
distribution, depending on whether there is some undefined product of
distributions. Supposing that the Lagrangian is a distribution, it is
unambiguously defined as the limit of a sequence of Lagrangians. But there
still remains the question: Is the distributional Lagrangian uniquely defined
by the limiting process for the fields themselves? In this paper a general
geometrical construction is advanced to address this question. We describe
certain types of singularities, not by distribution valued tensors, but by
showing that the action functional for the singular fields is (formally)
equivalent to another action built out of \emph{smooth} fields. Thus we manage
to make the problem of the lack of a derivative disappear from a system which
gives differential equations. Certain ideas from homotopy and homology theory
turn out to be of central importance in analyzing the problem and clarifying
finer aspects of it.
The method is applied to general relativity in first order formalism, which
gives some interesting insights into distributional geometries in that theory.
Then more general gravitational Lagrangians in first order formalism are
considered such as Lovelock terms (for which the action principle admits
space-times more singular than other higher curvature theories).
Tue, 01 Dec 2009 00:00:00 GMThttps://ktisis.cut.ac.cy/handle/10488/142352009-12-01T00:00:00ZIntersecting hypersurfaces in AdS and Lovelock gravityhttps://ktisis.cut.ac.cy/handle/10488/14222Title: Intersecting hypersurfaces in AdS and Lovelock gravity
Authors: Gravanis, Elias; Willison, Steven
Abstract: Colliding and intersecting hypersurfaces filled with matter (membranes) are
studied in the Lovelock higher order curvature theory of gravity. Lovelock
terms couple hypersurfaces of different dimensionalities, extending the range
of possible intersection configurations. We restrict the study to constant
curvature membranes in constant curvature AdS and dS background and consider
their general intersections. This illustrates some key features which make the
theory different to the Einstein gravity. Higher co-dimension membranes may lie
at the intersection of co-dimension 1 hypersurfaces in Lovelock gravity; the
hypersurfaces are located at the discontinuities of the first derivative of the
metric, and they need not carry matter.
The example of colliding membranes shows that general solutions can only be
supported by (spacelike) matter at the collision surface, thus naturally
conflicting with the dominant energy condition (DEC). The imposition of the DEC
gives selection rules on the types of collision allowed.
When the hypersurfaces don't carry matter, one gets a soliton-like
configuration. Then, at the intersection one has a co-dimension 2 or higher
membrane standing alone in AdS-vacuum spacetime \emph{without conical
singularities.}
Another result is that if the number of intersecting hypersurfaces goes to
infinity the limiting spacetime is free of curvature singularities if the
intersection is put at the boundary of each AdS bulk.
Mon, 11 Sep 2006 00:00:00 GMThttps://ktisis.cut.ac.cy/handle/10488/142222006-09-11T00:00:00ZIntersecting hyper-surfaces in dimensionally continued topological density gravitationhttps://ktisis.cut.ac.cy/handle/10488/14223Title: Intersecting hyper-surfaces in dimensionally continued topological density gravitation
Authors: Gravanis, Elias; Willison, Steven
Abstract: We consider intersecting hypersurfaces in curved spacetime with gravity
governed by a class of actions which are topological invariants in lower
dimensionality. Along with the Chern-Simons boundary terms there is a sequence
of intersection terms that should be added in the action functional for a well
defined variational principle. We construct them in the case of Characteristic
Classes, obtaining relations which have a general topological meaning. Applying
them on a manifold with a discontinuous connection 1-form we obtain the gravity
action functional of the system and show that the junction conditions can be
found in a simple algebraic way. At the sequence of intersections there are
localised independent energy tensors, constrained only by energy conservation.
We work out explicitly the simplest non trivial case.
Description: 20 pages, 3 figures. Accepted for Journal Math. Phys. Some minor
changes and corrections
Mon, 25 Oct 2004 00:00:00 GMThttps://ktisis.cut.ac.cy/handle/10488/142232004-10-25T00:00:00Z