Abstract—An analytical framework of error propagation for diffusion tensor imaging (DTI) is presented. Using this framework, any uncertainty of interest related to the diffusion tensor elements or to the tensor-derived quantities such as eigenvalues, eigenvectors, trace, fractional anisotropy (FA), and relative anisotropy (RA) can be analytically expressed and derived from the noisy diffusion-weighted signals. The proposed framework elucidates the underlying geometric relationship between the variability of a tensor-derived quantity and the variability of the diffusion weighted signals through the nonlinear least squares objective function of DTI. Monte Carlo simulations are carried out to validate and investigate the basic statistical properties of the proposed framework. Index Terms—Cone of uncertainty, covariance structures, diffusion tensor imaging, diffusion tensor representations, error propagation, invariant Hessian. I.

We show that it is possible to formulate gravity with a complex vierbein based on SL(2, C) gauge invariance. The proposed action is a four-form where the metric is not introduced but results as a function of the complex vierbein. This formulation is based on the first order formalism. The novel feature here is that integration of the spin-connection gauge field gives rise to kinetic terms for a massless graviton, a massive graviton with the Fierz-Pauli mass term, and a scalar field. The resulting theory is equivalent to bigravity. We then show that by extending the gauge group to GL(2, C) the formalism can be easily generalized to apply to a noncommutative space with the star product. We give the deformed action to second order in the deformation parameter, and derive the Seiberg-Witten map for the complex vierbein and gauge fields.

Abstract. Drawing an analogy between gravity dynamical equation of motion and that of Maxwell electrodynamics with an electric source we outline a way of appearance of a dual to graviton field. We propose a dimensional reduction ansatz for the field strength of this field which reproduces the correct duality relations between fields arising in the dimensional reduction of D-dimensional gravity action to D-1 dimensions. Modifying the PST approach we construct a new term entering the action of D=11 duality-symmetric gravity and by use of the proposed ansatz we confirm the relevance of such a term to reproduce the correct duality-symmetric structure of the reduced theory. We end up extending the results to the bosonic sector of D=11 supergravity. Contents

A new geometry, called General geometry, is constructed. It is proven that its the most simplest special case is geometry underlying Electromagnetism. Another special case is Riemannian geometry. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. It is shown that equation of motion for a particle interacting with electromagnetic field coincides exactly with equation for geodesics of geometry underlying Electromagnetism. It is also shown that Electromagnetism can not be geometrized in the framework of Riemannian geometry. Using General Geometry we propose a unified model of electromagnetism and gravitation which reproduces Electromagnetism and Gravitation and predicts that electromagnetic field is a source for gravitational field. This theory is formulated in four dimensional spacetime and does not contain additional fields.

It is shown that Electromagnetism creates geometry different from Riemannian one. General geometry containing Riemannian and geometry underlying Electromagnetism as special cases is constructed. Action for electromagnetic field and Maxwell equations are derived from curvature function of Geometry of Electromagnetism. PACS: 13.40.-f

Introduction Einstein's theory of general relativity predicts correctly all postNewtonian experiments in our solar system as well as some effects of strong gravitational fields observed in binary systems of neutron stars [1]. The theory has, however, two unsatisfactory properties. One is the somewhat academic fact that gravity cannot be quantized in a renormalizable way [2], but only as an effective theory which is unable to kleinert@physik.fu-berlin.de, http://www.physik.fu-berlin.de/~kleinert y pelster@physik.fu-berlin.de, http://www.physik.fu-berlin.de/~pelster 2 9801030 printed on February 19, 1998 predict short-distance gravitational phenomena at a scale much shorter than the Planck length l P ß 10 \Gamma32 cm [3]. Since this length scale is extremely small, the problem is not very serious for present-day physics. The other unsatisfactory property which ha

We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1

2. Differential manifolds

The motion of a conducting electron in a quantum dot with one or several dislocations in the underlying crystal lattice is considered in the continuum picture, where dislocations are represented by torsion of space. The possible effects of torsion are investigated on the levels of classical motion, nonrelativistic quantum motion, and spin-torsion coupling terms derivable in the non-relativistic limit of generalizations of the Dirac equation in a space with torsion. Finally, phenomenological spin-torsion couplings analogous to Pauli terms are considered in the non-relativistic equations. Different prescriptions of classical and non-relativistic quantum motion in a space with torsion are shown to give effects that should in principle be observable. Semi-classical arguments are presented to show that torsion is not relevant for the classical motion of the center of a wave packet. The correct semi-classical limit can instead be described as classical trajectories in a Hamiltonian given by the band energy. In the special case of a spherically symmetric band this motion reduces to straight lines, independently of local crystal orientations. By dimensional analysis the coupling constants of the possible spin-torsion interactions are postulated to be proportional to a combination of the effective mass of the electron, meff, the lattice constant, a, and ¯h. The level-splitting is 1 then very small with transition frequencies on the order of 1 kHz or smaller. PACS numbers: 61.72Bb, 73.20.Dx, 85.30.Vw I.

We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud’s identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud’s identity (which is a decomposition of the Einstein indexed 3-forms ⋆G a in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud’s identity is usually applied in the formulation of energy-momentum “conservation laws ” in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the