Direct Manifold Research Articles

Risk forecasting in (T)GARCH models with uncorrelated dependent innovations

(G)ARCH-type models are frequently used for the dynamic modelling and forecasting of risk attached to speculative asset returns. While the symmetric and conditionally Gaussian GARCH model has been generalized in a manifold of directions, model innovations are mostly presumed to stem from an underlying IID distribution. For a cross section of 18 stock market indices, we notice that (threshold) (T)GARCH-implied model innovations are likely at odds with the commonly held IID assumption. Two complementary strategies are pursued to evaluate the conditional distributions of consecutive TGARCH innovations, a non-parametric approach and a class of standardized copula distributions. Modelling higher order dependence patterns is found to improve standard TGARCH-implied conditional value-at-risk and expected shortfall out-of-sample forecasts that rely on the notion of IID innovations.

Maxwell equations and the direction of the electromagnetic field

Let M0 be the Minkowski space, let Λ2(M0) be the space of bivectors in M0, and let G1 ⊂ Λ2(M0) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ2(M0), U ⊂ ℝ4, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G1 is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G1 are considered. Bibliography: 7 titles.

Classification of Totally Geodesic Surfaces in the Manifold of Directions in Physical Space

The Grassmannian of bivectors over the pseudo-Euclidean Minkowski 4-space is considered and its two-dimensional totally geodesic submanifolds are classified. The family of such surfaces is described in the language of the affine geometry of three-space. Bibliography: 5 titles.

Totally Geodesic Subsets in the Manifold of Directions of Physical Space

Let M0 be the Minkowski 4-space, let \(\Lambda \)2(M0) denote the second exterior power of M0 equipped with a structure of a pseudo-Euclidean space with signature (3,3), let K0(M0)\( \subset \Lambda \)2M0 be the light cone, and let G1\( \subset \Lambda \)2(M0) be the set of the oriented 2-planes meeting the interior of K0(M0). Four types of totally geodesic two-manifolds in G1 are described such that manifolds of one type are pairwise congruent as subsets in \(\Lambda \)2(M0), while manifolds of different types are not. Models of such manifolds in the disk D3 are constructed. An explicit formula for the curvature tensor of G1 is given. Bibliography: 6 titles.

Absence of Three-Dimensional Totally Geodesic Submanifolds in the Manifold of Directions of Physical Space

It is proved that the manifold of directions of physical space contains no three-dimensional totally geodesic submanifolds. Bibliography: 3 titles.

Topology and a lorentz-invariant pseudo-riemannian metric of the manifold of directions in the physical space

In the mathematical model of the special relativity theory, a two-dimensional Minkowski subspace is treated as a one-dimensional direction in the physical space. The manifold of such planes is naturally endowed with the structure of a pseudo-Riemannian manifold on which the group of isochronous Lorentz transformations acts transitively by isometries. In this paper, the topology and the metric geometry of this manifold are studied. Bibliography: 4 titles.

Binocular correspondence and visual direction.

Two classic theories of direction vision, one by Hering, the other by Wells, are expressed in mathematical form and compared. The Hering disparity field differs considerably from the Wells disparity field, but if both are scaled for the change of acuity with eccentricity their differences are much more subtle. This explains why it is hard to determine which theory predicts direction perception best, although the tests favour Hering's theory. It is proved that Wells's construction (his rule 3) follows directly from his first two rules and Aguillonius's assumption that the horopter in the fixation plane is a frontoparallel line. Wells's theory is clearly outdated and does not mesh well with modern three-dimensional geometry of binocular vision, which Hering's theory does. Moreover, Wells inextricably mixes distance and direction vision right from the start, whereas Hering properly treats the two-dimensional manifold of directions and the depth-gauging principles separately. The use of terms such as 'Wells-Hering' rules should be discouraged and both Wells and Hering should be remembered separately for their clearly distinct and independent contributions. The work of Hering is still relevant to modern theory and praxis of binocular vision. The extension of Hering's approach to vertical disparities is treated for stimuli in frontoparallel planes. It is shown that acuity-scaled vertical-disparity information sampled at a single glance is below resolution beyond about arm's length. It can only be used if eye movements are allowed. Throughout, the simplest derivations of the geometrical relations that it was possible to find are given, so that the review of binocular geometry might also be of some didactical use. Finally it is indicated in which direction it might be necessary to modernise the concept of binocular correspondence.