We show, in this first part, that the maximal number of singular points a normal quartic surface $X \subset \mathbb {P}^{3}_{K}$ defined over an algebraically closed field K characteristic 2 is at most 16. produce examples with 14, respectively 12, and show that, under several geometric assumptions ( $\mathfrak S_{4}$ -symmetry, or behaviour Gauss map, structure tangent cone one P, separability...

In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end such manifold admits a torus fibration over an ALE end. addition, prove Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds curvature decay and As application, complete metric on $4$-manifold which is homeomorphic to $\mathbb R^4$ must be isometric Euclidean or ...

Abstract A dynamic mechanical thermal analyser operating in the single cantilever mode was used to examine properties of unaged and hygrothermally aged discontinuous asymmetric helicoidal (Bouligand) carbon fibre reinforced plastic (CFRP) composites as a function architecture. The Bouligand manufactured using two major pitch angles independent variables: 90° 120° from each angle, minor interply...

1. Write the formula for the Liouville measure Λ on the unit tangent bundle T 1 H n. Prove that Λ is invariant under the actions by Isom(H n) and the geodesic flow g t on T 1 H n. 2. Let δ be a geodesic in the hyperbolic space H n and let π : H n → δ denote the nearest point projection map. Suppose that for some x, y ∈ H n and some r > 0 we have d(x, δ), d(y, δ) ≥ r. Prove that d(π(x), π(y)) ≤ ...

We introduce a notion of rectifiability modeled on Carnot groups. Precisely, we say that a subset E of a Carnot group M and N is a subgroup of M , we say E is Nrectifiable if it is the Lipschitz image of a positive measure subset of N . First, we discuss the implications of N-rectifiability, where N is a Carnot group (not merely a subgroup of a Carnot group), which include N-approximability and...

The goal is to perform geometric studies on cone-beam CT scanning along a three-dimensional (3D) spiral of variable radius. First, the background for variable radius spiral cone-beam scanning is given in the context of electron-beam CT/micro-CT. Then, necessary and sufficient conditions are proved for existence and uniqueness of PI lines inside the variable radius 3D spiral. These results are n...

In Reference [1] the authors have shown that numerical differentiation is a competitive alternative to analytical derivatives for the computation of consistent tangent matrices. Relatively simple models were treated in that reference. The approach is extended here to a complex model: the MRS-Lade model [2,3]. This plastic model has a cone-cap yield surface and exhibits strong coupling between t...

Totally geodesic property of the Hopf vector field. Abstract We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation us...