Lines on K3 quartic surfaces in characteristic 3

We investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field of characteristic 3. We prove that if X contains 112 lines, then X is projectively equivalent to the Fermat quartic surface; otherwise, X contains at most 67 lines. We improve this bound to 58 if X contains a star (ie four distinct lines intersecting at a smooth point of X). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

A robust numerical method for single and multi-asset option pricing

<abstract><p>In this work, we numerically solve some different single and multi-asset European options with the finite difference method (FDM) and take the advantages of the antithetic variate method in Monte Carlo simulation (AMC) as a variance reduction technique in comparison to the standard Monte Carlo simulation (MC) in the end point of the domain, and the linear boundary condition has been implemented in other boundaries. We also apply the grid stretching transformation to make a non-equidistance discretization with more nodal points around the strike price (K) which is the non-smooth point in the payoff function to reduce the numerical errors around this point and have more accurate results. Superiority of our method (GS&amp;AMC) will be demonstrated by comparison with the finite difference scheme with the equidistance discretization and the linear boundary conditions (Equi&amp;L), the grid stretching discretization around K with linear boundary conditions (GS&amp;L) and also the equidistance discretization with combination of the standard Monte Carlo simulation at the end point of the domain (Equi&amp;MC). Furthermore, the root mean square errors (RMSE) of these four schemes in the whole region and the most interesting region which is around the strike price, have been compared.</p></abstract>

Signatures of Topological Branched Covers

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $\sigma (X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $\Xi (K)$ to $\sigma (X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a topological four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of points on $B$ which are not locally flat, $X$ in this setting is a stratified pseudomanifold, and we use the intersection homology signature of $X$, $\sigma _{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $\Xi (K)$, providing a new technique to potentially detect slice knots that are not ribbon.

Sparse Regularization-Based Approach for Point Cloud Denoising and Sharp Features Enhancement

Denoising the point cloud is fundamental for reconstructing high quality surfaces with details in order to eliminate noise and outliers in the 3D scanning process. The challenges for a denoising algorithm are noise reduction and sharp features preservation. In this paper, we present a new model to reconstruct and smooth point clouds that combine L1-median filtering with sparse L1 regularization for both denoising the normal vectors and updating the position of the points to preserve sharp features in the point cloud. The L1-median filter is robust to outliers and noise compared to the mean. The L1 norm is a way to measure the sparsity of a solution, and applying an L1 optimization to the point cloud can measure the sparsity of sharp features, producing clean point set surfaces with sharp features. We optimize the L1 minimization problem by using the proximal gradient descent algorithm. Experimental results show that our approach is comparable to the state-of-the-art methods, as it filters out 3D models with a high level of noise, but keeps their geometric features.

Mechanism Singularities Revisited From an Algebraic Viewpoint

It has become obvious that certain singular phenomena cannot be explained by a mere investigation of the configuration space, defined as the solution set of the loop closure equations. For example, it was observed that a particular 6R linkage, constructed by combination of two Goldberg 5R linkages, exhibits kinematic singularities at a smooth point in its configuration space. Such problems are addressed in this paper. To this end, an algebraic framework is used in which the constraints are formulated as polynomial equations using Study parameters. The algebraic object of study is the ideal generated by the constraint equations (the constraint ideal). Using basic tools from commutative algebra and algebraic geometry (primary decomposition, Hilbert’s Nullstellensatz), the special phenomenon is related to the fact that the constraint ideal is not a radical ideal. With a primary decomposition of the constraint ideal, the associated prime ideal of one primary ideal contains strictly into the associated prime ideal of another primary ideal which also gives the smooth configuration curve. This analysis is extended to shaky and kinematotropic linkages, for which examples are presented.

Smooth point-to-point trajectory planning for robot manipulators by using radial basis functions

SUMMARYThe paper introduces the use of radial basis functions (RBFs) to generate smooth point-to-point joint trajectories for robot manipulators. First, Gaussian RBF interpolation is introduced taking into account boundary conditions. Then, the proposed approach is compared with classical planning techniques based on polynomial and trigonometric models. Also, the trajectory planning problem involving via-points is solved using the proposed RBF interpolation technique. The obtained trajectories are then compared with those synthesized using algebraic and trigonometric splines. Finally, the proposed method is tested for the six-joint PUMA 560 robot in two cases (minimum time and minimum time-jerk) and results are compared with those of other planning techniques. Numerical results demonstrate the advantage of the proposed technique, offering an effective solution to generate trajectories with short execution time and smooth profile.

The exponential map at a cuspidal singularity

We study spaces with a cuspidal (or horn-like) singularity embedded in a smooth Riemannian manifold and analyze the geodesics in these spaces which start at the singularity. This provides a basis for understanding the intrinsic geometry of such spaces near the singularity. We show that these geodesics combine to naturally define an exponential map based at the singularity, but that the behavior of this map can deviate strongly from the behavior of the exponential map based at a smooth point or at a conical singularity: While it is always surjective near the singularity, it may be discontinuous and non-injective on any neighborhood of the singularity. The precise behavior of the exponential map is determined by a function on the link of the singularity which is an invariant of the induced metric. Our methods are based on the Hamiltonian system of geodesic differential equations and on techniques of singular analysis. The results are proved in the more general natural setting of manifolds with boundary carrying a so-called cuspidal metric.