Riemann Surface Structure for a Curved Surface with Punctured Features

In this paper, a generic procedure for the development and subsequent validation of the Riemann surface structure (RSS) for a punctured curved surface lying on a Riemann surface is discussed. The proposed procedure differs from the existing methods involving triangular meshes and rectangular grids that rely on induced patches on surfaces. This procedure can be applied to non-punctured surfaces as well as to surfaces with irregularly located punctures. Further, by defining appropriate transition functions, the proposed procedure eliminates the requirement for smooth transitions across the boundaries of adjacent patches. The analytic formulations of the RSS for an ellipsoid and a sphere are elaborated using the proposed procedure. Moreover, the RSS of a sphere defined through a family of conformal unit discs is proven equivalent to that defined by an existing method based on stereographic projection. This study proves that a smooth projection between the surface and (a subset of) the complex plane , can be remapped to the original surface.

Vector Arithmetic in the Triangular Grid

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

Axiomatizing Rectangular Grids with no Extra Non-unary Relations

We construct a first-order formula φ such that all finite models of φ are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆ ℕ is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t(n) and space s(n), where t(n)s(n) ≤ n and t(n); s(n) = Ω(log(n)).

MULTIVARIATE AFFINE FRACTAL INTERPOLATION

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the [Formula: see text] convergence of this type of interpolants for [Formula: see text] extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuous functions defined on a multidimensional compact rectangle is studied.

Finite-element EM modelling on hexahedral grids with an FD solver as a pre-conditioner

SUMMARY The finite-element (FE) method is one of the most powerful numerical techniques for modelling 3-D electromagnetic fields. At the same time, there still exists the problem of efficient and economical solution of the respective system of FE equations in the frequency domain. In this paper, we concentrate on modelling with adapted hexahedral or logically rectangular grids. These grids are easy to generate, yet they are flexible enough to incorporate real topography and seismic horizons. The goal of this work is to show how a finite-difference (FD) solver can be used as a pre-conditioner for hexahedral FE modelling. Applying the lowest order Nédélec elements, we present a novel pre-conditioned iterative solver for the arising system of linear equations that combines an FD solver and simple smoothing procedure. The particular FD solver that we use relies on the implicit factorization of the horizontally layered earth matrix. We assessed runtime and accuracy of the presented approach on synthetic and real resistivity models (topography of the Black Sea continental slope). We further compared performance of our program versus publicly available Mare2DEM, ModEM and MUMPS programs/libraries. Our examples involve plane-wave and controlled source modelling. The numerical examples demonstrate that the presented approach is fast and robust for models with moderate contrast, supports highly deformed cells, and is quite memory-economical.