Computable Constants for Korn’s Inequalities on Riemannian Manifolds

A method is presented for the explicit construction of the non-dimensional constant occurring in Korn’s inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface.

Modelling Secular Variation in the Southwest Pacific for the last 400 years

<p>A spherical cap harmonic analysis (SCHA) model has been used to derive a high-resolution regional model of the geomagnetic field in the southwest Pacific region over the past 400 years. Two different methods, a self-consistent and the gufm1 dipole method, have been used to fill in gaps in the available data. The data used in the analysis were largely measurements of the magnetic field recorded in ships logs on voyages of exploration in the region. The method chosen for the investigation used a spherical cap of radius 𝜃₀ = 50° centered at co-latitude and longitude of (115°, 160°). The results of each method used for SCHA are presented as contour plots of magnetic field declination, inclination and intensity and are compared with similar plots for a global model, gufm1. The root mean square misfit of the self- consistent and gufm1 dipole model to the actual data were around 2900 nT and 23000 nT respectively. Overall, the results suggest that the self-consistent model produces a more reliable model of the geomagnetic field within the area of interest than does the gufm1 dipole model. With more data included the self-consistent model could be further improved and used to develop a high-resolution mathematical model of the geomagnetic field in the southwest Pacific region.</p>

Modelling Secular Variation in the Southwest Pacific for the last 400 years

<p>A spherical cap harmonic analysis (SCHA) model has been used to derive a high-resolution regional model of the geomagnetic field in the southwest Pacific region over the past 400 years. Two different methods, a self-consistent and the gufm1 dipole method, have been used to fill in gaps in the available data. The data used in the analysis were largely measurements of the magnetic field recorded in ships logs on voyages of exploration in the region. The method chosen for the investigation used a spherical cap of radius 𝜃₀ = 50° centered at co-latitude and longitude of (115°, 160°). The results of each method used for SCHA are presented as contour plots of magnetic field declination, inclination and intensity and are compared with similar plots for a global model, gufm1. The root mean square misfit of the self- consistent and gufm1 dipole model to the actual data were around 2900 nT and 23000 nT respectively. Overall, the results suggest that the self-consistent model produces a more reliable model of the geomagnetic field within the area of interest than does the gufm1 dipole model. With more data included the self-consistent model could be further improved and used to develop a high-resolution mathematical model of the geomagnetic field in the southwest Pacific region.</p>

The Prospect of Spatially Accurate Reconstructed Atom Probe Data Using Experimental Emitter Shapes

Reliable spatially resolved compositional analysis through atom probe tomography requires an accurate placement of the detected ions within the three-dimensional reconstruction. Unfortunately, for heterogeneous systems, traditional reconstruction protocols are prone to position some ions incorrectly. This stems from the use of simplified projection laws which treat the emitter apex as a spherical cap, although the actual shape may be far more complex. For instance, sampled materials with compositional heterogeneities are known to develop local variations in curvature across the emitter due to their material phase specific evaporation fields. This work provides three pivotal precursors to improve the spatial accuracy of the reconstructed volume in such cases. First, we show scanning probe microscopy enables the determination of the local curvature of heterogeneous emitters, thus providing the essential information for a more advanced reconstruction considering the actual shape. Second, we demonstrate the cyclability between scanning probe characterization and atom probe analysis. This is a key ingredient of more advanced reconstruction protocols whereby the characterization of the emitter topography is executed at multiple stages of the atom probe analysis. Third, we show advances in the development of an electrostatically driven reconstruction protocol which are expected to enable reconstruction based on experimental tip shapes.

Indonesian Earth’s Lithospheric Magnetic Field modelling using Spherical Cap Harmonic Analysis Method

The earth’s lithospheric magnetic field is part of the main earth’s magnetic field. The lithospheric field has a very small value compared to the Earth’s main magnetic field, approximately less than 1%, and this field is generated at the earth’s crust and upper mantle. Modelling of lithospheric field is useful mainly for predicting the distribution of the value of lithospheric fields and to determine the magnetic anomaly. In this research, modelling the Earth’s lithospheric magnetic field uses Spherical Cap Harmonic Analysis (SCHA) method and this method can do modelling using regional magnetic data. The data used for the modelling are magnetic repeat station data in Indonesia region (BMKG’s Epoch) and SWARM satellite data. The results of the modelling using integrated SWARM satellite and repeat station data produce RMSE values of 64.0834 nT and the expansion of index K is 70. In addition, the results of the modelling resolution is 1.50. The value’s range of modelling’s result are -987.192 – 998.239 nT for X component, -968.189 – 949.438 nT for Y component, -981.266 – 608.676 nT for Z component, and -904.151 – 997.389 nT for total intensity are.

How Accurate Geomagnetic Regional Modeling Using Ordinary Kriging For Clustered Distributed Data?: New Insight From Indonesian Archipelago

Indonesia relies only on the limited number of repeat station networks due to the archipelago setting with the extensive sea with the clustery distributed pattern. This paper explored geostatistical modeling to overcome that typical data characteristic. The modeling used repeat station data from the 1985 to 2015 epoch. The research used ordinary kriging (OK) compared to the Spherical Cap Harmonic Analysis (SCHA) and Polynomial. The results show that the root means square error (RMSE) of each declination, inclination, and total intensity vary among epochs. OK method for declination component produces smaller average RMSE (7.67 minutes) than SCHA (9.26 minutes) and Polynomial (7.97minutes). For the inclination component, OK has an average RMSE of 9.55 minutes, smaller than SCHA (10.05) but slightly higher than Polynomial (9.36 minutes). For the total intensity component, OK produce an average RMSE of 63.58 nT, smaller than SCHA (82.24 nT) and Polynomial (68.97 nT). The finding shows that the kriging method can be a promising method to model the regional geomagnetic field, especially in the area of limited available data and clustered distributed data.

How Accurately Can a Spherical Cap be Represented by Rational Quadratic Polynomials?

This paper discusses the incapability of a tensor product rational quadratic patch to accurately represent a spherical cap. It was analytically found that there is no combination of control points and associated weights to accurately represent the spherical cap. On top of that, an optimization technique has revealed that for a unit sphere the computed radii in the parametric space may reduce within the interval [0.999999994, 1.000104146]. This study makes sense as a preparatory stage in relation with the isogeometric analysis (IGA), which may be applied in conjunction with either the Finite Element Method (FEM) or the Boundary Element Method (BEM).