A note on the boundary of the Birkhoff-James ε-orthogonality sets

The Birkhoff-James $\varepsilon$-sets of vectors and vector-valued polynomials (in one complex variable) have recently been introduced as natural generalizations of the standard numerical range of (square) matrices or operators and matrix or operator polynomials, respectively. Corners on the boundary curves of these sets are of particular interest, not least because of their importance in visualizing these sets. In this paper, we provide a characterization for the corners of the Birkhoff-James $\varepsilon$-sets of vectors and vector-valued polynomials, completing and expanding upon previous exploration of the geometric propertiesof these sets. We also propose a randomized algorithm for approximating their boundaries.

The Development of Log Aesthetic Patch and Its Projection onto the Plane

In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry.

Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

<p style='text-indent:20px;'>We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.</p>

PDE Surface-Represented Facial Blendshapes

Partial differential equation (PDE)-based geometric modelling and computer animation has been extensively investigated in the last three decades. However, the PDE surface-represented facial blendshapes have not been investigated. In this paper, we propose a new method of facial blendshapes by using curve-defined and Fourier series-represented PDE surfaces. In order to develop this new method, first, we design a curve template and use it to extract curves from polygon facial models. Then, we propose a second-order partial differential equation and combine it with the constraints of the extracted curves as boundary curves to develop a mathematical model of curve-defined PDE surfaces. After that, we introduce a generalized Fourier series representation to solve the second-order partial differential equation subjected to the constraints of the extracted boundary curves and obtain an analytical mathematical expression of curve-defined and Fourier series-represented PDE surfaces. The mathematical expression is used to develop a new PDE surface-based interpolation method of creating new facial models from one source facial model and one target facial model and a new PDE surface-based blending method of creating more new facial models from one source facial model and many target facial models. Some examples are presented to demonstrate the effectiveness and applications of the proposed method in 3D facial blendshapes.

Strata and Surface Influence Range of Deep Coal Mining for Mine Land Reuse

The rational assessment and determination of strata and surface influence range of underground coal mining is straightly associated to the safe production of the wellbore, the reuse of mine land and the regional development. With the depth of coal mining in the world increasing, if the boundary curve of strata and surface movement continues to be considered as a straight line, there will be a great deviation from the real situation, which will seriously waste the land resources of mining area. To solve this issue, the numerical simulation methods were employed to investigate the stratum and surface movement boundary curves of deep caving and backfilling mining in this paper. The findings indicated that: 1) The strata and surface movement boundary of deep caving and backfilling mining were all curves, and they were in accordance with the exponential function, but the influence range of strata and surface movement of deep different mining methods were different; 2) The backfilling rate of deep backfilling mining had an influence on movement boundary of strata and surface. With the backfilling rate decreasing, the influence range of strata and surface movement boundary were increased. 3) The research results were applied to a case in order to confirm the new methods for determining the influence range of strata and surface movement of deep mining. Example application shows that the safe production of the wellbore not only can be guaranteed, but also the reuse area of the mine field can be enhanced.

Edge detection with trigonometric polynomial shearlets

In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

Tool Wear Assessment Approach Based on the Neighborhood Rough Set Model and Nearest Neighbor Model

In order to assess the degree of wear of tool for milling process quantitatively, a new assessment approach is proposed. Firstly, making full use of the neighbor information, two sensitive features are selected by using the neighborhood rough set model, and then, boundary curves are established by using the nearest neighbor model with noncounter data in two dimension spaces. Secondly, the intersection area or expanding area is used to describe the difference between two boundary models because the intersection area or expanding area can consider the effect of distance and angle simultaneously in two dimension spaces. Thirdly, after determining a baseline state, a new quantitative assessment indicator (QAI) can be calculated based on the intersection area or expanding area. The QAI can directly measure the difference between the model of baseline state and the model of unknown state and indirectly measure the degree of wear of tool. Finally, the effectiveness of the assessment approach is proven by using the Milling Dataset which was provided by the NASA Ames Research Center.

Extended SQ-Coons Surface and Its Application on Fairing Automobile Surface Design

Aiming at car design, a parametric design method is proposed in this paper. There are two contents in this method, a novel parametric surface, a shape-adjustable automobile styling template, and a design method of integrating the two contents. The surface termed extended SQ-Coons surface constructed in this paper has the advantage of being fine adjusted while always interpolating at the given boundary curves, which is suitable for its application in automobile modeling design. Matching with the surface, a car model template built by multiple quadrilateral surfaces is proposed. The car models built by the template could achieve the parametric adjustability of all modeling features on the premise of maintaining G 1 continuity between subsurfaces. Finally, after the integration of the surface and the template, a whole set of parametric automobile surface modeling design method is proposed. In this method, the overall shape of the car body is determined through multi angle hand drawing, and curve control points, segmentation parameters, and shape parameters are used to adjust the detail modeling. The final results show that the novel surface and template proposed could be used to parametrically establish the vehicle models of various shapes and improve the design efficiency in the conceptual design stage.

Curvature effects on phase transitions in chiral magnets

Periodical equilibrium states of magnetization exist in chiral ferromagnetic films, if the constant of antisymmetric exchange (Dzyaloshinskii–Moriya interaction) exceeds some critical value. Here, we demonstrate that this critical value can be significantly modified in curved film. The competition between symmetric and antisymmetric exchange interactions in a curved film can lead to a new type of domain wall which is inclined with respect to the cylinder axis. The wall structure is intermediate between Bloch and Néel ones. The exact analytical solutions for phase boundary curves and the new domain wall are obtained.

Comparison of 316LN Ratcheting Boundary at Room Temperature and 350 °C

The ratcheting boundaries of 316LN austenitic stainless steel under room temperature and 350 °C were determined by efficiency curve method. The iso-cumulative plastic strain curves of the material were obtained through numerical interpolation method. It is indicated that the primary stress and secondary stress range increase with the increase of cumulative plastic deformation. Under the same cumulative plastic strain, the iso-cumulative plastic strain curve at 350 °C is higher than that at room temperature. The effective primary stress was introduced to obtain ratcheting boundary curves. The results show that the material at 350 °C has a higher ratcheting boundary compared with that at room temperature. It is indicated that, under equivalent stress condition, the bearing capacity of 316LN at 350 °C is stronger than that at room temperature. This is due to the dynamic strain aging of the material at high temperature. At 350 °C, the solid solution atoms and dislocations are pinned together, which leads to a strengthening effect on the material. The ratcheting boundary curves determined in this study are compared with relevant standards. The test results suggest that the material served in different environments should be checked by different ratcheting boundary curves.