Nonparallel Planes Research Articles

Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

A new interactive software tool is described, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) originated as a script for the open source visual CLUCalc, which fully supports geometric algebra computation. Selected generators (Hestenes and Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implementation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rotation, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes. We first give some insights into the Clifford geometric algebra description of space groups. We relate the choice of symmetry vectors and the origin of cells in the geometric algebra description and its implementation in the SGV to the conventional crystal cell choices in the International Tables of Crystallography (T. Hahn, Springer, 2005). Finally we briefly explain how to use the SGV beginning with space group selection. The interactive computer graphics can be used to fully understand how reflections combine to generate all 230 three-dimensional space groups.

Characterization and reduction of saturation banding in multiplanar coherent and incoherent steady‐state imaging

Hypointense band artifacts occur at intersections of nonparallel imaging planes in rapidly acquired MR images; quantitative or numerical analysis of these bands and strategies to mitigate their appearance have largely gone unexplored. The magnetization evolution in the different regions of multiplanar images was simulated for three common rapid steady-state techniques (spoiled gradient echo, steady state free precession, balanced steady state free precession). Saturation banding was found to be highly dependent on the pulse sequence, acquisition time, and phase-encoding order. Encoding the center of k-space at the end of the acquisition of each slice (i.e., reverse centric phase encoding) is demonstrated to be a simple and robust method for significantly reducing the relative saturation in all imaging planes. View ordering and resolution dependence were confirmed in multiplanar abdominal images. The added importance of reducing the artifact in accelerated acquisition techniques (e.g., parallel imaging) is particularly notable in multiplanar balanced steady state free precession images in the brain.

Formation of ultrafine and dense α-Al2O3 nanoparticles via kinetic phase change in a dynamic process

Ultrafine (5 nm) Al2O3 nanoparticles having a predominant α-type structure and with an internal compressive stress up to ca. 15 GPa were synthesized by pulsed laser ablation on Al target under a very high power density (1.8 × 1012 W/cm2) with oxygen flow in vacuum. The ultrafine α-Al2O3 was alternatively formed from the minor γ-Al2O3 nanocondensates upon electron irradiation. In such a case, the polymorphs follow a special crystallographic relationship [110]γ//[2 $$ \bar{1}$$ $$ \bar{1} $$ 0]α; ( $$ \bar{1} $$ 1 $$ \bar{1} $$ )γ//(0 $$ \bar{1} $$ 14)α with a mixed mismatch strain yet nonparallel close-packed planes indicating a reconstructive-type transformation. The formation of metastable α-Al2O3 in the dynamic processes can be rationalized by the kinetic phase change from the amorphous lamellar and/or γ-Al2O3 depending on their free energy versus cell volume curves. The dense and ultrafine-sized Al2O3 polymorphs with a rather low minimum band gap of 3.7 eV shed light on their natural occurrence in dynamic settings and abrasive as well as catalytic/optoelectronic applications.

Response to “Shear Impossibility—Comments on ‘Void Growth by Dislocation Emission’ and ‘Void Growth in Metals’”

We thank the authors of Ref. [1] for their comments and interest in our work [2–6]. We agree that the emission of shear loops on one single shear plane cannot lead to void growth. However, our simulations suggest that coordinated shear on non-parallel planes can lead to void growth/shrinkage [3–5]. Detailed descriptions of this process are still being worked out and represent an exciting area for future research. Thus, there seem to be two separate dislocation mechanisms operating either sequentially or simultaneously: shear and prismatic loop emission. Whereas prismatic loops are well known and have been extensively documented, the postulation of “special” shear loops is novel. The authors [1] have correctly pointed out that special restrictions need to be applied to these shear loops, something not mentioned in Refs. [2–6]. On the subject of shear impossibility, the authors [1] bring some good points in their comments of our papers. Indeed, the closed loop emitted from the surface of the void does not by itself expand the void. This is obvious from our papers, in which we considered a growth of a cylindrical void by emission of an edge dislocation [2], and a collapse of the void by an emission of an opposite-signed edge dislocation [5]. Thus, if both (positive and negative) dislocations were emitted (which is a 2-D counterpart of a dislocation loop), no net effect on the void growth would take place. Void growth takes place by the outward transfer of the material, which is possible even in an incompressible case due to the outward movement of the remote boundary of the body (the plastically deformed region around the void is surrounded by an elastically deformed region). Eq. (1) of the Comments on our papers [1] nicely states that the net amount of added material associated with the crea-

Proximal support vector machine using local information

Instead of standard support vector machines (SVMs) that classify points to one of two disjoint half-spaces by solving a quadratic program, the plane classifier GEPSVM (proximal SVM classification via generalized eigenvalues) classifies points by assigning them to the closest of two nonparallel planes which are generated by their corresponding generalized eigenvalue problems. A simple geometric interpretation of GEPSVM is that each plane is closest to the points of its own class and furthest to the points of the other class. Analysis and experiments have demonstrated its capability in both computation time and test correctness. In this paper, following the geometric intuition of GEPSVM, a new supervised learning method called proximal support vector machine using local information (LIPSVM) is proposed. With the introduction of proximity information (consideration of underlying information such as correlation or similarity between points) between the training points, LIPSVM not only keeps aforementioned characteristics of GEPSVM, but also has its additional advantages: (1) robustness to outliers; (2) each plane is generated from its corresponding standard rather than generalized eigenvalue problem to avoid matrix singularity; (3) comparable classification ability to the eigenvalue-based classifiers GEPSVM and LDA. Furthermore, the idea of LIPSVM can be easily extended to other classifiers, such as LDA. Finally, some experiments on the artificial and benchmark datasets show the effectiveness of LIPSVM.

TSVR: An efficient Twin Support Vector Machine for regression

The learning speed of classical Support Vector Regression (SVR) is low, since it is constructed based on the minimization of a convex quadratic function subject to the pair groups of linear inequality constraints for all training samples. In this paper we propose Twin Support Vector Regression (TSVR), a novel regressor that determines a pair of -insensitive up- and down-bound functions by solving two related SVM-type problems, each of which is smaller than that in a classical SVR. The TSVR formulation is in the spirit of Twin Support Vector Machine (TSVM) via two nonparallel planes. The experimental results on several artificial and benchmark datasets indicate that the proposed TSVR is not only fast, but also shows good generalization performance.

Fuzzy Twin Support Vector Machine 개발 및 전리층 레이더 데이터를 통한 성능 평가

Support Vector machine is the classifier which is based on the statistical training theory. Twin Support Vector Machine(TWSVM) is a kind of binary classifier that determines two nonparallel planes by solving two related SVM-type problems. The training time of TWSVM is shorter than that of SVM, but TWSVM doesn`t shows worse performance than that of SVM. This paper proposes the TWSVM which is applied fuzzy membership, and compares the performance of this classifier with the other classifiers using Ionosphere radar data set.

Optimal kernel selection in twin support vector machines

In twin support vector machines (TWSVMs), we determine pair of non-parallel planes by solving two related SVM-type problems, each of which is smaller than the one in a conventional SVM. However, similar to other classification methods, the performance of the TWSVM classifier depends on the choice of the kernel. In this paper we treat the kernel selection problem for TWSVM as an optimization problem over the convex set of finitely many basic kernels, and formulate the same as an iterative alternating optimization problem. The efficacy of the proposed classification algorithm is demonstrated with some UCI machine learning benchmark datasets.

Formulation of the rotational transformation of wave fields and their application to digital holography

Rotational transformation based on coordinate rotation in Fourier space is a useful technique for simulating wave field propagation between nonparallel planes. This technique is characterized by fast computation because the transformation only requires executing a fast Fourier transform twice and a single interpolation. It is proved that the formula of the rotational transformation mathematically satisfies the Helmholtz equation. Moreover, to verify the formulation and its usefulness in wave optics, it is also demonstrated that the transformation makes it possible to reconstruct an image on arbitrarily tilted planes from a wave field captured experimentally by using digital holography.

Effect of heat treatment on the stress and structure evolution of plasma deposited boron nitride thin films

Boron nitride (BN) thin films are deposited at 573 K by plasma enhanced chemical vapor deposition (PECVD) with ammonia (NH 3) and hydrogen diluted diborane (15% B 2H 6 in H 2) source gases. UV–visible and Fourier transform infrared (FTIR) spectroscopies together with surface profilometry are used for the film characterization. These films are hydrogenated (BN:H) whose hydrogen content is pursued following the 1.5 h annealing process at 748 K, 923 K and 1073 K under nitrogen atmosphere. Hydrogen escape with the rising annealing temperature is observed together with increases of the compressive stress, band gap and Urbach energies. Films are composed of the hexagonal BN (h-BN) clusters that grow dominantly parallel to the substrate surface with some non-parallel planes at the edges of the clusters, which are embedded in an amorphous tissue (the so-called turbostratic structure, t-BN). Annealing seems to promote non-parallel planes, thus creating more stressful and distorted network. Most of hydrogen atoms are removed from the film annealed at 1073 K and wurtzite BN (w-BN) phase is formed with volume fraction of 57%. As a consequence or in parallel of hydrogen reduction, high compressive stress causes the cracking of the films.

Continuous digitization in Khalimsky spaces

A real-valued function defined on R n can sometimes be approximated by a Khalimsky-continuous mapping defined on Z n . We elucidate when this can be done and give a construction for the approximation. This approximation can be used to define digital Khalimsky hyperplanes that are topological embeddings of Z n into Z n + 1 . In particular, we consider Khalimsky planes in Z 3 and show that the intersection of two non-parallel Khalimsky planes contains a curve homeomorphic to the Khalimsky line.

Developing and Benchmarking MH4D, A Tetrahedral Mesh MHD Code

Periodic boundary conditions, applicable to parallel and non-parallel planes have been developed. Mode selection is achieved by modeling a “slice” of a domain. For instance, modeling a 1/4-slice of a cylindrical Z-pinch geometry eliminates the m = 1,2, and 3 modes. M = 4 and higher modes will still be simulated. An insulating boundary condition has been developed. External circuits can be modeled by specifying electric and magnetic surface fields. A specified magnetic field simulates a circuit with constant current where Ampere’s law dictates the current flow. A specified electric field simulates a constant applied voltage. A constant current model is applicable to experiments with relatively high-impedance power supplies. To function with the insulating boundary condition, implicit and semi-implicit solvers require inclusion of surface terms. These solvers are being modified. Also, ionization physics capability is being developed. Progress in boundary condition development is presented, and a benchmarking and verification strategy is described.

SU-FF-J-118: A Method for Defining Segmentation Surfaces Using Nonparallel Contours

Purpose: Segmentation of anatomical structures from CT images for treatment planning has typically involved the process of manual delineation of contours on image slices. However, the original image slices are not always the best planes that allow accurate delineation of structures, and as a result, software tools that permit drawing of contours on oblique or even non-parallel planes are gradually becoming available. Unfortunately, what is missing still is the accompanying computational method for extracting surfaces from such contours. We investigate a novel approach for automatically generating a triangular surface mesh from contour lines defined on arbitrary, possibly intersecting planes. Method and Materials: Using contour delineation programs, we first obtain the defining equations of each contour plane as well as contour line segments delineated on each plane. The contour planes partition the entire space into convex sub-spaces. We compute a surface mesh within each subspace by extending our previously developed method for surface extraction from contours defined on two parallel planes. The entire surface is then constructed by merging surfaces within each subspace. Finally, a normal smoothing step is employed to fair the appearance of the surface. This has been implemented in CERR. Results: Testing the initial implementation on synthetic data reveals that our method is capable of extracting smooth, water-tight surface meshes from complex contour curves with arbitrary shape (e.g., either convex or non-convex) and arbitrary topology (e.g., consisting of either a single loop or multiple loops) that reside on arbitrarily oriented planes. Conclusion: We have developed a novel computational approach for extracting surfaces from contours delineated on planes that may have nonuniform spacing and may have arbitrary orientations. Our method meets the critical need for segmentation of anatomical structures in treatment planning systems using contours delineated from oblique or non-parallel planes.

Twin Support Vector Machines for Pattern Classification

We propose Twin SVM, a binary SVM classifier that determines two nonparallel planes by solving two related SVM-type problems, each of which is smaller than in a conventional SVM. The Twin SVM formulation is in the spirit of proximal SVMs via generalized eigenvalues. On several benchmark data sets, Twin SVM is not only fast, but shows good generalization. Twin SVM is also useful for automatically discovering two-dimensional projections of the data.

The role of α- to ɛ-Co phase transformation on strain hardening of a Co–Cr–Mo laser clad

A laser clad Co–Cr–Mo alloy has been analysed by the electron backscatter diffraction (EBSD) technique to examine the microstructural evolution during tensile deformation and to study the role of the evolved microstructure on strain hardening of the clad. The allotropic phase transformation from α- to ɛ-Co, that did not take place during cooling from the solidification temperature, occurs in the form of a strain-induced transformation during plastic deformation. Combined slip on non-parallel planes in parallel bands of α- and ɛ-Co prevents the formation of cracks and contributes to the strain hardening of the material. Due to twin formation at intersecting ɛ-Co bands, { 1 1 ¯ 0 1 } planes become almost parallel to {1 1 1} planes of α-Co, and facilitate further glide of dislocations. Owing to the strain-induced transformation of α- to ɛ-Co, the laser clad Co–Cr–Mo investigated in this study can accommodate high deformation before cracks are formed at the “wavy slip-lines” and the precipitates.

Fuzzy multi-category proximal support vector classification via generalized eigenvalues

Given a dataset, where each point is labeled with one of M labels, we propose a technique for multi-category proximal support vector classification via generalized eigenvalues (MGEPSVMs). Unlike Support Vector Machines that classify points by assigning them to one of M disjoint half-spaces, here points are classified by assigning them to the closest of M non-parallel planes that are close to their respective classes. When the data contains samples belonging to several classes, classes often overlap, and classifiers that solve for several non-parallel planes may often be able to better resolve test samples. In multicategory classification tasks, a training point may have similarities with prototypes of more than one class. This information can be used in a fuzzy setting. We propose a fuzzy multi-category classifier that utilizes information about the membership of training samples, to improve the generalization ability of the classifier. The desired classifier is obtained by using one-from-rest (OFR) separation for each class, i.e. 1: M -1 classification. Experimental results demonstrate the efficacy of the proposed classifier over MGEPSVMs.

Multisurface proximal support vector machine classification via generalized eigenvalues

A new approach to support vector machine (SVM) classification is proposed wherein each of two data sets are proximal to one of two distinct planes that are not parallel to each other. Each plane is generated such that it is closest to one of the two data sets and as far as possible from the other data set. Each of the two nonparallel proximal planes is obtained by a single MATLAB command as the eigenvector corresponding to a smallest eigenvalue of a generalized eigenvalue problem. Classification by proximity to two distinct nonlinear surfaces generated by a nonlinear kernel also leads to two simple generalized eigenvalue problems. The effectiveness of the proposed method is demonstrated by tests on simple examples as well as on a number of public data sets. These examples show the advantages of the proposed approach in both computation time and test set correctness.

Spin Rotation inα−Fe2O3Nanoparticles by Interparticle Interactions

Nanoparticles of alpha-Fe2O3 (hematite) typically have the sublattice magnetization directions in the hexagonal (001) plane below the Ne el temperature. By use of Mo ssbauer spectroscopy we have found that for agglomerated particles the sublattice magnetization may be rotated of the order of 15 degrees out of plane, depending on the particle size. The spin rotation can be explained by exchange interaction between neighboring particles with nonparallel (001) planes. The results imply that interparticle interactions can lead to spin directions deviating from the easy axis defined by the magnetic anisotropy.

Zero Moment Point—Measurements From a Human Walker Wearing Robot Feet as Shoes

The anthropomorphic biped robot Bip is equipped with sensors for measuring the ground/feet forces in order to localize the center of pressure (CoP) and zero moment point (ZMP). This paper focuses on experimental results regarding the evolution of the ground contact forces, obtained from a human walker wearing the robot feet as shoes. First, one determines the influence of heavy and rigid metallic shoes on the gait of the subject, during experiments carried out on flat ground. Second, the evolution of the contact forces is studied while walking on parallel planes with different elevations (stairs), then one addresses the case where the feet are supported by two nonparallel planes (uneven terrain). The corresponding analysis is founded on the concepts of virtual supporting surface and pseudo-CoP-ZMP introduced in a companion paper, discussing the theoretical aspects of CoP and ZMP (Sardain and Bessonnet, 2004). Beyond the academic contribution, the data analyzed in this paper could constitute an incitement to design truly anthropomorphic feet, for Bip as well as for other biped robots, because all the current robot feet are functionally rather poor.

Influence of non-glide stresses on plastic flow: from atomistic to continuum modeling

The Schmid law, which is accurate for face-centered-cubic (fcc) metals, assumes that only the shear stress acting in the slip plane in the slip direction controls the plastic deformation. Hence, it is implicitly assumed that the critical resolved shear stress (CRSS) for the slip is not affected by any other components of the applied stress tensor. This rule is almost ubiquitously utilized in large-scale continuum computations of plastically deforming single and polycrystals. On the other hand, in materials with more complex structures and for some orientations of the dislocation line the cores can spread onto several non-parallel planes. The most widely-known example is the screw dislocation in bcc metals, though this phenomenon is quite universal in structures that are not close packed. Signatures of such core configurations commonly include unexpected deformation modes and slip geometries, strong and unusual dependence of flow stresses on temperature and strain rate, a high Peierls stress, and, in general, a breakdown of the Schmid law. In this paper, we first summarize results of atomistic computer simulations of the response of 1/2〈1 1 1〉 screw dislocations to the applied shear stresses. The calculations have been made using central-force many-body potentials and tight-binding based bond-order potentials for molybdenum. While the core structure found is not the same for the two descriptions of atomic interactions, both lead to a very similar orientation dependence of the critical resolved shear stress for the dislocation motion which takes place along the most highly stressed {1 1 0} plane. This dependence reveals the break down of the Schmid law invoked by the effect of shear stresses acting in another {1 1 0} plane, which are called non-glide stresses. These results are then transferred to macroscopic level by formulating single crystal yield criteria that include the effects of non-glide components of the stress tensor. These criteria form a basis for multislip yield criteria and flow relations for continuum analyses. Using this approach we demonstrate that the effects of non-glide stresses that have their origin at the level of individual dislocations also have significant effect on polycrystalline response, including a significant tension-compression asymmetry.