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Mineral Crystals
Published in Dexter Perkins, Kevin R. Henke, Adam C. Simon, Lance D. Yarbrough, Earth Materials, 2019
Dexter Perkins, Kevin R. Henke, Adam C. Simon, Lance D. Yarbrough
The unit cell shapes in Figure 4.23 demonstrate several kinds of symmetry. The top four, for example, have vertical mirror planes of symmetry (shown as dashed lines). The right side is a reflection of the left side; if we observed these unit cells in a mirror, they would appear the same. The top halves of the square, rectangle, diamond, and hexagon unit cells also reflect to the bottom halves. So, these four examples have both vertical and horizontal mirror planes. Additionally, the square, hexagon, and rhombus have diagonal mirror planes. The parallelogram, however, does not have any mirrors of symmetry at all.
MATLAB® for Chemical Engineering
Published in Mariano Martín Martín, Introduction to Software for Chemical Engineers, 2019
Mariano Martín, Luis Martín de Juan
The boundary conditions are CA = 0, but at the top of the region CA = 25. With these data we need to build our system of equations by filling the matrix. Let's consider a mesh of 10 × 10 elements. As we can see from the previous equation, we have a system of linear equations where all the coefficients of the nodes surrounding the point i, j have a coefficient equal to 1. We use a counter “a” to fill the ones in the diagonals and a counter “q” so that all the internal lines of the matrix are filled. Next, we fill in the boundary conditions of the geometry at the maximum of the x and y axis. Finally we take care of the vertex of the geometry and the diagonal. Thus our model is as follows:
On Scattered Data Representations Using Bivariate Splines
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
To convert any triangulation to be a type-O triangulation, we introduce a so-called edge swapping algorithm. Every interior edge e of a triangulation Δ is the diagonal of a quadrilateral Qe which is the union of two triangles of Δ with common edge e. Following [63], we say that e is a swappable edge if Qe is convex and no three of its vertices are collinear. If an edge e of a triangulation Δ is swappable, then we can create a new triangulation by swapping the edge. That is, if v1, ···, v4 are the vertices of Qe ordered in the counterclockwise direction, and if e has endpoints v1 and v3, then the swapped edge has endpoints v2 and v4. Two vertices in Δ will be called neighbors of each other if they are the endpoints of the same edge in Δ. Hence, while v1 and v3 are neighbors in the original triangulation Δ, v2 and v4 become neighbors in the new triangulation after the edge e is swapped.
Non-fragile stabilisation of nonlinear switched systems with actuator saturation
Published in Journal of Control and Decision, 2023
Representing rowof matrixby, we define the following symmetric polyhedron:Letbe thediagonal matrix, where the diagonal elements are either 1 or 0. For example, if, then, It is easy to know that there areelements inSuppose that each element is labelledand denote
Output feedback robust MPC using general polyhedral and ellipsoidal true state bounds for LPV model with bounded disturbance
Published in International Journal of Systems Science, 2019
Baocang Ding, Jie Dong, Jianchen Hu
For k>0, the following three TSBs will be refreshed in order to handle (8): ellipsoid, i.e. ( lies in the ellipsoid centred at ), where ;polyhedron with plane representation, i.e. where is a bias item, a pre-specified transformation matrix with being nonsingular, a diagonal matrix, and with and (for all ) being pre-specified;polyhedron with vertex representation, i.e. .
Pattern graph for sparse Hessian matrix determination†
Published in Optimization Methods and Software, 2018
Shahadat Hossain, Nasrin Hakim Mithila
The sparsity pattern of a symmetric matrix A with non-zero diagonal is conveniently represented by its adjacency graph: for each column index j of A, there is a vertex , and for , with , there is an edge . There is a path of length between vertices and , denoted and abbreviated , if there exist distinct vertices such that . A starp- colouring of is a mapping such that andfor each path of length 3, vertices in the path assume at least three distinct colours.