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The Directional Discrete Cosine Transform
Published in Humberto Ochoa-Domínguez, K. R. Rao, Discrete Cosine Transform, 2019
Humberto Ochoa-Domínguez, K. R. Rao
Cartesian product of two graphs Recall that the Cartesian product of two sets X and Y is X×Y={(x,y)|x∈X,y∈Y}. The Euclidean plane Euclidean plane is an example of a Cartesian product of two sets of real numbers where R×R is the Cartesian productCartesian product of the two sets (i.e., X∈R and Y∈R). Another simple example is A={x,y} and B={−1,0,1} the A×B={(x,−1),(x,0),(x,1),(y,−1),(y,0),(y,1)}.
Optimal junction localization minimizing maximum miners’ evacuation distance in underground mining network
Published in Mining Technology, 2023
Zhixuan Shao, Maximilien Meyrieux, Mustafa Kumral
This section introduces three methods dedicated to solving the single junction point MiniMax localization problem in the underground mining network Section ‘Elzinga–Hearn algorithm’ shows how the Elzinga–Hearn algorithm (Elzinga and Hearn 1972) is used to solve the minimum covering circle problem. The minimum covering circle problem can be described as the process of obtaining the circle with the minimum radius in which the given points in the plane are covered. Note that the problem is valid in the two-dimensional Euclidean plane (i.e. the Euclidean distance is utilized as the distance metric in the plane). Specifically, define the function . The function is to be minimized by: where is used to denote the and coordinate of the new facility point and is used to denote the and coordinate of the surrounding point(s)-any known point that is not the new point.
Parallel curves
Published in International Journal of Mathematical Education in Science and Technology, 2022
Richard Dexter Sauerheber, Tony Stewart
Figure 1 shows the relationship between parallel lines, where vectors are drawn from one line to corresponding points on the parallel line where the connecting vector is perpendicular to both lines at the same time. Each connecting vector is the same length because the lines are themselves parallel and cannot intersect. Parallel lines must share the same derivative while being shifted in the Euclidean plane. Geometry courses routinely and correctly teach proofs indicating that parallel lines have linear transversals that must form congruent alternate interior angles, congruent alternate exterior angles, and pairs of interior angles and exterior angles on the same side of the transversal that must be supplementary ( = 180°) (Ratti and McWaters, 2010). This also means that any segment or vector Eextending from one line to the other that is perpendicular to one line must also be perpendicular to the other parallel line. All perpendicular segments intersecting any two parallel lines must always form only right angles and must be of equal length. If not, then the lines are not parallel.
Exploiting low-rank structure in semidefinite programming by approximate operator splitting
Published in Optimization, 2022
Mario Souto, Joaquim D. Garcia, Álvaro Veiga
Problem instances: In a set of numerical simulation, we randomly generate anchor points and distances measurements. Each anchor and sensor has its position in the two-dimensional Euclidean plane, i.e. d = 2. In this sense, if the relaxation is exact, the optimal solution must have a rank of two. This property justifies the ProxSDP (low-rank) performing well compared to other solvers when the number of sensors grows, as it can be seen in Table 2. The low-rank implementation give the same results of the full-rank version in less time. Interestingly SCS is still the fastest solver for this kind of problem. SDPNAL+ was not able to solve any of these instances because of some internal error.