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Linear maps on matrices preserving parallel pairs arxiv.org/abs/2408.06366

Linear maps on matrices preserving parallel pairs

Two (real or complex) $m\times n$ matrices $A$ and $B$ are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm $\|\cdot\|$ if $\|A+ μB\| = \|A\| + \|B\|$ for some scalar $μ$ with $|μ|=1$ (resp. $μ=1$). We study linear maps $T$ on $m\times n$ matrices preserving parallel (resp. TEA) pairs, i.e., $T(A)$ and $T(B)$ are parallel (resp. TEA) whenever $A$ and $B$ are parallel (resp. TEA). It is shown that when $m,n \ge 2$ and $(m,n) \ne (2,2)$, a nonzero linear map $T$ preserving TEA pairs if and only if it is a positive multiple of a linear isometry, namely, $T$ has the form $$(1) \quad A \mapsto γUAV \quad \quad \text{or} \quad \quad (2) \quad A \mapsto γUA^{t} V \quad (\text{in this case}, m = n),$$ for a positive number $γ$, and unitary (or real orthogonal) matrices $U$ and $V$ of appropriate sizes. Linear maps preserving parallel pairs are those carrying form (1), (2), or the form $$ (3) \ A \mapsto f(A) Z$$ for a linear functional $f$ and a fixed matrix $Z$. The case when $(m,n) = (2,2)$ is more complicated. There are linear maps of $2\times 2$ matrices preserving parallel pairs or TEA pairs neither of the form (1), (2) nor (3) above. Complete characterization of such maps is given with some intricate computation and techniques in matrix groups.

arxiv.org

A multi-objective mixed integer linear programming model for supply chain planning of 3D printing arxiv.org/abs/2408.05213

A multi-objective mixed integer linear programming model for supply chain planning of 3D printing

3D printing is considered the future of production systems and one of the physical elements of the Fourth Industrial Revolution. 3D printing will significantly impact the product lifecycle, considering cost, energy consumption, and carbon dioxide emissions, leading to the creation of sustainable production systems. Given the importance of these production systems and their effects on the quality of life for future generations, it is expected that 3D printing will soon become one of the global industry's fundamental needs. Although three decades have passed since the emergence of 3D printers, there has not yet been much research on production planning and mass production using these devices. Therefore, we aimed to identify the existing gaps in the planning of 3D printers and to propose a model for planning and scheduling these devices. In this research, several parts with different heights, areas, and volumes have been considered for allocation on identical 3D printers for various tasks. To solve this problem, a multi-objective mixed integer linear programming model has been proposed to minimize the earliness and tardiness of parts production, considering their order delivery times, and maximizing machine utilization. Additionally, a method has been proposed for the placement of parts in 3D printers, leading to the selection of the best edge as the height. Using a numerical example, we have plotted the Pareto curve obtained from solving the model using the epsilon constraint method for several parts and analyzed the impact of the method for selecting the best edge as the height, with and without considering it. Additionally, a comprehensive sensitivity and scenario analysis has been conducted to validate the results.

arxiv.org
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