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We study ground states of a relativistic Fermi system involved with the pseudo-differential operator - c 2 Δ + c 4 m 2 - c 2 m in the L 2 -subcritical case, where m > 0 denotes the rest mass of fermions, and c ≥ 1 represents the speed of light. By employing Green's function and the variational principle of many-fermion systems, we prove the existence of ground states for the system. The asymptotic behavior of ground states for the system is also analyzed in the non-relativistic limit where c → ∞ . [ABSTRACT FROM AUTHOR]

Draglines are among the largest earthmoving machines in surface mining, where they are used to remove the waste material (overburden) sitting above a target mineral or coal deposit. The effectiveness of their operation is highly dependent on decisions made by their operators, including the sequence of positions at which they operate and the bulk material movement (what to dig and where to dump) during excavation at each position. In this paper, we formulate a mixed-integer linear program that captures the operational constraints imposed in dragline excavation to determine the optimal material movement for a prescribed sequence of dragline positions. Through a simulation study, we show that the dragline productivity can be improved by more effective assignment of material movement tasks to individual dragline positions. We find that the improvement made compared to using a 'greedy'digging and dumping strategy depends on the dragline positioning sequence and the terrain profile in the specific environment. The solutions in terms of what to dig and where to dump provide insights into the optimal digging and dumping patterns for different types of dragline positioning sequences, and contribute towards solving the overall dragline operation planning problem that includes the planning of the dragline positioning sequence. [ABSTRACT FROM AUTHOR]