For a given tree tensor network $G$, we call a tuple of bond dimensions minimal if there exists a tensor $T$ that can be represented by this network but not on the same tree topology with strictly smaller bond dimensions. We establish necessary and sufficient conditions on the bond dimensions of a tree tensor network to be minimal, generalizing a characterization of Carlini and Kleppe about existence of tensors with a given multilinear rank. We also show that in a minimal tree tensor network, the non-minimal tensors form a Zariski closed subset, so minimality is a generic property in this sense.

This study examined the effects of robot speed, number, and spatial orientation on operator stress, mental workload, attention, and excitement during human–robot collaboration. As collaborative robots (cobots) become increasingly prevalent in modern manufacturing, operator safety concerns persist. Despite Industry 5.0’s focus on human-centered production, the impact of robot configuration on operators’ cognitive and emotional states remains underexplored. Thirty participants completed eight sessions of a Lego block stacking task using a mixed-factor (3 × 2 × 2) within-subjects design that varied robot speed (slow, fast, or mixed), robot number (one or two), and left–right spatial orientation. Physiological data were collected continuously via an Empatica E4 wristband and analyzed using linear mixed models; subjective data were collected via a post-session questionnaire and analyzed using non-parametric tests. Slower robot speeds reduced stress, workload, attention, and excitement relative to higher speeds. Two robots increased stress, mental workload, attention, and excitement relative to one robot, with consistent elevations in electrodermal activity, shorter inter-beat intervals, and lower skin temperature confirming sympathetic arousal. The fast two-robot condition produced the highest subjective demand across all measures. Notably, one robot at fast speed and two robots at slow speed produced statistically comparable subjective workload, suggesting both configurations offer productive alternatives with manageable operator strain. Robot spatial orientation did not significantly affect any outcome measure. These findings suggest that, under the conditions of this laboratory paradigm, a configuration of one robot at fast speed may offer a favorable balance of workload and engagement for safe and effective collaboration, with two robots at slow speed as a viable secondary alternative; direct objective productivity data were not collected, and these recommendations should be validated against task throughput in future work. Design implications for collaborative workstations are discussed.