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The Koopman operator theory provides a global linearization framework for general nonlinear dynamics, offering significant advantages for system analysis and control. However, practical applications typically involve approximating the infinite-dimensional Koopman operator in a lifted space spanned by a finite set of observable functions. The accuracy of this approximation is the key to effective Koopman operator-based analysis and control methods, generally improving as the dimension of the observables increases. Nonetheless, this increase in dimensionality significantly escalates both storage requirements and computational complexity, particularly for high-dimensional systems, thereby limiting the applicability of these methods in real-world problems. In this paper, we address this problem by reformulating the Koopman operator in tensor format to break the curse of dimensionality associated with its approximation through tensor decomposition techniques. This effective reduction in complexity enables the selection of high-dimensional observable functions and the handling of large-scale datasets, which leads to a precise linear prediction model utilizing the tensor-based Koopman operator. Furthermore, we propose an optimal control framework with the tensor-based Koopman operator, which adeptly addresses the nonlinear dynamics and constraints by linear reformulation in the lifted space and significantly reduces the computational complexity through separated representation of the tensor structure. 

We investigate the controllability of an origami system composed of Miura-ori cells. A substantial volume of research on folding architecture, kinematic behavior, and actuation techniques of origami structures has been conducted. However, understanding their transient dynamics and constructing control models remains a formidable task, primarily due to their innate flexibility and compliance. In light of this challenge, we discretize the origami system into a network composed of interconnected particle masses alongside bar and hinge elements. This yields a state-space representation of the system's dynamics, enabling us to obtain the system's controllability attributes. Informed by this computational framework, we explore the controllability Gramian-based method for finding the most efficient crease line for Miura-ori cell deployment using an actuator. We demonstrate that the deployment efficiency guided by this theoretical method agrees with the empirical results obtained from the energy consumption to deploy the origami structure. This investigation paves the way toward designing and operating an efficient the complex actuation system for origami tessellations. 

Abstract Bioturbation can increase time averaging by downward and upward movements of young and old shells within the entire mixed layer and by accelerating the burial of shells into a sequestration zone (SZ), allowing them to bypass the uppermost taphonomically active zone (TAZ). However, bioturbation can increase shell disintegration concurrently, neutralizing the positive effects of mixing on time averaging. Bioirrigation by oxygenated pore-water promotes carbonate dissolution in the TAZ, and biomixing itself can mill shells weakened by dissolution or microbial maceration, and/or expose them to damage at the sediment–water interface. Here, we fit transition rate matrices to bivalve age–frequency distributions from four sediment cores from the southern California middle shelf (50–75 m) to assess the competing effects of bioturbation on disintegration and time averaging, exploiting a strong gradient in rates of sediment accumulation and bioturbation created by historic wastewater pollution. We find that disintegration covaries positively with mixing at all four sites, in accord with the scenario where bioturbation ultimately fuels carbonate disintegration. Both mixing and disintegration rates decline abruptly at the base of the 20- to 40-cm-thick, age-homogenized surface mixed layer at the three well-bioturbated sites, despite different rates of sediment accumulation. In contrast, mixing and disintegration rates are very low in the upper 25 cm at an effluent site with legacy sediment toxicity, despite recolonization by bioirrigating lucinid bivalves. Assemblages that formed during maximum wastewater emissions vary strongly in time averaging, with millennial scales at the low-sediment accumulation non-effluent sites, a centennial scale at the effluent site where sediment accumulation was high but bioturbation recovered quickly, and a decadal scale at the second high-sedimentation effluent site where bioturbation remained low for decades. Thus, even though disintegration rates covary positively with mixing rates, reducing postmortem shell survival, bioturbation has theneteffect of increasing the time averaging of skeletal remains on this warm-temperate siliciclastic shelf. 

Abstract Understanding how time averaging changes during burial is essential for using Holocene and Anthropocene cores to analyze ecosystem change, given the many ways in which time averaging affects biodiversity measures. Here, we use transition-rate matrices to explore how the extent of time averaging changes downcore as shells transit through a taphonomically complex mixed layer into permanently buried historical layers: this is a null model, without any temporal changes in rates of sedimentation or bioturbation, to contrast with downcore patterns that might be produced by human activity. Assuming stochastic burial and exhumation movements of shellsbetweenincrements within the mixed layer and stochastic disintegrationwithinincrements, we find that almost all combinations of net sedimentation, mixing, and disintegration produce a downcore increase in time averaging (interquartile range [IQR] of shell ages), this trend is typically associated with a decrease in kurtosis and skewness and by a shift from right-skewed to symmetrical age distributions. A downcore increase in time averaging is thus the null expectation wherever bioturbation generates an internally structured mixed layer (i.e., a surface, well-mixed layer is underlain by an incompletely mixed layer): under these conditions, shells are mixed throughout the entire mixed layer at a slower rate than they are buried below it by sedimentation. This downcore trend created by mixing is further amplified by the downcore decline in disintegration rate. We find that transition-rate matrices accurately reproduce the downcore changes in IQR, skewness, and kurtosis observed in bivalve assemblages from the southern California shelf. The right-skewed shell age-frequency distributions typical of surface death assemblages—the focus of most actualistic research—might be fossilized under exceptional conditions of episodic anoxia or sudden burial. However, such right-skewed assemblages will typically not survive transit through the surface mixed layer into subsurface historical layers: they are geologically transient. The deep-time fossil record will be dominated instead by the more time-averaged assemblages with weakly skewed age distributions that form in the lower parts of the mixed layer.