A Polynomial Time Algorithm for Spatio-Temporal Security Games

An ever-important issue is protecting infrastructure and other valuable targets from a range of threats from vandalism to theft to piracy to terrorism. The "defender" can rarely afford the needed resources for a 100% protection. Thus, the key question is, how to provide the best protection using the limited available resources. We study a practically important class of security games that is played out in space and time, with targets and "patrols" moving on a real line. A central open question here is whether the Nash equilibrium (i.e., the minimax strategy of the defender) can be computed in polynomial time. We resolve this question in the affirmative. Our algorithm runs in time polynomial in the input size, and only polylogarithmic in the number of possible patrol locations (M). Further, we provide a continuous extension in which patrol locations can take arbitrary real values. Prior work obtained polynomial-time algorithms only under a substantial assumption, e.g., a constant number of rounds. Further, all these algorithms have running times polynomial in M, which can be very large

Composition-Dependent Mechanisms of Multiscale Tendon Mechanics

Tendons serve as an integral part of the musculoskeletal system by transferring loads from muscle to bone and providing joint mobility and stability. From the physiologically-loading perspective, while progress has been made in evaluating mechanical behavior of different types of tendons in tension, further work is needed to relate tendon mechanics to compositional and microstructural properties, particularly under non-tensile loading modalities (i.e., shear, compression). This information is vital to explore mechanisms of how mechanical signals lead to changes in tendon structure and composition to enable these tissues to function properly, including in in vivo multiaxial loading conditions. From the structural perspective, tendon exhibits a hierarchical organization as collagen is bundled into fibrils, fibers, fascicles, and finally full tissue. Within this hierarchy, linking components are believed to act as connections to maintain mechanical integrity. Three linking components have been proposed, namely elastic fibers, proteoglycans, and collagen crosslinks, however conclusions about their specific mechanical roles, assessed using experimental and computational approaches, are inconsistent. In addition, it remains unknown whether/how these linking components regulate tendon microscale behavior (i.e., at the level of cells) and mechanical signal transfer across length scales. Therefore, this study aimed to (1) develop a protocol that combined a biomechanical test device with two-photon microscopy to measure tendon mechanical strength and multiscale deformation; (2) apply this experimental approach to evaluate region-dependent biomechanics of tendons and related physical mechanisms governing their microscale behavior; (3) determine the role of proteoglycans and elastic fibers in tendon multiscale mechanical behavior using enzyme-treated tendons; and (4) elucidate the contribution of collagen crosslinks to tendon mechanics using in vivo treatment and in vitro culture. We found that different regions of bovine flexor tendon exhibited distinct elasticity, but not viscosity, when subjected to shear and compression, and that fiber sliding and reorganization were the primary modes of microscale deformation. Elastic fibers contributed to supraspinatus tendon (SST) mechanical strength in shear, while proteoglycans appeared to not contribute to SST multiscale biomechanics. Rat SST with decreased collagen crosslink density showed inferior mechanical properties, demonstrating the role of collagen crosslinks on tendon mechanical behavior. Taken together, these results have illustrated tendon composition-mechanics relationships by evaluating mechanical contribution of specific linking components at different length scales. In addition, this work provides insight into mechanical consequences that may accompany extracellular matrix changes during tissue injury/degeneration, and as well provides useful data to aid the design of biomimetic engineered tissues with appropriate structure and composition