We construct cellular resolutions for monomial ideals via discrete Morse theory. In particular, we develop an algorithm to create homogeneous acyclic matchings and we call the cellular resolutions induced from these matchings Barile-Macchia resolutions. These resolutions are minimal for edge ideals of weighted oriented forests and (most) cycles. As a result, we provide recursive formulas for graded Betti numbers and projective dimension. Furthermore, we compare Barile-Macchia resolutions to those created by Batzies and Welker and some well-known simplicial resolutions. Under certain assumptions, whenever the above resolutions are minimal, so are Barile-Macchia resolutions.

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<div>Improvised Explosive Devices (IEDs) have been developed over the years across many nations around the world. IEDs used by terrorist actions and in warfare cause devastating death, injuries and damage. To protect the public, many emergency responders have to risk their lives by performing extremely hazardous tasks such as interacting with suspected IEDs. To prevent the emergency response teams from being negatively impacted by IEDs, many different kinds of response robots have been deployed in many locations worldwide – allowing first responders a safe way to interact with these menaces from a distance. This thesis contributes to the understanding of using robot arms with a Leader–Follower (LF) approach to help humans with performing dexterous operations like those which are inevitably required for manipulating IEDs remotely. The LF approach allows operators to remotely manipulate a robot arm without putting operators’ lives in danger. By physically controlling one arm from a safe distance, operators can successfully copy its movements to a second arm. As a result, we argue, this approach can be helpful for minimizing operator risk when interacting with suspicious devices while at the same time facilitating more intuitive remote control.</div>