Toward the Rectilinear Crossing Number of KnK_n: New Drawings, Upper Bounds, and Asymptotics

Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersect in a point unless that point is an endpoint of all three. The rectilinear crossing number of K_n is the fewest number of edge crossings attainable over all rectilinear drawings of K_n. For each n we construct a rectilinear drawing of K_n that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of drawings of K_n with good asymptotics. Finally, we mention some old and new open problems.Comment: 13 Page

Nonlinear Methods for Spacecraft Guidance and Trajectory Optimization

Many future spacecraft missions are planned to operate far from Earth in highly nonlinear environments, while performing complex navigational maneuvers. This increased complexity in spacecraft trajectories will necessitate the development of new guidance, maneuver design, and trajectory planning algorithms that are suitable for these intricate mission designs. In particular, there is a need for new methods that strike a balance between computationally expensive, full-fidelity trajectory optimization algorithms, and simplified, linearized guidance methods. This dissertation seeks to bridge this gap by developing computationally efficient, accurate, and flexible algorithms using state transition tensors (STTs) to model the nonlinear spacecraft dynamics. First, a higher-order impulsive spacecraft guidance scheme with both a fixed and variable time-of-flight is developed using the STTs of a reference trajectory. Next, these methods are extended to consider continuous-thrust trajectory optimization by combining STTs with differential dynamic programming, a second-order optimization method. STTs are also shown to be useful for accounting for the effects of state uncertainty propagated through nonlinear dynamics, with application to impulsive statistical maneuver design. Building on this, a method is developed to accurately and efficiently model probabilistic constraints on non-Gaussian state distributions, which are frequently encountered in spacecraft dynamics. Finally, a strategy to approximate the higher-order STTs without losing important information is introduced, which improves the efficiency of the underlying algorithms. The STT-based methods are applied to a variety of complex trajectory scenarios, with a particular emphasis on spacecraft operating in cislunar space. These algorithms are shown to be computationally efficient while accurately capturing the effects of nonlinear dynamics. Altogether, this research provides the mathematical and computational tools to use higher-order STTs to achieve a variety of different objectives in spacecraft guidance and trajectory optimization.</p

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

A Polyhedral Study of Mixed 0-1 Set

We consider a variant of the well-known single node fixed charge network flow set with constant capacities. This set arises from the relaxation of more general mixed integer sets such as lot-sizing problems with multiple suppliers. We provide a complete polyhedral characterization of the convex hull of the given set

A Dynamical System Approach for Resource-Constrained Mobile Robotics

The revolution of autonomous vehicles has led to the development of robots with abundant sensors, actuators with many degrees of freedom, high-performance computing capabilities, and high-speed communication devices. These robots use a large volume of information from sensors to solve diverse problems. However, this usually leads to a significant modeling burden as well as excessive cost and computational requirements. Furthermore, in some scenarios, sophisticated sensors may not work precisely, the real-time processing power of a robot may be inadequate, the communication among robots may be impeded by natural or adversarial conditions, or the actuation control in a robot may be insubstantial. In these cases, we have to rely on simple robots with limited sensing and actuation, minimal onboard processing, moderate communication, and insufficient memory capacity. This reality motivates us to model simple robots such as bouncing and underactuated robots making use of the dynamical system techniques. In this dissertation, we propose a four-pronged approach for solving tasks in resource-constrained scenarios: 1) Combinatorial filters for bouncing robot localization; 2) Bouncing robot navigation and coverage; 3) Stochastic multi-robot patrolling; and 4) Deployment and planning of underactuated aquatic robots. First, we present a global localization method for a bouncing robot equipped with only a clock and contact sensors. Space-efficient and finite automata-based combinatorial filters are synthesized to solve the localization task by determining the robot’s pose (position and orientation) in its environment. Second, we propose a solution for navigation and coverage tasks using single or multiple bouncing robots. The proposed solution finds a navigation plan for a single bouncing robot from the robot’s initial pose to its goal pose with limited sensing. Probabilistic paths from several policies of the robot are combined artfully so that the actual coverage distribution can become as close as possible to a target coverage distribution. A joint trajectory for multiple bouncing robots to visit all the locations of an environment is incrementally generated. Third, a scalable method is proposed to find stochastic strategies for multi-robot patrolling under an adversarial and communication-constrained environment. Then, we evaluate the vulnerability of our patrolling policies by finding the probability of capturing an adversary for a location in our proposed patrolling scenarios. Finally, a data-driven deployment and planning approach is presented for the underactuated aquatic robots called drifters that creates the generalized flow pattern of the water, develops a Markov-chain based motion model, and studies the long- term behavior of a marine environment from a flow point-of-view. In a broad summary, our dynamical system approach is a unique solution to typical robotic tasks and opens a new paradigm for the modeling of simple robotics system