Compactifications of moduli spaces of (1,p)-polarized abelian surfaces with level structures of canonical type have been described in great detail by Hulek, Kahn and Weintraub. The aim of this paper is to determine some invariants of smooth models of these moduli spaces. In particular, a geometric description of their canonical divisors is given and their Chern numbers are computed.

J. Zintl

In Early Transcendentals (The American Mathematical Monthly, Vol. 104, No 7) Steven Weintraub presents a rigorous justifcation of the "early transcendental" calculus textbook approach to the exponential and logarithmic functions. However, he uses tools such as term-by-term differentiation of infinite series. We present a rigorous treatment of the early transcendental approach suitable for a first course in analysis, using mainly the supremum property of the real numbers.

Simon Cowell Philippe Poulin

We have obtained 1.64, 1.90 and 2.15 micron narrow-band images of five T Tauri stars in the TW Hya Association (TWA) using the Near-Infrared Camera and Multiobject Spectrometer aboard the Hubble Space Telescope. Most of the T Tauri stars in our study show evidence of absorption by H2O vapor in their atmospheres; in addition, the low-mass brown dwarf candidate, TWA 5B, is brighter at 1.9 microns than predicted by cool star models that include the effects of H2O vapor but neglect dust. We conclude that the effect of atmospheric dust on the opacity is important at 1.9 microns for TWA 5B, the coolest object in our sample. The available evidence suggests that the TWA is 5-15 MY old. Comparison of the colors of TWA 5B with theoretical magnitudes as a function of age and mass then confirms previous claims that TWA 5B is substellar with a mass in the range 0.02-0.03 solar masses. The accurate single-epoch astrometry of the relative positions and separation of TWA 5A and TWA 5B reported here should permit the direct measurement of the orbital motion of TWA 5B within only a few years.

David A. Weintraub Didier Saumon Joel H. Kastner Thierry Forveille

In this paper, an optimal control problem is considered where a target vehicle aims to reach a desired location in minimum time while avoiding a dynamic engagement zone. Using simple motion, four potential approaches are considered. First, the min-time strategy which ignores the engagement zone is posed and solved. Second, the min-time strategy which avoids the engagement zone entirely is considered. Third, the min-time strategy which allows for some time in the engagement zone; but, still strives to stay away from the center of the engagement zone is posed. Lastly, a fixed final-time strategy is considered, wherein the target tries to avoid the engagement zone; but, is required to arrive at the desired location at a specific time. Using a nonlinear program solver, the optimal strategies are numerically solved. From the results of the numeric solutions, the optimal strategies are discussed and comparisons are drawn.

Isaac E. Weintraub Alexander Von Moll Christian Carrizales Nicholas Hanlon Zachariah Fuchs

The maximum surveillance of a target which is holding course is considered, wherein an observer vehicle aims to maximize the time that a faster target remains within a fixed-range of the observer. This entails two coupled phases: an approach phase and observation phase. In the approach phase, the observer strives to make contact with the faster target, such that in the observation phase, the observer is able to maximize the time where the target remains within range. Using Pontryagin's Minimum Principle, the optimal control laws for the observer are found in closed-form. Example scenarios highlight various aspects of the engagement.

Isaac E. Weintraub Alexander Von Moll Eloy Garcia David W. Casbeer Meir Pachter

This work develops feasible path trajectories for a coordinated strike with multiple aircraft in a constrained environment. Using direct orthogonal collocation methods, the two-point boundary value optimal control problem is transcribed into a nonlinear programming problem. A coordinate transformation is performed on the state variables to leverage the benefits of a simplex discretization of the search domain. Applying these techniques allows each path constraint to be removed from the feasible search space, eliminating computationally expensive, nonlinear constraint equations and problem specific parameters from the optimal control formulation. Heuristic search techniques are used to determine a Dubins path solution through the space to seed the optimal control solver. In the scenario, three aircraft are initiated in separate directions and are required to avoid all constrained regions while simultaneously arriving at the target location, each with a different viewing angle. A focus of this work is to reduce computation times for optimal control solvers such that real-time solutions can be implemented onboard small unmanned aircraft systems. Analysis of the problem examines optimal flight paths through simplex corridors, velocity and heading vectors, control vectors of acceleration and heading rate, and objective times for minimum time flight.

Michael D. Zollars David J. Grymin Isaac E. Weintraub

This paper considers an M-pursuer N-evader scenario involving virtual targets. The virtual targets serve as an intermediary target for the pursuers, allowing the pursuers to delay their final assignment to the evaders. However, upon reaching the virtual target, the pursuers must decide which evader to capture. It is assumed that there are more pursuers than evaders and that the pursuers are faster than the evaders. The objective is two-part: first, assign each pursuer to a virtual target and evader such that the pursuer team's energy is minimized, and second, choose the virtual targets' locations for this minimization problem. The approach taken is to consider the Apollonius geometry between each pursuer's virtual target location and each evader. Using the constructed Apollonius circles, the pursuer's travel distance and maneuver at a virtual target are obtained. These metrics serve as a gauge for the total energy required to capture a particular evader and are used to solve the joint virtual target selection and pursuer-evader assignment problem. This paper provides a mathematical definition of this problem, the solution approach taken, and an example.

Isaac E. Weintraub Alexander Von Moll David W. Casbeer Satyanarayana G. Manyam

This paper explores a multi-agent containment problem, where a fast evader, modeled having constant speed and using constant heading, attempts to escape a circular containment region that is orbited by a slower pursuer with a nonzero capture radius. The pursuer is constrained to move along the edge of the containment region and seeks to capture the evader. This paper presents an in-depth analysis of this pursuer-evader containment scenario. First, multiple types of capture conditions for a single-pursuer case are analyzed defining the worst-case initial position for the pursuer. Second, a parametric study is performed to demonstrate the effects of speed ratio, capture radius, and initial location of the evader. Finally, a reachability analysis is performed to investigate the viable escape headings and reachable regions by the evader. This work provides a foundation for the analysis of escape under more general evader inputs as well as a multiple-pursuer version of the scenario.

Braulio Mora Alexander Von Moll Isaac Weintraub David Casbeer Animesh Chakravarthy

This paper establishes a more formal definition for an Engagement Zone (EZ) and derives some basic EZs associated with fundamental engagement models associated with pursuit-evasion and turret-evasion. The basic EZs presented in this paper capture the most salient aspects of the Pursuer-Agent and Turret-Agent engagements: namely the geometry of the aspect angle and the relative differences in capability (i.e., maximum speeds, range, etc.). One of the main advantages of utilizing EZs for path planning is that they encode an overall desire for Agent to go somewhere without requiring an aggressive maneuver or active evasion should the Pursuer or Turret begin its pursuit. It is shown that there is some advantage, in terms of time savings, in EZ-based navigation around a single range-limited Pursuer as compared with circumnavigating the capturability region.

Alexander Von Moll Isaac E. Weintraub

In classical works on a planar differential pursuit-evasion game with a faster pursuer, the intercept point resulting from the equilibrium strategies lies on the Apollonius circle. This property was exploited for the construction of the equilibrium strategies for two faster pursuers against one evader. Extensions for planar multiple-pursuer single-evader scenarios have been considered. We study a pursuit-evasion game on a sphere and the relation of the equilibrium intercept point to the Apollonius domain on the sphere. The domain is a generalization of the planar Apollonius circle set. We find a condition resulting in the intercept point belonging to the Apollonius domain, which is the characteristic of the planar game solution. Finally, we use this characteristic to discuss pursuit and evasion strategies in the context of two pursuers and a single slower evader on the sphere and illustrate it using numerical simulations.

Dejan Milutinovic Alexander Von Moll Satyanarayana G. Manyam David W. Casbeer Isaac E. Weintraub Meir Pachter