This paper investigates algebraic and combinatorial properties of the set of linear orders on the algebra of subsets of a finite set that are representable by positive measures. It is motivated by topics in decision theory and the theory of measurement, where an understanding of such properties can facilitate the design of strategies to elicit comparisons between subsets that, for example, determine an individual's preference order over subsets of objects or an individual's qualitative probability order over subsets of states of the world. We introduce a notion of critical pairs of binary comparisons for such orders and prove that (i) each order is uniquely characterized by its set of critical pairs and (ii) the smallest set of binary comparisons that determines an order is a subset of its set of critical pairs. The paper then focuses on the minimum number of on-line binary-comparison queries between subsets that suffice to determine any representable order for a set of given cardinality n. It is observed that, for small n, the minimum is attained by first determining the ordering of singleton subsets. We also consider query procedures with fixed numbers of stages, in each stage of which a number of queries for the next stage are formulated.1. Introduction. This paper investigates algebraic and combinatorial properties of additive linear orders on the Boolean algebra n of subsets of an n-item set n = 1 2 n . We refer to a binary relation ≺ on n as an additive linear order if there exists a measure on n such that i = i > 0 for all i ∈ n , and with A = i∈A i for all A ∈ n ,
This paper presents the closed-loop experimental framework and dynamic model validation for a 1/12-scale underwater kite design. The pool-based tow testing framework described herein, which involves a fully actuated, closed-loop controlled kite and flexible tether, significantly expands upon the capabilities of any previously developed open-source framework for experimental underwater kite characterization. Specifically, the framework has allowed for the validation of three closed-loop flight control strategies, along with a critical comparison between dynamic model predictions and experimental results. In this paper, we provide a detailed presentation of the experimental tow system and kite setup, describe the control algorithms implemented and tested, and quantify the level of agreement between our multi-degree-of-freedom kite dynamic model and experimental data. We also present a sensitivity analysis that helps to identify the most influential parameters to kite performance and further explain remaining mismatches between the model and data.
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