A dynamic systems model of dyadic interaction

We develop a differential equation model of dyadic interaction that embodies the basic assumption that members of intimate couples form an interactive system in which the behavior of each member of a couple is influenced by the other's behavior and by goals that each person has for herself or himself. The dynamic solutions of this system suggest that when each person in the dyad is “cooperative”, then an equilibrium can be approached. The equilibrium represents a compromise position between the individuals’ own ideals and those of the partner. On the other hand, if one individual, or both, is uncooperative, then this system often, but not always, becomes unstable. One paradoxical deduction from the model is that, through mutual cooperation, couples can experience periods of stability, but such stable situations are not necessarily satisfying.     

Reconfiguration of connected graph partitions

Motivated by recent computational models for redistricting and detection of gerrymandering, we study the following problem on graph partitions. Given a graph $G$ and an integer $k\ge 1$, a $k$-district map of $G$ is a partition of $V\left(G\right)$ into $k$ nonempty subsets, called districts, each of which induces a connected subgraph of $G$. A switch is an operation that modifies a $k$-district map by reassigning a subset of vertices from one district to an adjacent district; a 1-switch is a switch that moves a single vertex. We study the connectivity of the configuration space of all $k$-district maps of a graph $G$ under 1-switch operations. We give a combinatorial characterization for the connectedness of this space that can be tested efficiently. We prove that it is PSPACE-complete to decide whether there exists a sequence of 1-switches that takes a given $k$-district map into another; and NP-hard to find the shortest such sequence (even if a sequence of polynomial lengths is known to exist). We also present efficient algorithms for computing a sequence of 1-switches that take a given $k$-district map into another when the space is connected, and show that these algorithms perform a worst-case optimal number of switches up to constant factors.     

Approaching Dianionic Tetraoxadiborecine Macrocycles: 10-Membered Bora-Crown Ethers Incorporating Borafluorenate Units

The addition of non-benzenoid quinones, acenapthenequinone or aceanthrenequinone, to the 9-carbene-9-borafluorene monoanion ( 1 ) affords the first examples of dianionic 10-membered bora-crown ethers ( 2 – 5 ), which are characterized by multi-nuclear NMR spectroscopy (¹H, ¹³C, ¹¹B), X-ray crystallography, elemental analysis, and UV/Vis spectroscopy. These tetraoxadiborecines have distinct absorption profiles based on the positioning of the alkali metal cations. When compound 4 , which has a vacant C₄B₂O₄ cavity, is reacted with sodium tetrakis[3,5-bis(trifluoromethyl)phenyl]borate, a color change from purple to orange serves as a visual indicator of metal binding to the central ring, whereby the Na⁺ ion coordinates to four oxygen atoms. A detailed theoretical analysis of the calculated reaction energetics is provided to gain insight into the reaction mechanism for the formation of 2 – 5 . These data, and the electronic structures of proposed intermediates, indicate that the reaction proceeds via a boron enolate intermediate.