Heterogeneous surface-to-air missile defense battery location: a game theoretic approach

In the context of an air defense missile-and-interceptor engagement, a challenge for the defender is that surface-to-air missile batteries often must be located to protect high-value targets dispersed over a vast area, subject to which an attacker may observe the disposition of batteries and subsequently develop and implement an attack plan. To model this scenario, we formulate a two-player, extensive form, three-stage, perfect information, zero-sum game that accounts for, respectively, a defender’s location of batteries, an attacker’s launch of missiles against targets, and a defender’s assignment of interceptor missiles from batteries to incoming attacker missiles. The resulting trilevel math programming formulation cannot be solved via direct optimization, and it is not suitable to solve via full enumeration for realistically-sized instances. We instead adapt the game tree search technique Double Oracle, within which we embed either of two alternative heuristics to solve an important subproblem for the attacker. We test and compare these solution methods to solve a designed set of 52 instances having parametric variations, from which we derive insights regarding the nature of the underlying problem. Enhancing the solution methods with alternative initialization strategies, our superlative methodology attains the optimal solution for over 75% of the instances tested and solutions within 3% of optimal, on average, for the remaining 25% of the instances, and it is promising for realistically-sized instances, scaling well with regard to computational effort.     

Studies on the synthesis of apoptolidin: synthesis of a C1-C27 fragment of apoptolidin D

Synthesis of a C(1)-C(27) fragment, a key intermediate in the synthesis of apoptolidin D, is reported. The synthesis involves a combination of Heck coupling and Horner-Wadsworth-Emmons reaction for the C(1)-C(7) trienoate portion and an efficient Suzuki cross-coupling protocol for the C(10)-C(13) diene portion.     

Dications of benzylidenefluorene and diphenylmethylidene fluorene: the relationship between magnetic and energetic measures of antiaromaticity

Oxidation of m- and p-substituted benzylidene fluorenes to antiaromatic dications was attempted by electrochemical and chemical means. Electrochemical oxidation to dications was successful for benzylidene fluorenes with p-methoxy, p-methyl, p-fluoro, and unsubstituted phenyl rings in the 3-position; attempts to oxidize the m-substituted derivatives via electrochemistry were unsuccessful. Chemical oxidation with SbF(5)/SO(2)ClF gave the dication of 9-[(4-methoxyphenyl)methylene]-9H-fluorene cleanly; oxidation of all other substituted benzylidene fluorenes resulted in mixtures of products. The excellent linear relationship between the chemical shifts calculated by the GIAO method and the experimental shifts for the p-methoxy-substituted benzylidene fluorene dication suggests that the calculations satisfactorily reflect the magnetic properties of this dication and potentially those of the other dications studied. The redox potentials from electrochemical oxidation, a measure of the stability of the dications, showed a good linear relationship with another measure of stability, the calculated difference in energy between each dication and its neutral precursor. The dications of benzylidene fluorenes were less stable than the dications of diphenylmethylidene fluorenes; within each type of compound, dications with p-substituted phenyl rings were more stable than dications with m-substituted phenyl rings and dications with phenyl rings substituted with electron-donating groups were more stable than dications with phenyl rings substituted with electron-withdrawing groups. The antiaromaticity of the fluorenyl system was assessed through the nucleus-independent chemical shift (NICS) that was also calculated by the GIAO method. The plot of the NICS values per square area versus the calculated energy difference for the dications showed a moderate degree of linearity; the plot of NICS values per square area versus the oxidation potentials was less linear. Thus, a suggestive, but not conclusive, relationship between magnetic and energetic measures of antiaromaticity was observed.     

On cyclic compositions of positive integers

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let ${\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}$ denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We show that the sequence ${\left\langle\begin{array}{c}n \\ k\end{array}\right\rangle}$ is log-concave and prove some results concerning ${\left\langle \begin{array}{c}n \\ k \end{array} \right\rangle}$ modulo two.     

Alternating sign matrices and descending plane partitions

An alternating sign matrix is a square matrix such that (i) all entries are 1, ?1, or 0, (ii) every row and column has sum 1, and (iii) in every row and column the nonzero entries alternate in sign. Striking numerical evidence of a connection between these matrices and the descending plane partitions introduced by Andrews (Invent. Math.53 (1979), 193–225) have been discovered, but attempts to prove the existence of such a connection have been unsuccessful. This evidence, however, did suggest a method of proving the Andrews conjecture on descending plane partitions, which in turn suggested a method of proving the Macdonald conjecture on cyclically symmetric plane partitions (Invent. Math.66 (1982), 73–87). In this paper is a discussion of alternating sign matrices and descending plane partitions, and several conjectures and theorems about them are presented.     

Formation of a pyrazopyrazolone by the pyrolysis of antipyrine diazonium fluoroborate

The pyrolysis of antipyrine 4-dizaonium fluoroborate gave an antipyrylazopyrazopyrazolone instead of the desired 4-fluoroantipyrine. This compound was formed by intermolecular and intramolecular coupling of the diazo compound at elevated temperatures. J. Heterocyclic Chem., 14, 1107 (1977)     

EuFCl—A new mixed-halide rare-earth compound

This paper reports the synthesis and bulk magnetic properties of a new rare-earth compound, EuFCl, isostructural with tetragonal PbFCl, and solid solutions of it with several LnOCl compounds, where Ln = Eu, Sm, Nd, Gd.