Bases,spanning sets,and the axiom of choice

Two theorems are proved: First that the statement “there exists a field F such that for every vector space over F, every generating set contains a basis” implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ?₂ has a basis implies that every well‐ordered collection of two‐element sets has a choice function. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Survivable network design under optimal and heuristic interdiction scenarios

We examine the problem of building or fortifying a network to defend against enemy attacks in various scenarios. In particular, we examine the case in which an enemy can destroy any portion of any arc that a designer constructs on the network, subject to some interdiction budget. This problem takes the form of a three-level, two-player game, in which the designer acts first to construct a network and transmit an initial set of flows through the network. The enemy acts next to destroy a set of constructed arcs in the designer’s network, and the designer acts last to transmit a final set of flows in the network. Most studies of this nature assume that the enemy will act optimally; however, in real-world scenarios one cannot necessarily assume rationality on the part of the enemy. Hence, we prescribe optimal network design algorithms for three different profiles of enemy action: an enemy destroying arcs based on capacities, based on initial flows, or acting optimally to minimize our maximum profits obtained from transmitting flows.     

Symplectic Yang–Mills theory,Ricci tensor,and connections

A Yang–Mills theory in a purely symplectic framework is developed. The corresponding Euler–Lagrange equations are derived and first integrals are given. We relate the results to the work of Bourgeois and Cahen on preferred symplectic connections.     

Sensitivity of hypersaline Arabian Gulf to seawater desalination plants

Due to the hot, arid nature of its bordering lands, seawater in the Arabian Gulf can have significant evaporation rates leading to hypersaline conditions. If additional desalination plants were to operate along its coast, then the extraction of desalinated water and returned brine waste stream into the Gulf would increase the salinity. This paper uses a tidally and cross-sectionally averaged mathematical model that reveals multiplicative dependence of the salinity on factors associated with river flow, evaporation rates and each of the desalination plants. Present-day desalinated water production rates are in the linear regime, but hypersalinity has exponential sensitivity to the position and volumetric rate of desalinated water extraction.     

Zeta regularized determinants for conic manifolds

We study (relative) zeta regularized determinants of Laplace type operators on compact conic manifolds. We establish gluing formulae for relative zeta regularized determinants. For arbitrary self-adjoint extensions of the Laplace-Beltrami operator, we express the relative ζ-determinants for these as a ratio of the determinants of certain finite matrices. For the self-adjoint extensions corresponding to Dirichlet and Neumann conditions, the formula is particularly simple and elegant.     

Territories,corridors, and networks: A biological model for the premodern state

When depicted on maps as homogenous territorial wholes, ancient states are visually summarized as static entities in a way that conceals the highly fluid dynamics of polity formation, maintenance, and growth. Models derived from studies of animal behavior show that “territory” does not consist of an undifferentiated use of the landscape. Instead, the concept of territory can be parsed into a series of resource‐rich nodes linked by corridors of access, surrounded by unutilized regions and boundaries marked at points of competition. Ancient human groups also can be analyzed as having perceived and occupied landscapes through strategies of flexible networks in which nodes and corridors were surrounded by unutilized spaces around which boundaries were selectively identified and defended. This strategy is identifiable in human social groups at different levels of complexity ranging from hunter‐gatherers through ancient chiefdoms and states. © 2007 Wiley Periodicals, Inc. Complexity 12: 28–35, 2007     

A six-step synthesis of (S)-5-ethenyl-3-(1-methyl-2-pyrrolidinyl)pyridine (SIB-1508Y) from (S)-nicotine

The anti-Parkinson’s agent SIB-1508Y was prepared in six steps from (S)-nicotine in 20% overall yield. The strategy involves a regioselective formylation at C-5 of a 1,4-dihydronicotine intermediate.     

Pure adaptive search in monte carlo optimization

Pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space. At any stage, the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence. We show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem. This compares to exponential growth in iterations required for pure random search.