Some duality relationships for the generalized Neyman-Pearson problem

In 1963, Kuhn presented a dual problem to a relatively well-known location problem, variously referred to as the generalized Fermat problem and the Steiner-Weber problem. The purpose of this paper is to point out how Kuhn's results can be adapted to provide a dual to the generalized Neyman-Pearson problem, a problem of fundamental interest in statistics, which has applications in control theory and a number of other areas. The Neyman-Pearson problem, termed the dual problem, is a constrained maximization problem and may be considered to be a calculus-of-variations analog to the bounded-variable problem of linear programming. When the dual problem has equality constraints, the primal problem is an unconstrained minimization problem. Duality results are also obtained for the case where the dual problem has inequality constraints.This work was partially supported by the National Science Foundation, Grant Nos. NSF-GK-1571 and NSF-GK-3038. The authors would like to acknowledge the very useful comments of one of the referees, which led to more direct and general proofs of Properties 2.3 and 2.6.     

Polyphosphate ions encapsulated in oxothiomolybdate rings: synthesis, structure, and behavior in solution

Cyclic oxothiomolybdates containing polyphosphate ions were prepared by simple reactions in aqueous medium of the corresponding polyphosphate ions and the cyclic precursor K(2)I(2)Mo(10)S(10)O(10)(OH)(10)(OH(2))(5).15H(2)O. K(5)[Cl(P(2)O(7)]Mo(12)S(12)O(12)(OH)(12)(H(2)O)(4)].22H(2)O (1) was isolated from concentrated chloride solution (2.5 mol.L(-1)). 1 reveals a remarkable complex containing two different substrates encapsulated in a dodecanuclear ring, a H-bonded Cl(-) ion, and a covalently bonded [P(2)O(7)] group. The chloride ion in 1 can be selectively removed for a monohydrogenophosphate group yielding K(6)[(HPO(4))(P(2)O(7))Mo(12)S(12)O(12)(OH)(12)(H(2)O)(2)].19H(2)O (2), a mixed species containing a [P(2)O(7)] and a [HPO(4)] group. The substitution is accompanied by a significant change of the ring, which adopts a "pear-shape" conformation. In the presence of triphosphate ion, the "heart-shaped" decanuclear ring Rb(3)[(H(2)P(3)O(10))Mo(10)S(10)O(10)(OH)(10)].17.5H(2)O (3) is formed containing a linear [P(3)O(10)] group intimately embedded in the inorganic cyclic host. The three compounds were structurally characterized by single-crystal X-ray diffraction. The behaviors of 1, 2, and 3 in solution were studied by (31)P NMR. Variable temperature experiments, supported by a two-dimensional COSY (31)P experiment, revealed that the supramolecular interaction existing between the chloride ion and the ring in solid 1 is maintained in solution. Nevertheless, 1 remains labile, and successive equilibria were evidenced and interpreted as an ion-pair association involving a halide ion (Cl, Br, or I), responsible for the conformational change of the [P(2)O(7)] group within the cavity. The influence of the nature of the halide guest (Cl(-), Br(-), and I(-)) on the successive equilibria was studied, and the thermodynamic constant related to the postulated equilibrium was determined. The stability of the supramolecular association decreases in the order Cl > Br > I. In solution, a phosphate exchange is observed for 2 while for 3 the absence of temperature dependence of the (31)P NMR spectrum confirms the conformation of the host-guest system is blocked. Elemental analysis and infrared characterizations are also supplied.     

Ag6(PMo10V2O40](CH3COO).8H2O: A 3D macrocationic polyoxometallic Keggin complex

Silver atoms and strongly interlinking Keggin units ensure the tridimensionnal macrocationic structure of [Ag(6)(PMo(10)V(2)O(40))](CH(3)COO).8H(2)O, which catalyzes O(2)-based oxidation of p-methoxytoluene in acetic acid by air.     

Electrocatalytic activity of ordered intermetallic phases for fuel cell applications

The electrocatalytic activities of a wide range of ordered intermetallic phases toward a variety of potential fuels have been studied, and results have been compared to those of a pure polycrystalline platinum (Pt(pc)) electrode. A significant number of the ordered intermetallic phases exhibited enhanced electrocatalytic activity when compared to that of Pt, in terms of both oxidation onset potential and current density. The PtBi, PtIn, and PtPb ordered intermetallic phases appeared to be the most promising electrocatalysts tested thus far for fuel cell applications. PtPb, in particular, showed an onset potential that was 100 mV less positive and a peak current density approximately 40 times higher than those observed for Pt in the case of methanol oxidation. The ability to control the geometric and electronic structures of the electrocatalytic material by using ordered intermetallic phases has been shown to be a promising direction of inquiry in the search for superior electrocatalysts for fuel cell applications.     

Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.     

Ion acceleration by collisionless shocks in high-intensity-laser-underdense-plasma interaction

Ion acceleration by the interaction of an ultraintense short-pulse laser with an underdense-plasma has been studied at intensities up to 3 x 10(20) W/cm(2). Helium ions having a maximum energy of 13.2+/-1.0 MeV were measured at an angle of 100 degrees from the laser propagation direction. The maximum ion energy scaled with plasma density as n(0.70+/-0.05)(e). Two-dimensional particle-in-cell simulations suggest that multiple collisionless shocks are formed at high density. The interaction of shocks is responsible for the observed plateau structure in the ion spectrum and leads to an enhanced ion acceleration beyond that possible by the ponderomotive potential of the laser alone.     

Laser-polarized (129)Xe NMR and MRI at ultralow magnetic fields

Laser-polarized (129)Xe and a high-T(c)superconducting quantum interference device (SQUID) are used to obtain magnetic resonance images in porous materials at a magnetic field of 2.3 mT, corresponding to a Larmor frequency of 27 kHz. Image resolution of 1 mm is obtained with gradients of only 1 mT/m. The resolution of xenon chemical shifts in different physicochemical environments at ultralow fields is also demonstrated. Details of the circulating flow optical pumping apparatus and the SQUID spectrometer are presented.     

Ion heating and thermonuclear neutron production from high-intensity subpicosecond laser pulses interacting with underdense plasmas

Thermonuclear fusion neutrons produced by D(d,n)3He reactions have been measured from the interaction of a high-intensity laser with underdense deuterium plasmas. For an input laser energy of 62 J, more than (1.0+/-0.2)x10(6) neutrons with a mean kinetic energy of (2.5+/-0.2) MeV were detected. These neutrons were observed to have an isotropic angular emission profile. By comparing these measurements with those using a secondary solid CD2 target it was determined that neutrons are produced from direct ion heating during this interaction.