Objective function approximations in mathematical programming

Mathematical programming applications often require an objective function to be approximated by one of simpler form so that an available computational approach can be used. An a priori bound is derived on the amount of error (suitably defined) which such an approximation can induce. This leads to a natural criterion for selecting the best approximation from any given class. We show that this criterion is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. This gains access to the rich legacy on Chebyshev approximation techniques, to which we add some new methods for cases of particular interest in mathematical programming. Some results relating to post-computational bounds are also obtained.This paper was partially supported by the National Science Foundation and by the Office of Naval Research, and was the basis for a plenary lecture delivered at the IX International Symposium on Mathematical Programming in Budapest, Hungary, August 1976.     

A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let U_i = (X_i, Y_i), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μ_x, μ_y) and covariance matrix

. Let X_i, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H₀ : μ = 0 vs. H₁ : μ ≠ 0. We prove that Hotelling's T² test, which uses (X_i, Y_i), i = 1, 2,…, n (and discards X_i, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T²-test cannot be beaten. A similar result is also proved for the problem of testing μ_x ? μ_y = 0. A Bayes test and other competitors which are similar tests are discussed.     

Structure and physical properties of ternary rare-earth cobalt bismuth intermetallics (RE)12Co5Bi (RE = Y, Gd, Tb, Dy, Ho, Er, Tm)

The rare-earth intermetallic compounds (RE)12Co5Bi (RE = Y, Gd, Tb, Dy, Ho, Er, Tm) were prepared by arc-melting and annealing at 600 degrees C. These compounds extend the previously known (RE)6M2+xX1-x (M = Co, Ni; X = Ga, In, Sn, Pb) series with the Ho6Co2Ga-type structure to X = Bi. The crystal structure of Ho12Co5Bi was refined by the Rietveld method from powder X-ray diffraction data obtained using synchrotron radiation (Pearson symbol oI36, orthorhombic, space group Immm, Z = 2, a = 9.37598(14) A, b = 9.37871(14) A, c = 9.85465(13) A). Unlike other Ho6Co2Ga-type compounds, the 2a site in Ho12Co5Bi is exclusively occupied by Co atoms. Four-probe electrical resistivity measurements on sintered polycrystalline samples of (RE)12Co5Bi indicated metallic behavior. Magnetic measurements revealed behavior ranging from frequency-dependent maxima in the ac susceptibility for Y12Co5Bi to possible ferrimagnetic ordering for Gd12Co5Bi to antiferromagnetic ordering with metamagnetic transitions for the remaining compounds. As confirmed by band structure calculations using Y12Co5Bi as a model compound, Y-Y and Y-Co interactions are the most important bonding components, but matrix effects are likely responsible for anomalously short Co-Co contacts in the structure.     

Solution NMR techniques for large molecular and supramolecular structures

Transverse relaxation-optimized spectroscopy (TROSY) or generation of heteronuclear multiple quantum coherences during the frequency labeling period and TROSY during the acquisition period have been combined either with cross-correlated relaxation-induced polarization transfer (CRIPT) or cross-correlated relaxation-enhanced polarization transfer (CRINEPT) to obtain two-dimensional (2D) solution NMR correlation spectra of (15)N,(2)H-labeled homo-oligomeric macromolecules with molecular weights from 110 to 800 kDa. With the experimental conditions used, the line widths of the TROSY-components of the (1)H- and (15)N-signals were of the order of 60 Hz at 400 kDa, whereas, for structures of size 800 kDa, the line widths were about 75 Hz for (15)N and 110 Hz for (1)H. This paper describes the experimental schemes used and details of their setup for individual measurements. The performance of NMR experiments with large structures depends critically on the choice of the polarization transfer times, the relaxation delays between subsequent recordings, and the water-handling routines. Optimal transfer times for 2D [(15)N,(1)H]-CRIPT-TROSY experiments in H(2)O solutions were found to be 6 ms for a molecular weight of approximately 200 kDa, 2.8 ms for 400 kDa, and 1.4 ms for 800 kDa. These data validate theoretical predictions of inverse proportionality between optimal transfer time and size of the structure. The proton longitudinal relaxation times in H(2)O solution were found to be of the order of 0.8 s for structure sizes around 200 kDa, 0.4 s at 400 kDa, and 0.3 s at 800 kDa, which enabled the use of recycle times below 1 s. Since improper water handling results in severe signal loss, the water resonance was kept along the z-axis during the entire duration of the experiments by adjusting each water flip-back pulse individually.     

Studies of layered uranium (VI) compounds. VI. Ionic conductivities and thermal stabilities of MUO2PO4 · nH2O,where M = H,Li,Na,K,NH4 or 12Ca, and where n is between 0 and 4

We have measured the ionic conductivities of pressed pellets of the layered compounds MUO₂PO₄ · nH₂O, and correlated the results with TGA data. The conductivities (in ohm^?1 m^?1), at temperatures increasing with decreasing water content over the range 20 to 200°C, were approximately as follows: Li⁺4H₂O, 10^?4; Li⁺, Na⁺, K⁺, and NH₄⁺3H₂O, 10^?4, 10^?2, 10^?4, and 10^?4; H⁺, Li⁺, and Na⁺1.5H₂O, 10^?2, 10^?4, and 10^?4; Na⁺1H₂O, 10^?5; H⁺, K⁺, and NH₄⁺0.5H₂O, all 10^?5; and H⁺, Li⁺, Na⁺, K⁺, NH⁺₄, and $\text{1}\text{2}\text{Ca}{}^{\mathrm{2+}}\text{}\text{OH}{}_2\text{O}$, 10^?5, 10^?5, 10^?4, 10^?5, 10^?5, and 10^?6. A ring mechanism is proposed to account for the high conductivity found in NaUO₂PO₄ · 3.1H₂O. The accurate TGA data showed that most of the hydrates had water vacancies of the Schottky type, and should be represented as MUO₂PO₄(A ? x)H₂O, where x can be between 0 and 0.3.     

Determination of standard pH values for sodium hydrogen diglycolate buffer (0.05m) from 5–65°C

Values of pa _H ^o for 0.05 mole-kg^?1 aqueous solutions of sodium hydrogen diglycolate in the temperature range 5–65°C have been obtained from cells without transport, and can be fitted to the equation $$\begin{gathered} pa^\circ _H = 3.5098 + 2.222 \times 10^{ - 3} ({T \mathord{\left/ {\vphantom {T {K - 298.15}}} \right. \kern-\nulldelimiterspace} {K - 298.15}}) \hfill \\ + 2.628 \times 10^{ - 5} ({T \mathord{\left/ {\vphantom {T {K - 298.15}}} \right. \kern-\nulldelimiterspace} {K - 298.15}})^2 \hfill \\ \end{gathered} $$ The analysis has been carried out by a multilinear regression procedure using a form of the Clarke and Glew equation. This buffer standard may be a useful alternative to the saturated potassium hydrogen tartrate buffer.     

Medium effects on fluorescence quantum yields and lifetimes for coumarin laser dyes

Fluorescence quantum yields and lifetimes of coumarin dyes are sharply reduced in polar solvents if amine substituent groups are free to rotate. The polar solvent effect is interpreted in terms of relaxation of excited dye from an initial planar conformation to a twisted zwitterionic state.     

La détermination rapide du calcium et du magnésium dans l'eau de mer

A rapid method is described for the determination of calcium and magnesium in sea. water. The classical gravimetric methods are very lengthy and are susceptible to errors if used for sea-water. The two alkaline earths are determined by complexometric titration using complexen III (disodium ethylenediaminetetra-acetate) as reagent and two indicators (murexide and ériochrome black T) for the determination of calcium and for the sum of Ca⁺² and Mg⁺² ions. The quantity of magnesium is obtained by difference. While the classical gravimetric methods take at least 24 hours, these two determinations can be effected in 1 hour, giving satisfactory accuracy.     

Unfolding the singularities in superspace

A method is described for unfolding the singularities in superspace, \(\mathcal{G} = \mathfrak{M}/\mathfrak{D}\) , the space of Riemannian geometries of a manifoldM. This unfolded superspace is described by the projection $$\mathcal{G}_{F\left( M \right)} = \frac{{\mathfrak{M} \times F\left( M \right)}}{\mathfrak{D}} \to \frac{\mathfrak{M}}{\mathfrak{D}} = \mathcal{G}$$ whereF(M) is the frame bundle ofM. The unfolded space \(\mathcal{G}_{F\left( M \right)}\) is infinite-dimensional manifold without singularities. Moreover, as expected, the unfolding of \(\mathcal{G}_{F\left( M \right)}\) at each geometry [g _o] ∈ \(\mathcal{G}\) is parameterized by the isometry groupIg _o (M) of g₀. Our construction is natural, is generally covariant with respect to all coordinate transformations, and gives the necessary information at each geometry to make \(\mathcal{G}\) a manifold. This construction is a canonical and geometric model of a nonrelativistic construction that unfolds superspace by restricting to those coordinate transformations that fix a frame at a point. These particular unfoldings are tied together by an infinite-dimensional fiber bundleE overM, associated with the frame bundleF(M), with standard fiber \(\mathcal{G}_{F\left( M \right)}\) , and with fiber at a point inM being the particular noncanonical unfolding of \(\mathcal{G}\) based at that point. ThusE is the totality of all the particular unfoldings, and so is a grand unfolding of \(\mathcal{G}\) .     

Metal-templated diyne cyclodimerization and cyclotrimerization

Reaction of [Pt(CH3)2(COD)] (COD = 1,5-cyclooctadiene) with Ph2PCCCCPPh2 led to a mixture of [{Pt(CH3)2}2(mu-Ph2PC4PPh2)2] (1) and [{Pt(CH3)2}3(mu-Ph2PC4PPh2)3] (2). Reaction of [PtCl2(COD)] with Ph2PCCCCPPh2 led to a mixture of the thermally unstable compounds [{PtCl2}2(mu-Ph2PC4PPh2)2] (3) and [{PtCl2}3(mu-Ph2PC4PPh2)3] (4) which transform into [{PtMe2}2{mu-C8(PPh2)4}] (5) and [{PtMe2}3{mu3-C12(PPh2)6}] (6) containing 8-membered diene-diyne and 12-membered triene-triyne rings, respectively. Compound 2 can be converted to [{PtMe2}3{C12(PPh2)6}] (7) by heating with CuCl at 80 degrees C, while 1 can be heated without significant cycloaddition.