On the order of singular optimal control problems

In singular optimal control problems, the functional form of the optimal control function is usually determined by solving the algebraic equation which results by successively differentiating the switching function until the control appears explicitly. This process defines the order of the singular problem. Order-related results are developed for singular linear-quadratic problems and for a bilinear example which gives new insights into the relationship between singular problem order and singular are order.Dedicated to R. BellmanThis work was supported by the National Science Foundation under Grant No. ENG-77-16660.     

Self-adjoint algebras of unbounded operators

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.Alfred P. Sloan Foundation Fellow.     

Inclusive reactions: A new parametrization

We have fitted the π^? spectra from the reactions π^±p → π^?+ anything using an expression which exhibits the differing forward-backward slopes of the x-distributions, the turnover near (but not exactly at) x = 0 and the correlations between P₁ and p²_T. The expression is a generalized Bose-Einstein distribution which is covariant with respect to Lorentz transformation along the incident beam direction.     

The general relativistic hydrogen atom

The general relativistic Dirac equation is formulated in an arbitrary curved space-time using differential forms. These equations are applied to spherically symmetric systems with arbitrary charge and mass. For the case of a black hole (with event horizon) it is shown that the Dirac Hamiltonian is self-adjoint, has essential spectrum the whole real line and no bound states. Although rigorous results are obtained only for a spherically symmetric system, it is argued that, in the presence of any event horizon there will be no bound states. The case of a naked singularity is investigated with the results that the Dirac Hamiltonian is not self-adjoint. The self-adjoint extensions preserving angular momentum are studied and their spectrum is found to consist of an essential spectrum corresponding to that of a free electron plus eigenvalues in the gap (–mc ², +mc ²). It is shown that, for certain boundary conditions, neutrino bound states exist.Supported in part by the National Science Foundation     

Fibre sizing using Fraunhofer diffraction

The light scattering pattern from absorbing cylinders is shown to be adequately described by Fraunhofer diffraction for cylinders with diameter greater than 8 μm for light of wavelength 632.8 nm. This permits the use of a simple inversion procedure based on a Fourier transform of the scattering pattern to obtain the size distribution of fibre widths. For fibres of diameter less than 8 μm the same inversion procedure may be used when an electron micrograph of the fibres becomes the scattering object. The maximum resolution of this inversion procedure is shown to be proportional to the largest angle scanned and its performance with experimental data is illustrated.     

Separation of water-soluble vitamins by reversed-phase high performance liquid chromatography with ultra-violet detection: application to polyvitaminated premixes

Nine water-soluble vitamins: [thiamine (B1), ascorbic acid (C), nicotinamide (PP), pyridoxine (B6), calcium pantothenate (B5), folic acid (B9), cyanocobalamin (B12), riboflavin (B2) and biotin (B8)] were separated on a YMC-Pack Pro C18 column (250 mm x 4.6 mm, 5 microm particle size) in a single run with a gradient elution of mobile phase consisting of 0.025% trifluoroacetic acid pH 2.6 (solvent A) and acetonitrile (solvent B). The separation was achieved within 17 min with a flow rate of 0.8 ml min(-1) and the detection was performed at two wavelengths (210 and 275 nm). The calibration graphs plotted with six concentrations of each vitamin were linear with a regression coefficient R2 > 0.995. The method was applied for the quantification of vitamins B1, C, PP, B6, B5, B9 B2 and B8 in polyvitaminated premixes (premixes) used for the fortification of infant nutrition products. The sample preparation involves an aqueous extraction of vitamins and two different samples dilution were used prior the LC-analysis. The specificity of the method was demonstrated by the retention characteristics, UV spectra and by comparing the peak purity with the standard of each vitamin. The repeatability of the method was evaluated at different level of concentrations on 12 premixes and the coefficients of variation (CVr) were below 6.5%. The values of the intermediate precision (CV1) were below 9.6% (n = 6). The concentrations of vitamins found in premixes with our method were comparable to the declared values, since no bias was found between the two sets of results at 95% confidence. The simplicity of the procedure should make it highly desirable for quality control of premixes in the food industry.     

Muonic Au: A test of the weak-coupling model

We report the measurement of the energies of various muonic atomic transitions above 500 keV in ¹⁹⁷Au. We have used the weak-coupling model in order to analyze the hyperfine structure of ¹⁹⁷Au. Using transitions to and from the 1s, 2s 2p and 3p levels as well as the fine structure of the 2p states and hyperfine splittings of the 2p, 3p and 3d states we have fitted our data to two deformed Fermi distributions: one with a constant skin thickness, the other with a skin thickness which varies. Both yield equally good fits. We have determined the quadrupole moment of ¹⁹⁷Au to be 0.547±0.016 b. The model dependence of our results is discussed. The weak coupling parameters of de-Shalit Ω_p, Ω₂₂, Ω₂₀ and A² have been determined and are in fair agreement with Coulomb excitation results. There is also good agreement between our results and those of electron scattering.     

Some Asymptotic ith Ramsey numbers

Let G be a graph with chromatic number χ(G) and let t(G) be the minimum number of vertices in any color class among all χ(G)-vertex colorings of G. Let H′ be a connected graph and let H be a graph obtained by subdividing (adding extra vertices to) a fixed edge of H′. It is proved that if the order of H is sufficiently large, the ith Ramsey number r_i(G, H) equals $\text{[}\text{((χ(G)?1)(|}\text{H}\text{|?1)+t(G)?1)}\text{i}\text{]+1}$.     

Andreev states near short-ranged pairing potential impurities

We study Andreev states near atomic scale modulations in the pairing potential in both s- and d-wave superconductors with short coherence lengths. For a moderate reduction of the local gap, the states exist only close to the gap edge. If one allows for local sign changes of the order parameter, however, resonances can occur at energies close to the Fermi level. The local density of states (LDOS) around such pairing potential defects strongly resembles the patterns observed by tunneling measurements around Zn impurities in Bi2Sr2CaCu2O8+x (BSCCO). We discuss how this phase impurity model of the Zn LDOS pattern can be distinguished from other proposals experimentally.