A remark on the domain of an unbounded derivation of a C1-algebra

It is shown that if A is an hermitian element of a C¹-algebra and A is in the domain of a closed unbounded derivation δ of the C¹-algebra and if ? is a continuously differentiable complex valued function defined on a closed interval containing the spectrum of A then $\text{?(A)}$ is in the domain of δ and the norm of $\text{δ(?(A))}$ is bounded by the norm of δ(A) times the least upper bound of the absolute value of the derivative of ?. This result is generalized to the case of continuously differentiable functions of several variables.     

Function-space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs

Two existing function-space quasi-Newton algorithms, the Davidon algorithm and the projected gradient algorithm, are modified so that they may handle directly control-variable inequality constraints. A third quasi-Newton-type algorithm, developed by Broyden, is extended to optimal control problems. The Broyden algorithm is further modified so that it may handle directly control-variable inequality constraints. From a computational viewpoint, dyadic operator implementation of quasi-Newton methods is shown to be superior to the integral kernel representation. The quasi-Newton methods, along with the steepest descent method and two conjugate gradient algorithms, are simulated on three relatively simple (yet representative) bounded control problems, two of which possess singular subarcs. Overall, the Broyden algorithm was found to be superior. The most notable result of the simulations was the clear superiority of the Broyden and Davidon algorithms in producing a sharp singular control subarc.This research was supported by the National Science Foundation under Grant Nos. GK-30115 and ENG 74-21618 and by the National Aeronautics and Space Administration under Contract No. NAS 9-12872.     

Average boundary conditions in Cauchy problems

We consider the general Cauchy problem with initial data in a Hilbert space and with a formal dissipative linear generator. A complete parametrization is known of the (abstract) boundary conditions which make this problem well set. We exhibit a distinguished subset $\text{B}$_E of the set $\text{B}$ of boundary conditions and demonstrate explicitly that the evolution associated with each B in $\text{B}$ can be represented as a (time independent) average over the evolutions associated with B′ in $\text{B}$_E. Applications are discussed to Schrödinger equations in bounded regions or with singular potentials.     

Measurement of neutron energy spectra and neutron-gamma angular correlations for the muon capture process 16O(μ−, νμxn)14, 15N

Neutron-gamma coincidence spectra have been measured for muon capture in ¹⁶O. The γ-spectrum is dominated by the ground state transitions from the $\text{3}\text{2}{}^{\mathrm{?}}\text{(6.32}\text{MeV}\text{)}\text{and}\text{1}\text{2}{}^{\mathrm{+}}\text{+}\text{5}\text{2}{}^{\mathrm{+}}\text{(5.3}\text{MeV}\text{)}$ states of ¹⁵N that are populated after the emission of one neutron. The neutron energy spectra and the neutron energy dependence of the ny angular correlation coefficient A₂ for these final states are presented. The observed transitions in ¹⁴N and the associated neutron spectra give direct evidence for the emission of two correlated neutrons. The data are discussed in terms of direct neutron emission plus emission via giant resonance intermediate states of ¹⁶N. The large yield for the emission of one and two fast neutrons is explained by a capture mechanism involving short-range nucleon-nucleon correlations.     

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.