Determination of sample sizes for setting β-content tolerance limits controlling both tails of the normal distribution

The problem of determining sample size for two-sided-β-content tolerance intervals which control both tails of the normal distribution is investigated. The tolerance limits are defined to assure that the tail proportions do not exceed specified values p₁ and p₂. We determine the minimum sample size so that, at a given confidence level, the tail proportions will not be too small, i.e., so that the tolerance interval will not be overly conservative. Tables of the sample size n and the corresponding factors are provided.     

Strong solutions of stationary equations in abstract kinetic theory

An abstract differential equation with partial range boundary conditions, modelling a variety of plane-symmetric stationary transport phenomena, is studied in Hilbert space. The collision operator is assumed to be a positive compact perturbation of the identity. A complete existence and uniqueness theory for the abstract equation is presented and two examples from rarefied gas dynamics are detailed.Work supported in part by D.O.E. grant No. DE-AS05 80ER10711 and N.S.F. grant No. DMS-8312451.The research was carried out while the author was visiting the Department of Physics and Astronomy of the Free University of Amsterdam, The Netherlands.     

Direct current conduction in ammonium perchlorate single crystals

The dc electrical conductivity of pure and doped ammonium perchlorate (AP) has been studied in two different crystal orientations, with the electric field applied perpendicularly to either (001) or (210) planes. The conductivity along the direction of the c axis was found to be lower than that normal to (210) by a factor of 5 to 10. The dc electrical conductivity of AP is decreased by Pb²⁺ ions but increased by SO^2?₄ and CrO^2?₄ ions. The conductivity of pure AP and of Pb²⁺-doped AP displays two regions with activation energies for conduction of 0.56 and 0.87 eV, respectively. The conductivity of the anion-doped crystals has a single activation energy, 0.66 eV for SO^2?₄ and 0.72 eV for CrO^2?₄. Exposure to ammonia enhances the conductivity of pure AP. A proton conduction mechanism is proposed that takes due regard of the structure of AP. The effect of the various additives on the conductivity are attributed to their influences on the formation of charge-carrying protons.     

A stationary criticality problem in generalL p -space for energy dependent neutron transport in cylindrical geometry

The energy-dependent neutron transport integral equation in a homogeneous cylinder of radiusR and infinite height with isotropic scattering is studied as an abstract equationf=K f in the spaceL ₁((0, 1)×(E _m ,E _M )). By means of techniques based on the theory of positive operators in Banach spaces, we prove that the eigenvalue problem for the integral operatorK admits as a solution a unique a. e. positive eigenfunction to which the leading eigenvalue _o corresponds.After establishing continuity and strictly increasing monotonicity of _o inR we discuss and solve the criticality problem under the assumption of subcriticality for a non-multiplying medium.The formulation of the eigenvalue problem forK is finally extended to anyL _p space, 1p< . Recalling thatK is a Riesz operator inL _p , we prove, as a general result, that the spectrum ofK, acting onL _p , is independent ofp.

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Résumé On étudie un faisceau de neutrons d'une énergieE (comprise entre deux bornesM _e ), dans un cylindre de rayon 0rR de hauteur infinie; on considère la diffraction comme isotrope. L'èquation intègrale du transport de neutrons est formuleeé abstraitement parf=K f, oùK est un opèrateur dans l'espaceL ₁ (r,E) des fonctions intégrables. La théorie des opérateurs positifs dans les espaces de Banach nous permet de démontrer que l'opérateur intégralK possède une fonction propre unique, positive presque partout, correspondant à la valeur propre dominante _o.Après avoir démontré que _o est continue et strictement croissante par rapport àR, on discute et résout le problème critique sous une hypothèse bien motivée physiquement.La formulation du problème aux valeurs propres est généralisée dans un espaceL _p ,p quelconque,K étant un opérateur de Riesz; on obtient comme résultat que le spectre deK dansL _p est indépendant dep.

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Work performed under the auspices of C. N. R. (Gruppo Nazionale per la Fisica-Matematica) and partially supported by M. P. I.

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The research leading to this article was completed while the third author was visiting the University of Florence in the summer of 1983.     

Total synthesis of 5,5′,6,6′-tetrahydroxy-3,3′-biindolyl, the proposed structure of a potent antioxidant found in beetroot (Beta vulgaris)

5,5′,6,6′-Tetrahydroxy-3,3′-biindolyl, the proposed structure of a phenolic antioxidant isolated from the red beetroot (Beta vulgaris), has been synthesised. The spectroscopic data of the synthetic material is not consistent with that reported for the natural product.     

Characterization of Scattering Data for the AKNS System

We characterize the scattering data of the AKNS system with vanishing boundary conditions. We prove a 1,1-correspondence between L ¹-potentials without spectral singularities and Marchenko integral kernels which are sums of an L ¹ function (having a reflection coefficient as its Fourier transform) and a finite exponential sum encoding bound states and norming constants. We give characterization results in the focusing and defocusing cases separately.     

From the AKNS system to the matrix Schrödinger equation with vanishing potentials: Direct and inverse problems

We relate the scattering theory of the focusing AKNS system with vanishing boundary conditions to that of the matrix Schrödinger equation. The corresponding Miura transformation, which allows this connection, converts the focusing matrix nonlinear Schrödinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method to derive the multisoliton solutions of the nonlocal integrable equation, thus proposing a new method to solve the matrix NLS equation.