Asymptotic expansions for renewal measures in the plane

Summary Let P be a distribution in the plane and define the renewal measure R=P ^*nwhere * denotes convolution. The main results of this paper are three term asymptotic expansions for R far from the origin. As an application, expansions are obtained for distributions in linear boundary crossing problems.Research supported by NSF grants MCS-8102080 and DMS-8504708     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

Application of Mie theory to determine the structure of spheroidal scatterers in biological materials

We present here the results of a numerical study on light scattering from nonspherical particles with relevance to detecting precancerous states in epithelial tissues. In previous studies of epithelial cell nuclei, the experimental light scattering data have been analyzed by comparison with Mie theory. However, given the spheroidal shape of many cell nuclei, the validity of this assumption demands a thorough investigation. We investigate this assumption by using the T-matrix method to model light scattered from spheroids with parameters relevant to epithelial cell nuclei. In our previous studies, we have developed a data analysis procedure that extracts the oscillatory component of the angular-scattering distribution for an ensemble of epithelial cell nuclei for comparison with Mie theory. We demonstrate that application of our analysis procedure to the predictions of the T-matrix method for spheroids, oriented such that their axis of symmetry is aligned with the incident light propagation direction, generally yields the spheroid dimension that is transverse to the incident light propagation direction with subwavelength accuracy.     

Smooth extensions of functions on separable Banach spaces

Let X be a Banach space with a separable dual X*. Let ${Y\subset X}Let X be a Banach space with a separable dual X*. Let Y ì X{Y\subset X} be a closed subspace, and f:Y?\mathbbR{f:Y\rightarrow\mathbb{R}} a C ¹-smooth function. Then we show there is a C ¹ extension of f to X.     

Nitride protonation and NH3 binding versus N–H bond cleavage in uranium nitrides

The conversion of metal nitrides to NH₃ is an essential step in dinitrogen fixation, but there is limited knowledge of the reactivity of nitrides with protons (H⁺). Herein, we report comparative studies for the reactions of H⁺ and NH₃ with uranium nitrides, containing different types of ancillary ligands. We show that the differences in ancillary ligands, leads to dramatically different reactivity. The nitride group, in nitride-bridged cationic and anionic diuranium(iv) complexes supported by –N(SiMe₃)₂ ligands, is resistant toward protonation by weak acids, while stronger acids result in ligand loss by protonolysis. Moreover, the basic –N(SiMe₃)₂ ligands promote the N–H heterolytic bond cleavage of NH₃, yielding a “naked” diuranium complex containing three bridging ligands, a nitride (N³⁻) and two NH₂ ligands. Conversely, in the nitride-bridged diuranium(iv) complex supported by –OSi(O^tBu)₃ ligands, the nitride group is easily protonated to afford NH₃, which binds the U(iv) ion strongly, resulting in a mononuclear U–NH₃ complex, where NH₃ can be displaced by addition of strong acids. Furthermore, the U–OSi(O^tBu)₃ bonds were found to be stable, even in the presence of stronger acids, such as NH₄BPh₄, therefore indicating that –OSi(O^tBu)₃ supporting ligands are well suited to be used when acidic conditions are required, such as in the H⁺/e⁻ mediated catalytic conversion of N₂ to NH₃.

Ancillary ligands alter the reactivity of U-nitrides with H⁺, relevant to N₂ conversion to NH₃. The amides lead to complete ligand loss and NH₃ activation, while for siloxides, the nitride is protonated to NH₃ leaving the ancillary ligands intact.