Spanning trees with many leaves

We show that if G is a simple connected graph with and , then G has a spanning tree with > t leaves, and this is best possible. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 189–197, 2001     

Numerical indefinite integration of functions with singularities

We derive an indefinite quadrature formula, based on a theorem of Ganelius, for functions, for 1$">, over the interval . The main factor in the error of our indefinite quadrature formula is , with nodes and . The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of in the constant of the exponential. We conjecture that our formula has the best possible value for that constant. The results of numerical examples show that our indefinite quadrature formula is better than Haber's indefinite quadrature formula for -functions.

     

The evolution of a finite rate periodic oscillation

Finite rate oscillations of a gas in a closed tube arise when the amplitude of the applied periodic piston velocity is small while its acceleration is unrestricted. The asymptotic form of the periodic motion for large acceleration is given. The evolution to the final periodic motion from the initial state of rest is constructed for a finite rate oscillation. Exact results for a piecewise linear piston velocity are used to illustrate the solutions.     

Carbon–Carbon Cross‐Coupling Reactions Catalyzed by a Two‐Coordinate Nickel(II)–Bis(amido) Complex via Observable NiI,NiII, and NiIII Intermediates

Recently, the development of more sustainable catalytic systems based on abundant first‐row metals, especially nickel, for cross‐coupling reactions has attracted significant interest. One of the key intermediates invoked in these reactions is a Ni^III–alkyl species, but no such species that is part of a competent catalytic cycle has yet been isolated. Herein, we report a carbon–carbon cross‐coupling system based on a two‐coordinate Ni^II–bis(amido) complex in which a Ni^III–alkyl species can be isolated and fully characterized. This study details compelling experimental evidence of the role played by this Ni^III–alkyl species as well as those of other key Ni^I and Ni^II intermediates. The catalytic cycle described herein is also one of the first examples of a two‐coordinate complex that competently catalyzes an organic transformation, potentially leading to a new class of catalysts based on the unique ability of first‐row transition metals to accommodate two‐coordinate complexes.     

Preconditioning cubic spline collocation discretizations of elliptic equations

Summary. This work considers the uniformly elliptic operator defined by in (the unit square) with boundary conditions: on and on and its discretization based on Hermite cubic spline spaces and collocation at the Gauss points. Using an interpolatory basis with support on the Gauss points one obtains the matrix . We discuss the condition numbers and the distribution of -singular values of the preconditioned matrices where is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator given by in with boundary conditions: on on . The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by Gauss points or the space of continuous functions which are linear on the triangles of the triangulation of using the Gauss points. When we obtain results on the eigenvalues of . In the general case we obtain bounds and clustering results on the -singular values of . These results are related to the results of Manteuffel and Parter [MP], Parter and Wong [PW], and Wong [W] for finite element discretizations as well as the results of Parter and Rothman [PR] for discretizations based on Legendre Spectral Collocation. Received January 1, 1994 / Revised version received February 7, 1995     

Formation of [C18H36]+ ˙ and [C7H7O2]+ in the mass spectrum of n-octadecyl benzoate

A series of deuterium-labeled n-octadecyl benzoates serves to extend the literature using lower homologues for study of the mass-spectral reactions characteristic of alkyl esters. The benzoic acid radical cation (a) arises largely via hydrogen migration from C(2), as expected for the conventional mechanism passing through a six-membered quasicyclic intermediate; the presence of contributions from other processes is simply noted in passing. The octadecene radical cation (b) and protonated benzoic acid (c) arise by paths seemingly closely related to each other but differing from the path dominant for a. The data are rationalized in terms of a reaction sequence passing through two ion-neutral complex intermediates. The first, (d), consists of octadecyl carbenium ion and benzoyloxy radical; the second, (e), consists of octadecene radical cation and benzoic acid. The ionized partner in d appears to undergo none to several hydride migrations, but most prominently one; that in e may dissociate promptly to yield b or may persist, undergoing multiple hydride migrations to effect essentially complete scrambling of hydrogens, before dissociating to yield c. The argument for this mechanistic sequence to form b and c rests basically on its ability to account plausibly for the experimental data, and literature precedents cited for specific aspects of it furnish additional support. The low critical energies and small kinetic energy releases observed in this work are in accord with the characteristics listed by McAdoo for decompositions mediated by ion-neutral complexes, and the selection of an octadecyl, as opposed to a smaller alkyl, ester predisposes the molecule to favor such low-energy processes.