A matrix lower bound is defined that generalizes ideas apparently due to S. Banach and J. von Neumann. The matrix lower bound has a natural interpretation in functional analysis, and it satisfies many of the properties that von Neumann stated for it in a restricted case. Applications for the matrix lower bound are demonstrated in several areas. In linear algebra, the matrix lower bound of a full rank matrix equals the distance to the set of rank-deficient matrices. In numerical analysis, the ratio of the matrix norm to the matrix lower bound is a condition number for all consistent systems of linear equations. In optimization theory, the matrix lower bound suggests an identity for a class of min-max problems. In real analysis, a recursive construction that depends on the matrix lower bound shows that the level sets of continuously differentiable functions lie asymptotically near those of their tangents.

Ultra-lean, hydrogen-air mixtures are found to support another kind of laminar flame that is steady and stable beside flat flames and flame balls. Direct numerical simulations are performed of flames that develop into steadily and stably propagating cells. These cells were the original meaning of the word "flamelet'' when they were observed in lean flammability studies conducted early in the development of combustion science. Several aspects of these two-dimensional flame cells are identified and are contrasted with the properties of one-dimensional flame balls and flat flames. Although lean hydrogen-air flames are subject to thermo-diffusive effects, in this case the result is to stabilize the flame rather than to render it unstable. The flame cells may be useful as basic components of engineering models for premixed combustion when the other types of idealized flames are inapplicable.

Very lean hydrogen-air mixtures experience strong diffusive-thermal types of cellular instabilities that tend to increase the laminar burning velocity above the value that applies to steady, planar laminar flames that are homogeneous in transverse directions. Flame balls constitute an extreme limit of evolution of cellular flames. To account qualitatively for the ultimate effect of diffusive-thermal instability, a model is proposed in which the flame is a steadily propagating, planar, hexagonal, close-packed array of flame balls, each burning as if it were an isolated, stationary, ideal flame ball in an infinite, quiescent atmosphere. An expression for the laminar burning velocity is derived from this model, which theoretically may provide an upper limit for the experimental burning velocity.

This report describes the implementation of a Runge-Kutta iterationboth for mixture-averaged and for multicomponent diffusion with Dufourand Soret effects in the low Mach number combustion code.

Comparatively little attention has been given to multicomponent diffusion effects in lean hydrogen-air flames, in spite of the importance of these flames in safety and their potential importance to future energy technologies. Prior direct numerical simulations either have considered only the mixture-averaged transport model, or have been limited to stabilized flames that do not exhibit the thermo-diffusive instability. The so-called full, multicomponent transport model with cross-diffusion is found to predict hotter, significantly faster flames with much faster extinction and division of cellular structures.

Min-max and max-min identities are found for inner products on the boundaries of compact, convex sets whose interiors contain the origin. The identities resemble the minimax theorem but they are different from it. Specifically, the value of each min-max (or max-min) equals the value of a dual problem of the same type. Their solution sets can be characterized geometrically in terms of the enclosed convex sets and their polar sets. However, the solution sets need not be convex nor even connected.

Numerical tests are used to validate a practical estimate for the optimal backward errors of linear least squares problems. This solves a thirty-year-old problem suggested by Stewart and Wilkinson.

This conference paper considers how to use reaction path diagrams to better understand the output of reacting flow simulations. Briefly, these diagrams have long been used to depict the reactants and products in networks of chemical reactions. The diagrams can be generated in several ways from computer simulations of chemically reacting fluids to depict how the fluid moderates the chemistry by determining which species are brought into contact to react in quantity. The concept of a conditional diagram is introduced which depicts the reactions occurring in only a portion of the fluid domain, thus enabling comparisons between different regions of the fluid and the overall reaction network. Several examples are provided of the paths occurring in methane diffusion flames.