Search

The purpose of this work is to extend the applicability of moderate or intense low oxygen dilution (MILD) combustion to solid biomass fuels. A laboratory-scale furnace fitted with a parallel jet bu...

yields however a similar estimate for the second group of zeros in the interval wr/2 oo. We show in the first part of this paper how the inequalities (2), (3) can be derived in a very simple way using the classical ideas of Sturm and the well known differential equation satisfied by the function Pn(cos 0). The second part contains some elementary facts about the zeros of a class of trigonometric polynomials and gives also some inequalities for the zeros of Pn(cos 0) derived on the basis of these facts.

In what follows we discuss certain inequalities involving harmonic polynomials of two and of three variables, that is, polynomials u(x, y) and u(x, y, z) which satisfy Laplace's equation uXX+u,,=O and uxx+uy+uZZ=0, respectively. The inequalities in question furnish bounds for these polynomials and for their derivatives under proper conditions. Of particular interest are inequalities of the type of S. Bernstein's theorem, as discussed by the second author in a recent paper [7 ] ('). The first part of the present paper deals with the two-dimensional and the second part with the three-dimensional case. Of fundamental importance throughout the paper is an interpolation formula which is stated and proved in ?2 of Part I. In fact the results of Part I may be regarded as systematic applications of this formula. Several inequalities of this part are generalizations and refinements of earlier theorems. Most of the problems of the second part are new. In an Appendix we consider a generalization of the main problem treated in Part II, and another problem which deals with ellipses and is only in loose relationship with the other topics considered in the present paper.