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Let $ f \in {L^1}(0,\infty ),\delta > 0$ $ ({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds$ $ P = \{ 0 = {t_0} < {t_1} < \cdots < {t_i} < {t_{i + 1}} < \cdots \} $ $ [0,\infty )$ $ {t_i} \to \infty $f by the step function $ {A_P}f$

$\displaystyle {A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text{for}}\;{t_{i - 1}} \leqslant t < {t_i},$ where $ {\delta _i} = {t_i} - {t_{i - 1}}$ $ {A_P}f \to f$ $ {L^1}(0,\infty )$ . An application to difference approximations of evolution problems is sketched.

Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initial-value problems almost never have global-time classical solutions, and one must deal with suitable generalized solutions. The correct class of generalized solutions has only recently been established by the authors. This article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates. Analogous results are proved by similar means for the method of vanishing viscosity.

Recently M. G. Crandall and P. L. Lions introduced the notion of "viscosity solutions" of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a " viscosity solution" admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper.

Complementary macromolecules were isolated from yeasts of opposite mating type. These cell-surface molecules neutralize each other as do antibodies and antigens. Both yeast factors are glycoproteins of low molecular weight. Other specific cell associations may be due to the interaction of such complementary macromolecules.

A Producer for established and maintaining an esophageal fistula is presented. The preparation has been used successfully to collect saliva and boluses from steers consuming pelleted or ground feeds. The type of cannula used has an integral role in preventing leakage and maintaining a healthy animal.

A fast method is described for determining the mating type of yeast spore clones independent of genetic complementation between auxotrophic markers.

THE expression of the same genes can be studied in both haploid and diploid cells of certain yeasts. If the genes control primarily haploid mating type traits, then it is interesting to know whether these genes are expressed in the diploid which does not mate. In the agglutinative yeast, Hansenula wingei, the two haploid mating types (strains 5 and 21) produce cell surface glycoproteins which are complementary and which are responsible for the strong adhesion between cells during sexual agglutination1–3. The diploid hybrid, however, is completely non-agglutinative. This lack of agglutination in the diploid could be the result of a mutual repression of the synthesis of both complementary agglutination factors, or synthesis followed by mutual neutralization or inactivation of these glycoproteins. We have evidence that the genes controlling the haploid agglutination types are mutually repressed in the diploid.