In this paper we use generalized Fourier-Hermite functionals to obtain a complete orthonormal set inis a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in L 2 (C a,b [0, T ]) has an integral transform F γ,β F also belonging to L 2 (C a,b [0, T ]).

In this paper, we establish several very basic formulas relating convolution products, integral transforms and inverse integral transforms for functionals in L 2 (C 0 [0, T ]). The expansion of functionals in L 2 (C 0 [0, T ]) in terms of Fourier-Hermite functionals plays a key role.

We define two sequential transforms on a function spaceCa,b[0,T]induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals onCa,b[0,T]. We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms onCa,b[0,T].

In this paper we establish a Fubini theorem for functionals on a function space. We then establish some relationships as applications of our Fubini theorem. Finally, we present some historical remarks.

We investigated the effects of an acidic environment on the G2/M-phase arrest, apoptosis, clonogenic death, and changes in cyclin B1-CDC2 kinase activity caused by a 4-Gy irradiation in RKO.C human colorectal cancer cells in vitro. The time to reach peak G2/M-phase arrest after irradiation was delayed in pH 6.6 medium compared to that in pH 7.5 medium. Furthermore, the radiation-induced G2/M-phase arrest decayed more slowly in pH 6.6 medium than in pH 7.5 medium. Finally, there was less radiation-induced apoptosis and clonogenic cell death in pH 6.6 medium than in pH 7.5 medium. It appeared that the prolongation of G2-phase arrest after irradiation in the acidic environment allowed for greater repair of radiation-induced DNA damage, thereby decreasing the radiation-induced cell death. The prolongation of G2-phase arrest after irradiation in the acidic pH environment appeared to be related at least in part to a prolongation of the phosphorylation of CDC2, which inhibited cyclin B1-CDC2 kinase activity.