Robert A. Phillips

Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.

Abstract A system for fractionating populations of living cells by velocity sedimentation in the earth's gravitational field is described. The cells start in a thin band near the top of a shallow gradient of 3% to 30% fetal calf serum in phosphate buffered saline at 4°C. Cell separation takes place primarily on the basis of size and is approximately independent of cell shape. A sharply‐defined upper limit, called the streaming limit, exists for the cell concentration in the starting band beyond which useful cell separations cannot be achieved. This limit, which varies with the type of cell being sedimented, can be significantly increased by proper choice of gradient shape. For sheep erythrocytes (sedimentation velocity of 1.6 mm/hour) it is 1.5 × 10 7 cells/ml. Measured and calculated sedimentation velocities for sheep erythrocytes are shown to be in agreement. The technique is applied to a suspension of mouse spleen cells and it is shown, using an electronic cell counter and pulse height analyzer, that cells are fractionated according to size across the gradient such that the sedimentation velocity (in mm/hour) approximately equals r 2 /4 where r is the cell radius in microns. Since cells of differing function also often differ in size, the system appears to have useful biological applications.