Some of the elements of the criminal justice system are expressed in mathematical terms. The function of the criminal justice system is postulated to be the minimization of all the losses to society resulting from crime. Losses considered include the direct and indirect loss due to a crime, the loss to the convicted criminal due to the punishment he receives, and the net cost of imposing the punishment. The expression for overall loss also includes the probability of a member of a population committing a particular crime as a function of the severity of the punishment for that crime and the probability of arrest and conviction for the crime. In minimizing the loss to society, the loss to the criminal from his punishment should be included in some circumstances, but not in others. Suggestions are made on how required parameters for the criminal justice model might be obtained, in some cases experimentally.

The theory of time-limited functions with minimum out-of-band energy is extended to apply to functions whose amplitude is held constant and hence, are limited to phase modulation. In the absence of the amplitude constraint, the desired functions are prolate spheroidal functions of time, of order zero, with time-bandwidth product as a parameter. With the amplitude constraint, the desired functions can be expressed in terms of sums of even-order prolate spheroidal functions. An algorithm suitable for computations of the amplitude constrained functions is derived.