A simple and quick method is proposed for one-dimensional multi-group calculations of multiregion fast reactors. This method is based on the condition of neutron-balance in each of the regions or subregions. Neutron flux distributions are represented by the simplest functions.
It has been proved by a series of test computations that the accuracy is comparable to the ordinary diffusion theory calculations, while the computing time is saved by a factor of ten.

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A sampled data time optimal control of nuclear reactors has been developed using discrete versions of the Maximum Principle. The reactor model used is a second order nonlinear system derived by one group delayed neutron and prompt jump approximations. The control variable of the system is the rate of change of reactivity, and for reasons of safety in reactor operation, this, as well as the reactivity, is restricted. In addition, the reactor power overshoot and undershoot are also restricted to devise a practical control.
The discrete version of the Maximum Principle has been extended to deal with restricted state space problem, and with the use of this extension, a time optimal control for nuclear reactors with pulse width modulated control input has been derived.
The optimal control law is modified in the region around the terminal state, to obtain smooth control, and the stability of this modified control is verified by the second method of Liapunov. The results are directly applicable to digital computer control systems for nuclear reactors.

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A mathematical analysis of the time-dependent neutron transport equation is presented for the case of a moderator slab. The scattering of neutrons is treated on the basis of the Van Hove theory and is assumed to be isotropic.
First, the spectrum of the corresponding transport operator is investigated and it is shown that the complex λ-plane is decomposed by the operator into a band spectrum Re(λ) ?? -(νΣt)min, a discrete spectrum on the real axis λ>-(νΣt)min, and a resolvent set Re(λ)>-(νΣt)min deleted by the discrete spectrum.
The discrete spectrum is carefully studied and is found to consist of isolated eigenvalues or otherwise empty. For all categories of moderator, there exists an upper limit to the thickness of a slab having empty discrete spectrum. A slab exceeding this limit has a finite number of eigenvalues if the moderator material is gas or solid. This is also true for a certain class of liquid moderators. For other liquids, there is another critical thickness for the slab thickness such that if it is exceeded, the set of eigenvalues turns out to he denumerable with an accumulation point at -(νΣt)min. Each eigenvalue possesses a finite multiplicity, and the index is one.
Finally the related initial value problem is considered, and it proves to have a unique solution upon application of the Hille-Yosida theorem. The contribution of the band spectrum to the solution is evaluated and is shown to decay faster than exp{-(νΣt)mint}.

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In this paper the effect on the neutron flux, in a homogeneous mixture of moderator and fuel, of scattering from low energy resonances in the fuel is considered. Particular attention is given to resonances for which slowing down theory is inappropriate. The kernel for a monatomic gaseous resonance scatterer is derived and compared with the 'slowing down' kernel commonly used. The effects of the two kernels on the flux near the resonance for the particular case of ²⁴⁰Pu and a light moderator are also compared.

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The transmission of γ-rays through slab was calculated by the method of invariant imbedding. The results are in good agreement with other reliable solutions obtained by the moments method and Monte Carlo calculations. The present method has the following advantage; (1) It permits determination of the energy and angular distributions of reflected or transmitted photons due to monoenergetic and monodirectional source photon; (2) the method can be applied to multi-layer slabs; (3) extension to thicker slabs can be easily made; (4) the execution time required for numerical solutions is short enough to permit systematic calculations for many elements.

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