Shaping codes can be obtained from optimal unequal cost coding algorithms due to Lempel, Even, and Cohn (1973) in the variable-length-to-block case and to Varn (1971) in the block-to-variable-length case. The former is related to ad hoc approaches to shaping previously described while the latter is novel in the shaping context.

The procedure of Knuth and Yao (1976) to simulate random numbers with specified distribution by parsing sequences of fair coin tosses is generalized to employ discrete distributions of particular form instead of fair coins. Each probability in these distributions is an integral power of some fixed value t. The parse tree for the simulation procedure is closely related to the code trees arising in Karp's optimal variable-length coding algorithm for code symbols of unequal cost.

The problem of designing a sequence of optimal binary tests for the identification of a single faulty component is addressed. For components in linear order this is equivalent to the classical alphabetic coding problem solved by Hu and Tucker (1971). For partially ordered components the problem is solved by reduction to a minimization over a set of alphabetic problems.>

Forests constructed by the binary Huffman (1952) and Hu-Tucker (1971) algorithms solve parallelized search problems. Bounds on the resulting minimum average search lengths for items occurring with given probabilities are established.>

Conditions on the probability distributions of two sources that describe their relative uniformity are shown to be sufficient for inequalities on their minimum average codeword lengths and entropies. The Huffman coding problem and variants involving Renyi entropy are addressed.>

Source distributions that can be encoded with zero redundancy for the case of unequal code symbol costs are examined. These distributions provide a natural generalization of the binary, equal costs case for which these distributions are the dyadic distributions. These zero redundancy codes have the property that the expected proportion of codeword symbols given by a particular letter is equal to an experimental function of the code letter cost. The converse is not true in general; however, partial converse results hold. Maximum-entropy zero-redundancy distributions are easily identified through their connection with unequal cost coding for uniform sources.>

A new lower bound, which is the tightest possible, is obtained for the redundancy of optimal bimuy prefix-condition (OBPC) codes for a memoryless source for which the probability of the most likely source letter is known. It is shown that this bound, and upper bounds obtained by Gallager and Johnsen, hold for infinite as well as finite source alphabets. Also presented are bounds on the redundancy of OBPC codes for sources satisfying the condition that each of the first several probabilities in the list of source probabilities is sufficiently large relative to the sum of the remaining probabilities.

The problem of achieving synchronization for variable-length source codes is addressed through the use of self-synchronizing binary prefix-condition codes. Although our codes are suboptimal in the sense of minimum average codeword length, they have the advantages of being generated by an explicit constructive algorithm, having minimal additional redundancy compared with optimal codes-as little as one additional bit introduced into the least likely codeword for a large class of sources-and having statistical synchronizing performance that improves on that of the optimal code in many cases.

An expression for the probability density on the unit interval of the time to first ramp crossing is derived for Slepian's process, a Gaussian process with a triangular correlation function.

Wong's method for the solution of the zero-crossing problem for a particular stationary Gaussian process is applied to the case of some nonstationary Gaussian processes.

Mehr and McFadden's conditionally Markov processes and Jamison's reciprocal processes are two independent extensions of Slepian's work on a particular stationary Gaussian process. The relationship between them and their relationship with Gauss-Markov processes are discussed.