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- Jorge Rubinstein, Paul Penfield, Mark Horowitz
- IEEE Transactions on Computer-Aided Design of…
- 1981

In MOS integrated circuits, signals may propagate between stages with fanout. The MOS interconnect may be modeled by an RC tree. Exact calculation of signal delay through such networks is difficult. However, upper and lower bounds for delay that are computationally simple are presented here. The results can be used (1) to bound the delay, given the signal… (More)

- Paul Penfield
- APL '81
- 1981

Complex numbers are useful in science and engineering and, through analogy to the complex plane, in two-dimensional graphics, such as those for integrated-circuit layouts. The extension of APL to complex numbers requires many decisions. Almost all have been discussed in detail in a recent series of papers. One topic requiring further discussion is the… (More)

- Paul Penfield
- APLQ
- 1977

Many APL primitive functions can be extended to the domain of complex numbers. In some cases the generalizations are not straightforward, and two or more design choices are considered. This paper is intended to stimulate a discussion in the APL community of the possible design choices before the first implementation of complex APL renders such a discussion… (More)

- Paul Penfield
- APLQ
- 1977

Five proposals are presented for denoting the real part, imaginary part, magnitude, and phase of complex numbers in <i>APL</i>. The purpose of this paper is to stimulate a discussion in the <i>APL</i> community before the appearance of the first implementation of complex <i>APL</i> renders such a discussion academic.

- Paul Penfield
- APLQ
- 1978

This is the third and last paper in a series discussing the major design choices for APL to handle complex numbers. These papers are intended to stimulate a discussion in the APL community before the first implementation of complex numbers in APL renders such a discussion academic.

- Paul Penfield
- APL '75
- 1975

An APL interpreter has been written in APL. It includes a set of interacting APL functions that accept quote-quad input and act on it in the same way as APL itself would. The interpreter will be used to test some proposed APL language extensions. Extensions already implemented on it include complex number arithmetic, and reduction and scan operators acting… (More)

- Paul Penfield
- 1970

Absfract-Among the theorems of circuit theory, Tellegen’s theorem is unusual in that it depends solely upon Kirchhoff’s laws and the topology of the network. The theorem therefore applies to all electrical networks that obey Kirchhoff’s laws, whether they be linear or nonlinear, time-invariant or time-variant, reciprocal or nonreciprocal, hysteretic or… (More)

- Paul Penfield
- APLQ
- 1978

Results of discussions within the APL community about how APL should be extended to complex numbers are presented.

- Berger 's Burgers, Paul Penfield
- 2003

In Chapter 9, we discussed the technique of estimating input probabilities of a process that is consistent with known constraints expressed in terms of averages, or expected values, of one or more quantities, but is otherwise as unbiased as possible. This technique was described there for the simple case of one constraint and three input events, in which… (More)

- Paul Penfield
- APL '79
- 1979

Many people have suggested that APL be extended to complex numbers. Since complex numbers have great importance in mathematics, APL as a mathematical notation is deficient without them. Furthermore, an implemented extension to the complex domain would be of value to scientists, engineers, and educators, who use complex numbers in their work. A proposal for… (More)