Richard James Duffin

Series and parallel addition of matrices

Abstract Let A and B be Hermitian semi-definite matrices and let A + denote the Moore-Penrose generalized inverse. Then we define the parallel sum of A and B by the formula A ( A + B ) + B and denote it by A : B . If A and B are nonsingular this reduces to A : B = ( A −1 + B −1 ) −1 which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation. Here the ordering A ⩾ B means A − B is semidefinite and the following inequality holds: ( A + B ) : ( C + D ) ⩾ A : C + B : D . If R ( A ) denotes the range of A then it is found that R ( A : B ) = R ( A ) ∩ R ( B ). Moreover if A and B are orthogonal projection operators then 2 A : B is the orthogonal projection on R ( A ) ∩ R ( B ). The norms are found to satisfy the inequality ∥ A : B ∥ ⩽ ∥ A ∥ : ∥ B ∥. Generalization to non-Hermitian operators are also developed.

Geometric programming with signomials

The difference of twoposynomials (namely, polynomials with arbitrary real exponents, but positive coefficients and positive independent variables) is termed asignomial.Each signomial program (in which a signomial is to be either minimized or maximized subject to signomial constraints) is transformed into an equivalent posynomial program in which a posynomial is to be minimized subject only to inequality posynomial constraints. The resulting class of posynomial programs is substantially larger than the class of (prototype)geometric programs (namely, posynomial programs in which a posynomial is to be minimized subject only to upper-bound inequality posynomial constraints). However, much of the (prototype) geometric programming theory is generalized by studying theequilibrium solutions to thereversed geometric programs in this larger class. Actually, some of this theory is new even when specialized to the class of prototype geometric programs. On the other hand, all of it can indirectly, but easily, be applied to the much larger class of well-posedalgebraic programs (namely, programs involving real-valued functions that are generated solely by addition, subtraction, multiplication, division, and the extraction of roots).

Linearizing Geometric Programs

A geometric program concerns minimizing a function subject to constraint functions, all functions being of posynomial form. In this paper the posynomial functions are condensed to monomial form by the use of the inequality reducing a weighted arithmetic mean to a weighted geometric mean. The geometric mean is a monomial and by a logarithmic transformation it becomes a linear function. This observation shows that the condensed program is equivalent to a linear program. Moreover by suitable choice of the weights it is found that the minimum of the condensed program is the same as the minimum of the original programs. This fact together with the duality theorem of linear programming proves that the maximum of the dual geometric program is equal to the minimum of the primal geometric program. With this result as a basis a new approach to the duality properties of geometric programs is carried through. In particular it is shown that a “duality gap” cannot occur in geometric programming.

Potential theory on a rhombic lattice

Abstract Of concern are complex valued functions defined on the lattice points of the complex plane. The lattice can be the usual lattice of square blocks but of main interest is an irregular lattice with squares replaced by rhombs. A function is defined to be discrete analytic if the difference quotient across one diagonal of a rhomb equals the difference quotient across the other diagonal. Based on this definition discrete analogs of the following concepts in classical function theory are developed: Laplace equation, Cauchy-Riemann equations, differentiation, contour integration, Moreras theorem, and harmonic polynomials. The theory is more than an analogy because, for a common class of boundary value problems, it proves possible to obtain upper and lower bounds for the classical Dirichlet integral in terms of discrete harmonic functions.

A passive wall design to minimize building temperature swings

It has been found in many countries with arid climates that massively walled buildings provide steady, comfortable inside temperatures even though the outside temperature fluctuations may be sizeable. The adobe houses of the American South-West, and the rondavels of southern Africa are particular examples. This phenomena is often termed the thermal flywheel effect. One explanation is that the temperature at the inside of a massive wall lags approximately out of phase with the outside, and so it partly offsets the direct, in phase, infiltration losses into the building. Thus the room temperature is kept approximately constant. In this paper the question of designing a non-homogeneous wall to optimize this effect is considered.

Algorithms for Classical Stability Problems

A differential equation is stable if the roots of the characteristic polynomial are in the interior of the left half plane. Likewise a difference equation is stable if the roots of the characteristic polynomial are in the interior of the unit circle. This paper concerns algorithms which test polynomials for these properties. Also of concern is the relationship between the two problems. In particular, special numerical integration formulas are developed which transform a differential equation into a difference equation. These formulas are such that the differential equation and the corresponding difference equation are both stable or else they are both unstable.

Duality in Semi-Infinite Linear Programming

For the case of a consistent semi-infinite linear program, we provide several hypotheses, which are both necessary as well as sufficient, that there be no duality gap between the program and its formal dual (with attainment of value in the dual), for every linear objective function. Earlier work provided sufficient conditions for no duality gap for all linear objective functions, or a necessary and sufficient condition for no duality gap for a fixed linear criterion.

Equipartition of energy in wave motion

Abstract Of concern are solutions of the classical wave equation in three-dimensions. It is shown that if a solution has compact support then after a finite time the kinetic energy of the wave is constant and equals the potential energy. The proof employs the Paley-Wiener theorem of Fourier analysis.

Use of layered walls to reduce building temperature swings

Abstract One of the commonest passive solar building architectures is the use of massive walls to reduce the temperature fluctuations inside a building. This is often called the thermal flywheel effect, and can be seen in the adobe houses in the American southwest. In this paper we suggest several wall designs which improve the thermal flywheel effect.

Yukawan potential theory

Abstract This paper concerns the Yukawa equation Δu = μ2u where μ is a real constant. Given a solution u(x, y) of this equation then there is a conjugate function v(x, y) satisfying the same equation and related to u(x, y) by a generalization of the Cauchy-Riemann equations. This gives rise to interesting analogies with logarithmic potential theory and with complex function theory. In particular there are generalizations of holomorphic functions, Taylor series, Cauchys formula, and Rouches theorem. The resulting formulae contain Bessel functions instead of the logarithmic functions which appear in the classical theory. However, as μ → 0 the formulae revert to the classical case. A convolution product for generalized holomorphic functions is shown to produce another generalized holomorphic function.