Saleem Watson

Prime and maximal ideals in subrings of C(X)

The structure of ideals in the ring C(X) of continuous functions on a completely regular space X and its subring C∗(X) consisting of the bounded functions is well known. In this paper we study the prime and maximal ideals in subrings A(X) of C(X) that contain C∗(X). We show that many of the results known separately for C(X) and C∗(X), often by different methods, are true for any such A(X). Our results put the problems of C(X) and C∗(X) in a common setting by exhibiting these as special instances of the subrings A(X). We characterize prime and maximal ideals in any A(X) in terms of their residue class rings and in terms of certain z-filters on X that correspond to these ideals. We also characterize the intersection of the free ideals and the free maximal ideals in any A(X).

The Flip-Side of a Lagrange Multiplier Problem

It is apparent that these problems are related, but what, exactly, is the relationship between them? Do other optimization problems have a flip-side? If so, how does one formulate the flip-side of a given problem? We give an answer to these questions by considering the more general problem of optimizing a function f of two variables subject to a constraint g (x, y) = c using Lagrange multipliers. As the fencing-a-field problem suggests, the flip-side of a problem involves interchanging the roles of f and g (a process that is meaningful because the Lagrange multiplier condition Vf = kVg is symmetric in f and g). In this note we define what is meant by the flip-side of a problem and prove a result that relates an extremum of a problem to an extremum of its flip-side. In following the steps of the proof, students can see how properties of the gradient-in particular the property that the gradient points in the direction of the greatest rate of increase in the values of a function-can be useful visual tools in analyzing optimization problems. Several articles on Lagrange multipliers have appeared in the CMJ (see for instance [1], [2], [3], [5]), but it seems that the general relationship between a problem and its flip-side (as we call it here) has not been discussed.