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Learning Objectives Students learn about the complex number system, metric spaces and the topology of C, elementary properties of analytic functions, conformal mappings, complex integration (including variations of Cauchy's Theorem and the Cauchy Integral Theorem) and singularities of analytic functions Assessment of Learning Outcomes Assessment will be(More)
A natural L ∞ functional calculus for an absolutely continuous contraction is investigated. It is harmonic in the sense that for such a contraction and any bounded measurable function φ on the circle, the image can rightly be considered asˆφ(T), wherê φ is the solution of the Dirichlet problem for the disk with boundary values φ. The main result shows that(More)
This paper is a contribution to the problem of approximating continuous functions F defined on a compact HausdorfF space X, where the value F(x) is a compact convex set in R" for every x in X. More specifically we show how to transfer Korovkin type approximation theorems for real-valued continuous functions to this set-valued situation. 1. Introduction. We(More)
President David Eisenbud presided over the EC and ECBT portions of the meeting (items beginning with 0, 1, or 2). Board Chair John Conway presided over the BT portion of the meeting (items beginning with 3). Items occur in numerical order, which is not necessarily the order in which they were discussed at the meeting. President Eisenbud convened the meeting(More)
A version of a theorem commonly referred to as Caristi's Theorem is given. It has an elementary constructive proof and it includes many generalizations of Banach's fixed point theorem. Several examples illustrate the diversity that can occur. X is a complete metric space and 4> is lower semicontinuous. If for each x in X, (C) d(x,Tx) <(p(x)-4>(Tx), then T(More)
Of concern are the minimal and maximal operators on L2(R") associated with the differential expression Te = J2{id/dxJ + qJ{x))2 + W{x) j=i where (q.# ") = gradQ for some real function W on R" and W satisfies ¿M-2 < W(Jf) < C|x|~2. In particular, for Q = 0, Xq reduces to the singular Schrödinger operator-A+ W{x). Among other results, it is shown that the(More)
Suppose A is a C*-algebra and B is a Banach algebra such that it can be continuously imbedded in B(H), the Banach algebra of bounded linear operators on some Hubert space H. It is shown that if 6 is a compact algebra homomorphism from A into B, then 6 is a finite rank operator, and the range of 0 is spanned by a finite number of idempotents. If, moreover, B(More)