A Caratheodory theorem for the bidisk via Hilbert space methods

If \ph is an analytic function bounded by 1 on the bidisk \D^2 and \tau\in\tb is a point at which \ph has an angular gradient \nabla\ph(\tau) then \nabla\ph(\la) \to \nabla\ph(\tau) as \la\to\tau nontangentially in \D^2. This is an analog for the bidisk of a classical theorem of Carath\'eodory for the disk. For \ph as above, if \tau\in\tb is such that the $\liminf$ of (1-|\ph(\la)|)/(1-\|\la\|) as \la\to\tau is finite then the directional derivative D_{-\de}\ph(\tau) exists for all appropriate directions \de\in\C^2. Moreover, one can associate with \ph and $\tau$ an analytic function $h$ in the Pick class such that the value of the directional derivative can be expressed in terms of $h$

Facial behaviour of analytic functions on the bidisk

We prove that if $\phi$ is an analytic function bounded by 1 on the bidisk and $\tau$ is a point in a face of the bidisk at which $\phi$ satisfies Caratheodory's condition then both $\phi$ and the angular gradient $\nabla\phi$ exist and are constant on the face. Moreover, the class of all $\phi$ with prescribed $\phi(\tau)$ and $\nabla\phi(\tau)$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.Comment: 18 pages. We have replaced an erroneous proof of Theorem 5.4(1) by a valid proo

Operator monotone functions and L\"owner functions of several variables

We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone

Taking a Free Ride in Morphophonemic Learning

As language learners begin to analyze morphologically complex words, they face the problem of projecting underlying representations from the morphophonemic alternations that they observe. Research on learnability in Optimality Theory has started to address this problem, and this article deals with one aspect of it. When alternation data tell the learner that some surface [B]s are derived from underlying /A/s, the learner will under certain conditions generalize by deriving all [B]s, even nonalternating ones, from /A/s. An adequate learning theory must therefore incorporate a procedure that allows nonalternating [B]s to take a «free ride» on the /A/ →[B] unfaithful map

What does comparative markedness explain, what should it explain, and how?

These seven commentaries treat a wide range of topics in interesting and insightful ways. It is not possible to write a coherent response that addresses all of the criticisms and suggestions, large and small, that the authors have brought up. Several main themes emerge, however, that transcend the individual commentaries, and these themes supply the structure for this reply. They include alternatives to comparative markedness, possible counterexamples, comparative markedness on other dimensions of correspondence, and questions about the authenticity of opaque phonological processes. These themes will each be addressed in turn