4,965 research outputs found
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 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 an analytic
function in the Pick class such that the value of the directional
derivative can be expressed in terms of
Facial behaviour of analytic functions on the bidisk
We prove that if is an analytic function bounded by 1 on the bidisk
and is a point in a face of the bidisk at which satisfies
Caratheodory's condition then both and the angular gradient
exist and are constant on the face. Moreover, the class of all with
prescribed and 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
variables is locally monotone on -tuples of commuting self-adjoint
-by- 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
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Review of A. Bell and J. B. Hooper, eds., (1978) Syllables and Segments
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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
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Review of Iggy Roca (ed.) (1997) Derivations and Constraints in Phonology
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