Stephen Clark
Mathematical Foundations for a Compositional Distributional Model of Meaning.
Wed, 03/24/2010 - 16:01 — SomNambulAuthors: Stephen Clark, Bob Coecke, Mehrnoosh Sadrzadeh
Subjects: Computation and Language (Computational Linguistics and Natural Language and Speech Processing)
AbstractWe propose a mathematical framework for a unification of the distributional
theory of meaning in terms of vector space models, and a compositional theory
for grammatical types, for which we rely on the algebra of Pregroups,
introduced by Lambek. This mathematical framework enables us to compute the
meaning of a well-typed sentence from the meanings of its constituents.
Concretely, the type reductions of Pregroups are `lifted' to morphisms in a
category, a procedure that transforms meanings of constituents into a meaning
of the (well-typed) whole.Boundary Data Maps for Schrodinger Operators on a Compact Interval.
Thu, 02/04/2010 - 16:01 — SomNambulAuthors: Fritz Gesztesy, Marius Mitrea, Stephen Clark
Subjects: Spectral Theory
AbstractWe provide a systematic study of boundary data maps, that is, 2 \times 2
matrix-valued Dirichlet-to-Neumann and more generally, Robin-to-Robin maps,
associated with one-dimensional Schrodinger operators on a compact interval
[0,R] with separated boundary conditions at 0 and R. Most of our results are
formulated in the non-self-adjoint context.Minimal Rank Decoupling of Full-Lattice CMV Operators with Scalar- and Matrix-Valued Verblunsky Coefficients.
Thu, 02/04/2010 - 16:01 — SomNambulAuthors: Maxim Zinchenko, Fritz Gesztesy, Stephen Clark
Subjects: Spectral Theory
AbstractRelations between half- and full-lattice CMV operators with scalar- and
matrix-valued Verblunsky coefficients are investigated. In particular, the
decoupling of full-lattice CMV operators into a direct sum of two half-lattice
CMV operators by a perturbation of minimal rank is studied. Contrary to the
Jacobi case, decoupling a full-lattice CMV matrix by changing one of the
Verblunsky coefficients results in a perturbation of twice the minimal rank.
The explicit form for the minimal rank perturbation and the resulting two
half-lattice CMV matrices are obtained.Weyl-Titchmarsh Theory and Borg-Marchenko-type Uniqueness Results for CMV Operators with Matrix-Valued Verblunsky Coefficients.
Wed, 02/03/2010 - 16:01 — SomNambulAuthors: Maxim Zinchenko, Fritz Gesztesy, Stephen Clark
Subjects: Spectral Theory
AbstractWe prove local and global versions of Borg-Marchenko-type uniqueness theorems
for half-lattice and full-lattice CMV operators (CMV for Cantero, Moral, and
Velazquez) with matrix-valued Verblunsky coefficients. While our half-lattice
results are formulated in terms of matrix-valued Weyl-Titchmarsh functions, our
full-lattice results involve the diagonal and main off-diagonal Green's
matrices.
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