4,693 research outputs found
On the complete integrability of the discrete Nahm equations
The discrete Nahm equations, a system of matrix valued difference equations,
arose in the work of Braam and Austin on half-integral mass hyperbolic
monopoles.
We show that the discrete Nahm equations are completely integrable in a
natural sense: to any solution we can associate a spectral curve and a
holomorphic line-bundle over the spectral curve, such that the discrete-time DN
evolution corresponds to walking in the Jacobian of the spectral curve in a
straight line through the line-bundle with steps of a fixed size. Some of the
implications for hyperbolic monopoles are also discussed
The performance of Seventh District food processing
Federal Reserve District, 7th ; Food industry and trade
Differential Galois Theory of Linear Difference Equations
We present a Galois theory of difference equations designed to measure the
differential dependencies among solutions of linear difference equations. With
this we are able to reprove Hoelder's Theorem that the Gamma function satisfies
no polynomial differential equation and are able to give general results that
imply, for example, that no differential relationship holds among solutions of
certain classes of q-hypergeometric functions.Comment: 50 page
Toric partial density functions and stability of toric varieties
Let denote a polarized toric K\"ahler manifold. Fix a
toric submanifold and denote by the
partial density function corresponding to the partial Bergman kernel projecting
smooth sections of onto holomorphic sections of that vanish to
order at least along , for fixed such that . We
prove the existence of a distributional expansion of as , including the identification of the coefficient of as a
distribution on . This expansion is used to give a direct proof that if
has constant scalar curvature, then must be slope semi-stable
with respect to . Similar results are also obtained for more general partial
density functions. These results have analogous applications to the study of
toric K-stability of toric varieties.Comment: Accepted by Mathematische Annalen on 13 September 201
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