157,084 research outputs found
Singular Chern Classes of Schubert Varieties via Small Resolution
We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class
of a Schubert variety in the Grassmannian using small resolutions introduced by
Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and
local Euler obstructions using small resolutions instead of the Nash blowup.
The algorithm obtained for CSM classes also allows us to prove new cases of a
positivity conjecture of Aluffi and Mihalcea.Comment: Addressed referee's comments, Section 6.2 contains new material; 35
pages, 3 figures, and 2 table
Quadratic Tangles in Planar Algebras
In planar algebras, we show how to project certain simple "quadratic" tangles
onto the linear space spanned by "linear" and "constant" tangles. We obtain
some corollaries about the principal graphs and annular structure of
subfactors
Conceptual Problems in Scattering from Localized non-Hermitian Potentials
We highlight the conceptual issues that arise when one applies the
quasi-Hermitian framework to analyze scattering from localized non-Hermitian
potentials, in particular complex square-wells or delta-functions. When treated
in the framework of conventional quantum mechanics, these potentials are
generally considered as effective theories, in which probability is not
conserved because of processes that have been ignored. However, if they are
treated as fundamental theories, the Hilbert-space metric must be changed. In
order for the newly-defined probability to be conserved, it must differ from
the standard one, even at asymptotically large distances from the scattering
centre, and the mechanism for this is the non-locality of the new metric, as we
show in detail in the model of a single complex delta function. However,
properties of distant bound-state systems, which do not interact physically
with the non-Hermitian scattering potential, should not be affected. We analyze
a model Hamiltonian that supports this contention.Comment: The emphasis has been changed from v1, recognizing that it makes
physical sense that the wave functions of scattering states are fundamentally
changed in the quasi-Hermitian framework. In contrast, bound states should
not be significantly affected by the introduction of a distant non-Hermitian
scattering potentia
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