88 research outputs found
Generalized uncertainty inequalities
In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are
obtained for a pair of positive-self adjoint operators on a Hilbert space,
whose spectral projectors satisfy a ``balance condition'' involving certain
operator norms. This result is then applied to obtain uncertainty inequalities
on Riemannian manifolds, Riemannian symmetric spaces of non-compact type,
homogeneous graphs and unimodular Lie groups with sublaplacians.Comment: 19 page
Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
In the present paper, we develop geometric analytic techniques on Cayley
graphs of finitely generated abelian groups to study the polynomial growth
harmonic functions. We develop a geometric analytic proof of the classical
Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic
functions on lattices \mathds{Z}^n that does not use a representation formula
for harmonic functions. We also calculate the precise dimension of the space of
polynomial growth harmonic functions on finitely generated abelian groups.
While the Cayley graph not only depends on the abelian group, but also on the
choice of a generating set, we find that this dimension depends only on the
group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo
Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds
Measure contraction property is one of the possible generalizations of Ricci
curvature bound to more general metric measure spaces. In this paper, we
discover sufficient conditions for a three dimensional contact subriemannian
manifold to satisfy this property.Comment: 49 page
Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A
with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}.
The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos
Centenary conferenc
The "Statinth" wonder of the world: a panacea for all illnesses or a bubble about to burst
After the introduction of statins in the market as effective lipid lowering agents, they were shown to have effects other than lipid lowering. These actions were collectively referred to as 'pleiotropic actions of statins.' Pleiotropism of statins formed the basis for evaluating statins for several indications other than lipid lowering. Evidence both in favour and against is available for several of these indications. The current review attempts to critically summarise the available data for each of these indications
- …