"Where is this money going to come from?" A less sharp rate of increase has been forecast for New York City, which has its own pension system, but only because it is more poorly funded than the state pension fund and already requires steeper contributions.
Although growing advertising revenue continued to fuel the company's earnings, the rate of ad revenue growth at both The Times and The Globe was still below the sharp rate of increase in the 1999 third quarter.
One of the starker recent trends in all job data is the sharp rate of inflation in benefits costs.
Izenman and Tran ([1], 1990) discussed the uniform consistency and sharp rates of convergence under strong mixing and absolute regularity conditions.
(And I don't know anyone who still uses traveler's checks. Do you?) In Mexico, Banco de México said that the number of counterfeit notes removed from circulation last year rose about 9 percent over 2010, and that the sharpest rate of increase in counterfeits was in the lowly 50-peso note.
A merging of the two pools of liquidity could set off a sharp re-rating of the European big.caps; a phenomena seen among the U.K. mid-caps when a new trading platform was introduced.
And so there were widespread predictions of sharp rate rises.
In this note, using only elementary techniques, we demonstrate that the sequence is strictly increasing and that (1.1a) holds; in addition, we establish a sharp estimate of the rate of convergence.
Under an explicit nondegeneracy condition, we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain.
