SPECIAL QUADRATURE ERROR ESTIMATES AND THEIR APPLICATION IN THE HARDY-LITTLEWOOD MAJORANT PROBLEM
The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexam- ples for all p = 2 2N. Montgomery conjectured that there exist counterex- amples even among idempotent polynomials. This was proved recently by Mockenhaupt and Schlag with some four-term idempotents. However, Mockenhaupt conjectured that even the classical 1 + e 2ix e 2i (k+2)x three-term character sums, should work for all 2k < p < 2k + 2 and for all k 2 N. In two previous papers we proved this conjecture for k = 0;1;2;3;4, i.e. in the range 0 < p < 10, p = 2 2N. Here we demonstrate that even the k = 5 case holds true. Renements in the technical features of our approach include use of total variation and integral mean estimates in error bounds for a certain fourth order quadrature. Our estimates make good use of the special forms of functions we encounter: linear combinations of powers and powers of log- arithms of absolute value squares of trigonometric polynomials of given degree. Thus the quadrature error estimates are less general, but we can nd better constants which are of practical use for us.
HOW TO CONCENTRATE IDEMPOTENTS
Call a sum of exponentials of the form f(x) = exp (2πiN1x) + exp (2πiN2x) + · · ·+ exp (2πiNmx), where the Nk are distinct integers, an idempotent. We have L interval concentration if there is a positive constant a, depending only on p, such that for each interval I ⊂ [0, 1] there is an idempotent f so that R I |f (x)| dx R 1 0 |f (x)| dx > a. We will explain how to produce such concentration for each p > 0. The origin of this question and the history of the development of its solution will be surveyed.
Numerical Simulation of Thermal Sensitivity for Explosives
In order to compare the thermal sensitivity of several typical explosives, this article solved numerically one-dimensional transient heat conduction equation including thermal decomposition for explosives with finite difference method. The explicit difference equations under different boundary conditions were established, and the corresponding stability criterion was derived. The ignition times for typical explosives were calculated, which was used to evaluate thermal sensitivity of explosives.