The distribution of the eigenvalues of Hecke operators

For each prime $p$, we determine the distribution of the $p^{th}$ Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution

Ultrasound enhancement of microfiltration performance for natural organic matter removal

Sonication of water at 1500 W power prior to microfiltration showed that short sonication times (60 s) gave a reduced flux decline. It is suggested that a less potent, smaller molecular form of the natural organic matter (NOM) was produced by sonication. Longer sonication times diminished this beneficial effect. This may be due to the formation of aggregates or compounds that are more readily adsorbed on the membrane. Where the sonication was preceded by an alum treatment, the flux loss showed a regular decrease with longer sonication times. It is suggested that the effects of sonication on the alum flocs and on the flocs; NOM interactions may play a critical role in regulating the flux. Where sand was present on sonication at 800 and 1400 W, the cavitational energy was focussed on adsorbed organic material, resulting in more efficient destruction and the formation of compounds that counteracted the flux enhancement

Annual Survey of Virginia Law: The Virginia Limited Liability Company Act

Since the passage of the sixteenth amendment\u27 in 1913 and the income tax legislation adopted thereunder, the development of new forms of business entities has been driven largely by the desire to harmonize two goals: limited liability for the owners of the entity and pass-through treatment of the entity for income tax purposes

Lower order terms in the full moment conjecture for the Riemann zeta function

We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these coefficients. We also provide some numerical evidence in favour of the conjecture.Comment: 37 pages, 4 figure

The Nontrivial Zeros of Period Polynomials of Modular Forms Lie on the Unit Circle

We show that all but five of the zeros of the period polynomial associated to a Hecke cusp form are on the unit circl