Abstract. The paper generalizes, for a wide class of elliptic curves defined over Q, the celebrated classical lemma of Birch and Heegner to quadratic twists by discriminants having any prescribed number of prime factors. In addition, it proves stronger results for the family of quadratic twists of the modular elliptic curve X 0 (49), including showing that there is a large class of explicit quadratic twists whose complex L-series does not vanish at s = 1, and for which the full Birch-Swinnerton-Dyer conjecture holds.
A simple, rapid (10 s) and scalable method to fabricate superhydrophobic polypropylene (PP) fabrics is developed by swelling the fabrics in cyclohexane/heptane mixture at 80 °C. The recrystallization of the swollen macromolecules on the fiber surface contributes to the formation of submicron protuberances, which increase the surface roughness dramatically and result in superhydrophobic behavior. The superhydrophobic PP fabrics possess excellent repellency to blood, urine, milk, coffee, and other common liquids, and show good durability and robustness, such as remarkable resistances to water penetration, abrasion, acidic/alkaline solution, and boiling water. The excellent comprehensive performance of the superhydrophobic PP fabrics indicates their potential applications as oil/water separation materials, protective garments, diaper pads, or other medical and health supplies. This simple, fast and low cost method operating at a relatively low temperature is superior to other reported techniques for fabricating superhydrophobic PP materials as far as large scale manufacturing is considered. Moreover, the proposed method is applicable for preparing superhydrophobic PP films and sheets as well.
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