The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)mo...The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).展开更多
The laser-generated surface acoustic wave（SAW） technique is an accurate,fast and nondestructive solution to determine the mechanical properties of ultra thin films.SAWs are dispersive during the wave propagation on ...The laser-generated surface acoustic wave（SAW） technique is an accurate,fast and nondestructive solution to determine the mechanical properties of ultra thin films.SAWs are dispersive during the wave propagation on the layered structure.The Young＇s moduli of thin films can be obtained by matching the experimentally and theoretically calculated dispersive SAW curves.A short ultraviolet laser pulse is employed to generate the broad spectral range of the dispersive SAWs.The frequency range of dispersive SAWs in this study reaches 180 MHz,which is adequate for the SAW technique applied for the investigated samples.In this work,the Young＇s moduli of a series of nanoporous Black Diamond^（TM） low dielectric constant（low-k） films deposited on a Si（100） substrate are characterized successfully by the SAW technique.展开更多
基金supported by National Basic Research Program of China (973 Project) (Grant No. 2011CB808003)the PCSIRT Project of the Ministry of EducationNational Natural Science Foundation of China (Grant No. 11231004)
文摘The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).
基金Project supported by the National Natural Science Fotmdation of China(No.60876072)the New Century Excellent Talents in University,China(No.NCET-08-0389).
文摘The laser-generated surface acoustic wave（SAW） technique is an accurate,fast and nondestructive solution to determine the mechanical properties of ultra thin films.SAWs are dispersive during the wave propagation on the layered structure.The Young＇s moduli of thin films can be obtained by matching the experimentally and theoretically calculated dispersive SAW curves.A short ultraviolet laser pulse is employed to generate the broad spectral range of the dispersive SAWs.The frequency range of dispersive SAWs in this study reaches 180 MHz,which is adequate for the SAW technique applied for the investigated samples.In this work,the Young＇s moduli of a series of nanoporous Black Diamond^（TM） low dielectric constant（low-k） films deposited on a Si（100） substrate are characterized successfully by the SAW technique.
