Canine Adenovirus type 1 antigen

In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3)4(1954), 84-106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan's lost notebook, Adv. Math. 190(2005), 278-299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan's lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan's identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.

Separation of asphalt from carbonate unconventional ores is a challenge due to the strong interactions between carbonate mineral surfaces and asphalt. Our previous results show that the reactive extraction process performs well in recovering both extra-heavy oil and minerals from carbonate oil ores (RSC Adv. 2019, 9: 14372-14381). Herein, further tests have been conducted to understand the kinetics of the reactive extraction. It is found that the reaction kinetics of carbonate minerals in asphalt rocks is beyond the description of traditional reaction shrinking-core model. This is mainly ascribed to the coating of asphalt on the carbonate mineral surface. To mathematically describe the reaction kinetics, a modified reaction shrinking-core model has been proposed considering the unreacted oil on the mineral surface by adding a coverage coefficient theta in the kinetic model. Results show that this modified shrinking-core kinetic model works well in describing the real reaction (R-2 > 0.96) under different conditions. In addition, the thermodynamic properties, including the apparent active energy, enthalpy change, entropy change, Gibbs free energy change, and pre-exponential factor of this reactive extraction process are also determined for this kind of ores. This work would provide a primary insight into predicting the reactive extraction with different kinds of reactive carbonate minerals or other similar oil ores with dissolved minerals. (C) 2020 Elsevier Ltd. All rights reserved.