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A synthetic adsorbent of crystalline calcium silicate hydrate, the product LRA by Advanced Minerals Corp., has been studied for endotoxin removal from aqueous solutions. This adsorbent removes endotoxin effectively, and the removal is greatly enhanced by the presence of an electrolyte such as NaCl, Tris-HCl, or Na2HPO4. It has an endotoxin removal capacity as high as 6 million endotoxin units (EU) per gram. Its endotoxin removal kinetics is fast, and for instance, over 99.9% endotoxin in a 5000 EU/mL solution was removed by mixing for 2 min at an adsorbent usage of 10 g/L. Using the chromatographic column method to treat a 5000 EU/mL solution, an endotoxin log-reduction factor of 6.2 was achieved with a single pass. This adsorbent also demonstrated significantly better performance when compared to many commonly used endotoxin removal agents, such as ActiClean Etox Endotoxin Removal Resin, Affi-Prep Polymyxin Support, Detroxi-Gel Endotoxin Removing Gel, Q Sepharose Fast Flow Media, and Sigma Endotoxin Removal Solution. Furthermore, it demonstrated a high selective removal of endotoxin from a solution of lambda DNA. This adsorbent provides opportunities for developing disposable, scaleable, and cost-effective methods for endotoxin reduction in many biotechnological and pharmaceutical processes.

Let K be an algebraically closed field of characteristic 0. When the Jacobian (∂f /∂x)(∂g/∂y) − (∂g/∂x)(∂f /∂y) is a constant for f, g ∈ K[x, y], Magnus' formula from [23] describes the relations between the homogeneous degree pieces f i 's and g i 's. We show a more general version of Magnus' formula and prove a special case of the two-dimensional Jacobian conjecture as its application.

This article is part of an ongoing investigation of the two-dimensional Jacobian conjecture. In the first paper of this series, we proved the generalized Magnus' formula. In this paper, inspired by cluster algebras, we introduce a sequence of new conjectures including the remainder vanishing conjecture. This makes the generalized Magnus' formula become a useful tool to show the two-dimensional Jacobian conjecture. In the forthcoming paper(s), we plan to prove the remainder vanishing conjecture.

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